Discrete Stochastic Models for FRET and Domain Formation in Biological Membranes

Audi ByrneMarch 1st

Biomath Seminar

Biomathematics Study Group

Anne Kenworthy LabVanderbilt University Medical Center

protein traffickingsignal transductionFCS, FRAP, FRET

lipid rafts, Ras

We investigate ways in which FRET could be used to determine the micro-organization of lipids and proteins.

How are bio-molecules (lipids and proteins)

organized in the cell membrane?

I. Membrane Biology: how are molecules organized within the cell membrane?

II. FRET A. Light microscopy with nanoscale

resolutionB. Segregation FRET: to measure domain

separation

III. Computational Model For Segregation FRET A. Comparison with Ripley’s K measureB. Comparison with Hausdorff Measure

Presentation Outline

A. Biological question: how are lipids organized within the cell membrane?

B. FRET: experimental tool

C. Models:a. FRET (Berney and Danuser, Biophys J, year)

b. Domain formation (loosely based Potts model)

D. Goal: Investigating the potential of FRET to identify domains and domain characteristics.a. Challenge: a highly underdetermined inverse

problemb. Results: delimiting the “power” of FRET

Talk Outline

I. How do lipids organize within biomembranes?

Biological Membranes

Structurally composed of phospholipids…

http://academic.brooklyn.cuny.edu/biology

Phospholipids have a hydrophobic part and a hydrophilic part so they naturally form bi-layers:

Biological Membranes

Structurally composed of phospholipids…

http://academic.brooklyn.cuny.edu/biology

Phospholipids have a hydrophobic part and a hydrophilic part so they naturally form bi-layers:

Within the sea of phospholipids, there is a diverse variety of proteins and other kinds of lipids.

Are lipids and proteins arranged randomly throughout the biomembrane, or is there micro-organization?

Hypotheses for lipid organization:

• Random / homogeneous distributions• Highly ordered or regular• Complexes/Domains• Exotic organizations

Applications/Relevance

Immune system: Lipid domains are putatively required for antigen recognition (and antibody production).

Vascular system: lipid domains are putatively required for platelet aggregation.

HIV: lipid domains are putatively required to produce virulogical synapses between T-lymphocytes that ennable replication

Cancer: Ras proteins, implicated in 30% of cancers, are thought to signal by compartmentalizing within different domains

The positions of “n” molecules is described by 2n numbers in continuous space, and no fewer number of parameters can describe the distribution of points

Prior, Muncke,Parton and Hancock

However,

The organization of lipids are determined by physical and biological parameters that may greatly constrain the set of possible distributions (modulo noise)

Example: if the distribution of lipids is genuinely random, the entire distribution can be described by 2 parameters!

Hee

tder

ks a

nd W

eiss

Lipid-Lipid Interactions

Gel Domains: Phospholipids with long, ordered chains

Fluid Domains: Phospholipids with short, disordered chains

Cholesterol : Gel domains form a liquid ordered phase

Domain Formation In Model Membranes

The Lipid Raft Hypothesis The cell membrane phase separates into liquid-ordered domains and liquid-disordered domains.

Liquid-Ordered Domains

- “lipid rafts”

- enriched in glycosphingolipids and cholesterol

- act to compartmentalize membrane proteins: involved in signal transduction, protein sorting and membrane transport.

Open Questions

Whether domains form.

How large are these domains?

How dense are these domains?

More obliquely:What is the domain separation?

We can measure domain separation for a point pattern using Ripley’s K measure, the Hausdorff measure, or using segregation FRET.

B. Direct Observation Yields 40-200 nm resolution

“focused light is the only way to examine living cells non-invasively” Westphal and Hell, 2005

Studying molecule organization by “looking” at the membrane.. Light Microscopy

Diffraction-Limited Resolution in Light Microscopes

Ernst Abbe, 1873

Diffraction limit ~200nm

s = λ /(2n sin α)

where n sin α = numerical aperture of the objective λ = light wavelength

The wavelength of visable light ~ .5 microns.

Latest technology…

Current best resolution is ~40 nm with STED technique.

Stimulated Emission Depletion (STED) microscopy

B. Indirect FRET Yields 1-10 nm resolution

What is FRET?

Fluorescence Resonance Energy Transfer

Two fluorophores:

One “donor” fluorophore One “acceptor” fluorophore

Energy Transfer

A fluorophore with an excited electron (the donor) may transfer its electronic energy to another fluorophore (the acceptor) by resonance if the the emission energy of the first molecule matches the excitation energy of the second.

This occurs by dipole-dipole interaction. (The fluorophores must be close but not too close.)

Resonance Energy Transfer

Fluorescence occurs when an electron becomes excited by absorption of photons.

The electron is excited to a higher energy level and the electron spin is preserved, so that the electron may relax at any time. The lifetime of this excited state is very short (less than 10-5 s).

Pauli Exclusion principle:

No two electrons in the same orbital may have the same spin.

Fluorescence occurs when an electron becomes excited by absorption of photons.

http://www.olympusfluoview.com

http://www.pfid.org/html/un_fret

Dipole-dipole interaction is highly dependent upon distance. In 1948, T.M. Förster calculated that the rate of resonance energy transfer between two fluorophores would depend on the inverse of the sixth power of their separation.

Since then, this has been borne out by rigorous experimental tests.

Kt =KD(R0/r6)

Kt 1/r6

FRET Rate

Due to the sensitive dependence of FRET on inter-molecular separation, FRET has been used as an amazingly accurate “spectroscopic ruler” [Stryer, 1967].

Example: Two amino acids in a protein P are tagged with GFP. However, they can’t be resolved with a microscope (separation less than 200nm).

Example: Two amino acids in a protein P are tagged with GFP. However, they can’t be resolved with a microscope (separation less than 200nm).

One amino acid is labeled with a donor (absorbs blue light, emits green) and one is labeled with an acceptor (absorbs green light, emits yellow).

Under blue light illumination, the protein reflects 75% green light and 25% yellow light.

The transfer rate is .25! The distance that gives that transfer rate can be calculated. kt = kD * (R0/r)6

Model for FRET

Berney and Danuser [Biophys J, 2004]

Modeling FRET (Forward Problem)

1. Begin with a space-point distribution of lipids.

1. From ECM data,

2. “drawn” from simple rules

3. generated by simulations

2. Lipids are randomly labeled with “donors” and “acceptors” that can undergo “FRET”.

3. Lipids are assigned “states”

Initially, all fluorophores are assigned state ‘0’ “off”.

0 Un-excited 0 → 1Excitation

1 Excited 1 → 0 Decay or Transfer

1. Donor Excitation

Transfer occurs between every unexcited acceptor and every excited donor at rate kT, which depends upon their molecular separation r :

2. Transfer

kt = kD * (R0/r)6

Donors excite with constant rate kE, which models constant illumination.

3. Donor and Acceptor Decay

Excited fluorophores decay with constant rate kD, which models exponential decay:

Y = Y0 e -t/KD

The lifetime

of the fluorophore

Is 1/KD=.

These processes occur simultaneously, and

thus compete over time. Small timesteps (<< ) must be chosen to model the rates accurately.

Donor Excitation

Donor and Acceptor Decay

Transfer

FRET for a Clustered Distribution

FRET Efficiency = (# Actual Transfers) / (# Possible Transfers)= (Acceptor Fluorescence) / (Acceptor + Donor Fluorescence)

Over 10 Nanoseconds

1 TS = .1 nsD = 5 nsA = 10 nskE = .25/ns

FRET for a Clustered Distribution

FRET Efficiency = (# Actual Transfers) / (# Possible Transfers)= (Acceptor Fluorescence) / (Acceptor + Donor Fluorescence)

Over 10 Nanoseconds

1 TS = .1 nsD = 5 nsA = 10 nskE = .25/ns

Zooming in on a Single Cluster

Zooming in on a Single Cluster

Goal: Investigating the potential of FRET to identify domains and domain characteristics.

Challenge: a highly underdetermined inverse problem

Results: delimiting the “power” of FRET

The challenge…

Lipid distributions are under-determined by FRET:

.And potentially complex!

Approach:Instances of the forward problem

Disk-shaped Domain Model

(i) domains of radius ‘r’

(ii)Each domain has N molecules

(iii) Molecules are stochastically labeled with donors and acceptors in different ways

(iv)fluorophores between domains do not interact

Results found in the literature:

Combined in single functional relationship:

Acceptor Density Within Domains

An important consequence:

Two distributions with the same acceptor density cannot be distinguished!

Average distance between probes within a domain are the same!

One set of domains is smaller => less “edge” interaction with probes between domains.

Acceptor Density Within Domains

However, this is for idealized domains!

Future Directions

Investigate how to quantitatively distinguish (a) from (b) below:

Future Directions

Investigate how to quantitatively distinguish (a) from (b) below:

Future Directions

Investigate how to quantitatively distinguish (a) from (b) below:

Investigate other models for domain formation:

Oligomerization (e.g., mass action)Cell-controlled organization Protein “corals”

Thank you!