Discrete Stochastic Models for FRET and Domain Formation in Biological Membranes Audi Byrne March 1 st Biomath Seminar Biomathematics Study Group

Anne Kenworthy Lab Kimberly Drake Brooks Alford Laurel, Chris and Alessandro Minchul Kang

The micro-organization of lipids and proteins within the cell membrane is an open question. We investigate: (1) ways in which FRET could be used to determine existence and properties of domains in cell membranes and (2) a simple, general model for domain formation based on lipid-lipid interactions Overview

I.Biological question: how are lipids organized within the cell membrane? II.FRET A.Experimental tool with nanoscale resolution B.Forward problem: Point Distribution → FRET Discrete stochastic model for FRET Berney and Danuser, Biophys J, year C. Inverse problem: FRET → Point Distribution preliminary results III. Model for Domain Formation loosely based on the Potts model Generates noisy, irregular domains

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 problem b. Results: delimiting the “power” of FRET Talk Outline I. How do lipids organize within biomembranes?

Biomembranes Many cell types Different types of cell membranes membranes of organelles plasma membrane Inner verses outer membrane layers

Hypotheses for lipid organization: Random / homogeneous distributions Complexes/Oligomers Exotic organizations

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!

Heetderks and Weiss 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.

B. Investigating the membrane with FRET

Fluorescence Two views of the hydromedusa Aequorea victoria from Friday Harbor, Washington, copyright Claudia E. Mills 1999.

Fluorescence Two views of the hydromedusa Aequorea victoria from Friday Harbor, Washington, copyright Claudia E. Mills 1999.

Fluorescence

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 s). Pauli Exclusion principle: No two electrons in the same orbital may have the same spin.

A fluorophore with an excited electron may transfer its electronic energy to another fluorophore non-radiatively (by resonance) if: 1. the second fluorophore is near and 2. the emission energy of the first molecule matches the excitation energy of the second. This occurs by dipole-dipole interaction. Resonance Energy Transfer

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. K t =K D (R 0 /r 6 ) K t  1/r 6 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].

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”. 0Un-excited0 → 1Excitation 1Excited1 → 0 Decay or Transfer

1. Donor Excitation Transfer occurs between every unexcited acceptor and every excited donor at rate k T, which depends upon their molecular separation r : 2. Transfer k t = k D * (R 0 /r) 6 Donors excite with constant rate k E, which models constant illumination. 3. Donor and Acceptor Decay Excited fluorophores decay with constant rate k D, which models exponential decay: Y = Y 0 e -t/K D The lifetime of the fluorophore Is 1/K D = .

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 ns  D = 5 ns  A = 10 ns kE =.25/ns

FRET for a Clustered Distribution FRET Efficiency = (# Actual Transfers) / (# Possible Transfers) = (Acceptor Fluorescence) / (Acceptor + Donor Fluorescence) Over 10 Nanoseconds 1 TS =.1 ns  D = 5 ns  A = 10 ns kE =.25/ns

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!

Model for Domain Formation

Model Components Different Lipid Species Different lipid species are assigned different labels ‘  ’. Plasma Membrane NxN Square Lattice Every node is occupied by a single lipid. ( lipids)

Plasma Membrane

Lipid 1

Plasma Membrane Lipid 2 Lipid 1

Plasma Membrane Lipid 2 Lipid 1 Lipid 3

Plasma Membrane Lipid 2 Lipid 1 Lipid 3

Model Components II Every pair of lipid types is assigned an “interface energy”. Lipid Interaction Lattice Energy The total energy of the system is defined as the sum of the interface energies of all adjacent nodes on the lattice. 0 <   1  2 < 1 Example Like lipids:  = 0 Unlike lipids:  = 1

Lipid Diffusion Lipids diffuse by stochastic random walk in a way which decreases system energy by the Metropolis algorithm: Neighboring lipids switch locations if switching decreases the energy of the system. Otherwise, the switch is permitted only if the local temperature is high enough. (random variable)

Example Unlike Lipids δ =1 Lipid-Lipid Interface Energies Like Lipids δ=0 Local Interface Energy = 6

Example Unlike Lipids δ =1 Lipid-Lipid Interface Energies Like Lipids δ=0 Local Interface Energy = 6 Neighbor Switch

Example Unlike Lipids δ =1 Lipid-Lipid Interface Energies Like Lipids δ=0 Local Interface Energy = 6 Neighbor Switch Local Interface Energy = 3

Simulation Results Random Initial Conditions

Simulation Results

Generated domains: irregular, noisy But otherwise similar to the idealized disk domain model.

Vary the density of ‘black’ lipids labeled with acceptors.... what is the FRET efficiency verses acceptor density?

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!