Introduction to Macromolecular X-ray CrystallographyBiochem 300Borden Lacy

Print and online resources:

Introduction to Macromolecular X-ray Crystallography, by Alexander McPherson

Crystallography Made Crystal Clear, by Gale Rhodeshttp://www.usm.maine.edu/~rhodes/CMCC/index.html

http://ruppweb.dyndns.org/Xray/101index.htmlOnline tutorial with interactive applets and quizzes.

http://www.ysbl.york.ac.uk/~cowtan/fourier/fourier.htmlNice pictures demonstrating Fourier transforms

http://ucxray.berkeley.edu/~jamesh/movies/Cool movies demonstrating key points about diffraction, resolution, data quality, and refinement.

http://www-structmed.cimr.cam.ac.uk/course.htmlNotes from a macromolecular crystallography course taught in Cambridge

Crystal -> Diffraction pattern -> Electron density -> Model

Overview of X-ray Crystallography

Resolution, Fourier transforms, the ‘phase problem’, B-factors, R-factors, R-free …

Diffraction: The interference caused by an object in the path of waves (sound, water, light, radio, electrons, neutron..)Observable when object size similar to wavelength.

Object

Visible light: 400-700 nmX-rays: 0.1-0.2 nm, 1-2 Å

Can we image a molecule with X-rays?

1) We do not have a lens to focus X-rays.

2) The X-ray scattering from a single molecule is weak.

Not currently.

Measure the direction and strength of the diffracted X-raysand calculate the image mathematically.

Amplify the signal with a crystal - an array of ordered molecules in identical orientations.

The wave nature of light

f(x) = Fcos2π(x + )f(x) = Fsin2π(x + )

F = amplitude = frequency = phase

a) f(x) = cos 2πxb) f(x) = 3cos2πxc) f(x) = cos2π(3x)d) f(x) = cos2π(x + 1/4)

x

=c

Interference of two waves

Wave 1 + Wave 2

Wave 1

Wave 2

In-phase Out -of-phase

Bragg’s Law

Sin = AB/d

AB = d sin

AB + BC = 2d sin

n = 2d sin

n = 2d sin

Practically:Assign a coordinate (h, k, l)and intensity (I) to every spot inthe diffraction pattern—Index and Integrate.Ihkl , hkl

The intensity of each spot contains

information about the entire molecule.

The spacing of the spots is due to the size and symmetry of your lattice.

Diffraction pattern

Fourier transform:

F(h)= ∫ f(x)e2πi(hx)dx

where units of h are reciprocals of the units of x

Reversible!

f(x)= ∫ F(h)e-2πi(hx)dh

Calculating an electron density function from the diffraction pattern

hkl, hkl

F(h) = Fcos2π(h+ )F(h) = Fsin2π(h + )

F = amplitude = frequency = phase

Fhkl ~ √Ihkl

(x) = ∫ F(h)e-2πi(hx)dh

Experimental measurements:

Overcoming the Phase Problem

Heavy Atom Methods (Isomorphous Replacement)Anomalous Scattering MethodsMolecular Replacement MethodsDirect Methods

Heavy Atom Methods (Isomorphous Replacement)

The unknown phase of a wave of measurable amplitude can be determined by ‘beating’ it against a reference wave of known phase and amplitude.

Combined Wave

Unknown

Reference

Generation of a reference wave:

Max Perutz showed ~1950 that a reference wave could be created through the binding of heavy atoms.

Heavy atoms are electron-rich. If you can specifically incorporate a heavy atom into your crystal without destroying it, you can use the resulting scatter as your reference wave.

Crystals are ~50% solvent. Reactive heavy atom compounds can enter

by diffusion.

Derivatized crystals need to be isomorphous to the native.

Native Fnat

Heavy atom derivativeFderiv

The steps of the isomorphous replacement method

Heavy Atom Methods (Isomorphous Replacement)

The unknown phase of a wave of measurable amplitude can be determined by ‘beating’ it against a reference wave of known phase and amplitude.

FPH

FP

FH and H

Can use the reference wave to infer P. Will be either of two possibilities.

To distinguish you need a second reference wave. Therefore, the techniqueis referred to as Multiple Isomorphous Replacement (MIR).

Overcoming the Phase Problem

Heavy Atom Methods (Isomorphous Replacement)Anomalous Scattering MethodsMolecular Replacement MethodsDirect Methods

Anomalous scattering

Incident X-rays can resonate with atomic electrons to result in absorption and re-emission of X-rays.

Results in measurable differences in amplitudeFhkl ≠ F-h-k-l

Advances for anomalous scattering methods

Use of synchrotron radiation allows one to ‘tune’ the wavelength of the X-ray beam to the absorption edge of the heavy atom.

Incorporation of seleno-methionine into protein crystals.

Anomalous scattering/dispersion in practice

Anomalous differences can improve the phases in a MIR experiment (MIRAS) or resolve the phase ambiguity from a single derivative allowing for SIRAS.

Measuring anomalous differences at 2 or more wavelengths around the

absorption edge: Multiple-wavelength anomalous dispersion (MAD).Advantage: All data can be collected from a single crystal.

Single-wavelength anomalous dispersion (SAD) methods can work if additional phase information can be obtained from density modification.

Overcoming the Phase Problem

Heavy Atom Methods (Isomorphous Replacement)Anomalous Scattering MethodsMolecular Replacement MethodsDirect Methods

1- Compute the diffraction pattern for your model.2- Use Patterson methods to compare the calculated and measured diffraction patterns.3- Use the rotational and translational relationships to orient the model in your unit cell.

4- Use the coordinates to calculate phases for the measured amplitudes.5- Cycles of model building and refinement to remove phase bias.

Molecular Replacement

If a model of your molecule (or a structural homolog) exists, initial phases can be calculated by putting the known model into the unit cell

of your new molecule.

Direct Methods

Ab initio methods for solving the phase problem either by finding mathematical relationships among certain phase combinations or

by generating phases at random.

Typically requires high resolution (~1 Å) and a small number of atoms.

Can be helpful in locating large numbers of seleno-methionines for a

MAD/SAD experiment.

Overcoming the Phase Problem

Heavy Atom Methods (Isomorphous Replacement)Anomalous Scattering MethodsMolecular Replacement MethodsDirect Methods

F = amplitude = frequency = phase

(x,y,z)electron density

FT

Electron density maps

Duck intensities and cat phases

Are phases important?

Does molecular replacement introduce model bias?

Cat intensities withManx phases

An iterative cycle of phase improvement

Solvent flatteningNCS averaging

BuildingRefinement

Model building

Interactive graphics programs allow for the creation of a ‘PDB’ file.Atom type, x, y, z, Occupancy, B-factor

ATOM 1 N GLU A 27 41.211 44.533 94.570 1.00 85.98ATOM 2 CA GLU A 27 42.250 44.748 95.621 1.00 86.10ATOM 3 C GLU A 27 42.601 43.408 96.271 1.00 85.99ATOM 4 O GLU A 27 43.691 42.865 96.065 1.00 85.71ATOM 5 CB GLU A 27 41.725 45.720 96.687 1.00 86.36ATOM 6 CG GLU A 27 42.804 46.349 97.563 1.00 86.44ATOM 7 CD GLU A 27 43.628 47.387 96.817 1.00 86.98ATOM 8 OE1 GLU A 27 44.194 47.051 95.754 1.00 87.40ATOM 9 OE2 GLU A 27 43.713 48.540 97.296 1.00 87.02ATOM 10 N ARG A 28 41.662 42.882 97.053 1.00 85.65ATOM 11 CA ARG A 28 41.839 41.607 97.739 1.00 85.29ATOM 12 C ARG A 28 41.380 40.458 96.835 1.00 85.31ATOM 13 O ARG A 28 42.184 39.619 96.424 1.00 85.09ATOM 14 CB ARG A 28 41.035 41.607 99.045 1.00 84.62ATOM 15 CG ARG A 28 39.564 41.944 98.851 1.00 84.07ATOM 16 CD ARG A 28 38.845 42.152 100.169 1.00 84.00ATOM 17 NE ARG A 28 37.423 42.439 99.980 1.00 84.27ATOM 18 CZ ARG A 28 36.945 43.413 99.208 1.00 84.53ATOM 19 NH1 ARG A 28 37.771 44.208 98.537 1.00 83.83ATOM 20 NH2 ARG A 28 35.634 43.598 99.111 1.00 84.38...

The PDB File:

Occupancy

B-factor

How much does the atom oscillate around the x,y,z position?

What fraction of the molecules have an atom at this x,y,z position?

Can refine for the whole molecule, individual sidechains, or individual atoms. With sufficient data anisotropic B-factors can be refined.

Refinement

Least -squares refinement

= whkl (|Fo| - |Fc|)2hkl

Apply constraints (ex. set occupancy = 1) and restraints (ex. specify a range of values for bond lengths and angles)Energetic refinements include restraints on conformational energies,

H-bonds, etc.

Refinement with molecular dynamicsAn energetic minimization in which the agreement between measured and calculated data is included as an energy term.Simulated annealing often increases the radius of convergence.

Monitoring refinement

R =||Fobs| - |Fcalc||

|Fobs|

Rfree: an R-factor calculated from a test set that has not been used in refinement.