44471235 Topics in Fluorescence Spectroscopy Vol 2 Principles

Topics inFluorescence SpectroscopyVolume 2Principles

Topics in Fluorescence SpectroscopyEdited by JOSEPH R. LAKOWICZ

Volume 1: TechniquesVolume 2: PrinciplesVolume 3: Biochemical Applications

Topics inFluorescenceSpectroscopyVolume 2Principles

Edited by

JOSEPH R. LAKOWICZCenter for Fluorescence SpectroscopyDepartment of Biological ChemistryUniversity of Maryland School of MedicineBaltimore, Maryland

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

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Contributors

Katalin Ajtai • Department of Biochemistry and Molecular Biology, MayoFoundation, Rochester, Minnesota 55905; permanent address: Department ofBiochemistry, Eötvös Loránd University, Budapest, Hungary

Marcel Ameloot • Limburgs Universitair Centrum, Universitaire Campus,B-3610 Diepenbeek, Belgium

Joseph M. Beechem • Department of Physics, University of IllinoisUrbana–Champaign, Urbana, Illinois 61801; present address: Department ofMolecular Physiology and Biophysics, Vanderbilt University, Nashville,Tennessee 37232

Ludwig Brand • Department of Biology, The John Hopkins University,Baltimore, Maryland 21218

Thomas P. Burghardt • Department of Biochemistry and Molecular Biology,Mayo Foundation, Rochester, Minnesota 55905

Herbert C. Cheung • Department of Biochemistry, University of Alabamaat Birmingham, Birmingham, Alabama 35294

Maurice R. Eftink • Department of Chemistry, University of Mississippi,University, Mississippi 38677

Susan G. Frasier-Cadoret • Departments of Pharmacology and InternalMedicine, University of Virginia Health Sciences Center, Charlottesville,Virginia 22908; present address: Office of Interdisciplinary Graduate Studies,University of Virginia Health Sciences Center, Charlottesville, Virginia 22908

Enrico Gratton • Department of Physics, University of Illinois Urbana–Champaign, Urbana, Illinois 61801

Michael L. Johnson • Departments of Pharmacology and InternalMedicine, University of Virginia Health Sciences Center, Charlottesville,Virginia 22908

Jay R. Knutson • National Heart, Lung, and Blood Institute, NationalInstitutes of Health, Bethesda, Maryland 20892

v

vi Contributorss

Nicolai A. Nemkovich • Institute of Physics of the B.S.S.R. Academy ofSciences, Minsk 220602, U.S.S.R.

Anatolyi N. Rubinov • Institute of Physics of the B.S.S.R., Academy ofSciences, Minsk 220602, U.S.S.R.

Robert F. Steiner • Department of Chemistry, University of Maryland–Baltimore County, Baltimore, Maryland 21228

Martin Straume • Departments of Pharmacology and Internal Medicine,University of Virginia Health Sciences Center, Charlottesville, Virginia 22908;present address: Department of Biology, The Johns Hopkins University,Baltimore, Maryland 21218

Richard B. Thompson • Department of Biological Chemistry, University ofMaryland School of Medicine, Baltimore, Maryland 21201

Vladimir I. Tomin • Institute of Physics of the B.S.S.R., Academy ofSciences, Minsk 220602, U.S.S.R.

Preface

Fluorescence spectroscopy and its applications to the physical and life scienceshave evolved rapidly during the past decade. The increased interest influorescence appears to be due to advances in time resolution, methods ofdata analysis, and improved instrumentation. With these advances, it is nowpractical to perform time-resolved measurements with enough resolution tocompare the results with the structural and dynamic features of macro-molecules, to probe the structures of proteins, membranes, and nucleic acids,and to acquire two-dimensional microscopic images of chemical or proteindistributions in cell cultures. Advances in laser and detector technology havealso resulted in renewed interest in fluorescence for clinical and analyticalchemistry.

Because of these numerous developments and the rapid appearance ofnew methods, it has become difficult to remain current on the science offluorescence and its many applications. Consequently, I have asked theexperts in particular areas of fluorescence to summarize their knowledge andthe current state of the art. This has resulted in the initial two volumes ofTopics in Fluorescence Spectroscopy, which is intended to be an ongoingseries which summarizes, in one location, the vast literature on fluorescencespectroscopy. The third volume will appear shortly.

The first three volumes are designed to serve as an advanced text. Thesevolumes describe the more recent techniques and technologies (Volume 1),the principles governing fluorescence and the experimental observables(Volume 2), and applications in biochemistry and biophysics (Volume 3).

Additional volumes will be published as warranted by further advances inthis field. I welcome your suggestions for future topics or volumes, offers tocontribute chapters on specific topics, or comments on the present volumes.

Finally, I thank all the authors for their patience with the delays incurredin release of the first three volumes.

Joseph R. LakowiczBaltimore, Maryland

vii

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Contents

1. Fluorescence Anisotropy: Theory and ApplicationsRobert F. Steiner

1.1.1.2.

1.3.

1.4.

1.5.

1.6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122568

1.2.1.1.2.2.1.2.3.1.2.4.1.2.5.1.2.6.1.2.7.

Meaning of Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Influence of Excitation Pulse Shape. . . . . . . . . . . . . . . . . . . . .The Time Decay of Anisotropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . .The Rotational Diffusion of Ellipsoids of Revolution. . . . . . . .The Anisotropy Decay of Ellipsoidal Particles . . . . . . . . . . . .Partially Immobilized Systems . . . . . . . . . . . . . . . . . . . . . . . . .The Influence of Internal R o t a t i o n . . . . . . . . . . . . . . . . . . . . . .

Experimental Analysis of Anisotropy Decay . . . . . . . . . . . . . . . . . . .1.3.1.1.3.2.1.3.3.

Analysis of Time-Domain Data . . . . . . . . . . . . . . . . . . . . . . . .Time-Domain Measurements of Anisotropy Decay. . . . . . . . . .Frequency-Domain Measurements of Anisotropy Decay. . . .

Anisotropy Decay of Heterogeneous Systems. . . . . . . . . . . . . . . . . . . . .1.4.1.1.4.2.

Anisotropy-Resolved Emission Spectra. . . . . . . . . . . . . . . . . . . . .The Meaning of Correlation Times for Associative andNonassociative Heterogeneity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Anisotropy Decay of Intrinsic Protein Fluorophores. . . . . . . . . . . . . . . .1.5.1.1.5.2.

1.5.3.

1.5.4.

Anisotropy Decay of a Rigid Protein: S. Nuclease. . . . . . . . . . .Rotational Dynamics of Flexible Polypeptides: Adreno-corticotropin and Melittin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Anisotropy Decay of a Tightly Bound Fluorophore:Lumazine Protein. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Anisotropy Decay of a Transfer RNA . . . . . . . . . . . . . . . . . . .

Anisotropy Decay of Biopolymers Labeled with an ExtrinsicFluorophore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.6.1.1.6.2.1.6.3.

Anisotropy Decay and Internal Flexibility of Myosin . . . . . .Anisotropy Decay of a Fibrous Protein: F-Ac t in . . . . . . . . . .Anisotropy Decay for Proteins Displaying Internal RotationInvolving a Well-Defined Domain: The Immunoglobulins..

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1.6.4. Anisotropy Decay of Calmodulin Complexes with TNS . . . .References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. Fluorescence Quenching: Theory and Applications

Maurice R. Eftink

2.1.2.2.

2.3.

2.4.

2.5.

2.6.

Introduction ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Basic Concepts.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.1.2.2.2.2.2.3.2.2.4.2.2.5.

The Stern–Volmer Equation . . . . . . . . . . . . . . . . . . . . . . . . . . .Quenching Mechanisms and Efficiency . . . . . . . . . . . . . . . . . .Diffusional Nature of Quenching . . . . . . . . . . . . . . . . . . . . . . . .Static Quenching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Various Quenchers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quenching Studies with Proteins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.1.2.3.2.2.3.3.2.3.4.2.3.5.2.3.6.

2.3.7.2.3.8.2.3.9.

Exposure of Fluorophores. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Effect of the Macromolecule’s Size . . . . . . . . . . . . . . . . . . . . . .Electrostatic Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Tryptophan Residues in Proteins . . . . . . . . . . . . . . . . . . . . . . .Ligand Binding and Conformational Changes . . . . . . . . . . . .Mechanism of Quenching in Proteins—Penetration versusUnfolding Mechanisms .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Interaction of Quenchers with Proteins. . . . . . . . . . . . . . . . . . . . .Transient Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Multiple Quenching Rate Constants and FluorescenceLifetimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Studies with Membranes and Nucleic Acids. . . . . . . . . . . . . . . . . . . . .2.4.1.2.4.2.2.4.3.2.4.4.2.4.5.

Partitioning of Quenchers into Membranes/Micelles......Two-Dimensional Diffusion in Membranes. . . . . . . . . . . . . . . .Quencher Moieties Attached to Lipid Molecules. . . . . . . . . . . . .Membrane Transport and Surface Potential. . . . . . . . . . . . . . . . .Nucleic Acids.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Uses to Resolve Other Fluorescence Properties. . . . . . . . . . . . . . . . .2.5.1.2.5.2.2.5.3.2.5.4.2.5.5.

Resolution of Steady-State Spectra. . . . . . . . . . . . . . . . . . . . . . . .Resolution of Fluorescence Lifetimes. . . . . . . . . . . . . . . . . . . . . .Resolution of Anisotropy Measurements. . . . . . . . . . . . . . . . . . . . .Resolution of Energy Transfer Experiments . . . . . . . . . . . . . . .Other Uses of Solute Quenching. . . . . . . . . . . . . . . . . . . . . . . .

Recent Developments in Data Analysis . . . . . . . . . . . . . . . . . . . . . . .2.6.1.2.6.2.2.6.3.2.6.4.

Simultaneous Analyses of Quenching D a t a . . . . . . . . . . . . . . .Nonlinear Least-Squares Fits . . . . . . . . . . . . . . . . . . . . . . . . . .Distribution of Lifetimes or Rate Constants . . . . . . . . . . . . . .Experimental Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . .

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100101102103105108109112112113114116

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2.7.2.8.

Phosphorescence Quenching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. Resonance Energy TransferHerbert C. Cheung

3.1.3.2.3.3.3.4.

3.5.

3.6.

3.7.3.8.3.9.

Long-Range Dipole–Dipole Interaction. . . . . . . . . . . . . . . . . . . . . . . . .Determination of Energy Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Proximity Mapping of Molecular Assembly . . . . . . . . . . . . . . . . . .Experimental Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4.1.3.4.2.3.4.3.

Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Measurement of Transfer Efficiency. . . . . . . . . . . . . . . . . . . . . . .The Orientation Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Selected Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.5.1.3.5.2.3.5.3.

Myosin and Actomyosin....................................Troponin Subunits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Ribosomal Proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Comparison of FRET Results with Results from OtherTechniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.6.1.3.6.2.

Comparison with Crystallographic Data. . . . . . . . . . . . . . . . . .Comparison with Cross-Linking Data. . . . . . . . . . . . . . . . . . .

Application of FRET to Enzyme Kinetics . . . . . . . . . . . . . . . . . . . .Time-Resolved Energy Transfer M e a s u r e m e n t s . . . . . . . . . . . . . .Distribution of Distances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.9.1.3.9.2.

Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.10. Summary and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. Least-Squares Analysis of Fluorescence DataMartin Straume, Susan G. Frasier-Cadoret, and Michael L. Johnson

4.1.4.2.4.3.4.4.

4.5.

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Basic Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Assumptions of Least-Squares Analysis . . . . . . . . . . . . . . . . . . . . . . .Least-Squares Parameter Estimation Procedures. . . . . . . . . . . . . . . . . .4.4.1.4.4.2.

Modified Gauss–Newton Algorithm. . . . . . . . . . . . . . . . . . . . . . .Nelder–Mead Simplex Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . .

An Example of the Least-Squares Procedures—CollisionalQuenching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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128130132132132133135140140145147

148148151152155157158161170171

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4.5.1.Example of the Gauss–Newton Procedure . . . . . . . . . . . . . . .4.5.2.Example of the Nelder–Mead Simplex Procedure . . . . . . . . .

4.6.

4.7.

4.8.4.9.

Joint Confidence Intervals—Estimation and Propagation. . . . . .4.6.1.4.6.2.4.6.3.4.6.4.

4.6.5.

4.6.6.

Asymptotic Standard Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Linear Joint Confidence Intervals. . . . . . . . . . . . . . . . . . . . . . . . . .Support Plane Confidence Intervals. . . . . . . . . . . . . . . . . . . . . . . .Approximate Nonlinear Support Plane Joint ConfidenceIntervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A Monte Carlo Method for the Evaluation of ConfidenceIntervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Propagation of Confidence Intervals . . . . . . . . . . . . . . . . . . . .

Analysis of Residuals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.7.1.4.7.2.4.7.3.4.7.4.4.7.5.4.7.6.

Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Trends. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Outliers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Influential Observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Common Quantitative Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Implementation Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .In Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. The Global Analysis of Fluorescence Intensity and Anisotropy DecayData: Second-Generation Theory and ProgramsJoseph M. Beechem, Enrico Gratton, Marcel Ameloot, Jay R. Knutson,and Ludwig Brand

5.1.

5.2.

5.3.

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.1.1.

5.1.2.

5.1.3.

Multiexcitation/Multitemperature Studies of AnisotropicRotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Multiexcitation/Emission Wavelength Studies of TotalIntensity D a t a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Double-Kinetic Studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Global Analysis Philosophy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.2.1.5.2.2.

Evolution of the Global Analysis Approach . . . . . . . . . . . . . .Global Analysis Implementation Strategy. . . . . . . . . . . . . . . . . . .

General Elements of the Global Analysis Program. . . . . . . . . . . . . . . . .5.3.1.5.3.2.5.3.3.5.3.4.

Mapping to the Physical Observables . . . . . . . . . . . . . . . . . . .Empirical Description of the Fluorescence Decay . . . . . . . . .Compartmental Description of Photophysical Events. . . . . . . .Overview of Nonlinear Minimization (The Basic Equations)

202205207208208209

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5.4.

5.5.

5.6.

5.7.

5.8.

In-Depth Flow Chart of a General-Purpose Global AnalysisProgram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.4.1.5.4.2.

Overview of the Global Analysis Procedure . . . . . . . . . . . . .Flow Chart for the LFD Global Analysis Program“Global”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Case Studies of the Application of Global Analysis to ExperimentalData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.5.1.5.5.2.

Case Study of a Two-State Excited State Reaction . . . . . . .Distributions of Distances and Energy Transfer Analysis...

Anisotropy Decay Data Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.6.1.5.6.2.5.6.3.5.6.4.5.6.5.5.6.6.

5.6.7.5.6.8.

General Equations and Experimental Linkages . . . . . . . . . . .Changes in Anisotropy Data Collection Schemes. . . . . . . . . . . .Associative versus Nonassociative Modeling of AnisotropyAnisotropy Decay-Associated Spectra (ADAS) . . . . . . . . . . .Multidye Global Anisotropy Decay Analysis . . . . . . . . . . . . .Distributed Lifetimes and Distributed Rotational CorrelationTimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Multiexcitation Anisotropy Experiments. . . . . . . . . . . . . . . . . . . . .Example of Distributed Rotations: Fluorophore RotationsGated by Packing Fluctuations in Lipid Bi laye r s . . . . . . . . . .

Error Analysis and the Identifiability Problem . . . . . . . . . . . . . . . . .5.7.1.5.7.2.5.7.3.

The Identifiability Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Identifiability Study Using Laplace Identifiability Analysis..Error Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6. Fluorescence Polarization from Oriented SystemsThomas P. Burghardt and Katalin Ajtai

6.1.6.2.

6.3.

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Theory and Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2.1.6.2.2.6.2.3.

6.2.4.

6.2.5.

The Angular Probability Density N...........................Fluorescence Polarization in Homogeneous Space . . . . . . . .Time-Resolved Fluorescence Depolarization Determinationof the High-Resolution Angular Probability Density. . . . . . . . .Relation of Electron Spin Resonance Spectra to FluorescencePolarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Biochemical Techniques of Specific Labeling . . . . . . . . . . . . .

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7. Fluorescence-Based Fiber-Optic SensorsRichard B. Thompson

7.1.7.2.7.3.7.4.7.5.7.6.7.7.7.8.7.9.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Fiber-Optic Fundamentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Sensor Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Sensing Tip Configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Fiber Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Separating Excitation and Emission . . . . . . . . . . . . . . . . . . . . . . . . .Launching Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Light Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Time-Resolved Fluorescence in Fibers . . . . . . . . . . . . . . . . . . . . . .

7.10.7.11.

Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8. Inhomogeneous Broadening of Electronic Spectra of Dye Molecules inSolutionsNicolai A. Nemkovich, Anatolyi N. Rubinov, and Vladimir I. Tomin

8.1.8.2.

8.3.

8.4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Theoretical Considerations of Inhomogeneous Broadening. . . . . . . . .8.2.1.8.2.2.8.2.3.8.2.4.8.2.5.8.2.6.

Solvate Configurational Energy . . . . . . . . . . . . . . . . . . . . . . . .Field Diagram of a Polar Solution. . . . . . . . . . . . . . . . . . . . . . . . .Solvate Distribution in Configurational Sublevels . . . . . . . . .Nonpolar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Selective Excitation with Vibrational Spectral Broadening..Absorption and Fluorescence Spectra: Dependence onExciting Light Frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Stationary Inhomogeneous Broadening . . . . . . . . . . . . . . . . . . . . . . . . . .8.3.1.

8.3.2.8.3.3.

Universal Relationship between Fluorescence and Absorp-tion Spectra of Polar Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . .Luminescence Spectra at Red-Edge Excitation. . . . . . . . . . . . . . .Directed Nonradiative Energy Transfer in Organic Solutions

Dynamic Inhomogeneous Broadening in Liquid Solutions . . . . . . .8.4.1.8.4.2.

8.4.3.8.4.4.

Analysis of Configurational Relaxation in Liquid SolutionsExperimental Study of the Luminescence Kinetics of LiquidSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The Solution Spectrochronogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . .The Effect of Light-Induced Molecular Rotation inSolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

345346349350353357359359361362362363

367369369371374376378

383387

387388390395395

401404

406

Contents xv

8.5.

8.6.

Selective Kinetic Spectroscopy of Fluorescent Molecules inPhospholipid Membranes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.5.1.8.5.2.

8.5.3.8.5.4.

Energy Levels of an Electric Dipole Probe in a MembraneInhomogeneous Broadening in Steady-State FluorescenceSpectra of Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Kinetics of Probe Fluorescence. . . . . . . . . . . . . . . . . . . . . . . . . . . . .Rotational Dynamics of the Probe in the Membrane . . . . . .

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

413413

416419422423425

429

!"#$%&'()%#*+)*+#,*'--.%-)/+%0-'*1

1

Fluorescence Anisotropy:Theory and Applications

Robert F. Steiner

1.1. Introduction

Fluorescence anisotropy decay belongs to the general class of relaxationmethods which monitor the time dependence of the transition of the systemfrom a biased to a random arrangement. In this case the transition is froma specific to a random orientation in space and occurs via Brownian rota-tional diffusion. Unlike some other techniques of this type, such as orientationby the application of an external force field, fluorescence anisotropy decaydepends upon the initial selection of a population of specifically orientedfluorophores from a large assembly of randomly oriented fluorophores.From the nature of the change in anisotropy with time, information may bederived as to the rotational mobility of the fluorophore. If the fluorophore isa fluorescent label linked to a biological macromolecule it may reflect boththe overall motion of the particle and any internal rotational modes whichmay be present, including the localized motion of the probe.(4)

Two basic kinds of information may be derived from fluorescenceanisotropy decay measurements. To the extent that the macromolecule andthe attached label rotate as a unit, anisotropy decay may provide informationas to the size and shape factor of the macromolecule. It is also a convenientmeans of studying any internal rotational motions present in the macro-molecule and of examining the nature of the molecular flexibility.

The basic theory of the depolarization of fluorescence through Brownianrotation was presented by Perrin. Equations were developed which relatedthe degree of polarization to the size and shape parameters of a dissolvedfluorophore, the average lifetime of its excited state, and the temperature and

Robert F. Steiner • Department of Chemistry and Biochemistry, University ofMaryland–Baltimore County, Baltimore, Maryland 21228.

Topics in Fluorescence Spectroscopy, Volume 2: Principles, edited by Joseph R. Lakowicz. PlenumPress, New York, 1991.

1

2 Robert F. Steiner

viscosity of the solvent. The theory of Perrin was tested using fluorophores ofknown molecular volume and yielded reasonably self-consistent results.(6)

Subsequently, the use of the polarization of extrinsic fluorescent labels tostudy proteins was introduced by Weber and applied to the characterizationof a number of proteins by Weber and others.(10, 11) The mathematical for-mulation was substantially simplified by the adoption, first suggested byJablonski, of a linear parameter, the fluorescence anisotropy.(12) The use ofanisotropy is particularly advantageous for the study of heterogeneoussystems.

The development of the single-photon counting technique(13–15) made itfeasible to examine the time dependence of anisotropy directly. The basictheory of anisotropy decay was described by Wahl(16, l7) and applied by Wahland co-workers to the study of the rotational dynamics of biopolymers(17–19).

The basic theory was expanded by Gottlieb and Wahl,(20) who con-sidered the effects of internal rotations, and by Belford et al., who presenteda description of the anisotropy decay of ellipsoidal particles.(21) Kinosita et al.,have extended the theory to the hindered rotation of fluorophores embeddedin membranes.(22)

The last decade has witnessed a progressive refinement of the techniqueof time-domain measurements of anisotropy decay and the application of thismethod to a wide variety of biopolymer and membrane systems. In recentyears frequency-domain determinations of anisotropy decay have beendeveloped as a useful alternative to time-domain measurements. The presentchapter will describe the current status of the technique and present someselected applications.

1.2. Theory

1.2.1. Meaning of Anisotropy

The radiation emitted by a fluorophore may be polarized to varyingextents, depending upon conditions. The polarization is conventionallycharacterized with reference to a system of laboratory coordinates defined bythe directions of observation and of the exciting beam. It is customary toobserve the fluorescence at an angle of 90° to the exciting beam (Figure 1.1).If the center of the irradiated volume is chosen as the origin, O, and the x andy axes are taken along the direction of observation and along the direction ofthe exciting beam, respectively, then the directions Ox and Oy define a planecontaining all the instrumental elements. The Oz axis is perpendicular to thisplane.

The components of the total fluorescence intensity along the three coor-

Fluorescence Anisotropy: Theory and Applications 3

dinate axes are designated The sum of these three componentsis equal to the total fluorescence intensity, S:

The exciting beam may correspond to either unpolarized, or natural, light orto linearly polarized light whose direction of polarization is along the Ox orOz axis. In the case of unpolarized light, the electric vector may have anyorientation within the x-z plane. Symmetry considerations therefore requirethat

so that the total intensity is given by, from Eq. (1.1),

When the exciting beam is polarized along the Oz axis, symmetry nowdictates that

and

Finally, if the exciting beam is linearly polarized along the Ox axis, wehave

and

4 Robert F. Steiner

It is usual to perform fluorescence anisotropy measurements with anexciting beam which is polarized in the z-direction or, somewhat less com-monly, with unpolarized light. For both cases, the measured quantities are thecomponents of fluorescence intensity which are polarized in the Oz and Oydirections. (The component which is polarized in the Ox direction is notaccessible to measurement, as this is the direction of observation.) In this caseit is useful to define

and

In terms of the usual laboratory arrangement, may be identifiedwith the vertically and horizontally polarized components, respec-tively.

The emission anisotropy, A, is defined by

where

Employing Eqs. (1.5) and (1.10), we obtain, for vertically polarized excitingradiation,

and, for unpolarized exciting radiation,

It may be shown that the emission anisotropy for the two cases are related by

For horizontally polarized exciting light, we have

and


If more than one fluorescent species is present, the observed anisotropy isgiven by(10)

where are the emission anisotropy of species i and its fractionalcontribution to the total intensity, respectively.

An alternative formulation, which prevails in the older fluorescenceliterature, employs polarization (P) rather than anisotropy.(10) The emissionpolarization is defined by

However, since the use of anisotropy leads to simpler expressions, it hasreplaced polarization in the recent literature and will be exclusively employedin this chapter.

1.2.2. Influence of Excitation Pulse Shape

Observations of the time decay of fluorescence intensity or anisotropyusually involve measurement of the summed response to a series of repetitivelight flashes.(23) If each excitation pulse were infinitely sharp, correspondingto a function, and if only a single emitting species were present, thenthe fluorescence intensity would decay with time according to a simpleexponential law:

where is the intensity at zero time, corresponding to the time of theexcitation pulse, t is the time after excitation, and is the average decay timeof fluorescence intensity.

It is at present feasible to obtain excitation pulses of picosecond width,which approach the infinitely sharp case, by employing pulsed laser sources.(24)

However, the majority of anisotropy decay studies have made use of repetitivespark discharges, whose widths are of the order of nanoseconds. In both casesand especially for the latter case, decay and excitation processes overlap overa finite time interval, so that it is not permissible to ignore the finite durationof the excitation pulse.

The observed time decay profile i(t) may be related to the profile I(t)

6 Robert F. Steiner

which would be observed if the excitation pulse were infinitely sharp by theconvolution integral(25):

Here, t is the time and E(u) is the time profile of excitation. (In this chapterthe convention will be followed of denoting experimentally observed quan-tities, such as i, s, d, etc., by lowercase letters, while capital letters will be usedfor the corresponding quantities I, S, D, etc., which have been corrected forconvolution effects so as to correspond to the behavior expected if the excita-tion pulse were infinitely sharp.) The mathematical procedures which havebeen developed for analyzing the time decay of fluorescence intensity in termsof discrete decay times also provide for the deconvolution of the experimentaldecay curve so as to remove the distortion caused by the finite duration of theexcitation pulse.

1.2.3. The Time Decay of Anisotropy

The measurement of fluorescence anisotropy depends upon the selectiveand nonrandom excitation of fluorescent molecules or groups. When afluorophore absorbs a quantum of radiant energy, it undergoes a transition tosome vibrational level of a higher electronic state.(26) This is followed by avery rapid (several picosecond) process of internal conversion which placesthe fluorophore in a low vibrational level of the lowest electronic excited state.This process generally attains completion prior to the emission of fluorescence.It results in a nonequivalence of the transition moments corresponding toabsorption and emission.(1–3)

In the most general case the polarization properties of a fluorophore maybe described by a fourth-order tensor.(27) If the fluorophore is immobilizedin a rigid isotropic medium, so that Brownian rotation cannot occur, theanisotropy, A0, is independent of time and is given by, for vertically polarizedexcitation(1, 27, 28):

where are the components of the molecular absorption andmolecular emission tensors, along the three molecular axes.

For the case of common interest where absorption and emission arestrongly permitted electronic transitions and the wavelengths of measurementcorrespond fairly closely to the 0–0 vibronic transition, the transitionmoments of absorption and emission may be represented as linear with a well-


defined direction.(1) This model has proved adequate for the interpretationof nearly all the available experimental data and will be assumed in thediscussion to follow. In this case,

and

where are the direction cosines of the linear transition moments ofabsorption and emission, respectively, with respect to the molecular axes.Equation (1.19) now becomes, if is the angle between the two linearoscillators,

When a sample containing a population of fluorophores whose orienta-tion is random is irradiated with vertically polarized light, a biased distributionof orientations of the excited fluorophores is achieved by a photoselectiveprocess. A fluorophore is excited with a probability which is proportional to

is the angle between the absorption transition moment andthe electric vector of the exciting beam. If the exciting beam is verticallypolarized, preferential excitation of those molecules whose transition momentsare oriented in the z-direction will occur. For unpolarized exciting radiation,those molecules whose moments lie in the x–z plane are preferentially excited.

For short times after excitation, before significant Brownian rotation hasoccurred, the relative magnitudes of reflect this biased distribution.However, because of the finite breadth of the excitation pulse, Brownianrotation occurs while excitation is still taking place, so that the maximumobserved value of A is less than the true value of For a fluorophorewith rotational mobility in a liquid medium, the direction of the transitionmoments becomes progressively randomized with time until a uniform dis-tribution is ultimately approached with and A =0 (Figure 1.2).

The time profile of anisotropy decay of a fluorophore is dependent uponits rate of rotational diffusion. This is in turn related to its molecular charac-teristics and, in the case of a fluorescent conjugate, to those of the biopolymerto which it is linked. Most commonly, one is interested in the behavior ofa fluorescent probe joined covalently, or noncovalently, to a larger macro-molecule. In order to obtain a manageable expression for the time dependenceof anisotropy, it is necessary to approximate the actual, somewhat irregular,shape of the macromolecule by a smoothed geometrical form, such as anellipsoid of revolution.

We will consider first the case in which the fluorescent label is rigidly

8 Robert F. Steiner

attached to the macromolecule, with a fixed orientation of its transitionmoments with respect to the coordinate axes of the latter. For a system of thiskind, the time decay of anisotropy depends not only upon the characteristicsof the macromolecule, but also upon the orientations of the transitionmoments of absorption and emission with respect to the molecular axes.

Prior to describing the theory of anisotropy decay, it is appropriate todiscuss the rotational motion of rigid macromolecules, approximated asellipsoids.

1.2.4. The Rotational Diffusion of Ellipsoids of Revolution

The rate of rotational diffusion may be characterized by a rotary diffusionconstant, which is analogous to the familiar translational diffusion coefficient.The rotational motion of a diameter of a rigid sphere with an arbitrarilychosen orientation at zero time may be described by an equation analogousto Fick’s second law:

Here, W is a probability density function describing the distribution ofrotational angles, w is the rotational angle of the spherical diameter, and Dis the rotary diffusion coefficient.

In the case of a spherical particle, the rotary diffusion coefficient, isgiven by

where k is Boltzmann’s constant, T is the absolute temperature, is thesolvent viscosity, and V is the effective hydrodynamic volume, being equal tothe anhydrous volume plus an increment corresponding to the bound waterof hydration.


In order to describe the hydrodynamic properties of proteins, it is oftenconvenient to approximate their actual, somewhat irregular shape by asmooth and symmetrical geometrical figure. Ellipsoids of revolution, whichare the three-dimensional bodies generated by rotating an ellipse about one ofits characteristic axes, are perhaps the most commonly used of these forms.An elongated, or prolate, ellipsoid of revolution is generated by rotationabout the long axis of the ellipse, while a flattened, or oblate, ellipsoid ofrevolution is formed by rotation about the short axis.

An ellipsoid has three principal axes, each of which is associated with acharacteristic rotational diffusion coefficient. The three diffusion coefficientsare designated where the subscript indicates the axis aboutwhich diffusion occurs. In the case of a symmetrical ellipsoid of revolution, ifaxis 1 corresponds to the axis of symmetry and axes 2 and 3 correspond tothe (equivalent) equatorial axes, then (Figure 1.3).

The rotary diffusion coefficients for rotation about the axisof symmetry and the equatorial axes, respectively, may be related to the axialratio of the ellipsoid, and to the rotary diffusion coefficient, of theequivalent sphere by, for a prolate ellipsoid of revolution(29):

It is also useful to define a set of three rotational correlation times,which are functions of the rotational diffusion coefficients(30, 31):

10 Robert F. Steiner

For a particle with spherical symmetry,

Introducing the value of

1.2.5. The Anisotropy Decay of Ellipsoidal Particles

1.2.5.1. Theoretical Treatment of Anisotropy Decay

The time decay of anisotropy for a rigid ellipsoidal particle (notnecessarily an ellipsoid of revolution) has been treated by Belford et al.(21)

The model assumes that the fluorophore is immobilized with respect to theparticle, so that its transition moments have well-defined orientations withrespect to the axes of the particle. The time decay of anisotropy is in thiscase governed by the magnitudes of the three rotational diffusion coefficients

corresponding to the three axes and by the directions ofthe linear transition moments of absorption and emission. Belford et al. havederived the following equation for the most general case:

Here, D (the mean rotational diffusion coefficient)

or 312) where are the cosines of the angles formed by thetransition moments of absorption with the three axes, and ' arethe corresponding direction cosines of the transition moments of emission.The other quantities are defined by:

In the limiting case of a particle with spherical symmetry, for whichreduces to

Fluorescence Anisotropy Theory and Applications 11

where For a symmetrical ellipsoid of revolution, Eq. (1.29)reduces to the sum of three exponentials:

Here,

where are the rotary diffusion coefficients for rotation about theaxis of symmetry and about either equatorial axis, respectively, and

where are the angles formed by the absorption and emission tran-sition moments, respectively, with the axis of symmetry of the ellipsoid, and

is the angle formed by the projections of the two moments in the planeperpendicular to the axis of symmetry.

For low values of corresponding to the initial slope of theanisotropy decay curve(1),

Here, is the harmonic mean of the three correlation times and is defined,in the general case, by:

For the special case where the transition moments are randomly oriented withrespect to the axes of the particle so that

then


In the case where the moments of absorption and emission coincide, thenso that Eqs. (1.36) become

For an ellipsoid of revolution for which the absorption or emissionmoment is parallel to the axis of symmetry the time decay ofanisotropy is described by Eq. (1.33), but with replaced by

For the case of an ellipsoid of revolution for which the transitionmoments of absorption and emission lie in a plane perpendicular to the axisof symmetry, a more complex expression is obtained:

Here,

where is the rotational diffusion coefficient about the axis of symmetry,and is the angle formed by the two transition moments.

In contrast to the first special case, the time decay of anisotropy is hereclearly nonexponential. Also, if the second term on the right-handside of Eq. (1.43) will be negative, leading to the possibility of negativeanisotropies at short times after excitation. Since, for a prolate ellipsoid of

revolution, is always greater than a transition to positive anisotropiesoccurs at longer times in this case, followed by a monotonic decay to zero.

While, in principle, Eqs. (1.29) and (1.34) would permit the estimation ofthe shape parameters of a rigid asymmetric protein whose shape can bereasonably approximated by an ellipsoid, in practice instrumental limitationsnormally make it difficult to monitor anisotropy decay over much more thanone decade. This is insufficient for the accurate detection of multiple correla-tion times arising solely from molecular asymmetry. Further improvements ininstrumentation will probably be required before these relations can beprofitably capitalized on.


1.2.5.2. Simulation of Anisotropy Decay

An alternative approach to the time decay of fluorescence anisotropy hasbeen presented by Harvey and Cheung, who have utilized computer simula-tion.(32) A large population of equivalent fluorescent molecules, each with itsown internal coordinates and orientations of the linear transition momentsof absorption and emission, is formally assembled by means of a random-number generator, which is also used to produce the random stepwise motionof Brownian rotation. The frame of reference of laboratory coordinates ischosen so that the z axis is parallel to the direction of polarization of theexciting beam, while the y and x axes lie in the direction of the exciting beamand in that of observation, respectively. The angular distribution of orienta-tions in the initial excited state is generated by setting the probability ofexcitation proportional to is the angle formed by theabsorption transition moment with the z axis.

The normalized components of fluorescence intensity which are polarizedparallel and perpendicular to the direction of polarization of theexciting beam are given by

where is the angle formed by the emission moment with the z axis.The simulation of Brownian rotation was done by increasing the time in

a series of small increments and rotating the molecule at each step abouteach of its axes by angular increments ( /= 1, 2, 3) with the sign ofchosen randomly. The magnitude of each rotational step was determined bythe rotary diffusion coefficient about the corresponding axis, as predicted bythe classical theory of Brownian rotation:

After a set of time increments, the anisotropy was computed using Eqs.(1.45)–(1.47). This was repeated for a series of such data sets, and the resultantanisotropies were plotted as a function of time. This was done for oblate andprolate ellipsoids of revolution with varying axial ratios and orientations ofthe transition moments. In all cases the simulated anisotropy decay corre-sponded closely to the predictions of the theory of Belford et al.(21)

A surprising prediction of the theory of Belford et al., which was verifiedby the simulations of Harvey and Cheung, is that, for certain orientations ofthe transition moments, the anisotropy actually increases with time for sortintervals after excitation, so as to pass through a maximum before decayingto zero at long times (Figure 1.4).


1.2.6. Partially Immobilized Systems

A special case of interest is that for which the rotational motion of alabeled biopolymer is restricted so that free rotation occurs only about a singleaxis. This might, for example, be the case for a labeled protein embedded ina lipid bilayer which constrains its rotation to a single axis perpendicular to


the plane of the bilayer. In this case, rotational diffusion cannot depolarize thefluorescence entirely, so that the anisotropy approaches a finite value at longtimes. The time decay of anisotropy is given by, for a spherical particle(22, 34):

where is the correlation time for rotation of the particle about the fixedaxis, and The most general expression for the limitinganisotropy at long times, is

where is the average value of the cosine squared for all possible anglesbetween the direction of the absorption transition moment at the time ofabsorption and the emission transition moment at the time of emission.

If rotational wobble of the fluorophore is absent, so that rotation isstrictly confined to a single axis of the labeled biopolymer, then the motionof the emission moment will be effectively confined to the surface of a cone ofsemiangle is the angle formed by the emission dipole with thedirection of the axis about which rotation of the particle occurs. (1, 22, 34)

If the absorption and emission moments are parallel, then

Kinosita et al.(22) and Lipari and Szabo(34) have considered the parallelcase for which rotational wobble of the fluorophore is allowed. The rotatingunit, which may be either an unattached fluorescent probe or a labeledbiopolymer, is assumed to have cylindrical symmetry and the medium to haveuniaxial symmetry. The orientations of the transition dipoles of absorptionand emission with respect to the symmetry axis of the rotating unit areassumed to be invariant, so that the equilibrium orientational distributionof the two dipoles depends solely upon the angle between the axis of therotating unit and that of the medium (Figure 1.5). Since, for this case also, themembrane-embedded fluorophore cannot assume all possible orientationswith equal probability, the anisotropy does not decay to zero at long timesbut instead approaches a finite value,

A simple solution is possible if either the absorption dipole or theemission dipole is parallel to the direction of the unique symmetry axisof the rotating unit. If are unit vectors in the directions of theabsorption and emission dipoles and the axis of symmetry, respectively, thenA(t) may be expressed in terms of a correlation function as

Here, is the angle between P2 is the second Legendre polynomial


and the angle brackets indicate an equilibriumaverage, defined as follows:

where is the equilibrium orientation distribution function. In Eq. (1.51)the unit vector specifies the orientation of the probe at time t with respectto a membrane-linked coordinate system (Figure 1.5).

In order to obtain an expression for the complete time dependence ofA(t), it is necessary to assume some model for the dynamics of the probe.However, may be computed independently of any model:

and


where are the angles between the symmetry axis of the probe andrespectively.

Equation (1.54) relates the limiting fluorescence anisotropy to the orderparameter S of the probe, defined as

The order parameter governs the magnitude of the first nontrivial term inthe series expansion of the orientational distribution function in terms ofLegendre polynomials(34):

The order parameter thus furnishes model-independent information about theorientational distribution function at equilibrium.

Kinosita et al. have developed an alternative formulation in terms of anexplicit model in which the probe can undergo free rotational diffusion withina cone of semiangle This corresponds to the distribution function

For this model,

If either is parallel to then, using Eqs. (1.53) and (1.54),

To the extent that the diffusion in a cone model is valid, Eq. (1.59) permitsevaluation of the semiangle of the cone.

In order to evaluate the time-dependence of A(t) according to equation(1.51), it is necessary to evaluate the correlation functionThis requires the choice of a dynamic model. For the model corresponding todiffusion in a cone, Kinosita et al. showed that(22)

where is the effective correlation time corresponding to the ith rotationalmode and is the corresponding amplitude. An approximate expression wasalso derived in terms of a single rotational mode:


It may be shown(34) that is equal to the area under the curve of

1.2.7. The Influence of Internal Rotation

For real biopolymer systems, an intrinsic or extrinsic fluorescent labelwhich is rigidly integrated into the three-dimensional structure, so as to havea fixed and well-defined orientation with respect to the coordinate axes of theparticle, is often not obtained. In many cases, some form of internal rotationis present, so that a rotational motion of the label is superimposed upon thatof the entire particle.(20, 33, 34) The various types of internal rotation which arepossible may be roughly grouped into the following categories:

1. rotation of the fluorophore about the bond linking it to thebiopolymer;

2. rotational wobble of that portion of the biopolymer in proximity tothe fluorophore;

3. rotation of a well-defined molecular domain as a unit about a flexiblehinge point.

In practice, more than one rotational mode may be simultaneously present(Figure 1.6).

The complexity of the general problem does not favor the developmentof a comprehensive theory which would encompass a wide range of cases. Inparticular, the data obtainable with the current instrumentation do notreadily lend themselves to analysis in terms of the more general theoreticaltreatments, especially if more than one internal rotational mode is present.(35)

However, it is possible to obtain tractable solutions for several special casesof interest.

In particular, if only a single internal rotational mode is present, then thetheoretical treatment outlined in Section 1.2.6 may be extended to the case ofa label attached to a large, freely rotating particle. If the assumptions citedearlier are retained, then, for a spherical particle and a label which is confinedto a cone of semiangle

where are the correlation time of the particle and the effectivecorrelation time of the label, respectively, and is given by Eq. (1.59). If

then Eq. (1.62) reduces to the following relationship:


Equation (1.63) is usually applied in the more general form:

Here, depends solely on the localized rotation of the label, and reflectsthe rotation of the entire macromolecule. The sum of is equal to theanisotropy at zero time, A0.

For the case where the bond linking the fluorophore to the macro-molecule has a sufficient degree of rotational wobble so that the fluorophoremay rotate freely within a cone, the magnitudes of a1 and a2 are given by anequation analogous to Eq. (1.59)(22,34).

where is the semi-apex angle of the cone. Lipari and Szabo have shownthat Eq. (1.65) is valid for a probe with cylindrical symmetry, for which eitherthe absorption or emission moment is parallel to the cylindrical axis.(34) Thismodel implies the existence of a square-well potential which restricts rotation.

If wobble is absent, so that the emission moment is confined to the sur-


face of a cone, as would be the case if the emission moment makes a constantangle with the axis of rotation, then we have

The preceding treatment is strictly valid only for the case when eitheris parallel to the axis of the probe. If this is not the case, rotational

diffusion about the axis of the probe, as well as the rotational wobble of theaxis itself, can contribute to the time decay of anisotropy. The theory nowbecomes considerably more complex. Lipari and Szabo have derived thefollowing approximate expression for the time dependence of anisotropy(34):

where are the respective angles between isthe difference between the azimuthal angles of are theeffective correlation times for rotation about the axis and for the rotation ofthe axis itself, respectively. The quantities are reduced Wigner rotationmatrices:

In a subsequent theoretical development, Szabo has treated in detail anumber of models.(35)

Equation (1.64) is often loosely used as an empirical equation which hasbeen widely used to interpret anisotropy decay data.(4) In practice, useof a two-term equation is usually compelled by the inability of availabletechniques to reliably recover more than two correlation times.

In actual systems the existence of completely free rotation of thefluorophore is somewhat improbable, and some degree of hindrance torotation is likely to be present. It is useful to consider the limiting casewhere the label undergoes strongly hindered rotation. For this model the labelmay exist in any of several orientations corresponding to potential energyminima; the time spent in the specific positions is long in comparison with


the time spent in transit between positions. In this case the rate-limiting factorfor rotation is the probability of a jump between specific orientations. Thefollowing equation was derived by Gottlieb and Wahl for this model(20):

where is the frequency of jumps between positions, K is a numericalconstant, and

Equation (1.69) may be rewritten in a form analogous to Eq. (1.64). Theparameters now have the meanings:

If the above model is strictly adhered to, w should be insensitive tosolvent viscosity, since it depends solely on the magnitude of the potentialenergy barrier encountered by a transition between different positions. Thisproperty may provide a potential means of ascertaining the validity of themodel. If the viscosity of the solvent is altered by the addition of sucrose,glycerol, or some other viscosity-increasing substance, the value of is nowgiven by, from Eq. (1.70),

where is the factor by which the viscosity is increased. If measurements aremade for a series of viscosities and is linearly extrapolated versus to

then the intercept and slope should yield w and respectively. Thisanalysis is based on the assumption that the effective microviscosity sensed bythe label is equivalent to the bulk viscosity of the solvent. The rotating labelmay be partially, or wholly, shielded from the solvent, so that the effectivemicroviscosity is less than the bulk viscosity. In this case, extrapolationaccording to Eq. (1.71) would lead to an underestimation of and an over-estimation of w.

The above simplified models do not adequately account for the moregeneral case where the independent motion of the probe is superimposed uponthe rotational wobble of the adjacent polypeptide or the rotation of a well-defined subelement within the overall protein structure. With the usualinstrumentation available today, it is questionable whether more than twocorrelation times can be reliably extracted from anisotropy decay data. Ifmore than two rotational modes are present, application of an equation ofthe form of Eq. (1.64) will result in poorly defined averages. However, in


favorable cases where the probe is firmly immobilized in the tertiary structure,the detection of a correlation time close to that expected for a well-definedmolecular domain is evidence for the free rotation of the domain.

1.3. Experimental Analysis of Anisotropy Decay

1.3.1. Analysis of Time-Domain Data

The analysis of single or multiexponential decay of anisotropy has mostcommonly made use of the relationship

For vertically polarized exciting light,

where g is a factor correcting for the (usually imperfect) optical system andis given by

where are the summed values over all times of respec-tively, when the exciting light is horizontally polarized. In the absence ofdistortion arising from grating effects, etc., should be equal. For thedetermination of g, the fluorometer is normally operated in the static mode.

The time decay of fluorescence intensity may be represented by a modelof the form

where are the amplitude and decay time, respectively, of the ithdecay component.

The values of the may be determined from the experimental databy a least-squares fitting procedure.(36) Trial values of the are used togenerate the intensity decay function S(t). For each set of the valuesof the are obtained by solution of the corresponding set of equations linearin the which are obtained for different times. The computed curve of


S(t) is convolved with the instrumental response function E(t), accordingto Eq. (1.18), to yield a computed intensity decay function This iscompared with the experimental decay function and a value of thecriterion for goodness of fit, is calculated from

where are the values of respectively,for the jth data point, and is the corresponding precision. Since single-photon counting obeys Poisson statistics, is equal to where Y(j)is the number of single-photon counts for the jth data point.

The set of values of the which correspond to a minimum value ofmay be located by an iterative procedure in which each is incremented util

passes through a minimum. The value of corresponding to a minimalmay be located by parabolic interpolation. The process is repeated for each ofthe in turn until an overall optimum fit is attained. For each set of valuesof the the corresponding values of the are identified by a linear least-squares fit.

The time-dependent fluorescence anisotropy, A(t), is given by

The time decay of anisotropy may be represented by a model of the form

Here, Ai and are the amplitude and rotational correlation time, respec-tively, corresponding to the ith rotational mode, and is the limiting valueof the anisotropy attained at very long times after excitation. For fluorescentmolecules which are not partially, or wholly, immobilized in a matrix,

In evaluating the the previously computed values of the andmay be used to generate an impulse response function which represents

S(t), according to Eq. (1.77).(37) Trial values of the Ai and are used tocompute an analogous function for the anisotropy, A(t), according toEq. (1.79).(37) A(t) is then multiplied by S(t) to yield a trial representation ofthe deconvoluted difference decay function D(t). D(t) is next convolved withthe time profile, E(t), of the excitation pulse, according to Eq. (1.18), togenerate a computed difference decay function is comparedwith the experimental difference decay curve and a value of iscomputed.


For each set of trial values of the Ai and the corresponding value ofis computed from

where are the values of these quantities for the jth datapoint. The precision is taken as equal to where V(j) is theassociated variance. The optimum values of the Ai and are identified by aniterative procedure similar to that used to determine the decay times.

The ultimate criterion of the quality of fit associated with a particular setof parameters is the normalized value of This is equal to whereF is the number of degrees of freedom and is equal to the number of datapoints (time channels) minus the number of determined parameters. For aperfect fit, would have a value of unity.

A second useful criterion of quality of fit for both intensity andanisotropy decay data is the distribution of residuals, that is, the differencebetween observed and computed values. For an optimum fit the distributionof residuals plotted as a function of time should be essentially random.

An alternative criterion of quality of fit is the Durbin–Watson parameter,which is defined by(38)

The Durbin–Watson parameter monitors the correlation between theresidual values in neighboring channels. An optimum fit corresponds to amaximum in this parameter, which implies a maximum randomness ofresiduals.

An empirical procedure has proved to be useful in determining whetheris a real operating minimum for the set of optimized parameters.(39) A small

(2%) increase is made in the value of one parameter, and the correspondinghigher value of is computed. By parabolic extrapolation the parametervalue for a 5% increase in is determined. The procedure is repeated for a2 % decrease in the value of the same parameter. If the spread in parametervalues is symmetrical, is an operating minimum. The process is repeated forthe other parameters in turn. The changes of 2% and 5% are of courseentirely arbitrary, but have proved satisfactory in practice.


1.3.2. Time-Domain Measurements of Anisotropy Decay

Time-domain measurements of anisotropy are generally made using asingle-photon-counting nanosecond fluorometer.(13–15) A pulsed light sourcewhich flashes at a frequency of 15kHz to 2 MHz is used. Both lasers andpulsed arc sources have been used. The electronic elements of instruments ofthis kind must supply the following basic functions:

1. a start channel which times the beginning of a light pulse;2. a stop channel which times the detection of single photons;3. a selection channel which selects true single-photon events;4. a data acquisition system whereby the accumulated photon counting

events are stored in the corresponding time channels in a multi-channel analyzer.

The net result is the generation of a cumulative histogram of single-photon counts stored in the time channels of the multichannel analyzer. Thisis equivalent to a time profile of intensity. In order to avoid distortion of thedecay curve, precautions must be taken to avoid emission of more than onephoton as a consequence of a single lamp flash. In practice, this may beachieved by limiting the rate of photon detection to about 2 % of that of lampflashing by a suitable adjustment of apertures.

Spectral selection for the excitation and emission beams may be achievedwith the use of either filters or gratings. The excitation beam is normallyvertically polarized for an anisotropy decay experiment. The orientation of theexcitation polarizer may be verified by comparing the intensities of lightscattered by a glycogen suspension with the emission analyzer in the verticaland horizontal orientations. The latter intensity should be less than about 1 %of the former.

The time profiles of intensity decay for the vertically and horizontallypolarized components of fluorescence intensity are accumulated and stored indifferent sectors of the memory of the multichannel analyzer. In moderninstruments the emission analyzer is automatically alternated between verticaland horizontal orientations for a series of short intervals, thereby minimizingany differential effects of instrumental drift.

The time profile of the excitation is usually obtained using a scatteringsolution, suxh as a Ludox silica suspension. Since fluorescence normallyoccurs at a longer wavelength than scattered light, the wavelength dependenceof the photomultiplier may result in a significant time frame shift between thefluorescence and reference scattering signals. This may be corrected for eitherby determining the shift directly by measurement of samples of accuratelyknown lifetimes or by treating the shift as a variable whose value is chosen soas to optimize the least-squares fit of intensity decay data.


1.3.3. Frequency-Domain Measurements of Anisotropy Decay

In recent years the measurement of anisotropy decay in the frequencydomain has been developed as an alternative to direct observation of the timedependence.(40–43) In frequency-domain studies, the measured quantities arethe phase angle and the modulation m of fluorescence, which is excited withlight whose intensity i varies sinusoidally with time:

Here where f is the frequency. The modulation is defined by

For complete modulation, b = a and m = 1.As a consequence of the finite duration of the excited state, the

modulated fluorescence emission is delayed in phase by an angle relative toexcitation. In addition, a decrease in modulation of the fluorescence occurs.The intensity I of the fluorescence is given by

and the relative modulation, or demodulation factor, by

If only a single intensity decay mode is present, the phase and demodulationfactor are related to the decay time by

and

If multiple decay modes are present, the apparent values of computed fromEqs. (1.86) and (1.87) will correspond to averages. Ih this case measurementsover a range of frequencies are necessary in order to obtain well-definedphysical parameters. The phase angle and modulation values may be pre-dicted as a function of frequency for any postulated decay law. The assumeddecay law contains the information necessary for this prediction, just as it alsocontains the information required to predict the directly observed timedependence of fluorescence.

For frequency-domain measurements, sine and cosine transformations of


the assumed decay function (Eq. 1.77) are used instead of convolution withthe excitation lamp profile. If the quantities are defined by

then the phase angle and modulation may be predicted for each frequencyfrom

and

where are the computed values of the phase angle and demodula-tion factor, respectively, for frequency

The quantities may be related directly to the intensity decayparameters by

The goodness of fit between the computed and observed values of phaseangle and modulation may be assessed from the value of

Anisotropy decay may also be measured in the frequency domain. Theindividual polarized components are related to the total intensity i by:

whereThe quantities N and D may be evaluated for the vertically and horizon-

tally polarized components, using Eqs. (1.88) and (1.89). The quantities which


are actually measured are the phase angle difference between the twocomponents and the ratio of the modulated amplitudes:

The corresponding calculated values of respec-tively) may be obtained using

and

The quality of fit is estimated from

where n is the number of degrees of freedom (the number of data points minusthe number of variable parameters), and are the estimateduncertainties at frequency respectively.

In fitting actual data, trial values of the correlation times, are used tocompute the anisotropy as a function of time. The previously determined setof intensity decay times, are used to compute the intensity, i(t), as a func-tion of time. The substitution of A(t) and i(t) in Eqs. (1.95) yields computedvalues of introduction of these into Eqs. (1.88) and (1.89)provides values of as a function of frequency. Finally,Eqs. (1.97) and (1.98) are used to generate trial values ofthese are compared with the observed values, and a value of is calculated.The set of assumed correlation times is varied systematically by an iterativeprocess until the set yielding a minimum value of is identified.

1.4. Anisotropy Decay of Heterogeneous Systems

1.4.1. Anisotropy-Resolved Emission Spectra

In a heterogeneous mixture of fluorescent species, fluorophores differingin correlation time may also have different emission spectra.(44) A simple


example is a small fluorescent ligand which is reversibly bound by a largebiopolymer, such as a protein or nucleic acid. If the emission spectra of thefree and bound ligand are different, the observed static emission spectrum,measured in the conventional way as the total intensity as a function ofwavelength, will be composite, corresponding to the sum of the individualspectra. However, if instead of total intensity, S, the difference in intensity,

between the vertically and horizontally polarized components offluorescence is measured as a function of wavelength, the observed spectrumwill reflect primarily the complex species since D will be very small for the freeligand. If D is measured as a function of wavelength for a series of times afterexcitation, then only at very short times will the free ligand make a significantcontribution to the emission spectrum. At long times after excitation thevalues of D for the free ligand will have decayed to zero, so that the observedspectrum arises entirely from the complexed species. (For a homogeneoussystem whose correlation time is long enough to permit measurement, thespectra obtained using D and S are equivalent, since the wavelengthdependence of is the same as that for

More generally, for a heterogeneous system containing « species, theanisotropy as a function of time and wavelength is given by(44)

Here, an overbar indicates a heterogeneous sum; represents the emissionspectrum of species i; and Di(t) and Si(t) are the values of D and S forspecies i at time t. What is desired is the form of each The data requiredare the time dependence of A, D, and S as a function of wavelength.

The necessary information is contained in

If is measured at n or more different times (tj), the following set oflinear equations is obtained(38):

These correspond to the matrix equation


where each matrix element is given by

Further progress requires a knowledge of the individual functionsfor each tj. These may be obtained through an analysis of the time decays ofanisotropy and intensity, as described in earlier sections. If each Di is known,then the set of may be obtained by matrix inversion. The inverse of M,G, is multiplied by the observed value of , The corresponding matrixequation is

and

where Fi corresponds to the magnitude of the emission spectrum at wave-length of component i.

In this way the correct linear combinations of the values of measuredat different times will yield the anisotropy decay-associated spectra of eachspecies, The method depends upon the obtaining of accurate values ofthe Di. In some simple cases where a single decay time of fluorescenceintensity is unambiguously associated with a single rotational correlation time,each individual Di is directly related to these quantities. Thus, for a binarymixture whose species have distinct spectra, intensity decay times androtational correlation times we have

and

In more complex cases there may not be an obvious correspondence oflifetimes and correlation times, and a direct analysis of in terms of a set ofdecay times, each of which corresponds to a particular Di, may benecessary.(44)

In practice, the above relationships must be modified to allow forconvolution effects and for the fact that time-resolved emission spectra aregenerally measured over a range of time channels. In what follows, lower case


quantities will refer to the instrument (convolved) time scale. Thus, Di isreplaced by

where E(t) represents the excitation pulse (cf. Eq. (1.18)).Equation (1.101) now becomes

where now refers to the jth time window, which extends from timechannel aj to bj for the jth time-resolved emission spectrum. Since convolutionaffects solely the time axis, the matrix elements now are

Also,

where Gki is the inverse matrix of is the windowed difference time-resolved emission spectrum, and is the desired anisotropy decay-associated spectrum. The presence of convolution effects thus does not preventthe extraction of anisotropy decay-associated spectra.

Davenport et al. have employed the preceding theory to resolve theanisotropy decay-associated spectra of 1, 6-diphenyl-l, 3, 5-hexatriene (DPH)which is distributed between vesicles of (DML) and

In this model system the DPH fluorophorerotates rapidly within the DML vesicles, but rotates more slowly within theDPL vesicles. The corresponding emission spectra are significantly different,while DPH in a mixture of the two types of vesicles yields a static emissionspectrum of an intermediate type. Analysis of the mixed system using thetreatment described above yielded individual spectra which corresponded tothose obtained for purified vesicle systems.

1.4.2. The Meaning of Correlation Times for Associative and NonassociativeHeterogeneity

Time-resolved decays of both intensity and anisotropy are often multi-exponential. Both types of heterogeneity may be intrinsic to a single kind of


emitting species. A heterogeneity of intensity decay may arise from excitedstate processes or from the existence of populations of distinct conformers;a heterogeneity of anisotropy decay may reflect the presence of multiplerotational modes, such as localized and global diffusion. Heterogeneity of theabove kinds, which is termed nonassociative, often permits a comparativelystraightforward interpretation of anisotropy. The anisotropy at a particulartime t is equal to is equal to A0, the limitinganisotropy at zero time, which is characteristic of the fluorophore in theabsence of rotational diffusion. In this case any observed correlation modescorrespond to rotational modes involving the fluorophore of interest.

If the heterogeneity of intensity and anisotropy decay arises from multi-ple emitting species, the situation is quite different. For heterogeneity of thiskind, which is termed associative, it is not usually possible to identify anobserved correlation time with a rotational mode of a particular fluorophore.While the observed anisotropy at all times corresponds to the intensity-weighted average of the anisotropies of the different species, according toEq. (1.15), the fractional intensities are themselves time-dependent (unless thedifferent emitting species have identical decay times). Thus, at long times afterexcitation the observed average anisotropies become increasingly influencedby the species of longest decay times.

In practice, a wide variety of anisotropy decay patterns, some of themvery bizarre, can arise in this way. For example, if two emitting species arepresent, one of which has much longer decay times of both intensity andanisotropy than the other, the anisotropy may actually pass through a mini-mum and increase at longer times. In other, less dramatic cases, effects of thiskind may lead to plausible, but erroneous, interpretations of anisotropydecay.

1.5. Anisotropy Decay of Intrinsic Protein Fluorophores

1.5.1. Anisotropy Decay of a Rigid Protein: S. Nuclease

Staphylococcus aureus nuclease B, whose molecular weight is 2.0 × 104,contains a single tryptophan residue. Time-domain measurements of itsanisotropy decay were made in an early study by Munro et al. whichemployed a synchroton light pulse as the excitation source.(4) The half-widthof this source, 0.65 ns, was substantially less than those of the flashlamps incommon use at that time and facilitated the detection of very rapid rotationalmodes.

The time decay of fluorescence intensity at pH 7.4, 20 °C, was found to beexponential, corresponding to a single decay time of 5.05 ns. The time decay


of anisotropy was also found to be exponential and could be described interms of a single rotational mode, of correlation time 9.85 ns. Thus, noevidence for any internal rotational modes was found in this study.

Lakowicz et al. have subsequently reinvestigated this system, employingfrequency-domain measurements.(46) In contrast to the results of the earlierstudy, two decay components were required for an optimal fit of the fluores-cence intensity decay, withrespectively.

The time decay of fluorescence anisotropy was also biexponential, withrespectively. The con-

tribution of the more rapid rotational mode, although minor, appeared to bereal, in view of a twofold decrease in relative to that for the one-componentfit (Figure 1.7).

The model which emerges from these studies depicts the tryptophan assensing a relatively rigid microenvironment, in which its rotational motion isconfined to a rather narrow cone. Both studies assign a correlation time near10 ns to the dominant rotational mode. The correlation time predicted for arigid hydrated sphere of the molecular weight of S. nuclease, assuming ahydration of 0.2, is 7.6 ns. The somewhat larger value observed can easily beattributed to the contribution of molecular asymmetry.

S. nuclease thus provides an example of a protein with only slightmobility of its intrinsic fluorophore.

1.5.2. Rotational Dynamics of Flexible Polypeptides: Adrenocorticotropin andMelittin

An example of the application of fluorescence anisotropy decay to a smallpolypeptide with internal flexibility is provided by the study by Ross et al. ofadrenocorticotropin (ACTH).(47) The fluorescence of the single tryptophangroup of ACTH, Trp-9, was examined. The time decay of fluorescenceintensity could not be adequately fit by a single-exponential decay law. Theassumption of at least two decay times was required to yield an acceptable fit;no further improvement occurred if three components were assumed. Twodecay components were found in the absence and presence of 20% sucroseand at temperatures ranging from 3.5 °C to 15 °C (Table 1.1).

Since only one tryptophan is present, heterogeneity can be ruled out asa source of the multiple decay components. As the decay times were inde-pendent of emission wavelength, it is also unlikely that a two-state exited statereaction is responsible. The most plausible explanation is in terms of a mobiletryptophan group which can sample different microenvironments by rotation.Contact with different amino acid side chains may result in different degreesof quenching of the indole excited state and hence in multiple lifetimes.


This model was supported by measurements of the time decay ofanisotropy. For all conditions examined, a satisfactory fit was obtained onlywith the assumption of two rotational modes (Table 1.2). The more rapidrotational mode corresponded to a correlation time in the subnanosecondrange and was attributed to a localized motion of the tryptophan. Themagnitude of the shorter correlation time was almost independent of viscosity.The longer correlation time, which was viscosity-dependent, had a magnituderoughly consistent with rotation of the entire molecule.

The failure of the more rapid rotational mode to respond to an alterationin viscosity is not unexpected if the rotation of the tryptophan involves a


transition between different potential energy minima and the rate-limiting stepis the probability of release from a position of minimum energy. Alternatively,if the tryptophan rotates freely within a cone, the solvent composition andeffective viscosity in its vicinity may be different from those of the bulksolution.

Lakowicz et al. have subsequently reexamined the ACTH system usingfrequency-domain measurements.(46) There was qualitative agreement with thetime-domain results in that the time decays of both fluorescence intensity andanisotropy were found to be biexponential (Table 1.1). As in the earlier study,the anisotropy decay could be described by a subnanosecond correlation time,reflecting the localized motion of the tryptophan, and a longer correlationtime corresponding to the motion of all, or a major portion of, the molecule.While the values of the correlation times are smaller than those found by Rosset al., a quantitative comparison is difficult in view of the different experi-mental conditions. However, the magnitude of the shorter correlation time for


both sets of data is such as to suggest that it reflects the motion of severalamino acids.

It remains uncertain whether the rapid rotational mode reflects solely themotion of the tryptophan with respect to the balance of the protein orwhether neighboring residues are involved. It also remains to be seen whetheronly two rotational modes are strictly present or whether additional modesmay be detected with the development of techniques of higher resolution.

The amphipathic peptide melittin, which is isolated from bee venom,consists of 26 amino acids.(48) The sole aromatic chromophore is a tryptophanresidue at position 19. In solution at low electrolyte concentration, melittin isbelieved to exist as a largely structureless monomer. At high ionic strengths, self-association occurs to form a tetrameric species, which is mostly

Lakowicz et al. have employed frequency-domain measurements toexamine the anisotropy decay of monomeric and tetrameric melittin.(49) Ineach case, determinations were made in the absence and presence of thequencher acrylamide. Dynamic quenching by acrylamide increases the fractionof the total emission which occurs on the subnanosecond time scale, therebyproviding increased information about rotational motions in the picosecondrange. This is of particular usefulness in studying the internal motions ofproteins and peptides.

In 0.01 M Tris, pH 7, at 20 °C, where melittin is monomeric, the intensitydecay was found to be multiexponential and characterized by decay times of0.2, 2, and 4 ns in the absence of quenching. The addition of increasing levelsof acrylamide resulted in a progressive reduction of the magnitude of thedecay times, which were equal to 20 ps, 260 ps, and 1.39 ns in 2 M acrylamide.

The anisotropy decay of monomeric melittin could be analyzedacceptably in terms of two rotational modes (Figure 1.7). The correspondingcorrelation times were essentially independent of the degree of quenching byacrylamide, indicating that the decays of intensity and anisotropy were notcoupled and that acrylamide does not modify the rotational modes. Almost60% of the anisotropy decay was associated with the more rapid rotationalmode, with a correlation time of about 160ps, while the remainder decayedwith a correlation time near 1.7 ns.

In the presence of 2 M NaCl, in which case melittin is tetrameric, theintensity decay was also multiexponential and required the assumption ofthree decay times for acceptable fitting. The decay times decreased from 0.2,2, and 5 ns in the absence of quencher to 0.04, 0.3, and 1.2 ns in 2 Macrylamide.

As in the case of monomeric melittin, the anisotropy decay wasbiexponential. In this case the dominant rotational mode, which accounted forabout two-thirds of the anisotropy decay, corresponded to a correlation timenear 3.5 ns, while the balance of the anisotropy decayed with a correlationtime close to 60 ps.


The longer correlation times observed for both monomeric and tetramericmelittin are in the range expected for rotation of the entire molecules, towhich they can probably be attributed. The more rapid rotational mode mustarise from some form of internal rotation involving tryptophan. It is of interestthat the correlation time associated with the more rapid rotational mode islonger for monomeric than for tetrameric melittin (160ps versus 60 ps),presumably reflecting the differing contribution of segmental motion involvingtryptophan in the two cases. It is also worthy of mention that no convincingevidence was found for the presence of a correlation time of magnitude 1–2 ps,which would arise from the unhindered rotation of a single indole group.

Time-domain measurements of the anisotropy decay of melittin have beenmade by Tran and Beddard.(50) Their findings agree qualitatively with thoseof Lakowicz et al. in that short and long correlation times were observed; themagnitude of the latter was in the range expected for rotation of theentire molecule. The principal difference between the two studies was in themagnitude of the shorter correlation time, for which Tran and Beddard founda value of 600–700 ps, which is substantially larger than that reported byLakowicz et al. An anomalously low value of 0.14 was found for A0 forexcitation at 300 nm. It is possible that the time resolution of this study wasinsufficient to recover the early portion of the anisotropy decay, resulting ina low value of A0 and an elevated correlation time.(49)

1.5.3. Anisotropy Decay of a Tightly Bound Fluorophore: Lumazine Protein

Visser et al. have reported time-domain measurements of the anisotropydecay of the lumazine protein from Photobacterium leiognathi.(39) This protein,which has a molecular weight of 21,000, contains fluorescent 6, 7-dimethyl-8-ribityllumazine as a noncovalently bound prosthetic group. The anisotropydecay of the lumazine fluorophore was monitored for both the free state andwhen combined with protein.

The free lumazine derivative has a quantum yield of 0.45 and afluorescence decay time of 9 ns. The expected rotational correlation time fora molecule of this molecular weight is of the order of 100 ps, which iscomparable to the width of the laser pulse employed This systemthus provides a rigorous test of the ability of time-domain measurements torecover very short correlation times. Simulated data indicated that nearly allof the decay of anisotropy occurred within the time course of the excitationpulse.

The measured correlation time for free lumazine derivative was 70–80 ps(20°C). Essentially the same value was obtained for the unquenchedfluorophore and for the fluorophore in the presence of 0.24 M KI, which


reduced the decay time of fluorescence intensity to 590 ps. If the aromaticportion of the molecule is approximated by an oblate ellipsoid with semiaxesof 4 and 2 an average correlation time close to 40 ps may be computed forthe unhydrated molecule. Since the ribityl side chain, as well as bound water,would be expected to increase the observed correlation time, the measuredvalue is in reasonable agreement with that expected. This study, whichprobably approaches the practical limits of time-domain measurements ofanisotropy decay, underlines the importance of rigorous deconvolution fromthe excitation profile, especially for short correlation times. In the presentcase, simulated data indicated that, in the absence of deconvolution, theobserved time profile of anisotropy is severely distorted by the finite durationof the excitation pulse.

The anisotropy decay of the lumazine-containing protein was alsoexamined. It could be adequately fitted in terms of a single correlation timeof 9 ns (19 °C). The addition of sufficient KI to reduce the decay time offluorescence intensity from 14 to 3 ns did not alter significantly the computedcorrelation time, although the shape of the observed anisotropy decay curvewas considerably distorted. This provided a confirmation of the adequacy ofthe deconvolution procedure.

The lumazine-containing protein has a molecular weight of 2.1 x 104. Ananhydrous particle of this molecular weight with spherical symmetry wouldhave a predicted correlation time of about 6 ns. It is probable that thedifference between the predicted and observed values may be attributed to thecombined effects of hydration and of deviation from a strictly spherical shape.

The lumazine-containing protein can associate with luciferase to form aprotein complex of total molecular weight 1.0 × 105. The quantum yield of thelumazine fluorophore is not altered by the complex formation. The dissocia-tion constant may be controlled by varying such parameters as ionic strengthand temperature. In view of the constancy of the quantum yield and theobservation that the anisotropy decay of the lumazine protein in both the freeand complexed states can be adequately described by a single correlationtime, the preexponential factors ai in the equation

should reflect to the fractions of lumazine protein in the two states.The correlation times of free and completely complexed lumazine protein

were found to be 20 and 60 ns, respectively, at 2 °C. Anisotropy decaymeasurements on mixtures thus provide an excellent test of the ability ofcurrent techniques to resolve correlation times of a similar order of magnitude.The results were not entirely reassuring. If the sum of a1 and a2 was heldconstant and equal to the (invariant) limiting anisotropy and the values of a1,


allowed to vary so as to minimize the recovered values ofwere much too short, being equal to 7 and 33 ns, respectively.

However, if was fixed at 20 ns, the computed value of which corre-sponded to a minimal was close to 60 ns, as predicted, although the rangeof values corresponding to acceptable fits was very broad.

In the present case the ambiguities of fitting probably arise from severalfactors. The contribution of the longer component is less than that of theshorter, so that the former predominates in the noisy portion of the data atlong times. Also measurements extended only over about 55 ns, so thatdepolarization is not complete.

It is clear from these careful studies that the precise resolution of correla-tion times which are not separated by orders of magnitude is a difficultproblem, which has yet to be completely solved.

In a related investigation, Kulinski et al. have monitored the time decaysof fluorescence intensity and anisotropy for the trytophan residue of lumazineprotein in both the apo and holo forms of the protein.(51) In both cases thetime decay of intensity was highly nonexponential, requiring the assumptionof a minimum of three lifetimes for acceptable fitting. The recovered lifetimesinclude a short (0.1–0.6 ns), a medium (1.1–3.6 ns), and a long (6.1–6.6 ns)component. The presence of the lumazine derivative reduces the relativecontribution of the long decay time, resulting in a decreased average decaytime, presumably because of radiationless energy transfer to lumazine.

The time decay of fluorescence anisotropy at 20 °C (excitation at 300 nm,emission at 337 nm) was biexponential, in contrast to the behavior of thelumazine fluorophore. A short correlation time of 0.3–0.5 ns was observed,corresponding to a localized motion of the tryptophan, plus a longer correla-tion time of 5–6 ns, arising from the rotation of a major fraction of themolecule. It is of interest that the latter value is significantly shorter than thatobserved for the lumazine fluorophore. This might arise from either a differentorientation of the transition moments with respect to the molecular axes, thepresence of an internal rotation sensed by the tryptophan, or a combinationof both.

1.5.4. Anisotropy Decay of a Transfer RNA

The presence of a natural fluorophore, the Y base, in the anticodon loopof several tRNA species permits dynamic fluorescence measurements to bemade directly upon the unmodified molecules. In an early pioneering study,Beardsley et al. carried out time-domain measurements of the decay offluorescence intensity and anisotropy for yeast tRNAPhe.(52)

The physical properties of tRNAPhe are modified by Mg2+ ligation,which results in changes in circular dichroism and a major increase in


quantum yield of the Y base. Beardsley et al. found that the intensity decaycould be described by a single decay time of 4.3 ns, which increased to 6.3 nsin the presence of 10mM Mg2+. However, the quality of the fits was poor,as judged from the values, suggesting the presence of unresolved multi-exponential decay.

Parallel time-domain measurements of anisotropy decay were analyzed interms of a single rotational mode. Correlation times of 9.2 and 9.8 ns werefound in the absence and presence of Mg2 + , respectively. The predictedcorrelation time for a rigid anhydrous spherical molecule of the samemolecular weight is about 15 ns. Since the effect of hydration, or any deviationfrom spherical symmetry, would be to increase the average correlation time,the above value is a lower limit. It was accordingly concluded that internalrotational modes involving the Y base were present and that tRNAPhe pos-sesses a significant degree of flexibility.

Wells and Lakowicz have recently reexamined this system by makingfrequency-domain measurements of intensity and anisotropy decay.(53) Withthe improved resolution of the current instrumentation, it is evident thatat least two decay times are required to fit the intensity decay (Table 1.3).In both the absence and presence of Mg2+, the intensity decay could beadequately described by a short decay time of about 2 ns and a longer timenear 6 ns. However, the relative contributions of the two depend on the Mg2 +

level. In the absence of Mg2 + , the amplitudes associated with the two decaytimes are approximately equal; the addition of Mg2 + increases the relativeamplitude corresponding to the 6-ns decay time to over five times that for theshorter decay time.

A plausible explanation for the above observations is that the Y baseexists in two microenvironments which result in different decay times. A Mg2 + -induced conformational change favors the microenvironment associated withthe 6-ns decay time.

Anisotropy decay studies also indicated a major influence of Mg2 +


(Table 1.4). In both the absence and presence of Mg2+, the anisotropy decaycould be accounted for in terms of two rotational modes, corresponding tocorrelation times near 18ns and 0.3–0.4 ns. The presence of Mg2+ substan-tially increased the relative contribution of the slower rotational mode, whichnow dominated the anisotropy decay. It is probably that the longer correla-tion time represents the global rotation of the molecule, while the shorterarises from a localized motion of the fluorophore. An obious explanation isthat the anticodon loop, which contains the Y base, is somewhat flexible inthe absence of Mg2 + , with a significant degree of mobility of the bases.The binding of Mg2+ constrains the anticodon loop into a relatively rigidconformation in which the mobility of the bases is substantially reduced.

1.6. Anisotropy Decay of Biopolymers Labeled with an ExtrinsicFluorophore

1.6.1. Anisotropy Decay and Internal Flexibility of Myosin

The muscle protein myosin is roughly Y-shaped, with a rodlike stem andtwo globular (S-l) units at the head. A requirement of currently popularmodels for the process whereby a myosin “cross-bridge” is linked to the thinactin filaments within a muscle fiber and causes a mechanical thrust is that aflexible hinge point be present within the myosin molecule.(54) Mendelsonet al. have employed anisotropy decay to obtain direct evidence for such ahinge point.(54)

There is present in each S-l unit a single highly reactive sulfhydryl group,which is convenient for the selective attachment of iodoacetamide derivatives.Mendelson et al. employed the iodoacetamide derivative 1,5-IAEDANS[N-iodoacetyl-N1-(5-sulfoamino-l-naphthyl)ethylenediamine], which reactswith sulfhydryl groups and whose fluorescence properties resemble those of


dansyl goups. The fluorescent conjugates prepared and studied by Mendelsonet al. contained about two AEDANS groups per molecule; labeling wasconfined to the S-l head units. The labeled S-l units were also separated bypapain digestion and subsequent chromatographic purification.

For conjugates of purified S-l the time decay of anisotropy could bedescribed by a single rotational correlation time near 220 ns. Using theequation of Belford et al. in the form it assumes for ellipsoids of revolution,a series of predicted curves for anisotropy decay were generated as a functionof axial ratio (Eqs. (1.34)-(1.36)) for an ellipsoid with themolecular weight (1.15 × 105) of S-l and an assumed hydration of 0.2. A regionwas thereby identified in space which was consistent, withinexperimental error, with the observed anisotropy decay. A lower limit for 3.5,was attained at very low values of Since this minimal value agreeswith independent estimates of from other experimental approaches,(55) it islikely that the label is preferentially oriented with its transition momentsroughly parallel to the axis of symmetry of S-l approximated as an ellipsoid.

The anisotropy decay found for myosin itself was also monoexponentialwith a correlation time of 400–450 ns. This value, which is about twice thatfor free S-l, is considerably less than that predicted for a rigid molecule withthe molecular weight (5 × 105) and asymmetry of myosin. Inasmuch as thecorrelation time of an aggregated form of myosin increased to 1800 ns withoutany significant change in the decay time of fluorescence intensity (20 ns), itwas felt to be unlikely that the value for monomeric myosin arose from someform of localized motion of the probe. The favored interpretation was that thelow correlation time reflected primarily an independent rotation of the S-lunits within myosin.(54)

This interpretation was reinforced by parallel studies with heavymeromyosin (HMM), which is formed by the action of trypsin upon myosin.This fragment, whose molecular weight is 3.4 × 105, consists of both globularS-l units joined by a fraction of the rodlike myosin stem. While the molecularweight of HMM is about three times that of S-l, the measured correlationtime (400ns) was less than twice that of the latter fragment.

Although the interpretation is not entirely clear-cut, there is a definiteindication of some type of independent rotation of the S-l units within themyosin molecule. However, this rotation appears to be somewhat hindered, inview of the elevated correlation time, which is twice that of the isolated S-lfragment. It is therefore unlikely that the S-l units are joined to the myosinby some form of highly flexible universal joint.

In a later study Mendelson and Cheung reexamined the question ofpossible interaction of the two S-l units as a possible factor in the elevationof the correlation time of labeled myosin.(56) It is possible to remove a singleS-l head protein by limited proteolysis with papain. The resultant single-headed myosin, in which the remaining S-1 unit was labeled with I AEDANS,


was found to have a correlation time essentially equivalent to that of nativemyosin. This observation tends to rule out mechanical interference betweenthe two S-1 units as the primary cause of the increased correlation time ofmyosin relative to that of S-1. A more likely origin is a significant degree ofstiffness of the polypeptide hinge joining each S-1 unit to the stem. The overallmodel which qualitatively accounts for these results represents the S-1 units astumbling independently about the partially flexible joints connecting themwith the balance of the myosin molecule.

These findings are very relevant to proposed models for muscle contrac-tion and, in particular, to the function of the S-l head proteins as cross-bridgesto the thin actin filaments. The ability of the S-l units to undergo segmentalmotion independently of the myosin stem would render plausible the transla-tion of the myosin molecules along the actin filaments by an “arm-over-arm”movement.

1.6.2. Anisotropy Decay of a Fibrous Protein: F-Actin

The contraction of muscle and the mechanical force which is therebygenerated arise from the cyclic interaction of the two proteins myosin andactin, which is coupled with ATP hydrolysis. The actin-myosin system passesthrough several distinct states in the course of the overall process. Thedependence of the conformation of F-actin upon complex formation and uponthe presence of the various modifiers involved in muscle contraction is a factorof central importance in any comprehensive model of muscle contraction atthe molecular level. Purified F-actin, the bihelical polymeric form of G-actin,exists in solution as very long fibers, whose average length is of the order ofmicrons. If these were rigid and devoid of internal flexibility, their predictedrotational correlation times would be immeasurably long.

The fluorescence dynamics of F-actin have been examined by Wahl andco-workers, who utilized 1, 5-AEDANS conjugates of F-actin.(57) In thepresence of 0.1 mM Ca2+ the time decay of fluorescence anisotropy could beacceptably fitted in terms of two rotational modes according to Eq. (1.64),with the following values of the parameters:

The addition of the myosin proteolytic fragments S-l or heavymeromyosin (HMM) (which consist of the globular head protein and of bothS-l units plus a portion of the stem, respectively) to F-actin in the presenceof 0.1 mM Ca2+ resulted in a progressive increase in σ2 to a limiting value of1100 ns in excess HMM. The decay time of fluorescence intensity varied onlyslightly, remaining close to 19 ns for all compositions. In the presence of10mM ATP, which corresponded to dissociating conditions, the correlation


time reverted to that of free F-actin, thereby ruling out an artifact arising fromthe denaturation of F-actin.

In the presence of the anisotropy decay parameters ofF-actin were substantially altered from their values in A biexponentialfit yielded in this case a1 =0.04, a2 = 0.25, =5.8ns, and = 682ns. Thelonger correlation time thus undergoes a major increase in the presence of

The addition of HMM in the presence of resulted in a biphasicresponse of the magnitude of the longer correlation time, which passedthrough a minimum at a mole ratio of HMM to actin of 0.02, followed by anincrease. The addition of S-l under these conditions produced a similarbiphasic pattern.

These findings suggest that some kind of segmental flexibility exists inF-actin; the degree of flexibility is dependent upon conditions. The shortercorrelation time, presumably arises from some form of localized rotationalmotion, while the longer time, corresponds to the concerted motionof a larger unit, probably a set of actin monomers. The initial decrease inresulting from complex formation with HMM or S-l perhaps arises from anincrease in the flexibility of the links joining actin monomers, while theincrease observed at higher levels of HMM in the presence of and atall levels of HMM or S-l in the presence of may reflect a stiffening ofthese contacts.

The molecular dynamics of F-actin have also been studied by otherphysical techniques, with results which differ quantitatively from thosesummarized above. Thus, correlation times of and have beencomputed from quasi-elastic light scattering and from saturation transferelectron spin resonance, respectively.(58,59) In both cases a substantial increasein correlation time occurred upon complex formation with HMM. While thereasons for the varying results obtained with the different techniques remainobscure, it seems clear that different molecular motions are being sensed bythese methods.

1.6.3. Anisotropy Decay for Proteins Displaying Internal Rotation Involving aWell-Defined Domain: The Immunoglobulins

The antibodies of the higher vertebrates may be grouped intothree classes: IgG, IgA, and IgM. The most common of these, IgG, which hasa molecular weight near is composed of two “heavy” chains ofmolecular weight and two “light” chains of molecular weight

The chains are joined by disulfide bridges to form a molecule whichis roughly Y-shaped. The IgA molecules are of similar structure, while theIgM class consists of polymers of these basic molecular units.

The immunoglobulins, which normally arise in response to exposure tomolecules (antigens) which are foreign to the circulation of the animal, have


the biological function of combining with these antigens so as to facilitatetheir removal. Potential antigens include the proteins and carbohydratespresent on the surface of invading microorganisms. Antibodies may also arisefrom exposure to small groups, or haptens, which are conjugated tohomologous plasma proteins, which are not themselves antigenic.

The IgG immunoglobulins are bivalent, with two equivalent antigen-combining sites per molecule. In the case of multivalent antigens, whichinclude protein and cellular antigens, a three-dimensional network is built up,leading ultimately to an insoluble precipitate.

Papain cleaves the IgG molecules to yield an Fc fragment consisting ofmost of the two “heavy” chains and corresponding to the stem of theY-shaped molecule, plus two Fab fragments, each of which consists of a“light” chain plus the remainder of a “heavy” chain. An intact antigen-combining site is present in each Fab fragment. The Fab fragments representthe arms of the Y.

The IgG molecule may be regarded as an example of a multidomainprotein structure. The susceptibility to papain cleavage of the region near thejunction of the Fab and Fc subunits suggests that randomly coiled sections ofthe “heavy” chains may occur here. This in turn raises the possibility of aflexible hinge point in this region, which would permit some degree ofindependent motion of the structural subelements.

Early static anisotropy studies upon dansyl-conjugated immunoglobulinprovided some indication of the presence of internal rotations, but werequantitatively somewhat discordant.(60) The problem was subsequently rein-vestigated by Wahl, who performed time-domain measuremnts of anisotropydecay for a dansyl conjugate of γ -globulin.(61) The degree of substitution washeld sufficiently low (less than one dansyl group per molecule) to renderunlikely any significant depolarization by radiationless energy transfer.

The convoluted curves of both d(t) and s(t) at pH 8 showed a time decaywhich was clearly nonexponential. Since adequate mathematical proceduresfor analyzing such a complex system were not available at that time, anempirical curve-fitting procedure was adopted. It was assumed that the timedecays of S(t) and D(t) could each be described in terms of two decaytimes. By trial-and-error curve fitting, values of the two sets of decay timeswere found such that the corresponding convoluted forms of S(t) and D(t)reproduced graphically the observed curves of s(t) and d(t). The ratioD(t)/S(t) then yielded the deconvoluted anisotropy A(t) as a function of time.This proved to be also nonexponential. A repetition of the graphical curve-fitting procedure was done to reproduce A(t) by an expression of the form ofEq. (1.64), with a1 =0.075, a2 = 0.14, σ1 = 7.7ns, and σ2= 123 ns The value ofσ2 is in the range expected for the rotation of the entire γ -globulin moleculeand may probably be loosely attributed to this origin. That of σ1 can onlyarise from some form of internal rotation. As the sites of attachment of the


dansyl label, as well as the possible contribution of hindered rotation confinedto the probe itself, were unknown, it was not possible to draw definitestructural conclusions.

Yguerabide et al.(62) subsequently made time-domain measurements ofanisotropy decay for IgG antibodies directed against the fluorescent haptenε-dansyl-L-lysine. Complexes of hapten with both intact IgG and its Fabfragment were studied. The binding of hapten was accompanied by a 25-foldincrease in the quantum yield of fluorescence, so that it was readily feasible toexamine the fluorescence properties of the complex alone. In contrast to theearlier studies, the sites of attachment of the fluorescent label were known andspecific.

The time decays of and were determined for the complex ofε-dansyl-L-lysine with IgG and its Fab fragment. From these data, curves ofs(t) and A(t) were constructed. The time decays of s(t) were found to beequivalent for IgG and Fab. These were nonexponential, indicating thepresence of multiple decay times. This is not unexpected, in view of the knownmolecular heterogeneity of antibodies.

To minimize complications arising from convolution effects, analysis ofA ( t ) was confined to times after complete decay of the excitation pulse. Thisapproach has the obvious disadvantage of minimizing or missing altogetherthe contribution of any rotational modes of very short correlation time, asmight arise from a localized motion of the probe. Unlike the case of s(t),the time profile of anisotropy was very different for Fab and intact IgG.The time decay of anisotropy was exponential for Fab, corresponding to asingle rotational mode of correlation time 33 ns. For a prolate ellipsoidalmolecule with the molecular weight of Fab (5.0 × 104) and a hydration of0.3 ml/g, this value would be consistent with an axial ratio of 2.5. There isthus no evidence for any internal rotation sensed by the probe, although, forthe reasons stated above, a rapid component of the anisotropy decay mighthave been missed.

In contrast to the behavior of Fab, the time decay of anisotropy forintact IgG was nonexponential and could not be fit on the assumption of arigid ellipsoidal shape. A least-squares fit of the observed curve of A(t)indicated that it could be accounted for by Eq. (1.64), with a1 =0.14,

The value of 1, is substantially smallerthan the minimum value (47 ns) predicted for a rigid, unhydrated sphericalparticle of the same molecular weight and is of similar magnitude to thatfound for the Fab fragment. The longer time, is roughly consistent withthat expected for the rotation of a major fraction of IgG.

The above results are compatible with a model in which the Fab sub-molecules rotate with respect to the balance of the molecule. However, someuncertainty remains as to the possible presence of rapid rotational modes ofshorter correlation time, arising from segmental motion within the Fab units.


The independent rotation of the Fab units may be a significant factor in theantibody function, facilitating the combination with antigen.

More recently, these studies have been continued by Lovejoy et al.,(63)

who utilized rabbit antibodies directed against the hapten pyrenebutyrate(PBA). This fluorophore has a much longer average lifetime than dansyl;the value for the free hapten is about 100 ns for an air-saturated solution andsomewhat larger for an O2-free solution. The binding of PBA by the anti-body resulted in a significant red shift of the primary excitation maxima(from 326.5, 341.5 nm to 330.5, 347 nm) and of the emission maxima (from375, 395 nm to 376, 396 nm). The average binding constant was sufficientlyhigh to permit virtually quantitative binding of hapten.

The time profile of fluorescence intensity decay was generally hetero-geneous, except in the case of antibody produced with very long (11-month)immunization times, for which a single fluorescence decay time of 157 ns wasfound. The time decay of fluorescence anisotropy, which was similar for twodifferent preparations, was fitted to Eq. (1.64) by a least-squares procedure.The values obtained wereThe magnitudes of the correlation times are similar to those reported for adifferent hapten by Yguerabide et al. Inasmuch as the time decay for PBAcould be monitored over a sufficient time interval to make possible theaccurate determination of the longer correlation time, these observationsstrengthen the conclusion that mobility of the Fab units is a general propertyof IgG molecules.

Holowka and Cathou have made analogous fluorescence dynamicsstudies on the macroglobulin (IgM) class of antibodies.(64) Immunoglobulinsof the IgM class generally occur in animal sera as disulfide-linked pentamersof total molecular weight near 900,000. Each monomer unit is somewhatsimilar in structure to an IgG molecule, containing two light (L) and twoheavy chains linked by disulfide bonds and noncovalent interactions. Athird unrelated (J) chain is also present and may be involved in the assemblyof IgM from its subunits. A total of ten antigen-combining sites occur on itsten Fab units.

Holowka and Cathou prepared IgM antibodies directed againstfrom horse, pig, and shark antisera obtained by immunization with a

dansyllysine streptococcal conjugate, which favors the formation of IgMantibodies in these species.(64) The time decay of fluorescence intensity of

complexes with purified IgM molecules varied with thespecies. For horse and pig IgM, the dominant component had a decay timenear 24 ns, while a secondary component had a decay time of 8–12 ns. In thecase of shark IgM the pattern was reversed, with the major component havinga short (4 ns) decay time. However, the contribution of the long decay timewas sufficient in all three cases to permit the monitoring of anisotropy fortimes up to 200 ns.


Both horse and pig IgM displayed nonexponential anisotropy decaywhich required at least two correlation times for fitting by Eq. (1.64). In bothcases a very long correlation time was observed, which presumablyreflected the global motion of the IgM pentamer, plus a shorter (61–69 ns)correlation time, which must arise from some form of internal rotation. In thecase of shark IgM a third rotational mode of quite short correlation time

was detected. Also, the intermediate correlation time was somewhatlonger (93 ns) than the short correlation time for the other two species.

In the case of IgM no correlation time was observed which was equiv-alent to that found for isolated submolecules. By pepsin or papain digestionthe fragments, of molecular weight 56,000, were obtained; these areanalogues to the Fab fragments of IgG. In parallel to the latter case, theiranisotropy decay was monoexponential, corresponding to a single correlationtime of 32–36 ns. By limited pepsin digestion the species wasobtained, corresponding to the two arms of the Y, plus a short connectingsegment of the stem; its molecular weight was 105,000–120,000. Its anisotropydecay was nonexponential and required the assumption of two rotationalmodes for fitting according to Eq. (1.64). The corresponding correlation timewere 38 and 211ns. The latter value is consistent with some degree ofindependent motion of the units within the species.

While it is clear that segmental mobility on the nanosecond time scale ispresent in IgM, in contrast to the IgG case it is not possible to make a clear-cut identification of a rotational mode with the motion of a well-definedsubmolecule. This may possibly be attributed to the hindrance of the rotationof the , units by the quaternary structure of IgM.

In a subsequent study, Siegel and Cathou examined the effects of thermaltreatment (30 min at 60 °C) upon horse IgM antibodies against

The thermally treated IgM, as well as its fragment,showed qualitatively a more rapid time decay of anisotropy than the corre-sponding native species. While the magnitudes of the correlation times werealmost unchanged, their relative amplitudes were altered; the contribution ofthe shorter correlation time increased substantially. The implication is that themobility of the unit is increased by limited thermal unfolding.

1.6.4. Anisotropy Decay of Calmodulin Complexes with TNS

Calmodulin, a Ca2+-binding protein of wide occurrence in eucaryoticsystems, is known to combine with, and regulate the activities of, a largenumber of enzymes.(66) The combination with the regulated enzyme isgenerally Ca2+-dependent.(66) Calmodulin functions as an initial receptor forbiological signals involving a change in the level of free Ca2+.(67)


The three-dimensional structure of the form of calmodulinhas recently been described.(68) It is roughly dumbbell-shaped, consisting oftwo globular lobes joined by an strand (residues 66–92). The N- andC-terminal lobes each contain two sites. The molecule, whosemolecular weight is 16,700, is rather asymmetric, being about long, whilethe two globular N- and C-terminal lobes each have dimensions of about

The crystallographic structural determination was carried out forcalmodulin which had been crystallized from an acid medium (50 mMcacodylate, It is of interest to determine whether thecrystallographic structure persists under more physiological conditions, aswell as whether any internal rotational modes are present.

Steiner and Norris have made time-domain measurements of anisotropydecay for complexes of calmodulin with 2-toluidinylnaphtha-lene-6-sulfonate (2, 6-TNS) under varying conditions of pH, ionic strength,and temperature.(69) Calmodulin contains two binding sites for TNS; theN-terminal (1–77) and C-terminal (78–148) half-molecules each contain abinding site. The interaction is Ca2+-dependent, with little or no bindingoccurring for the apoprotein. The TNS fluorophore is almost nonfluorescentin aqueous solution, but acquires an intense fluorescence when bound tocalmodulin. Its use had the advantage of minimizing the contribution ofrotation confined to the fluorescent label, in view of its probable contact withseveral amino acid side chains on the protain surface.

Time-domain measurements were made using a nitrogen pulse lamp assource. While the pulse half-width was substantially longer thanthose of the laser sources employed in other recent studies, this was probablynot a source of serious error, in view of the relatively long fluorescence decay


times encountered. Deconvolution and fitting by a least-squares procedurewere carried out by the methods outlined in Section 1.3.1.

The time decay of fluorescence intensity was multiexponential for allconditions examined. The assumption of two or three components, dependingupon conditions, was required to obtain satisfactory fits. In each case a longerdecay time (12–16ns) and a shorter one (6–8 ns) were observed; at pH 6.5,but not at pH 5.0, a third component of decay time was detected(Table 1.5).

The time decay of fluorescence anisotropy was multiexponential, requiringthe assumption of at least two rotational modes for acceptable fitting(Table 1.6). When corrected to the standard conditions of H2O, 25 °Cby multiplying by the ratio according to Eq. (1.28), themagnitudes of the two correlation times did not vary greatly with temperature(Table 1.6). In each case a longer correlation time of 10–13 ns and a shorterone of 1–3 ns were detected. However, the relative amplitudes of the two rota-tional modes showed a significant dependence upon conditions. At pH 5.0,12°C, the anisotropy decay is clearly dominated by the slower rotationalmode. An increase in pH to 6.5 or an increase in temperature augments therelative amplutide of the more rapid rotational mode.

It is logical to associate the shorter correlation time with a localizedmotion of the label, while the longer correlation time may reflect the globalmotion of the molecule. According to this model, the localized motions sensedby the label are largely suppressed at pH 5 and low temperatures, whichapproximate the crystallization conditions, and become more important atmore alkaline pH and higher temperature.

It is of interest to compare the magnitude of the longer correlation timewith that predicted from the theory of Belford et al. if the actual shape ofcalmodulin is approximated by a prolate ellipsoid of revolution.(21) Such anellipsoid with the same length and molecular volume as calmodulin wouldhave an axial ratio near 3. From Eqs. (1.24)–(1.26), the computed value


of for an assumed hydration of 0.2 ml/g is 13 ns. This is in the range ofthe observed values of the longer correlation time at pH 5.0. These results arethus consistent with the crystallographic structure, provided that one of thetransition moments is roughly parallel to the axis of symmetry of the equivalentellipsoid, so that the anisotropy decay is controlled by this rotational mode,since A2 = A3 = 0 from Eqs. (1.36). Superimposed upon this overall rotation isa localized motion of the probe, which is dependent upon conditions.

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(1985).40. E. Gratton and M. timkeman, Biophys. J. 44, 315 (1983).41. J. R. Lakowicz and B. P. Maliwal, Biophys. Chem. 21, 61 (1985).42. J. R. Lakowicz, H. Cherek, B. P. Maliwal, and E. Gratton, Biochemistry 24, 376 (1985).43. J. R. Lakowicz, G. Laczko, I. Gryczynski, and H. Cherek, J. Biol. Chem. 261, 2240 (1986).44. J. R. Knutson, L. Davenport, and L. Brand, Biochemistry 25, 1805 (1986).45. L. Davenport, J. R. Knutson, and L. Brand, Biochemistry 25, 1811 (1986).46. J. R. Lakowicz, I. Gryczynski, H. Szmacinski, H. Cherek, and N. Joshi, unpublished.47. J. B. A. Ross, K. W. Rousslang, and L. Brand, Biochemistry 20, 4361 (1981).48. T. C. Terwillinger and D. Eisenberg, J. Biol. Chem. 257, 6016 (1982).49. J. R. Lakowicz, H. Cherek, I. Gryczynski, N. Joshi, and M. L. Johnson, Biophys. J. 51, 755

(1987).50. C. D. Tran and G. S. Beddard, Eur. J. Biochem. 13, 59 (1985).51. T. Kulinski, A. J. W. G. Visser, D. J. O’Kane, and J. Lee, Biochemistry 26, 540 (1987).52. K. Beardsley, T. Tao, and C. R. Cantor, Biochemistry 9, 3524 (1970).53. B. D. Wells and J. R. Lakowicz, Biophys. Chem. 26, 39 (1987).54. R. A. Mendelson, M. F. Morales, and J. Botts, Biochemistry 12, 2250 (1973).55. I. Miller and R. T. Tregear, J. Mol. Biol. 70, 85 (1972).56. R. A. Mendelson and H. C. Cheung, Biochemistry 17, 2140 (1978).57. M. Miki, P. Wahl, and J. C. Auchet, Biochemistry 21, 3662 (1982),58. S. Fujime and S. Ishiwata, J. Mol. Biol. 62, 254 (1971).59. D. D. Thomas, J .C . Seidel, and J. Gergely, J. Mol. Biol. 132, 257 (1979).60. J. A. Weltman and G. M. Edelman, Biochemistry 6, 1437 (1967).61. P. Wahl, Biochim. Biophys. Acta 175, 55 (1969).62. J. Yguerabide, H. F. Epstein, and L. Stryer, J. Mol. Biol. 51, 573 (1970).63. C. Lovejoy, D. A. Holowka, and R. E. Cathou, Biochemistry 16, 3668 (1977).64. D. A. Holowka and R. E. Cathou, Biochemistry 15, 3373, 3379 (1976).65. R. C. Siegel and R. E. Cathou, Biochemistry 20, 192 (1981).66. W. Y. Cheung, Science 207, 19 (1980).67. C. B. Klee, T. H. Crouch, and P. G. Richman, Annu. Rev. Biochem. 49, 489 (1980).68. Y. S. Babu, J. S. Sack, T. G. Greenbough, C. E. Bugg, A. R. Means, and W. J. Cook, Nature

315, 37 (1985).69. R. F. Steiner and L. Norris, Biophys. Chem. 27, 27 (1987).

2

Fluorescence Quenching:Theory and Applications

Maurice R. Eftink

2.1. Introduction

Solute fluorescence quenching reactions were first applied to biochemicalproblems in the late 1960s and early 1970s,(1– 7) and since that time they havebeen a very valuable research tool for studies with proteins, membranes, andother macromolecular assemblies. Quenching reactions are easy to perform,require only a small sample, usually are nondestructive, and can be applied toalmost any system that has an intrinsic or extrinsic fluorescence probe. Themost important characteristic, however, is the value of the information thatthese reactions can provide. Solute quenching reactions, using quencherssuch as molecular oxygen, acrylamide, or iodide ion, provide informationabout the location of fluorescent groups in a macromolecular structure.A fluorophore that is located on the surface of a larger structure will berelatively accessible to a solute quencher that is dissolved in the aqueousphase. A fluorophore that is removed from the surface of a structure will bequenched to a lesser degree by the quencher. Thus, the quenching reactioncan be used to probe topographical features of a macromolecular assemblyand to sense any structural changes that may be caused by varying conditionsor the addition of reagents. In addition, quenching reactions can, in somesituations, provide information about conformational fluctuations. In Sections2.3 and 2.4 I will discuss several examples of the use of solute quenchers instudies with proteins, membranes, and nucleic acids.

Solute fluorescence quenching reactions can also be used to selectivelyalter the fluorescence properties of a sample in order to resolve contributionsor aid in the measurement of data. To elaborate on this point, consider thedifferent characteristics of fluorescence: the quantum yield, excitation and

Maurice R. Eftink • Department of Chemistry, University of Mississippi, University,Mississippi 38677.


53

54 Maurice R. Eftink

emission spectral positions, anisotropy, the time dependence of the intensity(fluorescence lifetime) and anisotropy decay, and the wavelength dependenceof these parameters. If only a single emitting center exists in a sample, thenthe interpretation of these fluorescence properties may be straightforward.If, however, ground state (i.e., a mixture of fluorophores) or excited stateheterogeneity (i.e., excited state reactions) exists, the interpretation of steady-state, time-domain, or frequency-domain fluorescence data will be difficult.Solute fluorescence quenching provides the experimenter with anothervariable (i.e., another variable axis, in addition to time and wavelength) whichmay enable the resolution of fluorescence contributions. Consider Figure 2.1.,in which a comparison is made of the dependence of fluorescence intensity ontime and quencher concentration. For the time axis, the relative intensity, F,will decay in an exponential manner (Eq. 2.1) with lifetime, For thequencher concentration axis, the Stern–Volmer equation (Eq. 2.2) describesthe drop in the steady-state intensity F with quencher concentration, [Q],where Ksv is the dynamic quenching constant.

These equations apply to a homogeneously emitting sample; if the fluo-rescence is heterogeneous, the various contributions will lead to a morecomplex variation of the intensity with both time and [Q]. Even for a simplesystem, however, Figure 2.1 shows that the drop in intensity along the timeand [Q] axes will be similar. In practice, a more extensive set of data isusually collected in time-domain measurements, and one will typically be

Fluorescence Quenching: Theory and Applications 55

limited by a [Q] range of 0 to ~ 0.5 M. On the other hand, with quenchingstudies the experimenter can select different quenchers (i.e., ionic, neutral,polar, etc.) in order to achieve resolution. Several examples of the ways inwhich solute quenching can aid in the resolution of fluorescence propertieswill be given in Section 2.5.

In this chapter, I will limit consideration to solute quenchers, such asmolecular oxygen, acrylamide, and iodide, which quench by coming intocontact (or very close approach) with the fluorophore. Other types offluorescence quenching, due to long-range energy transfer, induced conforma-tional changes, and various intramolecular reactions, will not be discussed.Some of these topics are discussed in other contributions to this volume.Phosphorescence quencing will be discussed, since it is a similar process.I also point out that this chapter is not intended to be a comprehensivereview. Very many researchers are using solute quenching reactions, and Icannot hope to cite all the work that has been done. Various aspects of themethod have been treated in reviews (8-12), and here I will focus primarilyon recent developments and applications of the method in biochemistry,

2.2. Basic Concepts

2.2.1. The Stern–Volmer Equation

The inverse of Eq. (2.2) is the classic form of the Stern–Volmer equation,a relationship which describes the effect of quencher on the steady-statefluorescence of a sample(13–15):

The dynamic Stern–Volmer constant, is equal to the product of aquenching rate constant, times the fluorescence lifetime in the absenceof quencher. If quenching occurs only by a dynamic mechanism, then theratio where is the lifetime in the presence of quencher, will also beequal (Note that this is true only if is a monoexponentialdecay time.)

The dynamic quenching rate constant, will be the product of thequenching efficiency, times the diffusion-limited bimolecular rate constantfor collision, k (the efficiency will be defined below). The value of k can be


theoretically calculated by use of the Smoluchowski equation (here we neglectthe transient term):

where D and are the sum of the diffusion coefficients and molecular radii,respectively, of the quencher and fluorophore, and N' is Avogadro’s numberdivided by 1000. The diffusion coefficient for each species can be predicted bythe Stokes–Einstein equation:


where is the solvent viscosity, is Boltzmann’s constant, T is absolutetemperature, and R is the radius of the species. Thus, for an efficientquencher, is expected to vary with However, slight deviations fromStokes–Einstein behavior are commonly observed.(14) For typical values of R

values in the range ofare expected for efficient quenching.

In Figure 2.2 is shown a plot of versus for the acrylamidequenching of indole in various solvents. A reasonably good Stokes–Einsteinpattern is seen. These data also demonstrate that acrylamide quenching ofindole is a very efficient process in solvents of different polarity. In water at25°C, the for the acrylamide quenching of indole has repeatedly been found


to be in the range of which is approximatelyequal to the theoretical calculated via Eqs. (2.5)and

Quenching may also occur by a static process, that is, a process that doesnot involve diffusion. Usually both dynamic and static quenching occurtogether, and a modified form of the Stern–Volmer equation is then used (seebelow). When both quenching processes occur, a plot of versus [Q] willusually be upward curving. According to the simplest theory, the plotshould represent only the dynamic quenching component. In Figure 2.3A isshown a plot of and for the quenching of the fluorescence of indoleby acrylamide.

When there is ground state heterogeneity (i.e., more than one fluorescentspecies), and only dynamic quenching is kinetically important, theStern–Volmer equation is

where is the dynamic quenching constant for the ith species andis the fractional contribution of the ith species to the total fluorescence inthe selected excitation and emission wavelength regions. If there are twocomponents I and the for one is times larger than that for thesecond, the Stern–Volmer plot of versus [Q] will be downward curving.By fitting Eq. (2.8) to the data, the values of and can be determined(see Section 2.6.2). In Figure 2.3B are shown data for the acrylamidequenching of a mixture of 3-methylindole and 2-methylindole

Data for three emission wavelengths are shown, for reasonswhich will be presented in Section 2.6.1. For each wavelength the plots curvedownward very slightly at low quencher concentration.

2.2.2. Quenching Mechanisms and Efficiency

A general kinetic expression for a fluorescence quenching reaction in anisotropic phase is as follows:

Scheme 1

where A* is an excited state of a fluorophore, ( A * . . . Q ) is an encountercomplex, and (A ...Q) is some resulting complex in which the excess energyhas been dissipated as heat, The rate constants and are diffusionalrate constants for the formation and breakdown of the encounter complex;


is the rate constant for the internal quenching process. The electronicmechanism for the internal quenching process is thought to be different fordifferent quenchers. Molecular oxygen and paramagnetic species, for example,are thought to quench aromatic fluorophores by an electron spin exchangeprocess (leading to rapid intersystem crossing and thus facilitating conversionthrough the triplet manifold to the ground state).(15,18,19) Acrylamide, otheramides, and amines appear to quench via an electron transfer process, that is,transfer of an electron from the excited singlet state to the quencher to forma transient charge transfer complex [in this case (A ... Q) above may bewritten as For other quenchers, the quencher may bethe electron donor, and the excited state may be the acceptor.(23) Quenchersthat possess halogens or other heavy atoms appear to quench by enhancingintersystem crossing via a spin–orbital coupling mechanism. (24,25) Still otherquenchers may act by a resonance energy transfer mechanism, when spectraloverlap exists. Except for energy transfer, the quenching mechanisms appearto involve close contact between the excited state and the quencher, and thusthe quenchers may be considered to be contact quenchers (see below). Orbitaloverlap is thought to be necessary for quenching by oxygen. Electron transferquenching may show a very slight distance dependence of expwhere is a measure of the size of the molecular orbitals, r is the actualseparation distance, and is a constant that is near unity.(243)

It is difficult to experimentally determine the quenching mechanism for aparticular quencher. This is especially true for cases in which the value ofis very large compared to that of and that is, when the efficiency isnear unity. The quenching efficiency, for the reaction in Scheme 1 is equalto

This expression for should apply for most situations; see Ref. 9 for furtherdiscussion and alternate expressions for then

This means that every time an encounter complex forms, quenchingfollows. When will be equal to [see Eq. 2.5)], the rate constantfor collision between the quencher and fluorophore. If ,then and the encounter complexes may dissociate before quenchingoccurs. For such inefficient cases, will be equal to which of course willbe less than When fluorescence quenching reactions are applied to bio-chemical systems, it is desirable to employ an efficient quencher–fluorophorepair, so that the interpretation of values will be more straightforward.Molecular oxygen seems to be efficient for virtually all aromatic fluoro-phores,(26) but acrylamide and iodide(22) are not efficient for all commonfluorophores. Inefficient systems need not be completely avoided, but more


caution must be exercised in interpreting data with these. For example, it isadvised that, for an inefficient system, one sould always compare studies onthe quenching of a biochemical system with studies on the quenching of thecognate fluorophore alone.(9)

The fact that the value of for an efficient quenching reaction is so largemakes it difficult to study structure–reactivity relationships that ordinarilywould aid in revealing the quenching mechanism. For example, we have triedto demonstrate that acrylamide is a change transfer quencher by studying how

varies with the ionization potential of a series of fluorophores.(22) Since thevalue of for acrylamide is so large (we estimate it to be forindole and other fluorophores), very little variation in is found for variousfluorophores. The related quencher, succinimide, is less efficient for mostfluorophores, and a crude relationship between and the ionization poten-tials of the series of fluorophores was observed.(22) In studies with otherfluorophore–quencher systems, correlations have been found betweenvalues and the ionization potentials, electron affinities, or reduction potentialsof the reactants.(28–30)

2.2.3. Diffusional Nature of Quenching

In addition to questions regarding the step, the nature of the stephas been considered by several groups. (13,14,16,30–33) The step is of course adiffusional process, and there has been discussion regarding the inclusion of atime-dependent (transient) term for this rate constant. For a system withspherical symmetry (with respect to one of the reactants) and for the case thata reaction between two molecules, A and B, occurs when one of the reactantsapproaches to within an interaction distance, of the fluorophore,Smoluchowski(34) derived the following expression for the bimolecular rateconstant:

This complete version of Eq. 2.6 includes the transient term in the squarebrackets; the symbols are defined above. In principle, is the sum of the vander Waals radii for the two reactants, but in reality it may be slightlylarger.(32) Sveshnik off(35) was the first to apply this rate expression to solutefluorescence quenching reactions, and, in doing so, he introduced a proba-bility factor (i.e., an efficiency term) to account for the possibility that onlya fraction of the collisions may be effective in quenching. Collins andKimball(36) modified this theory to include the possibility that not everyapproach to results in quenching. Instead, Collins and Kimball used the


so-called “radiation boundary” assumption to derive an expression for k(t).They assumed that the reaction rate is proportional to the probability that onereacting species is at a distance of between and from the secondspecies, and they introduced the specific rate constant k (in units of cm/s) forthe reaction. The latter can be considered to be the effective rate constantwhen the species are close (i.e., within from one another); theparameter is essentially the same as our in Scheme 1, when units areadjusted (i.e., is converted to units of by multiplying byThe resulting expression for k(t) is

where and This expression is morecomplex than that of Smoluchowski (Eq. 2.10), but for ordinary values of

and Eq. (2.11) simplifies to a form that is similar to Eq.Yguerabide et al.(37) and Nemzek and Ware(32) showed how the above

radiation boundary rate constant applies to fluorescence quenching reactions.For either Eq. (2.10) or (2.11), the effective quenching rate constant, willbe time-dependent. The transient terms [i.e., the second part of Eq. (2.10)]relate to the very rapid reaction between A* and Q molecules that happen tobe near one another when A* is excited. In other cases, newly created A* willnot have neighboring Q molecules, and the quenching rate constant will bedescribed by the steady-state rate of diffusion given by Eq. [2.6].

The time-dependent bimolecular rate constants (Eqs. 2.10 and 2.11) thuspredict that fluorescence decay measurements, in the presence of quencher,may be nonexponential. From Eq. (2.10), the apparent decay time in thepresence of quencher will be

Due to the transient term, the decay time will be time-dependent. At timesimmediately following an excitation pulse, the decay will be rapid, as a resultof the term. For typical values of andthe transient term will be larger in magnitude than the steady-state term

until At the transient term will be only 5 % ofthe steady-state term, and the transient term can be neglected at times longerthan this. This transient term will be folded into the steady-state decay rateand will often be difficult to observe. For smaller D values, or for fluorophoreswith smaller the transient term will be of greater significance.

Several workers have experimentally demonstrated the existence of sucha transient term in fluorescence quenching reactions. Nemzek and Ware(32)


demonstrated a nonexponential decay in the quenching of 1,2-ben-zanthracene in a viscous solvent. Van Resandt(38) did likewise for the iodidequenching of N-acetyl-L-tryptophanamide in water, using a picosecond laser.Recently, Lakowicz et al.(39) have used a phase fluorometer, operating overthe range 10–2000 MHz, to demonstrate transient effects in the quenching ofindole by acrylamide and iodide. In Figure 2.4A are shown their data for the


acrylamide quenching of indole. In the pesence of quencher the frequency-domain data indicate a nonexponential decay. Fits of Eqs. (2.10) and (2.11)are shown in Figure 2.4B. Lakowicz et al. have found that the radiationboundary relationship (Eq. 2.11) yields a much better fit to the lifetime datathan does the Smoluchowski relationship (Eq. 2.10). In fact, Lakowicz and co-workers have suggested that the radiation boundary model may even be aninadequate model to describe the data. Nevertheless, their fits with Eqs. (2.10)or (2.11) were achieved with reasonable values of for the quenchers and


indole ring. The fitted D values were somewhat larger than expected withEq. (2.11), but were reasonable for Eq. (2.10). Lakowicz and co-workers havealso recently extended this treatment to the analysis of lifetime fluorescencequenching data with proteins, and they believe that in some cases thetransient term makes a significant contribution to the quenching rate.(41)

More on this will be presented in Section 2.3.8.

2.2.4. Static Quenching

The transient effects described above will lead to upward curvature inStern–Volmer plots from steady-state fluorescence data.(32,33) In fact, upward-curving Stern–Volmer plots have been routinely observed for efficientquenchers, and several explanations (summarized in Refs. 40 and 43) havebeen given over the years. In addition to (i) the above transient term in therate constant, the upward curvature could be due to (ii) the formation of achemically distinct ground state, nonfluorescent complex, and/or ( i i i ) theprobability that a quencher and chromophore happen to be adjacent (withoutnecessarily interacting) at the instant that the latter becomes excited. Complexformation can be easily included in the Stern–Volmer equation as follows(12):

where is the association constant for the one-to-one complex. (In fact,there are some cases in which complex formation is the dominant quenchingprocess; see, for example, Ref. 244 and references therein.) For the probabilityof the nonspecific occurrence of quencher–chromophore neighbors, one candefine an “active volume” element, V, surrounding the chromophore. If aquencher (one or more molecules) exists within this volume at the instant thatthe chromophore becomes excited, “static” quenching is assumed to occurinstantaneously. In this case, the modified Stern–Volmer equation is(40)

Note that Eqs. (2.13) and (2.14) are similar, since is simply the expan-sion of exp(x) when x is small. Either modified Stern–Volmer equation hasbeen used; we prefer the former only when a one-to-one complex is believedto form.

Yguerabide et al.(37) and Nemzek and Ware(32) have shown that the


transient effect discussed above leads to the following modified form of thesteady-state Stern–Volmer equation, which is similar in form to Eq. (2.14):

where

The factor embodies the quenching that is caused by the transient term.The term will lead to a slight upward curvature in a steady-stateStern–Vomer plot. Furthermore, Andre et al.(33) included a factor for truestatic quenching to give the following complete form of the Stern–Volmerequation:

The combined factor exp(V[Q])/Y corresponds to an upward curvature, andEq. (2.14) will usually be adequate to describe data; using Eq. (2.14), theapparent V value will then include contributions from both transient and truestatic quenching.

To illustrate how the transient effect can lead to apparent static quenchingin intensity data, I show in Figure 2.5 simulations in which the Smoluchowskiequation (Eq. 2.10) is used for Shown are simulations for two values of thediffusion coefficient The calculationswere performed by numerically integrating, over 50-ps intervals, the followingequation:

For both D values, upward-curving intensity Stern–Volmer plots arepredicted, even though no true static term was included. The solid lines inFigure 2.5 show fits of Eq. (2.14) to the simulated data. The good fits

demonstrate that Eq. (2.14) can describe the data, regardless of the cause ofthe apparent static effect. I also show calculated lifetime Stern–Volmer plots,for lifetime values that would be measured by the phase lag method. Note thatthe lifetime plots are also predicted to curve upward, although to a lesserdegree than the intensity plot, due to the transient effect. The dashed linegives the dynamic quenching component, as given by the steady-state



Smoluchowski equation (Eq. 2.6). The that is obtained by fittingEq. (2.14) turns out to be reasonably close to the steady-state dynamic

Modified forms of the Stern–Volmer equation have been derived byothers who have taken into account certain other aspects of the mutualcorrelation of reacting pairs, particularly at high concentrations of thereactants.(43–46) We find the article of Peak et al.(43) to be very lucid. Thesemodified Stern–Volmer equations all provide for an upward curvature at high[Q], and, again, Eq. (2.14) can be considered to be an approximation in eachcase.

If there is ground state heterogeneity in a system and if one considers anapparent static quenching constant for each component, then Eq. (2.8) mustbe expanded as follows:

2.2.5. Various Quenchers

A list and description of useful solute quenchers is given elsewhere(Table III in Ref. 9). To this list we can add trifluoroacetamide, which wasreported to quench tryptophanyl fluorescence in proteins,(47) and thalliumion, which has been used as a quencher of various extrinsic fluorescent probeson proteins.(48,49) Thallium ion also is an efficient quencher of indolefluorescence and may prove to be a useful cationic counterpart of iodideion.(9) However, the poor solubility of certain thallium salts must be con-sidered (i.e., the chloride salt has very poor solubility in water), the tendencyfor thallium to precipitate proteins may be a problem, and its toxicity mustbe recognized. Xenon gas or nitric oxide gas may prove to be useful.(50)

For any quencher–fluorophore pair, the efficiency, of the quenchingreaction must be determined in a model system. If the efficiency is much lessthan unity (100%), downward-curving Stern–Volmer plots and unusual tem-perature, viscosity, and solvent dependencies may be observed.(9,22,27) Forexample, we found succinimide to be only about 70% efficient in quenchingthe fluorescence of indole in water. This degree of inefficiency is not large, but,on varying the solvent, we found succinimide to be a very inefficient (~ 10%)quencher of indole in aprotic solvents.(17) It is necessary to appreciate this sol-vent dependency of the quenching efficiency. In a protein or membranesystem, the fluorophore may be in a nonaqueous microenvironment. If so, theability of an inefficient quencher, such as succinimide, to quench fluorescencewill depend not only on the accessibility of the fluorophore, but also on itsmicroenvironment. Interpretation of quenching data can then be difficult. Onthe other hand, a strong solvent dependence may make the quencher moreselective in quenching surface fluorophores.


2.3. Quenching Studies with Proteins

2.3.1. Exposure of Fluorophores

A common use of solute quenchers is to determine the degree of exposureof intrinsic and extrinsic fluorophores in biochemical assemblies. The most fre-quently studied systems are globular proteins and their fluorescent tryptophan(Trp) residues. This will be the primary focus of this section. Some mentionwill be made of other fluorophores, and other types of biochemical assemblies(e.g., membranes and nucleic acids) will be discussed in Section 2.4. Theexposure of fluorophores will be related to the magnitude of the quenchingrate constant, (for an efficient quencher–fluorophore system). The termaccessibility is related to the quenching constant, That is, the accessibilityof a fluorophore depends not only on its exposure but also on itsfluorescence lifetime since

The charged quencher iodide can be used to selectively quench surfacefluorophore residues(2) (charge effects are also important; see below). Theneutral and polar quencher acrylamide can usually quench internal residues toa small degree and shows a very large range of values(51) (see Table 2.1).Molecular oxygen is a very efficient quencher, and, due to its small size andapolar nature, it may penetrate into globular protein structures more readilythan the other quenchers.(52) Acrylamide is also an efficient quencher oftyrosine fluorescence in proteins,(53) but its quenching efficiency for extrinsicfluorophores is sometimes lower.(22,27) In Table 2.1 are given the range ofvalues of that have been observed for the quenching, by iodide, acrylamide,and oxygen, of the Trp fluorescence of several single-Trp-containing proteins.

I briefly note that certain energy transfer methods(54,55) can providesimilar topographical information about the positioning of fluorophores inproteins and membranes, but these methods are beyond the scope of thischapter.

2.3.2. Effect of the Macromolecule's Size

The maximum value that is observed for proteins having an exposedTrp residue, or other fluorophore, is about for iodide andacrylamide(51); for oxygen, larger values are observed, due to its largerdiffusion coefficient in water.(52) The maximum value of found for thequenching of a fluorophore attached to a macromolecule is expected to belower than that for the quenching of the free fluorophore. This of course isdue largerly to the reduced translational diffusion coefficient of the fluorophorewhen attached to the macromolecule. The rotational mobility of the macro-


molecule also must be considered. Johnson and Yguerabide(68) have extendeda general treatment of Shoup et al.(69) to predict the dependence of a quenchingrate constant on the size of a macromolecule. Shown in Figure 2.6 is a modelfor a quenching reaction between a fluorophore that is attached to a sphericalmacromolecule, of radius and a uniformly reactive, spherical quencher, ofradius The cone angle describes the portion of the surface area on themacromolecule that is occupied by the fluorophore. The full equation for thequenching rate constant for such a model will not be presented here (see Eq. 6of Ref. 68), but it is a function of the translational diffusioncoefficients, and and the rotational diffusion coefficient of the macro-molecule, In Figure 2.7A is shown the predicted quenching rate constantfor the macromolecule-associated fluorophore divided by the rateconstant for quenching of the free (unattached) fluorophore as afunction of the size (in molecular weight units) of the macromolecule. As canbe seen, begins at 1.0 for an infinitely small macromolecule andquickly drops to a plateau value of about 0.4–0.5 for a large macromolecule.Thus, a fully exposed fluorophore attached to a macromolecule is expected toshow a quenching rate constant that is about 50% of the value for the freefluorophore. This reduction in the maximum for a macromolecule-associated fluorophore is important with regard to the proper evaluation of

experimental values. Only if the observed is less than ~50% of that forthe free fluorophore should one consider any additional shielding to exist.Johnson and Yguerabide(68) further predicted the dependence of on

the fraction of the fluorophore’s surface area that is exposed on thesurface of the macromolecule. This dependence is shown in Figure 2.7B. It isinteresting that the dependence of on is not linear. At very lowdegrees of true exposure (i.e., low a steeper slope is predicted.


2.3.3. Electrostatic Effects

In addition to the above constraints due to the size of the macro-molecule, charged quenchers may experience additional limitations (orenhancements) due to electrostatic effects. This has not been exploited veryoften in quenching reactions, other than in a very qualitative manner, andusually researchers attempt to minimize electrostatic effects by working athigh and fixed ionic strength. Comparison of the quenching by charged (i.e.,

and and neutral (i.e., acrylamide) quenchers can reveal the sign ofan electrostatic potential near a fluorophore attached to a macromolecule.Ando and Asai(48) showed that by varying ionic strength one can, in principle,determine the number of adjacent charges and their distance from the fluoro-phore. The apparent quenching rate constant, for a charged quencher, will be

where is the rate constant in the absence of electrostatic effects, andare the charge on the quencher and macromolecule (near the fluorophore),

respectively, is the elementary charge, D is the dielectric constant, isdistance of closest approach of the quencher and fluorophore, C is a constant,equal to 1.02 for aqueous solution at 25°C, and is ionic strength. Thus, by


determining for a quencher of charge as a function of ionic strength,one can in principle determine the charge near the fluorophore.

Ando, Asai, and co-workers(48,49) have applied this strategy to determinethe electric potential near two extrinsic fluorescence probes on heavymeromyosin. The probes were covalently attached to two specific sulfhydrylgroups on the protein, and acrylamide, iodide, and thallium were used as aneutral, on anionic, and a cationic quencher, respectively. In a similar mannerthis group also characterized the electric potential around ATP bound tothis protein.(70)

2.3.4. Tryptophan Residues in Proteins

Among the single-Trp-containing proteins listed in Table 2.1, the oneswith the least exposed Trp residues, to each type of quencher, are apoazurinfrom Pseudomonas aeruginosa (Pae) (Refs. 71, 72, and 77), asparaginase fromEscherichia coli (Ref. 71), ribonuclease T, from Aspergillus oryzae (Refs. 51,56, 61, 71, 73, 74, and 78), and cod or whiting parvalbumin (Ref. 71, 75, 76,and 237). Among proteins having two or more Trp residues, Trp-314 of horseliver alcohol dehydrogenase (Refs. 56, 57, and 79–84), Trp-48 of apoazurinfrom Alcaligenes denitrificans (Ade) (Refs. 59 and 77), and Trp-109 ofalkaline phosphatase from E. coli (Ref. 77) are found to be among the leastexposed to quenchers. For each of these Trp residues, little or no quenchingby iodide can be observed. James et al.(6l) reported a very small of

for the iodide quenching of ribonuclease at pH 5.5.Very small values of are found for the acrylamidequenching of the Trp residues in these proteins. For asparaginase, theacrylamide is estimated to be less than or equal to

The oxygen quenching values for these proteins range fromto When compared to values for other proteins, these

oxygen are small. The value of for asparaginase is thelowest that has been reported. Due to its small size, it is generally acceptedthat oxygen can diffuse through proteins to quench internal Trp residues.For the above-mentioned four proteins (and also Trp-314 of alcoholdehydrogenase), the Trp residues apparently are buried, and some degree ofresistance to oxygen diffusion is afforded by the surrounding protein matrix.The Trp emission is quite blue for asparaginase, apoazurin, ribonucleaseand parvalbumin. The ranges from about 308 (apoazurin) to 322 nm

and thus there is independent evidence to suggest that theTrp residues of these proteins are in an apolar microenvironment. X-ray crys-tallographic data for Pae azurin,(85) ribonuclease carp parvalbumin(87)

(a homologous protein, in which a Phe is substituted for the single Trp in thecod protein), and alcohol dehydrogenase(88) are available and show the Trp


residues of these proteins to be buried with in the globular structures. Earlierwe noted that there is a crude relationship between the acrylamide quenching

for a protein and the for its Trp emission.(51) In Figure 2.8 we presentan updated version of this plot, which includes several new single-Trpproteins. While some outliers exist (e.g., the for HSA is lower than expectedfrom its red , the general correlation of the acrylamide withholds. Thus, the magnitude of can be considered a measure of the dynamicexposure of these Trp residues.

The for oxygen quenching also seem to correlate with Trp exposurealthough the range of is smaller. This is best seen from Figure 2.9, wherewe plot the log for oxygen quenching versus the log for acrylamidequenching for various single-Trp proteins. A good correlation is seen, with aslope of 0.44. When oxygen quenching was first applied to proteins, it wassuggested that all Trp residues in proteins could be quenched by oxygen witha very narrow range of That is, it was held that oxygen is not veryspecific in its quenching ability. Figure 2.9 leads to a different conclusion. Thegood correlation indicates that oxygen can be somewhat selective in sensing


the exposure of Trp residues in proteins. Figure 2.9 also contains severalpoints for the iodide quenching of single-Trp proteins. The slope is larger(~1.6), and the plot indicates, as expected, that iodide is a more selectivequencher of surface Trp residues than is acrylamide. (Of course, electrostaticeffects play a role in the selectivity of iodide quenching.)

Static quenching is sometimes seen for the quenching of these single-Trpproteins. This is most often seen for acrylamide as quencher, but someexamples with oxygen have also been reported.(57) Figure 2.10 shows data forthe oxygen quenching of asparaginase and ribonuclease These plotsshow a comparison of and Stern–Volmer plots. The larger slope (andupward curvature that is sometimes discernible) in the former is indicative ofa contribution from static quenching. Generally, such static contributions arefound to be smaller than the dynamic contribution. Human serum albuminhas an exceptionally large degree of static quenching by acrylamide(51) andoxygen,(52) and the static component for oxygen quenching of asparaginase(Figure 2.10), while small in magnitude, is relatively large when compared tothe dynamic quenching component.

In simplest terms, such static components must mean that there is a finiteprobability that the quencher exists near a Trp residue at the instance ofexcitation. One must keep in mind that the static quenching constants seen inproteins are generally smaller in magnitude than those seen for quenching ofindole or tryptophan in water. Thus, the static quenching for proteinsapparently does not represent a large partitioning of quenchers into the


protein matrix next to the Trp residues, but it also indicates that quenchersare often not excluded from a steady-state existence near Trp residues insideproteins. In a following section, I will comment on transient quenching effectsin proteins. This phenomenon may also contribute to an apparent staticquenching, particularly when one compares intensity data and averagelifetime data that are measured at a single frequency (via phase fluorometry).

2.3.5. Ligand Binding and Conformational Changes

Knowledge of the value for the solute quenching of a particularfluorophore in a biological structure is of interest, but quenching experimentsare especially useful for enabling the study of changes in the conformation ofmacromolecules that may be induced by ligand binding or by changing pH,degree of aggregation, etc. When there are several fluorophores, then theinterpretation of a change in the quenchability is difficult; one is usuallylimited to concluding that there is a general increase or decrease in theaccessibility of the several fluorophores. Changes in the relative quantum yieldand lifetime of various fluorophores complicate any interpretation. If there isonly a single fluorescent group, interpretation of changes in quenchability ismore straightforward, particularly if fluorescence lifetimes are also available.

In Table 2.2 are given several examples of recent applications of solutequenching to study changes in the conformation of proteins. The examplesrange from small single-Trp proteins, such as melittin, ribonuclease

76M

auriceR

.E

ftink

Fluorescence

Quenching:T

heoryand

Applications

77

78 . Maurice R. Eftink

azurin, and phospholipase and the interaction of these with small ligands,to very large lipoprotein and nucleoprotein complexes.

An elegant application of solute quenching, and other fluorescencemethods, is the study by O’Neil et al.(91) of complexes between calmodulinand a series of basic, amphiphilic, peptides. These peptides containeda single Trp residue, which was systematically positioned throughout thesequence. In the resulting complexes, the accessibility of the Trp residues toacrylamide was found to vary in a periodic manner (repeat unit of 3 to 4residues), consistent with the periodicity of the

2.3.6. Mechanism of Quenching in Proteins—Penetration versus UnfoldingMechanisms

To enable collisional quenching of internal fluorophores in proteins, it isaccepted by many, but not all, workers that some type of conformationalfluctuations in the proteins must occur. Since oxygen is so small, it is easyto imagine that it will be able to penetrate into the crevices in a protein’sstructure and that only small structural fluctuations would be necessary foroxygen to diffuse throughout a protein.(52,84) For larger quenchers, suchas acrylamide, there has been some question regarding the mechanism ofquenching of internal fluorophores. We suggested that, like oxygen, acryl-amide can penetrate into the matrix of a globular protein, with such inwarddiffusion being facilitated by rapid, small-amplitude fluctuations in theprotein’s structure.(73,110) Some have argued that, instead of the quencherdiffusing inward, a segmental unfolding of the protein occurs to increasethe exposure of the fluorophore to the aqueous phase and hence to thequencher.(111–113) Others have raised the possibility that some types ofquenching occur over a distance, so that physical contact may not be requiredfor quenching internal fluorophores.(74) Also, the extent to which quenchers,such as acrylamide and oxygen, are associated with proteins, and the effectthat such association would have on the interpretation of quenching ratecontants, has been discussed.(114) Here I will comment on these variousalternative interpretations.

The rate constant for an electron transfer reaction, is believed todepend on distance as , where r is the center-to-center separa-tion distance between the donor and acceptor, and is the contact distance.If 7 is the contact distance for a typical quencher–fluorophore pair, then at

beyond van der Waals contact), the value of would beonly 5 % of that at This would correspond to a fluorophore that is com-pletely shielded by an impenetrable layer of methylene groups. If quenching ata distance of a few angstroms were to occur to a significant degree, one wouldexpect the value of calculated from experimental values via Eq. (2.6),


to be larger than the van der Waals In fact, for efficient quenchers suchas acryfamide, an of about 7 is calculated for the quenching of indole inwater.(40) This is close to the sum of the molecular radii of indoleand acrylamide Also, analysis of multifrequency phase/modulationlifetime data for the indole–acrylamide reaction, in terms of the radiationboundary form of the time-dependent Smoluchowski equation, yieldsreasonable values.(39) Of course, the calculated value may be com-promised by a slight degree of inefficiency, but the point is that model systemstudies are consistent with requirement of contact for the quenching reaction.

Recently, we have prepared covalent adducts containing an indole ringand an acrylamide moiety, which are separated by one or two bonds. Wefind that intramolecular quenching occurs. Further study is needed toevaluate the extent to which this represents quenching over a distance, orquenching by intramolecular collisions between the groups.

At this time we cannot eliminate the possibility that some electron transferover a distance occurs in the acrylamide quenching of Trp fluorescence inproteins, but it seems likely that quenching must involve very close approach,if not contact, between the reactants.

The question of an unfolding process versus an inward penetrationprocess can be expressed by the two following kinetic schemes(84,117,123).

Here M and A represent a macromolecule and an attached fluorescenceprobe, is the rate constant for diffusion of the quencher through thesolution to approach the surface of the protein (or to quench a surfacefluorophore; in the latter case the discussion in Section 2.3.2 applies), isthe rate constant for diffusion away from the surface, is the rate constant fordiffusion (penetration) of the quencher through the protein matrix to quenchan internal residue, is the rate constant for segmental unfolding of themacromolecule, M, to form an altered conformation, M', and is the rateconstant for the reverse process. Figure 2.11 gives a visual depiction of thesetwo extreme kinetic models. In the penetration model, the final quenching(e.g., electron exchange) step occurs within the protein matrix. Small-amplitude fluctuations in the protein structure must facilitate the penetrationof quencher.(110,115–117,120) In the unfolding model, the final quenching stepoccurs in the aqueous environment, at the surface of the protein. The symbols


for the penetration model are taken from Gratton et al.,(117) who provided athorough kinetic description or this model. The apparent rate constants forsolute quenching will be, for the penetration model,

Thus, expressions for the rate constant are of slightly different form. (Thesemodels are analogous to the unfolding and penetration kinetic models forhydrogen exchange in proteins.)(112,118) If the term in the denominatorof Eq. (2.20) is larger than then the two rate expressions are dis-tinguishable, for Eq. (2.20) predicts a downward-curving Stern–Volmer plot(for a single type of fluorophore!). If, however, in Eq. (2.20), thenthe two rate expressions are not easily distinguished. In Eq. (2.19), may belimited either by diffusion of the quencher through the solvent (whenor by penetration through the protein matrix (when In Eq. (2.20),will be the product of the rate constant for diffusion through the solvent timesthe equilibrium constant for the segmental unfolding transition.

The two models represent extreme kinetic mechanisms (in one thequencher goes in, in the other the fluorophore come out), and in reality thedifference may be subtle. We suspect that for some buried fluorophores apenetration mechanism may be a better model, while for others an unfoldingmechanism may be required. For some of the proteins with single internal Trp


residues we have tried to determine the most appropriate model by varyingthe following parameters: temperature, bulk viscosity, quencher type, andhydrostatic pressure. The basis for these studies is that, for a deeply buriedTrp residue, the penetration model gives (when whereas theunfolding model gives In the former case, variation of withtemperature, pressure, viscosity, etc., will reflect variations in the penetrationstep, For the unfolding model, variations in with these conditions willreflect variations in both diffusion through the solvent and the unfoldingtransition

Temperature dependence studies with ribonuclease and codparvalbumin(75) give apparent activation energies of 6–9 kcal/mol for acryl-amide quenching. Figure 2.12 shows a recent redetermination, via lifetimemeasurements, of the Arrhenius plot for the acrylamide quenching ofribonuclease These activation energies are larger than expected fordiffusion through the solvent and could reflect the thermal activation of small-amplitude fluctuations needed for the step. For an unfolding mechanism,one might expect Kun to have a larger temperature dependence, if theunfolding is similar to the thermal denaturation of the protein. However,


if the segmental unfolding is less extensive in nature, it is reasonable thatthe apparent activation energy would be less, possibly as small as the6–9 kcal/mol that we observe for quenching. As shown in Figure 2.13,pressure dependence studies (up to 2600 bar) show essentially no variationin the for the acrylamide quenching of ribonuclease and cod par-valbumin.(119) This is reasonable for a penetration mechanism, where theconformational fluctuations may be small in amplitude, like the mobiledefects described by Lumry and Rosenberg(120). Based on other studies ofthe pressure-induced unfolding of proteins,(121,122) one would expect toincrease with pressure. Thus, an increase in would have been observed ifthe unfolding model were appropriate for these proteins.

The acrylamide for ribonuclease and parvalbumin is found to showlittle variation with bulk viscosity, between 1 and 10 cP, but to then show agreater drop as viscosity is increased from 10 to Figure 2.14


illustrates viscosity dependence studies for ribonuclease and parvalbumin,as well as the model systems NATA (N-acetyltryptophanamide) andglucagon. The results with ribonuclease and parvalbumin are easily inter-preted in terms of the penetration model. The rate-limiting step is at lowviscosity, but becomes at high viscosity. This will occur becausedecreases, in a Stokes–Einstein manner, with increasing viscosity, whereasmay be relatively independent of bulk viscosity. We have adapted the idea ofFrauenfelder and co-workers(126) and have employed the Kramers equation todescribe the viscosity dependence of the and steps:

where is the magnitude of the rate constant (either or ) at andx, the exponent of the viscosity, is some value between 0 and 1.0. Ifthe rate constant varies inversely with (Stokes–Einstein behavior). Ifthe rate constant is independent of bulk viscosity. Table 2.3 shows parametersfor a fit of Eqs. (2.19) and (2.21) to the acrylamide quenching data inFigure 2.14 for ribonuclease and parvalbumin. In fact, the step is foundto have little viscosity dependence. We interpreted this as indicating that thesmall-amplitude fluctuations needed for penetration of the quencher arecoupled very weakly to the viscosity of the bulk.

These viscosity dependence data are not easily interpreted in terms of theunfolding model. For this model should vary inversely with and, for theplateau region of versus to be explained, one would have to rationalizethat has the opposite viscosity dependence (i.e., decrease with fromthat of In view of our understanding of protein unfolding refoldingtransitions, such a dependence of on is not reasonable.(124,125) Thus, theviscosity dependence studies strongly favor a penetration mechanism foracrylamide quenching, at least for the two proteins studied.


In studies with different quencher types, one can vary the charge, size,and efficiency of the quencher. One should, of course, avoid comparing“contact” quenchers (see Section 2.2.2 for qualification of the term “contact”)quenchers, such as oxygen, acrylamide, and iodide, with those which probablyquench over a distance (i.e., via resonance energy transfer), such as nitrite andmethyl vinyl ketone.(111) Figure 2.9 shows a comparison, for several single-Trpproteins, of the quenching rate constants for oxygen, acrylamide, and iodide.For proteins with buried Trp residues, such as ribonuclease and par-valbumin, the rate constants vary in the order > acrylamide > iodide. Hereagain, the pattern is indicative of a penetration quenching process, withoxygen being very effective (large ) and the iodide ion being very poor (small

) at penetrating into the globular structures. If an unfolding mechanism wereto hold for all quenchers except oxygen, one would expect to see a similarfor most quenchers. This is because the would be the same for allquenchers. This clearly is not the case for most proteins.

In studies with the slightly inefficient quencher succinimide, we obtainedwhat may be the strongest evidence for a penetration model.(17) Succinimidehas a quenching efficiency of about 0.7 in aqueous solution and is about 20%larger in diameter than acrylamide. In comparative studies with several single-Trp proteins, the ratio of the apparent quenching constants for succinimideand acrylamide, was found to range from ~ 0.1 to~0.7 (see Figure 2.15). Proteins having relatively buried Trp residues werefound to have small values of That is, succinimide quenches these witha smaller rate constant than does acrylamide. Proteins with relatively solvent-exposed Trp residues, such as glucagon and adrenocorticotropin, were foundto have larger values. This wide range of values could be due to thecritical size dependence of the dynamic penetration of quencher through aprotein matrix. However, we also discovered another explanation, that beingthe inherent dependence of succinimide quenching on the microenvironmentof the indole ring. Whereas acrylamide is found to be ~100% efficient atquenching the fluorescence of indole in all solvent ( being dependent onlyon the inverse of the solvent viscosity, as shown in Figure 2.2), we found thatsuccinimide is a relatively inefficient quencher in aprotic solvents. For example,succinimide is 70% as efficient as acrylamide in water, but is only 15% asefficient in dioxane, 10% as efficient in acetonitrile, and 1% as efficient indimethylformamide. Thus, a low for a Trp residue in a protein could bedue to the aprotic microenvironment in which the quenching takes place.While there are two possible explanations (i.e., a critical size dependence oran aprotic microenvironment dependence) for the small for buried Trpresidues in proteins, both explanations are consistent with the dynamicpenetration model for solute quenching reactions. That is, low for buriedTrp residues can be explained as being due to the fact that succinimide isslightly larger than acrylamide and thus will not penetrate as well or as being


due to the fact that succinimide, upon penetrating to reach an internal Trp,experiences an aprotic microenvironment and thus will not quench well. Theunfolding mechanism, on the other hand, offers no means of explaining thewide range of since in this mechanism the quenching reaction is assumedto occur when a Trp is periodically exposed to the aqueous environment andthus no dependence on the size of the quencher is predicted.

2.3.7. Interaction of Quenchers with Proteins

A concern in the interpretation of solute quenching studies is whether ornot a quencher interacts with a protein in such a way that the local concentra-tion of quencher is increased. If there is a high local quencher concentration,


then the apparent values (obtained with reference to the bulk quencherconcentration) will give an overestimate of the kinetic exposure of thefluorophore. Blatt et al.(114) have recently emphasized this point, in com-parison with quenching studies with micelles (see Section 2.4.1). Based onmeasurements of the acrylamide quenching of proteins as a function ofprotein concentration, they calculated partition coefficients in the range of 30to 100 for the interaction of acrylamide with serum albumin, monellin, and

-lactoglobulin. We have investigated this matter using both fluorescencelifetime and intensity measurements and find no significant dependence of theacrylamide quenching of serum albumin and monellin on protein concentra-tion.(127) Furthermore, equilibrium dialysis measurements show no significantinteraction of acrylamide with serum albumin. Thus, we do not find evidencefor a high local concentration of this quencher in the globular structure ofthese proteins.

The evidence presented above for static quenching suggests that there isa certain probability that a quencher molecule (particularly neutral quencherslike acrylamide and oxygen) can exist adjacent to a Trp residue at the instantthat excitation occurs. The magnitude of static quenching constants for thequenching of Trp residues in proteins is less than that for aqueous indole,however, for all proteins studied. Thus, the small degree of static quenchingin proteins does not indicate a strong binding of the quencher to the Trpresidues. The observed static quenching may be more correctly attributed totransient effects as discussed below.

There is some evidence that certain quenchers may interact specificallywith certain proteins. For example, acrylamide is an inhibitor of the enzymaticactivity of alcohol dehydrogenase, which is not surprising in view of itsstructural similarity to another strong inhibitor, isobutyramide.(81) Acrylamideis also a weak competitive inhibitor of chymotrypsin,(127) a weak activator oftrypsin,(127) and a weak competitive inhibitor of cytochrome P450C-21.(128) Forseveral other enzymes, there is little or no effect of acrylamide on theiractivity.(51 ,104 ,110) Acrylamide will covalently react with lysine and cysteineside chains at high pH,(129,130) but there is no indication that the adductsproduced will act as quenchers.

The specific interaction of the charged quencher iodide with serumalbumin is well known,(67,131) and nonspecific electrostatic interactionsbetween a quencher and a macromolecule-associated fluorophore must alwaysbe considered (see Section 2.3.3).

It is reasonable to suspect that the nonpolar quencher oxygen will parti-tion weakly into the oily core of proteins. However, the degree of staticquenching by oxygen is not unusually large for most proteins.(52,71) Jamesonet al.(132) for example, fitted their data for the oxygen quenching of theporphyrin (iron-free) fluorescence of myoglobin and hemoglobin with modestpartition coefficients of 0.3 to 0.6. Trichloroethanol is another nonpolar


quencher which one would expect to show a tendency to interact with oilyregions in proteins. This interaction may occur with human serum albumin,but, for most proteins, quenching by trichloroethanol (at low quencherconcentration) shows a pattern similar to that by acrylamide.(133) At high(0.2 to 0.5 M) concentrations, trichloroethanol appears to induce a change inthe conformation of some proteins. This solvent-induced transition is thoughtto be similar to that induced by 2-chloroethanol, which involves interactionwith the nonpolar side chains of the protein and the subsequent unfolding ofthe globular structure.(135)

It is always necessary to consider the possibility that a quencher interactswith the system being studied. However, we believe that the evidence indicatesthat the commonly used neutral quenchers, oxygen and acrylamide, do notpartition into proteins to a significant degree.

In cases where specific quencher–protein binding occurs, it does notnecessarily follow that this leads to enhanced quenching. If the quencher inter-acts near a Trp residue, this would probably produce quenching, but interac-tion at a remote site may not cause a change in fluorescence. An examplein which quenching can be attributed to the specific binding of quencheris cytochrome P450C-21.

(128) The binding of acrylamide results in a staticquenching of the Trp fluorescence of this protein, and the association constantand static quenching constant are both found to be about 10

2.3.8. Transient Effects

There are data that demonstrate that the transient term of theSmoluchowski equation (Eq. 2.10) must be included to fit time- and fre-quency-domain measurements of the solute quenching of indole and othersimple fluorophores in isotropic solution.(32,33,37,39) Lakowicz et al.(39) havestudied whether such transient effects can be observed in the solute quenchingof Trp fluorescence in proteins. One protein that they studied was nucleasefrom Staphylococcus aureus.(41) This single-Trp protein shows a fluorescencedecay which is nearly a single exponential. When acrylamide or oxygen isadded, the fluorescence decay becomes more nonexponential. This is illustrated(for oxygen) by the phase–modulation data in Figure 2.16A. Notice thatLakowicz et al. were able to use a modulation frequency as high as 2000 MHz(as in Figure 2.4A) due to their novel application of a microchannel platedetection system.(134) Such high frequencies are necessary to enable the shortlifetime contributions to be revealed.

The phase–modulation data for acrylamide-quenched nuclease can befitted by a double-exponential decay law. However, Lakowicz et al.(41)

demonstrated that the data can also be fitted by a transient effects model,using either the time-dependent Smoluchowski equation (Eq. 2.10) or the“radiation boundary” form of the Smoluchowski equation (Eq. 2.11). In fact,


Lakowicz et al. found that the radiation boundary transient equation providesa superior fit for the solute quenching of nuclease by both oxygen andacrylamide. This is illustrated in Figure 2.16B by the lower and the moreuniform deviation plots for the fits of the radiation boundary model. Thefitting parameters obtained for this model were D, the effective diffusion coef-ficient for the quencher–fluorophore reaction, and k, the intrinsic quenching


rate constant (which is in units of cm/s, but can be converted to units ofas explained in Section 2.2.3). Tables 2.4 and 2.5 give such fits for

the quenching of nuclease and other proteins by oxygen and acrylamide.Parameters for both the Smoluchowski equation (called the model) andradiation boundary model fits are given. Note that for both fits the interactionradius, can be a fitted parameter or it can be fixed.

This work of Lakowicz et al. is provocative and should spur furtherexperimental and theoretical work. Some may question the application of


theories derived for isotropic, homogeneous systems to diffusional processes inan asymmetrically structured protein. Some steric factors must be considered,and, in cases where penetration by the quencher occurs, the two-step natureof the process (see Scheme 2 and Ref. 117) should be included in the inter-pretation. These studies may also stimulate molecular dynamics calculationsto simulate the movement of a quencher toward Trp residues in proteins.

The relationship between transient effects and static quenching of steady-


state fluorescence is illustrated in Figure 2.5. It may prove to be valuable tosimultaneously analyze both steady-state and time- (or frequency-) domaindata for such transient effects. The possible existence of multiple quenchingrate constants and multiple fluorescence lifetimes, discussed below, mayfurther complicate the analysis for transient effects.

2.3.9. Multiple Quenching Rate Constants and Fluorescence Lifetimes

Both the rate constant for solute quenching and the fluorescence life-time of a fluorophore in a protein may not be discrete values. Insteadthere may be multiple or even pseudo-continuous distributions of orvalues.(137,138,141) Consider the model for a protein in Figure 2.17. Three routesare shown for the penetration of a quencher to an internal fluorophore. If thisis a reasonable model, then the apparent will have contributions form thethree routes. Furthermore, if the different routes have different activationenergies, then one would expect to see a curved Arrhenius plot. We havecarefully studied the temperature dependence of the acrylamide quenching ofribonuclease (Figure 2.12), and we find a linear Arrhenius plot from 10 to

45°C. This indicates that either (a) there is one dominant rate for the collisionof the quencher with Trp-59 of this protein, or (b) that all routes have aboutthe same thermal activation energy. Figure 2.17 gives a model with multiple

but one could also argue for multiple values by consideration of theexistence of multiple conformational states of a protein.

There has also been much discussion recently about the fact that thefluorescence decay of individual Trp residues in proteins is not usually mono-exponential (see Refs. 66, 74, 137, and 138-143). This again may be due to theexistence of multiple conformations of proteins and may be best described asa pseudo-continuous distribution of decay times.(140,141) In Section 2.6.3 I willdiscuss the consequences of such nonexponential lifetimes on Stern–Volmerquenching plots.


2.4. Studies with Membranes and Nucleic Acids

Solute quenching reactions have been used quite often in studies withmicelles and membrane systems. Much less has been done with nucleic acids.Here we will focus on the significant differences in the application of thesolute quenching method to such structures. With micelles and membranes,quenching reactions are controlled by the extent to which the quencher entersthe hydrocarbon-like subphase. Polar or charged quenchers do not enter, toa significant extent, into most lipid subphases. As demonstrated by the workof Shinitzky and Rivnay(44) or Pownall and Smith,(145) charged quenchers,such as N-methylpicolinium, iodide, or cesium ions, can be used to determinethe aqueous surface accessibility of fluorophores in lipid assemblies. (Manysimilar applications of charged and polar quenchers to assess surfaceaccessibility are given as entries 1–17 in Table 2.6.) Nonpolar quenchers canpartition into the lipid subphases, as discussed in Section 2.4.1 below, and thiscan lead to enhanced quenching. Upward-curving Stern–Volmer plots areoften seen when nonpolar quenchers are used with micelles and membranes,and this is probably due to the transient term in the Smoluchowski equation,as well as true static quenching. Since diffusion is often limited to two dimen-sions, a different form of the Smoluchowski equation must be considered(Section 2.4.2). Some quenchers have been made to include a quencher moietyas part of a fatty acid or phospholipid molecule. These “quencher lipids”provide advantage in assessing the location and lateral mobility of membrane-associated fluorophores and the quencher itself (Section 2.4.3).

Below we will expand on these aspects of quenching reactions applied tomicelles and membranes.

2.4.1. Partitioning of Quenchers into Membranes/Micelles

Solute quenching studies in micelles and bilayer membranes are oftencontrolled by the degree to which small solute quenchers are partitioned intothe hydrocarbon-like subphases of these structures, as well as the diffusioncoefficient of the quencher within these subphases. Apolar quenchermolecules, such as oxygen, chloroform, dimethylaniline, and nitromethane,are favorably partitioned into micelles and membrane, and this enhances theirquenching effectiveness. Relationships describing the partitioning of solutequenchers into micelles or membranes and the resulting Stern–Volmerquenching pattern have been presented by several groups.(146–150) If afluorophore is found only in a micelle (or membrane vesicle; in the followingwe will use the term lipid phase) and if the solute quencher also associateswith and quenches in the lipid phase, then a modified Stern–Volmer Equation

FluorescenceQ

uenching:Theory

andA

pplications93


(Eq. 2.22) applies, but the quencher concentration term will be that in thelipid phase,

where is the bimolecular rate constant in the lipid phase, and V is a staticquenching constant. If one defines a partition coefficient aswhere is the quencher concentration in the aqueous solvent phase, then,from Eq. (2.23), which is a conservation of mass relationship, one can derivean expression (Eq. 2.24) for

where is the total quencher concentration over both phases, andand are the volumes of the lipid phase and the solvent and the total

volume, respectively. Substituting this expression for into Eq. (2.22),one obtains the following general Stern–Volmer equation for a phase parti-tioning system:

This equation contains three unknown parameters andpresumably can be determined, or one can consider the productas an unknown). When the lipid concentration is low theabove equation greatly simplifies. However, if one collects quenching data asa function of the ratio (i.e., as a function of micelle/membrane concen-tration), these three unknown parameters can be obtained by simutaneousnonlinear least-squares analysis of the data sets. Alternatively, Blatt et al.(149)

have described a graphical fitting procedure.Blatt, Sawyer, and co-workers(149,150) have also considered the possibility

that the interaction of the quencher with the micelle/membrane subphase maybe better described as a saturable binding process. In this case the expressionfor becomes

where is the association constant, and n is the number of saturable bindingsites. By substituting this expression for into Eq. (2.22) a Stern–Volmerrelationship (not shown) is obtained for this case of quencher binding. Thevalue will then be the rate constant for reaction of the bound quencher with


the fluorophore and may involve jumping of the quencher from its bindingsite to strike the fluorophore. Blatt et al.(149) discussed the possibility thatbound quenchers may quench only by a static mechanism. Regardless, theresulting Stern–Volmer equation will have a maximum of four unknownparameters ( and ), and simultaneous nonlinear least-squaresanalysis of data sets at different ratios may enable fits to be obtained.Again, a graphical procedure is described by Blatt et al.(149) for determiningthe fitting parameters. These researchers have also pointed out that bothpartitioning and binding of the quencher may occur together in a system, andthey have provided some interesting simulations. It should also be pointed outthat Eq. (2.25), and the discussion that follows, assumes that quenchingoccurs only by lipid-associated quencher. If some quenching occurs byquencher from the aqueous phase, an extra term must be added toEq.(2.25).

The advantage of a complete analysis of quenching data as a functionof the ratio is that one can (a) obtain intrinsic rate constants forquenching, and (b) determine the way in which the quencher associates withthe lipid phase. If one were to work at a single ratio, only an apparent

value could be obtained, and it would be a function of (or nand ), and that is, it would be incorrect to interpret such (app)in terms of the microviscosity of the lipid phase. By resolving andparameters are obtained that can be related to the physical characteristics ofthe quencher and the lipid phase.(151) When values are large, they may bedifficult to determine. Omann and Glaser(148) have developed a protocol,in which excess nonfiuorescent membrane vesicles are added, to aid in thedetermination of large values.

In Figure 2.18 are shown typical data for the quenching in a compart-


mentalized system. The fluorophore is 2-(9-anthroyloxy)palmitic acid, whichis incorporated into egg phosphatidylcholine (pc) vesicles. The quencher is5-nitroxide stearate. Stern–Volmer plots at four different lipid concentrationsare shown. These data were fitted to a model in which there is both parti-tioning and binding of the quencher, with both dynamic and static quenchingoccurring for each type of associated quencher molecules.(149)

Lakowicz et al.(146) studied the quenching of carbazole-labeled phospho-lipids in vesicles by chlorinated hydrocarbons and found that the processoccurs primarily by the partitioning of the quencher into the lipid phase.

The fluorescence quencher oxygen has often been used to quench lipid-associated fluorophores. Oxygen is favorably partitioned into lipid phases,and thus an equation such as Eq. (2.25) is needed to describe the data.Mantulin et al.(154) have recently studied the oxygen quenching of the Trpfluorescence of apolipoprotein A-I complexes with dimyristoylphosphatidyl-choline. They concluded that quenching occurs primarily by oxygen moleculesthat are partitioned into the lipid phase.

2.4.2. Two-Dimensional Diffusion in Membranes

The Smoluchowski equation (Eq. 2.10) describes diffusional processes inthree-dimensional space. For collisional quenching reactions within a bilayermembrane, the movement of the quencher and the fluorophore may beconstrained. The bilayer faces may act as boundaries. Consider a bilayerof thickness h. If the diameter of the quencher and fluorophore is smallcompared to h, then the normal Smoluchowski equation is probably adequate.If, however, the size of the quencher and fluorophore approaches the bilayerthickness, h, then diffusion will occur primarily in the plane of the bilayer.For this case, Owen(155) and Blackwell et al.(l56) have presented versions ofthe Smoluchowski equation for two-dimensional diffusion. The equation iscomplex and will not be given here. The important feature, as pointed outby Blackwell et al., is that the transient term can be very significant, incomparison with the “steady-state” rate term, for two-dimensional diffusion.This can partially explain why Stern–Volmer plots for membrane quenchingsystems often appear to curve upward. Also, the “steady-state” quenching rateconstant is given by

(2.27)

as compared to for three-dimensional diffusion.Fato et al.(157) performed a careful study of the quenching of the

fluorescence of a membrane-associated fluorophore, 9-anthroyloxy stearic acid,by membrane-associated quenchers, ubiquinone homologues, in phospholipid


vesicles and mitochondrial membranes. They first measured quenching plotsas a function of phospholipid volume, to correct for incomplete partitioningof the quencher and to determine its partition coefficient (see Section 2.4.1above). Knowing the concentration of the quencher in the lipid phase, theythen calculated quenching rate constants and, using Eqs. (2.27) and (2.26) thediffusion coefficients according to the two-dimensional and three-dimensionaldiffusion models. For this system there was essentially no difference betweenthe calculated diffusion coefficients for the two models.

2.4.3. Quencher Moieties Attached to Lipid Molecules

A group of lipid-like molecules to which quencher groups havebeen covalently attached have been employed in studies with membranes.Among the latter category are nitroxide-labeled fatty acids and phospho-lipids(151,158–161,164,165) and brominated fatty acids(136,162) and long brominatedhydrocarbons.(163) These molecules participate in the bilayer arrangement ofmembrane systems and have proved to be very useful probes, particularlysince the quenching group can be attached at various positions along the fattyacid chain.

For the most part such lipid-like quenchers can be considered to becompletely incorporated into membrane systems, but Chatelier et al.(161)

demonstrated with the quencher n-DOXYL-stearic acid that partitioning intoa purple membrane system is not always complete. These workers found thatquenching studies at various lipid concentrations were necessary to enable thequencher partition coefficient to be determined and to enablecorrected Stern–Volmer plots to be constructed.

The nitroxide-labeled stearic acid derivatives are commercially availablewith the nitroxide moiety at carbons 5, 7, 12, and 16. These derivatives havealso been included into phospholipids,(159) and a nitroxide-labeled cholesterolderivative is available.(151) Holloway and co-workers(136,162) have prepareddibrominated phosphatidylcholines with the bromines at carbons 6 and 7, 9and 10, 11 and 12, and 15 and 16. The efficiency of these various quenchersis difficult to assess. However, they all seem to require contact with thefluorophores in order to quench. Also, the fatty acid and phospholipidderivatives seem to align in a bilayer like normal lipids. For these reasons,studies with a series of quenchers can be used to locate the position of afluorescing group. For example, Markello et al.(162) used the series ofbrominated phosphatidylcholines to determine that the fluorescent Trpresidue of cytochrome b5, in phospholipid vesicles, is located approximately7 beneath the bilayer’s surface (see more on this below).

With spin-labeled or brominated phospholipids, quenching appearsto involve a combination of static and dynamic processes.(159,164,166) As an


alternative to the Stern–Volmer equation, the following relationship has beenfound to be useful for analyzing quenching data with these quenchers(159):

Here is the minimum fluorescence when the sample is fully quenched,is the mole fraction of the quencher lipid in the vesicle, and n is the numberof quencher lipid molecules that are close enough to the fluorophore to resultin quenching. If n is as large as 6, this indicates that the fluorophore can becompletely surrounded by six quencher lipid molecules when (Notethat in a hexagonal lattice, a given phospholipid molecule will have sixneighbors.) Smaller values of n indicate that there is less overlap betweenthe fluorophore and the lipid quencher. In Figure 2.19 is shown a plot ofdata, according to the above equation, for the quenching by a spin-labeledphospholipid of the fluorescence of diphenylhexatriene, gramicidin, tryptophanoctyl ester, and -ATPase embedded into phospholipid vesicles.(159)

Chattopadhyay and London(250) have demonstrated that, by the use oftwo such lipid quenchers with the quenching moiety at different locationsalong the alkyl chain, the penetration depth of the fluorophore can be deter-mined. By assuming (i) a random distribution of fluorophores and quenchingmoieties and (ii) a static quenching process, these workers presented thefollowing relationship for the degree of fluorescence quenching by a lipidquencher of concentration C (in units of molecule per unit of membranesurface area):


Here is the effective quenching encounter distance, and Z is the verticaldistance, within the bilayer, between the fluorophore and the quenchingmoiety (the equation assumes that if not, no quenching is predicted).By using two similar moieties, placed at different positions along a lipid chain,a parallax method is possible to determine the penetration depth of thefluorophore (assuming that the penetration depth of the two quenchingmoieties is known). This penetration depth, given as the distance,between the fluorophore and the center of the bilayer, can be determinedby comparison of the relative fluorescence, in the presence of anequal concentration, C, of quencher lipids #1 and #2, via the followingrelationship:

where is the vertical distance between the bilayer center and the moreshallow quencher, and is the vertical distance between the quencherson the two quencher lipid molecules. Using pairs of nitroxide-labeledphospholipids, Chattopadhyay and London(250) employed this parallaxmethod to determine the penetration depth of various membrane-boundfluorophores.

The above type of lipid quenchers are also useful because they can par-ticipate in phase transitions and phase separations like other phospholipids.Measurement of the quenching by these agents as a function of temperaturecan reveal differences in the fluidity of the gel and liquid-crystalline states.(166)

Also, London and Feigenson(159) have shown that the relative affinity of othertypes of phospholipids for proteins can be measured via their displacement ofsuch lipid quenchers from the boundary region about embedded proteins.These workers used this method to study the interaction of phospholipidswith -ATPase from sarcoplasmic reticulum(159) also see Ref. 165 for anearlier, less specific application).

Another interesting strategy is to selectively incorporate a quencher lipidmolecule into one monolayer of a vesicle and to then observe the degree ofquenching of a fluorophore (i.e., protein) that is incorporated into theopposite monolayer.(167)

2.4.4. Membrane Transport and Surface Potential

Two other interesting applications of solute quenching reactions are instudies of the transport of species across membranes and the determination ofthe surface potential of membranes.

Moore and Raftery(181)performed an elegant transport study by using


as a quencher and transport species. has about the same ionic radiusas , and they found that the acetylcholine receptor, embedded in mem-brane vesicles, will facilitate the uptake of into the vesicles. A fluorescentprobe (8-amino-l,3,6-naphthalenetrisulfonate) was loaded into the inneraqueous volume of the vesicles, and the inward flux of was thenmonitored by quenching of the fluorophore, following stopped-flow mixing.The inward flux was described by the relationship

where is the final, equilibrium concentration of inside the vesicles,and k is an apparent rate constant for the flux. Combining this kineticrelationship with the basic Stern—Volmer equation, the following equationwas obtained for the time dependence of the fluorescence signal:

The value of k, determined by analysis with the above equation, was furtherrelated to a transport number per channel per second, from knowledge of thevesicle size and the number of receptors per vesicle.

The electrostatic potential on a membrane surface can be estimated usingionic quenchers, as demonstrated by Winiski et al.(182) These workers usedand tempamine (4-amino-2,2,6,6-tetramethylpiperidine-l-oxyl) as cationicquenchers of the fluorescence of 2-(N-hexadecylamino)-naphtalene-6-sulfonateincorporated into phospholipid vesicles (which were neutral or negativelycharged by inclusion of phosphatidylglycerols). The apparent for thisreaction will be

where . is the quenching constant when charge effects are absent (i.e.,neutral vesicles), z is the valency of the quencher, F is Faraday's constant, and

is the electrostatic potential sensed by the quencher adjacent to thefluorophore.

2.4.5. Nucleic Acids

Most nucleic acids do not have intrinsic fluorophores (at room tem-perature), and very few solute quenching studies have been performed with thisclass of biomolecules. The Y base of phenylalanine tRNA does fluoresce,(183)

and the quenching of this Y base by acrylamide has been studied.(246)


Intercalating dyes and drugs, such as ethidium bromide provide extrinsicfluorescence probes. Since some of these are pharmacologically important,knowledge of the details of their binding is of interest. Lakowicz andWeber(26) found that ethidium bound to DNA is very well shielded fromquenching by oxygen. The in this case is only aboutThis is lower than that observed for oxygen fluorescence quenching in anyprotein system, and this indicates that intercalation of the dye between DNAbase pairs provides much steric protection. Zinger and Geacintov(247) havestudied the quenching, by oxygen and acrylamide, of three classes of DNA-binding chromophores: those that intercalate (proflavin), those that bind intothe minor groove (Hoechst 33258), and bulky polycyclic aromatic hydrocar-bons that partially intercalate (coronene). Oxygen was found to preferentiallyquench the more exposed, groove-binding chromophore. Acrylamide wasfound to be a relatively poor probe for the accessibility of the bound chromo-phores, due to its low efficiency of the quenching. Atherton and Beaumont(184)

have studied the quenching of intercalated ethidium by the metal ionsand A fluorescent group can be produced in DNA by reaction

with malondialdehyde. Fluorescent cross-links are formed, and Summerfieldand Tappel(185) have studied the quenching of these cross-links by iodideand Montenany-Garestier et al.(248) have used iodide quenching toshow that the Trp residue of a model tetrapeptide is not intercalated in acomplex with dsDNA. In studies with tRNA, Ferguson and Yang(186) havecovalently attached various extrinsic fluorescent probes at different places on

They then used iodide and acrylamide quenching to determine theaccessibility of these probes in the tRNA and its complex with Met rRNAsynthetase. Other fluorescent tRNAs have been made by forming a bimanederivative and by incorporating (188) into a specificposition; iodide quenching studies of these derivatives have been performed.

Most other applications of solute quenching to systems containingnucleic acids involve the fluorescence of a protein that interacts with a nucleicacid. Examples of this type of application are given as the last four entries inTable 2.2.

2.5. Uses to Resolve Other Fluorescence Properties

Another general use of solute quenching reactions is to enable the resolu-tion of heterogeneous emission from systems. As mentioned in Section 2.1,solute quenching reactions provide the experimenter with a means of con-trolling the fluorescence intensity and lifetime of a sample. Below we will giveexamples in which quenching reactions have been used to dissect contribu-tions to the steady-state, time-domain, and frequency-domain fluorescence ofa sample and to modulate excited state reactions.


2.5.1. Resolution of Steady-State Spectra

When there is ground state heterogeneity in the fluorescence of a sample,such as is expected for a protein that contains more than one Trp residue,solute quenching can be used to resolve the fluorescence spectra of the com-ponents. This is easily done for the case in which one class of fluorophores isaccessible and another class is completely inaccessible to the solute quencher.Lehrer(1) demonstrated this strategy by using iodide to selectively quenchcertain Trp residues in lysozyme. With the addition of quencher, the fluo-rescence of lysozyme shifted to the blue, and at high quencher concentrationthe remaining fluorescence was attributed to the inaccessible class of Trpresidues. By difference the spectrum of the accessible class can be determined.In Figure 2.20 are shown the emission spectra of apoazurin from Alcaligenesdenitrificans in the absence and presence of 0.45 M KI obtained in ourlaboratory. This protein has two Trp residues, one of which is deeply buriedand the other of which lies on the surface of the protein.(77) Quenching by KI


allows the emission of the two residues to be separated. One of the bestapplications of this use of quenchers to resolve spectra is the iodide quenchingof horse liver alcohol dehydrogenase. This protein also possesses only twotypes of Trp residues; one type (Trp-15) lies on the surface of this dimericprotein, and the other type (Trp-314) is buried at the intersubunit interface.Laws and Shore(80) and Abdallah et al.(79) have shown that iodide selectivelyquenches Trp-15, allowing the emission spectrum of the two residues to beresolved.

As mentioned in Sections 2.2.1 and 2.6.1, solute quenching can be usedto determine the relative contribution (in terms of fractional fluorescenceintensities) of components (two or, at most, three) at any choice of excitationand emission wavelengths, provided that the components have differentaccessibilities to quencher. Recently, we have compared the dissection ofcomponent spectra by solute quenching with that obtained by phase-resolvedspectral measurements for various two-Trp proteins.(59)

2.5.2. Resolution of Fluorescence Lifetimes

The analysis of fluorescence decay (time or frequency domain) of aheterogeneous sample can also enable the determination of the fractionalcontribution from components. This, of course, is only possible if the fluo-rescence lifetimes of the components are sufficiently different. By combiningfluorescence decay measurements with solute quenching, one can oftenachieve resolution of the components and obtain both the lifetimes andexposure of the components. In fact, in cases where the initial componentlifetimes are similar, the detection and resolution of heterogeneity may only bepossible by the addition of a selective solute quencher.

In studies with proteins, Ross et al.(l89) combined iodide quenching andpulse-decay measurements to obtain the individual Stern–Volmer quenchingplots for the Trp residues of alcohol dehydrogenase. Demmer et al.(190) haverecently repeated this study and have found that one of the Trp residuesof alcohol dehydrogenase (Trp-15) decays in a nonexponential manner.Torgerson(191) and Robbins et al.(192) have also employed acrylamidequenching to obtain individual Stern–Volmer plots for the fluorescencecomponents of myosin S-l and terminal transferase. Wasylewski and Eftink(93)

have used phase fluorometry in a similar manner to obtain the values foriodide quenching of the individual Trp residues of metalloprotease. Even forthe single-Trp protein, ribonuclease Chen et al.(74} have determined thevalues for acrylamide quenching for the two lifetime components of the Trpat pH 7, using the time-correlated single-photon counting technique. Theresults of their study are shown in Figure 2.21.


Use of global analyses, to be discussed in Section 2.6.1 (see also Chapter 5in this volume), should greatly enhance this type of combined measurement.

2.5.3. Resolution of Anisotropy Measurements

For a system with ground state heterogeneity, the steady-state anisotropy,r, of the emission will be the weighted average of the component values:

If one can selectively quench the emission of one of the components, thecomponent values can be obtained. For example, by selectively quenchingTrp-15 of alcohol dehydrogenase with acrylamide, a limiting anisotropy ofabout 0.265 is reached at high [Q] (at =300nm and 20°C). This can beassigned to the anisotropy of the inaccessible Trp-314 residue. From the rvalue at [Q] = 0 and the values, the anisotropy for Trp-15 (r = 0.210) canalso be calculated.(193)

For systems in which there is a single fluorophore, measurement of r asa function of [Q] can be used to construct a Perrin plot. From the slope ofthis plot one can determine the rotational correlation time, for the emittingcenter. This is because a dynamic solute quencher will cause a lowering of thefluorescence lifetime. The r value is dependent on the ratio as given by thePerrin equation:

Here is the limiting fluorescence anisotropy of the fluorophore in the absenceof motion. Equation (2.35b) is a combination of the Perrin equation and theStern–Volmer equation, where we have substitutedAccordingly, a plot of 1/r, obtained as a function of [Q], versus

will have a slope/intercept ratio of If is known,can be calculated. Thus, from steady-state quenching measurements one

can determine the rotational correlation time of a fluorophore. The termmay seem complicated, but it is just a factor proportional to

the fluorescence lifetime, when the fluorescence is homogeneous. If the staticquenching constant, V, is zero, then this term is just For thisreason, this method has been referred to as lifetime-resolved anisotropymeasurements, even though it is a steady-state method.(56,60)


In Figure 2,22 are shown plots of Eq. (2.35b) for the acrylamidequenching of several single-Trp proteins.(60) Lakowicz and co-workers(56,58)

have used oxygen as a quencher of a large number of peptides and proteins.In Figure 2.23 is an example of their use of oxygen quenching and anisotropymeasurements to study the monomer tetramer equilibrium in melittin.

The ^'-intercept in Figures 2.22 and 2.23 is The limitinganisotropy, of the fluorescence of the fluorophore will usually depend onexcitation wavelength, when there is more than one absorption oscillator. Fortryptophan and indole in a low-temperature, vitrified solvent, for example,the anisotropy shows a distinct dependence on and the reaches aplateau at 300 nm due to selective absorption into the band.(194,195) The

value at 300 nm is about 0.31 ± 0.01 for immobilized indole.(194–196) If the


that is found from a modified Perrin plot (Figures 2.22 and 2.23)is much less than the limiting value of this indicates the occurrenceof very rapid motion of the fluorophore that is independent of global rotationof the macromolecule. One can reasonably assume that this rapid motionwill be limited within a cone.(56) The cone angle, can be calculatedas For the data in Figure 2.22, the N form ofhuman serum albumin and the -deficient form of parvalbumin are foundto have much smaller than the other forms of these respectiveproteins. In Table 2.7 are summarized anisotropy data obtained for severalsingle-Trp proteins, by use of oxygen and acrylamide as solute quencher.

Solute quenching can also aid in the analysis of anisotropy decaymeasurements. This has been demonstrated by Lakowicz et al.,(197) whoprogressively quenched samples of the peptide melittin with acrylamide andmeasured the resulting intensity and anisotropy decays (frequency-domainmeasurements). The dynamic quencher reduces the mean lifetime of the singleTrp residue in melittin. By enabling fluorescence data to be collected atshorter times (higher frequencies), the contribution to the anisotropy decayfrom rapid, picosecond motion becomes enhanced. In the study with melittin,in both its monomeric and tetrameric forms, Lakowicz et al. were able toresolve 60- and 160-ps rotational correlation times, in addition to the longer


correlation times for global motion of the peptide, by quenching with up to2 M acrylamide. The picosecond rotational correlation times certainly exist inthe absence of quencher, but the accurate determination of these small valueson unquenched samples is difficult. Lakowicz and co-workers(246,249) havealso used this quenching/differential phase method to aid in the evaluationof the rotational correlation time of smaller molecules, such as indole andY base. This strategy of using solute quenchers to lower the mean decaytime can also be useful with pulsed single-photon counting anisotropy decaymeasurements.(198)

2.5.4. Resolution of Energy Transfer Experiments

Resonance energy transfer, from a donor fluorophore to an acceptor(which may or may not fluoresce), is an excited state reaction that cancompete with solute quenching reactions in certain systems. An experimentercan take advantage of this competition to try to resolve energy transfer pathsbetween multiple donors and acceptors.

Consider the simple system in Figure 2.24 where there is a single acceptor(A) and three donor ( and ) fluorophores (e.g., they may be Trpresidues). Let be so close to A that its transfer efficiency is >90%. Letbe located at a distance approximately equal to the distance for 50%


transfer efficiency. Let be located well beyond the distance so that itstransfer efficiency is <10%. Further, let A be fluorescent so that sensitizedemission can be observed, and let A not be directly quenched by a particularsolute quencher. If one were to excite exclusively into the donor fluorophores,the following patterns would result when the solute quenching of sensitized Aemission is observed. Energy that is transferred from to A would not besignificantly affected by the quencher. Since transfers little energy to A,solute quenching of would cause only a small, indirect quenchingof A. Solute quenching of would, however, lead to a significant quenchingof A. The point is that a solute quencher will most effectively quench thesensitized emission of an acceptor by quenching those donors that are at adistance of from the acceptor (not by quenching those donors that areclosest to the acceptor, as has been claimed by some).

The analysis of the time- (or frequency-) domain kinetics of resonanceenergy transfer reactions can, in some instances, be aided by employingsolute quenching. To illustrate this usage, consider the frequency-domaindata in Figure 2.25 for the Trp-to-NADH energy transfer in an alcoholdehydrogenase–NADH complex. There are two Trp residues that can transferenergy to bound NADH. The surface Trp residue, Trp-15, can be selectivelyquenched by iodide, leaving only Trp-314 as an energy donor. Figure 2.25shows that the multifrequency phase and modulation data are altered by thequenching of Trp-15. In another example of the use of solute quenching to aidin the analysis of energy transfer reactions, Gryczynski et al.(218) have usedsteady-state solute quenching to evaluate the distribution of end-to-enddistances of flexible molecules having a fluorescence donor and acceptor ateither end.

2.5.5. Other Uses of Solute Quenching

The competition between dynamic solute quenching and various excitedprocesses makes possible other practical uses of these reactions in addition tothe ones mentioned above. Solute quenching may compete with excited statechemical reactions, and thus quenchers can sometimes be used to protectchromophores from undergoing photolysis.(204) Of course, a quencher may


also potentiate photolysis if the quenching mechanism involves, for example,irreversible electron transfer.(205) Solute quenchers can compete with tripletstate decay (see Section 2.7), and some quenchers (i.e., heavy atom quenchers)can promote intersystem crossing and thus can lead to an increase in thepopulation of the triplet state.

Solute quenching reactions can compete with excimer formation. Forexample, Chong and Thompson(206) used oxygen quenching to attenuate theextent of excimer formation by pyrene-labeled sphingomyelin in phospholipidvesicles. From steady-state spectra, as a function of relative oxygen concentra-tion, and the fluorescence lifetime of the unquenched pyrene monomer, theseresearchers were able to determine the apparent rate constant for excimerformation. The latter rate constant provides information about the fluidityof the bilayer.(207) This quenching method offers several advantages over thestandard method for determining the pyrene excimer formation rate constant.The quenching method does not require the measurement of the lifetime orintensity of the excimer and requires knowledge only of the relative quencherconcentration.


There can also be competition between solute quenching reactions wheretwo different types of quenchers are used. In fact, one of the early means ofdetermining that a quencher is a dynamic quencher was to determine theapparent value for the quencher in question as a function of the concen-

tration of a second quencher that was known to be a dynamic quencher. If theapparent for quencher no. 1 drops as quencher no. 2 is added, then thefirst quencher must also be dynamic. Examples of the use of two quenchers instudies with proteins can be seen with liver alcohol dehydrogenase. One canquench most of the fluorescence of the surface Trp residue, Trp-15, of theprotein by adding iodide or acrylamide. Hagaman and Eftink(57) addedacrylamide to this protein to then enable quenching of the internal residue,Trp-314, by oxygen to be studied. Similarly, Laws and Shore(80) quenchedTrp-15 with iodide to enable the alkaline quenching of Trp-314 to becharacterized. Somogyi et al.(208) have presented a general discussion of theuse of two quenchers (so-called double quenching) to further resolve theheterogeneous emission of proteins. It is best to use a very selective quencher(i.e., ionic quenchers such as iodide), which will strike primarily surfacefluorophores, in combination with a less selective quencher (i.e., oxygen orperhaps acrylamide), which can quench internal fluorophores to a certaindegree. By performing quenching studies with one quencher as a function ofthe concentration of the other, the accessibility of the various classes offluorophores (surface and internal) can be quantitated, in optimal cases.

Another practical use of solute quenchers is to produce a fluorescencechange, where one did not exist before, to enable thermodynamic or kineticstudies to be performed. For example, consider a macromolecule–ligandcomplex in which the fluorescence of the macromolecule is unchanged by thebinding of the ligand. In this case, fluorescence measurements will normally beof little value in studying the binding of the ligand. However, suppose that theaccessibility of fluorophores on the macromolecule to solute quenchers isaltered by ligand binding. If so, the addition of the quenchers will enableligand binding titration curves (or kinetic binding data) to be obtained usingfluorescence spectroscopy. Messmer and Kagi(209) developed this argumentin their study with creatine kinase. The Trp fluorescence of this enzyme isquenched by only 9–15% upon the binding of ATP. The Trp residues ofcreatine kinase are partially accessible to iodide, but, when ATP is bound, theaccessibility drops to essentially zero. For such a system at intermediateconcentrations of the ligand, L, the following Stern–Volmer equation applies:

where is the dynamic Stern–Volmer constant for quencher, Q, for theprotein alone, is the ligand dissociation constant, n is the ratio of the


Stern-Volmer constant for the protein–ligand complex to that for the proteinalone and R is the ratio of the fluorescence intensity of thecomplex to that of the protein alone. The apparent for this systemwill be

If R = l .0 (i.e., the binding of ligand does not quench or enhance thefluorescence of the protein) and at the ratio of the initialin the presence of L to that in the absence is

and it can be seen that can be obtained from a plot of versus[L]. Messmer and Kagi(209) obtained dissociation constants for the inter-action of ATP, ADP, and AMP with creatine kinase in this manner, usingeither iodide or acrylamide as the. solute quencher.

The reduction of the fluorescence lifetime of fluorophores by addedquenchers can also be used to establish lifetime reference standards.(210)

2.6. Recent Developments in Data Analysis

It is always risky to entitle a section “Recent Developments...,” especiallyin the area of fluorescence spectroscopy, because one's comments can becomequickly outdated. Developments in picosecond flash and phase fluorometrywill be reviewed by other authors in this series. Here I will comment primarilyon developments in data analysis.

2.6.1. Simultaneous Analyses of Quenching Data

While others have certainly contributed,(211) Brand and his proteges(212–217)

have been foremost in promoting the strategy of overdetermining fittingparameters by simultaneously analyzing multiple, linked data sets (see alsochapters in this volume). This has been applied primarily to time- and fre-quency-domain fluorescence lifetime measurements,(212,213) but global analysescan be performed for any cases in which data sets can be linked. For example,if Stern–Volmer quenching data for a sample have been obtained at a set ofemission wavelengths, or at various temperatures, or pressures, or in thepresence of various ligand concentrations, etc., one could simultaneously fit


the data sets to obtain quenching constants plus other constants that dependon the nature of the linkage (i.e., when emission wavelength is varied, theother fitting parameters are the wavelength dependence of the fractional inten-sities of the component spectra). Beechem and Grattan(215) have generalizeda global fitting routine that will apply to many types of fluorescence data,including solute quenching. We believe that in the near future global analysisof solute quenching data sets will become a standard procedure for extractingquenching constants and other fitting parameters. This will particularly be sowith the expanding use of on-line data acquisition and other equipment, suchas photodiode array detectors, which will enable large bodies of data to berapidly acquired.

The three data sets in Figure 2.3B (the acrylamide quenching of amixture of 2-methyl- and 3-methylindole) were actually fitted simultaneouslyby Eq. (2.17), using global dynamic and static quenching constants and thefractional contributions of the components at each emission wavelength. Ifonly a single data set (single emission wavelength) had been analyzed alone,the heterogeneity could easily have been missed. This is because the downwardcurvature is slight, since the difference between the is not great, and sincestatic quenching further masks the downward curvature. By linking the threedata sets, the components are easily recovered. We have also applied suchsimultaneous analysis of quenching data to a two-Trp-containing protein.(219)

2.6.2. Nonlinear Least-Squares Fits

Practitioners of solute fluorescence quenching are often guilty of the samelax data analysis practices as are scientists in other areas. Like many otherequations used in biochemistry, the simplest form of the Stern–Volmer equa-tion is a linear function. Before the advent of computers, scientists favoredlinear transformations of equations, and equations like the Stern–Volmerequation have become traditionally accepted. Actually, a more statisticallysound plot would be of versus [Q]. As shown in Figures 2.1 and 2.26such a direct plot is nonlinear, but with nonlinear least-squares (NLLSQ)programs and microcomputers, fits to a direct plot can be obtained veryeasily. Acuna et al.(220) and Stryjewski and Wasylewski(221) have describedNLLSQ programs for Stern–Volmer quenching. Both programs will fitversions of the equation for more than one fluorescent component. In our labwe have extended the program of Stryjewski and Wasylewski to includegraphics capabilities, record cataloging, and the simultaneous fitting of data atdifferent wavelengths. Figure 2.26 shows various plots ( versus [Q] and

versus [Q]) of data for the quenching of a protein's fluorescence. Alsoshown is a deviation pattern, which demonstrates the goodness of the fit. Withsuch NLLSQ programs it should no longer be acceptable to use less rigorous


procedures, such as taking the limiting slope of plots of versus 1/[Q],to analyze quenching data.

2.6.3. Distribution of Lifetimes or Rate Constants

In recent years there has been discussion as to whether the nonexpo-nential fluorescence decays that are commonly observed for fluorophores inbiochemical assemblies are best described in terms of a small number ofdistinct lifetimes or an essentially continuous distribution of lifetimes.(138–141)

We do not wish to fuel this debate here, but we simply raise this issue in orderto discuss the implications in solute quenching studies. If an individualfluorophore attached to a macromolecule is characterized by two or morelifetimes (whether these are discrete values or a distribution) and if there is asingle rate constant for the solute quenching of each decay state, then therewill be two or more quenching constants, since Consider, for thesake of discussion, the pseudo-Lorentzian distribution of lifetimes centeredaround 5 ns in Figure 2.27A. A single rate constant (Figure 2.27B) will give adistribution of values (Figure 2.27C) and this will result in a slightly


downward-curving Stern–Volmer plot (Figure 2.27D). This effect of a lifetimedistribution on Stern–Volmer plots was pointed out by James et al.(139)

Ludescher et al.(66) considered this effect in fitting their data for the iodidequenching of phospholipase (for which they reported four decay times).

There is, of course, evidence that individual Trp residues in proteinsdecay in a nonexponential manner, and the above simulations may be ofgeneral significance. That is, the Stern–Volmer plots for the quenching ofindividual Trp residues may not be linear. In actuality, the Stern–Volmerplots for the quenching of most single-Trp proteins by acrylamide and oxygenare not downward curving. Such Stern–Volmer plots often curve upwardslightly, due to static quenching or the transient effect discussed in Sections2.2.3 and 2.3.8. Also, the predicted downward curvature in Figure 2.27D isbarely detectable, even when the width of the distribution of values is verylarge.

In Figure 2.29 is shown a Stern–Volmer plot for the acrylamidequenching of the single Trp in a highly purified sample of cod parvalbumin.The plot is not perfectly linear. The fluorescence decay of this protein is non-exponential.(222) A distribution of values was fitted to this data set asshown in Figure 2.29B. The fit shown is not necessarily better than a fit to twodiscrete values, and this model must be given due consideration.

A distribution of values may also result from the existence of a dis-tribution of quenching rate constants, as shown in Figure 28B and C, eventhough the initial fluorescence decay may be a pure exponential. A slightly


downward-curving Stern–Volmer plot is also predicted. A distribution ofvalues may be due to the existence of different conformations of the protein(see Section 2.3.1). It is also possible to have a distribution of both lifetimesand rate constants, which would further broaden the distribution andproduce a greater departure from linearity.

The difference between having a distribution of lifetimes and a distributionof rate constants is that, in the former case, the solute quenching process willresult in a decrease in the width of the lifetime distribution (Figure 2.27A).If there is a distribution of values and an initial exponential lifetime,quenching will result in a distribution of lifetimes (Figure 2.28A).

The above general discussion focuses on the effect of a pseudo-continuousdistribution of lifetimes or values, but also applies when there are twolifetimes or values for a single fluorophore. These considerations are alsoin addition to the transient effects that are discussed in Section 2.3.8.

2.6.4. Experimental Improvements

Many researchers have interfaced fluorometers with personal computersto acquire steady-state emission data on-line. The use of automated deliverysystems enables quencher- and fluorophore-containing solutions to be mixeddirectly in a fluorometric cell. Such systems minimize human handling errors.Desilets et al.(223) have described a continuous-flow (linear quencher gradient)system which uses solvent-delivery equipment of the kind commonly used in


liquid chromatography. We are using a computer-controlled syringe-typedispenser to deliver aliquots of a concentrated quencher solution into a cellcontaining the fluorescing solution.

2.7. Phosphorescence Quenching

The triplet state of aromatic chromophores is usually much longer livedthan the lowest singlet state. The phosphorescence lifetime of Trp residuesin proteins, for example, is known to range from to 1 s in the absenceof oxygen at room temperature.(224–226) These long lifetimes have made it


desirable to perform solute quenching studies of phosphorescence. With thelonger lifetimes, it should be possible to measure lower values for tripletquenching rate constants, (as compared to for the fluorescent state).For deeply buried chromophores (i.e., Trp residues of alcohol dehydrogenase,alkaline phosphatase, or apoazurin), it may be possible to better characterizethe kinetic exposure of such groups by their values than by theirvalues for a particular quencher (since prohibitively high quencher concentra-tion may be required for fluorescence quenching).

Horse liver alcohol dehydrogenase (LADH) and alkaline phosphatase(AP) are among the proteins which show long-lived phosphorescence at roomtemperature.(227) Their phosphorescence is believed to arise from the deeplyburied Trp residues (Trp-314 and Trp-109, respectively) in these proteins.Molecular oxygen is the triplet state quencher which has most often beenemployed. Of course, the removal of dissolved oxygen is necessary to observeroom temperature phosphorescence of Trp residues in proteins. When oxygenis added back, a decrease in phosphorescence lifetime results, which enables

to be determined. Because very low oxygen concentrations are needed,there have apparently been experimental problems regarding the best degas-sing procedures, means of introducing oxygen, and photodepletion of oxygenby intense light sources.(228) Consequently, there is some inconsistency in

values for oxygen quenching of Trp residues in proteins. For AP,Strambini(227) has reported an oxygen as comparedto a value of reported by Calhoun et al.(84) Likewise, thereis over a ten fold difference in the published values of the oxygen forLADH.(84, 227)

If the low values of found by Strambini are true, then the rateconstants for triplet quenching of these buried Trp residues are very small.The interest, of course, is to compare oxygen quenching and valuesto see if both quenching processes are reported the same degree of exposureof the Trp residue. LADH and AP have more than one Trp residue, whichmakes it difficult to determine the for the most buried residue. Usingdouble-quenching experiments, the oxygen value for Trp-314 of LADHhas been estimated to be (57) For AP, the Stern–Volmerplot for oxygen quenching of its fluorescence is nonlinear, and it is difficultto estimate the for Trp-109. Nevertheless, Strambini’s values of to

for the oxygen for these proteins are much smaller thanthe estimated for LADH and AP. Strambini discussedthis discrepancy and offered explanations, including the possibility that thecontact distance for fluorescence quenching by oxygen may be largerthan for triplet quenching.

Ghiron et al.(229–231) have monitored room temperature triplet–tripletabsorption of Trp residues in nine single-Trp proteins, following a 265-nmlaser pulse, and have determined values, for oxygen and acrylamide, for


the “quenching” of this triplet absorption. These workers confirmed theirvalues by making phosphorescence decay measurements for those proteinswith long decay times. While their study did not include LADH and AP, theiroxygen values are more in line with those of Calhoun et al.(84) Oncomparing their oxygen values with values, Ghiron et al. found a

ratio in the range of 0.1 0.6.(231)

With acrylamide, Ghiron et al.(230) also determined values forseveral single-Trp proteins. These values were found to range fromto The ratios were found to be as low as 0.001 forthe internal Trp residue of ribonuclease Calhoun et al. had previouslynoted that acrylamide (and several other quenchers) has a very low

value for the internal Trp of LADH. These workersargued that this indicates that such quenchers cannot penetrate into thematrix of proteins on the nanosecond time scale. However, Ghiron et al. havepointed out that the acrylamide quenching of indole and N-acetyl-L-tryp-tophanamide phosphorescence does not appear to be completely efficient (theefficiency is about 0.3 in water) and that the efficiency of phoshorescencequenching by acrylamide decreases significantly as the polarity of the solventdecreases (i.e., the efficiency is 0.002 in acetonitrile). As with the fluorescencequenching studies with succinimide (see Section 2.3.6), the fact thatacrylamide is not an efficient quencher of Trp phosphorescence in all micro-environments makes it tenuous to try to interpret the magnitude of valuesfor buried Trp residues in proteins. Also, Ghiron et al.(230) found thetemperature and viscosity dependence of the acrylamide in proteins to bevery difficult to rationalize. This led Ghiron et al. to cast doubt on the use-fulness of phosphorescence quenching by acrylamide. Oxygen, on the otherhand, has a quenching efficiency of about for indole phosphorescence,and this efficiency does not seem to change with solvent.(231,232) These oxygen

values are larger than the predicted spin statistical factor of one-ninth(231,233,234); however, they are consistent with other models for oxygen-triplet quenching reactions.(235,245) Ghiron et al. also found the oxygen tonot show Stokes–Einstein behavior for the quenching of model lumiphores.(231)

Barboy and Feitelson(236) have substituted a Zn-protoporphyrin intomyoglobin, in place of the nonfluorescent Fe-protoporphyrin. This Zn-protoporphyrin shows delayed fluorescence, and they have used oxygen andanthraquinonesulfonate to quench its triplet state. With both quenchers thequenching rate constant for the protein was found to be about tenfold lowerthan that for Zn-protoporphyrin that is free in aqueous solution. Also, theapparent activation energy for quenching in the protein was found to beabout 6.0 kcal/mol, for both quenchers, as compared to about 3.0 kcal/mol forthe free lumiphore. Thus, the protein matrix shields the protoporphyrin ring,and Barboy and Feitelson interpreted their results in terms of a “gated”penetration model.


2.8. Conclusion

Solute fluorescence quenching reactions are a poor man's means toobtain kinetic information about luminescing systems. The method shouldcontinue to enjoy widespread use in the determination of the accessibility offluorophores, and, as discussed in Section 2.5, quenching reactions havebeen found to be useful in conjunction with other types of fluorescencemeasurements.

The intrinsic luminescence of biological systems is usually complex, andso quenching data may be complex as well. With nonlinear least-squaresanalyses, and especially with simultaneous analyses of linked data sets, itshould be possible to extract meaningful quenching parameters. We look toimproved data analysis procedures to be the most important advance in themethodology in the upcoming years.

Acknowledgments

I wish to express my appreciation to my mentor, Dr. Camillo A. Ghiron,for his helpful advice and many stimulating discussions.

Some of the unpublished work presented here was performed under thesupport of grant DMB 85-11569 from the National Science Foundation.

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3

Resonance Energy Transfer

Herbert C. Cheung

The mechanism of fluorescence resonance energy transfer (FRET) was firstelucidated by Förster some four decades ago.(1,2) This radiationless transferof excitation energy from a donor to an acceptor chromophore is governedby a long range dipole–dipole interaction, and the Förster formulationis generally applicable to solution and the aromatic chromophores of bio-chemical interest. FRET offers an experimental approach to the determinationof molecular distances in the range 10–80 Å through measurements of theefficiency of energy transfer between a donor and an acceptor located at twospecific sites. Because of the sensitive inverse sixth power dependence of thetransfer efficiency on the donor-acceptor distance, FRET is also a sensitivetechnique for detection of global structural alterations.

It has long been recognized that a major uncertainty in the determinationof molecular distances by energy transfer is in the orientation factor fordipole–dipole coupling. Since this parameter cannot be determined by anycurrent solution technique, the distances calculated from energy transfer datausually are not unique except for cases where an appropriate average value ofthe orientation factor can be applied. Early energy transfer studies on macro-molecular systems were largely qualitative in nature, and most of the transferdata were used to demonstrate structural perturbations. It was not until themid-1970s that the potential usefulness of FRET became widely appreciatedand the literature on molecular distances based on energy transfer began tobuild up. This transition was due, in large part, to advances in instrumenta-tion for accurate determination of fluorescence lifetimes and to suggestionsmade by several groups to minimize the uncertainty in the orientation factor.These suggestions included (1) using polarized emission data to define themobility of donor and acceptor bound to a macromolecular substrate and to

Herbert C. Cheung • Department of Biochemistry, University of Alabama at Birmingham,Birmingham, Alabama 35294.

Topics in Fluorescence Speclroscopy, Volume 2: Principles, edited by Joseph R. Lakowicz. PlenumPress, New York, 1991.

127

128 Herbert C. Cheung

estimate a range of the orientation factor, (2) choosing donor and acceptorchromophores that have mixed polarizations and exhibit small limitingpolarization properties, and (3) using statistical interpretations of energytransfer data to define the limits for the donor–acceptor distance and the mostprobable distance.

The Förster formalism of FRET is based on the assumption that thedonor and acceptor chromophores are stationary on a time scale comparableto the lifetime of the excited state and, as a consequence, the donoracceptorseparation is static and can be described by a single distance. The dynamicnature of globular proteins and polymers has long been recognized, and thepossible existence of a distribution of the donor–acceptor distances in suchsystems was first explored over 15 years ago. Very little progress was madein the experimental investigation of such distributions for specific donor–acceptor probes attached to proteins until 1987. The recent progress in thisdirection indicates that FRET should be useful in monitoring structural tran-sitions. Only a few physical methods can yield direct structural information onmacromolecules in terms of molecular distances, and FRET is one of them.The distribution of energy transfer distances offers a potential experimentalapproach to the investigation of the mechanism of protein folding and unfoldingand the intermediates involved, flexibility of nucleic acids, and distribution oflipids in membrane.

3.1. Long-Range Dipole–Dipole Interaction

Transfer of electronic excitation energy from one atom or molecule toanother may involve different electronic states of both the donor and acceptorand may be governed by different mechanisms. In all these mechanisms, thetransfer takes place under the condition of conservation of total energy andthus occurs as a resonance process. The conservation of energy requires thatthe energy of the electronic state of the acceptor molecule must be either thesame or less than that of the donor molecule. The excess of electronic energyafter transfer has taken place may be dissipated into vibrational energy.The transfer process in effect competes with other modes of deexcitationincluding direct emission of photons with energy appropriate to the electronicstate of the donor. In this chapter we are primarily concerned with energytransfer between short-lived singlet electronic states in which (1) the energy ofinteraction is classified as very weak, meaning that the rate of transfer isproportional to the square of the vibronic interaction energy, and (2) theinteraction between donor and acceptor chromophores is purely dipolar sothat overlap between electronic wave functions of the donor and acceptor asa means of deexcitation is excluded. Under these stringent conditions, the rate

Resonance Energy Transfer 129

of transfer of excitation energy by the weakly coupled resonance mechanismis given by the Förster formulation(1,2)

where k 2 is the orientation factor for dipole–dipole interaction and is deter-mined by the angle between the donor and acceptor dipoles, is thefluorescence quantum yield of the donor in the absence of the acceptor, n isthe refractive index of the medium between the donor and acceptor, N isAvogadro’s number, is the fluorescence lifetime of the donor in the absenceof the acceptor, R is the distance between the centers of the donor andacceptor chromophores, and J is the normalized spectral overlap integral,given by

where is the fluorescence intensity of the donor in the absence of theacceptor at wavelength and is the molar absorption coefficient of theacceptor at Equation (3.1) is valid regardless of whether the donor andacceptor molecules are of the same or different kind and whether or not thespectra of these molecules have vibrational structure. Equation (3.2) representsthe overlap of the emission spectrum of the donor with the absorptionspectrum of the acceptor, modified by the factor

An alternative expression of the transfer rate is given by

where R0 is the Förster critical distance at which 50 % of the excitation energyis transferred to the acceptor (50% transfer efficiency). R0 then defines thespatial relationship of the donor and acceptor chromophores at which theprobability of donor deexcitation by energy transfer equals the probability ofdeexcitation by other processes that occur in the absence of the acceptor.Equations (3.1) and (3.3) can be combined to yield

In Eqs. (3.l)–(3.3) the fluorescence intensity is in arbitrary units, andthe other parameters are expressed in fundamental units: R, R0, and in cm,

and J in cm6/mol. Equation (3.4) can be rewritten as


There are several useful forms of Eq. (3.5):

can be determined from experiments independent of energy transfer. Onceit is determined, the donor-acceptor separation R can be calculated from theexperimental value of transfer efficiency (E) through the expression

Förster’s equation of energy transfer is proportional to and the overlapintegral J and inversely proportional to the sixth power of the donor–acceptorseparation R. By using donor and acceptor chromophores attached to theends of fused steroids of a known dimension, Latt et al.(3) showed that theobserved transfer efficiency was in agreement with that predicted by Förster’sequations. A more quantitative test of the dependence of energy transferwas provided by Stryer and Haugland,(4) who used a series of oligomers ofpoly-L-proline ranging from 12 to 46 and containing a donor and anacceptor at the ends. They found the observed transfer efficiencies to vary withthe inverse power of R, in excellent agreement with the predictedvalue. The dependence was also confirmed by Buecher et al.(5) inmolecular sheet experiments in which the donor and acceptor were separatedby multilayers of fatty acids of known dimensions. Finally, by changing themagnitude of J over a 40-fold range, Haugland et al.(6) showed that thetransfer rate is in fact proportional to J. These studies have established thegeneral validity of Förster’s theory, except for the angular dependence givenby

3.2. Determination of Energy Transfer

The transfer efficiency is defined as the ratio of the transfer rate to thesum of the rates of all deexcitation processes and is given by


where kf is the rate of fluorescence decay, and k' represents the sum of therates of all other deexcitation processes. Experimentally, E can be calculatedfrom either the relative quantum yields (or fluorescence intensities) orlifetimes of the donor determined in the presence and absence of the acceptor:

where Fd is the donor fluorescence intensity determined at a given wavelengthin the absence of the acceptor, Fda is the corresponding quantity determinedin the presence of the acceptor, and are the donor lifetimes in theabsence and presence of the acceptor, respectively. If the acceptor is alsofluorescent (which is not a requirement for energy transfer to occur), itsemission can also be used to determine E:

where is the absorbance of the acceptor in a sample containing bothacceptor and donor, and is the absorbance of the donor in the samesample containing both donor and acceptor, both measured at a wavelength

at which the donor absorbs strongly; Fad is the fluorescence intensity of theacceptor in the sample containing acceptor and donor, and Fa is the acceptorintensity in the absence of the donor, both excited at the donor wavelengthand measured at an acceptor wavelength This procedure based on enhan-cement of acceptor fluorescence is considerably more complicated than thedonor intensity method (Eq. 3.13a) because with most donor–acceptor pairssome direct excitation of the acceptor occurs upon irradiation in the donorabsorption range. It is also possible to determine E from the decay kinetics ofthe acceptor.(7) Because of the complexity of the decay pattern, this methoddoes not appear to have been used with success.

The orientation factor is given by

where is the angle between the emission dipole of the donor and theabsorption dipole of the acceptor, and are the angles between thevector joining the donor and acceptor and the emission and absorptiondipoles, respectively. k2 can range from 0 to 4.0. The minimum value obtainswhen the donor emission and acceptor absorption dipoles are perpendicular


to each other, and the maximum value corresponds to parallel and aligneddipoles. If both dipoles sample all orientations during the interval of theexcited state (isotropic, dynamic averaging),

3.3. Proximity Mapping of Molecular Assembly

A single distance between two specific sites within a macromoleculeprovides very little information on molecular geometry, but several suchdistances can be used to define the general shape and dimensions of themolecule. It is easy to visualize that to specify the relative configuration offour noncoplanar points requires six distances. This configuration is notunique because of an unresolvable ambiguity as to whether the fourth pointis above or below the plane defined by the first three points. For eachadditional point, four more distances are required to relate it uniquely tothe first four points. Thus, for a total of N points the number ofdistances required to determine the relative locations of the specific points bytriangulation is 6 + 4(N – 4) = 4N – 10. This three-dimensional map iscomplete except for the handedness alluded to above.

When FRET distances are used to construct a molecular model, severalpractical problems must be considered. First, it may not be feasible todetermine all 4N – 10 distances because of labeling difficulties. With proteinassemblies, reconstitution with labeled components may not always bepossible. Second, even if all 4N – 10 distances can be determined, there isstill the uncertainty concerning the orientation factor. Third, the FRETdistance refers to the distance between the centers of two chromophores whichfrequently are not intrinsic components of the molecule. If the number ofmeasured distances is one or two less than 4N – 10, the distances are stilluseful and can provide constraints on alternate models. The examples selectedfor discussion in this chapter are confined to proteins and protein assemblies.Individual nucleic acids and membranes will not be covered because of thelack of space.

3.4. Experimental Strategy

3.4.1. Sample Preparation

An ideal system for FRET measurements is one that contains a singleintrinsic donor chromophore and a single intrinsic acceptor chromophore. Anexample would be a peptide containing a single Tyr (donor) and a Trp(acceptor). Usually, it is necessary to introduce extrinsic fluorophores through


either noncovalent binding or chemical modification of well-defined functionalgroups. Chemical modification always poses potential experimental complica-tions because of (1) possible labeling heterogeneity, (2) possible structuralalteration resulting from the modification per se, and (3) nonstoichiometriclabeling of one or both sites. It is important to minimize these potentialcomplications by using appropriate control samples. When soichiometriclabeling of both donor and acceptor sites is obtained or when single intrinsicchromophores are used. Eq. (3.13a) can be used without modification todetermine E by measurements of the steady-state emission intensity of thedonor. If the donor decays monoexponentially, Eq. (3.13b) is also directlyapplicable in its simple form.

To use the donor emission properties for FRET measurements, threeseparate samples are generally required. A donor-containing or donor-labeledsample is needed for determinations of donor quantum yield and donorintensity (Fd) or donor lifetime at a given emission wavelength at whichthe acceptor chromophore has negligible or no detectable emission. Themeasurement of donor emission properties is then repeated at the sameexcitation and emission wavelengths with a second sample which containsboth the donor and acceptor chromophores to obtain the donor intensity(Fda) or lifetime in the presence of the acceptor. A third sample contain-ing only the acceptor is required for determination of the absorption spectrumof the attached acceptor, and for selection of an appropriate emissionwavelength for measurements of donor emission properties. If possible, thedonor-containing sample should also contain a nonabsorbing acceptoranalogue attached to the specific acceptor site. This strategy provides anassurance that the observed spectral properties of the donor more closelyreflect its properties in the donor-acceptor-labeled sample in the absence ofenergy transfer.

3.4.2. Measurement of Transfer Efficiency

An example of a study based on the procedure outlined in Section 3.4.1is shown in Figure 3.1. We modified the heavy chain of myosin ATPasesubfragment-1 (S-l) at Cys-707 with the donor probe N-iodoacetyl-N´-(5-sulfo-l-naphthyl)ethylenediamine (1,5-IAEDANS) and at Cys-697 with theacceptor 5-(iodoacetamido)fluorescein (IAF). Labeling specificity was par-tially verified by functional tests, digestion of the modified S-l with trypsinand hydroxylamine followed by gel electrophoresis, and detection of fluorescentfragments by UV illumination. The separation between the two sulfhydrylgroups (commonly referred to as SH1 and SH2) in the doubly labeled proteinwas determined by measurements of the donor fluorescence intensity.(8)

Curve A in Figure 3.1 is the emission spectrum of a sample of S-1 (sample A)


which was modified with the donor at the donor site and labeled at theacceptor site with the nonabsorbing sulfhydryl reagent N-ethylmaleimide(NEM). Sample B was an acceptor-labeled protein which was also labeled atthe donor site with NEM. Curve B is the spectrum of this sample. Curve C isthe spectrum of S-1 doubly labeled with the donor and the acceptor. Curve Dis the spectrum of a mixture of samples A and B. This spectrum agrees wellwith the sum of curves A and B. The quenching of the donor fluorescencein the donor–acceptor-labeled protein was accompanied by a concomitantenhancement of the acceptor fluorescence. These reciprocal spectral changesare qualitative evidence of resonance energy transfer. Since the labeling ofdonor and acceptor was not stoichiometric, Eq. (3.16) was used to determinethe transfer efficiency:


where is the observed degree of labeling of the acceptor site. andwere measured at several emission wavelengths below 475 nm, and theaverages of the intensities from these measurements were used to calculate E.

As a check on the results obtained by the donor intensity method, wealso measured the acceptor enhancement for one series of experiments. Thevalue of E was found to be 0.64 when determined by the donor quenchingmethod and 0.60 by the acceptor enhancement method. The close agreementprovided further confidence that FRET was being investigated.

When lifetime is used to determine E, Eq. (3.13b) can be employed if thedonor decays monoexponentially. This method is not affected by non-stoichiometric labeling. If the degree of acceptor labeling equals or exceedsthat of donor labeling, every donor molecule can be expected to participate inenergy transfer. The lifetime of these donor molecules will be reduced,but the decay will remain monoexponential. If acceptor labeling is less thandonor labeling, the decay pattern will become biexponential because afraction of the donor molecules do not participate in energy transfer. The longlifetime of the biexponential decay is which is the lifetime of the non-participating donor molecules, and the short lifetime is The fractionalamplitudes associated with the two lifetimes should approximate the propor-tions of the two chromophores. Lifetime measurements for energy transferstudies have the advantage of being independent of concentrations. If lifetimescan be determined with a high accuracy, E obtained by this method shouldhave a smaller uncertainty than that obtained by the steady-state intensitymethods. If the donor decay is not single exponential, a weighted average(e.g., of the lifetimes of the donor is usually takenas and the corresponding weighted average determined from the samplecontaining both donor and acceptor is In this case, stoichiometry labelingof both donor and acceptor sites will facilitate the analysis.

3.4.3. The Orientation Factor

The orientation factor has been widely discussed in numerous FRETstudies because, except in rare cases, it cannot be uniquely determined insolution. Early studies on molecular distances were frequently based on theassumption that both donor and acceptor dipoles randomize rapidly(dynamic averaging) and sample all orientations (isotropic condition) duringthe short interval when energy transfer occurs. Under these conditions,Frequently, these conditions are not met because the chromophore orienta-tions are limited by the surrounding macromolecular structure. The resultantdistribution of dipole orientations is likely restricted to a narrow range ofgeometries. A variation of from 4 to results in only a 35% error in R,but a variation from to 0.01 results in a twofold decrease in R. Several


protocols have been proposed to estimate the possible range of or toexperimentally minimize the uncertainty. These will be summarized belowwith emphasis on the protocols proposed by Dale and co-workers.(9,10)

3.4.3.1. Estimation of from Depolarization Factors

Polarization spectroscopy provides information on both the orientationalfreedom of donor and acceptor chromophores attached to a macromoleculeand on their relative orientations. This information does not normally yield aunique average value for the orientation factor, but may provide a realisticestimate of the range of If both donor and acceptor have reorientationaldistributions that have at least approximate axial symmetry with respect to astationary substrate, average depolarization factors can be derived that arerelated to the maximum and minimum values of This relationship isindependent of the particular reorientation distribution functions. In a seriesof elegant papers, Dale and co-workers(9,10) have shown that in the presenceof an energy acceptor, the absorption of excitation energy by a donor can befollowed by three depolarizing steps: those due to reorientation of donor andacceptor and that brought about by energy transfer between them. Theaverage depolarization factor due to energy transfer is given by the productof three axial depolarization factors:

where and are the axial depolarization factors for the donor andacceptor, respectively, and is the axial transfer depolarization factorassociated with the average axial orientations of donor and acceptor. Theaxial depolarization factors are related to observable depolarizations by

where is the donor depolarization factor, is the acceptordepolarization factor, is the experimentally determined limiting anisotropyof either the attached donor or the attached acceptor, each excited at itsabsorption band, and is the observed fundamental anisotropy of either thefree (unbound) donor or free acceptor. The subscripts D and A for the ratio

refer to the donor and acceptor, respectively. The transfer depolariza-tion factor is determined by


where is the limiting anisotropy of the acceptor determined in thepresence of the donor, and is the fundamental anisotropy of the freedonor, both excited at the same wavelength in the lowest absorption band ofthe donor. The axial transfer depolarization factor can be obtained from thethree observable depolarization factors in Eq. (3.17). Dale and co-workers(10)

have published contour plots for obtaining the upper and lower limits offrom observed values of

The limiting values of thus obtained represent the best estimate of theuncertainty in the relative orientations of donor and acceptor chromophores.In the absence of additional structural information, the uncertainty cannot befurther reduced.

Frequently, it is not feasible to determine the transfer depolarizationbecause of extensive overlap between the donor and acceptor emission bands.When is unknown and if are positive, simple expres-sions(10) are available for direct calculation of the limits of

If both donor and acceptor do not reorient after excitation, andare each equal to 1 (limiting anisotropy equals fundamental anisotropy) and

If the donor and acceptor orientations arecompletely randomized immediately after excitation (limiting anisotropy = 0),

Thus, for the isotropic dynamic averaging approximation to hold,the axial depolarization factors must be zero or at least very small (i.e.,limiting anisotropies very small and approaching zero).

The limiting anisotropy can be determined from the Perrin equation:

where k is Boltzmann's constant, T the absolute temperature, the solventviscosity, V the volume of the hydrated macromelecule, and r the observedanisotropy. The value of r0 is obtained from the intercept in a plot ofversus or merely if the anisotropy measurements are made iso-thermally. The viscosity is usually varied by addition of sucrose. The lifetime

should be included as a variable because it usually is temperature-dependentand may also change upon addition of sucrose. The limiting anisotropy isindependent of rotational motion of the macromolecule and reflects orientationof the attached chromophore immediately following excitation. Alternatively,this parameter can be obtained in a single experiment from measurement oftime-resolved anisotropy decay, r(t). If the decay is monoexponential, the


anisotropy at zero time, r(0), is the limiting anisotropy. If the decay isresolved into two components reflecting the rapid local motion (short correla-tion time) of the attached chromophore and the overall macromolecularmotion (long correlation time), is the amplitude of the anisotropy decayassociated with the long correlation time.

The fundamental anisotropy is an intrinsic property of thechromophore. It is 0.4 if the absorption and emission dipoles are collinear andless than 0.4 if the angle between them is greater than 0°:

Since the theory developed by Dale and co-workers assumes single transitionmoments, it may be appropriate to use 0.4 for for both donor and acceptor.However, absorption and emission transition dipoles are in general notcollinear, and the experimentally observed values for a number ofchromophores are indeed considerably less than 0.4. A second origin for

is mixed polarizations; that is, the absorption or emission across aspectral region is characterized not by a single transition dipole moment,but by a combination of two or more incoherent dipole moments.(11) Forchromophores with observed it is not entirely clear whether theobserved value or 0.4 is more appropriate for calculation of depolarizationfactor. Both values have been used in published studies.(8,12) The use of theobserved value offers a more conservative estimate of because this choiceyields a slightly wider range in than the use of 0.4.

Probes that possess spherical symmetry with triply degenerate transitionspolarized along mutually perpendicular directions approximate isotropicoscillators. Chromophoric metal ions such as and the lanthanide ionssuch as belong to this group. If such a metal ion isused as either an energy donor or energy acceptor, the uncertainty in isreduced to 12% or better. If these isotropic probes are used as both the donorand the acceptor, the uncertainty in is completely removed andThe lanthanides compete for sites in several -binding proteins andin Ca-ATPase. Their use can yield highly accurate intersite distances for thesesystems.

It has been frequently suggested that if both attached donor and attachedacceptor have unconstrained, isotropic, and rapid motions a reversalof the points of their attachment should have little effect on the isotropic andrapid motions. To obtain an “experimental” verification of the validity ofthe assumption of many investigators in the past performed reverselabeling with the same donor–acceptor pair in which the donor and acceptorsites were interchanged. If the interchange did not result in a gross alterationof the observed value of E, the assumption was then considered valid. This


protocol was used in our previous study(8) of the distance in S-1.A reversal of the labeling sites produced a very similar transfer efficiency, butthe depolarization factors were large, clearly indicating that the donor andacceptor were not in randomizing motions. In another study,(12) essentiallyidentical transfer efficiencies were obtained between two sites in the actin .S-lcomplex when the donor and acceptor sites were interchanged. Yet the axialdepolarization factors were above 0.5, leading to a range of 0.155 to 2.999,again showing that the attached fluorophores were in restricted, not ran-domizing, motions. For these systems (and possibly some other systems thathave been studied with reverse labeling), the assumption of isotropic anddynamic averaging was clearly not valid, and it was necessary to describethe donor–acceptor separation not by a single distance, but by a range ofdistances.

3.4.3.2. Statistical Interpretation of

In a treatment that is formally equivalent to that by Dale et al., Haaset al.(11) have proposed a method to evaluate the range of on the basis ofthree-dimensional transition dipole moments. Their procedure also yieldsstatistical prediction of the most probable distance. They also showed thatchromophores that have mixed polarizations and are characterized by multipledipole transitions can have a narrow range. For such chromophores theassumption of may be more acceptable than for those with a singletransition dipole. Chromophores with the dansyl moiety (e.g., IAEDANS)exhibit mixed polarizations across the absorption band, and in some FRETstudies involving such chromophores it has been assumed that the range islikely narrow and Such an argument has been used to justify, at leastin part, the use of with these probes. It was recently pointed out(14) thatthe reduction of the range by this mechanism would be possible only if theacceptor chromophore has multiplicity in its absorption transition dipole.Multiple transition dipoles in the donor are not likely to affect the rangemainly because directly excited vibronic states are rapidly relaxed to the samelowest energy level of the excited singlet state before emission occurs. It wouldseem that such a donor always emits from a single emission dipole. Furtherwork will be required to address this problem.

Statistical treatments were developed for frozen but randomly orienteddonors and acceptors, each characterized by a single transition moment.These methods(15,16) enable calculations of the most probable donor-acceptordistance from an observed transfer efficiency. The validity of these statisticalapproaches has been questioned(10) on the grounds that the orientations ofattached chromophores are generally constrained by the surrounding environ-ment. It remains an open question as to whether such statistical methods canyield physically meaningful distance parameters.


3.5. Selected Applications

3.5.1. Myosin and Actomyosin

The first contractile protein studied by energy transfer was myosinATPase.(17) We determined the transfer efficiency from the tryptophanylresidues to an extrinsic probe (8-anilino-l-naphthalenesulfonate) bound to asingle hydrophobic site by measurement of the quenching of the donor emission.No transfer distance was calculated because of the multiple donors andbecause of the uncertainty at that time of the number of tryptophanyl residuesin the enzyme. A small time-dependent change in the transfer efficiency wasdetected upon addition of ATP, and this change was correlated with ATPhydrolysis. These early results suggested that the conformation of myosinduring the steady-state ATP hydrolysis was different from that generatedby mixing myosin with ADP.(18) The existence of more than one myosinconformation in the myosin ATPase pathway was subsequently elucidated byother spectroscopic and rapid kinetic methods.

Several lines of physical evidence suggested that there was no significantchange in the gross shape of myosin ATPase subfragment-1 (S-1) duringinteraction with nucleotides or actin. This finding and other considerationshave led to the notion that an important aspect of the energy transductionmechanism in muscle involves communication between functionally importantsites located in S-l, namely, the ATPase and actin sites. This interactivebinding of substrate and actin can lead to localized changes in the S-lstructure. A detailed investigation of such a model will require full knowledgeof the three-dimensional structure of S-l. While crystallography will ultimatelyyield the structural information, X-ray data from single crystals will not beavailable for some time. FRET studies can provide preliminary informationon the approximate arrangement of selected points.

The heavy chain of myosin S-l contains two reactive thiols, which arecommonly referred to as (Cys-707) and (Cys-697). Chemicalmodification of results in a three- to fourfold activation of theCa-ATPase activity with a concomitant loss of the EDTA-ATPase and actin-activated ATPase activities. The additional modification of abolishes allenzymatic activities. We(8) used the donor intensity method to determine theenergy transfer distance between the two thiols with 1,5-IAEDANS attachedto as the energy donor and IAF linked to as energy acceptor(Figure 3.1). Very similar results were obtained when was labeled withthe acceptor and with the donor. The following distance parameterswere obtained: is thedistance based on and R(max) and R(min) are the maximum andminimum distances, respectively, based on calculated


from Eqs. (3.21) and (3.22). The presence of actin shortened the distance bywhereas MgADP had no effect on the distance. A different acceptor,

N-(4-dimethylamino-3,5-dinitrophenyl)maleimide (DDPM), was also used tomodify The distance from 1,5-IAEDANS at to DDPM atwas determined by the lifetime technique using phase fluorometry(19) andpulse fluorometry.(20) This second donor–acceptor pair yielded a value of27–28 for shorter than from the 1,5-IAEDANS–IAF pair. The1,5-IAEDANS–DDPM distance in S-l decreased by 7-8 upon addition ofMgADP, demonstrating a perturbation of the localized structure bynucleotide binding. The difference in structure between the two acceptors maybe responsible for the different results.

Of the five cysteines in actin, the penultimate residue (Cys-374) isaccessible for chemical modification under mild conditions. In the monomericform (G-actin) the protein contains a single bound ATP. Upon polymeriza-tion to F-actin, the bound ATP is split, converting the bound nucleotide toADP. We previously reported that replacement of bound ATP in G-actin by1, -ethenoadenosine triphosphate did not impair the ability of actinto polymerize(21) and determined the distance from bound (donor) toCys-374 modified with 4-[N-(iodoacetoxy)ethyl-N-methyl]amino-7-nitro-benz-2-oxa-l,3-diazole (IANBD) as the acceptor to be hasalso been used by other investigators to determine the separation betweenthe nucleotide site and Cys-10 and between the nucleotide site and twospecific points on the myosin S-l heavy chain in the acto · S-1 complex.Figure 3.2 is a space lattice for S-l and acto · S-1 constructed from FRETdistances.(23) The seven specific points in S-l include three residues on theheavy chain, one residue on each of the two light chains, and two pointswithin the nucleotide binding site that is involved in ATP hydrolysis. Tendistances have been reported between these seven points. Two distances inactin and eight distances across actin and S-l in acto · S-1 are also included.The distances shown in Figure 3.2 are far short of the 4N – 10 requirement forcomplete spatial specification of acto · S-l or the individual proteins, but havealready provided some important insight into the transduction mechanism.

Filamentous actin (F-actin) is comprised of two strands of actin polymerthat are assembled into a suprahelical structure. Individual actin monomer(G-actin) is a single-polypeptide protein which is bilobar. The orientation of themonomer in the filamentous structure based on FRET was first investigatedby Taylor et al.,(38) who measured the transfer between chemically equivalentsites located on different monomeric units within the suprahelical structure.In this study two preparations of labeled G-actin were used: one preparationin which Cys-374 was labeled with IAF serving as the energy donor, and theother in which the same residue was labeled with iodoacetamidoeosin ortetramethylrhodamine serving as the energy acceptor. Copolymers of pureactin filaments were obtained by using various molar ratios of the donor-


labeled and acceptor-labeled G-actins. From the known geometry of theF-actin filament, which was approximated as a 13/6 helix, and the FRET dataobtained from lifetime measurements, the radial coordinate of the modifiedresidue was calculated to be about Since this coordinate is the per-pendicular distance between the residue and the helix axis, the result indicatesthat Cys-374 is located near the outer surface of the filament. Three additionalradial coordinates have since been reported: Cys-10(39) and the nucleotide-binding site,(40) both obtained from donor steady-state intensity, and Gln-41,determined from time-domain lifetime data.(41) The latter study also showed


from a theoretical analysis that, within certain symmetry limits, four radialcoordinates (N = 4) and six intramolecular distances (4N – 10 = 6) arerequired to completely define the orientation of the actin monomer in thefilament. In the same study Kasprakz et al.(41) also found that myosin S-lbinding to F-actin increased the radial distance of Gln-41 frombut had negligible effect on the radial distances of Cys-374 and the nucleotide-binding site. These results do not rule out conformational changes which maybe induced on actin monomer in the region of the nucleotide site or Cys-374by interaction with S-l, but probably can eliminate certain models of inter-action which would require global rotation of actin monomer with largeamplitudes. Of special interest is the effect of binding of activator calcium toregulated actin filament (actin filament decorated with a full complement ofthe regulatory proteins troponin and tropomyosin) on the radial coordinates.We (Censullo and Cheung, unpublished results) have recently shown thatthe radial coordinate of Cys-374 decreases by 9% when theregulatory proteins are incorporated into the actin filament. In response tocalcium binding the coordinate decreases further to suggesting acompression of the actin filament. While four radial coordinates have beenreported, complete characterization of the orientation of actin monomerin the filament must await additional intramolecular distances because4N – 10 = 6 for this system and only two such distances are at hand (seeFigure 3.2).

The method developed by Taylor et al.(38) was intended for the use ofFRET to investigate actin assembly and disassembly during motile events innonmuscle cells. The authors also showed that the emission decay of thelabeled actin microinjected into living Chaos carolinensis was essentiallyunaltered and unquenched by the cytoplasm. This observation suggested thepossibility of quantitative measurements of energy transfer in living cells.FRET was also used recently to follow assembly of synthetic myosin thickfilaments and demonstrate exchange of myosin molecules between thefilaments.(42) Donor (IAEDANS)-labeled and acceptor (IAF)-labeled myosinswere preincubated in 0.5 M KCI. The assembly into filaments was initiated byreduction of ionic strength by dilution and monitored by following thedecrease of donor intensity. From these data the concentration of myosin thatremained unassembled (critical concentration) was calculated and found to be40 nM in the range of myosin concentration This determinationwas possible because of the high sensitivity of the FRET assay and themeasurement was considerably easier to carry out than that based on analyti-cal ultracentrifugation. Dynamic exchange of myosin molecules between thickfilaments was demonstrated by following the quenching of donor intensityresulting from donor-labeled thick filaments and acceptor-labeled thickfilaments. The extent of exchange was found to be 75% after 180 min and wasindependently confirmed by using 125I-labeled myosin and ultracentrifugation.


These approaches are applicable to investigations of macromolecularassembly and disassembly in other systems.

The hydrolysis of ATP by actomyosin ATPase can occur via both thedissociating and nondissociating pathways, in which the hydrolytic stepoccurs on myosin subfragment-1 (S-l) that is dissociated from actin orattached to actin, respectively. The dissociating pathway can be representedby

where A is actin and A · S-1 (acto · S-1) is the complex formed between actinand the myosin heads (S-l). The question of which chemical step is coupledto force generation during muscular contraction has been extensivelyinvestigated. It is believed that S-l attaches to actin in two states. Weaklybound states (e.g., A · S-l · ATP, A · S-1 · ADP · Pi) are associated with thebeginning of the power stroke, and the strongly bound states (A · S-1,A · S-1 · ADP ) are associated with the end of the power stroke. Transitionfrom the weakly bound states to the strongly bound states may be accom-panied by large-scale structural changes within the acto · S-1 complex. Thesechanges are particularly relevant in deciding whether the myosin head or aportion of the myosin rod is the energy-transducing element that is involvedin converting chemical free energy to mechanical work. The structures of thestrongly bound states have been extensively studied by various techniques,but not much is known about the weakly bound states. To investigate thissecond attached state requires a method to arrest the contractile process atthe pre-power stroke, where S-l is weakly attached to actin. One way toobtain stable analogous of the weakly bound state (A · S-1 · ATP) is throughtrapping of ATP at the active site by cross-linking the thiolsWhen distance 14 between actin Cys-374 and S-1 light-chain Cys-177(Figure 3.2) was measured with non-cross-linked S-l (A · S-1, A · S-1 · ADP),

was found to be in the range When cross-linked S-l was usedto mimic the weakly bound species (A · S-1 · ATP), Thislarge decrease has been taken as direct evidence for a change in the structureof myosin heads that could account for tension generation. Qualitativelysimilar, but less pronounced, results have also been reported recently ondistance 10 between actin Cys-374 and S-1 Cys-707 by using acto · S-1that was cross-linked by a carbodiimide in the presence of ATP.(31) was

for non-cross-linked acto S-1 (rigor) and increased to 54 Å for cross-linked acto · S-l. Not resolved in these studies, however, is to what extent theobserved distance change occurs within S-l as opposed to being due toreorientation of S-l relative to the actin filament. There is no information on


the kinetics of the distance change that occurs during the transition fromspecies associated with the pre-power stroke to those associated with the endof the stroke.

3.5.2. Troponin Subunits

Troponin consists of three nonidentical subunits: troponin C (TnC),troponin I (TnI), and troponin T (TnT). In vertebrate skeletal and cardiacmuscles these proteins and tropomyosin play a key role in the regulation by

of actomyosin ATPase and the contractile cycle. The initial molecularevent in the regulation is binding of to TnC. The signal is thentransmitted to specific sites on actin via the three other regulatory proteins.FRET has proved a useful tool for understanding the mechanism of this signaltransmission on the basis of global structural perturbations induced in theproteins by binding. Our strategy was to measure the separationsbetween specific sites in each of the troponin subunits and across the subunitsin reconstituted troponin. In one study(43) four specific residues were selectedfor the complex TnI · TnC: two (Cys-133 and Trp-158) in TnI and two(Met-25 and Cys-98) in TnC. These four points yielded six distances. TheTrp in TnI served as an intrinsic donor and the other three residues were


modified by three extrinsic probes: dansylaziridine (DNZ) for Met-25 and1,5-IAEDANS and IAE for Cys-98 and Cys-133. The donor intensity methodwas used to determine the transfer efficiencies for all six distances, and thelifetime method was used to confirm the results obtained from donor intensityfor one of the distances. A tentative proximity map was constructed from thesix measured distances for TnI · TnC (Figure 3.3). In Table 3.1 are listed the

values for the distances that were determined in the presence of EGTA,

Skeletal TnC has four sites. Two sites bind with a highaffinity also binds competitively at these sites.

The other two sites bind specifically with a low affinitySince the intracellular concentration is in the millimolar

range, the two high-affinity sites are likely occupied by in relaxedmuscle, where the cytosolic level is in the submicromolar range.activation involves binding of to the two low-affinity sites. The FRETdistances determined in the presence of and provide a basis forunderstanding structural changes that are induced when muscle is activatedby With the exception of the intersubunit distance C(25)-I(158), theother five distances were significantly perturbed when the ionic medium waschanged from to The changes were in both directions in therange of 5–10Å . A decrease of 5 Å in C(98)–I(133) in fully reconstitutedtroponin was also observed(44) when the medium was changed from to

Although there are inherent uncertainties in the individual measureddistances, large changes in the distances must reflect substantial structuralperturbations. These results suggest large-scale movements of two regions in


each protein toward or away from each other when muscle is switched on.These movements must play a role in the transmission of the signal.What is not resolved at this time is whether these changes occur with timeconstants that are compatible with the known kinetics of force generation.

The crystal structure of TnC is of considerable interest. It shows theprotein to be dumbbell-shaped,(45) with the N- and C-terminal regions foldedinto two globular domains. The two domains are connected by a nine-turn

(32 residues). The middle third of this long helix is not in contact withother regions of the molecule and is exposed to solvent. Since the proteincrystals were obtained at the question arises as to whether the proteinis also dumbbell-shaped at neutral Our previous study(46) showed that

for the distance between Met-25 (labeled with DNZ) and Cys-98 (labeledwith IAE) was 39 A The actual separation between the points ofattachment is likely smaller because of the size of the probes. When thewas reduced from 7.5 to 5.0, the transfer efficiency was reduced 13-fold,corresponding to an increase of the donor–acceptor separation by a factor of2.(47) These results are interpreted in terms of an acid-induced dimerization ofthe protein,(48) and suggest that the conformation of TnC in neutral solutionmay be different from that predicted by crystallography.

3.5.3. Ribosomal Proteins

In early studies(49) the proximity of the component proteins of the 30SEscherichia coli ribosomal particle was estimated by using 1,5-IAEDANS asthe energy donor and fluorescein isothiocyanate (FITC) as energy acceptor.Individual proteins were labeled either at a specific site or randomly onthe protein surface. Each donor-labeled protein was reassembled in parallelexperiments into two 30S samples, one containing no other labeled proteinand the other containing a single acceptor-labeled protein. Transfer efficien-cies between 20 pairs of proteins were determined by measurement of donorintensity. Sufficient transfer data were collected to yield 12 distances (15–78 Å)on a subset of six of the 30S proteins (S4, S13, S15, S16/17, S19, S20). Since4N – 10 = 14 for this case, two additional distances would be required tospecify the geometric arrangements of the six proteins within the 30S subunit.By assuming the proteins to be reasonably spherical and the probe dipoles tobe isotropically and randomly oriented, the investigators limited the arrange-ment of the proteins to four possible models and concluded that the sixproteins were not coplanar. The center-to-center distance between S7 and S9was 31 Å, in agreement with the value of deduced from neutronscattering. More recent FRET studies involved labeling specific sites such assulfhydryl groups and the 3´ end of RNAs. These studies included (1) thedistances between a thiol group of the elongation factor EF-Tu from Thermus


thermophilus bound to reconstituted Escherichia coli ribosome and the 3́ endsof I6S RNA, 5S RNA, and one of the two cysteines of Sl,(51) and (2) thedistances from the 3' end of 16S RNA to S1 and S21.(52) In the latter work,three probes were used: 3-(4-maleimidylphenyl)-4-methyl-7-(diethylamino)-coumarin (CPM) as energy donor, and fluorescein-5-rnaleimide (FM) andfluorescein thiosemicarbazide (FTS) as acceptors. The distance betweenCPM-S21 (the single cysteine at position 22) and FTS-16S RNA at the 3´ endwas 51 Å, the distance from FTS-16S RNA to CPM-S1 (at one of the twocysteines) was 68 Å, and that between CPM-S21 and FM-S1 labeled at bothCys-292 and Cys-349 was 68 Å. Since there were two acceptor sites in S1, theactual distance from the donor in S21 to either of the two acceptors in S1would be longer than 68 Å, dependent upon their geometric relationship withrespect to the donor. Binding of poly(uridylic acid) to the 30S subunitincreased the S1–16S RNA distance to at least 80 Å and the S21–16S RNAdistance to 56 Å, but had no effect on the S21–S1 distance. The effect of the50S subunit on the first distance was negligible, but an increase of 8–10 Å wasfound for the other two distances. The combined effect of poly(U) and 50Swas an increase of all three distances in the range of 10–12 Å. In spite of thefact that poly(U) is known to interact with S1 in 30S subunits, the interactionleaves the S21–S1 distance unperturbed. The other results suggest that 50Ssubunits affect the conformation of S21 in 30S subunits.

Among the proteins of the large SOS subunits, FRET has been used todetermine the distance from L11 to L6 (46 Å) and from L11 to L10 (56 Å).These results(53) suggested that the L6/L10/L11 domain is tightened duringactivation. Energy transfer data(54) were recently used to locate the position ofthe single –SH group (Cys-38) of L11 in the E. coli ribosome. The distancefrom the –SH group labeled with CPM to the 3´ end of 5S RNA labeled withFTS was 76 Å and that to the 3´ end of 23S RNA was 69 Å. From these andother distances within the 30S subunits and between the 30S and 50S sub-units, it was concluded(54) that L11 is located in the 50S subunit below thelateral protuberance characterized by L7/L12.

3.6. Comparison of FRET Results with Results from Other Techniques

3.6.1. Comparison with Crystallographic Data

3.6.1.1. Use of Lanthanide Ions

Because of the uncertainty in and because of the finite size of donorand acceptor chromophores, comparison of FRET distances with thosederived from other methods is generally hazardous. However, the uncertainty


in can be eliminated if both donor and acceptor chromophores areapproximate isotropic oscillators. A number of reports have appeared inwhich lanthanide ions were used as replacements in the determinationof distances between metal-binding sites in several -binding proteins.Such results for four -binding proteins given in Table 3.2 can becompared with the corresponding distances obtained from crystallography.The agreement is excellent in two cases, but only reasonable in others. In thefirst studies of this kind,(55,56) the FRET distances determined with asthe donor were as much as 3 Å shorter than those determined with asthe donor for the same sites. The possibility was raised by the investigators(55)

that as a donor would in general sense a higher transfer efficiency than

150 Herbert C. Cheng

This effect was particularly pronounced with the pair.The more recent studies of troponin and calmodulin(59) did not showa systematic trend for anomalous transfer efficiency that was sensed by

, although the FRET distances were smaller thanIt should be noted that in the crystals, lanthanide ions are not always

located at positions identical to those for bound . For TnC, andlocated at site III were within 1 Å of the position, but the same

ions bound to site IV were 5.5 Å away from the positions.(45) In a morerecent study(63) based on a higher resolution (2.2 Å), the distances between

and lanthanide ions in sites III and IV wererefined to 0.3 and 0.8 Å, respectively. These results could very well account forthe 2-Å difference between the interlanthanide and intercalcium distances evenif fluctuations in intersite distances arising from protein dynamics is assumedto be negligible. The difference between and the shorter observedfor parvalbumin and thermolysin was interpreted(64) as indicating thatexchange interactions involving electron overlap (which leads to a highertransfer efficiency than dipole–dipole interaction) became important fortransfer between lanthanide ions approximately 11.8 Å apart. This interpreta-tion was based on a general consideration(64) for the relative contributions ofthe two types of transfer mechanism involving chelated andHowever, the results shown in Table 3.2 are insufficient for such a generaliza-tion at this time.

3.6.1.2. Energy Transfer from Tyrosyl to Tryptophanyl Residues

Comparison of FRET distances with crystallographic or other data ismore difficult with aromatic chromophores (either intrinsic or extrinsic) thanwith isotropic oscillators. The distance between the two tyrosyl residues(residues 99 and 138) of calmodulin was reported(65) by measuring thetransfer efficiency from one Tyr to the other, which was nitrated withtetranitromethane. The best estimate of this separation (from Tyr-99 to thenitro moiety of nitrotyrosine-138) was for -saturatedcalmodulin. The center-to-center distance between the two phenol rings is11.27 Å.(62) The presence of a nitro group attached to the phenol ring as theacceptor may, in part, account for the longer FRET distance, but the limitedmobility of one or both phenol rings also may contribute to the observedtransfer distance.

The lens protein calf gamma-II crystallin offers an interesting example. Itcontains 15 Tyr and 4 Trp residues. The transfer efficiency from the tyrosylresidues to the tryptophanyl residues was found to be atOn the basis of the crystal structure, all Tyr–Trp distances were calculatedbetween the centers of the two aromatic rings; they ranged from 5 to 30 Å.


From these distances and the assumption of the transfer efficiency waspredicted to be 83%,(66) in reasonable agreement with the value observed insolution. The difference almost certainly arises from the possibility that some,if not many, of the donor and acceptor chromophores have limited motionalfreedom. Since only the short distances contribute significantly to E becauseof the dependence, the question is whether the side chains of the residuesthat are separated by short distances are highly polarized or depolarized. Thisquestion cannot be answered without other types of information.

A final example is the C-terminal octapeptide of cholecystokinin,(Asp-Tyr-Met-Gly-Trp-Met-Asp-Phe- ). NMR studies(67) showed that

exists preferentially in a folded conformation with and turnsaround the sequence Gly-Trp-Met-Asp and Met-Asp..Phe- . This foldedconformation results in a structure in which the side chains of Tyr and Trpmust be pushed away from each other, resulting in a relatively large separa-tion between them. This structural feature was confirmed by energy transfermeasurements,(67) which yielded The choice of for thiscase is reasonable because indole (acceptor) is characterized by two lineartransition moments and the Trp in the peptide is relatively free to rotate, asdemonstrated by NMR results.

3.6.1.3. Transfer between Aromatic Probes

The unfolding in the N-terminal region of RNase A was recently studiedby FRET(68) from a donor (ethylenediamine monoamide of 2-naphthoxyaceticacid) attached to carboxyl groups of Glu-49 and Asp-53 to a nonfluorescentacceptor (2,4-dinitrophenyl) on the -amino group. Since Glu-49 and Asp-53are close in both the sequence and the X-ray structure, heterogeneity in donorlabeling was unimportant. The interprobe distance was found to be

A under folding conditions. The contributions of the finite size of thedonor and acceptor moieties to the transfer distance were estimated bycombining the conformational statistics of the donor linkage with geometricalconsiderations. These contributions were then combined with structuralinformation from unmodified RNase to give an estimated average interprobedistance of 36 Å. This approach provides an expedient comparison of FRETresults with X-ray structural information. The excellent agreement in this casejustified the use of

3.6.2. Comparison with Cross-Linking Data

Chemical cross-linking of specific groups by bifunctional reagents isgenerally taken as evidence for close proximity between the two groups. Theiraverage separation, however, may or may not correspond to the span of the


bifunctional reagents. The FRET distance between the two thiols andof myosin subfragment-1 (distance 6, Figure 3.2) is in the range of

30–45 Å, dependent upon which acceptor is used. It has been known that thesame two thiols can be cross-linked by a variety of bifunctional reagents withwidely different spanning lengths (2–14 Å). The segment of the polypeptidechain between the two thiol groups is thought to be flexible. There must exista wide distribution in the separations between the two –SH groups. FRETmeasurements yield an average of these distances. The average contains notonly a population weighting, but also weighting by the inverse sixth power ofthe distances, which favors shorter distances. The cross-linking experimentsinvestigate instantaneous separations that fall into a certain narrow range,and the method has a very narrow distance window, whereas FRET resultsreflect the entire distance range.

Another example is the TnI • TnC complex. The transfer distance in thepresence of between Cys-133 of TnI and Cys-98 of TnC is in the rangeof 41–74 Å.(43) These two groups have been cross-linked by l,3-difluoro-4,6-dinitrobenzene in the presence of ,(69) providing evidence for their veryclose proximity. Even allowing for the finite size of the donor and acceptorprobes, it is unlikely that the average transfer distance could be as small as10 Å. The binding constant(70) of the labeled TnC for unlabeled TnI or of thelabeled TnI for unlabeled TnC is in the presence ofThe proteins are not very tightly held together. Some conformational fluctua-tions within each protein and in the complex as a whole can be expected.In consequence, the FRET distance is an average value. These examplesillustrate the inadequacy of using a single average distance to describe aproximity relationship. Section 3.9 will address the experimental determina-tion of distribution of FRET distances from both lifetime and steady-statedata.

3.7. Application of FRET to Enzyme Kinetics

Many enzymes contain tryptophanyl residues. If their substrates carrychromophores that can accept excitation energy from Trp, energy transfer canbe expected between enzyme and substrate as the enzyme-substrate (ES)complex is formed. Formation and breakdown of the ES complex can bedirectly monitored in stopped-flow experiments at submicromolar enzymeconcentration. First proposed in the early 1970s, this kinetic approach toresolve mechanistic details of catalysis has been developed mainly by Auldand co-workers(71,72) during the past decade and applied to a number ofproteases with dansylated substrates. Both pre-steady-state and steady-statekinetics can be observed in one experiment. Analysis of the pre-steady-statekinetics enables determination of the number of intermediates and individual


rate and binding constants.(73) Direct observation of the ES complex byenergy transfer under steady-state conditions simplifies and supplementsconventional initial-rate kinetic studies. The method is sufficiently sensitiveto allow measurements to be carried out under the kinetic conditions

and A steady state is thus achieved rapidly understopped-flow conditions and maintained until the reaction is complete.

The time-dependent fluorescence changes for the binding and hydrolysisof by chymotrypsin are shown in Figure 3.4. Curve Ashows the change of acceptor (dansyl) fluorescence intensity, which rapidlyreached a maximum, indicating attainment of the steady state. This rapid,pre-steady state was followed by a slower decay of the fluorescence as thesubstrate was hydrolyzed. Reciprocal changes were observed (curve B) whenthe tryptophan (donor) emission of the enzyme was monitored. Also shownin Figure 3.4 are schematic illustrations of the steady-state portion of thekinetic tracing. The acceptor emission at time zero when the steady state isachieved is and at any time t thereafter is The area described by theentire stopped-flow trace is

If the fluorescence properties of the acceptor are unchanged in substrate andproduct, is proportional to the steady-state concentration of ES and thearea A under the curve is inversely proportional to the maximum catalytic


rate (turnover number), The reaction velocity during the steady state isrelated to and by where is the initial substrateconcentration. The Michaelis–Menten equation can be expressed in terms ofthe two stopped–flow parameters:

where is the Michaelis–Menten constant. The steady-state kineticparameters can be determined from stopped-flow traces such as those shownin Figure 3.4 in several ways: (a) from measured values of and(b) from the dependence of on substrate concentration, and (c) from thedependence of on substrate concentration at time and atany time t can be determined directly from kinetic traces similar to thatshown in Figure 3.4, and can be readily determined from the area underthe kinetic trace when the initial concentrations of enzyme and substrate areknown. Thus, a double-reciprocal plot of versus can be con-structed from the kinetic trace obtained from a single substrate concentration.The values of and obtained by this method for the hydrolysis of

by chymotrypsin are in excellent agreement withthose obtained by the usual initial–rate method.(72) The action of an inhibitorcan be readily visualized. A competitive inhibitor reduces and does notaffect the area under the kinetic trace. In contrast, a noncompetitive inhibitoris expected to increase the area A but has no effect on

The rapid fluorescence change during the pre-steady-state time intervaldisplayed in Figure 3.4 is shown in Figure 3.5 on a faster time scale. Duringmixing and within the first 3 ms an increase in acceptor fluorescenceoccurred. The subsequent exponential increase to the final level,was characterized by the rate constant This two-step change reflectsthe presence of two intermediates, and these results are consistent with the


accepted acyl intermediate mechanism for ester hydrolysis by chymotrypsin asrepresented by the usual scheme

The initial increase in fluorescence to signals the formation of ES, and theexponential increase to reflects conversion of ES to the acyl intermediateEA. This example illustrates how both pre-steady-state and steady-statekinetic parameters can be extracted from one experiment on the basis ofenergy transfer.

Several anions including chloride, sulfide, and carboxylate have beenknown for over 30 years to inhibit carboxypeptidase A (CPA). The nature ofanion inhibition has been difficult to investigate partly because of the limitedsolubility of CPA in the absence of salt and hence the necessity to work withfairly high concentrations of anions. This difficulty has recently been over-come(74) by using dansylated di- and tripeptides. Chloride was found tobe a competitive inhibitor as qualitatively indicated by the changes of theshape of the stopped-flow kinetic traces and by detailed analysis of the traces.Since for chloride inhibition was strong (40–120 mM) and not stronglypH-dependent, the investigators concluded that the site of anion interaction isunlikely to be the metal atom.

3.8. Time-Resolved Energy Transfer Measurements

A major structural information that can be obtained from energy transferstudies is global perturbation resulting from ligand interactions. Such infor-mation provides a static picture of what has happened after the interaction.An important question is whether the perturbation is kinetically competent tobe a component of an overall biological mechanism. Up to this time, verylittle attention has been paid to the kinetics of energy transfer changes.

As we have discussed, the distance between the two thiols, andin myosin subfragment-1 is sensitive to the presence of MgADP. Th transferefficiency between the donor (IAEDANS) attached to and the acceptor(DDPM) linked to was increased upon addition of nucleotide, resultingin a decrease of the donor–acceptor distance by 7–8 Å. We(75) haveinvestigated the kinetics of this energy transfer change in stopped-flowexperiments by rapidly mixing the doubly labeled S-l withMgADP at 20°C. Since the acceptor was not fluorescent, the reaction wasmonitored by following the decrease in donor fluorescence. Steady-statemeasurements had shown this decrease to be about 15%. Kinetic traces couldnot be fitted with a single exponential, but were well fitted with a two-


exponential model. The two apparent first-order rate constants increasedfairly smoothly with increasing ADP concentration and appeared to level offbeyond 1 mM ADP. At the highest ADP level (0.94 mM) studied, was

and was When extrapolated to zero ADP concentration,was ca. and was The initial slope of the versus [ADP]plot was and that of the versus [ADP] plot

The anisotropy of the attached donor was also monitored in stopped-flow experiments over the same time interval used in the intensity exeriments.In all cases the donor anisotropy remained essentially unaltered during thechange in donor intensity, with an average value that was in agreement withthat determined with the steady-state fluorometer. These results suggest thatthere is no change in probe mobility during the course of reaction andindicate that the observed increase in energy transfer is due entirely to abiphasic decrease in the donor–acceptor distance from about 29 to 22 Å.The amplitude of the fast phase, at high nucleotide concentration, accountsfor approximately 70% of the total change of intensity. If the Förster criticaldistance remains constant during the reaction, the fast phase reflects adecrease in donor–acceptor distance of 4–5 Å and the slow phase an additionaldecrease of about 2 Å. The use of FRET in conjunction with anisotropysignals in stopped-flow studies to investigate the kinetics of structural pertur-bations has advantages over the use of signals from single chromophores. Theamplitude changes associated with the signal from a single chromophoreprovide little useful information, but the amplitudes of the change in theFRET signal can in principle be related to physical changes. Because thetransfer efficiency depends on the inverse sixth power of the donor–acceptorseparation and on the relative orientation of the donor and acceptor, smallstructural perturbations can be more readily detected by this method thanwith single chromophores. In the present case the physical changes inducedby nucleotide binding, naemely, the movement of different domains of S-ltoward each other, have been time-resolved. In a previous study Garland andCheung(76) showed that, on the basis of the emission of single fluorophores,the kinetics of the interaction of myosin S-l with nucleotides is best describedby a sequential three-step mechanism in which formation of the protein–nucleotide complex is followed by two first-order steps. The physical origin ofthese steps was undefined. The recent kinetic results based on time-resolvedFRET measurements of the same reaction are kinetically compatible with thethree-step mechanism and may provide a physical basis for the previouslyproposed reaction scheme.

While the work described above was in progress, the first full report(77)

on time-resolved stopped-flow measurements of energy transfer changesappeared. The three tRNA binding sites (P, A, E) on E. coli ribosomes havebeen investigated by various methods. The separation between the tRNA


anticodon loops in the P (peptidyl) and A (aminoacyl) sites was estimated byusing wybutine as the donor and proflavin as the acceptor, both located tothe anticodon of This distance was 23 Å.(78) A similar method wasused to determine the separation of tRNA bound to the E site from thatbound to the P site (34 A) and the A site (42 Å). These results describe thetRNA topography in the pre-translocative and post-translocative complexes.A more interesting problem is determination of the topography during trans-location. This was accomplished in stopped-flow experiments. Pre-translocativecomplexes were prepared by binding and atthe P and A sites, respectively, of poly(U)-programmed ribosomes. Thesecomplexes were then rapidly mixed with elongation factor G (EF-G) andGTP. Since the tRNA–tRNA distance in post-translocative complexes waslonger than the distance in pre-translocative complexes, a time-dependentdecrease in energy transfer between wybutine and proflavin in the two tRNAmolecules can be expected during translocation. This was found to be thecase: the acceptor (proflavin) fluorescence decreased and the donor fluo-rescence increased. The kinetic trace of the proflavin fluorescence was resolvedinto three phases with apparent first-order rate constants in the range 1 (fast),0.1 (intermediate), and The changes in the transfer efficiencyduring each kinetic step were calculated from the changes in amplitudes. Theinitial (pre-translocative) efficiency was 0.88. It decreased to 0.77, 0.35, and0.15 during the fast, intermediate, and slow steps, respectively. The fast stephas been assigned to a coordinated displacement of the two tRNA moleculesfrom their respective sites. Because the transfer efficiency during this phasewas still quite large no significant change of the distance betweenthe two anticodon loops occurred during the fast step. However, a significantalteration of the donor–acceptor relationship occurred during the intermediatekinetic step, which has been ascribed to the release of the deacylated tRNAmolecule. Because translocation as revealed by the puromycin assay wasfound to be complete within 60 s after addition of EF-G, the slow kinetic step,taking several minutes to complete, is unlikely to be a part of translocation.Paulsen and Wintermeyer(77) further suggested that the E site-bound tRNAmay not be in contact with the mRNA and that the tRNA in the E site maynot be functionally important for fixing the mRNA in the proper position, ashad been previously proposed.(79)

3.9. Distribution of Distances

The end-to-end distance of a flexible polymer chain is characterized by adistribution among all possible configurations. If one end is labeled with anenergy donor and the other with an acceptor, the observed energy transferefficiency must reflect this multiplicity in configuration. Because of the dynamic


nature of macromolecular structure, the observed efficiency of energy transferbetween a donor and an acceptor attached to specific sites within a macro-molecule or a membranous system can be expected to reflect a distribution intheir separations. Unfortunately, it is not possible to obtain the distributionfrom measurements of the steady-state emission intensity of the donor in asingle experiment. A protocol(80) based on theoretical considerations wasproposed in 1971 to obtain the distribution by measurements of steady-stateintensity in a series of experiments in which was varied. It was proposedto achieve this variation by using different donor–acceptor pairs that werecharacterized by different values and were specific for the same donor andacceptor sites. It was shown that the distribution could be extracted from thedifferent sets of transfer data by an iterative procedure. In practice, such anexperimental procedure is difficult to apply and has not been used because ofthe requirement of many labeled proteins. Since is proportional to donorquantum yield, it should be possible to vary for a given donor chromo-phore attached to a macromolecular substrate by using solvent perturbationin which the donor emission intensity is progressively quenched by additionof an external quencher molecule such as iodide or acrylamide. This approachwas briefly mentioned(80) and appears considerably less complicated than theuse of different donor–acceptor pairs.

An experimental method based on lifetime measurements was proposedin 1972 by Grinvald et al.,(81) who outlined various approaches to recover thedistribution in donor–acceptor distances from the monoexponential decaycurve of donor emission. Several model systems(82,83) were used in the inter-vening years to demonstrate the applicability of this general method, butprogress on its application to proteins has been slow until the past severalyears.

3.9.1. Theory

3.9.1.1. Recovery of Distribution from Lifetime Data

If the decay of the donor emission is single exponential, it is given by

where is the donor intensity at zero time, and is the donor lifetime. Inthe presence of a single acceptor located at a unique distance R from thedonor, the donor decay is given by


with as the Förster critical distance. The donor still decays monoexpo-nentially, but with a composite rate constant of corre-sponding to a lifetime The second term in theexponent of Eq. (3.28), is the rate constant of energy transferto the acceptor.

In a system containing donor–acceptor pairs of different distances, thedonor decay will not be single exponential, but will be dependent upon theprobability distribution of donor–acceptor distances, P(R). Equation (3.28)becomes

If the donor decay in the absence of the acceptor is described by a sum of iexponential terms, Eq. (3.29) is modified to

where the are the i donor lifetimes. Thus, direct determination of the donordecay curve will allow extraction of P(R). Equations (3.29) and (3.30) arevalid for a distribution that remains unchanged during the lifetime of thedonor. Perturbation of the distribution by diffusion may become significant ifthe donor lifetime is long and if the donor and acceptor can diffuse relativelyunconstrainedly toward or away from each other.

Fluorescence lifetimes can now be determined with high accuracy bymultifrequency phase fluorometry. In the frequency domain, one measures thephase shift and the modulation as a function of modulationfrequency The sine and cosine transforms of Eq. (3.30) yield the followingfrequency response functions(84,85):

where is the ith donor lifetime observed in the presence of the acceptor.If the donor emission in the absence of the acceptor is single exponential, nosummation is needed under the integral sign in Eqs. (3.31) and (3.32).

To evaluate P(R) from time-domain decay data, one selects anappropriate distribution function P(R) that will yield the best fit between thecalculated according to Eq. (3.29) or (3.30) and the experimental decay

The constants and are obtained from independent experiments.


When frequency-domain data are used, the chosen P(R) is used to calculatethe frequency response function, Eqs. (3.31) and (3.32), for comparison withexperimental and

3.9.1.2. Recovery of Distribution from Steady-State Data

The experimentally determined transfer efficiency (Eqs. 3.13a and 13b) isconventionally used to calculate a single donor–acceptor separation R byusing Eq. (3.11) once the Förster critical distance is known. If there is adistribution of distances P(R) for the donor–acceptor pair, the transferefficiency is given by(80)

The dependence of the measured transfer efficiency on is different from thatindicated by Eq. (3.11). For each of the donor–acceptor distances, the transferefficiency will depend upon the value of used in the experiment. Theproblem now is to find different experimental conditions that will yield arange of values for a given donor–acceptor pair and to recover thedistribution function from the transfer efficiencies measured over the rangeof values. Since is proportional to the sixth root of donor quantumyield (Eq. 3.4), one convenient way to obtain a range of values is throughcollisional quenching of the donor quantum yield. This can be accomplishedby using neutral quencher molecules.

The steady-state emission intensities of a donor determined in the absenceand presence of quenching are related to the quencher concentration

[Q] by

where and are the dynamic and static quenching constants, respec-tively. A plot of versus [Q] should yield a straight line andallow determination of both and Alternatively, can be determinedfrom a plot of versus [Q], where and are the donor lifetimes inthe absence and presence of quenching, respectively. The dynamic quenchingconstant is then used to obtain the quenched donor quantum yield atvarious values of [Q]:

where is the donor quantum yield in the absence of quenching. (A different


symbol is used here for quantum yield to avoid confusion with that forquencher.) Once a range of values are obtained, they can be used togetherwith the Förster critical distance determined in the absence of quenchingto calculate the corresponding range of quenched Förster critical distances

It is necessary to use the dynamic quenching constant to calculate thequenched quantum yield and because the observed steady-state quenchinggenerally contains both static and dynamic components. The staticallyquenched species do not contribute to the emission.

3.9.2. Examples

3.9.2.1. Model System

A model system was recently used by Lakowicz et al.(84) to demonstratethe feasibility of using the frequency-domain data to recover the distributionof FRET distances. The molecule N-dansyl undecanoyl tryptamide (TUD)shown in Figure 3.6 contains a donor (indole) at one end and an acceptor(dansyl moiety) at the other, connected by a flexible alkyl chain. The decayand quantum yield of the donor in the absence of the acceptor were deter-mined independently with N-myristoyl tryptamine (TMA). The frequencyresponse of the donor emission is shown in Figure 3.7. In the absence of the


acceptor, the donor decay was adequately described by a single exponentialIn the presence of the acceptor, the donor decay in TUD clearly

was not single exponential, but could be fitted to a triexponential function.The donor decay in TUD is expected to remain single exponential if thedonor–acceptor pair is characterized by a single separation. The observeddeparture from monoexponential decay is evidence for a range of transferefficiencies caused, at least in part, by a distribution in donor–acceptor


distances. The data were then analyzed by using Eqs. (3.31) and (3.32) and aGaussian distribution function


where is the standard deviation of the distribution, is the average dis-tance, and 1, or 2. The half-width (HW) of the distribution (full widthat half-maximum) is The best fit of the data was obtained with

and (Figure 3.8). When HW was fixed at 2 Å, nosatisfactory fit could be obtained. The distributions shown in Figure 3.9 wererecovered from data obtained at several temperatures and using multipliers

1, and 2. The use of a Gaussian distribution must be regarded as anapproximation at this time. It is unclear which multiplier is appropriatealthough is appropriate for an infinite flexible chain in three dimensions.

Regardless of which model is used, the recovered distribution is broad andindicates a wide range of distances.

In a recent study, Gryczynski et al.(85) have demonstrated the validity ofthe procedure outlined in Section 3.9.1.2 to recover distance distribution by thesteady-state quenching method. These workers measured values and transferefficiencies for TUD and two related peptides in the presence of acrylamide atseven different concentrations (Table 3.3). As expected, and E decreasedwith increasing acrylamide concentration. The distance distributions wererecovered from least-squares analysis of transfer efficiencies. In Figure 3.10 thefirst distance distributions recovered from steady-state quenching data foroligopeptides are compared with the distributions obtained from frequency-domain lifetime data. The good agreement suggests that the steady-statemethod could be a very useful tool for determination of the distribution ofdonor–acceptor separations. The uncertainty in the recovered distributiondepends, among other factors, upon the range of values that were used


in the calculations. Although it is generally not possible to obtain a widerange of values due to the sixth root dependence of on quantum yield,simulation and experimental data obtained by Gryczynski et al. indicate thateven a 20 % range in values can yield good estimates of the donor–acceptordistribution. The method is considerably simpler than the previously proposedmethod requiring labeling of macromolecules at the same two sites withseveral different donor–acceptor pairs and does not require advanced


instrumentation and analysis methods that are essential with the frequency-domain or time-domain lifetime techniques.

3.9.2.2. Distributions in Proteins

The general procedure used for TUD has been applied to the small proteinTnI (see Figure 3.3). We(86) reinvestigated the distance between Trp-158 andCys-133 labeled with IAE by multifrequency phase fluorometry over the rangeof 10–300 MHz and extracted the donor–acceptor distance distributionunder different experimental conditions. The analysis of the data was morecomplicated than that for TUD because the decay of the single Trp of TnIwas triexponential,(86,87) ranging from below 1 to 6–9 ns. In the presenceof the acceptor, the donor decay was also approximately triexponential, butthe decay was more heterogeneous than in its absence as judged by statisticalcriteria. This heterogeneity was taken as manifestation of a distribution intransfer distances. Shown in Figure 3.11 are the recovered Gaussian distribu-tions of donor–acceptor distances for native and denaturated TnI. The dis-tribution for the native protein was characterized by an average distanceof 23 Å, in good agreement with the value previouslyreported(43) on the basis of a single donor–acceptor distance and a HW of12 Å with Upon denaturation increased by about 4 Å, whereas theHW increased fourfold to 47 Å.

Table 3.4 summarizes additional distance distribution parameters thatwere determined for TnI with the complex formed between TnI and TnC.(88)

In the complex the average distance between Trp-158 and Cys-133 wasslightly longer than in isolated TnI, also in agreement with previousresults. However, was not affected by binding to the sites of the


TnC in the complex. This finding differs from previous results based on asingle distance. The reason for the discrepancy is unclear at this time. It is ofinterest to note that binding to TnC reduced the half-width of thedistance distribution in Tnl, and the decrease was shown to be statisticallysignificant. This narrower distribution shows an effect of binding on thedynamics of TnI, and this effect may be increased dynamic fluctuations. Thealtered dynamics may play an important role in the transmission of thesignal from TnC via TnI to activate actomyosin ATPase.

Simulation studies and examination of the ratio surfaces for thesimulated frequency-domain data indicated that values of and HW can berecovered to within 2 Å.(88) Probe mobility and the uncertainty in the orienta-tion factor can contribute to the observed distribution width. Theseproblems were considered in detail, and the possible effect of on HW wasinvestigated by measurements of anisotropy decay.(85,88) While it was not yetpossible to assign unequivocally the various factors contributing to anyobserved distribution width, it was concluded that experimental half-widths inexcess of 10 Å, such as those observed with TnI in the native state, mustreflect to a certain extent local dynamic fluctuations of the protein. The dis-tribution was used to follow structural changes as a function of denaturationby guanidine hydrochloride (GuHCl). The half-width increased progressivelyand dramatically with increasing denaturant concentration from about 12 Åto a final level of about 50 Å, which was observed at Themidpoint of the transition was near 1.5 M GuHCl. The change in followeda similar pattern. TnI is known to be stabilized against denaturation bycomplexation with TnC in the presence of This protection was alsoevident from the recovered distributions determined under these conditions.The large HW observed in the presence of GuHCl is compatible with theexpectation of a wide range of distances and decreased dynamic fluctuationsin the denatured protein. These results demonstrate that the distributions ofFRET distances in protein can be recovered with good accuracy and are use-ful in studies of protein folding and unfolding.


The steady-state quenching method was also used to determine thedistance distribution of TnI in a manner similar to that described inSection 3.9.2.1 for model systems. The recovered distributions(85,88) are inexcellent agreement with those shown in Figure 3.11. The good agreementprovides further confidence in the steady-state method.

We recently recovered the distribution of the distances between Met-25and Cys-98 in TnC. Of interest is the reduction of the mean distance and thehalf–width of the distribution induced by calcium binding to the two calcium-specific sites. These decreases are observed with both isolated TnC and TnCcomplexed to TnI(89). The results from this study and those on the distancedistributions of TnI in the TnI–TnC complex indicate that the binding ofactivator calcium to its receptor sites (TnC) in muscle alters the conforma-tional dynamics of both proteins. The alternate interaction of TnI with actinand TnC during cycles of relaxation–contraction may be closely related to thechanges in dynamic fluctuations in these proteins, and these fluctuations mayprovide a mechanism by which the calcium signal is transferred from itsreceptor via TnI to distant sites on the actin filament.

A fourth example of distance distribution is the distances ofmyosin S-l (distance 6, Fig. 2). The distribution parameters recovered underseveral experimental conditions are listed in Table 3.5.(90) The previouslyobserved decrease in induced by MgADP was reproduced from therecovered distribution. In contrast to TnI, denaturation of S-l in 6 Mguanidine hydrochloride resulted in a small increase in the half-width and adecrease of only 3 Å in the mean distance. The lack of a significant denatura-tion effect on the distribution half-width indicates that the polypeptide segmentis question has very similar conformational dynamics regardless of whetherthe protein is native or denatured. The conformation of the segmentin native S-l approaches a random coil. This is the first direct physicalevidence for an extreme segmental flexibility in this region of the S-l heavychain. It is not possible to obtain this information from other types ofexperiments.

Recovery of the distribution of FRET distances in proteins from time-


domain data has also been accomplished in several studies. Amir and Haas(91)

determined the distribution of several intramolecular distances in bovinepancreatic trypsin inhibitor (BPTI). They labeled the N-terminal aminogroup with the energy donor (2-methoxy-l-naphthyl)methyl (MNA) and the

group of one of the four lysine residues (positions 15, 26, 41, and 46)with the acceptor [7-(dimethylamino)coumarin-4-yl]acetyl (DA-coum). Foreach of the four derivatives with attached donor and acceptor, themonoexponential decay curve of the donor emission was used torecover the distance distribution. The half-widths of these distributions werein the range 9–11 Å, and the average distances ranged from 21 to 34 Å. Whenthe sizes of the donor and acceptor probes were estimated and taken intoaccount, the recovered average distances were very close to those expectedfrom the crystal structure of BPTI.

In another study of the distribution of distances based on time-domaindata, Haas et al.(92) labeled Glu-49 and Asp-53 of bovine pancreatic RNasewith ENA (ethylenediamine monoamide of 2-naphthoxyacetic acid) as energydonor and the N-terminal amino group with l-fluoro-2,4-dinitrobenzene asenergy acceptor. The attached ENA decayed monoexponentially in theabsence of the acceptor. The distribution of distances recoverd from dataobtained in 50% glycerol showed an average distance of and a half-width of The average distance compares favorably with the valuepreviously determined in aqueous solution on the basis of a single donor–acceptor separation.(68) Upon denaturation by 6 M guanidine hydrochloridein 26% glycerol the average distance increased to and the half-widthto These authors also discussed in detail possible contributionsto the observed half-width. Since the attached donor probe used in this studyhas a considerable length (eight or nine single bonds and two amide bonds),probe flexibility (as opposed to protein dynamics) may have contributedsignificantly to the observed distribution recovered for the protein in thenative state. Nevertheless, these results qualitatively corroborate those wehave obtained with TnI in that the parameters of the distribution of FRETdistances are sensitive to transition from an ordered state to a partially dis-ordered or completely disordered state.

In two recent reports Brand and co-workers reported the distributions ofthe distances between two specific residues of a staphylococcal nucleasemutant that were recovered from time-domain data as a function of tem-perature(93) and guanidine hydrochloride concentration.(94) Bimodal distribu-tions were needed to fit the data obtained at high temperature or in highguanidinium concentration, providing evidence for two denatured states of themutant protein consistent with a circular dichroism study(95) thatstaphylococcal nuclease mutants can exist in two denatured states. Theseresults clearly suggest potential applications of the distribution of FRETdistances to protein folding/unfolding problems.


In closing, mention should be made of the possibility of extractingdistance distributions from measurements of the fluctuation of either donor oracceptor steady-state fluorescence intensity. Haas and Steinberg(96) presenteda very interesting theoretical treatment in which the autocorrelation functionof the fluorescence intensity is derived for a flexible polymer chain in terms of(1) the diffusion coefficient of one set of chain ends relative to the other and(2) the equilibrium distribution function of distances between donor andacceptor chromophores. Simulations were carried out for a series of oligomerchain molecules which were labeled with an energy donor at one end and anenergy acceptor at the other. The equilibrium distance distributions of thesemolecules had been previously obtained from the nonexponential decay curveof the donor emission.(82) The results showed that the intensity fluctuationcan be large enough to be experimentally useful in the study of moleculardynamics of macromolecules. In spite of the simplicity of the derivedautocorrelation function, no experimental studies have been reported becauseof the enormous instrumental problems associated with such fluctuationmeasurements.

3.10. Summary and Prospects

The foundation for FRET studies of biological systems is well estab-lished. For simple macromolecules the determination of energy transfer bymeasurements of donor steady-state intensity is relatively simple to carry out,but for macromolecular assemblies the determination is more complicated.Lifetime measurements for energy transfer should offer advantages over thesteady-state method because they are independent of concentrations, but theyare considerably more difficult to perform. The accuracy of the measureddonor–acceptor distances are frequently limited to the uncertainty of theorientation factor, except for special cases where isotropic oscillators (e.g.,lanthanide ions) are used as the donor and acceptor. However, by usingdonor–acceptor pairs that meet certain spectroscopic criteria and by alsoperforming ancillary experiments on fluorescence depolarization, one canplace an estimate on the error in distance resulting from the uncertainty inIn many reported studies, the error in assuming is not more than 30%.

The general method has already provided answers to a number ofquestions in certain biological systems. Among these is the demonstration ofglobal structural perturbation resulting from a biological event. Individualdonor–acceptor distances within a macromolecule or in an assembly of suchmolecules can be useful in providing an approximate molecular shape of thesystem in solution. Most donor–acceptor pairs are useful in the distance range10–60 Å, which is the range of molecular dimensions for many globular


proteins. The large FRET distances are complementary to the considerablysmaller distances that are determined by NMR for construction of solutionmolecular structure.

Recent applications of FRET include stopped-flow studies of enzymecatalysis and the kinetics of changes in energy transfer that occur during abiological event. The time constants of these changes can provide an insightinto the biological relevance of global structural perturbations detected byFRET.

The recent success in recovering distributions in donor–acceptor distancesby both pulse and phase fluorometry has opened a new direction for experi-mental investigation of intramolecular dynamics. The feasibility of usingsteady-state emission intensity data to determine distance distributions inpeptides and proteins has now been experimentally demonstrated. Becauseof the relative ease with which such data can be generated, the steady-statequenching method should be of widespread usefulness in studies of macro-molecular dynamics. The experimental distribution may serve to test thevalidity of distance distributions predicted from theoretical considerations.Because the distance distribution is determined by the overall dynamics of themacromolecule, it should be useful in studies of the mechanism of proteinfolding and elucidation of the sequence of formation of its intermediates alongthe folding pathway. The protocols for recovering distance distributions bymultifrequency fluorometry have been extended to cases where the decay ofthe donor emission is multiexponential, but are still limited to the situationwhere there is a single acceptor for each donor. When the algorithms areextended to multiple acceptors, it should be possible to determine clusteringof lipids or proteins in membranes and ion distribution around polyelectro-lytes such as nucleic acids.

Acknowledgment

The preparation of this chapter and the research carried out in mylaboratory and described herein were supported in part by NIH grants AR31239 and AR 25193.

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47. C. K. Wang and H. C. Cheung, Effect of pH on the distance between Met-25 and Cys-98 oftroponin C, Biochemistry 24, 3364 (Abstract) (1985).

48. C. K. Wang, J. Lebowitz, and H. C. Cheung, Acid dimerization of skeletal troponin CProteins: Structure, Function and Genetics 6, 424–430 (1989).

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50. J. A. Langer, D. M. Engelman, and P. B. Moore, Neutron-scattering studies of the ribosomeof Escherichia coli: A provisional map of the locations of S3, S4, S5, S7, S8, and S9 in the30S subunit, J. Mol. Biol. 119, 463–485 (1978).

51. W. Rychlik, W. Odom, and B. Hardesty, Localization of the elongation factor Tu binding siteon Escherichia coli ribosomes, Biochemistry 22, 85–93 (1983).

52. O. W. Odom, E. R. Dabbs, C. Dionne, M. Muller, and B. Hardesty, The distance betweenS1, S2, and the 3´ end of 16S RNA in the 30S ribosomal subunits, Eur. J. Biochem. 142,261–267 (1984).

53. K. B. Steinhauser, P. Wooley, J. Dijk, and B. Epe, Distance measurement by energy transfer.Ribosomal proteins L6, L10, and L11 of Escherichia coli, Eur. J. Biochem. 156, 497–503(1986).

54. H.-Y. Deng, O. W. Odom, and B. Hardesty, Localization of L11 on the Escherichia coliribosome by singlet–singlet energy transfer, Eur. J. Biochem. 156, 497–503 (1986).

55. M. J. Rhee, D. R. Sudnick, V. K. Arkle, and W. D. Horrocks, Jr., Lanthanide ionluminescence probes. Characterization of metal ion binding sites and intermetal energytransfer distance measurements in calcium-binding proteins. 1. Parvalbumin, Biochemistry 20,3328–3334 (1980).

56. A. P. Snyder, D. R. Sudnick, V. K. Arkle, and W. D. Horrocks, Jr., Lanthanide ionluminescence probes. Characterization of metal ion binding sites and intermetal energytransfer distance measurements in calcium binding proteins. 2. Thermolysin, Biochemistry 20,3334–3339 (1980).

57. C.-L. A. Wang, T. Tao, and J. Gergely, The distance between the high affinity sites oftroponin C measured by interlanthanide ion energy transfer, J. Biol. Chem. 257, 8372–8375(1982).

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4

Least-Squares Analysis ofFluorescence Data

Martin Straume, Susan G. Frasier-Cadoret,and Michael L. Johnson

4.1. Introduction

The evaluation of mechanistic parameters, such as fluorescence lifetimesand amplitudes, by the application of nonlinear least-squares methods for theanalysis of experimental data is a mathematically ill-posed problem. In effect,this means that it is much more difficult to derive fluorescence lifetimes fromexperimental data than it is to calculate synthetic data from known valuesof the fluorescence lifetimes and relative amplitudes. The reader should beaware that nonlinear least-squares parameter estimation techniques are notguaranteed to find a set of mechanistic parameters which will provide a gooddescription of any particular set of experimental data. Furthermore, even ifa set of mechanistic parameters is found to fit the experimental data withinreasonable statistical precision, there still is no guarantee that those parameterswill be the unique, or the best, set of answers. However, when the assumptionswhich govern the parameter estimation techniques are carefully obeyed,experimental data can be analyzed to yield useful information about anexperimental system.

Nonlinear least-squares analysis is a numerical procedure for estimatinga set of parameters, of an equation, such that this equation willdescribe a particular set of data points, and These procedures work bysuccessive approximation; that is, given an estimate of the values of thenonlinear least-squares algorithm returns a more accurate approximation of

Martin Straume, Susan G. Frasier-Cadoret, and Michael L. Johnson • Departmentsof Pharmacology and Internal Medicine, University of Virginia Health Sciences Center,Charlottesville, Virginia 22908. Present address for M.S.: Department of Biology, The JohnsHopkins University, Baltimore, Maryland 21218.Topics in Fluorescence Speciroscopy, Volume 2: Principles, edited by Joseph R. Lakowicz. PlenumPress, New York, 1991.

177

178 Martin Straume et al.

This approximation procedure is then applied iteratively until the values ofthe parameters being estimated, do not change within some specified limit.

Linear least-squares analysis is a special case of the more general non-linear least-squares procedures which requires only a single iteration todetermine correct values for the parameters. A system of equations is linearif all of the second and higher order derivatives of the function withrespect to all of the individual parameters which are being estimated, areequal to zero. Examples of linear equations are straight line and quadraticequations. Because the only significant difference between linear and nonlinearleast-squares techniques is the number of iterations required to reach a stablesolution, we will only discuss the more general nonlinear least-squaresparameter estimation procedures in this chapter.

All linear and nonlinear least-squares parameter estimation techniquesare based on a series of assumptions. Only when these assumptions are strictlyobeyed can the use of least-squares parameter estimation techniques bejustified or the results obtained be considered valid. When these assumptionsare not obeyed, the least-squares techniques are, at best, approximate andtheir applications somewhat empirical. Nevertheless, least-squares techniqueshave proven to be extremely useful for the analysis of data from fluorescenceexperiments.

The collection of experimental data and the analysis of these data by atechnique such as nonlinear least squares is not a process of two independentsequential steps. The method of choice for the analysis of the experimentaldata is dependent on the inherent random and systematic uncertainties andexperimental limitations of the actual experimental data. Consequently, thedata need to be collected in such a manner that the concomitant experimentaluncertainties of the data are compatible with the assumptions inherent to themethod of analysis.

The purposes of this chapter are, first, to review the assumptions uponwhich nonlinear least-squares parameter estimation techniques are basedand discuss the consequences of these assumptions; second, to describe themathematical details of some of the nonlinear least-squares techniques forparameter estimation, confidence interval evaluation, and confidence intervalpropagation, and, third, to review other measures of goodness of fit withparticular reference to the information which can be obtained from the dif-ferences betwen the actual experimental data and the “maximum likelihood”fitted function. We discuss only those techniques which we have found to beparticularly useful and do not attempt to review all of the nonlinear least-squares methodologies in the literature. We will also provide a series of specificexamples of the application of these techniques to the analysis of fluorescencedata.

The primary emphasis of this chapter will be on the analysis offluorescence lifetime data. However, the techniques and methodologies of least-

Least-Squares Analysis of Fluorescence Data 179

squares analysis were developed for, and are commonly applied to, a widerange of types of experimental data in fields such as physics, chemistry,biology, sociology, and economics.

4.2. Basic Terminology

Before we proceed with the mathematical basis of the assumptionsand methodologies of nonlinear least-squares analysis, terms such as datasets, dependent variables, independent variables, fitting functions, and theparameters of a fitting function must be defined. In each case examples arepresented to clarify these concepts.

For time-domain fluorescence lifetime measurements a data set consistsof a series of observations of fluorescence intensity and lamp intensity as afunction of time. For frequency-domain fluorescence lifetime measurements adata set consists of a series of observations of amplitude modulation andphase shift of the emitted light as a function of the modulation frequency ofthe excitation light. A typical data set for steady-state fluorescence intensity isa series of observations of fluorescence intensity as a function of emissionwavelength. For the mathematical discussions we refer to a data set as a seriesof observations, and where the subscript refers to an individual datapoint within the data set.

Independent variables are those quantities whose values we can controlby appropriate settings of the instrumentation or sample preparation. Forexample, time is the independent variable for time-domain fluorescencelifetime measurements, frequency is the independent variable for frequency-domain fluorescence lifetime measurements, and wavelength is the independentvariable in the steady-state intensity example above. We refer to independentvariables as

Dependent variables are those quantities whose values are actuallymeasured by the experimental protocol. For example, fluorescence intensity isthe dependent variable for time-domain fluorescence lifetime measurements,amplitude modulation and phase shift are the dependent variables forfrequency-domain fluorescence lifetime measurements, and intensity is thedependent variable in the steady-state fluorescence example. We refer todependent variables as

For convenience in the discussion of the numerical methods, we presentthe independent and dependent variables in two dimensions as a series ofscalar quantities, and However, all of the mathematical proceduresapply equally well when and are vectors in multiple dimensions. Forexample, for frequency-domain fluorescence lifetime measurements thereare actually two dependent variables, amplitude modulation and phase shift.A two-dependent-variable problem such as this can easily be handled by


considering each of the as a vector of length two, the first element beingthe amplitude modulation of the emitted light and the second being the phaseshift of the emitted light.

The mathematical relationship between the independent and dependentvariables is referred to as the fitting function:

where α is a variable-length vector of parameters of the fitting function,The difference between the individual data points, and the fitting

function evaluated at the "optimal" values of is expressed in Eq. (4.1) asThese differences are sometimes referred to as

the residuals of the fit.In fluorescence lifetime measurements the simplest representation of the

fluorescence intensity, as a function of time, t, is given by

where is the fluorescence amplitude at time zero, and is the fluorescencelifetime. For time-domain experiments the corresponding fitting function,

, is a convolution integral of the intensity decay, and the lampintensity, L(t):

where t is the variable of integration over time from 0 to the vector offitting parameters, consists of and and corresponds to the timevalue of a particular data point.(1) The lamp intensity as a function of time,L(t), is experimentally determined for each experiment.

Since frequency-domain experiments have two dependent variables,amplitude modulation and phase shift, there are actually two fitting functions,

and where the “am” and “ps” subscripts refer to eitheramplitude modulation or phase shift(1):

and

where is equal to the excitation modulation frequency, and


For frequency-domain experiments where is given by Eq. (4.2), thevector of fitting parameters, has only one element, the fluorescence life-time, The total amplitude, has been canceled by normalization of theintegrals [the denominator of Eqs. (4.6) and (4.7)].

It should be noted that when the intensity decay law consists of multipleexponentials, or some nonexponential form, Eqs. (4.3)–(4.7) are still valid,and the fitting parameters, are the parameters characteristic of a differentintensity decay law, Some examples of more complex decay laws arepresented later in this chapter.

4.3. Assumptions of Least-Squares Analysis

The desired result of the analysis of any set of data, and is toevaluate a set of parameters, of a fitting function, The analysisprocess should yield the parameter values that have the maximum likelihoodof being correct. Furthermore, the function evaluated at these optimalparameter values must be a reasonable description of the actual experimentaldata in the absence of experimental uncertainties. In order to correctly obtainthe maximum-likelihood parameter values, the function and thedistribution of experimental uncertainties must satisfy a series of assumptions.These assumptions are presented here, and their effects on the analysis processare discussed.

All nonlinear least-squares parameter estimation procedures have sixinherent basic assumptions. These are:

1. All of the experimental uncertainties of the data must be attributableto the dependent variables,

2. The experimental uncertainties of the dependent variables, mustfollow a Gaussian (normal or bell-shaped) distribution.

3. No systematic uncertainties can exist in either the dependent, orindependent, variables.

4. is the correct mathematical description of the data.5. There must be a sufficient number of data points to yield a good

random sampling of the parent population of residuals.6. The data points must be independent observations.

The first assumption means that the values of the independent variablesare known to much greater precision than the values of the dependentvariables, so that the parameter estimation algorithm can ignore anyexperimental uncertainties in the independent variables. For a two-dimensionalY versus X analysis the least-squares parameter estimation algorithm willminimize the sum of the squares of the vertical (Y axis or ordinate) deviations,as shown in Figure 4.1. It is particularly important to realize that it is not


the perpendicular or horizontal distance from each data point to the best-fitcurve which is minimized. Therefore, in order for a least-squares analysis tocorrectly predict the maximum-likelihood parameter values, the analyticalproblem must be formulated, and the data collected, such that the X axis(abscissa) is known to much greater precision than the Y axis (ordinate).Experimental uncertainties in the independent variables cannot be correctlycompensated for by “appropriate weighting factors.” In general, there is noway to circumvent this requirement and still correctly utilize a least-squaresprocedure. A different method of analysis which does allow Gaussiandistributed experimental uncertainty on both the dependent and independentvariables is presented elsewhere.(2)

The first assumption allows transformations of the independent variables,the X axis or abscissa. For example, in frequency-domain fluorescence lifetimeexperiments the independent variable is frequency, but it is usually moreconvenient to graphically represent the experimental data as a function of thelogarithm of frequency. Such a logarithmic transformation of the independentvariables does not invalidate the least-squares procedure.

The data obtained from fluorescence experiments are usually consistentwith the first assumption. For time-domain fluorescence lifetime measurementsthe uncertainty in the time measurement (the X axis or abscissa) is small.For frequency-domain fluorescence lifetime measurements the modulationfrequency of the excitation light (the X axis or abscissa) is known to excellentpresicion. The precision of the wavelength (the X axis or abscissa) for thesteady-state fluorescence case is also sufficient for the least-squares method ofanalysis.

The second assumption states that experimental uncertainty of thedependent variables, must follow a Gaussian distribution. This meansthat if a particular data point was measured an infinite number of times,a frequency histogram of the observed values for that point would followa Gaussian distribution. Figure 4.2A depicts a distribution of this form.This assumption is met by many, but not all, experimental procedures.In particular, the validity of this assumption, and the consequences of


violating this assumption, have been discussed with reference to the use ofthe method of moments for the analysis of time-domain fluorescence lifetimemeasurements.(3)

Nonlinear transformations of the dependent variables, the Y axis orordinate, violate the second assumption. While Figure 4.2A shows a Gaussianfrequency distribution of a dependent variable, Figure 4.2B illustrates theperturbation of the same frequency distribution introduced by performing alogarithmic transformation on the dependent variable. It is clear that thefrequency distribution in Figure 4.2B is not Gaussian and thus violates thesecond assumption of least-squares parameter estimation. For example, intime-domain fluorescence lifetime experiments the counting uncertainty in thedependent variable, the frequency of counts within a given time range, followsa Poisson distribution. When the frequency of counts is large, these Poissondistributions can be approximated by a Gaussian distribution with a standarddeviation equal to the square root of the mean. Consequently, the secondassumption is usually reasonable for time-domain fluorescence experimentswhen the data are represented as counts as a function of time. A commonmethod employed to analyze this type of data, which violates this assumptionof least-squares analysis is to plot the logarithm of the number of countsas a function of time. For the case of a single exponential decay this plotshould be a straight line. However, such a logarithmic transformation ofthe dependent variables produces a skewed distribution of experimentaluncertainties which is no longer a Poisson distribution and cannot beapproximated by a Gaussian distribution, as shown in Figure 4.2B, and thuscontradicts the assumptions of the least-squares procedure. Consequently, forthis example, it is not valid to use any least-squares procedure to determinethe slope of a plot of the logarithm of counts as a function of time.

The third assumption states that the experimental data have no systematicuncertainties. All parameter estimation procedures make this assumption.Systematic errors are a problem that must be corrected in the experimentalprotocol rather than by the analysis procedure. The only way to circumventthis assumption in the analysis procedure is to include a series of terms intothe fitting function, to describe all of the systematic uncertainties. The


vector of fitting parameters, will then include parameters which describe thesystematic uncertainties. A possible systematic uncertainty for time-domainfluorescence lifetime measurements might be a slight difference in the initialtime value between the lamp intensity profile, L(t), and the observed fluo-rescence intensities, If such a difference exists, and it cannot be removedby instrument calibration, then a time offset term, could be included inEq. (4.3) such that

where the vector of fitting parameters, now includes and as inEq. (4.2), and the time offset term, The least-squares estimation procedurethen involves evaluation of the three parameters implicit in Eq. (4.8) ratherthan the two parameters of Eq. (4.3).

The fourth assumption states that the choice of the fitting function,is the correct mathematical description of the phenomenon being

observed. This seems obvious, but an example is worth noting. Consider thecase of fluorescence lifetime measurements of a solution containing a randomdistribution of fluorescence donors and fluorescence acceptors such thatenergy transfer occurs. In this case the form of the fluorescence intensity as afunction of time, can sometimes be approximated by(4)

where b is proportional to the concentration of the fluorescence acceptor andan encounter radius. Analysis of data of this type could also be performed byassuming that the intensity decay law is the sum of a series of k exponentialdecays of the form

where the vector of fitting parameters, consists of a series of amplitudes,and lifetimes, If k is sufficiently large, then

Eq. (4.10) is capable of describing the intensity decay to any desired precision.However, in that case, the values of and will havelittle, or no, actual physical meaning. However, if the data were analyzed byEq. (4.9), then the resulting parameters would be physically meaningful.It is a necessary, but not a sufficient, condition for a physically meaningfulanalysis of experimental data that the functional form, be capable ofdescribing the experimental data to within the precision of the data.

The fifth assumption requires that the number of data points be sufficientto ensure a good statistical sampling of the random experimental uncertainties.


The exact number of data points required to satisfy this assumption is difficultto specify and differs for each set of experimental conditions. Most time-domain fluorescence lifetime instruments collect a minimum of several hundreddata points, which is usually sufficient for this type of experiment. However,some of the older frequency-domain fluorescence lifetime instruments donot produce an adequate number of data points for a proper least-squaresanalysis since they collect data at only three fixed modulation frequencies.The newer multi-frequency fluorometers can collect data at any number offrequencies and can thus provide ample data for the analysis.

The last assumption states that each experimental observation,is independent of any other data point. This means that the concomitantexperimental uncertainties in a particular data point do not influence theexperimental uncertainties in any other data point. For example, if the multi-channel analyzer used for time-domain fluorescence lifetime experiments didnot provide stable time windows, then the uncertainties in the data pointswould appear to be correlated with each other; that is, too many counts inone channel would probably be correlated with too few counts in an adjacentchannel. Error dependence between data points might cause the analysisprocedure to produce erroneous parameter values.

Utilizing these six basic assumptions, it is relatively easy to demonstratethat the method of least squares does produce values for the parameters,with the highest probability, that is, maximum likelihood, of being correct. Ifthe parameters, are correct and all six assumptions are met, we can writethe probability of observing a particular dependent variable, at aparticular independent variable, as being equal to

where is the standard deviation of the random experimental uncertainty ata particular data point, and Z is a proportionality constant. Equation (4.11)also gives the probability, that the values of the fitting parameters,are correct for a data point, and

The total probability that the values of the fitting parameters, are correctfor a complete data set is the product of the probabilities for the individualdata points:


where the product, and summation, apply to each of the n data pointswith subscript i. The maximum probability that the fitting parameters arecorrect, will occur when the summation in the exponential of Eq. (4.13)is a minimum. Consequently, a procedure which will minimize this summationwill yield the set of fitting parameters which have the maximum likelihood ofbeing correct. We can thus define a NORM of the data to be minimized toyield the maximum-likelihood parameter values as

where NDF is the number of degrees of freedom. NDF is defined to be thenumber of independent experimental observations minus the number ofconstraints. Typically, NDF is evaluated as the number of data points timesthe number of dependent variables per data points minus the number offitting parameters. Equation (4.14) is the standard weighted least-squaresNORM of the experimental data points, which is commonly written as thesample variance, Consequently, with the six assumptions stated above, theparameter values, with the maximum likelihood of being correct will begiven by the method of least squares.

It is possible, of course, that a minimum may not exist for the least-squares NORM, s2, and consequently the values of the fitting parameterscannot be determined by any least-squares method. It is also possible thatmore than one minimum of the least-squares NORM exists, in which case thevalues of the fitting parameters will not be unique.

If any of the assumptions are not rigorously met, then the least-squaresmethod may yield a set of parameter values which do not have the maximumlikelihood of being correct. It is impossible to state a priori how much of anerror in the estimated parameters will be introduced if one, or more, of theassumptions is not correct. Therefore, it is important to try to abide by theassumptions if a least-squares method is to be used for the analysis ofexperimental data. Least-squares methods are commonly utilized for theanalysis of fluorescence data because, in most cases, the assumptions can bereasonably applied.

4.4. Least-Squares Parameter Estimation Procedures

The nonlinear least-squares minimization process can be performed by anumber of numerical procedures. All of the methods will yield equivalentresults for the majority of problems. Consequently, we will only present twoof the methods, a modified Gauss–Newton procedure(5) and the Nelder–Mead


simplex procedure,(6) which we have found to be particularly useful. We willalso discuss the advantages and disadvantages of each of these methods.

The reader is reminded that although the algorithms will be presented interms of a two-dimensional problem, Y versus X, they are easily expanded tomore dimensions. This expansion allows the analysis of experimental data withmultiple dependent variables, such as amplitude modulation and phase shift forfrequency-domain fluorescence lifetime measurements. This scalar-to-vectorexpansion also allows for the analysis of experimental data with multipleindependent variables, for examples, steady-state intensity measurements as afunction of emission wavelength, temperature, quencher concentration, etc.

4.4.1. Modified Gauss–Newton Algorithm

The Gauss–Newton nonlinear least-squares algorithm is a procedure thatstarts with approximate values for the fitting parameters, and generates abetter estimate of the fitting parameters. The algorithm is applied iterativelyuntil the values of the fitting parameters do not change within a specifiedprecision. Once the fitting parameters are stable, that is, do not change withina specified tolerance, they will correspond to a minimum of the least-squaresNORM, (Eq.4.14).

In the Gauss–Newton algorithm the data points are assumed to beapproximated by the fitting function evaluated at the values of the fittingparameters after iterations:

The fitting function evaluated after iterations is expressed as a first-orderseries expansion of the function evaluated after k iterations:

where the i subscripts refer to a particular data point, the j subscripts refer toa particular fitting parameter, and the k and superscripts refer to theiteration number. Equation (4.16) is actually a series of equations, one foreach data point. This series of equations can be used to evaluate the valuesof the fitting parameters for the iteration, because the form of thefitting function, is known and we know the values of the fittingparameters from the previous iteration, For the first iteration we mustarbitrarily assign the initial values of the fitting parameters, The procedureis repeated until the fitting parameters, do not change significantly forseveral iterations.


It should be noted that the only difference between linear and nonlinearleast-squares analysis is the number of significant terms in Eq. (4.16). If all thesecond and higher order derivatives of the function with respect to theFitting parameters are zero, then the first-order series expansion, Eq. (4.16), isexact. If Eq. (4.16) is exact, then the solution for the fitting parameters,will be exact, and the process requires only a single iteration to reach stablevalues of the fitting parameters, If the second and higher derivatives are notzero, then Eq. (4.16) is only approximate, the parameter values produced willbe only approximately correct, and the process will require multiple iterationsbefore stable parameter values are attained.

Some authors have claimed that linear least-squares analysis does notrequire an initial estimate, or guess, of the values of the fitting parameters,This is not strictly correct. All of the standard equations for the solution oflinear least-squares problems can be derived from the Gauss-Newton non-linear least-squares procedure by a careful selection of the initial values of thefitting parameters and the use of a single iteration. When zero is used as theinitial value for the fitting parameters, then all terms which are explicitlymultiplied by the fitting parameters become zero and can be eliminated. So,although it seems that no initial estimate of the parameter values is requiredfor the linear case of least-squares analysis, actually the initial estimate hasbeen assumed to be zero.

The evaluation of the fitting parameters at the next iteration, ismost easily explained in matrix notation such that Eq. (4.16) can be writtenas

where is a vector of weighted residuals whose individual elements are

P is a matrix of weighted partial derivatives whose elements are

and is a correction vector whose elements are

A weighting factor, has been included in Eqs. (4.18) and (4.19) toallow for a variation in the magnitude of the random experimental uncertain-ties of the individual data points. Here, represents the standard deviationof the experimental uncertainty in the dependent variable and has the sameform and meaning as the weighting factor described in the least-squares


NORM (Eqs. 4.11, 4.13, and 4.14). If the value of the dependent variable,represents the mean of a group of observations, then is the standard errorof the mean (SEM) of that group of observations, not the standard deviation(SD) of the group of observations.

The matrix form of the series expansion, Eq. (4.17), can be solved for thecorrection vector, if both sides of Eq. (4.17) are multiplied by the transpose,

of the partial derivative matrix, P, as in Eq. (4.21). Standard matrixinversion techniques can then be applied to evaluate the correction vector,as in Eq. (4.22):

Equation (4.17) can also be solved directly by singular-value decompositiontechniques. The singular-value decomposition method for the solution ofsystems of linear equations can be implemented with the FORTRAN subroutineSVD (Ref. 7, p. 192) or the EISPACK† FORTRAN subroutines SVD and MINFIT.(8, 9)

Once the correction vector has been evaluated, then the values of the fittingparameters for the next iteration, can be evaluated:

The Gauss–Newton process, Eqs. (4.15)–(4.23), is repeated until the valuesof the fitting parameters, do not change with successiveiterations to within some specified precision.

In the Gauss-Newton procedure the magnitude of the correction vector,after iterations is proportional to the square of the magnitude of the

correction vector after k iterations. This is referred to as a “quadraticconvergence.” This means that once the parameter values, are close to themaximum-likelihood values, convergence is extremely rapid. Table 4.1 showsthe Gauss-Newton procedure used to evaluate the square root of nine. In thisexample, the correction vector, decreases by a factor of 2.5 for the firstiteration, by a factor of 4.25 for the second iteration, by a factor of 16 forthe third iteration, by a factor of 256 for the fourth iteration, and by morethan 18,000 for the last iteration. It is interesting to note that because of thisquadratic convergence property most computer systems, and hand calculators,use the Gauss-Newton procedure to evaluate square roots.

Several different criteria are commonly used to decide if the Gauss–Newton procedure has converged. For example, it is commonly assumed thatif the least-squares NORM, that is, the sample variance does not change

† The EISPACK software package is a library of FORTRAN subroutines to perform eigensystemmanipulations. It is available from the International Mathematical and Statistical Library(IMSL) Distribution Service, P.O. Box 4605, Houston, Texas 77210-4605, for a nominal charge.


by one part per thousand then the process has converged. A much bettercriterion is that both the values of the fitting parameters, and the variancechange by less than one part per hundred thousand between successiveiterations. It is realistic to apply this more stringent criterion to the Gauss–Newton procedure because usually only a few additional iterations are requiredto produce the more rigorous convergence.

The derivation of the Gauss–Newton procedure presented in Eqs.(4.15)–(4.23) has not been based on the minimization of the least-squaresNORM (i.e., sample variance ) in Eq. (4.14). However, from a carefulexamination of Eq. (4.21), it can be demonstrated that the Gauss–Newtonprocedure is a least-squares minimization procedure. It can be shown that theelements of the vector on the left-hand side of Eq. (4.21), are propor-tional to the derivative of the least-squares NORM, with respect to thecorresponding fitting parameter, The Gauss–Newton process is repeateduntil the correction vector, on the right-hand side of Eq. (4.21) is equal tozero. Thus, at convergence the terms on the left-hand side of Eq. (4.21), theelements of the vector, are proportional to the respective first derivativesof the least-squares NORM, and the right-hand side of Eq. (4.21), isequal to zero. The standard method to locate a minimum of any function,such as the least-squares NORM, is to set the first derivative of the functionequal to zero and to solve for the variable parameters, The Gauss–Newtonprocedure is, consequently, a least-squares minimization procedure since thefirst derivatives of the function are equal to zero at convergence.

For typical parameter estimation problems to which the Gauss–Newtonprocedure is applied, the matrix (P'P) in Eq. (4.21) is difficult to invertbecause it is almost singular. Consequently, the numerical procedure is verysensitive to numerical truncation and round-off errors. For example, a typicalset of fitting parameters for time-domain fluorescence lifetime experiments iscomposed of an amplitude of tens of thousands of counts per time channel


and a relaxation time of five nanoseconds. If the fitting function is writtenin units of counts and seconds, the magnitude of the fitting parameters willdiffer by 13 orders of magnitude, and consequently the elements of the (P´P)matrix could vary by approximately 26 orders of magnitude. Single-precisioncomputer computations are accurate to about five orders of magnitude, anddouble precision is accurate to approximately 12 orders of magnitude. There-fore, it is important to scale the fitting parameters so that the individualelements of the (P´P) matrix are of comparable size within the precision of thecomputer calculations. For the time-domain fluorescence example, a betterchoice of units is percentage of full-scale counts and nanoseconds. All of thecalculations should be performed in double precision, not just the matrixinversion. It is also important to choose a computer algorithm for the solutionof Eq. (4.21) which is proven to work with near-singular matrices. The SquareRoot Method(10) and the LINPACK FORTRAN subroutines DPOSL and DPOFA(11)

work well for near-singular matrices. The singular-value decompositionmethod for the solution of systems of linear equations also works well underthese conditions (see Ref. 7, p. 192, and Refs. 8 and 9).

The advantage of the Gauss–Newton procedure is that with reasonableinitial values of the fitting parameters it will usually converge very rapidly.The disadvantages are that it does not always converge, that it requiresderivatives of the function with respect to the fitting parameters, and that itcan only be used for a least-squares minimization process. The failure to con-verge is a consequence of neglecting the higher order derivatives in Eq. (4.16)and thus is a problem which might occur with any nonlinear problem. Thisproblem is minimized if the initial values of the fitting parameters, are closeto the maximum-likelihood answers. Only one of the numerous methods toimprove the convergence properties of the Gauss–Newton algorithm will bediscussed here.

One simple way to guarantee convergence of the Gauss–Newton algo-rithm is a small modification of Eq. (4.23). If the value of the least-squaresNORM, the sample variance evaluated at the estimated fitting parametersfor the next iteration, is greater than or equal to the value of the samplevariance evaluated for the current iteration, then each element of thecorrection vector, is divided by two and a new set of values is calculated forthe next iteration, Otherwise, the analysis continues using the standardGauss–Newton procedure as outlined above. This cycle may be repeatedseveral times for each iteration of the Gauss–Newton procedure to ensure thatthe sample variance for the next iteration is always less than the samplevariance for the current iteration.

The LINPACK software package is a library of FORTRAN subroutines to perform linear algebramanipulations. It is available from the International Mathematical and Statistical Library(IMSL) Distribution Service, P.O. Box 4605, Houston, Texas 77210-4605, for a nominal charge.


The Gauss–Newton algorithm uses the first derivatives of the fittingfunction, with respect to each of the fitting parameters, Thisimplies that first derivatives of the function exist, that the first derivatives arecontinuous, and that the second derivatives are continuous so that the firstderivatives vary smoothly. However, the algorithm needs only a numericalvalue for the derivative of the fitting function evaluated at the current fittingparameters, and the values of the independent variables, The algorithmdoes not use, or need, explicit equations for the derivatives. It is thereforerecommended that the values of the derivatives be evaluated by a three- orfive-point Lagrange differentiation formula.(12) The five-point formula for thederivative, of a function f(z) evaluated at z is

where is a small numerical increment of z. The value of should be chosensuch that

The error in the evaluation of the derivative is proportional to the fourthpower of the increment, and to the fifth derivative of the functionevaluated at where is in the range

Once convergence has been reached the information matrix, (P´P) inEqs. (4.21) and (4.22), contains some useful information about the parameterestimation process. For example, an estimate of the cross-correlations betweenthe estimated parameters can be obtained from the correlation matrix. Thecross-correlation coefficients, between any two fitting parameters and

are defined as

where the k and j subscripts refer to elements of the correlation matrix andthe matrix.

The cross-correlation between fitting parameters, is particularlyuseful to test the reliability of the determined parameters, If any of thevalues of approach plus or minus one, then any variation in can bealmost totally compensated for by a variation of As a consequence, uniquevalues of and cannot be evaluated without additional information.Unfortunately, the degree to which the can approach plus or minus onewithout indicating a serious problem with the parameter estimation is not welldefined for nonlinear least-squares problems. The definition of the isbased on a number of limiting assumptions, such as the assumption that thefitting equation was linear in the fitting parameters. Consequently, the use of


the cross-correlation coefficients is only approximate for nonlinear problems.However, in a number of nonlinear problems it has been empirically observedthat an acceptable value for appears to be within approximately

Another use of the information matrix, (P´P), is to evaluate the variance–covariance matrix whose elements are

One of the most common uses of the variance–covariance matrix is to useits diagonal elements, as estimates of the variance of thedetermined parameters, The definition of the variance–covariance matrixis also based on a number of limiting assumptions, such as the assumptionthat the fitting equation is linear in the fitting parameters. It has also beenassumed that the number of data points is large enough that the samplevariance, is a close approximation to the parent population variance,Furthermore, the use of only the diagonal elements of the variance–covariancematrix is equivalent to assuming that the parameters are orthogonal oruncorrelated, that is, that a variation of one of the parameters, does noteffect the evaluation of the other parameters, for Consequently, theuse of the diagonal elements of the variance–covariance matrix as estimates ofthe variance of determined parameters is only approximate for nonlinearproblems and for linear problems where the parameters are not orthogonal.The square roots of these diagonal elements of the variance–covariance matrixare sometimes called the asymptotic standard errors of the determinedparameters. A survey of methods for the evaluation of the joint nonlinearconfidence intervals of the determined parameters will be presented later inthis chapter.

4.4.2. Nelder–Mead Simplex Algorithm

The Nelder–Mead parameter estimation algorithm(6,13) provides a methodfor minimizing, or maximizing, any NORM of the data and does not requireevaluation of derivatives or inversion of matrices. In the case of least-squaresminimization, this NORM is the sample variance, in Eq. (4.14). TheNelder–Mead procedure involves a multidimensional search for the minimumvariance by performing a series of carefully selected one-dimensional searches.This method always converges if a minimum exists. For the followingdescription of this method it is assumed that the variance of the residuals,

in Eq. (4.14), is being minimized.The determination of n parameter values requires that n + 1 different sets

of parameter values be arbitrarily selected such that an n-dimensional simplex


with n + 1 vertices is defined. Each of these vertices is represented by a vectorof parameter values, which defines its coordinates. The superscript specifiesa particular vertex of the simplex. For example, for a problem in whichtwo parameters are being estimated, the simplex which is manipulated is atriangle: three (n + 1) vertices in a two (n) dimensional space of parametersbeing estimated.

Initial estimates for parameter values are required to begin the parameterestimation procedure. A set of n + 1 simplex vertices is generated. Thesevertices must be chosen such that they define an n-dimensional figure ratherthan a line in n dimensions. One way to accomplish this is to choose the n + 1vertices such that the arithmetic mean of the vertices corresponds to the initialestimates of the parameters being determined and such that the vectors fromthe initial estimates to the vertices are maximally orthogonalized. A judgmentmust be made, however, when assigning appropriate ranges for each of theparameters during generation of the initial simplex. An initial simplexexhibiting either too closely spaced or too distantly spaced a distribution ofvertices may present the parameter estimation algorithm with a variance spacewhich causes insensitive or hypersensitive response behavior. The algorithmmay therefore incorrectly assume that convergence has been achieved, or it maymigrate to an undesired local minimum quite distant from the initial estimatevalues. Some rationale, whether theoretical or empirical, must therefore beinvoiced to define the range of values which the initial simplex may span.

The first step in the estimation procedure involves calculation of thesample variance at each of the n + 1 simplex vertices. The vertex with thehighest variance, is identified, and the centroid of the remaining vertices iscalculated. The centroid, is the average value of all vertices except thevertex with the highest variance, A one-dimensional search is then con-ducted along the line defined by the points and The immediate goal ofthis search is to identify a new set of parameter values that yield a variancelower than the maximum observed among the current simplex vertices. Thevertex of the current simplex exhibiting the highest variance, is thenreplaced by this newly identified vector of parameter values, and then theentire Nelder–Mead process is repeated.

Three operations are available for this one-dimensional search: reflection,expansion, and contraction. Figure 4.3 presents a graphical depiction of thesethree operations as applied to a two-parameter estimation problem.

The reflection operation generates a vector of parameter values, as

where is the reflection of the current simplex vertex with the highestvariance, relative to the centroid of the remaining vertices, and is thereflection coefficient The reflected vector is thus on the line joining


and on the side opposite from If the variance, associated withis smaller than the two highest variances and higher than the smallest vertexvariance, then is replaced with This generates a new simplex from whichthe Nelder–Mead parameter estimation process is repeated.

Should the variance, of the reflected vector, be smaller than thesmallest vertex variance, then rv is expanded to as

where is the expansion coefficient If the variance, associatedwith the expanded vector, is less than the variance associated with thereflected vector, then is replaced with to generate a newsimplex and the process is begun again. If is greater thanthe expansion operation has failed, is replaced with and then theNelder–Mead estimation procedure is repeated.

The reflection operation, however, may not yield a parameter vector withan associated variance smaller than that of two of the current simplex vertices.On such condition, a new is defined as either the old or asif The contraction operation is then performed on the resultantsimplex as

where is the contracted vector, and is the contraction coefficient


If is replaced with thus generating an improvedsimplex, and the Nelder–Mead parameter estimation process is repeated.If the contraction also fails to reduce the value of a new simplex isgenerated by replacing all of the simplex vertices, with fori = 1, 2, 3, ..., n + 1, and then the parameter estimation procedure is repeated.The vertex of the current simplex with the smallest associated variance isrepresented by Upon encountering a failed contraction, the current simplexis replaced by one in which all vertices have been contracted in the directionof the vertex with the smallest currently identified variance.

The reflection, expansion, and contraction operations just describedinvolve one-dimensional searches over relative distances that are defined bythe magnitude of the respective operation coefficients. Optimal values for thecoefficients cannot be assigned based on any theoretical considerations. Anempirical approach was applied by Nelder and Mead(6) to a number ofanalytically challenging response surfaces. The conclusion of their attempts isthat values of and produce optimum behavior fromthe algorithm.

The criterion for convergence, that condition which signals the satis-factory completion of the parameter estimation process, can conceivably bedefined in a number of ways. For example, when relative or absolute changesin the values of the variable parameters among successive iterations becomesmaller than some specified tolerance limit, convergence may be considered tohave been achieved. Such a criterion, however, implicitly requires knowledgeabout the absolute or relative values of the variable parameters at or nearthe minimum in variance space as well as about sensitivity of the varianceto changes in these parameters. A too stringent criterion may not signalconvergence until an unnecessarily large number of iterations is performedor, in the extreme, may not permit termination of the parameter estimationprocess at all, due to round-off and truncation errors. A too lax criterion,on the other hand, may prematurely cause the algorithm to decide thatconvergence has been achieved and perhaps report parameter values that area poor approximation to those characteristic of the true minimum in variancespace.

A criterion for convergence that may generally be applied, independentof the values of the parameters being estimated or of the sensitivity of thevariance to changes in these parameter values, involves terminating theestimation procedure when the relative change in the variance is less thansome specified value among successive iterations. Use of such a criterion forconvergence is a theoretically sound choice because, depending upon the extentof curvature of the variance space at or near the minimum, the values of theparameters in the fitting vector may be well defined (marked curvature) orpoorly defined (slight curvature). Terminating the estimation procedurewhen the fractional change in the variance is 0.0001 or less generally ensures


identification of a variance minimum. This mechanism implicitly identifies theminimum with sufficient accuracy to permit confidence in the derived parametervalues relative to their statistical uncertainty.

The Nelder–Mead simplex algorithm exhibits linear convergence behavior.This simply means that the relative changes in parameter value for thecurrent iteration are approximately proportional to the changes in theprevious iteration. This is in contrast to the quadratic convergence behaviordisplayed by the Gauss–Newton method in which the relative changes inparameter values improve geometrically among successive iterations (seeTable 4.1). The more rapid convergence characteristic of the Gauss–Newtonalgorithm arises from the use of derivatives of the function withrespect to the variable parameters, Calculation of derivatives and manipu-lation of matrices, as in the Gauss–Newton procedure, are time-consumingprocesses but permit convergence with fewer iterations than does the simplexapproach. A trade-off therefore occurs between the time required forindividual iterations and the number of iterations required before convergenceis achieved. The net computational time required to converge is almost alwaysless for the Gauss–Newton parameter estimation procedure as compared tothe Nelder–Mead simplex algorithm. However, derivatives of the function andmatrix inversions are not required by the Nelder–Mead procedure, thusmaking this algorithm, in principle, somewhat easier to encode.

A major advantage of the simplex algorithm over most other parameterestimation protocols is that any NORM may be employed in the estimationprocess, not just the sample variance, This quality makes the Nelder–Meadalgorithm potentially useful in a variety of applications, even those in whichno explicit functional form is known. The following example describes anapplication of the Nelder–Mead algorithm to maximize a NORM which is anexperimental response, instead of minimizing the least-squares NORM,

Identification of the conditions at which an enzyme exhibits its maximumcatalytic velocity as a function of the three variables pH, temperature, and saltconcentration may be accomplished by exhaustively assaying as a functionof each of these variables. This could be accomplished by searching pH,temperature, and salt concentration at all possible combinations of tendifferent values each, that is, as a search of a three-dimensional grid that isdimensioned 10 × 10 × 10. This approach would require performing the assay athousand times to fully characterize the catalytic velocity with respect to eachvariable, a process that would almost certainly involve expenditure of muchtime and laboratory resources. An alternative approach would utilize theNelder–Mead simplex operations to guide the experimenter in deciding whichsets of conditions to examine. The experimenter can arbitrarily choose foursets of values ( n + 1 points in an n-dimensional space) for the three variablespH, temperature, and salt concentration. The NORM of each of these setsof values is evaluated by actually performing a laboratory experiment to


determine the enzyme’s catalytic velocity for each set of conditions. TheNelder–Mead parameter estimation algorithm can then be applied to arriveat the next set of conditions (pH, temperature, and salt concentration) foran experimental determination of the catalytic velocity. The Nelder–Meadprocess is continued with actual experimental measurements as the method ofevaluation of the NORM, catalytic velocity, to be maximized. Convergence isachieved when the observed catalytic activity of the enzyme no longer changeswithin the precision of the experimental observations. Such an approach willpermit identification of the desired optimal conditions with much less effortthan that involved in performing the assay at all of the possible one thousandsets of conditions.

The Hessian matrix of second derivatives of the NORM with respect tothe parameters being estimated is analogous to the (P´P) matrix of theGauss–Newton procedure. This matrix can be utilized to evaluate the samevariance–covariance matrix and the cross-correlation coefficients of theestimated parameters as with the Gauss–Newton procedure. A convenientmethod for estimating the shape of the variance surface near thinvolves using a quadratic approximation.(14) The n + 1 vertices characteristic

e minimum

of the final, converged simplex are used to create “half-waypoints” defined as A quadratic surface is then estimatedfrom this combined set of (n + 1)(n + 2)/2 points. The original vertices of thefinal, converged simplex are used in defining a set of obliqueaxes with coordinates thus permitting the points to be represented as:

The quadratic approximation of this variance surface, near the minimumis then given by

where the are elements of the vector A and the are the elements ofthe matrix B, which is analogous to the (P´P) matrix of the Gauss–Newtonprocedure. Equation (4.32) can be written in matrix notation, as


where the prime refers to the transpose of the matrix. The coefficients areestimated as:

A set of parameters corresponding to the minimum variance may be estimatedby

In general, given a convergence criterion sufficiently stringent (less than0.0001 relative change in variance, for example), any improvement in theestimate of realized by Eq. (4.40) will not be significant relative to thestatistical uncertainty inherent in the derived parameter values. Additionally,the information contained in the matrix B is based upon the assumption thatthe variance near the minimum (as defined by the final simplex vertices and theassociated “half-way points”) may be described by a quadratic surface. Anynonlinear deviations of the true variance surface from the assumed quadraticbehavior will introduce a systematic error into the statistical uncertaintiesderived for the estimated parameter values.

Quadratic approximation demands careful consideration of possiblecomputer-associated round-off and truncation errors. Calculation of the coef-ficients in the matrix B may be substantially biased by round-off errors whenthe final simplex vertices (and “half-way points”) are too closely spaced.It will therefore be necessary to enlarge the final simplex by expanding thevertices away from the centroid until the values calculated for the andare a factor of 1000 or more greater than the round-off error. Of course, atrade-off occurs as a result of this expansion in that a more distantly spacedsimplex increases the probability that the variance space encompassed by thesimplex will deviate from a quadratic surface.

4.5. An Example of the Least-Squares Procedures—Collisional Quenching

Collisional quenching of fluorescence requires contact between thefluorophore and quencher during the lifetime of the excited state. In the


presence of quenching the decays of fluorescence intensity are expected tobecome more complex than a single exponential due to transient effects whichoccur immediately following excitation of the fluorophore. The detailed formof the decay law is expected to depend upon the interaction radius, R, the sumof the donor and acceptor diffusion coefficients. D, and the specific rateconstant, K, for quenching [see Eqs. (4.44)–(4.46) below]. Consequently, thetime-resolved decays reveal details of the interactions between donor andacceptor in solution. Table 4.2 presents some actual experimental measure-ments for the transient effects in collisional quenching of 1,2-benzanthracene(BA) by 0.1 M carbon tetrabromide as measured by frequency-domainfluorometry.(4) These data were taken from a concentration series which


was used to compare the classical square root of time decay law, Eq. (4.9),with the more complete radiation boundary condition decay law. For moredetails about the experimental protocol, refer to Joshi et al.(4) In the presentexample we will analyze these data by the classical square root of time decaylaw and the more complete radiation boundary condition formulation of thedecay law.(4)

The independent variable for this example is excitation frequency,Each of the 21 data points has two different dependent variables,

amplitude modulation and phase shift, for a total of 42 experimental observa-tions. The form of the fitting equations for these two dependent variables,

and is given by Eqs. (4.4)–(4.7) in terms of the intensitydecay law,

The fluorescence intensity decay, for the square root of timeapproximation of the decay law is given by Eqs. (4.9) where

where is the fluorescence decay lifetime in the absence of quencher, R is theinteraction radius (the sum of the radii of the fluorophore and the quencher),D is the sum of the diffusion coefficients, N' is (Avogadro’snumber/1000), and [Q] is the concentration of the quencher. Consequently,for the square root of time intensity decay law, Eqs. (4.9), (4.42), and (4.43),the vector of fitting parameters, has three elements: R, and D.

For the more complete radiation boundary model of collisional quenching,the intensity decay, is given by

where the time-dependent quenching rate constant, k(t), is given by

where

erfc( ) is the error function complement, K is a specific rate constant forquenching and has units of cm/s, is the fluorescence decay lifetime in theabsence of quencher, R is the interaction radius (the sum of the radii of thefluorophore and the quencher), D is the sum of the diffusion coefficients, N'is (Avogadro’s number/1000), and [Q] is the concentration of


the quencher. Consequently, for the radiation boundary condition intensitydecay law, Eqs. (4.44)–(4.46), the vector of fitting parameters, has fourelements: R, D, and K.

The reader should note that the actual fitting function, the combinationof Eqs. (4.44)–(4.46) for with Eqs. (4.4)–(4.7) for frequency-domainfluorescence lifetime measurements, is a system of equations requiring evalua-tion of a double integral. Neither of these integrals can be done explicitly, sothe integrals were performed numerically. The integral in Eq. (4.44) can bewritten in the form

where x is defined by Eq. (4.46). A series of values of f(x) were evaluated onceand stored for 12 logarithmically spaced values of x. The error functioncomplement, erfc( ), was evaluated by the method of Hastings et al.(15) Theevaluation of Eqs. (4.44)–(4.47) utilized a cubic spline interpolation (Ref. 7,p. 70) of the values f(x) and x for each iteration of the parameter estimationprocess. The integrations in Eqs. (4.6) and (4.7) were performed by an 8-panelNewton–Cotes adaptive quadrature routine, DQUANC (Ref. 7, p. 97).

The choice of units for the fitting parameters, is important to avoidtruncation and round-off errors during the calculations. Consequently, theintrinsic fluorophore lifetime, was calculated in nanoseconds, the unit forthe interaction radius, R, was angstroms, the sum of the donor and acceptordiffusion coefficients was expressed as the logarithm (base ten) of D in cm2/s,and the specific rate constant, K, was expressed as cm/s.

For the purposes of this example we have actually only performed theparameter estimation process on two parameters, the interaction radius, R,and the log of the diffusion coefficient, log D. This was done to simplify theproblem so that two-dimensional graphs of the process could be presented.The donor lifetime was assigned the value 38.8 ns and the quenching rateconstant was assumed to be 150 cm/s, as in the paper by Joshi et al.(4) Givensufficient data, the least-squares procedures would, of course, be capable of anevaluation of more than two parameters. The number of degrees of freedomfor this analysis is 40, that is, 21 data points times 2 dependent variables perdata point minus 2 parameters being estimated.

4.5.1. Example of the Gauss–Newton Procedure

In Table 4.3 the values of the interaction radius, R, and the logarithm ofthe sum of the donor and acceptor diffusion coefficients, log D, are presentedfor successive iterations of the Gauss–Newton procedure applied to the data in


Table 4.2 and the radiation boundary intensity decay law, Eqs. (4.44)–(4.46).These results were presented in Table 2 of Joshi et al.(4) The rapid, quadraticconvergence properties of the Gauss–Newton procedure are clearly shownin Table 4.3. The sample variance, decreased from 5605 to 98 in the firstiteration. By the fourth iteration the procedure has converged to where thefractional changes in the parameters were 0.00005 for log D and 0.00018 forR while the variance decreased by a fractional change of only 0.00018. Thechanges for the fifth iteration were below the level of the single-precisionarithmetic of the computer. At each iteration the derivatives of the fittingparameters were evaluated by a three-point Lagrange differentiation for-mula,(12) so each iteration required five evaluations of the fitting function ateach data point.

Once the Gauss–Newton procedure has converged, the (P'P) matrixcan be used to evaluate several measures of the reliability of the parameterestimation procedure. The cross-correlation coefficient between R and log D,for this example, is –0.90280. This value is within an acceptable range,indicating that the parameter estimation procedure has worked reasonablywell. The asymptotic variance of the estimate of the interaction radius, R, is

which corresponds to an asymptotic standard error ofThe asymptotic variance of the estimate of the logarithm of the sum of thedonor and acceptor diffusion coefficients, log D, is whichcorresponds to an asymptotic standard error of The asymptoticcovariance between R and log D was

The eigenvalues of the correlation matrix were 1.90280 and 0.09719545.Neither eigenvalue is zero, indicating that the analysis procedure hasproduced reasonable values. The largest eigenvalue is proportional to theerror in the asymptotic standard errors of the fitted parameters when cross-


correlation with other fitted parameters is neglected. In this case, the actualstandard errors of the determined parameters will be about 1.9 times theasymptotic standard errors.

Table 4.4 presents a comparison of the values of R, D, and for thesquare root of time and the radiation boundary intensity decay laws.(4) It isclear that the values of the interaction radius and the total diffusion coefficientas obtained from the two fitting functions, Eqs. (4.44)–(4.46) as compared toEqs. (4.9) and (4.42)–(4.43), are quite different. This is an example of a conse-quence of the fourth assumption of least-squares parameter estimation. Thephysical meaning of the determined parameters is based on an assumptionthat the fitting equation, is correct. This assumption applies to allparameter estimation procedures, not just least-squares. Consequently, thedetermined mechanistic parameters should be reported only with reference tothe assumed mechanism.

It is also clear from this example that the variance of fit, is sub-stantially larger for the square root of time intensity decay law as comparedto the radiation boundary condition intensity decay law. Thus, the radiationboundary condition intensity decay law appears to be a better description ofthe actual experimental data. If the two sample variances, in Table 4.4,were only different by 10% instead of 480%, the statistical validity of thisstatement would be questionable. An F test(16) can be applied to evaluate theprobability that the radiation boundary condition intensity decay law is abetter description of the actual experimental data than the square root of timeintensity decay law.

In the present example the F test is used as a measure of the probabilitythat the data analysis by the two independent intensity decay laws yieldsresiduals—the differences between the calculated values and the actual datapoints—which are different. The F statistic is defined as the ratio of the twovariances:


where is the number of degrees of freedom of the variance in thenumerator, is the number of degrees of freedom of the variance inthe denominator, and PROB is the probability that and are dif-ferent. For the present example, and

Most statistics textbooks contain tables of valuesof the critical F statistic for different values of and PROB.(17)

Table 4.5 presents the value of the F statistic for for sucha series of probabilities. This table shows that the probability that the squareroot of time and radiation boundary condition intensity decay laws yieldequivalent fits, for this particular data set, is vanishingly small.

An alternative to the use of tabulated values found in statistics books isto generate a computer program which explicitly calculates the values in thetable. The reader is referred to Zelen and Severo(18) for the mathematicalformulas with particular reference to their equations 26.2.15, 26.2.16, and26.2.23. A second alternative is to use the IMSL† subroutine MDFI, which findsthe inverse of the F probability distribution function.

4.5.2. Example of the Nelde–Mead Simplex Procedure

Table 4.6 and Figure 4.4. demonstrate the use of the Nelder–Mead simplexprocedure for the same radiation boundary condition example as used for theGauss–Newton procedure above. Points A, B, and C are the initial simplex

†The IMSL is the International Mathematical and Statistical Library. It is the most completelibrary of subroutines of its type and was written in standard FORTRAN. It is available from theInternational Mathematical and Statistical Library (IMSL) Distribution Service, P.O. Box4605, Houston, Texas 77210-4605.


vertices in this example, and the ellipse is a contour of constant variance,about the maximum-likelihood values which are marked with a plus sign.Point D is the reflection of vertex A, the current simplex vertex with thehighest associated variance, about the centroid of the remaining simplexvertices, B and C. The variance associated with point D is lower than thatassociated with any of the current simplex vertices, A, B, or C; consequently,an expansion of point D to point E is attempted. Point E produces a furtherimprovement in the variance so that the expansion is successful and thesimplex for the next iteration is defined by the points BCE.


Vertex B now corresponds to the vertex with the highest associatedvariance of the current simplex vertices. Point F is then calculated as thereflection of point B about the centroid of the remaining vertices, C and E.The variance associated with point F is less than that associated with two ofthe current simplex vertices but greater than that of the third. The reflectionis therefore successful, and the simplex CEF becomes the simplex for the nextiteration.

Vertec C now has the highest associated variance, and point G is itsreflection about the centroid of E and F. The variance associated with pointG is greater than any of the variances associated with the current simplexvertices. The reflection operation therefore fails, and a contraction is per-formed. Point H is the contraction of vertex C toward the centroid of Eand F. The variance associated with point H is smaller than that associatedwith two of the current simplex vertices and greater than that of the third.Therefore, the contraction successfully leads to the generation of a newcurrent simplex, EFH.

This procedure is continued until the fractional change in variancesassociated with the simplex vertices varies by less than 0.0001 (the convergencecriterion), thus signaling the end of the estimation process. Ultimately, thefinal, converged simplex will closely surround the end point (marked by +).The spacing between the vertices of this final, converged simplex will dependupon the tolerance required to achieve convergence as specified by theconvergence criterion. To demonstrate the “average” improvement in theestimated parameters per iteration, the centroids of the successive simplexesdeduced above are presented in the lower portion of Table 4.6. The varianceassociated with these centroids is seen to be approaching steadily thatassociated with the end point.

Once convergence has been reached the information matrix, B, can beevaluated from Eqs. (4.36) and (4.37). This information matrix can then beused to calculate the cross-correlation matrix, the variance–covariance matrix,etc., as shown above for the Gauss–Newton procedure. A comparison of thetwo methods indicates that both procedures reach the same values, withinreasonable precision, but the Nelder–Mead technique is slower.

4.6. Joint Confidence Intervals—Estimation and Propagation

The analysis of a set of experimental data by a procedure such as least-squares involves not only the evaluation of the parameter values, which havethe maximum likelihood of being correct, but also the evaluation of a reasonableestimate of the joint confidence intervals of those determined parameters. It isthe joint confidence intervals of the determined parameters which provide ameasure of the overall precision of the parameters and thus the reliability of


the estimated values. The joint confidence interval for a determined parameterincludes the effects of variations of all of the other parameters, within theirrespective joint confidence intervals, while simultaneously maintaining astatistically valid fit of the data. The maximum-likelihood parameter valueshave little meaning without a reasonable estimate of their precision. Conse-quently, the maximum-likelihood parameter values and the associated jointconfidence intervals should always be reported as a single entity; that is, neverreport the parameter values without a reasonable estimate of their uncertainty.

4.6.1. Asymptotic Standard Errors

The simplest estimates of the precision of the determined parameters arethe square roots of the diagonal elements of the variance–covariance matrix,Eq. (4.27). These provide an estimate of the asymptotic standard errors of thedetermined parameters. However, as described earlier, the derivation of theasymptotic standard errors is based on two limiting assumptions: (1) thatthe fitting equation is linear, and (2) that the determined parameters areorthogonal. The first of these assumptions means that the shape of thevariance space, that is, the values of the variance in the neighborhood of themaximum-likelihood values, can be predicted from the minimum variance andthe (P'P) matrix from the Gauss-Newton procedure or the B matrix from theNelder–Mead procedure. When parameters are orthogonal, the determinationof a particular parameter, is independent of the determination of all of theother parameters, and all of the off-diagonal elements of the (P'P) matrixare zero; that is, the parameters are not correlated. In general, these twoassumptions are unlikely to be satisfied. Consequently, a more realisticestimate of the confidence intervals of the determined parameters is needed.

4.6.2. Linear Joint Confidence Intervals

A well-known linear approximation to the joint confidence interval isprovided by the solutions, . of

where n is the number of parameters, s2 is the estimated variance of theminimum, F-statistic is based on the desired confidence probability and thenumber of degrees of freedom, and P, P', and are as previously defined.(19-22)

The solutions, of Eq. (4.49) predict a multidimensional elliptically shapedcontour in a parameter space which corresponds to a constant confidenceprobability. The elliptically shaped region in Figure 4.4 is a two-dimensional


example of a joint confidence contour. It should be noted that the ellipse inFigure 4.4 was actually evaluated by exhaustively searching the parameterspace for a 67% confidence level, rather than by evaluating the solutions ofEq. (4.49), and thus represents the exact confidence contour. In this case,however, the ellipse in Figure 4.4 is a close approximation to the solution ofEq. (4.49). Any pair of values, log D and R, which are inside this contour areacceptable within the specified probability level, and any pair of parametersoutside the contour are excluded at that probability level. Consequently, themaximum range over which either of the parameters, log D or R, can deviatefrom the optimal values and still remain within the joint confidence regionis used as an approximation of the confidence region for that parameter.This approximation to the joint confidence interval correctly considers thenonorthogonal nature of the determined parameters, but fails to consider thenonlinear nature of the fitting equation and consequently the nonlinearity ofthe variance space.

4.6.3. Support Plane Confidence Intervals

It is possible to determine a confidence interval for each of the deter-mined parameters by the support plane procedure. For this procedure a seriesof values for a particular determined parameter are assumed. For each ofthese assumed values the optimal values of all of the other determinedparameters are redetermined by nonlinear least squares, and the variance ateach of the series of assumed values is evaluated. This procedure thus createsa table of values for the optimal variance as a function of assumed values forone of the determined parameters. This series of variance values can now beinterpolated to determine where a statistically significant increase in thevariance occurs. The points at which these statistically significant increasesoccur mark the ends of the confidence interval for that determined parameter.The process is repeated for each of the determined parameters.

A significant increase in the variance is defined by partitioning thevariance into two terms:

where S2minimum is the sample variance at the maximum-likelihood parameter

values, s2 is the sample variance at any other set of parameter values, andS2

parameters is that portion of the total variance which can be attributed to avariation in the parameter values. The number of degrees of freedom forS2

minimum is the number of observations minus the number of constraints,that is, the number of data points times the number of dependent variablesper data point minus the number of determined parameters, N – NDP. The


number of degrees of freedom for S2parameters is just the number of determined

parameters, NDP. The critical ratio of these variances, F-statistic, for anyprobability level, PROB, is determined by

where F(NDP, N – NDP, 1 – PROB) is the F-statistic for NDP degrees offreedom, N – NDP degrees of freedom, and a probability of 1– PROB. Thistranslates into a fractional increase in the overall variance due to a variationof the parameters as

For a one-standard deviation confidence interval the value of PROB isapproximately 67%. For a two-standard deviation confidence interval thevalue of PROB is approximately 95 %.

Figure 4.5 presents a support plane analysis of the radiation boundarycondition collisional quenching example presented in Figure 4.4 andTables 4.2–4.4 and 4.6. This figure is a rotation of the ellipse in Figures 4.4and 4.6 into a third dimension, showing a cross-sectional cut. A series ofvalues of the interaction radius, R, were chosen to span the maximum-likelihood value. For each of these values the maximum-likelihood value ofthe logarithm of the sum of the donor and acceptor diffusion coefficients,log D, and the corresponding variances were determined by nonlinear leastsquares. Figure 4.5 presents a graph of the variance at the maximum-likeli-hood value of log D for each of the assumed values of R. For this exampletwo parameters were determined, there were 21 data points, each with twodependent variables, and F(2, 40, 1.0–0.67) 1.14. Consequently, the criticalvalue of the fractional increase in the overall variance, from Eq. (4.52), is


approximately 1.057. The horizontal line in Figure 4.5 corresponds to a frac-tional change of 1.057 in the overall variance of fit introduced by a variationof the interaction radius. The points of intersection of this horizontal line withthe variance curve correspond to the end points of a 67 % confidence interval.The vertical lines represent the projection of these points of intersection ontothe interaction radius axis; that is, 8.386 interaction radius 8.562.

In recapitulation, the support plane procedure correctly treats nonlinearvariance spaces and nonorthogonal determined parameters. Its major limita-tion is the large amount of computer time required to evaluate the confidencelimits. For example, consider a hypothetical complex problem where sixparameters are simultaneously being estimated. If ten values of the variance asa function of each of the determined parameters are required to determine theconfidence interval of each determined parameter, then the group of sixparameters will require 60 variance values. Remember that each of thesevariance values is determined by a five-parameter least-squares parameterestimation procedure. Therefore, for this hypothetical example the amount ofcomputer time required to evaluate the confidence intervals by the supportplane procedure is approximately 60 times the amount of computer timerequired to determine the actual maximum-likelihood values of the parameters.Clearly, a method which requires less total computer time is desirable forcomplex problems.

4.6.4. Approximate Nonlinear Support Plane Joint Confidence Intervals

The only disadvantage of evaluating the confidence intervals by thesupport plane method is the large amount of computer time required fora complex multiparameter analysis. The disadvantage of the linear jointconfidence interval method is that it does not consider the nonlinear natureof the fitting equations and consequently can, in some cases, provide incorrectestimates of the confidence intervals of the derived parameters. We havedeveloped an approximate method for the evaluation of nonlinear joint con-fidence intervals which is, in effect, a combination of the support planemethod and the linear joint confidence interval method.(5,20–22)

The support plane method is time-consuming because each of the fittedparameters must be searched independently for a critical variance ratio(Eq. 4.52), and each point on this search requires that the least-squares fittingprocedure be repeated in order to account for the possible variation of all ofthe other fitting parameters. If the variation required to maintain the lowestpossible variance in the other fitting parameters can be predicted a priori, thenonly a single search would yield a reasonable approximation of the jointconfidence intervals for all of the parameters in question. Furthermore, eachof the variance estimates required for this search can be directly evaluated by


Eq. (4.14), and the entire least-squares procedure need not be repeated. Thus,if the joint variation in the fitting parameters for a minimum variance can bepredicted a priori, then the support plane method for the evaluation of theconfidence intervals is practical for multiparameter problems.

The advantage of the linear joint confidence interval method is that, fora linear fitting equation, it provides a method of predicting the shape ofthe variance space for any variation of the parameters. The major axis ofthe multidimensional elliptically shaped solution of Eq. (4.49) is a goodapproximation, even for most nonlinear problems, of how the variationof a single fitting parameter will induce a variation in the remaining fittingparameters and still maintain the lowest possible variance. Consequently, thesolutions of Eq. (4.49) can be used to vary the parameters jointly in thesupport plane method for multiparameter problems. The reader is cautionedthat this approach is only an approximation to the complete support planemethod, but our experience has been that it is almost always a very goodapproximation.

The solutions, of Eq. (4.49) define a new set of fitting parameters.These new fitting parameters, NEW i, are formed as

where the summation is over each of the n fitting parameters, and the SFij area set of scaling factors. These scaling factors are chosen such that the newparameters, NEW i, are all orthogonal. Since these new parameters are allorthogonal to each other, the variation of one of these new parameters,NEW i, will not affect the values of the other new parameters, NEWj, forwhich correspond to a minimum variance.

The geometric interpretation of this new set of parameters is relativelystraightforward. Sets of parameter values which can be considered as “correct”answers to within some statistical probability lie within the multidimensional


elliptical region (Figures 4.4 and 4.6) while sets of parameter values out-side this elliptical region are considered to be “incorrect.” The axes of thismultidimensional ellipse do not, in general, coincide with the axes of ourcoordinate system. The process of choosing a new coordinate system is simplya rotation of the coordinate system such that the new coordinate system,NEW i, coincides with the major and minor axes of the multidimensionalellipse (as in Figures 4.4 and 4.6).

The support plane method can be applied to the variation of each ofthese new parameters, NEWi, in order to find the value of each of the newparameters which corresponds to the critical variance ratio (Eq. 4.52). Thisis equivalent to searching each of the major and minor axes of the multi-dimensional elliptically shaped confidence region (see Figures 4.4 and 4.6) forthe critical variance ratio. The values of the original fitting parameters, atthe ends of the axes of the elliptically shaped region can be evaluated from thescaling factors, SFij, and the new parameters, NEWi, which correspond tothe critical variance ratio defined in Eq. (4.52). The maximum variations ofthe original fitting parameters, are then taken as an approximation to thenonlinear joint confidence intervals.

Figure 4.6 graphically depicts an example of this search procedure. Itcorresponds to the analysis of the confidence intervals for the example whichwas previously used to demonstrate the Gauss-Newton and the Nelder–Meadprocedures (see Figure 4.4 and Tables 4.2–4.6). The example involved theevaluation of log D and R by the radiation boundary model based on thedata presented in Table 4.2. The elliptical region in Figure 4.6 is the same asthe region in Figure 4.4 and corresponds to a one-standard deviation, 67%confidence region. The diamonds are the results of the search of the newparameters NEWi which corresponded to the major axis of the solution ofEq. (4.49). The triangles are the results of the search of the new parametersNEWi which corresponded to the minor axis of the solution of Eq. (6.49). Theplus signs correspond to the values found by searching the original fittingparameters, independently of each other, that is, without repeating thefitting procedure at each of the values of currently being evaluated.

It is recommended that confidence regions be defined by the evaluationof all of the points shown in Figure 4.6. This will require searches for thecritical variance ratio in 4n directions, where n is the number of parametersbeing estimated, that is, each in both directions and each NEWi in bothdirections. The reader is reminded that these 4n searches involve only theevaluation of the variance; that is, they do not require additional least-squaresparameter estimation at every point in the search. Consequently, even thoughthis represents twice as many searches as the support plane method, each ofthe searches is much faster. It is important to perform each of the searches inboth directions, because for nonlinear functions the joint confidence intervalis usually asymmetrical.(5, 20–22) The net result of performing all 4n searches is


that the entire shape of the joint confidence region is characterized, even whenit is asymmetrical.

The evaluation of the scaling factors, SFij, is relatively easy. The eigen-vectors of the correlation matrix, Eq. (4.26), define the directions of the axesof the multidimensional elliptical region in the cross-correlation space. Bymultiplying the elements of these eigenvectors times the square roots of therespective diagonal elements of the variance–covariance matrix, the eigen-vectors become direction vectors in the original parameter space. It is thesedirection vectors starting with the maximum-likelihood values of the fittingparameters, which are to be searched for a critical variance ratio. Theeigenvalues provide the relative lengths of the axes. Thus, in order to searchthe major axis for a critical variance ratio Eq. (4.52), the following steps needto be performed. First, find the eigenvector of the correlation matrix thatcorresponds to the largest eigenvalue of the correlation matrix. Second,convert the eigenvector into the parameter space by multiplying its elementsby the particular eigenvalue and by the square roots of the respective diagonalelements of the variance–covariance matrix. Third, search along this eigen-vector for the set of parameter values which yield the critical variance ratiocalculated from Eq. (4.52). It should be noted that the shape of the variancespace for this search is proportional to the square of the distance along theeigenvector. Be sure to search the eigenvector in both directions since thevariance space will, in general, not be symmetrical. Furthermore, it isimportant to search all of the eigenvectors, that is, the eigenvectors whichcorrespond to all of the eigenvalues of the correlation matrix (see Section4.6.6). The reader is referred to the EISPACK software package(8,9) for a groupof FORTRAN subroutines, such as TQL2 and TRED2, which can be used toevaluate the eigenvalues and corresponding eigenvectors of the correlationmatrix.

In recapitulation, this approximate method for the evaluation of theconfidence intervals of determined parameters has the advantages of beingrelatively fast, of considering the nonlinear nature of the fitting function,and of considering the cross-correlation between the fitting parameters.Although it is only an approximation of the support plane method, it is highlyrecommended.

4.6.5. A Monte Carlo Method for the Evaluation of Confidence Intervals

The last method for the evaluation of confidence intervals which we willpresent is a Monte Carlo method. This method has the advantage that itdirectly generates the entire probability distribution for each of the determinedparameters without making an assumption about the shape of the probabilitydistribution. However, this method requires a knowledge of the shape of thedistribution of the experimental uncertainties associated with each of the data


points. It should also be noted that this method requires a large amount ofcomputer time and, as a consequence, probably should only be used in caseswhere the entire probability distribution of the fitted parameters is required.

The basic method is very simple. The first step is to analyze the data byany method, such as least-squares, which will yield a set of fitted parameters,

which have the maximum likelihood of being correct. Given this set ofanswers, and the original distribution of independent variables, Xi, a set ofsynthetic data is generated. This set of data simply consists of the evaluationof the fitting function at and each of the Xi, The next step is to take thisperfect simulated data set and add to it realistic simulated experimentaluncertainties such that we have a data set which is based on a known setof parameters and contains realistic experimental uncertainties. The con-sideration of uncertainties requires the generation of pseudo-random noiseto be superimposed upon the simulated dependent variable values, Yi. Theparticular distribution of this pseudo-random noise should be consistent withthe experimental uncertainty distribution of the actual experimental data. Thisdistribution of experimental uncertainties is required to follow a Gaussiandistribution by the second basic assumption of the least-squares method.For a discussion of how to generate Gaussian distributed pseudo-randomnumbers, see Ref. 7, p. 240. This last step is repeated a few hundred times suchthat a few hundred simulated data sets are generated, each with a different setof random uncertainties included. The least-squares analysis is then performedon each of the few hundred simulated data sets, and frequency histograms ofeach of the parameters, are generated from the results of the few hundredleast-squares parameter estimations. These histograms will represent areasonable approximation of the probability distributions for the determinedparameters based on an analysis of the original experimental data.

In effect this process has asked the question, “How does a particular dis-tribution of experimental uncertainties affect the analysis of a particular dataset?” The results are dependent on the assumed magnitude and distribution ofexperimental uncertainties. The results are also dependent on the particulardistribution of independent variables, Xi. As the number of simulated datasets is increased, the precision of the resulting probability histograms willincrease. The exact number of simulated data sets which are required isimpossible to predict a priori, but a reasonable number seems to be a fewhundred. This, of course, means that the computer time required to performthis process will be a few hundred times what was required for the originalleast-squares analysis.

4.6.6. Propagation of Confidence Intervals

Quite often an investigator will wish to use a set of determinedparameters, and their associated confidence intervals to calculate some


other derived quantity. For example, for the square root of time descriptionof collisional quenching the investigator may wish to calculate the value offrom Eq. ( 4.42 ) or the value of b from Eq. (4.43). For the radiation boundaryexample the investigator might want the value of the quenching rate at timezero, k(0), from Eq. (4.45). It is relatively easy to calculate the maximum-likelihood values of these quantities from the maximum-likelihood values ofthe fitting parameters, However, the method for the propagation of theconfidence intervals is not obvious since the individual fitting parameters,are expected to be correlated with each other and their respective confidenceintervals are expected to be asymmetrical.

The confidence intervals of the fitting parameters can be evaluatedrelatively simply by the method outlined in Section 4.6.4.(22) The mainobjective of this method is to characterize the shape of the one-standarddeviation joint confidence intervals by evaluating 4n points on the ellipticallyshaped confidence region. A similar confidence region exists for any derivedquantity. All that is required to determine confidence intervals for any derivedparameter is to map each of the 4n points which characterize the originalconfidence ellipse into a parameter space for each of the derived quantities.This is done by evaluating the derived quantity at each of the 4n points whichdescribe the original confidence contour. The extreme spread of the resulting4n values of each of the derived quantities is simply the one-standarddeviation confidence interval for that derived parameter.

Evaluating confidence intervals of derived parameters from the proba-bility distributions obtained by the Monte Carlo method is also simple. Thederived parameters are evaluated from each set of the maximum-likelihoodfitted values, obtained from each of the few hundred sets of synthetic data.A complete probability distribution of the resulting derived parameters canthen be constructed.

4.7. Analysis of Residuals

Estimation of parameters characteristic of some mathematical model bya least-squares procedure produces those parameter values, which have thegreatest probability (i.e., maximum likelihood) of correctly accounting for theexperimental data being analyzed when certain assumptions are made aboutthe distribution of random experimental uncertainties. Before any quantitativeleast-squares analysis of experimental observations is possible, however,some reasonable fitting function, G must be generated. The processof defining such a mathematical model involves theoretical considerationsand, often, previous experimental conclusions when the understanding of thephenomenon being examined is evolving to a level of greater complexity.


Recent advances responsible for expanding the base of biophysical knowledgeas well as the ongoing development of more sophisticated instrumentationhave made necessary this evolution of mathematical models to account for themore accurate and precise experimental observations now possible. Feweranalytical approximations are necessary, and, in fact, approximations areoften no longer tolerable as progress in experimental and analytical methodsproceeds at an accelerating pace. Higher quality biophysical data obtainedat greater levels of determination often point out deficiencies in currentanalytical models as evidenced by their inability to satisfactorily describe theavailable data. The current section of this chapter deals with the methodsavailable for determining the appropriateness of an analytical model. Thismay be accomplished most easily by examining the residuals.

Residuals are the differences between the observed experimental data andthe fitting function evaluated at the maximum-likelihood values of theparameters, The residuals may be thought of as the observed experimentaluncertainties when the model is correct. As stated earlier (see Section 4.3),certain assumptions are implicit and must necessarily hold if a least-squaresestimation procedure is to be a valid method for obtaining estimated modelparameter values. A way of demonstrating the consistency of a model in termsof accounting for observed experimental behavior is to examine the residualsin order to confirm or deny these assumptions. After examining the residuals,a decision should be possible as to whether the assumptions are consistentwith the analysis. If examination of the residuals indicates that the assumptionsappear to be satisfied, they provide no justification for rejecting the currentmodel, G as an accurate description of the observed experimentalbehavior. This is not the same as saying that examination of the residualsjustifies accepting the current model as necessarily being correct. The best onecan say in this case is that the model is consistent with observation. If, onthe other hand, the residuals are clearly inconsistent with the experimentaluncertainties of the experimental data, then the current model may correctlybe rejected.

Current analytical models and numerical methods are rapidly moving ina direction of increasing complexity. As a result, very sophisticated means areoften employed to ultimately obtain estimated parameter values and residuals.An investigator who is involved in analyzing complex biophysical phenomenais obviously in a highly quantitative environment. However, when consideringthe properties of residuals the investigator is required to operate in a largelyqualitative or, at best, semiquantitative domain. This arises due to the natureof residuals and the means by which information may be extracted from them.Plots of residuals versus a number of relevant parameters can be inspectedvisually to recognize significant behavior. Cumulative frequency plots, plotstesting for serial correlation, and plots making possible detection of outliers(bad points) all require such visualization as a basis for arriving at conclusions.


Some quantitative tests also exist permitting examination of the number ofruns (the number of consecutive positive and negative residuals) or detectionof influential observations (those which strongly influence the value of oneor more model-dependent parameter values). There are also the familiarchi-square tests (for testing frequency distributions and for determining levelsof significance), the Durbin–Watson test (a test for a particular type of serialcorrelation), and the Kolmogorov–Smirnov test (a goodness-of-fit test that isan alternative to the chi-square test for frequency distributions). These arejust a few of the means available for analyzing residuals, some qualitativeand others quantitative. A number of these methods should be applied inanalyzing residuals in order to thoroughly examine the displayed behavior.

4.7.1. Plots

Visualization of residuals plotted as scatter diagrams against variousexperimental parameters addresses two important points: whether one ormore of the assumptions of least-squares parameter estimation are violatedand whether correlations or trends are suggested in the behavior of theexperimental observations which have not been accounted for by the fittingfunction used to analyze the data. Plots of the residuals, the observed minusthe calculated values of the dependent variable, are commonly generatedversus (a) Ycalc, the calculated values of the dependent variable, (b)Xi , the


values of the independent variable(s), some potentially signifi-cant functional relationship(s) of one or more of the independent variable(s),and (d)X', one or more new independent variable(s) that potentially may becorrelated with the observed phenomenon but were not explicitly consideredin the current analytical model (Ref. 23, pp. 316–319).

Visual inspection of scatter diagrams of residuals versus and[cases (a) and (b) above] permits immediate recognition of obviously non-Gaussian residual distributions. The pseudo-random distributions presentedin Table 4.7 represent examples of approximately Gaussian and clearly non-Gaussian distributed residuals as a function of an arbitrary abscissa (X axis).Figure 4.7 is a graphical representation of the scatter diagrams for these same


two distributions. The point distribution in the scatter diagram of the upperpanel of Figure 4.7 shows no obvious non-Gaussian behavior, whereas thelower panel of Figure 4.7 clearly exhibits an upward trend in the residualvalues with increasing abscissa (X axis). If the variability in the residuals isnot constant, the experimenter is readily made aware of the presence of somepotential systematic behavior in the observed data. Some inadequacy of theanalytical model is implied if the variance of the residuals (proportional to thewidth of the scatter band) remains approximately constant but the meanresiduals appear to show some trend with or This phenomenon isshown in the lower panel of Figure 4.7. Differences in the variance of residualsas a function of or on the other hand, imply that some potentialsystematic behavior is arising from the measuring process itself rather thanfrom an inappropriate model used in analysis. Of course, combinations ofboth of these effects may be present. This very simple, visual tool may there-fore yield substantial insight into difficulties with the experimental procedureand/or the fitting function.

Plots of the residuals versus [case (c) above] offer a test of thehypothesis that the functional relationship(s), of one or more independentvariables, represents implied, or necessary, functional behavior thatshould have been, but was not, included in the analytical model currentlyemployed in estimating the observed behavior. For example, suppose ourmodel is given by

where the two independent variables, and are assumed to be relatedlinearly to the observed data, by the variable parameters and Theresiduals may be plotted against the product to test for any apparentcorrelation of the data with this particular functional relationship (theproduct) of the two independent variables. If, for example, a linear correlationof the residuals with the product is apparent (as illustrated by the lowerpanel of Figure 4.7), an additional term should be included in the currentfitting function, thus generating the modified function:

Although this example is a particularly elementary one, it serves todemonstrate the potential usefulness of checking for correlation of theresiduals with some functional relationship(s) of the independent variable(s).The mathematical complexity of analytical models will vary depending uponthe phenomenon, or combinations of phenomena, occurring and implicit in theexperimentally observed parameters with respect to the independent variable(s).A thorough appreciation of the experimental system and phenomena beingstudied may, however, suggest relationships worth considering in an exami-


nation of residuals based on theoretical and/or intuitive reasons and mayconceivably involve rather complicated mathematical formulations. Thereader is cautioned that including relationships which are based solely onobserved correlations but are without theoretical and/or intuitive reasons willlikely produce erroneous and/or irrelevant results with little physical meaning.Therefore, the use of these plots represents a rather convenient, qualitativeway to test for some additional relevant effects not accounted for by a currentfitting function.

A concept closely related to the one just presented involves examining ascatter diagram of the residuals plotted against some new independentvariable(s), X' [case (d) above]. The parameter X' is not explicitly consideredin the current analytical model but may play a role in producing the behaviorobserved in the experiment. Any correlation apparent in a plot of residualsversus X' suggests that X' should not be excluded from consideration.

A brief comment should be made about quantitative interpretations thatmay be implied by the above scatter diagrams. Quantitative informationextracted from plots of residuals versus or X', for example, must beviewed with caution. Such an examination will almost certainly accuratelysuggest a valid qualitative relationship overlooked by the current analyticalmodel. However, the best quantitative estimate of the influence of anyparticular independent variable should be considered simultaneously with allother independent variables in one mathematical formulation of the analyticalmodel. Returning to the example used previously, suppose Eq. (4.54) is usedto analyze a set of experimental data and the residuals suggest the need foran additional term such as that introduced in the modified function ofEq. (4.55). If the third term, is completely uncorrelated with eitherof the first two terms, and then estimating by fittingto a plot of the residuals versus will yield an accurate estimateof while and remain unchanged. If, however, some correlation existsamong and and/or (which in practice will almost always be the caseto at least some extent), this process will not yield the best least-squaresestimate of all three parameters. Instead, the modified fitting function must beused to reanalyze the original data set while simultaneously estimating allthree variable parameters, and This process will permit effects ofcorrelation among the three fitting parameters to be accounted for duringthe least-squares estimation procedure. Only this latter approach will permitvalid estimation of the three maximum-likelihood values for andconsistent with the expanded fitting function.

4.7.2. Distributions

Another graphical method useful for examining residuals is the cumulativefrequency plot.(24) This technique permits investigation of the probability


distribution of the residuals. The residuals must be ordered and numberedsuch that:

with

The quantity is an estimate of the probability of encountering a residualwith an absolute value less than or equal to Plotting versusapproximates the cumulative distribution function of the residuals. Such aplot, when generated on normal probability paper, will produce a straight line


if the residuals follow a Gaussian distribution. Converting the cumulativeprobability, to its associated Z value (standard normal deviate) makesgenerating the cumulative frequency plot more convenient in that the ordinate(Y axis) is linear in Z-value space (for Gaussian distributions). Z values andtheir associated probabilities may be found tabulated in most statistics books.Table 4.8 and Figure 4.8 present the cumulative frequencies, associated Zvalues, and ordered residuals for the residual distributions of Table 4.7 andFigure 4.7. The left panel of Figure 4.8 (corresponding to the approximatelyGaussian distributed residuals of Table 4.7) shows all of the points residingvery near the theoretical cumulative distribution function line (for Gaussiandistributed residuals exhibiting a standard deviation of 0.92). This behavioris expected if the residuals are truly Gaussian distributed. The right panelof Figure 4.8 presents the cumulative frequency distribution for the non-Gaussian distributed residuals of Table 4.7. The points for this case are not asclosely superimposable onto the theoretical line (again for a standard devia-tion of 0.92) as was observed in the left panel. However, there would be littlejustification for claiming that the distribution represented in the right panelof Figure 4.8 is not Gaussian distributed on the basis of this test alone. Thepresence of a few points distant from what is otherwise a good representationof a straight line by the majority of points suggests the occurrence of outliers,or bad points, a topic to be discussed later. Inability to generate a straight linein a cumulative frequency plot indicates that the residuals are not Gaussian


distributed and thus represents a violation of one of the assumptions of theleast-squares method.

Another more quantitative method of assessing whether or not residualsfollow a Gaussian distribution involves evaluating the statistical probabilitythat the observed distribution is different from a Gaussian distribution(Ref. 23, pp. 391–393, and Ref. 25, pp. 340–345). The range of residual valuesis divided into mutually exclusive, continuous intervals, and the frequency ofoccurrence of residuals possessing values within each of these intervals isnoted. The variance of the expected population of residuals is approximatedby the computed variance of the residuals, and the mean of the residuals isassumed to be zero. This expected behavior of the distribution of residualsthen permits calculation of the expected frequencies in each of the above-mentioned intervals if the distribution of residuals is truly Gaussian.

With this information in hand, the expected frequency of occurrence foreach interval may now be calculated. This involves determining the relativefrequency for each interval and multiplying it by the total number of residuals.This is most easily achieved by working with standard normal deviates, Z,and finding appropriate values from published tables of areas under standardnormal curves. Here, Z is defined as

where is the mean value (in this case, zero), s is the standard deviation ofthe residuals (the square root of the sample variance), and X is one end of theinterval being considered. The absolute value is used to eliminate theoccurrence of negative Z values which would arise when For example,suppose that the relative frequency of occurrence is desired within the interval

that is, for residuals with values greater than and less thanValues for and are calculated as per Eq. (4.59) by substituting orfor X. The relative frequency of occurrence over the interval will thenbe defined by the expression where freq correspondsto the frequency found from a table of areas under standard normal curves at

The absolute value is again used to eliminate the occurrence ofnegative values. This process is carried out for all intervals, and the relativeexpected frequencies are multiplied by the total number of residuals to yieldthe absolute expected frequencies.

The next step involves quantitatively characterizing the discrepanciesbetween observed and expected frequencies by calculating a chi-square statisticas


where and are the observed and expected frequencies in interval i,respectively. If the residuals are actually Gaussian distributed, then this statisticwill be distributed approximately as given in chi-square tables (located in moststatistics texbooks) with (n – c) degrees of freedom, where n is the total numberof intervals and c is the number of constraints on the system. One constraintarises from restricting and additional constraints occur for eachparameter estimated. In this case, only one additional constraint occurs fromestimation of the variance of the residuals, thus defining the value of s inEq. (4.59). The value of this chi-square statistic is then compared withtabulated values of degrees of freedom] to quantify theprobability that the residuals do not follow a Gaussian distribution at thelevel of confidence specified in the table used.

Table 4.9 shows the procedure and calculations necessary to apply thistest to the residual distributions of Table 4.7 and Figure 4.7. The range ofresidual values was divided into nine equally spaced intervals, each of whichwas (s/2) units wide. Here, s corresponds to the calculated standard deviationof the residuals, which was approximately 0.92 for each of the distributions ofTable 4.7. The results of this analysis indicate that the approximatelyGaussian distribution has a probability of between 1% and 2.5% ofbeing non-Gaussian whereas the non-Gaussian distribution has a 75-90%probability of being non-Gaussian. These probabilities are derived from thecalculated values with seven degrees of freedom.


A point must be made about small expected frequencies, that is, thoseless than one. This approximation of is not strictly valid when the expectedfrequencies are small. Generally, adjacent intervals should be combined toachieve a desired minimum frequency of not less than one.

4.7.3. Trends

Another quality of residuals to be concerned with is the presence oftrends in the residuals with respect to dependent or independent variables.Theoretically, it is desirable to have no correlation of residuals with respect toany of these variables. The presence of trends suggests that some systematicbehavior is influencing the experimental observations and that this systematicbehavior is not being accounted for either due to (1) inappropriate experi-mental design or (2) deficiencies in the fitting function.

An easy and convenient test for randomness of residuals with respect toany variable is accomplished by counting the number of runs.(24) A run heremeans a consecutive sequence of residuals of equal sign. If is the totalnumber of positive residuals and is the total number of negative residuals,the number of runs expected, if the sequence of residuals is random is givenby

The variance of the number of runs, is given by

A parameter, Z, may then be defined as

where is the actual number of runs observed. The value of Z will be dis-tributed approximately as a standard normal deviate if both and aregreater than 10. The value of 0.5 is a continuity correction which partiallycompensates for the approximation of a discrete distribution by a continuousdistribution. If too many runs, instead of too few, are being tested for, a valueof –0.5 is used. This approach therefore permits one to estimate the statisticalprobability that the number of runs actually encountered is different from thatexpected if the residuals had occurred randomly with respect to the variableunder consideration. In other words, the greater the value of the greater


the probability that some correlation exists in the residuals relative to thevariable being considered.

Table 4.10 demonstrates an application of the runs test to the residualdistributions presented in Table 4.7 and Figure 4.7. The derived Z values(between 0.40 and 1.24) are not sufficiently large to necessarily conclude thateither of these two distributions is not Gaussian distributed, based on this testalone. On the other hand, values of Z greater than approximately 2.5 willcorrespond to probabilities of less than about 1 % that the distributions areactually random. It is important to note that a Z value greater than 2.5implies that one may justifiably reject the hypothesis that the residuals areGaussian distributed. However, if a Z value less than or equal to 2.5 isobtained with the runs test, one cannot necessarily conclude Gaussiandistribution of residuals without further confirmation.

Results of an analysis of the number of runs may indicate that too fewruns are present (implying possible positive serial correlation) or that toomany runs are present (implying possible negative serial correlation). Theterm “serial correlation” implies a repeated pattern among residuals. Serialcorrelation is most frequently applied to time series experiments. Variablesother than time, however, may conceivably offer practical significance withregard to the concept of serial correlation under appropriate conditions.

Correlation among residuals may be determined 1,2,..., n units apart,thus characterizing the lag1, lag2,..., lagn serial correlations, respectively.(26)

An examination of the lagn serial correlation of residuals is accomplished bygenerating a plot of the value of each residual versus the value of the residualoccurring n units before the one currently being considered. Of course, thefirst n residuals therefore cannot be plotted. Positive lagn serial correlation will

228 Martin Straume et al..

be apparent in a lagn serial plot as a band of points generally exhibiting atrend from the lower left (negative, negative) quadrant to the upper right(positive, positive) quadrant. Negative lagn serial correlation, on the otherhand, will be visualized as a band of points distributed from the upper left(negative, positive) quadrant to the lower right (positive, negative) quadrant.Residuals displaying no lagn serial correlation will produce a lagn serial plotwith no apparent trend. The residual distributions of Table 4.7 and Figure 4.7are presented as lag1, lag2, lag3, and lag4 serial plots in Figure 4.9. Only thelag1 serial plot for the non-Gaussian distributed residuals (panel A of thelower group of serial plots) shows any indication of serial correlation(positive, in this case), as may be expected given the pattern displayed bythese residuals in the lower scatter diagram of Figure 4.7.

4.7.4. Outliers

Occasionally, a residual (or residuals) may be encountered that possessesan unusually large (absolute) value. A data point giving rise to such aresidual, termed an outlier, suggests that some error occurred during acquisi-tion of the experimental data to generate the erroneous observation. Thepresence of such a point in the data set may substantially affect the values ofderived parameters characteristic of the fitting equation used in analysis. Thispossibility makes it imperative to identify such a point or points and, perhaps,to reanalyze the data set without considering the outliers during the estima-tion process.


The presence of potential outliers may be conveniently detected by visualinspection of residual plots.(24) Scatter diagrams will frequently permit readyidentification of residuals which possess particularly large absolute values. Acumulative frequency plot of residuals graphed on normal probability paperwill suggest the occurrence of outliers by the obvious presence of points whichreside distant from the majority of linearly arranged points.

Reanalyzing the data set without outliers may result in significantlyaltered fitting parameter values relative to those derived with the bad pointspresent. A more accurate characterization of the observed experimentalphenomenon is likely to be obtained from such a reanalysis. It is sometimesdifficult, however, to have confidence that a particular point should berejected as an outlier rather than being retained as a point which just has arelatively low probability of being valid. A commonly employed method formaking this decision involves rejecting any points whose residuals haveabsolute values greater than 2.5 or 3 standard deviations from zero. However,even with residuals that follow a perfect Gaussian distribution, this criterioncan sometimes incorrectly suggest the presence of outliers. For example, theprobability that any particular data point will yield a residual 3 standarddeviations from zero is about 0.0025. The probability of encountering at leastone data point with a residual 3 standard deviations from zero in a data setconsisting of n points is therefore approximately n times 0.0025. So, as nincreases, the probability of encountering a point falsely indicated to be anoutlier increases proportionately. It is expected that this criterion will result inone point being incorrectly judged to be an outlier for approximately every400 (or 1/0.0025) data points.

4.7.5. Influential Observations

As discussed in the previous section, outliers may cause derivation ofmodel parameter values that are inappropriate due to bias introduced by thepresence of a bad point in the data set being analyzed. Tests for such aninfluential observation may be applied in order to determine the sensitivity ofderived parameter values to outliers.(26) It is possible that an outlier may exertlittle influence on derived model parameter values if many data points aredistributed closely around the outlier in independent variable space. A pointrelatively isolated from others in independent variable space, on the otherhand, may not appear as an outlier by the conventional methods of identi-fying such points but may nonetheless be an influential observation withregard to derived parameter values. This is a very important point! Theformer case will lead to conclusions which are not substantially affected by abad point. The latter situation, however, will probably not identify the point


in question as an outlier but could very significantly influence the values ofderived parameters.

An observational oversight giving rise to a point which strongly influencesderived parameter values but is not detectable as an outlier presents aunique challenge for identification. An approach in which suspected influentialobservations are deleted from the data and reanalysis is performed on thismodified data set may permit identification of potential influential pointswhich are inconsistent with the rest of the experimental observations, given aparticular analytical model. It is possible, however, that the points deletedalmost exclusively define certain of the parameters of the analytical model.Their exclusion may thus lead to inability to define these particularparameters. Such a situation may seem inconclusive but actually should makeapparent to the experimenter that greater experimental determination isnecessary in this region of independent variable space, that is, that moreexperimental observations are needed around these values of the independentvariable(s).

A number of quantitative protocols have been presented in the literatureto deal with situations involving influential points.(26) The details of these willnot be dealt with here. A common feature of these methods, as applied toleast-squares analysis, involves evaluating the fitting parameters from datasets in which individual data points which are suspected influential observa-tions are omitted. The resulting variations of these fitting parameters thenindicate the influence that a particular experimental observation has indetermining the fitting parameter values. When data points are identified asbeing particularly influential, then more data should be obtained in theneighborhood of those particular points to minimize the chances of derivingsignificantly biased parameter values.

4.7.6. Common Quantitative Tests

4.7.6.1. The Chi-Square Test

Residuals that appear to follow a Gaussian distribution about zero withno detectable trends or systematic behavior with respect to any relevantvariables, dependent or independent, may be used to test the adequacy ofan analytical model for accurately estimating observed experimental data.A commonly employed test for this purpose involves calculation of the chi-square statistic.(16) It can be shown that increases very nearly linearly withthe number of degrees of freedom when the analytical model is indeed correct.Therefore, the reduced value, the value divided by the number of degreesof freedom, will be approximately equal to one under these conditions.A potential problem that makes implementing this test very difficult is that


estimates of the uncertainties associated with the values of the dependentvariable must be known with great confidence. These uncertainties in effectcontribute to normalization of the squared residuals such that the valuemay have significance.

When reliable estimates of the uncertainty in the dependent variable areavailable, a chi-square statistic is calculated as

where and are the observed and model-dependent calculatedvalues of the ith dependent variable, respectively, is the standard errorassociated with and n is the number of data points, or dependentvariable values, contained in the data set being analyzed. The value of isdivided by the number of degrees of freedom to obtain the reduced form ofthis parameter as:

where n is the number of data points, and c is the number of parametersestimated in analysis. The value of should not be much greater than oneif the model used accurately estimates the experimental data. Statisticalprobabilities that the analytical model employed is incapable of adequatelycharacterizing the data may be obtained by comparing the derived values of

with statistical tables of reduced values for (n – c – 1) degrees offreedom.

An example of this analytical tool is presented in Table 4.11 as applied tothe approximately Gaussian distributed residuals of Table 4.7 and Figure 4.7.The experimental uncertainties, presented in Table 4.11 produce thereported values and the associated probabilities that the model used in

analysis provides a statistically valid characterization of the experimentaldata. For purposes of this example, it was assumed that four parameter valueswere derived from a least-squares analysis of 25 experimental observations, thusproducing 20 degrees of freedom (NDF = n – c – 1, where NDF = 20 becausen = 25 and c = 4). There would be little justification for rejecting this(hypothetical) analytical model as long as was actually estimated to begreater than or equal to approximately 0.74 (at a 1 % level of confidence).

4.7.6.2. The Durbin–Watson Test

The Durbin–Watson test is a test commonly employed for determiningwhether a certain type of serial correlation exists in residuals.(26) The nullhypothesis is that no correlation exists in the residuals. This implies that the


residuals follow a Gaussian distribution with some constant variance, Thealternative hypothesis, in this case, is formulated as

where and are the ith and (i – l ) th residuals, respectively, and andare parameters defining the form of serial correlation hypothesized to be

present in the residuals. The are assumed to follow a Gaussian distributionwith some constant variance, and are independent of all and when

It is also assumed that the have a constant mean and variance and areindependent of when Therefore, the follow a Gaussian distributionwith the variance given by When no serial correlation is apparent,

and the situation reduces to that stated in the null hypothesis.To decide whether or not any serial correlation is present in the residuals,

a parameter, d, is calculated as

Two critical values of and specifying a range, are involved indeciding whether or not to reject the null hypothesis. Tables of these critical


values, at various levels of confidence, are used to decide whetheror In case (a), testing a value of implies

significance, at the level specified in the table used, and is accepted. Ifthen d is not significant and is rejected. If the test is

considered inconclusive. In case (b), testing the value of d above isreplaced by (4 – d), and the same criteria as in case (a) are applied. In case(c), testing a value of or of is significant and attwice the significance level specified in the table (because it is a two-sidedtest). If and is said to be equal to zero, again at twicethe significance level of the table used. Otherwise, the test is inconclusive.

Avoiding the inconclusive cases may be approximately accomplished byexclusively using the critical value in place of for all of the above-mentioned comparisons. This simplified test should, however, be used only ifthe original test is in the inconclusive region. The significance level associatedwith the simplified test is then somewhat higher than that specified in thetable used.

This test, when applied to the residual distributions of Table 4.7 andFigure 4.7, yields a value of d = 2.27 for the approximately Gaussian dis-tributed residuals and a value of d = 0.93 for the non-Gaussian distributedresiduals. Tables of and values reveal that, for 25 observations andone independent variable, (at 5% confidence), (at 2.5%confidence), and (at 1% confidence) and that (at 5% con-fidence), (at 2.5% confidence), and (at 1% confidence).The conclusions from this exercise are that the approximately Gaussiandistributed residuals show no apparent serial correlation, that is, andthat the non-Gaussian distributed residuals display serial correlation with

4.7.6.3. The Kolmogorov-Smirnov Test

Testing whether residuals follow a Gaussian distribution is critical to anevaluation of the adequacy of an analytical model for accurately estimatingobserved experimental behavior. One way in which this may be accomplishedis by applying the chi-square test of goodness of fit to distributions asdescribed earlier. The chi-square procedure, however, requires grouping theresiduals into discrete intervals, thus discarding some details of the informa-tion contained in the original, more continuous distribution of residuals. Thechi-square procedure also requires a minimum population in each interval inorder to permit valid characterization of all regions of the distribution. Togenerate more continuous derived distributions, more data points are neededso that a greater number of more closely spaced intervals may be used inanalysis.

The Kolmogorov–Smirnov test for goodness of fit to distributions is an


alternative to the more familiar chi-square procedure and does not suffer fromthe aforementioned limitations. This test permits a statistically quantifiablecomparison to be made between a sample cumulative distribution andsome theoretical cumulative distribution function. A cumulative distributionfunction is that which gives the probability that a value less than or equal tosome specified value will occur. The statistic calculated in the Kolmogorov–Smirnov test, D, is the maximum difference between the observed andtheoretical cumulative distribution functions over all values of the variableunder consideration, X (Ref. 25, p. 387):

I

where and are the sample and theoretical cumulative distributionfunctions, respectively, and means “the supremum (largest) over all X.”Tables of this statistic, as a function of significance level and size of sample(Ref. 25, p. 483), are then consulted in order to decide whether and

are statistically different at some specified confidence level.In practice, the parameter D will be obtained by considering discrete

points of and that is, at values of i = 1, 2, ..., n. Becauseis continuous with X and is discrete over finite intervals of X, both


endpoints of in each interval must be considered relative toso as not to overlook any of the relevant distances between the sample andtheoretical functions. Therefore (Ref. 25, p. 391),

The distances between and each endpoint of each discrete portion ofare thus considered by this expression.

Figure 4.10 demonstrates the process implied by Eq. (4.69). The five pointspresented in this example produce a mean value of zero and a calculatedstandard deviation of one. A cumulative frequency plot is generated withcumulative frequency (expressed as probability, i.e., ranging from zero to one)as the ordinate (Y axis) and the value of the ordered points (in increasingmagnitude) as the abscissa (X axis). The continuous line corresponds tothe theoretical distribution function for a Gaussian distribution with astandard deviation of one. The Kolmogorov–Smirnov D parameter for thisexample has a value of 0.241, the maximum vertical distance between theobserved and theoretical cumulative probability functions. Examination of astatistical table for this parameter indicates that there is no justification forrejecting the hypothesis that these five points are Gaussian distributed with astandard deviation of one at any reasonable level of confidence (Ref. 25,p. 483).

4.8. Implementation Notes

Before proceeding to code a computer program, an investigator must firstaddress a simple question which has very subtle implications. The question is“What computer language should be used for this type of analysis?” Ouranswer is almost always FORTRAN-77. There are six primary considerationswhich need to be addressed before the computer language is chosen. They are:

1. The computer language must be universal.2. Only the “language standard” should be used†; i.e., no non-standard

extensions.3. The investigator should consider the language features required for

the particular problem.4. The computer program should be efficient.5. Options inherent within certain languages can make the particular

problem easier.

† A “language standard” is the internationally recognized definition of the computer language.For example, the American National Standards Institute defined FORTRAN-77 as their X3.9-1978standard.(28)


6. There is no need to reinvent the wheel by rewriting software whichhas already been written and is available at a lower cost than that ofpaying a programmer to do it again.

The optimal choice of a language is one which meets all of the investigator’sabsolute requirements and is available on almost every computer system.For example, ten years ago when at the National Institutes of Health, oneof us (MLJ) asked their computer specialists which language should be used.He was told that if he wanted to use their Digital Equipment Corporationcomputer (DEC System 10), he should use the SAIL language, and if he wantedto use their International Business Machines computer (IBM 360), he shoulduse PL/1. This investigator instead chose FORTRAN. Later, after movingto the University of Virginia, he discovered that SAIL programs can onlyrun on DEC System 10/20 computers. Furthermore, the University ofVirginia’s Control Data Corporation computer (CDC 855) did not have aPL/1 compiler. However, all three computers had FORTRAN compilers, and,consequently, there was no need to translate any programs. Translationscan be very time-consuming and expensive, especially when the computerprograms are long and complex. Almost every computer currently in use hasa FORTRAN-77 compiler available. Most computers now also have Pascal, C,and BASIC available. Other languages like ALGOL, BLISS, FOCAL, and Modula-2are not commonly available. Therefore, in this discussion of relative merits wewill consider only the most popular languages: FORTRAN-77, Pascal, C, andBASIC.

The reader is cautioned not to use the convenient language extensionswhich most manufacturers have included in their compilers. For example,some versions of BASIC allow the direct manipulation of matrices while othersand the language standard do not. Consequently, if a BASIC program whichuses these matrix features on one computer is transfered to another computerwhose compiler does not have these extensions, the program will not run.Another example is the number of attractive services provided by the UNIXoperating system which are exclusive to particular versions of UNIX. If acomputer program is developed which relies on operating system features onone computer, it will be difficult or impossible to transfer it to a differentcomputer system! To make matters more confusing, a number of computerlanguages, such as Pascal and BASIC, have such poorly defined “standards”that they vary substantially among computers from different manufacturers.A computer program should use only those features of a language whichare the same on all computers which implement that language, that is, theminimal language standard.

There are a number of computer language features that are required forthe development of least-squares analysis programs which are not commonamong FORTRAN-77, Pascal, C, and BASIC. BASIC has a number of unattractive


features for this application. It does not always permit the easy use of sub-routines (or subprocedures), and it does not always permit more than single-character variable names. A potential problem with Pascal relates to the“Pascal standard,”(27) which does not provide for double-precision floating-point variables. Matrix manipulations, such as dot products, must be performedwith greater precision than that offered by single-precision floating-pointcalculations, which typically store variables as 32-bit numbers. Furthermore,the “Pascal standard” also does not allow subprocedures to be compiledseparately from the main program. It is inefficient and time-consuming torecompile thousands of lines of Pascal code in order to make a simple changein a ten-line subprocedure. The Pascal enthusiast can argue that most Pascalcompilers are not restricted by the “Pascal standard.” However, bewarethat different compilers implement these features in different ways and thustranslation from one computer system to another may be difficult.

The efficiency of a computer program is important for some of theprocedures which we have presented, which require hundreds of evaluationsof complex functions. Inefficiencies can be grouped into two categories:(a) the inherent slowness of a language, for example, BASIC, which is usuallyinterpreted† rather than compiled, and (b) inherent inefficiencies of somecompilers. For example, we have two different Pascal compilers for ourDEC PDP-11/73 running the operating system. We used a primenumber generation program to compare our Pascal-2 compiler with ourNBS-Pascal compiler†† and found that the NBS-Pascal compiler requiredalmost five times as much computer time for the same calculation. However,the Pascal-2 compiler was substantially more expensive.

Of less importance, but still worth considering, are the optional featureswhich the languages do not have in common. None of these are required forthe least-squares application but are needed in some other applications. TheFORTRAN “labeled common blocks” are very useful and are not allowed by thePascal, C, and BASIC standards. The FORTRAN “equivalence statements” canalso be very useful. Recursion, the ability of a procedure, or subprocedure, to

† An “interpreted” computer language is one in which code is translated to machine instructionsline by line. Each translated line is executed and discarded before the next line of the computerprogram is compiled. This process is usually substantially slower than that followed in the caseof a “compiled” computer language, whereby the entire computer program is translated intomachine instructions, then executed in a separate subsequent step.

is a time-sharing operating system for DEC PDP-11 computers which is availablef r o m S and H Computer Systems, Inc., 1027 I7th Avenue South, Nashvill, Tennessee 37211.This operating system is essentially an extension of the DEC RT-11 operating system.

†† Pascal-2 is a commercial Pascal compiler which is available from Oregon Software, 2340 S.W.Canyon Road, Portland, Oregon 97201. NBS-Pascal is available from the Digital EquipmentComputer Users Society (DECUS), 219 Boston Post Road, Marlboro, Massachusetts 01752,for a nominal fee. The NBS-Pascal compiler is included in the Spring 198S RT-11 SpecialInterest Group Distribution Tape.


either directly or indirectly call itself, can also be useful. The andBASIC standards do not allow recursion, but some implementations of theselanguages do allow recursion. Another language feature which is attractive,but not essential, is the ability to write “structured” programs,and allow structured programming, while Pascal and C require struc-tured programming. It should ne noted that it is sometimes very convenientnot to be required to use structured programs, even though it is usually anexcellent way to program.

The last important point is to avoid writing routines which have alreadybeen programmed. Subroutine libraries exist to perform most of the linearalgebra and eigensystem operations required for least-squares parameterestimation. Examples are the andlibraries of subroutines. These are all written in and/or

Some computers will allow linking a routine written in onelanguage to a program written in another language if certain specific condi-tions are met, but this is a difficult process at best. A better way to approachthe encoding of an analysis procedure is to begin coding in the language ofchoice, that is, the standard, so that implementing preexistingroutines to handle complicated, but standard operations is straightforward.There is little justification for ever converting a functional computer programfrom one language to another unless the computer system being used does nothave a compiler for the first language. Again, we recommend that computerprograms be written in either the standard(28) or, sometimes,if the computer is running the operating system, the C languagestandard.(29)

4.9. In Summary

Computers are not oracles. The results of any computer analysis aredependent on the quality of the data, on the assumptions of the analysismethod, and on the expertise of the computer programmer who generated thesoftware. Consequently, computer users should always be aware that com-puter programs might give them wrong answers. Computer programs shouldalways be tested with various types of “test” data and the possibility thatanswers derived from the analysis of actual experimental data may have nophysical meaning should always be considered.

Two of the cited references are excellent entry-level numerical methods

and comprise a library of subroutines to perform least-squares functionminimization and parameter estimation problems. They are available from the InternationalMathematical and Statistical Library (IMSL) Distribution Service, P.O. Box 4605, Houston,Texas 77210-4605, for a nominal charge.


textbooks.(7,16) Explicit definitions of several computer languages are providedin the following sources: FORTRAN-77 in Ref. 28, C in Ref. 29, and Pascal inRef. 27. In addition, several of the references provide subroutine librariesprinted in the text.(7–9, 11,16) Some subroutine libraries can be obtained onmagnetic media for a nominal fee.(8,9,1l)

The nonlinear least-squares procedures presented in this chapter have asound statistical basis if certain assumptions are met. A number of authorshave questioned the validity of the assumptions and, as a consequence, ofleast-squares procedures(3) However, no other procedure has as good abasis in statistics as least-squares, which is a special case of the more generalmaximum-likelihood method. As long as the assumptions are reasonable,least-squares analysis is the best available method because it yields answerswhich have the highest probability of being correct.

References

1. J. R. Lakowicz, Principles of Fluorescence Spectroscopy, pp. 65–82, Plenum Press, New York(1983).

2. M. L. Johnson, The analysis of ligand binding data with experimental uncertainties in theindependent variables, Anal. Biochem. 148, 471–478 (1985).

3. I. Isenberg, Robust estimation in pulse fluorometry: A study of the method of moments andleast squares, Biophys. J. 43, 141 (1983).

4. N. Joshi, M. L. Johnson, I. Gryczynski, and J. Lakowicz, Radiation boundary conditions incollisional quenching of fluorescence: determination by frequency domain fluorometry, Chem.Phys. Lett. 13S, 200–207 (1987).

5. M. L. Johnson and S. G. Frasier, Nonlinear least-squares analysis, Methods Enzymol. 117,301–342 (1985).

6. J. A. Nelder and R. Mead, A simplex method for function minimization, Comput. J. 7,308–313 (1965).

7. G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methods for MathematicalComputations, Prentice-Hall, Englewood Cliffs, New Jersey (1977).

8. B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, andC. B. Moler, in: Matrix Eigensystem Routines—EISPACK Guide, Second Ed. (G. Goos andJ. Hartmanis, eds.), Springer-Verlag, New York (1976).

9. B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler, in: Matrix EigensystemRoutines—EISPACK Guide Extensions, (G. Goos and J. Hartmanis, eds.), p. 69, Springer-Verlag, New York (1977).

10. V. N. Faddeeva, Computational Methods of Linear Algebra, p. 81, Dover, New York (1959).11. J. J. Dongarra, C. B. Moler, J. R. Bunch, and G. W. Stewart, LINPACK Users' Guide, Society

for Industrial and Applied Mathematics, Philadelphia (1979).12. F. B. Hildebrand, Introduction to Numerical Analysis, p. 82, McGraw-Hill, New York (1956).13. M. S. Caceci and W. P. Cacheris, Fitting curves to data, Byte Magazine 9(5), 340–362

(1984).14. W. Spendley, G. R. Hext, and F. R. Himsworth, Sequential application of simplex designs in

optimization and evolutionary operation, Technometrics 4, 441 (1962).15. C. Hastings, J. T. Hayward, and J. P. Wong, Approximations for Digital Computers, p. 187,

Princeton University Press, Princeton, New Jersey (1955).

240 Martin Straume et at.

16. P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences, pp. 187–203,McGraw-Hill, New York (1969).

17. G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for Experimenters, p. 636, Wiley-Interscience, New York (1978).

18. M. Zelen, and N. C. Severo, Probability functions, in: Handbook of Mathematical Functionswith Formulas, Graphs, and Mathematical Tables, Ninth Printing (M. Abramowitz andI. A. Stegun, eds.), pp. 925–997, National Bureau of Standards Applied Mathematics Series#55, Washington, D. C. (1970).

19. G. E. P. Box, Fitting empirical data, Ann. N.Y. Acad, Sci. 86, 792–816 (1960).20. M. L. Johnson, H. R. Halvorson, and G. K.. Ackers, Oxygenation-linked subunit interactions

in human hemoglobin: Analysis of the linkage functions for constituent energy terms,Biochemistry 25, 5363–5371 (1976).

21. M. L. Johnson, J. J. Correia, D. A. Yphantis, and H. R. Halvorson, Analysis of data from theanalytical ultracentrifuge by nonlinear least-squares techniques, Biophys. J. 36, 575-588(1981).

22. M. L. Johnson, Evaluation and propagation of confidence intervals in nonlinear, asymme-trical variance spaces: Analysis of ligand binding data, Biophys. J. 44, 101–106 (1983).

23. P. Armitage, Statistical Methods in Medical Research, Fourth Printing, pp. 316–319,Blackwell Scientific Publications, Oxford (1977).

24. Y. Bard, Nonlinear Parameter Estimation, pp. 201–202, Academic Press, New York (1974).25. W. W. Daniel, Biostatistics: A Foundation for Analysis in the Health Science, Second Ed.,

John Wiley & Sons, New York (1978).26. N. R. Draper and H. Smith, Applied Regression Analysis, Second Ed., pp. 153–174,

John Wiley & Sons, New York (1981).27. K. Jensen and N. Wirth, Pascal User Manual and Report, Second Ed., Springer-Verlag,

New York (1978).28. L. P. Meissner and E. I. Organick, FORTRAN 77: Featuring Structured Programming,

pp. 427–482, Addison-Wesley, Reading, Massachusetts (1980).29. B. W. Kernighan and D. M. Ritchie, The C Programming Language, Prentice-Hall,

Englewood Cliffs, New Jersey (1978).

5

The Global Analysisof Fluorescence Intensity andAnisotropy Decay Data:Second-Generation Theoryand Programs

Joseph M. Beechem, Enrico Gratton, Marcel Ameloot,Jay R. Knutson, and Ludwig Brand

5.1. Introduction

Time-resolved fluorescence spectroscopy has proven to be a powerfulphysical technique for the studies of fast reactions and dynamics on the sub-picosecond to microsecond time scales. Examination of the decay kinetics ofspecific excited state photophysical processes (e.g., resonance energy transfer,solvent relaxation, excimer/exciplex formation, dynamic quenching, protontransfer, rotational diffusion, etc.) can provide important information concer-ning reactions approaching equilibrium. When the fluorescent molecule is anintegral part of a larger biological system (e.g., proteins, membranes, macro-molecular structures), the decay kinetics can reveal information concerningthe structural dynamics of these complicated systems.

There have been significant advances in the instrumentation used to

Joseph M. Beechem and Enrico Gratton • Department of Physics, University of IllinoisUrbana–Champaign, Urbana, Illinois 61801. Marcel Ameloot • Limburgs UniversitairCentrum, Universitaire Campus, B-3610 Diepenbeek, Belgium. Jay R. Knutson • NationalHeart, Lung, and Blood Institute, National Institutes of Health, Bethesda, Maryland 20892.Ludwig Brand • Department of Biology, The Johns Hopkins University, Baltimore,Maryland 21218. Present address for J.M.B.: Department of Molecular Physiology andBiophysics, Vanderbilt University, Nashville, Tennessee 37232. Information concerning acommercial version of the program described in this chapter can be obtained by writing to:Globals Unlimited, Laboratory for Fluorescence Dynamics, University of Illinois, Urbana,Illinois 61801. No endorsement by the U.S. Government is implied.Topics in Fluorescence Spectroscopy, Volume 2: Principles, edited by Joseph R. Lakowicz. PlenumPress, New York, 1991.

241

242 Joseph M. Beechem et al.

obtain time-resolved fluorescence data during the past ten years. The use ofhigh-repetition-rate picosecond dye lasers and fast detectors (such as micro-channel plate photomultipliers) has had a major impact on both time- andfrequency-domain measurements. Data acquisition time has been greatlydecreased, especially for single-photon counting experiments, which used to belimited by the repetition rate of the excitation light source. Accurate single-photon counting decay data may now be collected in under a second(1–3) andwill probably approach the millisecond time scales very soon. These fastcollection rates are beginning to make it possible to perform “double-kinetic”experiments: multiple picosecond-microsecond fluorescence experiments per-formed on the millesecond time scale. Reports of similar efforts to decreasedata acquisition time in the frequency domain have also appeared.(4) Parallel(“multifrequency”) data collection(5) (i.e., where many frequencies are acquiredsimultaneously) should soon result in a drastic reduction in data acquisitiontime in the frequency domain. Alternative stroboscopic data acquisitionschemes in the time domain are also being developed.(6)

Many numerical techniques, such as nonlinear least squares,(7–9) methodof moments,(10,11) Laplace transforms,(12,13) and modulating functions,(14)

have been used to analyze fluorescence decay data. It has been found thatfor the accurate recovery of closely spaced exponentials (or distributionsof exponentials), the analysis of individual fluorescence decay experimentsis usually insufficient. For the accurate recovery of complex fluorescencedecay phenomena, it is advantageous to combine more than one fluorescencedecay experiment into a single analysis.(15–20) A number of nonlinear last-squares(19–26) and Laplace transform(27) software routines have been developedwith multiexperiment capabilities. The simultaneous analysis of multiplefluorescence decay experiments is frequently referred to as “global” analysisand has proven useful for both time- and frequency-domain data.

Global analysis procedures are of significant advantage when someunknown parameter(s) of interest is linked between two or more fluorescencedecay experiments obtained under different conditions. A “condition” in thiscontext refers to temperature, pressure, excitation/emission wavelengths, orany other independent variable which can be experimentally manipulated. Inmany situations, the parameters are not linked directly between experiments,but rather through a mathematical function. The overall strategy behind atypical global analysis can best be illustrated with a few examples.

5.1.1. Multiexcitation/Multitemperature Studies of Anisotropic Rotation

For nonspherical molecules, the decay of the emission anisotropy isdescribed by a sum of exponentials† with rotational correlation times inde-

† For cases of depolarization due to rotational diffusion.

Global Analysis of Fluorescence Intensity and Anisotropy Decay Data 243

pendent of the excitation wavelength and preexponential terms dependent onthe relative direction of the absorption/emission oscillators with the principaldiffusion axes of the molecule.(28) In this case, global analysis of experimentsdone at several excitation wavelengths provides a useful overdeterminationof the rotational correlation times. The multiple excitation experimentsare analyzed for an internally consistent set of correlation times, whosepreexponential factors are allowed to vary as a function of the excitationwavelength (example of a simple global “linking” of the rotational correlationtimes over the multiple excitation wavelengths).

Experiments can also be performed at multiple temperatures (T) and/orviscosities The rotational correlation times cannot be directly linkedbetween these experiments, because they change with T and To link rota-tional rates over multiple temperatures and viscosities requires a mathe-matical function which predicts how these quantities will change. Many simplemolecules in solution show classic Stokes–Einstein type of behavior overparticular ranges. In these cases, one can link rotational parameters overtemperature using this relationship. This allows one to explore those regionswhere the lifetime may be much faster/slower than the rotational relaxationrate.

5.1.2. Multiexcitation/Emission Wavelength Studies of Total Intensity Data

Global analysis has also proven useful in the evaluation of a mixture oftwo or more noninteracting fluorescent species, each with a unique decayfunction and excitation/emission spectrum. Global analysis of a data setobtained as a function of emission wavelength, with linkage of decay con-stants, allows more closely spaced lifetimes to be recovered than single-curveanalysis alone. In addition, the preexponential terms may be used to recoverthe individual spectra associated with the particular decay time (decay-associated spectra, DAS).(29,30,17) The occurrence of discrete DAS forindividual tryptophan/tyrosine residues in many different proteins(31) isproving to be useful for monitoring structural changes occurring in differentregions within a single protein molecule in solution. These DAS have alsoproven to be useful in resolving macromolecular interactions and sitespecificquenching.(l,32) The assignment of specific spectra to different fluorescencelifetimes may be further aided by utilizing additional independent variables,such as quencher concentration. If DAS can be related to a specific tryp-tophan residue (or conformation), then one might expect that differentdynamic quenching constants may be "linked" to each spectrum (quenching-decay associated spectra, QDAS). Indeed, spectra associated with distinctquenching constants have been recovered and can be obtained by eitherglobal or nonglobal methods; usually, both are combined.(1,33–35) Just as for


lifetimes, closely spaced quenching decay constants are more easily resolvedby global multiwavelength analysis (either steady state or time-resolved).

5.1.3. Double-Kinetic Studies

Another example where global analysis has been helpful is in studies ofthe kinetics of protein unfolding/refolding on the second to minute timescales.(1–3) Complete fluorescence decay curves were collected every fewseconds, and the effects of unfolding on the lifetime component(s) wereobserved. The individual decay curves are too noisy to support complexanalysis. However, when the entire data surface (collected during the proteinconformational change) is globally analyzed using various linkage schemes,time-dependent changes in lifetimes and amplitudes can be recovered. In thismanner, a direct correlation of the picosecond/nanosecond fluorescence decaykinetics with macroscopic millisecond/kilosecond biological reactions is madepossible.

5.2. The Global Analysis Philosophy

5.2.1. Evolution of the Global Analysis Approach

The global analysis examples described above (and many others) haveproven very useful for many different studies. However, these examples mainlyrepresent an adaptation of classical single-curve analysis algorithms to performa global analysis. The global analysis approach has since been expanded, ina manner that is farther removed from simple curve fitting, to an emphasis onphysical-model evaluation.

For instance, in the rotational diffusion global analysis example (see Sec-tion 5.1.1), the quantities of physical interest are not the preexponential factorsand characteristic rotational correlation times, but rather the shape and sizeof the molecule and its interactions with the solvent. There is ample theorywhich describes the relationships between the size and shape of molecules andthe exponential relaxation times which are observed. An important resultwhich often arises from theory is the fact that the observed set of rotationalcorrelation time(s) and preexponential factor(s) are in general notindependent of each other. If the data analysis is performed in space,then the recovered values for these parameters may mathematically representthe data very well, but may not have any physical analogue. Abondoning thistype of curve fitting and directly analyzing the rotational data in terms


of the physical parameters of the system has been termed “target analysis”( physical model fitting) by Arcioni and Zannoni(36) and has been rigorouslyadopted by the current global analysis programs. See Ref. 37 for a recentstudy of anisotropic rotations using a “global-target” analysis of differentialphase/modulation data obtained at multiple temperatures.

In the current global analysis programs, every effort has been put forthto allow the user to select the physical invariants of the system as fittingparameters, obviating multistep analyses. In the case of rotational diffusion,the physical invariants of the system are the lengths of the major axes of themolecule, the set of angles which describe the orientation of the absorption/emission oscillators and the principal diffusion axes of the molecule, and the“slip” or “sticking” boundary conditions. Of course, from a single anisotropyexperiment, one should not expect to recover all the values of the physicalinvariants of a system, and, generally, a global analysis proceeds using thefollowing approach.

Initially, the emission anisotropy may be fit to a series of exponentialterms. From these results, some of the possible physical models for themolecular shape can be disregarded. For example, if the data fit very well toonly a single rotational correlation time, then there is little reason to performan analysis in terms of lengths and multiple diffusion coefficients. However,upon excitation of the molecule into different absorption oscillators, it may befound that the single recovered rotational correlation time varies. One shouldthen consider performing a global analysis of the two data sets in terms of aninternally consistent molecular shape with varying angles describing therelative orientation of the absorption and emission oscillators and the principaldiffusion axes of the molecule.

Experiments performed at multiple temperatures and viscosities canfurther assist in unraveling complex decay. In this case, one needs to utilizea physical model which will relate how these variables affect rotationalbehavior. For instance, one can model rotational diffusion as an activatedprocess and directly analyze the data in terms of the energy of activation forrotational diffusion (i.e., the physical invariant of the system).

From this example, a general pattern for performing a global analysisemerges:

1. Examine the individual data sets in a semiempirical manner (e.g.,using sums of exponentials, parameterized or maximum–entropylifetime distributions). Utilize this information as a preliminary stepin establishing a set of possible physical models to be applied to thesystem.

2. Change the fitting parameter space from the semiempirical fittingparameters (amplitudes and relaxation times) to more “target”parameters, which represent the physical invariants of the system.


3. In order for this “target” fitting space to be useful, one requires amapping of the physical model that has been chosen to the space inwhich the experimental data have been collected. This mappingscheme represents the section(s) of code which embody, in an algo-rithmic fashion, the theory of this particular physical model (e.g.,compartmental modeling of pholophysical reactions, partial differen-tial diffusion equation solver for energy transfer, etc.).

4. Once the physical model has been completely specified (i.e., thephysical invariants of the system, relationships to the experimentalobservables), one can perform a global analysis (parameter optimiza-tion) of the multidimensional data surface. In this type of analysis, allof the implicit linkages which exist between the fitting parameters andthe data sets are utilized.


5. Rigorous error analysis† is now applied to the recovered parametersof the model.

6. Alternative physical models are applied to the data surface.

This process is schematically represented in Fig. 5.1.Often, it is found that the global analysis error surface is much better

defined than the individual curve error surfaces (item 5 above; see Figure 5.2).Therefore, the fitting parameters can be recovered with much smaller uncer-tainties than in individual curve analysis. This is often proclaimed as themajor advantage of global analysis. Actually, an equally important aspect ofglobal analysis is its model-testing capability.

It may be desirable to apply several different physical models to aparticular data surface (item 6 above). In this case, a whole series of physicalmodels, each with its own distinct set of linkages, would be applied to thedata. The global analysis error statistics can now be invoked to “rank”the possible sets of models as to how well they represent the experimentalobservables (see Figure 5.1).

It is consistently found that both parameter recovery and model-testingcapabilities are enhanced upon performing global analysis. As will be

† To be described in Section 5.7.3.


described in more detail in subsequent sections, the increased discriminationcapability arises from the drastic reduction in the number of fitting parametersobtained throughout the global model specification process.

It should be emphasized that global analysis is in no way limited toone type of experimental data. Fluorescence decay may be combined withsteady-state fluorescence, absorbance, or other types of experimental data.The only requirement is that the equations representing the various types ofexperiments have some common terms as fitting parameters.

5.2.2. Global Analysis Implementation Strategy

There are three basic methods with which one can implement a globalanalysis program:

1. Specific case programs: If the number and type of experiments whichare going to be examined are relatively limited, one may wish todirectly hardwire specific global linkages directly into the software.For instance, combining multiple excitation/emission wavelengths interms of an internally consistent set of lifetimes is a specific case, andseveral special-purpose routines have been written for these types ofexperiments.

2. User subroutines: Nonlinear minimization routines can be writtenwhich only contain the logic needed to perform a global minimizationover multiple experiments, but there is no explicit linking mechanismwithin the analysis program. In this case, each particular user (ofthe program) would be required to write a subroutine, which wouldcontain all of the logic required to link the various experimentstogether.

3. General-purpose linking algorithms: In a general-purpose linkingnonlinear least-squares algorithm, a mechanism within the programallows the user to explicitly enter the number and types of linkagesdesired over the entire data surface. A general-purpose numericalalgorithm is utilized to calculate the observed fluorescence response;all derivatives are calculated numerically, so that a wide variety ofcomplex models can be examined.

Of the above-described methods, no single methodology is advantageousin all cases. Specific-case algorithms suffer from a lack of flexibility, but areoften very fast and produce relatively compact programs. This type ofmethodology should definitely be pursued if only a single type of globalanalysis needs to be performed. Programs which rely on external user-writtensubroutines are more useful, but incur the added expense of requiring thatthe program be recompiled for each new type of analysis. The user mustalso supply the necessary linking logic, which usually requires a reasonably


sophisticated understanding of the structure of the main program. Thismethodology allows a set of subroutines to be developed which can analyzea wide variety of data. These type of algorithms have an advantage overgeneral-purpose algorithms, because a large region of code and data spacedevoted to all of the other models is not wasted (in memory) for the analysisof one particular special case. General-purpose linking algorithms provide a setof indirect addressing matrices (which are user-defined at run lime), whichcompletely specify both the set of fitting functions to be used and the set oflinkages desired between the multiple experiments. The major advantage ofthis technique is that very complicated sets of fitting functions and linkagetypes can be used in a single program.

The original set of global analysis routines† (19,20) were specifically designedto link fluorescence lifetimes as a function of excitation/emission wavelengths.Analysis programs were then developed for excited state reactions(24,27)

and anisotropic rotations.(23,25) Each of these programs was specific to a par-ticular photophysical model and data structure. These programs have sincebeen enhanced, so that all of these models (and many more) can be examinedwithin a single global data analysis environment,(26) and constitute the set ofprograms currently operating at the Laboratory for Fluorescence Dynamics(LFD). These algorithms are of the general-purpose linking type and will bedescribed in detail below.

Although both transform techniques and standard nonlinear minimizationroutines can be adapted to perform a global analysis, the nonlinear least-squares algorithms are far easier to implement and are flexible. In thischapter, only the nonlinear minimization algorithms will be described. Theemphasis in this chapter will not be on explicit technique for solving the least-squares equations; the global analysis methodology can be implemented usingmany different minimization algorithms. Instead, emphasis will be placed onthe “mechanics” of developing a multidimensional global analysis and onspecific examples of overdetermination which are important for fluorescenceexperiments.

5.3. General Elements of the Global Analysis Program

The general scheme of the global analysis program as implemented at theLFD is shown in Figure 5.3. The general elements of the program are asfollows:

• For the construction of the model matrix (Figure 5.3), the presentprogram implements both direct physical models and empirical

†No attempt has been made to reference all the laboratories which have developed a set of globalanalysis algorithms. Instead, only the algorithms with which the authors have had personalinvolvement are described.


models. The selection consists of forming a list of keywords that willbe later interpreted and bound to specific analytical functions. SeeSection 5.4 for a full list of the keywords currently used by the presentprogram (this list is constantly expanding). Finally, for the construc-tion of the model matrix, there is a possibility to specify other elementswhich represent specific experimental conditions, such as the fractionof photons absorbed for each species, current emission wavelengthsettings, etc. In this manner, a physical model is completely specifiedby combinations of keywords which fully describe the parameters ofthe model. A supervisory editor monitors the input of the model forself-consistency. The parameter descriptor list contains limits for the


numerical values of the various fitting parameters (e.g., the program“knows” that distances, rates, spectra, must be positive, etc.).

• The linking matrix (Figure 5.3) is built using a list of pointers. Agraphics-based linking mechanism allows parameters of complemen-tary type to be combined over multiple experiments. Once the physicalmodel has been built, the linking process consists only in the specifica-tion of the invariants of the system.

• The data matrix (Figure 5.3) is constructed by specifying which datafiles are to be used for a specific analysis. The number of elements ofthe data matrix depends on the experiment to be analyzed, the numberof data points for each data curve, and the type of data examined.

• Once the three basic structures (described above) have been constructed,the global analysis program proceeds as follows (see Figure 5.4):

• The keyword descriptor for each fitting parameter for this experi-ment is read.

• A “parser–decoder” routine recognizes the keywords of the list andloads the appropriate functions.

• The fluorescence intensity is calculated for all the frequencies (ortimes) of a given experiment.


• Each experiment is processed in the above manner, until all of thedata surface has been examined.

• A nonlinear minimization algorithm modifies the values of the fittingparameter to best fit the experimental data.

In the next section we illustrate (in detail) some of the most commonlyused models and algorithms for the calculation of the fluorescence intensity(empirical sums-of-exponentials and compartmental models). We also providean overview of the minimization method utilized by the global analysis.

5.3.1. Mapping to the Physical Observables

In any fitting procedure, one needs to be able to calculate the experi-mental observables, given a set of fitting parameters. This process will transferthe parameters of the model into the decay-associated parameters. The currentprogram has a choice of methodologies available to perform this mapping.This first method to be described (which is also the most commonly used) isbased upon the methods of compartmental analysis and systems theory. Thereis a very large number of models and processes which can be examined usingthis methodology (e.g., empirical multiexponential decay, classical excitedstate reactions, some energy transfer processes, dipolar relaxations, rotationalmotions, collisional and static quenching, etc.). There are, of course, systemsthat cannot be represented using this mapping procedure, and alternativemethodologies must be utilized. For instance, in the calculation of thefluorescence observables for transient effects in energy transfer and quenching,one can utilize the numerical solution of a set of second-order parabolicpartial differential equations (an example will be described in Section 5.5.2).What should be kept in mind, however, is that the global analysis metho-dology is not limited in any way by the particular numerical procedureutilized to calculate the observed fluorescence.

5.3.2. Empirical Description of the Fluorescence Decay

All the information content in a single fluorescence decay experiment(performed in the linear regime) is contained in the impulse response function(time domain) or the transfer function (frequency domain) of the sample. Theimpulse response function and transfer function are directly related to eachother through the Fourier transform. For instance, if the time-domain impulseresponse function is being modeled as a sum of exponentials, one has


The frequency-domain transfer function in complex notation (i.e., the Fouriertransform of Eq. 5.1) is

where (v is the frequency of the excitation source). In the timedomain, the observed experimental quantities are the impulse responseconvolved with an excitation profile l(t):

In the frequency domain, the experimental observables are the phaseand demodulation (M ) of the fluorescence emission with respect to theexcitation source:

where Im is the imaginary part of and Re is the real part ofTherefore, it would appear that the most natural set of fitting parameters

for the analysis of fluorescence experiments would be the elements of theimpulse response function or transfer function of the sample. However, as willbe shown, confining the fitting process to and space may be inadequateand is often an undesirable method of analysts (taken alone).

5.3.3. Compartmental Description of Photophysical Events

The use of systems theory for the analysis and interpretation offluorescence decay data was pioneered by Eisenfeld and Ford.(15) Thismethodology is very general, and flexible, and is implemented in the currentglobal analysis program. A diagrammatic representation of a fluorescencedecay experiment in terms of a systems theory view of a photophysical systemis presented in Figure 5.5. The excitation light source enters afluorescence cuvette which contains a mixture of ground state species. Thelight partitions into the various ground state species depending upon theirconcentrations and extinction coefficients. This ground state absorbance forthe ith species is denoted as A population of n excited fluorescence species,

is now produced. The interaction of the excited molecules can bemodeled using a wide variety of user-definable functions, fij.

In Figure 5.5, only a sequentially linked set of n different excited statespecies (or compartments) are depicted, but any looped or branching variation


of this diagram is also allowed. The excited state species may emit fluorescencewith a functional form depicted as These emitted photons are observed bya detector through an emission monochromator/filter which operates as a“gain-discrimination” device, weighting the photon from each particularspecies by a factor proportional to the emission spectra of that state. Thesegain factors are denoted as The fluorescence actually detected will thensimply be the weighted sum of the individual emissions.

Consider the case of a simple two-state excited state reaction. Theseschemes have been extensively utilized to model excited state proton transfer,exciplex and excimer formation, etc. Historically, these types of systems havebeen analyzed utilizing the simple solution to the coupled two-compartmentlinear differential equations, which yields


Naturally, the individual and are functions of all the rate constants of thesystem and the initial boundary conditions. Previously, systems of this typewere first analyzed in terms of individual and as a function of pH,concentration of species, etc. These results were then analyzed graphically, toobtain the rate constants of the system (and spectra).

With the systems theory formulation, this sequential type of analysis isno longer needed. The kinetic system described above is fit directly in termsof the rate constants and spectra associated with each particular system. Forthis simple case, analysis with either methodology may yield comparableresults. However, in more complicated systems (e.g., many more states,distributed or time-dependent rate constants, etc.), the analytical forms of theimpulse response functions may be very difficult (impossible?) to obtain, and,hence, analysis using impulse response equations [e.g., Eqs. (5.6) and (5.7)]is not possible.

The kinetics of systems shown in Figure 5.5 can be written in terms of thestandard systems theory equations as(43, 44)

Here, X(t) is the (n x 1) vector of the concentrations of each excited statespecies; X(t) is the time derivative of X(t); T is the (n x n) transfer matrixwhich describes the connectivity between different compartments, with

and for where fji is thefunction being used to describe the transfer of photons (molecules/species)from compartment i to compartment j; is the parameter vector whichcharacterizes the function associated with the transfer between (or into)compartments; a is a vector of the zero-time absorbances of each ground statespecies; and l(t) is the measured excitation function.

In a fluorescence experiment one does not directly observe the decay ofeach excited state species; instead, the observed emission is weighted by thespectral contours of each particular state. The emission spectral dependencecan be added to the “discrete-case” solution of Eq. (5.8) to yield the predictedtime course of the total-intensity (i.e., nonpolarized) fluorescence:

where is the jth eigenvalue of the matrix T (Eq. 5.8), V is the matrix ofeigenvectors of T (modal matrix), n is the number of excited state species, and

is the emission spectral contour of the kth species at the ith emissionwavelength.

Note that the final amplitude associated with a particular decay rate(commonly denoted as is actually formed as an inner product of the “s”


vector and the eigenvector associated with that particular eigenvalue. In thecurrent global analysis programs, the individual can be actual fittingparameters. The are referred to as because they represent aparticular chemical’s “species-associated spectrum” (i.e., the emission spectrumof this species if it could be observed separately). This is in contrast to

(decay-associated spectra), a term which has come to represent the-set of amplitudes associated with a particular decay time.† In the case of strictground state heterogeneity, the eigenvectors are just the normal basis vectors,and therefore the amplitude associated with a particular lifetime is directlyproportional to its emission spectrum (i.e., for ground state heterogeneity,DAS = SAS). However, for excited state reactions, the final observedamplitude is a linear combination of the SAS and each particular eigenvector(Eq. 5.9) and hence the DAS are not the same as the SAS. The advantage ofbeing able to fit in an “SAS” space is that, since the represent speciesemission spectra (a physical invariant), they can be linked over experimentswhere the kinetics of interconversion have been altered (e.g., by changing pH,concentration of quenchers, etc.) whereas simple amplitude (DAS) termscannot.

The “distributed-case” solution to Eq. (5.8) [i.e., where the parametervector is distributed according to can be written as

In the frequency domain one has

These equations can be rewritten in the more common form

Note that the final represent the “end result” of a complex set ofoperations: the inner product of the eigenvectors of and the emissionspectral contours of each state (Eq. 5.9) weighted by the multidimensional

† Knutson et al.(30) originally defined DAS to encompass both ground state and excited statesystems.


probability function Although Eqs. (5.12) and (5.13) are of the correctmathematical form for fitting fluorescence data, Eqs. (5.10) (convoluted) and(5.11) are the form actually utilized in the global analysis because it allowsone to “link” specific subsets of the “lumped” parameters betweenexperiments.

In the case of excited state reactions, the set of all andare not independent of each other (due to eigenvector–eigenvalue system-wide relationships) so that these distributed cases are very difficult (if notimpossible) to solve analytically, and a numerical integration procedure mustbe performed. However, for cases where there are no excited state interactions(i.e., the T matrix is diagonal), then the complete set of anddecouple, and one can rewrite Eqs. (5.12) and (5.13) as

The integrals given by Eqs. (5.14) and (5.15) need to be calculated only overthe parameters used to characterize each individual distribution, because theyare not a function of the entire set of constants and distributions describingthe total kinetic system

Equation (5.14) is the definition of the Laplace transform of thedistribution function, Within the context of fluorescence decay experi-ments, this integral is also similar to what is known in various other fieldsas the “moment-generating function” or “characteristic function” of thedistribution.(46) Since there exist large tables of Laplace transforms, moment-generating functions, and characteristic functions, a wide variety of analyticalforms now exist to analyze time- and frequency-domain data given a specificanalytical form for the distribution function (in addition to standard sums-of-exponentials models).

An approach which can be taken to analyze fluorescence decay datais to perform an inverse Laplace transform of the data to directly recoverthe probability distribution associated with each lifetime. Many different“regularized” solutions of the inverse transform have been applied with varyingamounts of success.(47, 48) In general, an entire family of distributions will beconsistent (i.e., have the same value) for any given data set. To chooseamong these different distributions, one can analyze for the distributionconsistent with the data that has the largest “entropy.”(49–5l) An importantadvantage of these techniques is that no analytical form needs to be assumedto describe the distribution function itself. However, an important point toconsider is the fact that the intrinsic distribution of the amplitudes as a


function of lifetime generally does not represent the final parameters of interestin a global analysis. These methods do not currently take advantage of multipleaxes of overdetermination.

For example, consider the case in which a particular fluorescence sampleis measured as a function of some independent variable and the distributionof versus versus independent variable is recovered perfectly. Is this result,in itself, of any use without a particular model which can predict such shapes?For this reason, regardless of which analysis methodology is employed torecover a set of versus eventually it is desirable to decompose therecovered versus in terms of an internally consistent physical model.It is advantageous not to operate on the versus themselves, but ratherto examine the original data surface in terms of an internally consistentphysical model. The amplitudes and lifetimes should be utilized as a guide toperforming physical modeling.

5.3.4. Overview of Nonlinear Minimization (The Basic Equations)

Nonlinear least squares involves the minimization of the chi-squarefunction:

where N is the number of data points in the entire data surface, is thestandard deviation of the ith data point in the qth experiment, m is thenumber of total fitting parameters, is the total number of experiments,and n(q) is the total number of data points taken in the qth experiment.Division by the degrees of freedom (N — m—1) does not vary during each fit,so that minimization of a nonreduced will produce exactly the same result.The reduction process, however, allows for some preliminary model testing.Using the definition of and assuming a Gaussian error distribution, onemay assemble F tests (or related criteria) to judge the “quality” of the fit. Areduced that significantly differs from unity calls into question the validity

of either the model or the weighting estimates In the time domain, onecan use single-photon statistics and show that the variance in each channelis directly proportional to the number of counts. In the frequency domain,no simple relationships exist to predict analytically what the error in themeasured phase and modulation should be. Instead, measurements arerepeated and an experimental variance is determined. Improperly weighteddata can have drastic effects on the final recovered parameters, especially foranisotropy decay.

The mechanics of minimization are many and varied. We have adopted


a popular and demonstrably reliable method attributed to Marquardt(40) andLevenberg.(41) Given an initial set of guesses in a vector (the superscriptdenotes the current iteration number), one obtains by forming:

The parameter improvement vector is determined from the shape of thehypersurface around the current fitting parameters, by solving the followinglinearized system of equations:

where

and is the parameter increment in fitting parameter k. In Eq. (5.19), is ascaling factor, and I is the identity matrix. The diagonal scaling matrixdetermines the “mode” in which the Marquardt–Levenberg algorithmoperates. Changing by decreasing it after each “successful” ( reducing)step and increasing it upon step failures represents a strategy that guaranteesfinding a minimum. The possibility of local minima is not obviated, however,so the analyst must remain skeptical and exhaust alternative minimizationpathways (multiple analyses employing different initial guesses, rigorous erroranalysis, etc.).

5.4. In-Depth Flow Chart of a General-Purpose Global Analysis Program

In this section, technical details concerning the implementation of thegeneral-purpose global analysis routine will be presented. The information inthis section should be sufficient for independent laboratories to develop theirown specific global analysis programs (if desired).

5.4.1. Overview of the Global Analysis Procedure

In relation to Figure 5.3, there are three structures which must be estab-lished before the global optimization can be performed:

• The physical model/solution process/experimental conditions matrix• The linking connectivity matrix• The data matrix.


The first task in any analysis procedure is to construct the calculateddata associated with the ith experiment. In single-curve analysis, there isonly one experiment (and hence one fitting function), so that one does nothave to perform any specific parameter indexing (i.e., every fitting parameterin single-experiment analysis is either directly used or is held as a fixedconstant). However, for a global analysis, many different experiments needto be combined into a single analysis. Therefore, certain specific indexingmethodologies need to be added to the fitting routines so that the correctfitting parameters and fitting functions can be obtained for each individualexperiment.

A global analysis algorithm must first recognize those elements of thetotal fitting parameter vector which are used for that particular experiment.Then it must pass these particular elements to the fitting function appropriatefor the experiment. The calculated data values for this experiment can then beobtained. By “looping” in this fashion over the entire set of experiments beingexamined, a global analysis can be performed.

A description of the logic implemented in the current global analysisprograms will now be presented. It should be emphasized that the followingsteps could be implemented in many different ways. We have chosen thefollowing implementation because it has proven useful for fluorescence decaydata and because it can be readily adapted for the analysis of many differenttypes of models.

5.4.2. Flow Chart for the LFD Global Analysis Program "Global"

The following text describes in detail a typical iteration of the general-purpose discrete/distributed global analysis program as currently implementedat the LFD. This process has been divided into 12 logical steps, which willnow be described. (Steps 1 through 3 need to be performed prior to thebeginning of the analysis.)

1. Physical Model Describing the Data Is Established

A wide variety of physical models appropriate for fluorescence spec-troscopic data have been coded into individual subroutines. These individualsubroutines are accessed through the use of specific “keywords” which theprogram “understands.” Some of these routines are very general purpose, suchas the compartmental equation solver (Eq. 5.8), and others are very specific toone type of model (e.g., anisotropy analysis). A list of the current setof keywords which the program understands is given in Table 5.1. This tablecontinues to grow, and the program evolves to the extent that it is able tointerpret (in a rigorous mathematical sense) most of the theory and conceptsthat fluorescence spectroscopists employ.


An in-depth description of how a model is specified will now be given.For convenience, the following description will emphasize the completespecification of models for the compartmental systems theory aspect of theglobal analysis program. An alternative tool for the calculation of fluorescenceobservables will be described in Section 5.5.2.

2. The Linkage Patterns Appropriate for This Combination of ExperimentsAre Established

The total number of fitting parameters used over an entire data surfaceare stored in a vector P of length n. An important consideration when


performing a global type of analysis is that one must provide some type ofsupporting logic to tell the fitting program which elements of the fitting vectorare appropriate for a particular experiment. What is often implemented insingle-experiment analysis programs is some type of standard “ordering” ofthe parameters within the fitting vector so that the program knows what typeof fitting parameter is in each element of the vector. For instance, when fittingdiscrete exponentials to a single experiment, the parameters could be enteredinto the fitting vector as

It must be emphasized that when a global multiexperiment analysis is per-formed, this type of ordering of the parameter vector is usually inedequate.In a global analysis one needs the following additional set of supporting logic:

(a) Logic must be supplied to tell the program which elements ofthe total surface fitting vector are being used for this particularexperiment.

(b) Logic must be supplied to tell the program what type of fittingfunction is appropriate for the fitting parameters.

The logic required to inform the program which elements of the fitting vectorare appropriate for a particular experiment can be presented using a matrixof integer values. This matrix ( M ) (mapping matrix) is dimensioned with itsfirst index equal to the maximum number of experiments which can besimultaneously analyzed and its second index equal to the maximum numberof fitting parameters which can be local to a single experiment, and a thirdindex (which is either 1 or 2) describes whether this number points to aparameter-vector element or a parameter-descriptor element (to be describedshortly). The third index will be set to 1 when pointing to the fittingparameter array element and 2 when pointing to a parameter descriptor.When performing a loop over the entire data surface, the current experimentindex counter (nc) is used as the row index of M [M(nc , · , ·)] . The numberswhich are then placed along a particular row of M will then be the indicesof the total parameter fitting vector P which are appropriate for the ncthexperiment. For instance, if the ncth experiment utilizes elements 5, 6, 8, and10 of the main fitting parameter vector, then M(nc, 1, 1) = 5, M(nc, 2, 1) = 6,M(nc, 3, 1) = 8, M(nc, 4, 1) = 10. Utilizing the logic in the M matrix, there-fore, allows one to pass to a subroutine only those vales of the fittingparameter vector which are appropriate for the jth experiment. The localfitting terms [LT( i)] for the jth experiment can be extracted from the totalsurface fitting parameter vector (P) by using the logic:


In this way the first level of support logic [item (a) above] has beenaccomplished in a completely general way. However, one additional logicalstep remains. Now that the elements of the total fitting vector appropriate forany single experiment within the data surface have been obtained, one mustdetermine what type of fitting parameter this element represents (e.g., is thefitting parameter a simple rate or perhaps a length distribution parameter oran activation energy, etc.). Again, there are many types of logic which can beimplemented to perform this function, but we have found the followingmethodology very flexible.

Again, an indexed vector can perform this function. One can define avector dimensioned to the maximum number of fitting parameter “descriptors”which can be used to define the entire data surface [D(np)]. Each element ofthis vector (D ) (descriptor vector) is a string of ASCII keywords whichuniquely describes the functional characteristics of this fitting parameter.Obviously, many different types of descriptor vectors could be used. Wecurrently utilize the above formalism because it assists both in makingthe source code of the program easy to read (and hence to modify) andestablishes a “natural-language interface” for describing the fitting functionsdesired. For instance, a typical keyword descriptor for a simple transfer ratebetween compartment 1 and 2 would appear as D(1) = “rate 21.” If that ratewere distributed as a Gaussian, then D(2) = “rate 21 distributed Gaussian.”If the transfer between compartments involved an energy transfer process,then D(3) = “length distributed Gaussian 21.” These descriptor terms wouldthen be “decoded” by the parser subroutine to map to the following fittingfunctions:

This type of methodology has proven very useful and flexible and allowsthe establishment of a “vocabulary” of keywords which can continuallyexpand and grow. Eventually, the program evolves to the extent that itbecomes essentially an “expert system” for the analysis of fluorescence decay


models. One of the advantages for an indexed list of fitting functions is thatone does not need to expend a large amount of programming effort to incor-porate new models into the analysis. In addition, since all of the old model“keywords” are maintained, a very general purpose nonlinear analysis fittingenvironment develops, whereby a wide variety of different models and linkagepatterns can be applied to a particular data surface. There is no longer anyneed to stop and redevelop the analysis programs each time one desires toanalyze the data using a new model.

3. Particular Forms for the Functions Linking the Compartments Are Chosen

The appropriate form for the function linking the compartments togetheris usually decided by choosing that functional form in which the physical

quantity of interest appears explicitly (eliminate multistep analysis). Forinstance, if a multiple-temperature study is being performed, instead of arate-space type of analysis, one may wish to globally analyze the data directlyin terms of activation energies and frequency factors. In an energy space,one useful functional form to describe the excited state interconversion fromcompartment 1 to 2 would be

If the activation energy linking the two compartments was distributed, onewould have

If one were performing a multiple quenching experiment (allowing compart-ment 2 to be the quenched fluorophore and compartment 1 the unquenchedfluorophore), one could define

If the quenching constant was not defined in terms of a discrete quenchingrate, but rather by a distribution of quenching rates, then one could allow

By establishing a wide variety of functional forms for the extremelycomplicated ground state and excited state reactions can be examined.


4. The Connectivity of the Compartmental Model Is Established

The connectivity of a photophysical system simply means those sets offluorescent molecules which are allowed to interact during the lifetime of theexcited state. For instance, complex decay of fluorescence from systems withno connectivity simply represents ground state heterogeneity. Mathematically,a lack of connectivity requires that the T matrix (Eq. 5.8) is completelydiagonal. For excited state reactions, however, there is connectivity betweenthe various species, and hence some off-diagonal elements of T are nonzero.Deciding which elements are nonzero establishes the pattern of connectivityof a system. If some connections between compartments are known to beabsent, the corresponding elements of the matrix T can be fixed at zero. If theconnectivity of the compartments is not known, then various experimentalconnectivities may be applied and their corresponding values examined.

Consider the case of a three-state excited state reaction. Three particularconnectivity diagrams are shown in Figure 5.6. These diagrams wouldcorrespond to patterns of zeros in the transfer matrix (Eq. 5.8) as follows:

where correspond to Figs. 5.6a, b, and c, respectively, and xis some nonzero value. Note how simple it is to examine a wide variety ofdifferent excited state reaction patterns, simply by specifying the connectivitybetween the various compartments. One does not need to derive the impulseresponses of these various models (analytically); the eigenvector–eigenvaluesolver simply calculates them numerically for both discrete and distributedtransfer rates.

5. Initiate Global Analysis over the Data Surface

A typical way of assembling the data structure is to treat any high-dimensional data surface as a series of two-dimensional slices of data. For


instance, within the analysis program there are characteristic data matricesof the form data(x, nc), where nc represents the experimental curve numberindex and x follows the number of “channels” of data collected. Data(5, 10)would therefore represent the fifth data point in the tenth experiment. Onecan now set up an index variable (nc) to keep track of the current experimentwhich is being examined. This index will therefore be incremented from oneto the total number of experiments which are being combined in the analysis.

6. Perform Parameter Linking and Function Mapping

Using the current experiment index (nc), the logic described in step 3 ofthis discussion is now employed.

The relevant ncth-experiment fitting parameters are extracted from thetotal fitting parameter vector and passed to a subroutine using mappingmatrix In addition, the field descriptor specifications of theseparameters are also passed using A typical implementation of theFORTRAN-77 to perform these operations is as follows:

do 100 i=l, nloctermsipointp=m(nc, i, 1)ipointd=m(nc, i, 2)LT(i)=p(ipointp)LD(i)=D(ipointd)

100 continuecc ipointp is a pointer into the fitting parameterc vector ‘‘p’’c ipointd is a pointer into the descriptor vector ‘ ‘D’ ’


c p=total data surface fitting parameter vectorc D=character array vector containing the ‘ ‘keyword’ ’c descriptors for each particular fitting parameterc LT() is a vector containing those fitting parametersc appropriate for the ncth experimentc (Lt: local terms)c LD() is a vector containing the descriptors associatedc with each particular local fitting parameterc (LD: local descriptor)cc one can now call a ‘ ‘parser’ ’ which decodes the parameterc values and their associated descriptors to assemble thec appropriate set of differential equations to solve.c Given LT and LD, parse returns a fully assembled set ofc systems theory differential equations in the matrix T andc vector b.c

call parse (nlocterms, LT, LD, T, b)ccc ...

The matrix T corresponds to the T in Eq. (5.8), while “b” corresponds to theelement in the same equation.

Within the subroutine “parse,” the field descriptor parameters are used to“decode” the local experiment fitting parameters into proper state space form.This process involves using the keyword descriptors to locate the appropriatefunction which will map the current parameter values to a rate space. Theposition identifier is now utilized to find which matrix element of the statespace equation this parameter value belongs.

Once all local experiment fitting parameters have been assembled into thestate space equations, this subroutine returns to the main program. Allelements of the state space equation for the ncth experiment are now fullydefined.

7. Solve the Eigenvalue/Eigenvector Problem and Compute Probabilities†

The matrices T and vector appropriate for this experiment are now sentto a nonsymmetric eigenvector–eigenvalue solver (e.g., EIGRF,(38) Eispack,(39)

† Some photophysical problems cannot be solved using an eigenvector/eigenvalue approach andmust be solved with some specialized routines (e.g., transient diffusion effects for energy transfermeasurements such as in Section 5.5.2). In these cases, the eigenvector/eigenvalue routines arebypassed, and control is passed to a specialized equation solver appropriate for that particularproblem.


or equivalent routine). The inverse negative eigenvalues of the matrix T nowbecome the observed fluorescence lifetimes for this particular realization of thefitting vector. For instance, for a two-state excited state reaction one candenote the current spectral output gain vector at a particular emission wave-length as SAS1 and SAS2. The eigenvectors associated with two eigenvaluesof this system can be denoted EIG(1, 1), EIG(2, 1) and EIG(1, 2), EIG(2, 2),respectively. The final amplitudes associated with the two relaxation timeswould therefore be [cf. Eqs. (5.6) and (5.7)]:

8. Perform the Integration over the Distributed Parameter (s)

If some parameters are distributed, one additional step needs to be per-formed before the final amplitude terms associated with the exponentials arecomplete. For distributed systems, steps 4– 6 will only represent one possiblerealization of the distributed system. A “realization” simply means one par-ticular response of the system out of the total set of the possible responses ofthe system (discrete systems have only one realization). Associated with eachrealization is a particular probability (e.g., for a Gaussian distributed variablethe realization of a value one standard devation from the means is assigneda probability of 0.242, while for a uniform distribution all realizations areweighted by unity, etc.). Therefore, the final amplitude terms from step 6 willneed to be further scaled by the probability of the realization. If a particularparameter that was Gaussian distributed was currently being sampled onestandard deviation from its mean, then the final amplitudes from step 6 wouldneed to be scaled as In the case where more than one param-eter is distributed, the probability associated with a particular realizationdepends upon the correlation between the individual probabilities. If thedifferent fitting distributions are independent of one another, then the finalrealization probability is just the product of the individual probabilities.However, if there is some type of correlation between the probabilities, thenthe final realization probability is calculated from the multidimensionalprobability function.

These final amplitude terms and their associated lifetimes are then usedto calculate the observed fluorescence using Eq. (5.10) (time domain) or (5.11)(frequency domain). Integration routines now iterate steps 5 and 6 until someconvergence criterion is met (e.g., the calculated phase and modulation valuesor intensity at a particular channel fall by a factor of 5–10 less than the noisein the actual data). The actual integration procedure occurs in the fittingspace, but the convergence criterion is maintained in the data space. For


instance, when integrating over a distance distribution, or an activationenergy, etc., there is not a simple map between the integration space and thedata space. Therefore, one integrates in energy/distance space, but examinesfor convergence criteria in the data space. The limits over which the distribu-tion function is integrated are user defined. In practice, one usually limits theintegration procedure to approximately 99 % of the area under the distribu-tion. Discrete summing of many exponential terms to approximate theintegral can be highly inaccurate, especially for short-lifetime distributions.This type of summing does not produce a truly continuous distribution oflifetimes. Certain integrals can be calculated analytically, and a new functionalform may be used to analyze the data.

One should note that the fitting space in which the distribution functionis defined is important. For instance, the integration of a uniformly distributedlifetime from to will yield different results than the integration of auniformly distributed rate from When performing analyses interms of some distributed parameter, it is very important to consider whatspace the distribution originates from. For instance, consider the decay froma single fluorophore in solution. The lifetime can be written as

If is found to be distributed, the origin of this distribution could residein a distributed radiative or nonradiative rate. The rate distribution itselfcould result from either a distribution of Arrhenius frequency factors or adistribution of activation energies, or even both. Any particular distributionfunction (e.g., uniform, Gaussian, etc.) applied in these different distribution“spaces” will each yield a very different set of distributed lifetimes. The choiceof where to place the distribution function is very important and should beguided both by the types of experiments which are being combined and thephysics of the system being examined. Being constrained to only fittingdistributed “empirical” lifetimes is a severe limitation and should be avoided.In particular, single-curve analysis via empirical distribution functions issubject to errors from parameter correlations. This behavior is just like thatseen in single-curve exponentials analyses, and overdetermination is usuallyrequired to validate mean and width estimates.(42, 26)

9. Calculate Derivatives and Set Up the Least-Squares Equations

The result of step 7 then becomes the calculated values for this particularexperiment and the chi-square and residual vector can be formed. The basicequations required to perform a nonlinear analysis are now assembled (seeSection 5.3.4).


The key distinguishing feature of a “global” analysis is that the nonlineardata analysis is simultaneously performed over the multiple experiments [seeEq. (5.16)]. To perform this type of analysis, one must supply an additionalamount of logic which will allow one to combine the elements of Eq. (5.18)(C and B) when using multiple fitting functions defined over multipleexperiments. There is no unique (or best) way to implement this logic.

The elements of Eq. (5.18) are then formed using numerical or analyticalderivatives (whichever are available for a particular model type). In addition,a chain-rule set of derivatives can be easily calculated for certain models.A combination of all three approaches is performed in the current analysisprogram. For numerical derivatives, either forward difference or centraldifference equations are used (and are user selectable). The least-squaresequations are then assembled. Note that the derivatives calculated fromdifferent experiments pertaining to linked parameters must be summedtogether. This can be performed quite easily utilizing the mapping matrix Min the following fashion (explained using FORTRAN-77 code):

c

c Section of LFD global analysis FORTRAN 77 code whichc assembles the normal least-squares equations using thec mapping matrix to index the various elements.cc DEFINITIONS OF TERMScc nlocterms=the number of fitting parameters for the ncthc experimentc derphase (i, j)=the derivative of the calculated ithc phase data with respect to the jth localc fitting parameterc dermod (i, j)=the derivative of the calculated ithc demodulation data with respect to the jthc local fitting parameterc residphase (i)=the ith residual in the phase of the ncthc experimentc residmod (i)=the ith residual in the modulation of thec ncth experimentc errorphase (i, nc)=the variance of the experimentalc measurement of the ith phase in thec ncth experimentc errormod (i, nc)=the variance of the experimentalc measurement of the ith modulation in thec ncth experimentc varphase=the variance of the ith phase data point


c varmod=the variance of the ith modulation data pointc ndata(nc)=the number of data points taken in the ncthc experimentc ipointl, ipoint2=perform the ''reverse-mapping'' so thatc the nonlinear least-squares equationsc can be assembledc

do 200 j=l, nloctermsipointl=mpoint(nc, j, 1)do 110 i=l, ndata(nc)

varphase=errorphase(i , nc)varmod=errormod(i ,nc)beta (ipointl )=beta( ipointl )+

1 residphase(i)*derphase(i, j)/varphase+2 residmod(i) * dermod(i, j)/varmod

110 continuedo 150 k=l, j

ipoint2=mpoint(nc, k, 1)do 120 i=l, ndata(nc)

C(ipointl, ipoint2)=C(ipointl, ipoint2)+1 derphase(i, j) * derphase(i, k)/varphase+2 dermod(i, j) * dermod(i, k)/varmod

120 continueC(ipoint2, ipointl)=C(ipointl, ipoint2)

150 continue200 continue

The matrices C and beta (above) correspond to C and B from the normalleast-squares equations (Eq. 5.18).

10. Finish Loop over All Experiments

Test to see whether the entire surface has been examined. If there aremore experiments in this data set, then increment the current experimentalcounter (nc) and go to step 5. Otherwise, proceed to the next step.

11. Get Proposed Parameter Increment for Global Fitting Vector

Solve system of equations for proposed delta parameter increment. Testwhether this new set of parameters decreased the value. If so, decrease(Eq. 5.19) and proceed until some convergence criterion is met. If increases,make larger and solve the modified set of normal equations untildecreases.


As an aside, one should note that Eq. (5.18) is seldom solved by matrixinversion. More efficient equation solvers are available for such symmetricsystems. We have found that the use of the Cholesky decomposition or“square-root” method(52) is sufficient for almost all data sets. The most robustnumerical method of performing nonlinear least squares is by by passing theformation of Eq. (5.18) altogether and using the singular-value decompositiontechnique (see, for example, Refs. 53 and 54). However, this techniquerequires the formation of a matrix of dimension as compared tomatrices with dimension in Eq. (5.18) (N represents the total numberof data points used in the analysis, and m the total number of fittingparameters). For many global analyses, the data densities are simply too great(often as many as a million total data points) for the use of the singular-valuedecomposition technique for solving the nonlinear least-squares problem.However, in cases where the analysis has been performed using both techni-ques (when applied to typical fluorescence decay data/models), identicalresults are usually obtained (J. M. Beechem, unpublished results). It should beemphasized that it is almost always not the mechanics of the minimizationthat causes problems in the data analysis, but rather the limited “informationcontent” of the experiments.

5.5. Case Studies of the Application of Global Analysis toExperimental Data

5.5.1. Case Study of a Two-State Excited State Reaction

Originally, global analysis procedures emphasized systems having groundstate heterogeneity.(19,20) In these cases, the observed lifetimes represent thedecay rates for each species in solution. The amplitudes associated with eachparticular lifetime are also directly related to the excitation/emission spectraof each emitting state. For excited state reactions, neither of these simplerelationships exists. Global analysts of excited state reacting systems in termsof lifetimes and their associated amplitudes, therefore, does not directly recoverthe “target” parameters of real interest: rate constants for interconversion andspecies-associated spectra (SAS). Various levels of empirical and target globalanalysis can be performed on these systems and will be described below.

Consider the very simple case of a two-state excited state proton transferreaction. Within this model framework, the following two types of overdeter-mination axes can be easily exploited: multiple emission wavelengths andmultiple pHs. For a global analysis of the data surface,one should examine which fitting space is most “natural” for this particularset of experiments. With a global analysis in mind, this question becomes,


which fitting space will minimize the number of fitting parameters neccessaryto describe the fluorescence surface and provide numerical values (andassociated error bounds) of the primary parameters of interest?

Consider the global analysis of two pH experiments with four differentemission wavelengths. This data surface can be represented in terms ofimpulse response parameters as:

From the completely empirical model of a sum of two unrelatedexponentials, this total decay surface can be described using totalfitting parameters. Although each data set would be examined in terms of afour-parameter minimization, the net result after the analysis of the eightexperiments would be the same as if a 32-parameter “nonlinked” minimizationwas being performed.

From the structure of a two-state excited state reaction, it is known thatthe fluorescence lifetimes are invariant as a function of the emissionwavelength. Therefore, one can perform a global analysis of each particularpH experiment by linking the observed lifetimes as a function of emissionwavelength. This reduces the dimensionality of the fitting space from 32 to 20(two sets of two lifetimes plus two sets of four values). The lifetimes andamplitudes between different pH experiments cannot be linked because theyare a function of the kinetic system as a whole and will change with pH.

If the fitting space is removed completely from space, a furtherreduction in the number of parameters results. Using the systems theoryequations (Eq. 5.8), a fitting space can be defined which consists of the rateconstants of interconversion the lifetime of eachemitting state and the spectra associated with eachemitting state For this type of analysis, knowledge ofthe boundary conditions may be required.(55) The dimensionality of thefitting space is now reduced to just 12 fitting parameters. Simply by imposingthe physical model “two-state excited state proton transfer,” the fitting spacehas been reduced from 32 to 12 dimensions. This radical reduction in fitting


space can result in a transformation of the error surface from a rather ill-defined flat surface to a very well defined, nearly quadratic error surface.

This type of analysis was applied to the excited state proton transfer of-naphthol.(24) Time-resolved data were collected at pH 2.15 and 3.0 at 75

emission wavelengths in each case (this number of emission wavelengths isonly necessary if high-resolution species-associated spectra are desired). Aglobal analysis was now performed by simultaneous analysis of a data surfaceconsisting of fluorescence versus time versus emission wavelength versus pH.The total number of fitting parameters are the four rate constants plus twospecies-associated spectra (a total of 154 fitting parameters!). It should benoted that the rate constants can be completely determined using only twoemission wavelengths, so that only six fitting parameters are needed for thatanalysis. However, it was desired to attempt to recover the high-resolutionSAS for this system so that all of the emission wavelengths were utilized. Therate constants recovered from this system are shown in Table 5.2.

One can see that the simultaneous analysis of only two pH values wassufficient to resolve the rate constants of this system, compared to theextensive pH study performed previously.(56) The global analysis results, ofcourse, only strictly pertain to the pH region in which the study was actuallyperformed.

One reason the 2-naphthol case was reexamined was because the species-associated spectra can be experimentally determined, and therefore the SASrecovered from the analysis (using 154 fitting parameters) can be directlycompared with the known spectra. The results of this comparison are shownin Figure 5.7. The spectrum associated with each emitting state recoveredfrom the analysis program is plotted along with the experimentally determinedSAS (pH = 0 yields only naphthol, pH = 13 yields only naphtholate). Fromthese results, it appears that not only all the rate constants for the reactioncan be recovered, but also each individual SAS. Therefore, all 154 fittingparameters are uniquely recoverable and can be shown to faithfully representphysical quantities.

For many biochemical systems of interest, it will not be possible to alter


conditions so as to obtain the individual SAS experimentally in a direct way(i.e., extreme pH regions will destroy sample). Application of a “target” globalanalysis as described above can decompose these complicated spectra intotheir time-invariant components. Davenport et al.(57) have demonstrated aDAS remixing scheme suitable for some of these cases. An alternative multi-step approach to obtain SAS has recently been described by Löfroth.(45)

Consider a hypothetical extension of the above two-state excited statereaction system, where the proton-transferring probe is now covalently labeledto a protein or biological membrane. It may be found that the observedlifetimes of the fluorescent probe as a function of pH and emission wavelengthare more complicated and are better fit using a distribution of lifetimes ratherthan discrete lifetime values. A whole series of empirical analyses can now beperformed using distribution functions to describe the lifetime components.The end result of these analyses, using either parameterized or maximum-entropy methods, would be a series of distributions as a function of pHand emission wavelength. However, the primary quantities of interest, rateconstantst and SAS, would not be determined by this type of empiricalanalysis.

One may feel that an empirical analysis is all that can be performed ona system as complicated as this. However, various physical models can still beapplied using the systems theory and global analysis framework. The conceptof a discrete pH value inside a protein may not be particularly valid, and,during the lifetime of the excited state, the probe may indeed experience adistribution of proton concentrations (activities). With this model in mind, aglobal analysis of the multiple pH and emission wavelengths can be performedby altering the function willrepresent the distribution of protons within the interior of the protein andwill be an actual fitting parameter. Within this model framework, one canmaintain a structure which is consistent with the probe’s solution behavior,yet allows an additional level of complexity due to its now more complicated(protein/membrane) environment. A similar reduction in parameter spacecompared to empirical fitting is still obtained; however, since is nowdistributed, both lifetimes and their associated amplitudes will also appeardistributed due to the nature of the eigenvector eigenvalue solution ofEq. (5.8). Negative-amplitude terms will appear “naturally” due to the factthat the excited state reaction distribution structure is being imposed at thelevel of the differential equations, not their solutions. The final results atthe end of the analysis would contain, in addition to all the discrete fittingparameters, a distribution parameter which would reflect the actual protondistribution in the interior of the protein.†

†Since pH is a macroscopic parameter (by definition), the use of proton concentration shouldperhaps be replaced with that of site-specific reactivities.


This type of analysis result clearly represents the philosophy behind theglobal analysis step: transform a series of empirical fittings into aninternally consistent physical result. The unlinked empirical analysesrepresent the best model-independent representation of a given set of experi-ments. However, the solutions, although mathematically correct, may haveno physical analogue. The global analysis procedures do not compete withthe empirical single-experiment analysis approaches; they simply representthe logical next step in a multistep process which transforms data into“biochemical information.”

5.5.2. Distributions of Distances and Energy Transfer Analysis

The advantages of the use of global/target analysis can also be illustratedfor the interpretation of resonance energy transfer experiments. Resonanceenergy transfer is well established as an experimental method for estimatingdistances or distance changes in macromolecules.(58–62) The time dependenceof the rates of nonradiative energy transfer between probes attached to uniquesites on a biopolymer is a function of both the distribution of the distancesand the rates of interconversion between the distances. One can model thisprocess, using the following equation(63)

where is the equilibrium distribution of the energy transfer pair,is the interprobe diffusion constant, r is the

interprobe distance, is the lifetime of donor, and is the associatedamplitude.

The first term on the right-hand side of Eq. (5.31) represents both thespontaneous decay of the donor and the decrease of excited donor concen-tration due to nonradiative energy transfer. The second term represents thereplenishment of the depleted fractions by the Brownian motion of the labeledsegments. The distance distribution, obtained by solving Eq. (5.31)represents the time-dependent reduction of the excited donor population[normalized to ] at each distance fraction, taking into account bothdistance distributions and interconversions. In spite of the fact that the timedependence of the rate of energy transfer is clearly a function of the donor–acceptor distance distribution and fluctuations between the distances that occurduring the lifetime of the excited state, attempts to recover both the distancedistributions and the diffusion rates have proved to be elusive. A method hasrecently been developed by Beechem and Haas,(64) which performs a globalanalysis of donor and acceptor decay in terms of both distance distributionsand diffusion rates as the target fitting parameters. This method has overcome


many problems associated with close correlation between fitting parameters.Both distance distributions and intramolecular diffusion parameters wererecovered with high statistical significance.

Initial time-resolved fluorescence experiments aimed at obtaining thespatial distribution and the diffusion rate information from the available dataemployed a multistep analysis process similar to that used for analysis ofother excited state reactions.(58–60) The decay time of the donor was obtainedin the absence of acceptor to determine the donor lifetime and amplitude. Thisdecay time then became a quantity that was fixed in the next stage of theanalysis involving the decay of the donor in the presence of the acceptor.Analysis of experiments at high viscosity was used to obtain distance distribu-tions in the absence of diffusion. This recovered distance distribution was thenfixed in a subsequent analysis of data obtained at low viscosity to yield thediffusion coefficient. This type of multistep analysis leads to difficulties in errorpropagation and involves restrictive assumptions involving extrapolation oftemperature- and viscosity-dependent data.

Upon treating these data with a global analysis algorithm, the very illdefined error surfaces from “donor-only” analysis are replaced by well-definedsingle minima (see Figure 5.8). The radical transformation of the error surfacecan be rationalized as follows. The interprobe distance distribution of anensemble of molecules with an excited donor probe at the time of excitationwill be identical to the equilibrium distance distribution. A very rapid decayof the donor excited states occurs for the fraction of molecules with shortinterprobe distances because of the steep dependence of transferprobabilities. Thus, the donor emission of the fractions of molecules with theshortest interprobe distances can decay very fast. It is difficult to obtain anexact quantitation of the amplitude contributions of these very fast decaycomponents (they disappear at quite early times). These parameters are thusrecovered with poor accuracy from analysis of the donor fluorescence decaycurve alone. The acceptor emission, however, recovers these parameters withhigh sensitivity, due to convolution of the transferred photons with theacceptor impulse response, which effectively “spreads” this information outover a much larger time interval. A similar argument holds for the weightingof the emission from the fractions with large (relative to ) interprobe dis-tances. These long distances contribute a weak signal to the acceptor emissionbut are well weighted in the donor emission. Thus, the simultaneous use ofboth energy transfer donor and acceptor decay data leads to a significantimprovement in the ability to recover the desired fitting parameters. A furtherenhancement of parameter recovery can be achieved by combining experi-ments done at different temperatures and/or viscosities or done in the presenceof different concentrations of quenchers. The combination of these variousexperiments into a single data analysis provides an ideal example of the“global philosophy.”


5.6. Anisotropy Decay Data Analysis

5.6.1. General Equations and Experimental Linkages

The emission anisotropy r(t) is defined as

where and are the time-dependent emission components measuredparallel and perpendicular to the polarization of the excitation light, respec-tively. From the definition of the emission anisotropy and the fact that thetotal fluorescence for a macroscopically isotropic sample is given by

the individual polarized components can be expressed as:

These expressions reveal that both and are determined by the sameset of fitting functions [i.e., f(t) and r(t)] and hence the same set of fittingparameters. This statement can be made for any polarized intensity obtainedunder any combination of polarization directions in the excitation and theemission. The intensity corresponding to excitation and emission polarizers atangles and with respect to the normal to the excitation–emission planeis given by

The time dependence of the anisotropy is expressed by the ensemble averageof the angle between the absorption dipole at time zero and the emissiondipoles at time

where is the second-rank Legendre polynomial, and the angle bracketsdenote the ensemble average; and are unit vectors along the transitionmoment of the absorption at time t = 0 and of the emission at time t, respec-tively.


From the general expression for the decay of the emission anisotropy(Eq. 5.37), various specific cases can be calculated. The anisotropy decayof a general rigid asymmetric body in an isotropic environment is describedby a sum of five exponentials.(66–68) In many cases, these five exponentials canbe approximated by a sum of three exponentials.(69) When approximatingthe hydrodynamic shape of a fluorescent system in terms of an ellipsoid ofrevolution, one has(28):

where

where and are the angles made by the absorption and emission dipoleswith respect to the symmetry axis and where is the angle between theirprojections in the plane perpendicular to the symmetry axis. The corre-sponding rotational correlation times are given by:

where is the rate of rotation about an axis perpendicular to the symmetryaxis, and is the rate of rotation about the symmetry axis. Note that thereare three rotational rates but only two physical parameters: and

The anisotropy decay of a body subjected to diffusion in a hinderingpotential of arbitrary form cannot be expressed in closed form (as describedabove). In these cases, the anisotropy decay is found to be described as aninfinite number of exponentially decaying functions. An empirical expressionconsisting of a limited sum of exponentials and a constant term has often beenused to approximate this decay:


By examining trends in the parameters and along various experimentalaxes, various conclusions can be made concerning the physical parameterswhich describe the system (e.g., critical temperatures, diffusion coefficients,etc.). The problems inherent in this type of two-step analysis are identical tothose already discussed for total intensity decay data.

A sequential method of analyzing polarized fluorescence decay datainvolves the formation of two related decay curves, S(t) (the “sum” curve)and D ( t ) (the “difference” curve):

with

where l(t) is the measured excitation function, and denotes the convolutionproduct.

The sum curve is proportional to the decay of the total intensity andcontains no fitting parameters for the anisotropy. In the sequential analysis,the total fluorescence decay parameters are determined first from nonlinearminimization of S(t). The results from the S(t) analysis are used as fixedparameters in the subsequent analysis of the D(t) curve.

As an alternative to this “sum-and-difference” analysis, and can besimultaneously analyzed for the parameters of S(t) and r(t) using completelinking of parameters.(70,71,23,25) This method has certain advantages withrespect to the “sums-and-difference” analysis approach and has been adoptedfor all of the current global analysis programs. What will be discussed inthe following sections will be the global analysis of multiple anisotropyexperiments, keeping in mind that each individual experiment requiressimultaneous analysis of a single and measured decay.

The actual fitting programs for both empirical and model-dependentglobal analysis can be implemented using the general methodologies describedin detail in Section 5.2. One merely defines an additional set of anisotropy-specific “keywords.” A description of global anisotropy analysis and essentialprogram code are also given elsewhere.(23)

In the following sections, we present a series of examples to illustrate howglobal analysis has been used to study some complex rotational behavior.Some of these cases are of general applicability, whereas other cases arespecific to the study of particular biological systems.


5.6.2. Changes in Anisotropy Data Collection Schemes

Since the development of the global analysis software, we have foundthat certain problems associated with the actual hardware collection offluorescence data have been eliminated. In the case of anisotropy data, theanalysis of data obtained using the simultaneous collection of and(“T-format” time-resolved anisotropy data collection) proved difficult becauseof the requirement that the timing characteristics of both phototubes beidentical. However, when performing a global analysis on andeach with its own excitation profile and timing calibration, there is no need tomatch the two phototubes.(23, 74) This matching is both difficult to accomplishand unnecessary.

In global total intensity decay analysis, experimental use of multipletiming calibrations has proved useful in determining multiple, widely varyingdecay rates.(72) A similar approach can also be used in anisotropy analysis. Toresolve multiple rotations occurring on widely differing time scales, multipleanisotropy experiments are performed under differing timing calibrations.These multiple timing calibration experiments can be combined into a singleglobal analysis in which one finds an internally consistent set of rotationalcorrelation times. In this way, one is not forced to fix various rotationalcorrelation times when performing a sequential analysis over differing timedomains. Also, instead of using very large single data sets, obtained overmany channels using a timing calibration which represents a compromise toresolve both fast and slow motions, it may be much more convenient toobtain a series of small sets collected at multiple timing calibrations.

The above are simply two examples in which the data analysis metho-dology has directly influenced the data collection aspects of anisotropyexperiments. In both cases, rather difficult “hardware”-related problems havebeen solved using a “software” approach.

5.6.3. Associative versus Nonassociative Modeling of Anisotropy

The problem of associative versus nonassociative decay is related to amolecular description of the species present in the experiment. This problemarises when a multiexponential decay of the fluorescence intensity is combinedwith a multiexponential decay of the fluorescence anisotropy. In the non-associative case, the set of correlation times in the anisotropy decay iscommon to each of the relaxation times in the fluorescence decay [i.e., allcross-terms in the formation are nonzero]. In the associativecase, each particular lifetime may be “associated” with a specific rotationalcorrelation time.


Both types of associations can be expressed mathematically by thefollowing expressions(1,74):

The matrix L is defined as

It is very difficult to discriminate associative from nonassociative decaybased on a single anisotropy experiment, as equally good fits result in manycases. The model-testing capability of global analysis, however, can be helpfulin the discrimination of these two cases. Associative and nonassociativemodels behave very differently when one performs a global analysis of thepolarization decays collected at multiple emission wavelengths. For instance,when the species of the polarization decays collected at multiple emissionwavelengths. For instance, when the species in an associative case havedifferent emission spectra, one can perform a global analysis by linking

(which may be independent of emission wavelength for a given model, e.g.,ground state heterogeneity) and only allowing the to vary. Using thislinkage, proved to be very sensitive to discriminating (in a statisticallyjustifiable manner) associative versus nonassociative behavior.(23,74,72)

5.6.4. Anisotropy Decay-Associated Spectra (ADAS)

Even in the absence of heterogeneity in total intensity decay, steady-statespectra can still be resolved into underlying components that the variousspecies have different anisotropy decays.(75) Consider a fluorescent probewhich partitions into a lipid bilayer having two distinct phases (a “fluid”phase and a “gel” phase). The emission spectra of the dye in the two phasesmay be distinct, yet the fluorescent lifetimes may be very similar. One wouldlike to be able to decompose the emission spectrum into the contributionsfrom the gel and fluid phases.

The different spectra can be obtained from a data surface that isconstructed by recording anisotropy decays across the emission band. Thiscan be demonstrated as follows. In an associative model, the parallel polarizedintensity component for species is given by


where is the total intensity decay of species k, factorizingspectral- and time-dependent parts. Similar expressions can be given for otherpolarized intensities. The desired spectra can be recovered fromdata of this type, even in the case that (i.e., no total-intensity decayheterogeneity), because the multiple species are associated with differentanisotropy functions.

To resolve these anisotropy decay-associated spectra (ADAS), one mayperform a global analysis of this data surface by linking the fluorescencerelaxation times and associated anisotropy functions acrossthe emission band, allowing only the spectral components tovary. A plot of the individual as a function of emission wavelength thenrepresents the emission spectra associated with a particular anisotropy decayfunction and hence the ADAS. ADAS have been successfully obtained forfluorophores incorporated into rotationally distinct environments.(75– 77)

5.6.5. Multidye Global Anisotropy Decay Analysis

The rotational motion of a protein is often monitored through the use ofextrinsic chromophores covalently attached or adsorbed to the protein.However, due to the geometry of dye binding to the protein, the measuredanisotropy from the extrinsic probe may preferentially emphasize particularaspects of the protein motion. For instance, extrinsic dyes whose dipoles alignpreferentially along the long axis of the protein tend to yield longer rotationalcorrelation times since they are primarily sensitive to rotation about theshort axes. The net result is that multiple dyes attached to a single proteinmay yield widely different rotational correlation times. An explicit example isthe change in mean rotational correlation times observed with variousfluorophores bound to horse liver alcohol dehydrogenase (HLADH).(25) Sinceeach particular dye may bind to HLADH with a different direction of theabsorption/emission oscillators with respect to the principal diffusion axes,each dye will report a mean rotational correlation time which will be adifferent weighted average of the correlation times of the protein. In the caseof HLADH, mean values were found to differ greatly as a function of theparticular reporter dye that was bound. A global analysis was then performedon the multiple dye experiments, in terms of resolving an internally consistentset of rotational correlation times over the different experiments. The multiplecorrelation times obtained from this global analysis were in reasonableagreement with those predicted from the crystal structure. In this way, theanisotropy character of the overall motion of a globular protein was uncovered.

5.6.6. Distributed Lifetimes and Distributed Rotational Correlation Times

The polarized intensity of a sample containing several species is just thesum of the individual polarized intensities. A natural extension to multiple


discrete lifetimes and rotation rates is distributions of these decay parameters.A general, empirical fitting of fluorescence anisotropy decay data surfaces isaccomplished by distributing the relaxation times in the expression for totalfluorescence and anisotropy. The parallel-polarized intensity component isthen given by

where is the multidimensional probability associated with theangular dependence, rotational dependence, and lifetime dependence of thedistribution function. For the nonassociative case, one can replacewith

Distributed rates may be expected for fluorophores in anisotropic mediafor a variety of reasons; the simplest would obviously be simply having a veryheterogeneous rotational environment. We will also show, however, that whenfluctuations in the environment occur during the lifetime of the excited state,a distribution of rotational rates will also result. A physical model for fluores-cent probes in membranes which leads to distributed rotational correlationtimes will be presented in Section 5.6.8.

5.6.7. Multiexcitation Anisotropy Experiments

Anisotropic rotations in small molecules were first determined by Weberet al.(78,79) It is often very difficult to detect anisotropic rotation of a moleculebecause the rotational correlation times do not differ by a large extent. As aresult, the analysis of a single anisotropy experiment often yields only theaverage of the multiple correlation times. Barkley et al.(28) described the use ofmultiple excitation wavelengths, which alter but yield the same to aidin unraveling complex anisotropic rotations. This useful overdeterminationarises because the are only functions of the relative orientation of thetransition moments in the molecular frame [see Eqs. (5.38)–(5.41)]. If onedoes not obtain the same values upon excitation into different absorptiontransitions, anisotropic rotations must be considered. The relaxation times ofthe anisotropic motion can be obtained accurately by global analysis of theanisotropy decays at different excitation wavelengths by linking the correla-tion times. The different preexponential factors at the various excitationwavelengths result in a different weighting of the correlation times. Thisapproach has been applied to recover the anisotropic motion of perylene and9-aminoacridine.(28,23) Perylene is very anisotropic rotator, so the use of


global analysis simply provides a rigorous way of recovering the most inter-nally consistent ratio of diffusion coefficients for the moleculeIn the case of 9-aminoacridine,(28,23) however, it was impossible to determine fromsingle-experiment analysis whether this molecule was an anisotropic rotator.However, with global analysis, one could determine, in a statistically signifi-cant manner, that this molecule was not an isotropic rotator and recover

Recent work (performed in the frequency domain) suggests thatthe rotational behavior of perylene may be even more complicated.(80)

5.6.8. Example of Distributed Rotations: Fluorophore Rotations Gated by PackingFluctuations in Lipid Bilayers

Davenport et al.(8l,82) investigated the fluorescence anisotropy of thelong-lifetime probe coronene in liposomes. The unusual behaviorobserved with this probe led to the suggestion that the emission anisotropy issensitive to “packing fluctuations” in lipid bilayers.

Two levels of modeling, one discrete and one distributed, were performed.For the discrete model the membrane system was treated in terms of a simplegel–fluid equilibrium, characterized by two rate constants, andThese rate constants represent the rate of lipid exchange from the gel to thefluid compartment and from the fluid to the gel compartment, respectively.It is assumed that little rotation of the probe takes place in the gel state and,also,

Within this framework, probe molecules found initially in gel arecharacterized by an “effective” rotational correlation time (83)

The emission anisotropy is “gated” by the transfer of probe into the fluidstate. Those probes will thus rotate only by going through a “local melting”process. If all probes participate in this equilibrium, the decay of the emissionanisotropy can be expressed as

where andWhen is large, can take the place of found in previous models.

One may extract and terms as a function of temperature. Trends forthe and terms can be analyzed in a sequential manner or by globalanalysis. Examination of these intermediate parameters revealed behaviornot consistent with this model framework. Thus, an alternative frameworkwas developed which predicts a distribution of melting rates. Jahnig(84) hasdiscussed the use of a Landau model for packing fluctuations. A model basedon rotation “gated” by such packing fluctuations was proposed. In this model,


the free energy potential shapes for dipalmitoyllecithin(85) were used byDavenport et al.(81) to model the emission anisotropy behavior of coronene inliposomes as a function of temperature. A target parameter space, consistingof free energy profiles, a gating factor, and a diffusion coefficient, was estab-lished for the analysis. Simulation of the observed emission anisotropy ofcoronene as a function of temperature was performed. This use of Landautheory provided a method whereby functional “linkages” could be constructedin a physically meaningful manner over the temperature axis.

5.7. Error Analysis and the Identifiability Problem

The result of a nonlinear fit to data is a series of numerical valuesassociated with the fitting parameters of the proposed model, the value of thereduced chi-square, a surface of residuals, and possibly a surface of auto-correlations. The recovered numerical values of the fitting parameters areessentially useless unless some type of error analysis is performed. Historically,only the errors associated with the recovery of the fitting parameters havebeen examined. What we will describe in the following sections is theexamination of two types of error:

1. Are the recovered parameters unique, or can they take on widelydiffering values and still yield the same identical response (and hencechi-square)?

2. What is the range of possible values that these parameters couldassume and still yield chi-square values consistent with that obtainedat the minimum?

The first item is directly related to the identifiability of the model, while thesecond addresses error analysis within a model. A description of both theidentifiability problem and the error analysis problem will be discussed inthis section. Although the error discussion will be completely general, theidentifiability study will focus only on nondistributed compartmental models.

5.7.1. The Identifiability Problem

The concept of identifiability (in terms of typical fluorescence decayexperiments) can be stated very simply: Does the series of experiments beingperformed contain enough information to provide a unique solution for theset of global fitting parameters (within the model structure) which is beingapplied to the data? Figure 5.9 schematically depicts an identifiable and anonidentifiable set of models. In the nonidentifiable model “A,” various sets of


numerical values for the fitting parameters (denoted as specific values2, 3, …, N) each yield identical impulse response functions. Therefore, onecould never differentiate these sets of parameters from each other, and hencethis model is said to be nonidentifiable. In contrast, all specific sets of numeri-cal values associated with model “B” yield unique impulse response functions.Therefore, this model is termed identifiable. Note that no statement is being


made concerning how “closely” the impulse responses resemble one another,only that they are not identical. From this figure, it is apparent that one doesnot need to discuss the concept of identifiability if one only fits each experi-ment in terms of the parameters of an impulse response function ( and ).However, with the use of global analysis, problems are often reparameterizedso that the actual fitting parameters are no longer a direct description of theimpulse response function. In these cases, it is important to determine whetherthe recovered parameters provide a unique mapping to a set of impulseresponse functions.

Consider the following simple two-state excited state reaction withand allow only

compartment 1 to be populated in the ground state (can be generalized to anycombination of initial ground state populations). The above system isexamined at two different emission wavelengths toward the blue and red edgesof the emission spectrum The impulseresponse functions for these two experiments are

The above hypothetical data set can be analyzed in terms of a sum ofunrelated exponentials. Depending on the noise in the data, one should beable to recover a total of four amplitudes and two lifetime terms. Fromexamination of the results of this empirical analysis, it should be evident thata two-state excited state reaction is occurring (note: negative amplitude onred edge).

In this case, the “target” parameters of interest are probably not theamplitudes and lifetimes, but rather the rate constants for interconversion,the decay times, and the spectra associated with the two emitting species.One therefore reparameterizes the problem, in terms of the “target” fittingparameters: When performing the globalanalysis in terms of these target parameters, it will be found that there is aninfinite set of target parameters which will yield the two impulse responsefunctions given by Eqs. (5.55) and (5.56). One element of this set is

and(compare with above synthesized values). Depending on the

type of error analysis performed on these target parameters, one would findvery large error bars associated with these values. These large error bars arenot due to “noise” in the observed data (infinitely accurate data would haveexactly the same error limits). Instead, these errors result from the fact thatthere is no unique set of target parameters which map into the impulseresponse functions (Eqs. 5.55 and 5.56). However, if one can experimentallyalter either or (by changing pH, concentration, etc.),


then analysis of the combined data surface of fluorescence versus versuspH (or concentration) versus time/frequency will yield a unique solution to theproblem (assuming knowledge of the ground state absorbances).

5.7.2. Identifiability Study Using Laplace Identifiability Analysis

The above example (Section 5.7.1) is a single example of a model whichis not identifiable. Identifiability analysis attempts to predict what combi-nation of experiments is required to uniquely determine a particular modelfrom a given set of data. There are numerous techniques which have beendeveloped (mostly within the field of systems theory and compartmentalanalysis) to determine the identifiability of a particular model. What will bepresented below will be one particular identifiability approach, to allowreaders to obtain a better understanding of how to determine the “informationcontent” of a given set of fluorescence experiments.

In the case of a linear problem, one can determine the possibility of aunique solution to a problem, by simply counting the number of unknowns(to be determined) and the number of independent equations. If the numberof unknowns exceeds the number of independent equations, then no uniquesolution is possible. This basic idea can be extended to examine theidentifiability of particular compartmental models of fluorescence decay datathrough the use of the Laplace transform. It should be emphasized that thefollowing analysis is completely independent of the noise on the data.

The basic systems theory equation (which is utilized by the globalanalysis program for all discrete/distributed compartmental models) can bewritten:

[see Eq. (5.8) for definition of terms]. The observed fluorescence can bewritten as

where F is the observed fluorescence, A is an output matrix of spectralcontours, and X is the time course of each fluorescent species in solutions[vector form of Eq. (5.9)]. Taking the Laplace transforms of Eqs. (5.57) and(5.58) yields (given a zero initial state):


where s is the Laplace transform parameter. Solving for the Laplace transformof the concentration of each individual species yields

(given that the inverse exists). The Laplace transform of the fluorescence cantherefore be written:

The factor is termed the transfer function matrix because itrelates the input of the system with the output of the systemThe analytical form of the transfer function will be useful in deciding thenumber of unknowns which can be determined from a particular set ofexperiments.

Consider the very simplified case of a two-state excited state reaction inwhich only species 1 is excited, and only species 1 is observed. This wouldcorrespond to and The transfer function is therefore

Expanding out this equation and substituting the explicit rates into yields

The usefulness of Eq. (5.64) is that it predicts what combinations of rateconstants will yield identical transfer function matrices, and therefore identicalimpulse response functions (see Figure 5.9). From Equation (5.64), one hasthat whenever the following relationships are satisfied, identical impulseresponse functions will occur:

Therefore, from our model (two-state excited state reaction, known inputand output vectors) one can derive only three independent relationships (Eqs.5.65–5.67), but there exist four independent fitting parameters (the four rateconstants). Therefore, one has determined a priori that a global target analysisof this particular experiment directly in terms of the rate constants of thesystem is not possible. This result is completely independent of the accuracyof the collected data.


One can proceed with this type of approach to examine the identifiabilityof the two-state excited state reaction, but in this case consider the combina-tion of two fluorescence experiments performed by altering one of the rateconstants for interconversion (e.g., changing the pH for excited state protontransfer). In this case, an additional transfer function is generated, with

altered to where [M] is the concentration of interactant. Thisadditional transfer function generates a new set of independent equations(similar to those above). By combining the number of independent equationsfrom experiments 1 and 2, one can determine that the number of unknownsin the system is now five.† The number of independent equations generatedfrom this combination of experiments is also five. From these two experiments,then, one has a set of five nonlinear equations in five unknowns. Since theseequations are nonlinear, one is not guaranteed that there is a solution;however, the possibility exists that this system is identifiable. In fact, we knowfrom the study of the excited state reaction of (see Section 5.5.1)that this type of system can be uniquely recovered.

More complex models with additional compartments are even less likelyto be identifiable than the two-compartmental model described above, asthey allow for many different connectivities between the compartments. Thenumber of compartments used in the analysis, of course, cannot exceed thenumber of relaxation times which can be resolved by the global analysis.‡

If the dependencies of the rate constants on concentration are unknown, thiswill create additional identifiability problems.

The structural identifiability problem is well known in the field of com-partmental modeling(86,44,87,88) and is still a subject of intensive study. For adetailed example of the use of identifiability analysis applied to fluorescencedecay data, see the study of Ameloot et al.(27) From this study, no single setof experimental conditions was found to uniquely determine the systemparameters for all types of two-state excited state reactions. In all cases,at least two different concentrations of interactants are needed. This is anecessary (but not a sufficient) condition for identifiability. The set of fluo-rescence lifetimes alone may, in some cases, resolve all of the rate constants.In these special cases, neither normalization between experiments nor know-ledge of the absorption vector is required. In most cases, the informationcontent of the preexponentials associated with each particular lifetime must beincorporated. Multi-emission or multi-excitation wavelengths may also beessential to obtain complete identifiability. Identifiability may also depend onthe particular values of the system parameters.

Fluorescent systems may also be “partially” identifiable. When only a

† If the effect of [M] on the rate constant is known, then there will be only four unknowns.‡ Hence, simple linkages over emission wavelength are still useful to identify the dimensionality

of the system.


single-ground-state compartment is excited directly, the SAS of the firstcompartment is always uniquely identifiable. This SAS can be obtained byjust summing at each emission wavelength the preexponentials resulting fromthe global analysis in terms of linked lifetime.(57) If several compartments areinitially excited, this sum reflects that initial mixture. This result also followsdirectly from an identifiability study using the technique described above.Upon performing that analysis, the SAS of compartment 1 is always uniquelyisolated from the other sets of coupled equations and, hence, can always beuniquely recovered.

Another example of a partially identifiable case is taken from ananisotropy study. It can be shown that fluorescence anisotropy data for aprobe in an anisotropic medium can be analyzed in terms of the rotationaldiffusion coefficients and the order parameters of second and fourth rank, thatis, and In some cases, there exist two values for at thesame values of the rotational diffusion coefficients and of Thisambiguity can only be resolved by performing the same kind of studies onoriented samples. Oriented sample measurements excited and measured atvarious angles in space may provide an ideal setting for a global targetanalysis of complex rotational motion.

Recent work on identifiability by Eisenfeld(90) has shown that byexamination of the determinant of (where J is the Jacobian matrix of thetransformation of the fitting parameters to the observable parameters), onecan determine whether a particular model is identifiable or not. Examinationof the rank (and correlation analysis) of the elements of the matrix “C”[Eq. (5.19) with can also yield important identifiability information.(91)

This recent work should provide the necessary framework to integrategeneral-purpose identifiability tests into the analysis software. Additions ofthis type are currently being entered into the global analysis software.

Therefore, when a global analysis in terms of a specific set of targetparameters is performed, the identifiability of this particular parameterizationshould be considered. This type of analysis is relatively “new” to the field offluorescence spectroscopy, but the insights which this technique can offer makethese types of studies very important. In the case of excited state reactions,experiments must be performed which can alter the equilibrium betweenthe various species. This condition, however, does not guarantee a uniquesolution. For many experimental configurations, the identifiability problemwill be very difficult to solve analytically, and one must resort to some typeof rigorous error analysis to test for nonuniqueness.

5.7.3. Error Analysts

The values of the parameters recovered by a particular analysis shouldalways be reported with their corresponding errors. The level of rigor on


which the error analysis is performed depends on only a single factor: the sizeof the region of the error surface which is explored.† One can classify thevarious error analysis algorithms as follows:

1. Linear approximation: None of the error surface is directly explored.The information contained in the curvature matrix at the chi-squareminimum is utilized to calculate the uncertainties in the recoveredparameters, assuming no correlation between the various fittingparameters.

2. Unidimensional search: Directed searches along each parameter axis,not allowing any of the other fitting parameters to vary.

3. Directed search: Directed searches along the eigenvectors of thecurvature matrix at the minimum.

4. Exhaustive search: Directed searches along each parameter axis,allowing all other parameters to vary so as to obtain a minimumchi-square at each point.

Each of these error analysis methods is diagrammatically depicted inFigure 5.10 (for a two-parameter analysis of “P” and “Q”).

In the linear approximation, the errors on the recovered parameters areestimated utilizing the square root of the diagonal elements of the inverse ofthe curvature matrix see Eq. (5.18)] at the chi-squareminimum. This type of error analysis is strictly valid only for linear modelsand assumes that there is no correlation between the individual fittingparameters. This methodology always produces a symmetric error resultand should always be regarded with a large amount of skepticism. These errorestimates will always predict the smallest amount of uncertainty (see shadedregion in Figures 5.10 and 5.11).

The unidimensional search algorithm simply calculates the observed chi-square by sequentially altering the recovered fitting parameters along eachindependent axis. None of the other fitting parameters are allowed to varyduring this process (no correlations between the parameters are taken intoaccount). The region of the error surface which is examined by this approachis schematically represented as the lines A–B and C–D in Figure 5.10.The error bars recovered from this analysis may be asymmetric. The majorlimitation in this error analysis technique is that no correlations between thefitting parameters are allowed.

In the directed search algorithms, an attempt is made to take into accountsome of the correlation between the fitting parameters. By calculating the

†Of course, knowledge of the chi-square surface over the entire domain of the fitting space wouldprovide all of the information necessary to establish the error bounds and identifiability.However, this type of error analysis is usually not possible (too computationally intensive), andone needs to resort to examination of the chi-square surface along particular axes.


eigenvectors of the curvature matrix at the minimum chi-square, one effectively“rotates” from the fitting parameter frame to a frame of reference directedalong the regions of greatest covariance between the fitting parameters.Directed searches in these directions can therefore utilized to examine theerrors in the recovered parameters. These directions are schematically depictedas the lines E–F and G–H in Figure 5.10. Note that asymmetric errorestimates can be obtained from this analysis, and by rotating to this particularframe, some of the correlations between the individual fitting parameters aretaken into account.

In the exhaustive search method, one systematically alters the ith fittingparameter at a series of values and performs a complete nonlinear minimiza-tion, allowing the remaining n–1 parameters to “adjust,” so as to minimizechi-square. One then records the series ofminimum values possible over aparticular range of the ith fitting parameter. This process is schematicallyrepresented in Figure 5.10 as a series of arrows (i.e., minimizations) originatingat regular intervals along the various parameter axes, which subsequentlyrelax to the region of lowest possible chi-square given this particular value ofthe ith parameter. This method of error analysis takes into account all of thehigher order correlations which may exist between a given set of fittingparameters. This method also provides the maximum possible variation of afitting parameter which is consistent with a particular chi-square range.

A flow chart for the exhaustive search process can be written as follows:

1. Denote the fitting parameter vector at the minimum chi-square value

2. Call an F-statistic-generating routine.(92) This routine, when given aspecific confidence level, degrees of freedom in the analysis, andwill calculate a which represents a value greater thanwhich one should use to determine when one has searched out “farenough” along a particular axis.

3. Initialize delta to some small value [e.g., delta = std. errorj (seeabove)]:

4. Perform nonlinear analysis of all of the remaining fitting paramtersexcept with the element fixed at the above value. Set equalto the minimum value of this particular fit.

5. If (within a specified number of decimal places), thenoutput the current value of and stop. This value of the jthfitting parameter will represent the farthest value that this parameter

†For clarity, only the negative “P” and “Q” exhaustive search calculations are shown inFigure 5.10.


could take and still be consistent with the data at a particularconfidence level. If is still less than then increase thedelta parameter increment and go to step 3.

The methodology requires a whole series of nonlinear minimizations to beperformed. It is also very easy to implement and is the method currentlyutilized by the global analysis program at the LFD. This methodology willalso establish whether a particular model is identifiable or not. For noniden-tifiable models, the error surface will be completely flat along a particularset of fitting parameters. Note that the error bars recovered from this type oferror analysis may certainly be nonsymmetric (i.e., not a simple representa-tion). A comparison of the error bars obtained using the linear approximationand the exhaustive search approach is shown in Figure 5.11. It is well knownthat the uncertainty in a recovered rotational correlation will be at a minimumwhen the rotational correlation time has the same value as the total-intensitydecay lifetime. In Figure 5.11, this result is also found numerically. As theratio of rotational rate to fluorescence lifetime becomes large theuncertainty in the recovered should also become very large. With linearlyapproximated error, this increased uncertainty occurs very slowly. The morerigorous error analysis, however, reveals a much more realistic representationof the actual uncertainties in the recovered rotational correlation time. Notealso that the recovered error bars are very asymmetric, in exactly the mannerwhich would be predicted: when the positive confidence interval is verylarge whereas the negative confidence interval is still bounded; whenthe exact opposite occurs. This result reveals that exhaustive search erroranalysis can not only assist one in determining a realistic representationof the uncertainties in the recovered parameters, but can also assist inexperimental design. Although this example is rather obvious (and wellknown), there may certainly be cases in which it is not at all obvious whichset of independent variables will be important to alter in order to reduce theerror in a given set of parameters. Examination of the asymmetry in therecovered fitting parameters can often reveal clues concerning which of theseindependent variables are important.

5.8. Conclusions

One of the chief “complaints” concerning the use of global analysis is thatit is model dependent. This is certainly true, and application of a physicalmodel which is inappropriate for a given data type will certainly yieldquestionable results. However, as far as the analysis is concerned, the globalapplication of a particular model over a multidimensional data surface is as


statistically rigorous a test of a particular model as can be performed. If theapplied model appears to be consistent with a given data surface (all of theerror criteria are satisfied), then there is nothing additional that can be doneto test this particular model. If there are two (or more) physical models whichappear to be appropriate for a given data surface, and both of them have beensuccessfully applied to the data in a global manner, then one must concludethat there is not enough information in this particular data surface to dis-criminate between these two models. To discriminate between these twomodels, the global analysis programs (as described in this chapter) allow theresearcher to easily examine, in a quantitative manner, new experimental axes,which may be capable of discriminating between these alternative models.These additional axes can be immediately incorporated into the analysis ofthe (ever-growing) data set, until (hopefully) only a single model prevails.

How does the global analysis approach relate to other numerical analysistechniques, such as the application of singular-value decomposition (SVD) tothe analysis of data of the form F(x, y)? A classic application of this type ofanalysis is when x = time, and y = excitation or emission wavelength. SVDacts to separate the spectral- and time-dependent parts of the data matrix.The spectra are decomposed into the minimum number of independent species(very useful information). The time-dependent part (a much smaller data set),now separated from the rest of the data matrix, can be analyzed usingstandard nonlinear least-squares techniques. This data reduction techniquehas shown itself to be valuable for this type of data. The global analysistechnique, however, can operate in exactly the same manner. With the globalanalysis, the time and spectral parts of the problem are not decomposed; theyare simply determined simultaneously in a single analysis. The SVD approachfor this type of data may certainly be less computationally intensive, but thenet result of both types of analysis should (in theory) be identical. The realdifference between the SVD and global analysis approach results when higherdimensional data surfaces are to be examined.

Data consisting of more than two dimensions cannot be operated on ina single step using the SVD approach. If the fluorescence data obtained alongthe various experimental axes all appear linearly (in the observed data),various multilinear models can be applied.(93) These types of analyses are ina very early stage of development (as applied to fluorescence data), yet theydo have the potential of being very valuable methods of analysis. In a globalnonlinear analysis, however, one can analyze as many dimensions as desiredin a single analysis, regardless of whether they relate linearly to the observedfluorescence or not. Just as importantly, since the analysis is being performedon the original data surface, there are no problems concerning how toproperly weight the data in the direct global analysis.

Therefore, one should view some of the alternative schemes (e.g., SVD,factor analysis, multilinear analysis, etc.) as useful analysis tools for particular


data types. By specializing in particular data types, one may obtain both areduction in computational time and, in some cases, additional informationwhich might not be immediately apparent when performing a global analysis.The global analysis methodology, as described in this chapter, does not haveany qualifiers attached to the types of data which can be combined or themodels which can be applied (other than those specific to all nonlinear least-squares analyses). As such, it is a general-purpose analysis methodology,which can be applied to a wide variety of fluorescence data. These featuresmake global analysis, with the “target” approach, an ideal vehicle for testingmodels and recovering fitting parameters. An important advantage of thenumerical procedures discussed here is that the original data are testeddirectly against the final model of interest. Multistep analysis is completelyeliminated.

Do the global analysis procedures described here provide a “black box”that can always be used to obtain system parameters of interest? Certainlythis is not the case. A reasonably good global fit over a large numbr ofdifferent experimental data sets may obscure a poor “local” fit for a particularexperiment(s). The regions where a global fit is not satisfactory can provide asmuch useful information as those areas in which the fits are very good. Theseregions should not be overlooked. Once trivial instrumental artifacts areeliminated as a source of the mismatch, the globally applied models should bereevaluated. The ability to disprove a particular model is often as importantas the determination of the recovered fitting parameters.

The exploration and development of new physical models (and theories)which will allow additional sets of overdetermination axes to be used in aglobal analysis is an important direction for future application. Determiningthe “information content” of these high-dimension data surfaces using theidentifiability approach may help in establishing which sets of experimentalaxes are worth investigating. The application of the widest possible variety ofphysical models to the highest dimensional data set that can be obtained stillprovides the experimenter with the best overall chance of understanding theobserved fluorescent system.

Acknowledgments

Much of the inspiration and many of the ideas in this work evolvedthrough interactions with scientists who have spent time in (or around) thelaboratories of Ludwig Brand and Enrico Gratton. These scientists include(from the Brand lab) Drs. D. W. Walbridge, R. P. DeToma, G. Ackers,B. Turner, L. Davenport, R. Dale, M. Barkley, A. Kowalczyk, J. B. A. Ross,W. W. Laws, M. K. Han, and M. Pritt and (from the Gratton lab) Drs. R.


Alcala, C. Royer, D. Piston, B. Feddersen, W. W. Mantulin, and N. Silva. Wethank Julie Butzow (of the LFD) for expert assistance in figure preparation.

JMB gratefully acknowledges support from the Lucille P. Markeyfoundation. JMB is a Lucille P. Markey scholar in biomedical science. LB issupported by NIH grant GM 11632. Laboratory for Fluorescence Dynamics(LFD) is supported jointly by the Division of Research Resources of theNational Institutes of Health (RR03155-01) and the University of IllinoisUrbana–Champaign.

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76. L. Davenport, J. R. Knutson, and L. Brand, Anisotropy decay associated spectra: 2. DPH inlipid bilayers, Biochemistry 25, 1811–1816 (1986).

77. L. Davenport, J. R. Knutson, and L. Brand, Studies of membrane heterogeneity usingfluorescence associative techniques, Faraday Discuss. Chem. Soc. 81, 81–94 (1986).

78. G. Weber, Theory of differential phase fluorometry: Detection of anisotropy molecularrotations, J. Chem. Phys. 66, 4081–4091 (1977).

79. W. Mantulin and G. Weber, Rotational anisotropy and solvent–fluorophore bonds: Aninvestigation by differential polarized phase fluorometry, J. Chem. Phys. 66, 4092–4099(1977).

80. J. R. Lakowicz, I. Gryczynski, and H. Cherek, Resolution of three-rotational correlationtimes for perylene by frequency-domain fluorescence spectroscopy, Biophys. J. 53, 87a (1988).

81. L. Davenport, J. R. Knutson, and L. Brand, Time resolved fluorescence anisotropy of mem-brane probes: Rotations gated by packing fluctuations, in: Time-Resolved Laser Spectroscopyin Biochemistry (J. R. Lakowicz, ed.), Proc. SPIE 909, 163–169 (1988).

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6

Fluorescence Polarization fromOriented Systems

Thomas P. Burghardt and Katalin Ajtai

6.1. Overview

The polarized emission from dipolar emitters rigidly embedded in theorganized molecular components of an assembled system can indicate some ofthe lower angular resolution features of the emitter angular probability density.This information is useful to indicate changes in the angular probabilitydensity (referred to from now on as N) as the result of perturbations.(1) It isgenerally less useful in ascertaining a detailed shape of N due to the limitedamount of angular information provided by dipolar emission, although thisshape determination can be accomplished under some circumstances.(2) Wesummarize in this review a general method to quantitate N in terms of aninfinite set of irreducible order parameters. We show that the number andrank of the order parameters. We show that the number and rank of the orderparameters that are detectable with a particular experimental form of thepolarized emission technique indicate the extent of the angular resolutiontheoretically obtainable by the technique.

The experimental techniques discussed in this chapter include time-resolved fluorescence depolarization from mobile components of a biologicalassembly and steady-state fluorescence polarization from immobilized orslowly moving components of a biological assembly. The time-resolvedtechnique detects higher resolution features of N. A closely related technique,electron spin resonance (ESR) of spin labels in ordered assemblies, detects thespin angular probability density with a resolution that exceeds that obtainablefrom fluorescence techniques. The mathematical relationship between ESR-and fluorescence polarization-detected angular densities is discussed, and an

Thomas P. Burghardt and Katalin Ajtai • Department of Biochemistry and MolecularBiology, Mayo Foundation, Rochester, Minnesota 55905. K. Ajtai was on leave from theDepartment of Biochemistry, Eötvös Loránd University, Budapest, Hungary. (Manuscriptsubmitted January 1, 1988.)


307

308 Thomas P. Burghardt and Katalin Ajtai

example using ESR data to supplement information from fluorescencepolarization is described.

The extraction of useful information from an emitted signal dependsultimately on the use of an appropriate probe molecule. The biochemicalmethods for placing fluorescent probes specifically and rigidly on molecularcomponents of biological assemblies are well developed. Equally important,biochemical methods for verifying the position and specificity of the probesare also elaborate. We summarize the principles of these methods to givesome idea of the biochemical difficulties associated with useful probe studies.

In each of the subsections of the main body of this chapter (Section 6.2),we have included examples, usually describing the experimental realization ofthe ideas presented in the subsections. In this way we wish to convey themessage that our general approach to the investigation of angular order inbiological assemblies is practical. Although all of the examples are taken fromprobe studies of muscle fibers, reference is made to other systems of interestsuch as lipid membranes or supported lipid monolaye-rs.

The investigation of angular order in muscle fibers is the focus ofthe applications discussed in this chapter (several good reviews of musclestructure exist(3–5)). The muscle fiber consists of two interdigitated proteinfilaments that interact to produce fiber shortening during muscle contraction.The thick filament (named for its appearance under the electron microscope)is composed of spatially assembled dimers of myosin proteins, and the thinfilament is two strands of polymerized actin proteins wound in an -helix. Themyosin molecules contain a globular enzymatic (ATPase) head region thatinteracts cyclically with an opposing thin filament while hydrolyzing ATPduring fiber shortening. Since a model for this interaction of the myosin headwith the actin was first proposed to involve a rotation of the head moietywhile attached to actin,(6,7) intensive study of the angular disposition of thehead has occurred. The careful determination of the angular disposition of thehead is clearly an important experimental objective, and the reliable data nowpublished are ambiguous enough to have caused the production of a confusionof data interpretations. This ambiguity is likely due in part to the complexityof the system and more importantly to the resolution limitations imposed bythe experimental techniques employed. This circumstance in muscle researchcaused us to undertake the systematic investigation of the angular resolutionin the fluorescence techniques and to develop both methods to reduce dataambiguities as well as a unified approach to the description of N that wouldallow relevant data from different techniques to be compared.

6.2. Theory and Application

The theoretical description of angular order of components of a biologicalassembly is centered on a model-independent approach in which we expand

Fluorescence Polarization from Oriented Systems 309

the angular probability density in terms of an appropriate set of completeangular functions with unknown expansion coefficients. In Section 6.2.1.we discuss this idea in detail. In Sections 6.2.2 and 6.2.3 we discuss someexperimental fluorescence techniques for measuring the angular probabilitydensity, N. In Section 6.2.4 we compare the fluorescence techniques with ESRtechniques for measuring N and show an example of the use of ESR data incombination with fluorescence data to form a more accurate description of N.In Section 6.2.5 we briefly discuss the biochemical techniques for attachingcovalent probes to the components of the biological assembly.

6.2.1. The Angular Probability Density N

The probability density function N describes the angular distribution ofa particular set of molecule-fixed coordinate frames in a biological assemblyof molecular components, relative to a laboratory-fixed coordinate frame. Anymolecular frame is related to the laboratory frame by a rotation. We chooseto describe this relationship using the Euler angles and which aredenoted collectively by It is convenient to expand in terms of Wignerfunctions that form a complete, orthogonal basis set of functions onthe domain and denoted by such that

The parameter is called an order parameter of rank j, and it representsthe contribution to N of the (j, m, n)th orthogonal component of the set ofbasis functions. The individual are complex and can be negative. N isreal and is greater than or equal to 0 as required for it to have meaning asa probability density.

The ’s are orthogonal on such that

where * indicates the complex conjugate, is the Kronecker delta, andis the integration element The orthogonality property ofEq. (6.2) shows the normalization of N, such that implies

The order parameters are unknowns that we measure experimen-tally. The rank of is given by j. The angular resolution of an experimentalmethod is theoretically limited by the rank and total number of the orderparameters detected. Consider, as an example, a system in which all of

310 ThomasP. Burghardt and Katalin Ajtai

the molecular frames have the unique polar angle and are randomlydistributed in the other Euler angles and The normalized angularprobability density for such a system is

where is a Dirac delta function. Equation (6.3) is inverted usingEq. (6.2), and the order parameters of this probability density are easilyshown to be

where is a Legendre polynomial. Shown in Figure 6.1 is the probabilitydensity computed using Eqs. (6.1) and (6.4) for and including inthe summation of Eq. (6.1) the order parameters and (the limitof resolution typical for fluorescence polarization; see Section 6.2.2). Forcomparison, we also show a probability density computed by including theorder parameters and (resolution more typical of


electron spin resonance) in the summation of Eq. (6.1). The plots demonstrateboth the best estimate obtainable by fluorescence polarization when describingthe probability density of Eq. (6.3) and the relationship between angularresolution and order parameter rank.

In the next section we derive the relationship between the orderparameters and the measurable fluorescence polarization signal.

6.2.2. Fluorescence Polarization in Homogeneous Space

In fluorescence polarization experiments a fluorescent probe is specificallyand rigidly fixed in the molecular frame of a component of the biologicalassembly. A propagating, linearly polarized, electromagnetic field excites theprobe. The fluorescent emission efficiency is detected as a function of one ormore parameters such as the polarization of the excitation or emission fields.The probability of a single probe absorbing light and subsequently emittingthe energy as fluorescence is denoted by K. K depends on the molecular frameEuler angles and can be expanded in terms of the Wigner functions just aswe did for N in the last section. We find

where the coefficients are calculated from the well-known expressiondescribing the interaction of a fluorophore with an electromagnetic field (seebelow). The fluorescence signal, F, for an ensemble of probes is given by

Substituting Eqs. (6.1) and (6.5) into Eq. (6.6) and using Eq. (6.2), we find

With known, a linear combination of the is determined by observing F.The expression for K appropriate for a probe in homogeneous space is

explicitly

where c is a constant, and are the unit electric dipole moments of thefluorophore for absorption and emission, respectively, E is the electric fieldvector of the excitation light, and v is the unit vector of polarization of theemitted light that is observed (using a polarizer).


The are calculated using Eqs. (6.2), (6.5), and (6.8) and thensubstituted into Eq. (6.7), and the summation is carried out. This calculationof F, using Eq. (6.7), is straightforward but tedious and has been done for thegeneral case, including corrections for high-aperture optics.(8) The readershould consult Ref. 8 for a slightly more detailed description of the calcula-tion. An important and general result for fluorescence polarization, however,is derived in this calculation where it is shown that the summation on jfor is limited to This result holds generally for fluorescencepolarization in homogeneous space and, as discussed in Section 6.2.1, setsa theoretical upper limit on the angular resolution obtainable from thetechnique.

We performed the summation of Eq. (6.7) for specific applications usinga symbolic manipulation computer program (SMP, Inference Corp.,Pasadena, California). This latter procedure is more convenient for all but themost simplified applications for the analysis of fluorecence polarization data.Below we apply this formalism to the study of

ethylenediamine (l,5-IAEDANS)-labeled myosin cross-bridges inmuscle fibers.

6.2.2.1. Example 1: Fluorescence Polarization from 1,5-IAEDANS-LabeledMuscle Fibers

Muscle fibers are made up of actin and myosin filaments that slide inopposite directions during muscle shortening. The myosin filament iscomposed of myosin molecules that have a globular head region that can beproteolytically cut from the rest of the molecule (subfragment-1 or S-l). TheS-l moiety spans the distance from the myosin to actin filaments and duringmuscle shortening is the site of the ATPase activity. A proposed model ofmuscle contraction holds that the S-l rotates while bound to the actin toproduce the filament sliding. Because of this model it is of interest to muscleresearchers to determine the angular probability density of S-l in the fiberwhen the fiber is subjected to various chemical and mechanical perturbations.

In this example single muscle fibers were washed in a buffer solutionand transferred to a quartz slide on the microscope stage. Extrinsic S-l,which binds rigidly to the actin filaments in the fiber at bare zones where theintrinsic myosin is not present, was specifically labeled with the fluorescentprobe 1,5-IAEDANS and allowed to diffuse into the single muscle fibers.

The experimental apparatus shown in Figure 6.2 consists of a lens thatfocuses an argon ion laser beam on the back image plane of theepi-illumination microscope holding the muscle fiber.(2) All of the optics inthe excitation light path were quartz to minimize background fluorescence.A Pockels cell rotates the laser polarization through 90° every 5 ms such thatthe polarization rotates from parallel to perpendicular with respect to the fiber


axis. The polarized fluorescence emission is collected in intervals synchronizedwith the laser polarization and quantitated by photon-counting electronics.The filter block (see Figure 6.2) contains a dichroic mirror and a barrierfilter. The dichroic mirror reflects the excitation light into the objectiveand transmits the longer wavelength emission collected by the objective.A rotatable film polarizer analyzes the polarized emission.

Known properties of the muscle fiber and a particular choice of themolecular frame greatly reduce the number of order parameters on which anappropriate angular probability density can depend. There is experimentalevidence indicating that a muscle fiber has azimuthal symmetry such that itis unchanged by an arbitrary rotation about the fiber axis.(1) If the fiber isplaced lengthwise along the lab frame z axis, then is the independentvariable for the probe angular probability density about the fiber axis, andazimuthal symmetry in the fiber requires N to be independent of This fact

314 Thomas P. Burghardtand Katalin Ajtai

is expressed implicitly in N by requiring in Eq. (6.1). It is wellknown that a 180° rotation relates opposite half-sarcomeres in the musclefibers. This symmetry property requires unless j–n = 0, 2, 4, .... Further-more, we choose to express the probe angular distribution relative to amolecular coordinate frame wherein the absorption dipole, , points alongthe z axis, and the projection in the x–y plane of the emission dipole,points along the x axis. This choice of molecular coordinate frames makesthe polar angle and the torsional degree of freedom of the molecular frameand renders the fluorescence polarization signal independent of orderparameters except when n = 0, ±2 (inspection of Eq. 17 in Ref. 8 verifiesthis result). This last restriction on n is not a result of fiber symmetry but ofthe properties of dipolar emission.

The theoretical fluorescence intensity, F, computed from Eq, (6.7) for themicroscopic application was shown to be(2)

where is the laser polarization and the emission polarization as shown inFigure 6.3, is a constant, and are the correction factors forhigh-aperture collection of polarized light,(9) and

The quantities appearing in Eqs. (6.12)–(6.14) are


Clebsch–Gordon coefficients,(10) and the parameter in Eqs. (6.13)–(6.17)is the polar angle of the emission dipole in the molecular frame. All otherquantities in the sum of Eq. (6.9) that are not explicitly mentioned inEqs. (6.10)–(6.17) are zero.

A fluorescence polarization experiment consists of the measurement offour intensities denoted by The symbols andmean parallel and perpendicular to the fiber axis while the first index on Frefers to excitation polarization and the second to emission polarization.Three ratios, defined below by Eqs. (6.18)–(6.20), combine the four intensitiesinto the quantities , which are independent of the absoluteintensity calibration of the optical instrument. The normalized ratios aredefined by


To solve Eqs. (6.18)–(6.20) for the unknown order parameters, we must alsoknow the optical correction factors and and the emission dipolepolar angle, The correction factors are computed from formulas derivedpreviously (see Eq. 20 in Ref. 9) and are and

(for glycerol immersion quartz objective with a numerical apertureof 1.25 and quartz interfaces). The angle between the absorption and emissiondipoles was estimated by measuring from 1,5-IAEDANS-labeled S-l inbuffer + 50% glycerol (1 :1 buffer to glycerol by volume, pH 7.0) solutioncooled to –20°C and solving Eq. (6.9) with Wefound for the excitation wavelength of 364 nm and filter block cutoffwavelength at 420 nm that

The restrictions on the order parameters for muscle fibers, shown pre-viously to require with j = 0, 2, and 4 and n = 0, causeEqs. (6.18)–(6.20) to depend only on the four parameters

and Because there are four unknowns with threeequations to constrain them, we must supply one new constraint. We show inSection 6.2.4 that we can obtain the additional constraint from the model-independent analysis of ESR spectra from muscle fibers.(2)

Fluorescence polarization experiments on 1,5-IAEDANS-labeled S-lbound to muscle fibers were performed on two states of the muscle fiber. Onestate, called MgADP, occurs when the fiber is bathed in buffer containingMgADP. In this state both the nucleotide and actin are bound to S-l. Thesecond state, rigor, occurs when the nucleotide is removed. With the help ofthe ESR data, as described in Section 6.2.4, we find for rigor fibers:


We summarize the results with a plot of the polar angular probabilitydensity found from by averaging over the Euler angles That is,

Plots of for fibers in rigor and in the presence of MgADP are shown inFigure 6.4. These plots of the probe orientation probability density indicatethat the probe rotates upon the binding of MgADP. To ascertain if this resultalso indicates that S-l rotates upon MgADP binding, we must undertakecontrol experiments to determine whether or not the probe remains rigidlyfixed in the S-l under these conditions. Generally, two time-resolved fluores-cence experiments are helpful as controls. First, the probe lifetime is measuredunder appropriate conditions since the lifetime iand may change if the probe rotates locally.(11, 12)

Second, if possible, the time-s often environment sensitive

resolved fluorescence depolarization from the labeled, isolated protein elementfrom the ordered ensemble (S-l in this case) tumbling by Brownian motionin solution is measured. This measurement senses motion of the probe ora domain of the protein containing the probe, if the motion is a rotationthat changes the probe orientation relative to the hydrodynamic frame ofthe protein element.(13, 14) In this experiment (in agreement with previous


s t u d i e s ( 1 5 , 1 6 ) ) both controls indicate that the 1,5-IAEDANS remained fixed inthe hydrodynamic frame of S-l in the presence and absence of MgADP,suggesting that the probe rotation observed with fluorescence polarizationwas due to S-1 rotation.

6.2.2.2. Example 2: Wavelength-Dependent Fluorescence Polarization

Fluorescence polarization studies (as in Section 6.2.2.1) established theability of the myosin cross-bridge (the S-l moiety of myosin in a muscle fiber)to bind to the actin filament at more than a single orientation.(1, 17, 18) Thisobservation is based on experiments using a single probe, iodoacetamido-tetramethylrhodamine (IATR), covalently linked to cysteine 707 (also calledsulfhydryl 1 or SH1) or myosin. The IATR probe showed that the cross-bridge maintains one orientation in rigor and a second orientation whenthe cross-bridge binds MgADP. Spin probes(19, 20) and 1,5-IAEDANS(17)

also linked to SH1, showed no angular reorientation upon the binding ofMgADP. The studies of Borejdo et al.(17) on the 1,5-IAEDANS probe wereperformed before the studies described in Section 6.2.2.1 and with the lowerresolution technique linear dichroism of fluorescence.These conflicting findings obtained with IATR on the one hand and with1,5-IAEDANS and spin probes on the other suggested that the probes havediffering orientations in the S-1 molecular frame such that rhodamine is poisedon the cross-bridge at an angle favorable for the detection of cross-bridgeangular displacement while 1,5-IAEDANS and the spin probes maintainan unfavorable orientation.(21) With wavelength-dependent fluorescencepolarization, we use the ability to change the direction of the 1,5-IAEDANSabsorption dipole by changing the excitation wavelength(22) to directlyascertain if differing dipole orientations could account for this variability ofresults.

The wavelength-dependent experiments were performed on a commercialfluorescence spectrometer with Glan–Thompson polarizers in the excitationand emission beam paths. A rectangular stainless steel support, made to fit ina standard fluorescence cuvette, held mounted fibers in the optical paths (seeFigure 6.5). The muscle fibers were mounted in the vertical configuration suchthat the fiber axis was perpendicular to the plane defined by the path of theexcitation and collected emission beams as in Figure 6.5. As in the exampledescribed in Section 6.2.2.1 S-l was specifically labeled with 1,5-IAEDANSand allowed to diffuse into the muscle fiber under rigor conditions. After ~ 30min the free labeled S-l was washed out with rigor buffer, and the experimentwas performed on the fiber decorated by S-1.

In this application we measured [see Eq. (6.19)] as a function ofexcitation wavelength, The results, shown in Figure 6.6, indicate a regionin the spectrum where for fibers in the presence of


MgADP and for fibers in rigor.(2) For a random distribution of probesat any These data indicate that the probe angular probability

densities for these two states are unmistakably different since the addition ofa random probability density to either probe density will not transform oneinto the other. As described in Section 6.2.2.1, the controls indicate that the1,5-IAEDANS remained fixed in the hydrodynamic frame of S-l in thepresence and absence of MgADP, suggesting that the probe rotation observedwas due to S-1 rotation.

6.2.3. Time-Resolved Fluorescence Depolarization Determination ofthe High-Resolution Angular Probability Density

Time-resolved fluorescence depolarization following polarized excitationis generally used to study the nanosecond rotational motion of molecules insolution. Several investigators also employed the technique to look atnanosecond motion of elementary subunits of biological assemblies. Theapplication of this method to biological assemblies yields information relatedto the time-averaged angular probability density, N, that augments theinformation obtained from the time-independent fluorescence polarizationmethods discussed in the last section.

The calculation of the time course of the emission decay from rotationalmotion of a fluorescent (labeled) protein element of a biological assembly canbe done using a suitable model. Kinosita et al.(23) developed a model theoryfor fluorescence polarization decay that was particularly suitable for lipidmembrane applications. The model theory, generally characterized asrotational diffusion in an angular potential, was subsequently generalizedto include more elaborate angular potential functions.(11, 24–27) Prior to thisfluorescence work, closely related calculations were performed to determinethe effect of probe rotational diffusion in a potential on the shape of ESRspectra.(28–30) We have subsequently introduced a model-independent approachto this problem that, among other attributes, demonstrates the contributionof the time-dependent signal to the determination of the time-averagedangular probability density, N.(31) We summarize the model-independentapproach and describe a biological application below.

The equation describing molecular rotational motion is assumed to be

where t is time, H is an operator, and is the time-dependent prob-ability density for molecular orientations. If H is not explicitly dependent ontime, Eq. (6.30) is separable in time and space coordinates so that


is the solution to Eq. (6.30) when The initial condition, determined bythe experimental application, is expressed in the expanded form

where is a constant coefficient. Equation (6.32) can be inverted usingEq. (6.2) to give

In a time-resolved fluorescence depolarization experiment, the intensityemitted by fluorophores specifically attached to elements of the biologicalassembly is observed. The fluorescence intensity that a fluorophore with emis-sion dipole emits into a polarizer oriented with its polarizing axis along vis . The fluorescence intensity from an ensemble of such fluorophores,F(t), is given by

where G(t) contains the time dependence of the signal due to the fluorescencelifetime of the probe. The explicit computation of F(t) using Eq. (6.34) wasdescribed in detail previously.(31) This calculation is performed usingEqs. (6.2) and (6.32) and the expression for in terms of Wignerfunctions that has been derived in several papers (e.g., Ref. 14). We find


and

where and are spherical unit vectors along the x, y, and z axes inthe laboratory frame. T is the time development operator relating and

through the linear operation such that

with matrix elements given by

In further discussions we assume that the fluorescence decay function G ( t ) isproportional to a single exponential with half-time Whenwhere is the maximum rotational relaxation rate of the orderedsystem, it is appropriate to expand in Laguerre polynomials,

such that (32)

The functions defined by

so that we can express the fluorescence signal following polarized excitationby a pulse of infinitesimal duration, F(t), as a sum of orthonormal functionsof time given by


where

and

The coefficients in Eq. (6.44) are determined by the initial conditionas given by Eq. (6.33). For the usual case of initial excitation by a

short pulse of polarized light, discussed in the following example, we have

where N is the steady-state angular probability density and satisfies thesteady-state condition From the general expression for N inEq. (6.1), the expansion of in Wigner functions shown elsewhere,(31)and Eq. (6.33), we solve for

We notice from Eq. (6.47) that the parameters depend on (orderparameters) of all ranks. Some of the order parameters can be determined bytime-independent methods as described in Section 6.2.2. Our purpose indescribing this method here is to show how it can approximate orderparameters of ranks not detected by the (standard) time-independent techni-ques (as in Section 6.2.2). This is done by solving for the coefficients inEq. (6.47) from the time-resolved signal that is related to the viaEq. (6.44). The are related to the order parameters by Eq. (6.47).We discuss an application of the time-resolved method that enabled thecalculation of rank 6 order parameters in Section 6.2.3.2 below.

With Eqs. (6.43)–(6.47) the time-resolved fluorescence depolarizationsignal is constructed in a model-independent manner. The unknownparameters are generally the matrix elements of the operator The linearcombinations of these matrix elements that are the coefficients in Eq. (6.44)are determined experimentally from the time-resolved curves.

6.2.3.1. Considerations for Real Instruments and Related Numerical Methods

On a real instrument the excitation lamp pulse has a finite width that isconvoluted with the fluorescence emission signal F(t). It could be essential toaccount for the finite width of the lamp pulse when the fluorescence lifetimeof the probe is much shorter than any of the motional relaxation times ofthe ordered systems Under these circumstances theobserved signal, F(t), is(3 l )

where is the shape of the lamp pulse. Using Eq. (6.43), we find

or

where

is the new function of time in which the observed fluorescent signal is



expanded. Unlike the of Eq. (6.41), the are not orthogonal. Thedeparture of the from orthogonality depends on the shape of the lamp pulsesuch that when the pulse is infinitesimal in duration, that is, it is a functionin time, Using a general numerical method described for an ESRapplication, we calculate the coefficients from F(t). The procedure isdescribed briefly below.

We define a projection operator by the equation

where is a constant independent of time. We require that have theproperty

Then,

We construct the operator using the constants which are found bysubstituting Eq. (6.52) into Eq. (6.53) and solving for We find

where is the (i, j)th element of the inverse of the matrixwith elements This method was used in both the ESR andtime-resolved fluorescence applications.

6.2.3.2. Example 3: Application to Fluorescent-Labeled Muscle Fibers

We measured the time-resolved fluorescence depolarization from1,5-IAEDANS-labeled cross-bridges in muscle fibers in the presence of ATP

without In this state, called relaxed, the myosin cross-bridges areknown to be in a rapid equilibrium between actin-attached and detachedstates(34, 35) so that the fiber can be slowly stretched with the minimum ofresistance. Borejdo and Putnam(36) showed that the myosin cross-bridge wasselectively labeled with 1,5-IAEDANS, using a method developed by Dukeet al.(37)

, such that of the label was covalently linked to Niheiet al.(38) showed that modification of with 1,5-IAEDANS does not impairfiber contractility when the degree of labeling is as high as 0.8 mole of


fluorophore per mole of myosin, implying that the modified cross-bridge isable to produce force in the same manner as the native cross-bridge.

In this example we employed low-aperture excitation and collectionoptics so that the excitation electric field, E, is linearly polarized and

We choose E to be polarized along the laboratory frame z axisso that E = (0, 0, 1). We collect the emission polarized along the laboratoryframe z and x axes [corresponding to and in Eq. (6.37)]. Theobserved signals are then

When This approximation is appropriate forthe muscle fiber application.

The parameters in Eqs. (6.56) and (6.57) are the following: the coeffi-cients which depend on the order parameters and the absorptiondipole moment , the emission dipole moment and the matrixelements of The values of and some of the can be determinedfrom independent observations. Mendelson et al.(39) estimated the directionsof and for myosin cross-bridges modified at by 1,5-IAEDANS. Theorder parameters are dependent on the physiological state of the muscle, andsome of them are known for relaxed cross-bridges.(27) Usually only the matrixelements of in Eqs. (6.56) and (6.57) are unknowns to be determined in thetime-resolved experiment.

The time-resolved data give supplemental information related to thesteady-state angular probability density, N, when we make use of a particularmodel for cross-bridge motion. When the operator H describes a probemoving with rotational Brownian motion in a potential, then

where is the rotational diffusion constant, is a differential operator


identical to the quantum-mechanical orbital angular momentum operator,and f is related to N by

with related to the angular potential V by

where k is Boltzmann’s constant, and T is temperature.The matrix elements of H are calculated using Eq. (6.2) and the relations

from which we find


where

Higher order matrix elements are calculated from Eq. (6.65) using the com-pleteness of the Wigner functions on the interval . Completeness requires

where is a Dirac function. With Eq. (6.67) we can show that thesecond-order matrix element, is

Higher order matrix elements can be similarly constructed.Time-resolved measurements were made on muscle fibers in two

configurations. In the vertical configuration the fiber axis (the z axis in thelaboratory frame) is parallel to the excitation light polarization so that thedepolarization curves are sensitive to rotational motions of the probes about


axes other than the fiber axis. In the horizontal configuration the fiber isrotated by 90° so that the excitation light polarization is perpendicular to thefiber axis, and its light propagation vector makes an angle of 45° with thefiber axis (see Figure 6.5). Depolarization curves from the horizontal fibers aresensitive to probe motion about axes parallel to the fiber axis. By requiringour model of Brownian motion in an angular potential to account for therelaxation rates of the cross-bridges in both fiber configurations, we acquiresufficient constraints on the free parameters to estimate the rank 6 orderparameters of N.

With the available steady-state fluorescence polarization data,(40) weshowed that in the vertical fiber configuration, N is described by the orderparametersFor the horizontal fiber configuration we can readily compute the newhorizontal order parameters, denoted by from the vertical orderparameters, using the relation

A similar relation holds for in Eq. (6.66). With the rank orderparameters given as known quantities, we estimate the rank 6 orderparameters using the model.

An interesting new feature of the proposed polar steady-state angularprobability density computed using Eq. (6.29) is introduced by the additionof the rank 6 order parameters. Shown in Figure 6.7 is a comparison of proba-bility densities with and without the inclusion of the rank 6 parameters. Thehigher resolution plot has a bimodal feature not observed at lower resolution.

6.2.3.3. Generalized Method for Evaluating Matrix Elements of T

The expansion of in terms of Laguerre polynomials, done inEq. (6.40), is not easily extended to the case in which the rotational relaxationhalf-times of the system are similar to or smaller than the probe lifetime. Wepropose here an untried method that makes use of Laplace transforms toisolate the lowest order relaxation rates from the time-resolved fluorescencedata. We also make use of some new notation to make the derivation simpler.

Let the ordered triplet of indices ( j , m, n) from be summarized bya single index i such that is equivalent to Then we can writeEq. (6.35) in the form


where summarizes the time-independent constants. Again assuming G(t) tobe a single exponential,

If we can neglect the width of the lamp pulse, then the observed signal is givenby Eq. (6.71). F(t) is Laplace transformed to give the quantity such that

so that

Equation (6.73) can be expanded in powers of to giveto lowest order in H,


On a plot of as a function of is the y-intercept, and the slopeis a function of the first-order matrix elements of H.

6.2.4. Relation of Electron Spin Resonance Spectra to Fluorescence Polarization

6.2.4.1. Rotation Connecting Spin and Fluorescent Probe Order Parameters

There are many well-developed methods available that use ESR spectrato determine spin probe angular probability densities.(42,43) As mentionedin Section 6.2.2, we made use of ESR data in the determination of orderparameters of the angular distribution of 1,5-IAEDANS-labeled cross-bridgesin muscle fibers in example 1. In example 1 we described the determination ofthe four unknown order parameters,from the three quantities measured by fluorescence polar-ization [see Eqs. (6.18)–(6.20)] with additional information from the ESRspectrum. In this section we wish to describe in more detail the connection ofthe ESR and fluorescence polarization techniques for determining order.

The fluorescence and ESR data are quantitated in terms of orderparameters, but the values of the order parameters are not interchangeablebecause in these applications both sets of order parameters describe the probeangular probability density. The probe angular probability density derivedfrom ESR data describes how the spin probe is oriented. Likewise, the probeprobability density derived from fluorescence data describes how the fluores-cent probe is oriented. A rotation can always be found to relate the two setsof order parameters such that(2,8)

where represents the Euler angles describing the rotationfrom the spin probe fixed frame to the fluorophore fixed frame. Equation(6.75) transforms the order parameters calculated in the principal magneticframe of the spin probe to the reference frame of the fluorescent probe (forexample, our choice with the z axis along the absorption dipole and the x axisalong the projection of the emission dipole in the x–y plane). Known sym-metry properties of the muscle fiber simplify Eq. (6.75), reducing it to theexpression

We estimate by initially guessing its value and then calculatingand (fluorescence) using Eq. (6.76) and known values for

(see, for example, Ref. 33). We then solve Eqs. (6.18)–(6.20) forand (fluorescence) and compare these values with the values of

and (fluorescence) computed directly from Eq. (6.76) and (ESR).On the basis of this comparison, an improved estimate of is made, andwe repeat the above steps until the difference between the fluorescence orderparameters computed by the two methods is minimized. By this method, infor-mation from ESR can augment fluorescence data to provide a more completedescription of the fluorescent probe angular probability density.

6.2.4.2. Example 4: Fluorescence Polarization from 1,5-IAEDANS-LabeledMuscle Fibers (Example 1, Continued)

ESR spectra measured from fibers in rigor and in the presence ofMgADP were analyzed by a model-independent method to yield the orderparameters of the spin probe angular probability density.(41) We foundin this investigation that

and for fibers in rigor, andand

for fibers in the presence of MgADP. By themethod outlined above (Section 6.2.4.1), we estimate and

These values for and (ESR) were used in Eq. (6.76) tocompute the order parameters (fluorescence) listed in Eqs. (6.21)–(6.28).

6.2.5. Biochemical Techniques of Specific Labeling

An important task in the study of oriented systems of biomolecules is tofind a reporter group that fits specifically at a well-defined point on thebiomolecule without altering its biological activity. One can introduce thefluorescent derivatives specifically using covalent or noncovalent interactionsbetween the probe and the biomolecule. After the modification, methods forverifying the location of the probe and for assaying the biological activity ofthe modified system must be developed.

6.2.5.1. Covalent Fluorescent Labels

A well-chosen covalent probe reacts specifically with the region of intereston the target biomolecule.(44) The method of making a specific fluorescentprobe is to synthesize a derivative from the dye by attaching a spacer with afree reactive group at one end that selectively modifies particular side chainson the target molecule. For proteins, the most often used probes are thethiol-selective (iodoacetamido, iodoaceto, maleimido, and bromoacetamido)or amine-selective (isothiocyanato, carboxyl, sulfonyl acid, and succinimidyl



ester) derivatives. Examples of these probes are drawn in Figure 6.8, where itis shown that tetramethylrhodamine can be used as both a sulfhydryl- and anamino-selective probe.

In most cases, the amino acid side chains of the protein that are involvedin the molecular aspects of biological activity have altered reactivities due totheir irregular values. This gives one the possibility to introduce labelsexclusively at these points. By lowering the pH of the reaction medium (towithin pH 6–7) far below the of the majority of the lysine groups

the ionized form of these groups becomes unfavorable forcovalent reactions while cysteinyls, especially those having altered values,can still react. The sulfhydryl (Cys-707) of myosin is an example of thisphenomenon. reacts specifically with iodoacetamide derivatives of dif-ferent fluorophores near pH 7 under mild conditions.(45,46) The remaining SHgroups available to the dye react slowly. The reactivity of is so enhancedthat it can serve as a single labeling point in a whole muscle fiber system.(47,48)

Using sulfhydryl-specific labels, we can also change the possible target sidechain by prereacting the system with certain reversible SH-blocking com-pounds. The reactive thiols are reversibly blocked, and the chemically less

reactive groups are exposed to the covalent by binding probe. After thecovalent binding procedure is completed, the blocking is reversed, restoringreactive thiols to their native state.(46,49)

The chemical characteristics of the probes themselves also affect thebinding of the label to the target molecule. Rings with apolar character reactpreferentially with lipids, membranes, or protein side chains in hydrophobiccrevices. Polar probes label groups on the more hydrophilic surfaces of theprotein.

We see that the reaction of a specific fluorescent probe with a proteindepends on the nature of the reactive group attached to the chromophoreand on the interaction between the dye and the environment of the sidechain to be labeled. The three-dimensional fit of an optical probe also couldbe influenced by other factors. The best fit of the chromophore ring on theprotein surface could vary for different stereoisomers of the label, leading tononhomogeneous labeling. This would be detected by nonhomogeneity in thefluorescence lifetime and would cause difficulties in the interpretation ofthe polarized fluorescence. The length of the spacer group from the dye to thereactive group also is an important factor with regard to the specificity andrigidity of the probe. With short spacers, dyes are introduced at the proteinsurface while longer side chains allow the probes to penetrate into proteinsand membranes. This latter property is facilitated by an apolar spacer group.An example of this is provided by the derivatives of pyrene shown inFigure 6.9. An increasing number of apolar groups allows the label tomodify points deep inside a folded macromolecule.

6.2.5.2. Noncovalent Labeling Techniques

Proteins and lipids can also be labeled using specific noncovalent associa-tions. This technique, called affinity labeling, uses ionic, hydrophobic, orhydrophilic interactions to complex the probe with the biomolecule. Theprobe–biomolecule interaction can subsequently be made more permanentusing photoaffinity labeling methods(50) or by probe trapping using cross-linkers.(51)



The affinity probes are a rather heterogeneous group varying from thefluorescent lanthanides (which substitute for calcium in calcium-bindingproteins such as actin(52) and calmodulin) to fluorescent derivatives ofnucleotides, coenzymes, enzyme substrate inhibitors, and other proteins.Fluorescent derivatives of adenine nucleotides were used for orientationstudies in muscle fibers,(53) as was rhodamine-labeled phalloidin, a moleculethat binds very tightly to actin.(54,55) Fluorescent-labeled antibodies providea whole class of probes that can be targeted to a particular sequence ofpolypeptides with three-dimensional structure.(56)

6.2.5.3. Labeling Methods

The careful fluorescent probe study of oriented biological systems requiresthat we introduce the probe while maintaining the native biochemical characterof the system. Usually, fluorescent dyes are introduced in the biologicalsystem at neutral pH, in mild salt buffers. We must be sure that the probe issolubilized in the labeling buffer and that it can penetrate the membranecovering the cells or tissues. The large aromatic rings of tetramethylrhodaminelabeling muscle fibers, for example, require that we first dissolve the probe athigh concentration in an apolar organic solvent (dimethylformamide) beforediluting it ~100-fold in the neutral buffer labeling solution. The muscle fiberis demembranated chemically by bathing it in a mild neutral detergent (TritonX-100, 0.1–0.2%) before the application of the labeling solution. After thelabeling is completed, care must be taken to remove the unreacted dye toprevent further (nonspecific) labeling and to ensure that the fluorescencesignal originates from the covalently attached probe. This can be done byextensive washing in probe-free buffer.

6.2.5.4. Probe Specificity

Once a protocol is worked out that places a sufficient amount of probeon the biological assembly such that a fluorescent signal with a high signal-to-noise ratio is detected, the next step is to localize the probe in the system. Thisis obviously important when a particular constituent of the system is ofexperimental interest and is essential for unambiguous interpretation of thepolarized fluorescence signal. An example of this process is described below.

The skeletal muscle fiber is an ordered molecular assembly of two majorproteins and eight minor proteins (see Figure 6.10). To study the orientationof the myosin head during changes in the physiological states of the fiber, wehave covalently labeled the thiols on S-l with 4-[N-(iodoacetoxy)ethy1-N-methyl]amino-7-nitrobenz-2-oxa-l,3-diazole (IANBD). The labeling proce-dure follows the general procedure of fiber labeling introduced by Borejdoand Putnam.(36) The labeling is carried out on fibers that have been demem-

branated by chemical skinning with Triton X-100, at 0°C in 100 dyeconcentration for 15 min at neutral pH in phosphate buffer. The reaction isstopped by exhaustive washing of the fiber in probe-free buffer.

The whole protein assembly in the fiber is dissolved in sodium dodecylsulfate (SDS), and SDS–polyacrylamide gel electrophoresis (SDS-PAGE)analysis is performed. The result of this analysis is shown in Figure 6.11. Ofthe ten different protein components, only two are fluorescently labeled. Asjudged from the fluorescent intensity measured from a scan of the gel, ~82%



of the dye was incorporated into the heavy polypeptide chain of myosin while18 % of the total fluorescence is associated with the troponin I protein.

The localization of the label in the primary sequence of the myosin iscarried out on myosin purified from labeled muscle fibers. The labeled proteinis proteolytically cut to yield S-l, and further tryptic digestion of S-l yieldsthree peptides of 20, 50, and 27 kilodaltons (kDa). Following the fluorescenceon the gel as shown in Figure 6.12 allows us to identify the labeled peptide asthe 20-kDa fragment, which contains the two fast-reacting sulfhydrylsand (57) More specialized methods are required to biochemically separate

and These methods are described elsewhere.(58)

6.2.5.5. Biological Activity

Of equal importance to the localization of the probe is the assessment ofthe damage caused by the incorporation of the probe into the biologicalsystem. The native biochemical activities of the system must be essentiallypreserved in the probe-modified system for data interpretation to be meaning-ful. This question can be complicated since some biochemical alteration by theprobes is inevitable, and one must be selective in deciding what deviation

from the native biochemical activities can be tolerated. Again an examplefrom muscle illustrates this point and is described below.

The enzymatic activities of the myosin cross-bridge are significantlyaltered by the probe modification of myosin at It is well known thatupon modification of the ATPase of myosin in the presence of isactivated 2–10-fold while the ATPase in the absence of divalent cations, theso-called ATPase, is inhibited linearly with the fraction ofmodified.(45,46) Surprisingly, the active isometric tension of muscle fibers isunaffected by probe modification of (36,59) Researchers have focused onthe fiber active tension result as proof that the system stays essentially intactand have reasoned that the altered ATPase activities reflect modifications inenzymatic steps of activated cross-bridges that are not usually rate-limitingsteps in muscle contraction.(59)

6.3. Discussion

Our intent in this chapter is to introduce a systematic description of N,the angular probability density of molecular elements in a biological assembly.The expansion in terms of order parameters in Eq. (6.1) is a description of Nthat is applicable to steady-state fluorescence polarization (Section 6.2.2),time-resolved fluorescence depolarization (Section 6.2.3), and electron spinresonance (Section 6.2.4). We find that we can distinguish experimentaltechniques based on their angular resolution. This property proved to be use-ful in our investigation of cross-bridge orientation during muscle contraction.

One important avenue for classical optical research, which could haveimmediate application in biophysics, is the investigation of experimentalmethods to increase the angular resolution of fluorescence polarization techni-ques. We are aware of two interesting possibilities, which we describe brieflybelow.

The first method comes from the application of total internal reflectionfluorescence polarization techniques to study the probe spatial and angularprobability densities in two-dimensional layers.(60–69) This work brought tolight several useful properties of fluorescent light emission from probes neardielectric interfaces.(9,70–74) When a fluorescent probe is near a dielectric inter-face (within a distance of approximately a wavelength of the interface), weshowed that the interface can perturb the near field of the dipole and causethe observed electromagnetic emission pattern of the fluorophore to be greatlyaltered from the typical dipolar emission pattern.(9) Axelrod and co-workershave since pointed out that the rate of total energy dissipated by a fixed-amplitude dipole varies with its orientation and distance from the interfaceand the nature of the interface coating.(75,76) They showed that a power-normalized dipole is the correct model of a fluorophore under constant



illumination. We derived the power-normalized fluorescence polarizationsignal from fluorophores near an interface and its dependence on the orderparameters and showed that the fluorescence polarization signal dependson an infinite number of parameters such that j = 0, 1, 2, 3, ....(77) Thisfinding proves formally that we can increase the angular resolution of fluo-rescence polarization measurements by observing the emission of a fluorophorenear an interface. We developed this idea and derived an analytical, model-independent method of obtaining this information from the fluorescencesignal. Experiments are under way to take advantage of this higher resolutiontechnique in the study of protein components in muscle fibers.

The second method comes from the suspected observation of electricquadrupole radiation from fluorescent probes near dielectric interfaces. Thisobservation raises the possibility of measuring polarized fluorescent emissionfrom the electric quadrupole transition.(61) The fluorescence polarizationsignal from such a transition should be sensitive to order parameters of rankgreater than 4. The difficulty associated with such an approach is the smallintensity of this second-order electronic transition compared to that of adipole transition. It is estimated for atomic electron transitions that the inten-sity of the electric quadrupole transition is a factor of weaker thanthat of the electric dipole transition, where is the effective nuclear charge.The value of is 1 for transitions of valence electrons, and theatomic number, for X-ray transitions.(78) It may prove to be experimentallyinfeasible to isolate the quadrupolar from the dipolar emissions for themolecular transitions giving rise to visible or near-visible fluorescence;however, some theoretical work is now under way in our laboratory to furtherinvestigate this idea.

The development of higher resolution fluorescence polarization methodsis an important research objective in biophysics. Presently the fluorescencedetection of on the order of 10 fluorescent probes in the visible region ispossible. This sensitivity allows good time resolution in the observation ofdynamical processes such that picosecond spectroscopy is not uncommon. (79,80)

Presently, however, angular resolution is severely limited as pointed out inSection 6.2.1. The combination of good time resolution and high angularresolution in a fluorescence technique would obviously be of benefit to muscleresearch and in other areas of biophysics.

Acknowledgment

The writing of this chapter was supported by a grant from the MayoFoundation.

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78. J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York (1975).79. P. M. Rentzepis, Advances in picosecond spectroscopy, Science 218, 1183–1189 (1982).80. T. M. Nordlund and D. A. Podolski, Streak camera measurement of tryptophan and

rhodamine motions with picosecond time resolution, Photochem. Photobiol. 38, 665–669(1983).

7

Fluorescence- BasedFiber-Optic Sensors

Richard B. Thompson

7.1. Introduction

Fiber-optic sensors represent an emerging technology that will haveimpact in fields as diverse as medical diagnostics, pollution monitoring,aeronautical engineering, oceanography, and navigation. Fiber-optic sensorsoperating on a variety of principles, and detecting a great variety of analytesand influences such as temperature or pressure, have been described in theliterature(1,2). An important subset of fiber-optic sensors utilizes fluorescence:that is, the signal is a change in fluorescence caused by the influence oranalyte. For our purposes, “fluorescence” will serve as a generic term forphotoluminescence of all types (e.g., fluorescence, phosphorescence, delayedfluorescence) except where distinctions are important. This chapter will con-sider such sensors and some technical issues of their design and construction.Since good reviews exist,(3–5) the prior art will not be comprehensivelyreviewed.

The advantages of fiber-optic sensors for many applications are worthsummarizing; we are concerned both with “real-world applications” and withthe use of fiber sensors in laboratory situations. Perhaps best known is theircapability for remote, continuous monitoring of analytes or influences in realtime. A central instrument (perhaps in a benign enclosure) can monitor anarray of fibers, which may be in hostile environments. Several early applica-tions have made use of the dielectric fiber’s immunity to powerful electro-magnetic fields and its lack of electrical current in flammable environments.Optical signals have an enormous inherent bandwidth (and thus informationcapacity), and fibers offer a multiplex advantage since signals of different

Richard B. Thompson • Department of Biological Chemistry, University of Maryland Schoolof Medicine, Baltimore, Maryland 21201. This work was performed while the author was with theBio/Molecular Engineering Branch, Naval Research Laboratory, Washington, D.C. 20375.

Topics in Fluorescence Spectroscopy, Volume 2; Principles, edited by Joseph R. Lakowicz. PlenumPress, New York, 1991.

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346 Richard B. Thompson

colors can propagate simultaneously in either direction. Fiber optics permitoptical signals to pass through tortuous passages, out of sealed chambers,or through reciprocating machinery; evidently, a fiber itself has none of thealignment problems of a system built using discrete optical components.Indeed, there are some tasks which can only be accomplished using fibers.Fiber optics can be very small and light, an important consideration in ships,airplanes, and spacecraft.

7.2. Fiber-Optic Fundamentals

From an optical standpoint, fibers are a subclass of optical waveguideswhich operate using the principle of total internal reflection. Light incident onthe interface between two dielectric media of different refractive indices will beeither reflected or refracted according to Snell’s law (see Figure 7.1). Light isthus confined within a transparent cylinder (or fiber) if the light is launchedinto it at a shallow enough angle and the cylinder is surrounded by a mediumof lower refractive index. Thus, optical fibers consist of a core of highrefractive index, surrounded by cladding of slightly lower refractive index, thewhole usually protected in a nonoptical jacket (see Figure 7.2). Fibers can bemade of fused silica, optical glasses, various plastics, even sapphire, dependingupon the desired optical properties.

Snell’s law serves to describe the macroscopic optical properties ofwaveguides, but breaks down on a microscale. In particular, waveguides ofsubmillimeter dimension are much better described by theory that considers

Fluorescence-Based Fiber-Optic Sensors 347

their ability to propagate particular modes of light; that is, the optical powerin the waveguide will have particular spatial, temporal, and polarizationproperties, which are determined by the geometry of the waveguide in afashion analogous to modes in a laser cavity or microwave waveguide. Fora few ideal cases, the mode distribution can be calculated exactly,(6) but inmany cases the modes a fiber will propagate are determined empirically. Thereader is referred to recent texts(6–8) for details, but some important conceptsare described below.

Some fibers are constructed with thin cores and a very smalldifference in refractive index between the core and cladding, so as to conductonly a single mode (the mode). These singlemode fibers have a numberof characteristics potentially important in fluorescence sensors. They have thelowest possible temporal dispersion; that is, pulse spreading is minimized,increasing the effective bandwidth (useful for telecommunications) and maxi-mizing the resolution of time-resolved techniques (see below). Because theyare singlemode, such fibers are insensitive to influences such as temperatureor bending, which change modal distribution, and therefore propagatedintensity. Some fiber-optic properties, such as polarization preservation, areonly available in singlemode fibers.

By comparison, fibers with larger cores and/or larger refractive indexdifferences are termed multimode since they will propagate several modes atonce. The physical properties which determine how many modes a fiber willaccommodate are expressed in the waveguide parameter, or V-number:


where is the wavelength, is the core diameter, and and are the coreand cladding indices, respectively; a V-number of 2.405 or less ensures thatthe fiber is singlemode. The waveguide parameter is wavelength-dependent,and thus a fiber that is singlemode in the infrared will typically carry a fewmodes at visible wavelengths. Among the advantages of multimode fibers isthe ease of coupling light into the fiber, since they have both larger cores and(because of the larger difference in refractive indices) higher numerical aper-tures (see Figure 7.2). While singlemode fibers have both core and claddingmade of glass or fused silica, plastic-clad fused-silica (multimode) fibers areavailable; the plastic cladding is easy to remove, permitting access to thecore (see below). Multimode fibers with large cores are also better suited topropagating high-powered laser (excitation) beams; a multiwatt argon laserbeam might create some interesting but unproductive effects if focused to amicron-sized spot on the end of a singlemode fiber.

Another important phenomenon not described by Snell’s law is theevanescent wave. When total internal reflection takes place at an interface, theelectric field strength of the incident light beam does not drop abruptly tozero at the interface; rather, it decreases exponentially from the interface intothe lower refractive index medium over a distance of a few hundrednanometers. The electric field present in the lower refractive index mediumcan be absorbed by fluorophores near the interface and thus permits surface-specific fluorescence spectroscopy. This phenomenon has been extensivelyexploited by Harrick (9, 10) and Axelrod and Burghardt.(11, 12) Others (13, 14) havemade use of the surface specificity to devise immunoassays. As a firstapproximation for multimode waveguides, the depth of penetration of theexcitation into the surrounding medium (if the cladding has been removed) isa function of wavelength, angle of incidence and refractive index ratio of themedia (where ):

The polarization of the incident light also is a factor, and for waveguidespropagating only a few modes, the above equation is inexact. As a ruleof thumb, however, the fewer modes a fiber optic carries, the greater theproportion of power propagated in the cladding. The evanescent wave isimportant because it represents an alternate means of coupling power out ofthe fiber to excite fluorescence, in addition to the distal tip.

From the standpoint of performing fluorescence experiments, perhaps themost important property of the fiber is its attenuation at the wavelengths ofinterest. Remember that in a fiber optic, one is essentially looking through awindow of glass several meters thick; the achievement of ultralow-loss fiberoptics is all the more impressive considering that exceptionally clear water is


required if we wish to see ten meters through it. Attenuation is expressed indecibels (dB), which are defined in terms of the intensities I:

Thus, a length of fiber with 40 dB of attenuation would decrease the lightintensity at the distal end ten thousand-fold, or, in more familiar terms, havean optical density of 4.0. Attenuation is wavelength- and fiber-dependent(see Figure 7.3) to an extent that has a large impact on any fluorescenceexperiment. For instance, a fluorophore that can be excited at 500 nm wouldreceive one hundred million-fold (80 dB) more light at the end of a kilometerof the fiber profiled in Figure 7.3 than one excited with equal incident inten-sity at 300 nm. This is one reason why most long-haul telecommunication isperformed at near-infrared wavelengths. Note that the main source of theattenuation in the visible and near UV is Rayleigh scattering, with itsdependence, and thus most fibers exhibit similar attenuation profile shapesat these wavelengths. Obviously, if the application requires a shorter fiber,these considerations become less important. Conversely, a remote sensingapplication might preclude the use of a fluorophore with an absorptionmaximum of less than 600 nm.

7.3. Sensor Design

A large number of fiber-optic sensors employing fluorescence have beendescribed in the literature; we shall consider those of particular didactic value.The reasons for using fluorescence (as opposed to other observables) in fiber-

optic sensors are well known: fluorescence is intrinsically more sensitive thanabsorbance due to the Stokes’ shift and is more flexible in that a great varietyof analytes and influences are known to change the emission of particularfluorophores. The operating principles (observables) used in several sensorswill be discussed, followed by design considerations of sensing tips, fibersthemselves, optical components of the system, and light sources.

The essence of fluorescence analysis is to understand and exploit changesin fluorescence. These changes may be in the intensity, lifetime, color,and/or polarization of the emission. Indeed, any interpretable observationof fluorescence might be made the basis of a sensor. Thus, temperaturesensors available commercially are based on changes in relative intensity ofemission lines of europium: gadolinium oxysulfide or changes in lifetime of aphosphor.(15) The former illustrates an important point, namely, that simplymeasuring changes in intensity is unsatisfactory for many applications. This isbecause interfering factors such as microbending, temperature, source fluctua-tion, fluorophore degradation, and detector aging affect accuracy. Moreover,recalibrating a sensor that may be in a remote or hostile environment couldprove difficult. Thus, several pioneering sensors that measured changes influorescence intensity due to oxygen quenching,(16) binding of metal ions,(17)

or pH changes(18) have yet to come into wide use. It is unfortunate that manyof the analytical applications of fluorescence(19–21) are based strictly onintensity changes and thus are less useful than they might be. By comparison,spectral shifts can be measured ratiometrically, thus minimizing some ofthe above problems. Use of external standards or monitoring scatteredexcitation in a “double-beam” arrangement can also improve accuracy.(22, 23)

Fluorescence lifetimes(15, 24, 25) are also largely intensity-independent and there-fore are increasingly used.

7.4. Sensing Tip Configurations

Fiber-optic fluorescence sensors have generally had only two types ofconfiguration of the “business end” of the sensor: the “distal cuvette” con-figuration (Figure 7.4) or the “waveguide binding” configuration (Figure 7.5).The distal cuvette configuration looks simple, but is seldom trivial toconstruct. The fluorescent material is generally immobilized in front of thefiber(s) within some transparent matrix that permits access of the influence(temperature, pressure) or analyte to the fluorophore. Due to differentrequirements a variety of matrices have been employed, including controlledpore glass,(18) cellulose,(17) polyacrylamide,(26) polystyrene resins,(22) sili-cones,(27) and liquid encapsulated in dialysis tubing.(28) The difficulties lie inthat few matrix materials are free from interfering fluorescence, permitimmobilization of the fluorophore without disturbing its properties, are


mechanically strong and durable under the experimental conditions, permitrapid diffusion of the analyte, and can be bonded somehow to the fiber tip.This last issue looms large when the tiny dimensions of the fiber tip (usually<0.5 mm) are considered. Indeed, the tiny cross-sectional area of the fibercore imposes a limit on the area (or volume) which may be illuminated;Wyatt et al.(29) have also pointed out the photobleaching problems which maybe encountered if the fluorophore is too close to the end of the fiber. The useof polymers which permit the fluorophore to be covalently bound in a matrixwhich is itself covalently bound to the glass of the fiber will find wide use,(26)

and clearly many of the techniques developed for affinity chromatography canbe adapted for biosensing applications.(5)

The waveguide binding configuration makes use of the evanescentwave propagating near the surface of the waveguide, as mentioned above;essentially, surface binding (or desorption) of an analyte is detected bychanges in fluorescence intensity as it enters or leaves the zone near thesurface (see Figure 7.5). This scheme has been the basis of severalimmunoassays,(l3,14,30,3l) exhibiting high sensitivity.(3) Issues that may havehindered wide use include speed, the difficulty of immobilizing antibodieswhile retaining activity and preventing nonspecific adsorption, and the sen-sitivity attainable in a fluorimeter where only a small fraction of the excitinglight reaches the fluorophores.(5)

Indeed, the problem of coupling the light out of a fiber to excite thefluorophore and coupling the fluorescence back into the fiber (or anotherfiber) seems to be central, and yet little explored. For instance, it is easy toshow that most of the optical power in a step index singlemode fiber actuallypropagates in the cladding. As we consider fibers that propagate a few, thenmore modes, an ever-decreasing proportion of the power propagates in thecladding (see Figure 7.6). Recall that for a fiber that guides a single mode atvisible wavelengths, the core diameter is less than ten microns, and thedifference in refractive indices between core and cladding is very small, lessthan 0.01, yielding a V-number of less than 2.405. The fibers (rods) that areoften used in waveguide binding configurations have diameters of 600 orgreater, and the refractive index difference between them and the solution thatbathes them (and serves as cladding) is perhaps 0.17, giving a V-number ofabout 4400 (Eq. 7.1). Thus, these fibers propagate a great many modes, andonly a tiny fraction of the optical power that they guide leaks out into themedium and is available to excite fluorescence (Figure 7.6).(32) Under thesecircumstances, the importance of launching the light near the critical anglehas been emphasized.(33)

There are some data that suggest that, for fluorophores near to thewaveguide surface (or fiber core), fluorescence is coupled back into the fibermore efficiently than would be anticipated from a strictly ray optics view-point.(34,35) If it were possible, a systematic study that examined the efficiency



of fluorescence being coupled back in as a function of refractive indices,fluorophore proximity and orientation, and mode structure would be mostwelcome. It may be that the fluorescence will be coupled back into the fiberto the extent that the fluorescence distribution mimics the distribution of lightin the mode(s) guided by the fiber.

7.5. Fiber Characteristics

The characteristics of the fiber itself play an important role in thedesign. The variety and quality of fibers made available due to their usefor communications and other applications has been an enormous benefit.Thus, single- and multimode fibers are available cheaply, and with very lowattenuation. We have discussed the importance of fiber attenuation above.

The background fluorescence of the fiber is also a consideration.(36) A repre-sentative spectrum is shown in Figure 7.7 for a common multimode fiber, withexcitation in the ultraviolet. The emission bands may be due to plasma linesfrom the laser discharge, fluorescing metal ion impurities in the fiber, colorcenters, and/or Raman scattering.(37) Evidently, as the fiber lengthens therelative intensity of these bands will increase, lowering sensitivity if the signalfluorescence overlaps the background bands. A several authors have pointedout, time-resolved methods offer the prospect of avoiding this background, asillustrated in Figure 7.8. Essentially, one sends a brief pulse of exciting lightdown the fiber and waits to turn on the detector until the fluorescence isexpected to return; photoluminescence originating elsewhere within the fiberwill arrive at the detector at different times depending on its intrinsic life-time and position and will not be detected. Note that this is one of the fewinstances where phase methods are likely to be inferior to time-resolvedmethods, since in order to utilize phase-sensitive detection methods( 38,39)

successfully, the photon transit time (fiber length) should be a harmonic of thelifetime.

A second fiber-optic characteristic, called dispersion, limits the fiberbandwidth and thus can affect time-resolved fluorescence measurementsthrough fibers. Dispersion is of two types, termed modal dispersion and


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material dispersion. Modal dispersion occurs in multimode fibers and arisesfrom differences in the group velocities of various modes propagating in thewaveguide; a detailed treatment is available.(8) More intuitively, rays launchedinto the fiber near the critical angle travel a longer path getting to the distalend than those launched parallel to the fiber axis, and thus a short pulse oflight will be broadened by the time it reaches the end of the fiber. Thisproperty is usually expressed as a frequency–distance product: that is, a fiberspecified as having a 300-MHz-km product would carry a 300-MHz signal1 km with less than 3-dB loss due to demodulation. Note that graded indexfibers are specifically constructed to minimize modal dispersion. (6, 8) By theirvery nature, singlemode fibers do not suffer from modal dispersion.

Material dispersion affects all fibers and is a limiting factor in singlemodefiber bandwidth. It arises from the variation of refractive index (light velocity)with wavelength of the glass used in the fiber. A red photon travels fasterthrough most fibers than a blue one, and thus a pulse of monochromatic lightwill be less broadened (temporally) than an equally short one having a larger(wavelength) bandpass (Figure 7.9). High data rate communications applica-tions therefore require sources which can be rapidly modulated and also arevery monochromatic. Since material dispersion is a property of the glass, thefiber may be constructed to minimize the material dispersion in a particularwavelength regime, usually one of the low-attenuation (IR) windows. Finally,carefully controlled material dispersion is important in fibers used in “pulsecompressors.”(40)



7.6. Separating Excitation and Emission

How exciting light is coupled into and fluorescence collected from thefiber is also an important design question. The simplest solution is to employdifferent fibers for guiding excitation to and fluorescence from the sensing tip.This has the advantage of simplicity in design and permits optimizing the fiberfor each wavelength. However, it also has drawbacks: the receiving fiber easilycollects scattered excitation, it precludes the use of the waveguide bindingconfiguration, and it multiplies the engineering problems of the distal cuvetteconfiguration. These last include the difficulty of registration: exciting thesame volume of sample that the emission fiber is capable of viewing, andattaching and sealing the sensing tip. Thus, this geometry is less favored bycurrent workers in the field.

By contrast, use of a single fiber to guide excitation and emissionsimultaneously avoids these problems and makes use of the fiber’s naturaladvantages. In this case, it is necessary to separate the excitation and emissionbased on their differing wavelengths or spatial characteristics; a number ofconfigurations have been described for doing this (Figure 7.10A, B). The twomost common make use of multilayer dielectric-coated mirrors or interferencefilters. In Figure 7.10A the interference filter (with the depicted spectrum)passes the exciting light, while reflecting the emission; the dielectric-coatedmirror in Figure 7.10B does the converse. The configuration in Figure 7.10Asuffers some loss because the interference filter (which may be angle-tuned toadmit the desired wavelength) seldom passes more than 30% of the light atthe peak. The configuration in Figure 7.10B takes advantage of the fact thatmirrors with multilayer dielectric coatings can be outstanding reflectors, withcoatings available that reflect 99 + % of particular laser lines, while passinglonger wavelengths. Problems can arise with this configuration, however,when powerful (laser) or short-wavelength sources are used. In both con-figurations the excitation hits the filter directly and often causes it to fluoresceitself. The importance of this problem varies with the particular filter, thewavelength of the signal fluorescence, and the sensitivity required. It is of coursegenerally true that the greater the sensitivity required, the more importantobscure sources of background fluorescence become. The configurations inFigure 7.10C (3) and D use a spatial filtering method for separating (laser)excitation from emission coming back out of the fiber. In both cases thehighly collimated laser beam passes through the hole in the mirror into thefiber; the fluorescence comes out with a conical distribution determined bythe fiber’s numerical aperture and is reflected by the mirror into the detector.The drawback to this method is that such mirrors must often be custom-made.The diameter of the laser beam, the numerical aperture of the fiber, and themagnification of the coupling lens all affect the placement of the mirror andthe size of the hole. Thus, a fiber with a low numerical aperture would need


the mirror placed far from the coupling lens to maximize the fluorescencecollected. The configuration in Figure 7.10D is simply an extension of that inFigure 7.10C which eliminates the collecting lens; we have found that a smallaspheric mirror produced by Polaroid(41) is satisfactory for this purpose.

7.7. Launching Optics

Up to this point, we have neglected the lens(es) used to couple light intoand out of the fiber. It is fortunate that microscope objectives often fulfill mostof the desiderata for such lenses, being highly corrected, compact, widelyavailable with a variety of characteristics, and relatively inexpensive, con-sidering their quality. Design, construction, and alignment of comparableoptical systems using discrete lenses would be slow and costly. In fact,microscope objectives of low power are well suited to coupling lightinto multi- or singlemode fibers. Practical concerns here include matching theangle of launching and spot size to the fiber numerical aperture and core sizeor matching mode distributions in singlemode fibers.(7) Most microscopeobjectives have low loss in the visible region of the spectrum, but transmissionfalls off abruptly in the near UV; also, the glass in the objective can fluoresceas well. Quartz objectives that minimize these problems are available, but arevery expensive. Reflective objectives are also expensive and are tricky to usewith lasers, but are largely wavelength-independent (see Figure 7.7). Otheroptics used for coupling light into fibers include spherical lenses and gradient-index (GRIN) rods,(7) which are akin to gradient-index fibers. These smalldevices are best suited for use with laser sources and are less convenient to usethan microscope objectives; they are widely used to couple diode laser outputinto fibers.

7.8. Light Sources

A great variety of light sources have been used with fluorescence-basedfiber-optic sensors. Incoherent sources such as lamps or light-emitting diodeshave been widely used until now because they are inexpensive, offer wavelengthvariability, can be very rugged, and are simple and reliable. Despite theseattributes, for many purposes lasers are better sources, mainly because theyare brighter and easier to couple into fibers. Readers unfamiliar with lasercharacteristics are referred to texts(42) and catalogs(43) for more information.An important advantage of lasers for fluorescence sensors is their narrowbandwidth compared with that of lamps; the lack of scattered stray lightat the fluorescence wavelength eliminates a common source of background.Many visible lasers operate at a single line, and multiline lasers can be easily

tuned. Most gas lasers also produce plasma lines at wavelengths differing fromthe laser wavelength, which may mimic the fluorescence to be detected(Figure 7.7). These plasma lines are incoherent and uncollimated, however,and can be eliminated by a monochromator or spatial filter; the perforatedmirrors in Figure 7.10C and D are well suited for this.

Several attributes of a laser must be taken into account for fluorescencesensors. Foremost is the wavelength of the laser, which is constrained by (orconstrains) the choice of fluorophores. In particular, a probe having a lowerpeak extinction coefficient or quantum yield might still provide more signal,if it could be excited more effectively by a particular laser line. (5,44) Thereare several lasers available with output at wavelengths above 500 nm, butcomparatively few with output in the blue and UV regions. The paucity ofUV laser types, along with the high attenuation of fibers in the UV (seeFigure 7.3), is unfortunate, given that many interesting and useful fluorescentprobes are excited in this wavelength range. Future probe development willlikely be at wavelengths ranging into the near IR, where attenuation is low,bandwidths are high, and a variety of sources are available.

In addition to wavelength, other attributes of the laser are important: itspower, stability, spatial and temporal characteristics, cost, and reliability areall factors in the laboratory. For laser-based sensors employed in the field orharsh environments, the cooling, space, and electrical power requirements mayalso be important. Output power must be great enough to provide adequatesensitivity, yet not so great as to damage components of the system or createnonlinear effects. Damage is wavelength-dependent: picosecond microjoulepulses (having very high peak powers) in the infrared may be compressedin singlemode optical fibers,(40) but coupling even modest output from anexcimer laser in the far UV into a large-core silica fiber remains difficult.Stability, both in intensity and beam pointing, is a concern, but most lasersare adequate in this respect. The temporal characteristics are primarily ofinterest for time-resolved experiments, and they are dealt with in the nextsection.

The recent trend toward development of low-power solid-state laserspromises to have a large impact on fluorescence-based fiber-optic sensors. Thediode-pumped YAG lasers with output frequency-doubled in the green† areindicative of this trend, being all solid state, small, lightweight, and rugged,and having low power consumption. At present, they remain expensive and ofmodest (< 5 mW) output, but both characteristics are certain to improve.Prototype continuous-wave (cw) diode lasers emitting frequency-doubled(blue) output have been shown.(45) The commercial availability of gas lasers

† Diode-pumped YAG lasers with frequency-doubled output in the green are currently availablefrom Spectra-Physics, Moutain View, California; Amoco Laser, Inc., Naperville, Illinois, andAB Lasers, Concord, Massachusetts.



emitting lines heretofore only demonstrated in laboratory prototypes is verywelcome: the recent availability of argon output in the 275–305-nm band is ofparticular interest to biophysicists. Similarly, helium–neon lasers emitting at543 nm are notable because they offer a wavelength useful for excitingfluorescence in a package that is rugged, convenient, and potentially batterypowered. The milliwatt output of most of these lasers is hardly a drawback,given the sensitivity of fluorescence, and their typically low power consump-tion frees the practitioner from the tyranny of water cooling.

7.9. Time-Resolved Fluorescence in Fibers

The utility of temporal resolution in fluorescence studies of all kinds ispractically self-evident, particularly to readers of these volumes. Two questionsarise: can one do the same sort of time-resolved experiments “through” fiberoptics, and what are the problems in using fibers for such experiments? In fact,time resolution is already employed in commercial fiber-optic temperaturesensors(15) and offers the prospect of substantial background reduction(Figure 7.8). In addition to the well-known pitfalls of frequency- and time-domain measurements,(46,47) fiber dispersion (discussed above) can poseproblems. Either modal or material dispersion can impose a limit on thetime resolution of a fluorescence measurement, or on how far away it can bemade. A typical value of modal dispersion in a multimode fiber might be300 MHz-km, making subnanosecond measurements through kilometers offiber difficult; however, use of a gradient-index fiber could reduce this substan-tially. A typical value for material dispersion in a singlemode waveguide at633 nm might be 300 ps/nm per kilometer. Thus, a short pulse from a synch-pumped dye laser having a linewidth of 0.01 nm would be lengthened by onlya few picoseconds on going through a kilometer of fiber, but fluorescenceemission over a 50-nm band passing back through that kilometer of fiberwould experience dispersion over 15 ns, which would have to be deconvolutedfrom the fluorescence lifetime itself. An expensive solution to this is todesign and construct the fiber to have zero (or negative) material dispersionin the wavelength range of interest. Note that the material dispersion is wave-length- and material-dependent,(8) typically decreasing going from the visibleto the infrared and becoming zero at or so. From the standpoint offluorescence measurements, this effect could be the source of artifacts similarto the timing errors introduced by the color effect in photomultiplier tubes.

An additional factor is the availability of light sources. The sort of pulsedlamp source which is convenient for time-correlated single-photon counting isdifficult to use with fibers because of its low intensity, poor collimation, andlarge working distance. Pulsed laser sources are better choices for fiberapplications for the reasons described above. In addition to the well-known

problems involved in using pulsed or cw mode-locked lasers for time-resolvedmeasurements,(47) there are additional caveats. In particular, if the spectrallinewidth of the laser is broad, the pulses will be temporally broadened dueto the material dispersion. Also, short pulses having very high peak power canexperience nonlinear effects in the fiber due to its long interaction length.Frequency-domain measurements would appear to be technically easier tocarry out than time-domain measurements, given that the intensity modulationof (more widely available) cw laser, lamp, and diode sources is straight-forward.(24) To date, relatively few reports of time-resolved measurementsthrough fibers have appeared in the literature; advances in technology and theobvious utility of the technique both suggest that it will play an importantrole.

7.10. Polarization

While fluorescence polarization remains an important tool in biochemicalstudies,(48) it has been little employed using fibers. Ordinary fibers becomebirefringent when stressed by bending, causing fluctuations in the propagationof polarized modes. “Polarization-preserving” singlemode fibers propagateorthogonally polarized modes at different speeds. Thus, linearly polarizedlight launched into the fiber at some angle to its axes of stress will be in turnlinearly, circularly, and elliptically polarized as it passes down the fiber andthe relative phases of the E-vectors change.(7) Using coherent sources, if somemode coupling between the orthogonally polarized modes in the polarization-preserving fiber occurs as a result of microbending or just long interactionlength, then interference and intensity fluctuations occur. Also, couplinguncollimated fluorescence into singlemode fibers may prove difficult. Despitethese concerns, the utility and ratiometric nature of fluorescence polarizationmeasurements argue that fiber-optic sensors will ultimately be designed to usethem.†

7.11. Conclusion

We have discussed some of the special problems and opportunities thatresult from employing fiber optics in fluorescence sensors and experiments. Itshould be evident that fiber optics do not represent a panacea, but rather ameans for solving particular problems—sometimes elegantly. One purpose ofthis chapter was to stimulate the community to apply this technology to new

† Companies offering polarization-preserving fibers include Andrew Corporation, Orland Park,Illinois, and York V.S.O.P., Princeton, New Jersey.



experiments in biochemistry and biophysics; from the experimentaldemonstration of such techniques come the practical applications oftomorrow.

Acknowledgments

I would like to thank Susan McBee and Lynne Kondracki for their helpin preparing the figures; Frances Ligler, David Kidwell, and Carl Villarruelfor helpful discussions and careful reading of the manuscript; Michael P. Edenof Polaroid Corporation for the gift of an aspheric mirror; and the Office ofNaval Technology for support.

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16. O. S. Wolfbeis, H. E. Posch, and H. W. Kroneis, Fiber optical fluorosensor for determinationof halothane and/or oxygen, Anal. Chem. 57, 2556–2561 (1985).

17. L. A. Saari and W. R. Seitz, Immobilized morin as fluorescence sensor for determination ofAl(III) , Anal. Chem. 55, 667–670 (1983).

18. L. A. Saari and W. R. Seitz, pH sensor based on immobilized fluorcsceinamine, Anal. Chem.54, 821–823 (1982).

19. D. M. Hercules (ed.), Fluorescence and Phosphorescence Analysis, Wiley-Interscience,New York (1965).

20. S. Udenfriend, Fluorescence Assay in Biology and Medicine, Vols. 1 and 2, Academic Press,New York (1962, 1969).

21. C. E. White and R. J. Argauer, Fluorescence Analysis: A Practical Approach, Marcel Dekker,New York (1970).

22. J. I. Peterson, R. V. Fitzgerald, and D. K. Buckhold, Fiber optic probe for in vivo measure-ment of oxygen partial pressure, Anal. Chem. 56, 62–67 (1984).

23. E. D. Lee, T. C. Werner, and W. R. Seitz, Luminescence ratio indicators for oxygen, Anal.Chem. 59, 279–283 (1987).

24. F. V. Bright, Remote sensing with a multifrequency phase-modulation fluorometer in: Time-Resolved Laser Spectroscopy in Biochemistry (J. R. Lakowicz, ed.), Proc. SPIE 909, pp. 23–28(1988).

25. O. S. Wolfbeis and M. J. P. Leiner, Recent progress in optical oxygen sensing, in: Proceedingsof the SPIE Conference on Optical Fibers in Medicine III, Proc. SPIE 906, pp. 42–48(1988).

26. D. M. Jordan, D. R. Walt, and F. P. Milanovich, Physiological pH fiber-optic chemicalsensor based on energy transfer, Anal. Chem. 59, 437–439 (1987).

27. J. L. Gehrich, D. W. Lubbers, N. Opitz, D. R. Hansmann, W. W. Miller, J. K. Tusa, andM. Yafuso, Optical fluorescence and its application to an intravascular blood gas monitoringsystem, IEEE Trans. Biomed. Eng. BME-33, 117–132 (1986).

28. J. I. Peterson, S. R. Goldstein, R. V. Fitzgerald, and D. K. Buckhold, Fiber optic pH probefor physiological use, Anal. Chem. 52, 864–869 (1980).

29. W. A. Wyatt, F. V. Bright, and G. M. Hieftje, Characterization and comparison of three fiber-optic sensors for iodide determination based on dynamic fluorescence quenching ofrhodamine 6G, Anal. Chem. 59, 2272–2276 (1987).

30. C. Dahne, R. M. Sutherland, J. F. Place, and A. S. Ringrose, Detection of antibody–antigenreactions at a glass–liquid interface: A novel fibre-optic sensor concept, in: Proceedings of theSecond International Conference on Optical Fiber Sensors (R. T. Kersten and R. Kist, eds.),pp. 75–79, VDE-Verlag, Berlin (1984).

31. J. D. Andrade, R. A. Vanwagenen, D. E. Gregonis, K. Newby, and J.-N. Lin, Remote fiber-optic biosensors based on evanescent-excited fluoroimmunoassay: Concept and progress,IEEE Trans. Electron Devices, ED-32, 1175–1179 (1985).

32. C. A. Villarruel, D. D. Dominguez, and A. Dandridge, Evanescent wave fiber optic chemicalsensor, in: Proceedings of the SPIE Conference on Fiber Optic Sensors II, Proc. SPIE 798,225–229 (1987).

33. W. F. Love and R. E. Slovacek, Fiber optic evanescent sensor for fluoroimmunoassay, in:Proceedings of the Fourth International Conference on Optical Fiber Sensors, Institute ofElectronics and Communication Engineers of Japan, Tokyo (1986).

34. T. P. Burghardt and N. L. Thompson, Effect of planar dielectric interfaces on fluorescenceemission and detection, Biophys. J. 46, 729–737 (1984),

35. E.-H. Lee, R. E. Benner, J. B. Fenn, and R. K. Chang, Angular distribution of fluorescencefrom liquids and monodispersed spheres by evanescent wave excitation, Appl. Opt. 18,862–868 (1979).

36. J. P. Dakin and A. J. King, Limitations of a single optical fibre fluorimeter system due to



background fluorescence, in: Proceedings of the First International Conference on OpticalFibre Sensors, pp. 195–199, Institution of Electrical Engineers, London (1983).

37. K. Newby, W. M. Reichert, J. D. Andrade, and R. E. Benner, Remote spectroscopic sensingof chemical adsorption using a single multimode optical fiber, Appl. Opt. 23, 1812–1815(1984).

38. J. R. Lakowicz and H. Cherek, Phase-sensitive fluorescence spectroscopy: A new method toresolve fluorescence lifetimes or emission spectra of components in a mixture of fluorophores,J. Biochem. Biophys. Methods 5, 19–35 (1981).

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42. O. Svelto, Principles of Lasers, 2nd Ed., Plenum Press, New York (1982).43. T. V. Higgins (ed.), Lasers and Optronics Buying Guide, Gordon Publications, Dover,

New Jersey (1988).44. R. B. Thompson and L. Vallarino, Novel fluorescent label for time resolved fluorescence

immunoassay, in: Time-Resolved Laser Spectroscopy in Biochemistry (J. R. Lakowicz, ed.),Proc. SPIE 909, pp. 426–433 (1988).

45. G. Tohmon, K. Yamamoto, and T. Taniuchi, Blue light source using guided-wave frequencydoubler with a diode laser, in: SPIE Conference on Miniature Optics and Lasers, Proc. SPIE898, pp. 70–75 (1988).

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8

Inhomogeneous Broadening ofElectronic Spectra ofDye Molecules in Solutions

Nicolai A. Nemkovich, Anatolyi N. Rubinov, andVladimir I. Tomin

8.1. Introduction

It is well known in spectroscopy of complex organic molecules that thelarge width of their absorption and emission bands is largely due to theexistence of a continuous set of vibrational sublevels in each electronic state.The spectroscopic properties of dye molecules in solution are in additioninfluenced by the surrounding medium.(1–3) For several decades it had beenbelieved that because of the fast energy exchange between the vibrational sub-levels the fluorescence spectrum of organic dye in solution was independent ofthe frequency of the exciting light. In 1970 it was shown(4) for the first timethat apart from molecular vibrations there is another cause of the substantialbroadening of electronic spectra of organic molecules in solution, namely, thefluctuations of the structure of the solvation shell surrounding the molecule.The variation of the local electric field caused by the fluctuation of the shellstructure leads to a statistical distribution of the frequencies of the electronictransitions of the molecules and, therefore, to inhomogeneous broadening ofthe dye spectrum.

This broadening was experimentally demonstrated(4–6) from thedependence of the fluorescence spectrum of a dye solution on the excitingradiation frequency at 77°K. Later, Personov et al.(7) observed inhomo-geneous broadening of electronic spectra from frozen solutions of complexmolecules at lower (liquid helium) temperatures. It was shown that at liquidhelium temperatures discrete fluorescence spectra of complex molecules with

Nicolai A. Nemkovich, Anatolyi N. Rubinov, and Vladimir I. Tomin • Institute of Physics ofthe B.S.S.R. Academy of Sciences, Minsk 220602, U.S.S.R.


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368 Nicolai A. Nemkovich et al.

resolved vibrational structure could be obtained by eliminating the inhomo-geneous broadening by means of selective monochromatic excitation of thesolution. This made it possible to gain a deeper insight into the nature of theelectronic spectra of complex molecules and led to new, more sophisticatedmethods for their study. It has been shown(8,9) that exposure of a frozensolution of organic molecules at liquid helium temperatures to sufficientlyintense monochromatic radiation results in a selective energy gap in theirinhomogeneously broadened absorption spectrum. Such an energy gap mayexist for a long time, which opens up new possibilities of studying very subtlespectroscopic effects in complex molecules and of using such systems asoptical memories. Subsequently, it was shown(10–12) that inhomogeneousbroadening of electronic spectra of complex molecules occurs not only in solidsolutions but in liquid solutions, too. In the latter case, it is dynamic in natureand can only be detected by nano- or picosecond time-resolved spectroscopictechniques.

The systematic study of inhomogeneous broadening offered an essentiallynew picture of the nature of the spectra of complex molecules in solutions atvarious temperatures and revealed new spectroscopic effects and features.Thus, the “up-relaxation” effect was discovered.(10) This effect manifested itselfas an increase in the energy emitted by a fluorescent molecule during con-figurational relaxation of the solvation shell surrounding it. In other work,(13)

the phenomenon of directional nonradiative energy transfer from the “blue” tothe “red” centers of an inhomogeneous ensemble of complex molecules insolid solutions was experimentally observed. This phenomenon explains suchwell-known effects as the Weber effect(14–17) and the red shift of fluorescencespectra with increasing concentration in such systems,(18) and it gives a deeperinsight into the mechanism of energy transport in the photosynthetic systemof plants.(19, 20)

Finally, optically induced dye molecule rotation due to the configurationalrelaxation of nonequilibrium solvates was observed in polar solutions.(21) Theeffect depends on the excess of configurational energy of selectively excitedsolvates and leads to specific dependencies of the kinetics of radiationanisotropy on the exciting light frequency and the frequency at which theemission is recorded.

Thus, the discovery of inhomogeneous broadening of electronic spectra ofcomplex molecules in solid and liquid solutions gave rise to a new field,namely, the selective spectroscopy of organic solutions. The main feature ofthis field is the selective excitation of some members of an inhomogeneousensemble of molecules, which enables them to be studied and treated selectively.

This chapter is not intended as a comprehensive review of studies ofinhomogeneous broadening. We will just consider the principal physicalfeatures of this phenomenon and the main spectroscopic effects associatedwith it.

Inhomogeneous Broadening of Electronic Spectra of Dye Molecules 369

8.2. Theoretical Considerations of Inhomogeneous Broadening

8.2.1. Solvate Configurational Energy

The spectroscopic properties of a dye solution may be analyzed by treatingits elementary cell (solvate), which includes a fluorescent organic moleculeand its immediate surroundings. Due to the statistical variation of the cellstructure, different cells may have different local electric fields. To describe thissituation phenomenologically, a solvate field diagram (representing the poten-tial energy of the solvate versus the reactive field strength) was introduced.(22)

The use of such a diagram makes it easier to analyze the inhomogeneousbroadening of electronic spectra of dye molecules in solution in order tounderstand its various spectroscopic manifestations. In this section we analyzethe inhomogeneous broadening characteristics of fluorescent organic moleculesin solid and liquid solutions using a classical solvate model suggested inRef. 12.

Without specifying the dimensions and spatial configuration of thesolvation shell, we will treat it in terms of its macroscopic characteristics, thatis, its susceptibility, just as various dielectric materials are treated.

First, consider a polar solution in which the solute molecules possessconstant dipole moments. In each elementary cell of such a solution theimmediate surroundings are polarized due to the dipole moment, of thedye molecule, thus giving rise to a reactive field, in the cell:

where is the susceptibility of the solution.Inhomogeneous broadening occurs because, as a result of the thermal

motion of molecules in a solution, different cells have different solvation shellsand, therefore, different reactive fields R, which differ somewhat from themean value of found in Eq. (8.1). As the 0–0 transition frequency ofthe dye molecule depends on the reactive field intensity, this means that in thesolution there is an inhomogeneous set of molecules specifically distributedin frequencies of the 0–0 transition. This distribution is the fundamentalcharacteristic of inhomogeneous broadening, hereinafter referred to as theinhomogeneous broadening function, The function can be found ifthe dependence of the 0–0 transition frequency on the strength of the reactivefield R and the elementary cell distribution with respect to field intensity areknown. Since as a function of R is well known,(1) the problem is one offinding the distribution over R.

Let us assume in the simplified model under consideration that two forcesare responsible for the elementary cell state: (1) the polarizing force which is


due to the presence of a constant dipole moment in the dye molecule andinduces the reactive field R in the cell; and (2) the restoring force which is dueto the action of the reactive field on the dipolar molecules of the solvation shell.The action of the restoring force becomes evident if we consider what wouldhappen if the dye molecule were extracted from the cell. In this case the cellwould tend to reach its initial state at under the action of the restoringforce. The cell is in its equilibrium state when the two forces cancel eachother. This state is the most stable one, and it is characterized by the reactivefield Any deviation from this state requires some work to be done. Thelatter will just be the measure of the potential energy of the nonequilibriumsolvate configurational interaction. Obviously, an absolute thermodynamicequilibrium must lead to a Boltzmann equilibrium distribution of cells withrespect to the energy of the configurational interaction. Thus, the problemis reduced to determining the configurational energy of the cell versus itsreactive field R.

Taking into account that the reactive field R is parallel to the dipolemoment inducing this field, the dipole–field interaction energy can berepresented as

This energy is essentially the work done by the dipole to reorient (polarize)an elementary cell of the solution. The work necessary to perform suchreorientation of the solvation shell and to change the reactive field by dR canbe written in differential form as

It can easily be seen that in the solvate and R may be treated as a polarizingforce and the generalized coordinate for the solvation shell configuration,respectively. The restoring force acts along with the polarization force

in the solvate. Thus, the total force affecting the solvation shell is

For the cell configuration corresponding to the reactive field bothforces are equal, and

Since we are considering the dipole–dipole interaction, the restoring forcemust obviously be proportional to the field:

Assuming that at the total force is zero and taking into accountEqs. (8.1) and (8.4), we obtain the proportionality factor


Consequently, the restoring force is of the form

Thus, according to Eqs. (8.1) and (8.5), the total force acting upon thesolvation shell is

It can be seen from Eq. (8.8) that the total force is zero only for cells withSuch cells are the most stable ones and possess minimum configura-

tional interaction energy.The deviation of the cell configuration from the equilibrium configuration

involves an increase in energy as compared to the minimum value. Themagnitude of this excess may be considered as a measure of the potential orfree energy of the nonequilibrium solvate. For a cell with the reactive field R,this energy can be determined as the work required to restructure the solvatefrom its equilibrium state with the field to the state with the field R:

As can be seen from Eq. (8.9), a plot of the potential energy of the solvateconfigurational interaction versus the internal reactive force has a parabolicshape with its minimum at

8.2.2. Field Diagram of a Polar Solution

It is convenient to introduce the total solvate energy, which is the sum ofthe configurational and electronic energies of the solvate molecules.† Since weare considering electronic transitions of the dye molecule only, the electronicenergy of all the solvation shell molecules can be assumed invariable and equalto zero. We will call elementary cells with the dye molecule in its groundand excited states unexcited and excited solvates, respectively. Assuming theelectronic energy of the ground state of the dye molecule to be equal to zero,the total unexcited solvate energy will be described by Eq. (8.9).

The electronic transition frequency of the dye molecule as a function ofthe reactive field of the solvate can be written(1) as‡:

† In thermodynamics it may be called “the free energy of the system.”‡ For simplicity, we will consider the electronic transition with a constant dipole moment of the

dye molecule changing its magnitude rather than orientation. We will also neglect the quasi-continuum of vibrational states of a complex molecule in this section.


where and are the dipole moments of the dye molecule inthe ground and excited states, respectively, and is the 0–0 transitionfrequency of a free (noninteracting) molecule.

Upon absorption of a quantum hv and transition of the solvate to theexcited state, the energy of the solvate becomes, according to Eqs. (8.9) and(8.10), equal to

This equation takes into account that the reactive field for an equilibriumsolvate in its excited state is equal to

As seen from Eq. (8.11), the potential energy of the excited solvate is repre-sented by the same parabola as for the unexcited state, but raised by

and it has its minimum at a value of the reactive fieldThus, we have arrived at the field diagram (Figure 8.1) that was intro-


duced earlier on a purely phenomenological basis.(11) This diagram representsthe dependence of u, the sum of the orientational interaction, and electronicenergies of the molecules in the first coordination sphere, or the potentialenergy E of the solvate on the electric field R. This dependence is similar tothat for the potential energy of a molecule, with the differences that, insteadof a molecule, the solvate as a whole is considered and that, instead of thevibrational energy, the molecular configurational interaction energy in thesolvate is considered. The solvate field intensity is naturally the generalizedcoordinate.

This diagram can conveniently be used to analyze the spectral propertiesof a solution, and it automatically takes into account the inhomogeneousbroadening. The shell structure and, therefore, the field remain unchangedduring the electronic transition. As in the case of molecules, the Franck-Condon principle is in force here; that is, quantum transitions between thepotential curves are given by vertical lines on the diagram. For unexcitedsolvates, the equilibrium Boltzmann distribution of the configuration energyis normally valid. The configuration energy distribution for excited solvatesdepends on the relation between the lifetime of the excited fluorescentmolecule and the configurational relaxation time, of the elementary cell. At

the equilibrium distribution of excited cells over the configurationalstates is attained, whereas in the reverse case the distribution function is non-equilibrium and depends on the exciting light frequency.

In Figure 8.1a, we may recognize two characteristic transition frequenciesfor the inhomogeneous ensemble of molecules under consideration: corre-sponding to the 0–0 transition of solvates which have the equilibriumconfiguration in the unexcited state, and corresponding to the 0–0transition of solvates whose configuration is the equilibrium one in the excitedstate. According to Eq. (8.10) and taking into account Eqs. (8.1) and (8.12),these frequencies are given by

The field diagram in Figure 8.1a corresponds to the case in which thedipole moment of the molecule increases on excitationIn this most common case, the solvent causes both inhomogeneous broaden-ing and a general spectral shift to ward longer wavelengths (Figure 8.10)(negative solvatochromism). When the solvent causes a spectral shiftto ward shorter wavelengths (positive solvatochromism), and the fielddiagram of such a solution has the form shown in Figure 8.1b. The followinganalysis of the inhomogeneous broadening characteristics holds equally forboth cases.

8.2.3. Solvate Distribution in Configurational Sublevels

If the lifetime of unexcited solvates is long enough (i.e., moderate excitinglight intensities are used), the distribution of solvates in energy of configura-tional interaction is of the Boltzmann form:

where c is a normalization factor, and is the statistical weight of thestate with energy Substituting Eq. (8.9) into Eq. (8.16), we obtain thesolvate distribution as a function of the local field R:

where is the width of the distribution function.Thus, the distribution of solvates with respect to fluctuations of the local

field is of a symmetric Gaussian form, and the width of the distribution isuniquely determined by the susceptibility of the solution and the temperature.It may be assumed that the local field fluctuations are caused bythe susceptibility fluctuations of different elementary cells of the solution.According to Eq. (8.1),

Substitution of Eq. (8.18) into Eq. (8.17) gives the susceptibility fluctuationdistribution function for the cells

where is the width of the distribution function.Of particular interest is the solvate distribution over the electronic

transition frequencies of the dye molecules. Using Eqs. (8.14) and (8.1), theelectronic transition frequency Eq. (8.10) can be given as

Using this expression, we obtain from Eq. (8.17)

where

Thus, the inhomogeneous broadening function characterizing the solu-



tion absorption spectrum is found to be of a symmetrical Gaussian form withits maximum at Inhomogeneous broadening (Eq. 8.22) depends on thedifference between the dipole moments of the dye molecule in its ground andexcited states as well as on the solvent susceptibility and temperature. Thetemperature dependence of exists until the freezing point of the solutionis reached. Then, the spatial configuration fluctuations of the elementarycells become fixed, and the inhomogeneous broadening function becomestemperature-independent. When calculating the width of the inhomogeneousspectrum for a frozen solution, the melting point should be substituted for Tin Eq. (8.22), irrespective of the real experimental temperature.

As can be seen from Eq. (8.22), inhomogeneous broadening depends onthe susceptibility of the medium. Employing the Onsager model for a cell inthe solution, the value of can be found from the formula

where is the dielectric constant of the medium, and a is the Onsager sphereradius. Using this expression, Eq. (8.22) may be written in the form

where

The Onsager sphere radius is governed by the size of the solutemolecules. Thus, according to Eq. (8.24), inhomogeneous broadening dependsnot only on the dipole moments of the dye molecule but also, and mostessentially, on its dimensions. The smaller the molecule, the stronger is theinhomogeneous broadening effect. This agrees with experimental data:inhomogeneous broadening is most pronounced for small organic molecules,for example, phthalimides.

The estimates of presented in Table 8.1 for typical polar solutions, for


example, 3-aminophthalimide in ethanol, show that, at rangesfrom 400 to and that it decreases to at the freezingpoint of the solution (158°K). In obtaining these estimates, the followingvalues were used in Eqs. (8.22) and (8.24):

8.2.4. Nonpolar Solutions

Above we have considered the case in which the fluorescent moleculehas a constant dipole moment which changes as the molecule goes to theexcited state. Inhomogeneous broadening will also manifest itself when theluminophore molecule does not possess a constant dipole moment either in itsground or in its excited state. The solvent may be polar or nonpolar and ischaracterized by its susceptibility

In this case the equilibrium state of the cell will obviously correspond tothe local field However, because of the thermal motion of molecules inthe solution, nonequilibrium states for the cells with will occuralongside the equilibrium ones. The configurational energy of such a cell, orrather of its solvation shell, can be found from Eq. (8.9) assuming

The local fluctuation field R acting on the dye molecule will polarize it, thuscreating an induced dipole moment:

where is the molecular polarizability. Thus, along with the configurationalenergy of the solvation shell, the nonequilibrium cell is also characterized bythe energy of interaction between the induced dipole moment and the localfluctuation field. The absolute value of this interaction energy is

The total configurational–interaction energy of the elementary cell is the sumof Eqs. (8.25) and (8.27):

On transition to the excited state, the molecular polarizability changes andbecomes equal to In addition, the energy of the cell increases by the energyof the absorbed light quantum. It can easily be seen that the total energy ofthe excited solvate will be


where is the electronic transition frequency of a free molecule. The solvatefield diagrams for and are given in Figures 8.2a and 8.2b,respectively. In both cases, the parabolas lie one above the other and havetheir minimum at but they differ in shape. The electronic transitionfrequency for the cells with the local field is

Taking into account that unexcited cells obey the Boltzmann law and usingEq. (8.28), we obtain the solvate distribution as a function of local fieldfluctuations:

where

(8.32)

As seen, in this case also, the cell distribution over local field fluctuations isof a Gaussian form. The distribution over electronic transition frequencies is,however, of a different form. We obtain from Eqs. (8.30) and (8.31)


where

Thus, for nonpolar dye molecules there is an asymmetric exponential distribu-tion of cells over electronic transition frequencies with a sharp boundary atthe frequency For molecules with the exponential rise is towardshorter wavelengths; for molecules with toward longer wavelengths.

Estimations of by Eq. (8.34) for typical systems similar to thoseinvestigated experimentally yield for a polar solvent with

(e.g., butyl bromide) and for a nonpolar solvent(e.g., n-hexane). A dye molecule of the coronene or perylene type wastested; its polarizability was estimated by the valence-optical scheme as

and and a were assumed to be and2 Å, respectively. The estimates agree fairly well with the experimental data

obtained for the inhomogeneous broadening function by themethod of double scanning for a perylene solution in n-hexane.(23)

8.2.5. Selective Excitation with Vibrational Spectral Broadening

The spectra of a molecule in solution can be appropriately described onlywith knowledge of both the inhomogeneous broadening function and thehomogeneous molecular spectra, that is, the absorption and fluorescencespectra of molecules in cells of the same configuration. If the homogeneousspectrum is essentially narrower than the inhomogeneous one, both theshape and the width of the experimentally observed spectra will mainly bedetermined by the inhomogeneous broadening effect and will be described byEqs. (8.21) and (8.31). In this case, monochromatic radiation providesselective excitation of a given group of elementary cells of an inhomogeneousensemble. When the homogeneous spectrum is wider than the inhomogeneousone, the inhomogeneous broadening effect can only weakly affect the shape ofthe experimentally observed spectra, and, generally speaking, the possibility ofselective excitation of individual solvate groups is lost.

The homogeneous spectra of complex molecules at moderate temperatures(essentially above liquid helium temperature) are defined by the presence in themolecule of a quasi-continuum of vibrational states. As is well known,(24, 25)

energy exchange between these states occurs in extremely short times (less than10–13 s), and this is responsible for the homogeneous nature of vibrationalbroadening.

Below, a specific situation is shown to obtain for complex molecules insolution. Selective excitation of solvates proves to be possible even when thewidth of the homogeneous spectrum exceeds that of the inhomogeneous


spectrum. Thus, the inhomogeneous broadening effect will be apparent inpractically all cases.

Let the solution be excited by radiation of frequency v. As seen fromFigure 8.1, photons of this frequency will be absorbed both selectively, bynonequilibrium solvates of type i, for which the frequency of the 0–0transition in the dye molecule coincides with the exciting light frequency, andnonselectively, by equilibrium solvates with the type I configuration. First,consider the case where In this case, the radiation of frequency v willonly be absorbed by those equilibrium solvates in which the dye molecule hasexcess vibrational energy equal to

To characterize the degree of excitation selectivity, we introduce the quantitywhich is the ratio of the number of selectively excited solvates of type i

to the number of nonselectively excited solvates of type I (i.e., solvates whichhad the equilibrium configuration in the ground state):

The values of and can be found from the balance equations if theBoltzmann distribution of the configurational and vibrational energies of theparticles is taken into account:

Since the configurational energy distribution is continuous, n(E) is thenumber of solvates per unit energy for the energy value of E.

For equilibrium solvates and excitation at a steady-stateradiation density U, the balance equation is of the following form:

where is the fluorescent lifetime of a molecule in the solvate with the typeI configuration, and is the Einstein coefficient defining the absorption ofsuch solvates at a frequency v. Using Eqs. (8.35) and (8.37), we obtain, fromEq. (8.38),

where n is the total concentration of unexcited solvates.For nonequilibrium solvates of the ith type the balance equation has the


form

and, consequently, the number of excited particles is

where is the Einstein coefficient for absorption, and is the fluorescentlifetime of a molecule in a cell of the ith type. Substituting Eqs. (8.39) and(8.41) into Eq. (8.36) and using Eqs. (8.9) and (8.20), we obtain for the degreeof selectivity:

Here is the width of the inhomogeneous broadening function, given forpolar molecules by Eq. (8.22). For simplicity, assume the homogeneousspectrum to be of the Gaussian form:

where is the 0–0 transition frequency for ith type solvates, and is thewidth of the homogeneous absorption band. For ith type solvates resonantlyexcited by frequency v, the condition is fulfilled, and henceFor the type I solvates, and the Einstein coefficient for the excitinglight frequency is equal to

Substituting these values of and into Eq. (8.42), we obtain for thespectral excitation range (corresponding to the long-wavelength slope ofthe total absorption band) the following expression for the degree of selectivity:

A similar analysis can be made for the exciting light frequency range(corresponding to the short-wavelength slope of the total absorption band).The difference between these two cases is that radiation of frequencycan be absorbed by all the equilibrium solvates irrespective of the vibrationalenergy stored by the dye molecule. Integration of the expression analogous


to Eq. (8.38) will therefore cover the range from 0 to For excitationfrequencies this results in

It follows from Eqs. (8.45) and (8.46) that the excitation selectivities for theblue and the red edge of the total absorption band of a solution are essentiallydifferent. Excitation can be regarded to be selective when since only inthis case is the number of selectively excited nonequilibrium solvates greaterthan that of nonselectively excited equilibrium solvates.

Following Eq. (8.46), in the case where and is assumedto increase with blue-edge selective excitation is only possible for

that is, under the trivial condition that the width of the inhomo-geneous spectrum is wider than that of the homogeneous spectrum. Forcomplex molecules, the width of the vibrational (homogeneous) spectrumamounts to hundreds of wave numbers; that is, it is of the same order ofmagnitude as the width of the inhomogeneous spectrum. Therefore, noselectivity will be observed with blue-edge excitation, as a rule.

The situation is, however, different for the long-wavelength, red-edgeabsorption. According to Eq. (8.45), in this case the degree of selectivity canbe greater than 1 even at In other words, if the incident lightfrequency is localized at the long-wavelength edge of the absorption band,selective excitation of nonequilibrium solvates will be ensured even whenthe width of the homogeneous spectrum exceeds that of the inhomogeneousspectrum. As seen from Eq. (8.45), however, this is true provided theexcitation frequency is within the frequency range given below:

The maximum value of the selectivity is found in the middle of thisrange. On shifting toward both shorter and longer wavelengths, the excitationselectivity decreases, and it is less than unity beyong this range. Thedependence of the selectivity on the relation between the widths of thehomogeneous and inhomogeneous spectra is illustrated for all the above casesin Figure 8.3. Note that the degree of selectivity is also a function of the ratio

that is, it depends on how the lifetime of a molecule’s excited statechanges depending on the elementary cell configuration.

As noted above, fluctuations of the local structure become fixed in afrozen solution and are no longer temperature-dependent (provided no phasetransitions occur with decreasing temperature in the frozen solution). There-fore, the inhomogeneous spectrum broadening function for a solution at liquidhelium temperature is the same as at liquid nitrogen temperature and can be


calculated using Eqs. (8.21) and (8.22) for polar molecules and Eqs. (8.33)and (8.34) for nonpolar molecules if the solution melting point is taken as thevalue of T. As shown above (see Table 8.1), the width of the inhomogeneousfunction has been estimated as for polar molecules and

for nonpolar molecules. These results are of the same order ofmagnitude as measurements made at liquid helium temperature.(23, 26) Thus,for the polar 1-iodonaphthalene molecule in butyl bromide atan inhomogeneous spectral width of was reported,(26) whereas, asmentioned above, for the nonpolar perylene molecule in n-hexane at the sametemperature, was measured to be

Thus, the characteristics of inhomogeneous spectral broadening ofcomplex molecules at liquid helium and liquid nitrogen temperatures prove tobe the same. However, the homogeneous spectra of molecules are essentiallydifferent at these temperatures. As shown in Refs. 7 and 27 for amorphousmatrices, fine-structure, rather than diffuse, absorption and fluoresenceespectra consisting of a set of narrow nonphonon lines are observed at liquidhelium temperature. Superposition of such spectra belonging to cells ofdifferent configurations leads to a structureless spectrum. Therefore, it is theinhomogeneous broadening effect that plays the determining role in thecommonly observed structureless appearance of absorption and fluorescencebands. If the homogeneous linewidth is rather small (experiments(28, 29) show


the nonphonon linewith to be to ), that is, provided theconditions and are fulfilled, the selective excitation

function becomes symmetric about the frequency and, following Eqs. (8.45)and (8.46), assumes the form

(8.48)

shown in Figure 8.3.However, in reality, due to the overlap of the phonon wings at short

wavelengths, selective excitation of nonequilibrium cells at liquid heliumtemperature can be realized only for long wavelengths of an inhomogeneouslybroadened absorption band.

8.2.6. Absorption and Fluorescence Spectra: Dependence on Exciting LightFrequency

Let us consider the influence of the configurational broadening on theabsorption and fluorescence spectra of a solution. To calculate the totalabsorption spectrum, we will integrate the transition probability over all theinitial configurational states of the solvates and over all the initial vibrationalstates of the dye molecule, taking into account the corresponding distributionfunctions:

Here is the coefficient for absorption of light of frequency v bydye molecules with the pure electronic transition frequency vel when in thevibrational state of energy

The vibrational state-averaged Einstein coefficient

characterizes the homogeneous molecular absorption spectrum:

Also, for simplicity, let the configurational interaction be weak compared to theintramolecular one. Then may be assumed to be independentof the stored vibrational energy that is, the shape of the spectrum is the

same for all the solvates. Thus, the configurationally broadened absorptionspectrum can be represented as

where the distribution function is given by Eq. (8.21).Note that such a distribution is usually observed experimentally in the

absence of strong excitation and enhanced inflow of molecules from excitedstates. An increase in the exciting radiation density leads to a shortening ofthe time that the molecule spends in the ground state. Thus, if the excitationdensity at is such that the inequality

is fulfilled, the solvate distribution in the ground state should differ from theequilibrium one. In this case, the solution becomes more transparent andbehaves as a spectrally inhomogeneous system.

With knowledge of the shape of the homogeneous absorption spectrumthe absorption band of the solution can be calculated from Eq.

(8.52). As shown in Ref. 30, for complex organic molecules the absorptionband can be nicely approximated by the function

The matching coefficients a and b are related to the width of the absorp-tion band (for the level determined by the pure electronic transition

frequency ) and the distance between the absorption maximum and thefrequency by the relations

Figure 8.4 presents calculated homogeneous and inhomogeneous absorp-tion spectra obtained from Eqs. (8.49)–(8.54) using values of the parameterstypical for phthalimide molecules. In this case, the width of the homogeneousspectrum, is close to that of the inhomogeneous spectrum,

The configurational broadening does not, therefore, causeany essential changes in the shape and position of the absorption band. Thedeformation is most pronounced at the long-wavelength slope of the band.The absorption in this region is mainly due to the nonequilibrium solvates. Inaddition, the absorption band is transformed with temperature due to thedependence of on T. It should be noted that with rising temperature thebroadening of the band prevails.



The steady-state fluorescence spectrum of a solution, in accordance withEq. (8.52), can be expressed as

where

is the Einstein coefficient for spontaneous emission integrated over vibrationalstates, and is the pure electronic transition frequency distributionfunction of excited cells. Using the same approximation as in Eq. (8.53), thespectral relationship can be expressed as

where the coefficients a and b are the same as in Eq. (8.53).As shown in Ref. 4, the relation between the configurational relaxation

time of the medium and the excited state lifetime of the molecule maygive rise to two different situations. If the condition of fast reorientation ofthe solution molecules, is fulfilled, then a steady-state equilibriumdistribution of the cells with respect to configurational energy is attainedduring the lifetime of the excited state. In this case, the solvate distribution isof the following form:


The homogeneous and configurationally broadened fluorescence spectracalculated from Eqs. (8.55)–(8.58) are also shown in Figure 8.4. In this case,the configurational spectral broadening is seen to result in some broadeningof the fluorescence band at its short-wavelength slope. The position and shapeof the spectrum are independent of the exciting light frequency.

The situation is essentially different for a rigid solution characterized byslow reorientation of molecules, In this case, the distribution function

depends on the exciting light frequency is the number of cellswith electronic transition frequency that are excited at excitation frequency

is the total number of excited cells.The quantities and can be found from the balance

equations for the steady-state condition. As a result, one can easily obtain

where is the excited state lifetime for molecules with electronic transi-tion frequency is the mean lifetime of excited molecules, and isdetermined from Eq. (8.21) and from Eq. (8.53) with the excitationfrequency substituted for v.

The solution fluorescence spectrum changes with the distribution functionFigure 8.4 shows the fluorescence spectra calculated from Eqs.

(8.55)–(8.60) as a function of the exciting light frequency. Upon excitation atthe long-wavelength slope of the absorption band, the fluorescence spectrumdepends on the excitation frequency; it is shifted toward longer wavelengthsas decreases. Initially, the change in does not alter the shape of thefluorescence band. Only for rather small incident light frequencies, that is,upon excitation of essentially nonequilibrium solvates, is spectral broadeningobserved owing to the above-mentioned decrease in the excitation selectivity

Thus, when the condition is satisfied, the configurational spectralbroadening of the solution is inhomogeneous, which is manifested as a strongdependence of the position of the fluorescence maximum on the exciting lightfrequency, the latter being localized at the long-wavelength slope of theabsorption band. The experimental verification of this conclusion was of greatimportance as it could provide evidence for the validity of the theoreticaltreatment.

In the next section we present results obtained from the nonhomogeneoussolution model.


8.3. Stationary Inhomogeneous Broadening

8.3.1. Universal Relationship between Fluorescence and Absorption Spectra ofPolar Solutions

Stepanov and Gribkovsky have established a general law relating theabsorption spectra of complex molecules to their fluorescence spectra.(31)

According to this law, the contours of the absorption K(v) and fluorescencebands of such species are related by

where D(T) is a temperature-dependent constant. Equation (8.61) is analogousto the well-known Kirchhoff law, which states the relationship between theabsorption and luminescence spectra of a blackbody, since luminescencepower is proportional to thermal emission power for complex molecules.

Sufficiently fast vibrational relaxation of molecules in both singlet statesand is the only condition for these relations to be fulfilled. As the

vibrational relaxation time in dye molecules is very short theuniversal relationship should be valid for practically all solutions of complexorganic compounds. Experiments(1) have shown, however, that for polar solu-tions the universal relationship holds at room temperature, but is violated atlow temperatures in viscous and especially in rigid frozen solutions.

It follows from the previous section that the absorption K(v) andemission of the solution should be described as integrals over theconfigurational energy continuum (Eqs. 8.52 and 8.55).

As can be seen from Eqs. (8.52) and (8.55), the relationship between theabsorption and fluorescence spectra depends on the configurational energycontinuum distribution function. Thus, if and are equilibriumBoltzmann functions, then can easily be shown(32) to obeyEq. (8.61). This is valid for liquid solutions at room temperature when fastreorientation of solvent molecules occurs and the Boltzmann distribution overconfigurational sublevels reaches steady state in times much shorter than thefluorescence lifetime. If the function differs from the Boltzmann form,which is encountered in low-temperature or high-viscosity solutions whenthe reorientation is delayed or absent during the excited state lifetime, theuniversal relationship of Eq. (8.61) is not obeyed.

Thus, the solution model that has been developed to account for theinhomogeneous broadening of solution electronic spectra predicts boundariesof validity of the universal relationship of Eq. (8.61) that are in agreementwith experimental data.


8.3.2. Luminescence Spectra at Red-Edge Excitation

As shown in Section 8.2.6, it follows from the proposed model forpolar solutions that in high-viscosity solvents (e.g., frozen ones), in whichmolecular reorientation is slow or absent and energy exchange between theconfigurational sublevels of the excited state is hindered or does not occur,luminescence will occur from the configurational states into which the solvateswere excited or from the states close to them. In the most common case wherethe electric dipole moment of a luminophore molecule increases ongoing from the ground state to the upper singlet state, in accordance withEqs. (8.10), (8.14), and (8.15) and as shown on the field diagram of Figure8.1, the steady-state fluorescence spectrum of the solution must be shiftedtoward wavelengths as the viscosity of the solution increases. Thisphenomenon, known as the blue S-shaped shift of the fluorescence spectrum,has received much study for frozen solutions(1) and is most pronounced forexcitation at the absorption band maximum.

For a long time there was a firm belief that in pure dye solutions theluminescence spectra are independent of the exciting light frequency. Thisstatement was formulated as a law.(33) Any deviation from this law wasattributed the presence of admixtures or aggregates in the solution.

However, it follows from the field diagram for polar solutions (Figure8.1) that, by varying the excitation frequency, we will selectively excitedifferent configurational states, and in this case inhomogeneous broadening ofthe solution spectra is expected.

A study of polar solution spectral properties for was undertaken.This condition can be realized either by increasing or decreasing Theformer can be achieved by freezing the solution or using polymer matrices,whereas the latter can be achieved by quenching the luminescence by usingadmixtures or a powerful light flux. AH four approaches were tested experi-mentally.(4, 6, 34–36) The excitation frequency dependence of the fluorescencespectrum was first revealed for frozen solutions of phthalimide derivatives.(4, 6)

Later, it was observed for rigid polymer solutions of these compounds atroom temperature as well as for other dyes.(34) As this effect consisted in thered shift of the fluorescence band, it was called “bathochromic luminescence”(BL). Actually, this term applies to luminescence from selectively excitedsolvate states with a large store of configurational energy.

It has been shown(35, 36) that BL is also observed when the conditionis fulfilled due to the reduction of the dye molecule excited state

lifetime as a result of fluorescence quenching by impurities or in a powerfullaser field rather than due to an increase in the configurational relaxationtime of the solution. These experiments have firmly established that BLof polar solutions is caused by inhomogeneous configurational broadeningof electronic energy levels. Later, it was found that not only singlet but


also triplet states of dyes were broadened configurationally and, under thecondition of slow reorientation of molecules in a solution, this broadeningwas inhomogeneous.(37)

The application of tunable dye lasers for excitation opens up newpossibilities of studying configurationally nonequilibrium solvates in asolution. Figure 8.5 gives BL measurement data for a frozen ethanol solutionof the well-known dye rhodamine 6G under fluorescence excitation by acontinuous-wave (cw) tunable dye laser. The solid curve (I) represents theabsorption spectrum. At the bottom of the figure, a set of luminescence curvescorresponding to excitation at different wavelengths (indicated by the arrows)in the far anti-Stokes spectral region is given. Absorption in this region isalready very small due to the small population of the upper configurationalstates. Nevertheless, the excitation of luminescence proved to be possible dueto the high laser radiation intensity employed. Excitation at frequencynear the absorption maximum yielded the fluorescence spectrum labeled 1.The position and the shape of this spectrum are practically independent ofthe frequency for varying within the main part of the absorptionband, where the broadening is largely due to the vibrational sublevels of theequilibrium unexcited configurational state. The situation changes, however,when is shifted to the anti-Stokes region. Because of the abrupt break inthe long-wavelength slope of the absorption spectrum of complex molecules in


solution, there exists a possibility of selective excitation of some solvategroups whose electronic transition frequencies differ from that of the equi-librium configurational state. It should be noted that with decreasing anappreciable red shift (up to several tens of nanometers) of the luminescencespectrum is observed. In liquid solutions at room temperature, this effect dis-appeared and the luminescence band did not depend on

Figure 8.5 also shows the BL excitation spectrum obtained by recordingrhodamine 6G fluorescence at a fixed wavelength, In essence, thisspectrum characterizes the absorption of nonequilibrium solvates of a givenconfiguration (determined by the wavelength at which the fluorescence isrecorded). It is seen that the excitation spectrum is similar in shape to theordinary absorption spectrum, but it is substantially red-shifted. This isconsistent with the suggestion that the two spectra belong to identical dyemolecules which differ only in configuration.

Thus, as follows from the model representations and experimental resultsreported in this section and Section 8.2.6, it is possible to selectively excite andstudy solvates of essentially nonequilibrium configuration whose fluorescencespectrum is shifted to lower frequency by many hundreds of wave numbersrelative to the ordinary fluorescence spectrum. The information on such non-equilibrium solution microstructures is of particular interest because cells ofabnormally large local electric field intensity are concerned here. In such cellsthe dye molecules may differ not only in their spectral frequencies, but, inprinciple, also in their nonradiative transition probabilities.(38)

8.3.3. Directed Nonradiative Energy Transfer in Organic Solutions

It has been known for many years that the influence of inductive-resonance energy transfer between similar molecules on the luminescenceproperties of rigid solutions is restricted by the fact that the degree ofpolarization, the quantum yield, and the decay time decrease with increasingconcentration.(39) The last two effects are believed to be caused by energymigration to low-luminescence centers(39); that is, they would not be observedif luminescence centers of only one type were present in the solution. Themajority of previous theories which described the influence of energy transferon the luminescence characteristics of solutions (see Refs. 39–41) did notpredict the existence of any concentrational effects besides those mentionedabove.

The presence of a continuous set of configurational solvate states presup-poses the possibility of inductive–resonance energy transfer between differentstates provided that the concentration of dye molecules in the solution is highenough. If such a transfer does occur, it will apparently be directed from thesolvates with a high frequency of the pure electronic transition to those with


lower values of this frequency, that is, from the “blue” solvates to the “red”ones (the reverse transfer must be hindered due to its low probability).

All this is illustrated by the diagram in Figure 8.6 of electronic–configura-tional states in solution. For simplicity, the diagram depicts a discrete, ratherthan a continuous, set of electronic–configurational states.

When the solution concentration is increased, the sublevel may be deac-tivated by both spontaneous emission and nonradiative energy transfer(dashed lines in Figure 8.6) to solvates with lower frequencies ofthe pure electronic transition, since the probability of energy transfer in thisdirection is higher than in the reverse direction because of different overlap


integrals of the absorption and emission spectra of “blue” and “red” solvates.For rigid solutions, gradual passage to excitation on the low-frequency slopeof the absorption spectrum, that is, decreasing the excitation frequency,results in selective excitation to the first singlet state of the solvates that, intheir ground state, populate configurational sublevels higher than the equi-librium sublevel (Figure 8.6). When excited, these solvates will, accordingly,populate the configurational sublevels lying below the Franck–Condon sub-level I* (up to the equilibrium sublevel II*). At the same time, the efficiencyof nonradiative energy transfer decreases, since energy transfer to more“red” solvates practically does not occur due to their insignificant proportionin the solution, and the reverse energy transfer to the sublevel I* is of lowprobability.

Directed nonradiative energy transfer (DNRET) in rigid polar solutionsand polymer matrices is responsible for a number of peculiar concentrationaleffects. These include the long-wavelength shift of luminescence spectra withincreasing luminophore concentration(13) and the dependence of emissionanisotropy on the luminescence and absorption spectra.(42) For the emissionspectra, luminescence anisotropy is higher on the blue slope and drops towardlonger wavelengths. In the case of red-edge excitation, the emission anisotropyvalues increase to close to the limiting values.

It was also found that in solutions of complex molecules with inhomo-geneously broadened spectra, at high concentrations some unusual timecharacteristics of the luminescence were observed: the luminescence decaytime increased toward longer wavelengths, and the luminescence decay on theslopes differed appreciably from single exponential.(38, 42) Finally, it has beenshown(43) that for red-edge excitation the concentrational effects disappear insystems with inhomogeneously broadened spectra, as follows from thediagram in Figure 8.6.

DNRET was directly observed for the first time using nanosecondspectrofluorimetry. This technique makes it possible to visualize the dynamicsof the process and measure the required kinetic parameters.(44) In Ref. 45,DNRET was observed between different configurational states of solvatedmolecules in a rigid solution. Poly(vinyl alcohol) was used as the solvent.At a low dye concentration, steady-state inhomogeneous broadening wasobserved: for 3-aminophthalimide molecules the luminescence spectrumshifted as the exciting light frequency was varied. In other words, in this caseordinary bathochromic luminescence was observed, consistent with selectiveexcitation of various configurational states of solvated molecules. The batho-chromic luminescence spectrum was time-independent. At high concentrations(Figure 8.7) the situation was, however, different. The instantaneous emissionspectrum shifted gradually to the red. In this case, each instantaneousspectrum corresponds to a specific solvate state whereas the entire sequenceof spectra represents the energy transfer dynamics for a set of configurational


states. The higher the dye molecule concentration, the faster is the energytransfer. As would be expected, the energy transfer direction is always thesame, from the “blue” to the “red” solvates. The time shift of the luminescencespectrum is responsible for the nonexponential luminescence decay in theemission spectrum, being most distinct on its slopes. In addition, the lumines-cence decay time depends on the recording wavelength (Figure 8.7, curve 5).Figure 8.8 shows the position of the luminescence spectrum maximum versustime for solutions with different concentrations of 3-aminophthalimide. In linewith theory, the rate at which the spectrum shifts is seen to depend on time.It is maximum just upon excitation and then decreases. With decreasingconcentration, the time shift amplitude decreases.

The unique feature of directed energy transfer in the systems involved isthat it occurs between chemically identical molecules with different shellsrather than between chemically different molecules with differently locatedenergy levels. This means that in this case the directed transfer results inspontaneous concentration of excitation energy on molecules with a specificsurrounding structure, that is, in some structurally specified local regions of therigid solution. Of particular interest is the fact that energy is transferred fromstructures with a low degree of molecular orientation to those characterizedby a high degree of orientation. Indeed, the dependence of the electronictransition frequency of a molecule on the local field intensity is defined byEq. (8.10). This suggests that the higher the solvate reactive field intensity,the lower is the transition frequency. In other words, the “blue” centerscorrespond to solvates with low R, that is, with a more chaotic orientation of


solvent molecules in the solvate shell, whereas the “red” centers representsolvates with a high-intensity reactive field and, therefore, with a higherdegree of orientation of their molecules.

Thus, the migration of electronic excitation from “blue” to “red” centerscan be regarded as directed energy transfer from structures of low order tohighly ordered structures in the solution. Such a process can naturally beexpected to play a certain part in the mechanism of directed energy transferin biological systems, in particular, in the transfer of electronic excitationenergy from the antenna chlorophyll molecules to the reactive center in thephotosynthetic system of plants. In Refs. 19 and 20 energy exchange betweenmolecules of the photosynthetic pigments chlorophyll a and pheophytin a wasstudied experimentally under model conditions with pigments introduced intothe polar matrix. Data for chlorophyll a are presented in Figure 8.9. Both theinstantaneous spectrum and the spectrum provide evidence for the existenceof DNRET in this case.

We conclude by noting that the theory of inductive–resonance energytransfer in solutions with inhomogeneous spectral broadening is given in Refs.17 and 42.

8.4. Dynamic Inhomogeneous Broadening in Liquid Solutions

8.4.1. Analysis of Configurational Relaxation in Liquid Solutions

In liquid solutions, unlike the rigid systems discussed in Sections 8.2.6and 8.3.2, the optically excited nonequilibrium distribution function of solvatesis easily converted into the equilibrium one due to the high rate of inter-molecular energy exchange. This process must involve the variation of thelight frequency with time, that is, a shift of the solution fluorescence spectrumduring emission. The verification of these conclusions was of great interest toprovide support for the theoretical model of the dynamic nature of configura-tional inhomogeneous spectral broadening in liquid polar solutions. Specialequipment was designed to selectively excite individual inhomogeneousbroadening states and to record the luminescence with the required timeresolution.(11)

Let us consider the basic luminescence characteristics of a liquid solution.Substitute the values of and as functions of the excitation parametersand time into the expression for luminescence power (Eq. 8.55):

Equation (8.62) is rather general as it takes into account not only the


change with time and the dependence of the excited state population on theexperimental conditions, but also the dependence of the spontaneous emissionspectrum on these conditions.

Analysis of Eq. (8.62) reveals all the experimental features of lumines-cence of polar solutions including the kinetics of the instantaneous emissionspectra [at a given /, we have the instantaneous spectrum for a given instantof time, and the decay law if the frequency is fixedand t is variable.

8.4.1.1. Instantaneous Spectra

As follows from Eq. (8.62), the configurational relaxation of luminescencespectra can be described if we know the type of distribution function thatdescribes the distribution of solvates over the configurational broadeningstates. It is not clear, however, how this function can be calculated in thegeneral case from the model considerations. Assuming weak dependence ofon Eq. (8.62) can be rewritten as

where is the half-width of the distribution function. It is seen fromEq. (8.63) that the process can be described approximately by taking intoaccount the frequency relaxation corresponding to the maximum of thefunction

In Ref. 22 the analytical expression for configurational relaxation of anexcited solvate is substantiated. Let us consider its derivation. Excitation of asolution at a radiation frequency involves selective excitation of solvateswith the local field intensity determined from Eq. (8.10). The excitedsolvates are reoriented, with tending to due to the total unbalancedforce (see Section 8.2.1) and the friction force which may be assumedto be proportional to the rate of change of the reactive field during relaxation:

If

substituting Eqs. (8.4) and (8.64) into Eq. (8.65) and integrating over t, weobtain

Expressing the electric field intensities in terms of frequencies of the excitinglight and the corresponding solvate transitions using Eqs. (8.10) and (8.14),


and taking into account the relationship between the frequencies andaccording to Eq. (8.15), one can obtain the molecular electronic transitionfrequency as a function of time and exciting light frequency:

Assuming for simplicity that the absorption and fluorescence spectra aremirror images and taking into account that for the cells with field R0 to beselectively excited it is necessary to irradiate the solution with radiation offrequency rather than determined from Eq. (8.67), we obtainthe explicit form of the dependence of the solvate fluorescence maximum ontime and the exciting light frequency

Let us rewrite this expression in a somewhat different form:

where is the deviation of the solvate fluorescence maximum at time tfrom its equilibrium position In Eq. (8.68),

that is, is the maximum absorption frequency of solvates with localfield

As seen from Eq. (8.69), at the initial instant of time linearlydepends on the exciting light frequency; that is, inhomogeneous spectralbroadening is present. Over time, becomes less dependent on that is,the inhomogeneous broadening decreases, and as it vanishes.

Thus, for liquid solutions the degree of inhomogeneous broadeningdecreases with time; in other words, it is dynamic in nature. As thefluorescence maximum tends to its equilibrium value, determined by thefrequency

The relaxation of the fluorescence spectrum depends essentially on theexciting radiation frequency, that is, on the type of selectively excited solvates.The following three cases are possible:

1. Excitation in the Stokes spectral region results in a redshift of the fluorescence spectrum with time. We

call(10,11) such a process the configurational down-relaxation of the


luminescence spectrum. This type of relaxation can easily beexplained in terms of molecular reorientation in solvates of similarconfigurations in the ground electronic state, and it was experi-mentally observed long ago.(46, 50, 51)

2. For excitation in the anti-Stokes spectral region the situa-tion is the reverse; that is, at the initial instant of time the spectra arered-shifted as compared to the steady-state spectra In thiscase, the return of the spectrum to its normal position during con-figurational relaxation will lead to a blue shift with time. From thephysical point of view, this means that the configurational interactionenergy excess which solvates possess during relaxation is partiallyconverted into emitted quantum energy, which just leads to anincrease in the radiation frequency with time. In this connection, theprocess has been termed the “up-relaxation” of the luminescencespectra.

3. Finally, the third case corresponds to pumping of the solution by thefrequency In this case, the luminescence spectrum immediatelyafter excitation must be close to the steady-state one and willnot vary with time. From the physical point of view, this casecorresponds to the situation where those solvates are excited whoseconfiguration is nonequilibrium in the ground state, but correspondsto the equilibrium configuration in the excited state (i.e., solvateswith a local field It should be noted that the frequencysimultaneously characterizes both the spectral properties of the dyemolecules and the effect of inhomogeneous broadening; that is, it isa complex characteristic of a solution. It separates the excitationfrequency ranges responsible for “down-” and “up-relaxation” of thefluorescence spectra and is called the equilibrium configurationalexcitation frequency.(22)

As seen from Eq. (8.69), the relaxation rate of the fluorescence spectrumis determined by the value of If the friction coefficient is a constant,then the decay obeys a simple exponential law. In a more general case, ofcourse, the coefficient must depend on the rate of change of the reactivefield, dR/dt, and increase with it. This, in turn, must inevitably lead tomore complicated decay laws which can be described by assuming to bea function of R or by replacing the exponent in Eq. (8.69) by a sum ofexponents with corresponding coefficients, as was done inRef. 46.

Note that the nature of all the implications of Eq. (8.69) can bequalitatively explained using the field diagram of a solvate.

The relationship is calculated from Eq. (8.69) for a typicalphthalimide molecule in n-propanol at different temperatures in Ref. 11.


We think that the advantage of the proposed model is its independenceof any particular diagram of polar solution energy levels, as none of them wasused to derive Eq. (8.69).

8.4.1.2. Fluorescence Decay

For configurational orientation at the laws of spontaneousemission decay as well as the instantaneous spectra are determined by theevolution of the distribution function of the solvates, and, hence, theydepend on the whole set of parameters which determine the inhomogeneousbroadening function,

To analyze the decay laws, it is necessary to specify the shape of thefluorescence contour and its variations with pumping frequency and time. Thecontour will be described, as before, in terms of the hyperbolic function [seeEq. (8.57)], but we will assume that the frequency shifts withtime according to Eq. (8.69). Let the decrease in the total population of allconformational states obey a simple exponential law,

and originate from its initial value of given by the excitationpower. As in the previous section, we will study the decay law in the regionsof down- and up-relaxation and at their boundaries.

Let us consider the case of Stokes excitation and assume, for simplicity,that It can easily be shown that, under such conditions, in theshort-wavelength range of luminescence frequencies the fluorescence decay isdefined by the expression

that is, the decay is faster than without relaxation. In the long-wavelengthspectral range,

which is indicative that the configurational relaxation is associated with anincrease in the fluorescence decay time. Finally, in the region of the maximumof the equilibrium fluorescence spectrum

that is, the fluorescence decay time does not vary due to the configurationalrelaxation. The above types of behavior are shown in Figure 8.10a (curve 1).

It is quite clear, from the physical point of view, that the drift of thefluorescence spectrum away from the recording frequencies is equivalent toa decrease in whereas the shift of the spectrum toward the recording


frequencies must lead to an increase in and even rise of luminescence intensityprovided the relationship between the parameters is favorable. The latter effectwas predicted in Ref. 48. However, it has not yet been clearly observed in thenanosecond region.

The situation is quite different for excitation in the far anti-Stokes regionof the absorption spectrum; beyond the equilibrium configuration excitation


frequency, the fluorescence decay time will be described not by Eq. (8.73), butby Eq. (8.72), whereas for the low-frequency portion of the spectrum,is specified by Eq. (8.72) rather than by Eq. (8.73). However, in the region ofthe maximum of the equilibrium spectrum the decay law is not distorted byrelaxation, as before. These results are illustrated by curve 3 in Figure 8.10a.

Light pumping at the equilibrium configuration excitation frequencyyields a purely exponential decay law (Eq. 8.74) for all regions of thefluorescence spectrum, since in this case the fluorescence spectrum does notundergo a shift with time (Figure 8.10a, curve 2).

Summarizing the results of the analysis of the spectral behavior, it isnecessary to describe three different types of dependences of the fluorescencedecay for variation of the excitation frequency over a wide range. Such adiversity of spectra is due to the existence in the solution of differenttypes of solvates, each having its own dynamic luminescence spectralbehavior, and the possibility of their selective excitation by monochromaticlaser light. If this fact is neglected, the picture obtained is poor and describesthe solvation shell reorientation kinetics for only one type of centers. Thiswas the situation that existed before the concepts of the configurationalbroadening of polar solution spectra were introduced. The fluorescence decaybehavior in a viscous homogeneous solution was considered for the firsttime in Ref. 48 in terms of the Onsage model under the assumption thatthe maximum shift of the fluorescence spectrum due to the reorientationof the molecules of the solvation shell followed an exponential law. Theauthors calculated the curves from the emission spectrum, assuminginstantaneous excitation, and showed their essential dependence on therecording frequency.

Somewhat later, identical results were obtained using both a discrete anda continuous model of the solution.(49)

All the theoretical dependences considered in this section have beenverified and supported by the methods of time-resolved spectroscopy. Theresults are given in the next section.

8.4.2. Experimental Study of the Luminescence Kinetics of Liquid Solutions

The study of the time shift of the spontaneous emission spectrum withvarying energy of the exciting radiation is of indisputable interest, since itenables the conclusions about the process of intermolecular relaxation thathave been made on the basis of the model presented here to be verified.

The first comprehensive measurements in both the Stokes and anti-Stokes regions were reported in Refs. 10 and 11. The studies were performedwith a nanosecond spectrofluorimeter (time resolution about 1 ns) using atunable pulsed dye laser as the excitation source.


Figure 8.11 gives the instantaneous fluorescence spectra (panel a) andthe position of their maxima at different times (panel b) as functions of theexcitation frequency for 3-ANMP. It is seen that a decrease in the excitationfrequency leads to a reduction in the fluorescence spectrum rangeThus, for whereas at

Thus, the frequency correspondsapproximately to the equilibrium configurational excitation frequencyFurther decrease of reverses the behavior of the relaxational spectral shift;that is, the shift of the spectrum with time is toward shorter wavelengths.Characteristically, in all cases the position of the maximum in the emissionspectra relaxes to that in the steady-state fluorescence spectrum (Figure 8.11b,dashed line) recorded at room temperature, whereas at short times afterexcitation the position of the maximum in the instantaneous emission spectrastrongly depends on the excitation frequency. The above result is a directmanifestation of inhomogeneous configurational broadening of electronicspectra of complex molecules in viscous solutions.

It is worth noting that in the course of spectral relaxation the shape ofthe instantaneous fluorescence spectra of fluorescent solutions is little affectedby the transformation of the inhomogeneous broadening function with time.This is illustrated in Figure 8.11a, which shows instantaneous fluorescencespectra with excitation in the Stokes region. At the position of thefluorescence spectra can be assumed to correspond to that at low tem-perature, that is, under conditions of slow molecular reorientation.

The experimental spectral/kinetic behavior in Figure 8.11a and b agreeswith the theoretical relationships shown in Figure 8.11c. The results obtainedwere observed for cooled, polar solutions in which the orientational relaxationis delayed. Exactly the same spectral behavior was observed(52) for low-viscosity liquid solutions at room temperature, in which the orientationalrelaxation rate is much higher. A 6-ps-resolution spectrometer was used in theexperiments.

Thus, it has been shown that the difference in behavior betweeninstantaneous fluorescence spectra of different groups of inhomogeneoussolution centers is indicative of the existence of dynamic inhomogeneousconfigurational broadening in polar liquid dye solutions. Previously, onlylong-wavelength shifts of emission spectra with time were observed underconfigurational relaxation with Stokes excitation by a continuous pulse-discharge spectrum(50,51) and a fixed nitrogen laser frequency.(46) The short-wavelength shift of fluorescence spectra (up-relaxation) with time is a radicallynew effect which was observed for the first time in Ref. 10. Us treatmentnecessarily involves inhomogeneous dynamic broadening of the solutionspectra. The experiments have revealed the existence of reorientation of excitedsolvate molecules due to the disturbance of the electrical equilibrium in theelectronic transition.

Inhomogeneous

Broadening

ofE

lectronicS

pectraof

Dye

Molecules

403


Interesting results have been obtained for the luminescence lifetimespectra presented in Figure 8.10b for a viscous 3-ANMP solution in n-propanolat different excitation frequencies It is seen that on excitation of dyemolecules in the Stokes region, the luminescence lifetime monotonicallyincreases with decreasing frequency from 2.5 ns to20.5ns for 3-ANMP solutions atWith decreasing excitation frequency, the lifetime spectrum is modified. Therange of sharp increase of on the frequency scale is decreased and shiftstoward shorter wavelengths. Beginning at some excitation frequencies (atwhich up-relaxation already appears), qualitatively different spectraare observed. These are characterized by a monotonic increase in withincreasing Thus, for excitation of 3-ANMP at 21,190 cm–1 the value of

is 4 ns at v rec = 16,670 cm–1 and increases to 16.5 ns at v rec = 21,740 cm–1

The above features of the behavior of lifetime spectra of viscous solutionswith variation in the excitation frequency are in good agreement with theconfigurational relaxation model considered in the previous section. Thefield diagram of the energy levels (Figure 8.1) can also contribute to theirqualitative explanation.

To conclude, it should be noted that simultaneous configurationalrelaxation of excited levels and their spontaneous deactivation results in non-exponential fluorescence decay. The rate of change of the intensity decreasesmonotonically with time at the high-frequency edge and increases at lowfrequencies for excitation in the Stokes region with The reverse isobserved at Only for is the decay rate constant over thetime.

8.4.3. The Solution Spectrochronogram

As mentioned above, it is not clear how the relaxation of the distributionfunction can be calculated in the general case from a model.However, to correctly describe the luminescence kinetics, it is necessary totake into account that the spectrochronogram when excited bya pulse, represents convolution of the solvate distribution function overfrequencies of the 0–0 electronic transition, and of the homogeneousluminescence spectrum, in agreement with Eq. (8.62). Equation(8.62), which assumes an identical lifetime for all sublevels, can be rewrittenin a more convenient form for practical applications as

where is a constant.


As shown in Section 8.2.3, the equilibrium inhomogeneous function of apolar solution is Gaussian in shape, which permits us to write it for groundand excited states as

Here the subscripts g and e indicate the ground and the excited singlet state,respectively; and are the root-mean-square deviation of thisdistribution and the position of its maximum during relaxation, respectively.

Since the width of the homogeneous absorption spectrum is large due tovibrational broadening, it is impossible to excite selectively only one type ofsolvates. Therefore, immediately after optical excitation the instantaneousluminescence spectrum is inhomogeneously broadened. Since the number ofsolvates with the 0–0 transition frequency at the zeroth instant of time in theexcited state is equal to the product of the number of solvates in the groundstate with the same frequency times the absorption coefficient of this type ofsolvate, the inhomogeneous broadening function at time t = 0 is equal to

where is the exciting light frequency, is the homogeneousabsorption spectrum, and is the equilibrium distribution function inthe ground state, which can be found from Eq. (8.52).

Equations (8.75)–(8.76) and (8.52) make it possible to find the functionsand exactly, as well as the solution correlation function

where is the position of thecenter of gravity of the distribution function at different times.

Figure 8.12 shows the instantaneous luminescence spectra of 3-ANMF inglycerin calculated at time t = 0 (curve 1) and experimentally recorded attimes t = 3 ns (curve 2) and t = 16 ns (curve 3), as well as the inhomogeneousbroadening functions (curve 4) and (curves 5 and 6) for


these same times; has been calculated from Eq. (8.77), andwas found by deconvolution of instantaneous luminescence spectra recordedon a laser spectrometer.

The steady-state luminescence and absorption spectra of 3-ANMF innonpolar n-hexane have been taken as the homogeneous spectra. The calcula-tions show that, with the assumptions of a Gaussian shape, is about

at t = 0, then increases to (t = 3 ns), and decreases againto (t =16 ns)

As follows from Eq. (8.75), the fluorescence kinetics at a fixed frequencyare determined by the variation of the inhomogeneous broadening functionwith time, which, in turn, depends on the functions and Calcula-tions using Eq. (8.75) have shown that experimental and calculated decaycurves are well matched (Figure 8.13) if is specified as

where are constants.Since in the initial stage of relaxation the luminescence kinetics are

mainly determined by the variation of the inhomogeneous function, for thesake of simplicity of calculation can be given by one exponential formwith a certain relaxation time.

Thus, for liquid polar solutions the fluorescence spectrochronogramis well described by a continuous model if we associate changes in the

half-width of instantaneous spectra with the evolution of the inhomogeneousbroadening function of the solution and express its relaxation as the sum oftwo exponents.

8.4.4. The Effect of Light-Induced Molecular Rotation in Solution

The configurational relaxation which takes place upon excitation insystems with dynamic inhomogeneous spectral broadening not only manifests


itself in a shift of the instantaneous luminescence spectra and a change in thedecay law over the spectrum, but it also affects the rate of Brownianmolecular rotation of the luminophore in solution. This effect depends on theinitial store of configurational energy of solvates in an excited electronic state,which, in turn, is determined by the excitation frequency, and can, therefore,be considered as light-induced rotation of molecules in the liquid (LIR). LIRleads to the dependence of the depolarization behavior of the solutionluminescence on the excitation and recording frequency.(21)

If the solvates are excited at a frequency and the fluorescence emissionis recorded at then in the course of relaxation the solvates giving rise tothe recorded emission have decreased their potential energy by (Figure8.1). This energy is mainly converted into rotational energy of solvation shellmolecules. The released energy can be estimated from the magnitude of theshift of the instantaneous solution fluorescence spectra. For 3-ANMP inglycerin excited at its absorption maximum, this shift is aboutTaking into consideration that the solvate includes one 3-ANMP moleculeand five or six glycerin molecules, each solvate molecule accounts for

of energy released during the configurational relaxation. This valueis comparable to the thermal energy of molecules in solution at roomtemperature. The fluctuational force acting on the solute molecules increasesas the average rotational energy of the neighboring solvent molecules isincreased. This must accelerate the luminescence depolarization of thesolution upon excitation by linearly polarized light.

Characterizing the depolarization due to the release of the excess energyby the value of and assuming its linear relationship with we have

Using Eqs. (8.10) and (8.11) and neglecting the vibrational spectralbroadening (in this case, we can write this expression as

where is a constant determined by the properties of thesolution.

Taking into account the fluorescence depolarization due to opticallyinduced molecular rotations, the emission anisotropy is

where is the emission anisotropy in the absence of intermolecular relaxation.Let us analyze the following three characteristic cases. In the first case,

the solution is excited near the absorption band maximum andThen, it follows from Eqs. (8.79) and (8.80) that with decreasingincreases and the emission anisotropy r decreases.

Thus, in the case of Stokes excitation, the fluorescence depolarization dueto configurational relaxation is most pronounced at the red edge.


When the solution is excited in the anti-Stokes region (configurationalup-relaxation), Then, it follows from Eqs. (8.79)and (8.80) that increases and decreases with increasing Hence, theaccelerated fluorescence depolarization is more pronounced at the blue edge.

Finally, when the solution is excited at the frequency of zero confi-gurational excitation and In this case, theexcitation does not induce molecular rotation.

The above model considerations have been confirmed by experiment.(21)

Different solutions (mainly, of phthalimide derivatives) in glycerin and normalalcohols as well as phospholipid bilayer membranes were investigated using ananosecond laser fluorimeter. The experiments revealed that in systems inwhich the configurational relaxation occurs on a time scale comparable to theexcited state lifetime, the fluorescence depolarization rate depends on therecording frequency. In addition, the emission anisotropy kinetics (EAK) arecharacterized by two components: a fast one in which the measured rate ofdecrease of the emission anisotropy depends on the excitation andrecording frequencies, and a slow component which is independent ofthe experimental conditions.

The experiments also revealed that with excitation at the absorptionband maximum, the decrease in the emission anisotropy with time is morerapid at the long- than at the short-wavelength edge. The red-edge decay ismultiexponential (Figure 8.14a). In this case, the instantaneous fluorescencespectra undergo a long-wavelength shift (Figure 8.14b). It is also seen that thedependence of the instantaneous anisotropy values on the recordingwavelength becomes stronger during the course of configurational relaxation.The relaxation energy decreases with decreasing excitation frequency. Thedependence of the EAK on thus weakens with decreasing excitationfrequency and, therefore, the instantaneous values of anisotropy vary lessrapidly with changes in With excitation of the solution at the frequencyof zero configurational excitation the EAK are no longer dependent onthe recording frequency.

Further decrease of the excitation frequency gives rise to ashort-wavelength shift of the instantaneous spectra (up-relaxation)(10) (Figure8.15). As a result, the blue-edge EAK are multiexponential (Figure 8.15a), andthe instantaneous anisotropy spectra (Figure 8.15b) are the reverse of thosefor The values of the molecular rotation times measured from doubleexponential fit of anisotropy decay for different excitation and recordingwavelengths are presented in Table 8.2.

The experimental data confirm that the random-force impulses rotatingthe luminophore molecules during relaxation following excitation depend onthe released configurational energy of the solvate. In the course of relaxation,the excess potential energy due to dipole–dipole interactions of solvatemolecules is expended in their rotational degrees of freedom. It should be

Inhomogeneous

Broadening

ofE

lectronicS

pectraof

Dye

Molecules

409

410 N

icolaiA. N

emkovich etal.


noted that this additional kinetic energy imparted to the molecules turns outto be significant and is sufficient to rotate them by a particular angle. Thevalue of this angle depends on the intermolecular solvate energy after opticalexcitation. Such an effect, as mentioned above, can be regarded as light-induced molecular rotation.

The emission anisotropy additivity rule has been used to estimate theangle of rotation of the luminophore molecule.(53) According to this rule, ifthere are several types of luminescent centers in the system, the emissionanisotropy of the whole system is equal to the sum of the values of theemission anisotropy of the individual components. If the emission only fromthe Franck–Condon and equilibrium sublevels is taken into account (Figure8.la), this rule can be written as

where the characterize the emission anisotropy kinetics for the Franck–Condon (i = 1) and equilibrium (i = 2) states, and is the parameter charac-terizing the contribution to the total fluorescence of the ith state, which isdetermined by the expression

where the characterize the decay kinetics for the Franck–Condon andequilibrium states calculated from the differential balance equations for thepopulations of the states, the are the spectra of the states, andis the integrated luminescence intensity of both types of states.


As in the case of spherical top type molecules, the emission anisotropykinetics of the individual states can be represented by

where are the limiting values of the emission anisotropy of thestates, is the Fitting parameter for the decrease in emission anisotropyduring configurational relaxation due to the light-induced rotation of solvatemolecule, and are the times of Brownian rotation of molecules inthe different states.

Equation (8.81) was used to estimate the value due to the effect oflight-induced molecular rotation. The calculation was performed for 3-ANMPin glycerin with experimentally determined parameters. The spectrum of3-ANMP in hexane, which does not display inhomogeneous broadening,served as the spectrum of the Franck–Condon state The equilibriumstate spectrum was taken as the instantaneous spectrum of 3-ANMP inglycerin after configurational relaxation.

The experimental results were fitted for both Stokes (up-relaxation) andanti-Stokes (down-relaxation) excitation.

Figure 8.16 gives the EAK calculated for excitation of the solution at theabsorption band maximum. Comparison with the experimental data (Figure8.12) shows that satisfactory agreement takes place at


Using the Levshin-Perrin formula(54) for the limiting value of theemission anisotropy,

where is the angle between the absorption and emission oscillators, one candetermine the angle of light-induced rotation of the luminophore moleculeduring configurational relaxation. For 3-ANMP in glycerin, this value was

at the excitation frequency

8.5. Selective Kinetic Spectroscopy of Fluorescent Molecules inPhospholipid Membranes

8.5.1. Energy Levels of an Electric Dipole Probe in a Membrane

The phospholipid bilayer (Figure 8.17) which makes up a membrane issuch that the hydrophilic parts of the phospholipid, the charged heads, projectoutward, whereas the hydrophobic parts of these molecules, the hydrocarbonchains, lie inside the bilayer. By virtue of such organization, the membranehas a significant polarity gradient—the polarity decreases as one moves fromthe periphery to the center of the bilayer. On embedding a probe, the outerpart of the membrane, with the probe incorporated into it, behaves as a polarsolvent and the inner part as a nonpolar solvent.

Many probe molecules, such as aminonaphthalenes and phenylnaphthyl-amines, have an electric dipole moment and are located in the membrane closeto its polar region.(55) Since the incorporation of a probe into the membraneis a statistical process, the probe is distributed throughout the membranethickness in a specific manner. This leads to a distribution of the energy ofintermolecular, configurational probe–environment interactions. Furthermore,the individual energies in this distribution fluctuate due to the thermal motionof the probe and, hence, its different orientations relative to the local field, aswell as due to the segmentary motion of the membrane. As in the case ofpolar solutions, the statistical distribution of the probe–membrane interactionenergy, according to Eq. (8.10), must cause inhomogeneous broadening of theluminophore’s electronic spectra.

The probe luminescence lifetime, which usually ranges from nanosecondsto tens of nanoseconds, can be used as the natural time scale. At roomtemperature the rotation time of the probe in the membrane is comparable toits luminescence lifetime whereas the radial motion of the probe, that is, itstranslational motion, is much slower. The inhomogeneous spectral broadeningfor the probe due to its localization at different depths in the membrane must,


therefore, be static, as in the case of rigid polar solutions. The broadening dueto fluctuations in the probe–environment interaction energy at a given depthwill be dynamic, as in the case of viscous solutions.

Let us construct, by analogy with polar solutions, a diagram of electronic-configurational states of the system consisting of the probe molecule and itsimmediate surroundings in the membrane (we shall call this system thesolvate). Let us plot on the abscissa the local electric field intensity R and onthe ordinate the solvate potential energy, which includes the electronic energyof the probe and the probe–environment interaction energy as well as theinteraction energy of the solvate membrane segments.

Such a diagram for two different probe localization depths is shown inFigure 8.18, where the left-hand pair of curves corresponds to the mostprobable localization. On each curve the energy minimum corresponds to the

Inhomogenaous Broadening of Electronic Spectra of Dye Molecules 415

most stable configuration of the probe and the solvate membrane segments,for which all the types of interaction are in relative equilibrium. Thermalmotion can also give rise to other solvate configurations with lower or highervalues of R. In this case, however, the relative equilibrium of intermolecularforces is disturbed and the potential energy of the solvate increases.

As the solvate configuration and, therefore, the field do not changeduring the transition of the probe molecule to another electronic state, theelectronic transitions in Figure 8.18 are represented by vertical lines (inter-molecular treatment of the Franck–Condon principle). In the excited statethe electric dipole moment generally increases. Thus, in the upper statethe potential energy minima are shifted toward higher R as compared to inthe ground state. It can be shown that, as in the case of solutions, the pureelectronic transition frequency of the probe in the membrane is unambiguously


related to the distance between the combining pairs of configurational sub-levels in the ground and excited states.

Upon transition to another electronic state, the equilibrium in the solvateis disturbed due to the change in the probe dipole moment. Therefore, as soonas the probe molecule finds itself in another electronic state, the process ofconfigurational relaxation and establishment of the equilibrium Boltzmanndistribution over configurational sublevels take place. As can be seen fromFigure 8.18, upon excitation at the absorption spectrum maximum or close toit, this process involves a decrease in the emitted energy.

The spectral properties of the entire system consisting of the membraneand the probe molecules should be analyzed using the diagram in Figure 8.18and taking into account the distribution of the membrane solvate over thedifferent states associated with the different depths of localization of theluminophore as well as the distribution (at each depth) over the states causedby local field fluctuations.

When analyzing the diagram shown in Figure 8.18, one should bear inmind that at room temperature no transitions are observed between differentlylocalized centers because of the low velocity of the translational motion ofthe probe during the excited state lifetime. Therefore, at room temperature theprocess of configurational relaxation that occurs within several nanosecondsmainly involves orientational motion of the probe and, to some extent,segmentary reconstruction.

8.5.2. Inhomogeneous Broadening in Steady-State Fluorescence Spectra ofProbes

As mentioned above, one of the characteristic features of inhomogeneousbroadening of electronic spectra of complex organic compounds in condensedmedia is the long-wavelength shift of the luminescence spectrum uponexcitation by narrow-band radiation at the red edge of the absorptionspectrum. Such an effect has been observed for dipolar fluorescent probemolecules in a number of cases(59) in both polar solutions and membranes.For 1-phenylnaphthylamine (1-PNA), bathochromic fluorescence is illustratedin Figure 8.19, which shows experimental data for the position of the emissionband in solutions (Figure 8.19a) and in membrane phospholipid bilayers(MPB) (Figure 8.19b) as functions of the excitation wavelength. A significantshift of the luminescence spectrum toward longer wavelengths upon red-edgeexcitation is observed for both rigid and viscous glycerin solutions and inlecithin liposomes. It is noteworthy that the spectral shift does not involve anysubstantial deformation of the emission bands. There are no isosbestic pointsor any other effects which could be ascribed to chemical or quasi-chemicalbonds between the luminophore and the environment.


The bathochromic fluorescence shift of the probe in MPB can easilybe interpreted in terms of the diagram given in Figure 8.18. With red-edgeexcitation, mainly solvates with low values of the 0–0 transition frequenciesare promoted to the upper singlet state. As configurational relaxation occurson the nanosecond time scale, that is, on a time scale comparable to theexcited state lifetime of the probe, the emission for each probe localizationdepth is from the same state as or a similar state to that in which the solvatefound itself after excitation. This causes a bathochromic fluorescence shift withdecreasing energy of the exciting radiation.


The influence of the membrane viscosity on the observed effect isdemonstrated by measurements of the position of the fluorescence spectrumfor 1-PNA in liposomes as a function of the temperature and the excitationwavelength (Figure 8.20). For edge (390-nm) excitation, the position of thespectrum slightly depends on temperature. When the membrane is heated,the positions of the spectra for ordinary and edge excitation come closertogether because of the accelerated configurational relaxation in the system.The temperature dependence of the position of the fluorescence maximum(Figure 8.20) is complicated. The monotonic fluorescence maximum versustemperature curve has a plateau in the temperature range 20 to 40 °C. Thiseffect can apparently be attributed to the complex nature of the inhomo-geneous broadening of the electronic spectra of the probes in the membrane.It is not improbable that the position of the fluorescence spectrum maximumin the range shifts only due to the rotational dynamics ofthe phospholipid segments and the probe and that, beginning at thefluorescence spectrum maximum is affected by the translational motion of theprobe. The plateau of the fluorescence maximum versus temperature curvefrom 20 to indicates that the configurational relaxation of the probeand membrane segments is already sufficiently fast and no longer affects theposition of the spectrum of the solvate located at a certain depth. At the sametime, within this temperature range the role of the translational motion of the


probe in mixing states which differ in depth within the membrane is notsignificant.

8.5.3. Kinetics of Probe Fluorescence

An understanding of the elementary mechanisms of the complex processesby which the numerous functions of biological membranes are realizedinvolves detailed information not only on the molecular structure of the latter,but also on their structural dynamics. It is therefore necessary to elucidate thekinds and the characteristic times of motions of the membrane molecules andsegments.

Time-resolved spectroscopy makes it possible to study the dynamics ofconfigurational relaxation, which are manifested by shifts in the instantaneousfluorescence spectra upon optical excitation.

Studies of the fluorescence decay curves, anisotropy, and nanosecondtime-resolved fluorescence spectra(56) of 2-PNA in lecithin liposomes revealedthe existence of nanosecond dynamics. However, the mechanism of proberotation was not specified but was assumed to be equilibrium Browniandiffusion in nature. Later, phase fluorometry was used(57) to observe thespectral kinetics of 1-PNA.

In Ref. 58, the fluorescence spectra of 1-PNA in liposomes measured ona nanosecond fluorometer were reported to show shifts toward longerwavelengths on the nanosecond time scale (Figure 8.21). The magnitude ofthe shift decreases as the excitation wavelength increases. However, even with410-nm excitation, for which a significant bathochromic shift of the steady-state fluorescence spectra is observed, the time dependence of the instan-taneous emission spectra was observed (2–8 ns after the excitation pulse thereoccurs a shift of the instantaneous spectra by 10 nm). In our opinion, thesedata can easily be interpreted in terms of the model considered here asfollows.

Upon excitation at 337 nm, all types of fluorescence centers at the differentlocalization depths pass to the excited state. After excitation, configurationalrelaxation takes place, during the course of which each type of fluorescencecenters tends toward its equilibrium configuration.

As is clear from the field diagram (Figure 8.18), the nanosecond spectraldynamics are associated with the configurational nonequilibrium of thesolvates. Their existence even at 410-nm excitation indicates that the distribu-tion function of the solvates in 0–0 transition frequency, which is associatedwith the static inhomogeneous broadening, decreases toward longerwavelengths less rapidly than does the distribution function associated withthe dynamic inhomogeneous broadening.

Directly associated with the long-wavelength shift of the probe emission


spectrum with time is the dependence of the fluorescence lifetime on the wave-length (see Figures 8.22 and 8.23). The parallel processes of configurationalrelaxation and deactivation of the excited state lead to essentially non-exponential decay kinetics for the luminescence at the red and the blue edgeof the spectrum.


8.5.4. Rotational Dynamics of the Probe in the Membrane

Let us now direct our attention to the diagram of energy levels of fluo-rescent centers in a membrane (Figure 8.18). In the case of excitation such that

(the subscript i indicates that the frequency of zero configurationalexcitation is different for centers with different depths of localization), energyrelaxation begins upon excitation. During relaxation, the excess configura-tional energy is expended in other degrees of freedom of the fluorescentcenter. As in the case of LIR in solution, this must lead to an increase in thefluctuation force acting on the probe and acceleration of the process offluorescence depolarization. It is clear that the effect is the stronger, thegreater the intermolecular interaction energy released in the course ofrelaxation. This conclusion is confirmed by experiment.

The studies reported in Ref. 58 have shown that the emission anisotropykinetics of 1-PNA in MPB essentially depend on the excitation and recordingwavelengths. For excitation within the long-wavelength absorption band thekinetics are nonexponential, and the depolarization rate decreases with time(Figure 8.23). The red-edge fluorescence depolarization is always faster thanthe depolarization at the blue edge.

With excitation in the region of the maximum of the long-wavelengthabsorption band (337 nm), the moment the excitation pulse causes, theemission anisotropy values are significantly lower than the limiting value. Thispoints to the existence of a subnanosecond component of the depolarizationthat is invisible under the experimental conditions. With excitation at the rededge of the absorption band, (390 nm, Figure 8.23), the values of the emissionanisotropy increase practically up to the limiting value; that is, the subnano-second component of the depolarization disappears.

The data indicate that for 1-PNA molecules in phospholipid membranesthe process of configurational relaxation of the luminescence spectrum,which follows the process of excitation, accelerates the probe fluorescencedepolarization. This depolarization is most pronounced when the recordingfrequency is at the red edge of the fluorescence spectrum, where the greatestcontribution to the luminescence is made by solvates that have undergonerelaxation.

The increase in the fluorescence anisotropy on going to longer excitationwavelengths (Figure 8.23) may be attributed to the lower configurationalenergy of the solvates absorbing the lower frequency radiation.

The dependence of the luminescence depolarization rate on the recordingwavelength influences the character of the steady-state polarization spectrum;that is, as the wavelength increases, the depolarization monotonicallydecreases. The wavelength dependence of the fluorescence decay time alsoessentially contributes to this effect.

As indicated above, light-induced rotation was also observed in polar


solutions of some phthalimide derivatives. Therefore, we may maintain thatthis phenomenon is universal, and it should be taken into account whenconsidering the spectroscopic and photochemical properties of different systemswhich include polar molecules that exhibit inhomogeneous configurationalbroadening of their electronic spectra. The specific character of excitation-induced probe rotation in a membrane is associated with the nature of themembrane structure, in particular, with the great length of phospholipidmolecules. Thus, the light-induced rotation of the probe in a membrane willbe more pronounced than in a solution with smaller, more mobile solventmolecules.

The dependence of the probe rotation rate in a membrane on theconfigurational energy of the luminescent center complicates the problem ofdetermining the microviscosity of membranes from the Levshin–Perrinformula, since this formula was derived for diffusion in a system at thermo-dynamic equilibrium without consideration of the influence of nonequilibriumsolvation in a polar solution molecular rotation. It is clear that the rotationaldiffusion time from which the microviscosity is calculated should be determinedfrom experiments unaffected by the light-induced rotation of the probe.Furthermore, one should bear in mind that in the case of edge excitation wedetermine the microviscosity of that portion of the polar membrane wheremost red fluorescent centers are located.

The value of for 1-PNA in a phospholipid membrane calculated from

Using the procedure described in Section 8.4.4, the angle of rotation of1-PNA in a membrane undergoing configurational relaxation was found to beabout 22°.

8.6. Conclusions

We will summarize here the results presented in this chapter. The datathat have been obtained to date enable us to draw the following conclusions:

1. Polar solutions of complex molecules are systems with inhomo-geneous configurational broadening. The differences in the spectraof the fluorescent centers in a polar solution are caused by thevariability in structure of the solvation shells surrounding the fluo-rescent molecule owing to the thermal motion of the molecules in thesolution.

2. In rigid solutions, inhomogeneous broadening is static in nature andshows up in steady-state spectra. Static inhomogeneous broadeningis observed in liquid solutions, too, when the condition of slowmolecular reorientation is satisfied. This condition can be


fulfilled by reduction of the luminescence lifetime under strongquenching by impurities or under powerful light.

3. In liquid solutions, inhomogeneous configurational spectral broad-ening is dynamic in nature. This is manifested, in particular, by thedependence of the position of the instantaneous fluorescence spectraon the excitation frequency and the time at which the spectra arerecorded. The dependence on the excitation frequency decreases withtime, and at the spontaneous spectra no longer depend onthe excitation frequency.

For polar solutions there exists a special parameter, the equi-librium configurational excitation frequency of the solution, Thisparameter corresponds to the excitation frequency at which thequasi-equilibrium distribution of excited solvates over configurationalstates subsequent to excitation is attained.

4. The difference between the solvates in a solution is manifested inwavelength dependences of the decay time as well as of the depolar-ization kinetics of fluorescence. The rotation rate of luminophoremolecules is found to depend on the configurational energy of thesolvate. This explains the change of the rate of fluorescence depolar-ization with the wavelength of excitation.

5. Inhomogeneous broadening is pronounced in electronic spectra ofdipolar fluorescent probe molecules in biological membranes, whereit is mainly associated with the heterogeneous character of thelocalization of the luminophore in the bilayer. This inhomogeneousbroadening is responsible for the bathochromic shift of the fluo-rescence band of the probe molecule and its temperature dependence,the kinetics of instantaneous fluorescence spectra, and light-inducedrotation of probe molecules.

The discovery of inhomogeneous configurational spectral broadening ofsolutions and the study of its fundamental properties not only proved to beof great utility in spectroscopy of organic liquid and frozen (300 to 77°K)solutions with broad-band spectra, but it has also aided in the understandingof nonphonon transitions of the same molecules at liquid helium temperatures.In the latter case, as indicated in Sections 8.2.3 and 8.2.5, inhomogeneousbroadening is of the same physical nature as at liquid nitrogen temperature.

It can be stated with confidence that the notions about inhomogeneousconfigurational broadening are of practical importance not only for spectro-scopy, but also for other sciences. These notions are already being used tostudy the structure and structural dynamics of biopolymers,(59) which isextremely important in elucidating the functional properties of biologicalmolecules, since the latter are determined, to a great extent, by these factors.The concept of inhomogeneous broadening requires that the interpretation of


existing data be revised from new standpoints, and it promises fundamentallynew results.

In chemistry, consideration of nonequilibrium solvation is of particularimportance in kinetics, since solvation conditions have a strong influence onthe rates and mechanisms of chemical reactions. By modifying the solvationconditions by selective optical excitation, it is possible to manipulate the ratesof chemical reactions.(60)

Acknowledgment

The results presented in Section 8.5 have been obtained in collaborationwith Prof. A. P. Demchenko and Dr. N. V. Shcherbatskaya from the Instituteof Biochemistry, Kiev, and Dr. D. M. Gakamsky from the Institute ofPhysics, Minsk.

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19. A. N. Rubinov, E. I. Zenkevich, N. A. Nemkovich, and V. I. Tomin, Directional energytransfer in photosynthetic pigment solution caused by orientational broadening of energylevels, Opt. Spektrosk. 50(6), 848–854 (1982).

20. A. N. Rubinov, E. I. Zenkevich, N. A. Nemkovich, and V. I. Tomin, Directed energy transferdue to orientational broadening of energy levels in photosynthetic pigment solutions,J. Lumin. 26, 367–376 (1982).

21. A. N. Rubinov, D. M. Gakamsky, N. A. Nemkovich, and V. I. Tomin, Dye MoleculeRotation in Solution Induced by Optical Excitation, Institute of Physics, BSSR Acad. Sci.,Minsk, Preprint N 476, p. 10 (1987).

22. A. N. Rubinov, V. I. Tomin, and B. A. Bushuk, Kinetic spectroscopy of orientational statesof solvated dye molecules in polar solutions, J. Lumin. 26, 377–391 (1982).

23. T. B. Tamm, Ya. V. Kikas, and A. E. Sirk, Measurement of nonhomogeneous distributionfunction of impurity centers by the method of double scanning of spectra, Zh. Prikl.Spektrosk. 24(3), 315–321 (1976).

24. G. Mourou and M. M. Malloy, The Stokes-shifted fluorescence risetime of dye molecules insolution, Chem. Phys. Lett. 45, 291–294 (1977).

25. C. V. Shank, E. P. Ippen, and O. Teschke, Sub-picosecond relaxation of large organicmolecules in solution, Chem. Phys. Lett. 45, 291–294 (1977).

26. J. Fünfschilling, Ha Biletationd dchrift. Inst. für Physik der Univ. Basel, 135 (1979).27. R. I. Personov, E. P. Altshits, L. A. Bykovskaya, and B. M. Kharlamov, Fine structure of

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28. A. A. Gorokhovskii, R. Kaarli, and L. A. Rebane, The homogeneous pure electronic linewidthin the spectrum of a N-phthalocyanine solution in n-octane at 5 K, Opt. Commun. 16,282–284 (1976).

29. H. de Vries and D. A. Wiersma, Homogeneous broadening of optical transitions in organicmixed crystals, Phys. Rev. Lett. 36, 91–94 (1976).

30. I. Dombi, I. Kechkemety, and L. Kosma, On the fluorescent solution spectra profile, Opt.Spektrosk. 18, 10–14 (1965).

31. B. I. Stepanov and V. P. Gribkovsky, Introduction to the Theory of Luminescence, Izd. Akad.Nauk Byel. SSR, Minsk (1963).

32. A. N. Rubinov and V. I. Tomin, On the condition for realizability of Stepanov’s universalrelation for complex molecules in solution, Opt. Spektrosk. 30(6), 859–862 (1971).

33. S. I. Vavilov, Light Microstructure, Collected papers in 2 volumes, Izv. Akad. Nauk SSSR 2,999 (1952).

34. V. I. Tomin and A. N. Rubinov, Bathochromic luminescence of organic dyes in alcoholsolutions and polymer matrices, Opt. Spektrosk. 32(2), 424–427 (1972).


35. V. I. Tomin, A. N. Rubinov, and V. F. Voronin, Quencher effect on fluorescence spectra ofpolar dye solutions, Opt. Spektrosk. 34(6), 1108–1111 (1973).

36. A. N. Rubinov, V. A. Zhivnov, and V. I. Tomin, Molecule spectrum fluorescence shift in thelaser nonresonance frequency light field, Opt. Spektrosk. 35(4), 778–779 (1973).

37. A. N. Rubinov, V. I. Tomin, L. Kosma, I. Hilbert, and B. Nemet, Homogeneous and non-homogeneous broadening of dye and β –phosphorescene spectra, Ada Phys. Chem. Szeged,21(1,2), 11–18 (1975).

38. L. Kosma, N. A. Nemkovich, A. N. Rubinov, and V. I. Tomin, The anti-Stokes luminescencequantum yield drop and nonhomogeneous broadening of dye spectra in solution, Opt.Spektrosk. 59(2), 311–316 (1985).

39. V. L. Ermolaev, B. N. Bodunov, E. B. Sveshnikova, and T. A. Shakhverdov, NonradiativeElectronic Excitation Energy Transfer, Nauka, Leningrad (1977).

40. M. D. Galanin, Resonance excitation energy transfer in luminescent solutions, Trudy FIAN12, 3–53 (1960).

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42. I. M. Gulis, A. I. Komyak, and V. I. Tomin, Electronic excitation energy transfer in theconditions of complex organic molecules level nonhomogeneous broadening, Izv. Akad. NaukSSSR, Ser. Fiz. 42, 307–312 (1978).

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44. N. A. Nemkovich, A. N. Rubinov, and V. I. Tomin, Directed energy transfer in single-component dye solutions, Pis’ma Zh. Eksp. Tear. Fiz. 6(5), 270–273 (1980).

45. I. M. Gulis and A. I. Komyak, Inhomogeneous broadening effect on temporal characteristicsof complex molecules luminescence at energy transfer, Vestn. Byel. Gos. Univ., Ser. I, 2, 3–6(1980).

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48. N. G. Bakhshiev, Yu. T. Mazurenko, and I. V. Piterskaya, On the emission decay in variousregions of molecule luminescence spectra in viscous solutions, Opt. Spektrosk. 21(5), 550–554(1966).

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52. B. A. Bushuk, A. N. Rubinov, and A. P. Stupak, Anti-Stokes excitation picosecondspectroscopy of dye polar solutions, Acta Phys. Chem. Szeged, 24(3), 387–390 (1978).

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Index

Actomyosin, 140ADAS: see Anisotropy decays, decay-associated

spectraAdrenocorticotropin, 33Alcohol dehydrogenase, 110Analysis: see Least-squares analysisAnisotropy, 1

fiber optics, 362fundamental, 7, 138oriented systems, 307quenching-resolved, 105wavelength-dependent, 318

Anisotropy decays, 1, 407associated, 28, 283, 411data analysis, 22decay-associated spectra (ADAS), 284definition, 2electron spin resonance, 331ellipsoids, 9frequency domain, 26global analysis, 280hindered rotation, 15instrumentation, 25membranes, 287multiexcitation wavelength, 286muscle fibers, 325order parameter, 17oriented systems, 320proteins, 32quenching-resolved, 105

Associated anisotropy decay, 29; see alsoAnisotropy decay

Azurin, 102

Broadening, 367; see also Inhomogeneousbroadening

Calmodulin, 48

Collisional quenching, 53; see also QuenchingConfidence intervals, 207

Monte Carlo, 214

Data analysisanisotropy, 22global, 241least-squares, 113, 177quenching, 112, 193

Diffusionlight-induced, 406rotational, 8two-dimensional, 96

Distance distributions, 157acceptor decay, 278frequency domain, 162global analysis, 277myosin, 168quenching-resolved, 160theory, 158troponin I, 166

Distributions, 261; see also Distancedistributions

distance, 271

Ellipsoids, 9Energy transfer, 127, 261

actomyosin, 140applications, 140distance distribution, 157efficiency, 130enzyme kinetics, 152Förster distance, 129inhomogeneous broadening, 390labeling, 133lanthanides, 148myosin, 134orientation factor, 131, 135

Italic numbers indicate a detailed description of the topic.

429

430

Energy transfer (Cont.)quenching-resolved, 160ribosomes, 147stopped-flow, 153time-resolved, 155troponin, 145, 166tyr-to-trp, 150

Error analysis, 207, 288; see also Data analysisExcited state reaction, 272

F-Actin, 43Fiber optics

light sources, 359mode, 347transmission, 349

Fiber-optics sensors, 345design, 349mode, 347polarization, 362time-resolved, 361transmission, 349

Fluorescence sensors, 345; see also Fiber-opticsensors

Förster distance, 129Frequency domain

anisotropy decays, 27distance distributions, 162global analysis, 253intensity decays, 26troponin I, 166

FRET, 127; see also Energy transfer

Gauss–Newton algorithm, 187Global analysis, 241; see also Least-squares

analysis; Data analysisanisotropy decay, 280associated anisotropy decay, 283decay-associated spectra, 284distance distributions, 277eigenvalue/eigenvector, 267error analysis, 288, 294excited state reaction, 272flow chart, 259frequency domain, 253identifiability, 288implementation, 248intensity decay, 252least-squares, 258multidye, 285multiexcitation wavelength, 286overview, 244physical models, 261

Index

Global analysis (Cont.)quenching, 112

Glucagon, 108

Horse liver alcohol dehydrogenase, 110

Immunoglobulin, 44Inhomogeneous broadening, 367

intensity decay, 399membranes, 413steady-state emission, 416stationary, 387time-resolved emission spectra, 396vibrational broadening, 376

Instrumentation, anisotropy decays, 25

Labeling, oriented systems, 332Lanthanides, 148Lifetime-resolved anisotropy, 105Least-squares analysis, 177

assumptions, 181chi-square test, 233confidence intervals, 207cumulative frequency plots, 223Durbin–Watson test, 231frequency domain, 199Gauss–Newton, 187, 203global analysis, 258implementation, 235Kolmogorov–Smirnov test, 233Neldes–Mead, 193, 205outliers, 228quenching, 113, 199residuals, 216Simplex, 193, 205theory, 180trends, 226

Lifetime distributions, 114, 261Light sources, fiber optics, 359Lumazine, 37

Melittin, 33, 107Membranes, 92, 413, 422

surface potential, 99Micelles, 92Microscopy, oriented systems, 313Monellin, 108Muscle fibers, anisotropy decay, 325Myelin basic protein, 108Myosin, 41, 134

Naphthol, excited state reaction, 274

Index

Nelder–Mead Simplex algorithm, 193Nitroxide quenching, 98Nucleic acids, 100Nuclease, 108

Order parameter, 17, 331Orientation factor, 131, 135Oriented systems, 307

electron spin resonance, 331instrumentation, 313, 319labeling, 332order parameter, 331time-resolved, 320

Perrin equation, 137Phosphorescence, 117Polarization, 5, 307; see also Anisotropy;

Anisotropy decaysProteins

actomyosin, 140adrenocorticotropin, 33alcohol dehydrogenase, 110azurin, 102calmodulin, 48F-actin, 43glucagon, 108immunoglobulin, 44lumazine, 37melittin, 33membrane-bound, 93monellin, 108myelin basic protein, 108myosin, 41, 134, 140, 168nuclease, 108quenching, 68red-edge excitation, 388, 417resolution of spectra, 102ribosomes, 147S. nuclease, 32, 88single tryptophan, 69troponin, 146, 146troponin I, 166

Quenchers, 67, 93interaction with proteins, 85partitioning, 92

Quenching, 53anisotropy, 105conformational changes, 76data analysis, 199distance distribution, 160efficiency, 59

431

Quenching (Cont.)electrostatic effects, 71energy transfer, 108, 160frequency domain, 88, 199heterogeneous systems, 58least-squares analysis, 113lifetime distributions, 114ligand binding, 75mechanisms, 78membrane-bound proteins, 93membranes, 92, 96membranes and micelles, 92micelles, 92nitroxide, 98nucleic acids, 100phosphorescence, 117proteins, 68resolution of lifetimes, 103resolution of spectra, 102quenchers, 67quenching-resolved, 105radiation boundary, 61Smoluchowski, 60static, 64Stern-Volmer, 55surface potential, 99transient effects, 60, 87

Red-edge excitation, 388, 417Resonance energy transfer, 127; see also Energy

transferRibosomes, 147Rotational diffusion, 8

correlation time, 31ellipsoids, 10

S. nuclease, 32Segmental flexibility, 18Selective excitation, 378, 383

red-edge excitation, 388vibrational levels, 378

Sensors, 345; see also Fiber-optic sensorsSmoluchowski quenching, 60; see also

QuenchingSolvent effects, inhomogeneous broadening, 367Solvent relaxation

inhomogeneous broadening, 395intensity decay, 399

Static quenching, 64Stern-Volmer, 53; see also QuenchingStopped-flow fluorescence, energy transfer, 153

Time-resolved anisotropy, 1

432

Time-resolved energy transfer, 155Time-resolved emission spectra, inhomogeneous

broadening, 396, 420Total internal reflectance, fiber optics, 346Transfer RNA, 39

Index

Transient effects, 87Troponin, 145, 146Troponin I, 166

Yt-base, 40

44471235 Topics in Fluorescence Spectroscopy Vol 2 Principles

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