Methods of Membrane Domain Investigation in Noisy Environments

GUIDING QUESTION: ARE THERE LIPID DOMAINS IN THE PLASMA MEMBRANE? While protein clustering in the plasma cell membrane has been well-established, determining the size, longevity and roles of lipid domains within the bilayer has proved elusive1. [DETAILS BELOW ]FRET A promising experimental technique for studying lipid organization is FRET. FRET provides nanoscale resolution and reports on lipid positions without altering the membrane. [DETAILS RIGHT ]

INVERSE PROBLEM FOR IDEALIZED CASES The individual-based model of Berney and Danuser2 solves the forward problem: given a distribution of fluorophores, all relevant aspects of membrane FRET can be simulated. However, of most interest experimentally is the inverse problem: given a specific FRET signature, what is the underlying distribution of fluorophores? We have found that FRET can be used to measure the total fraction and average diameter of ideal, disk-shaped lipid domains. [DETAILS RIGHT ]

NOISY ENVIRONMENTS The inverse problem is easiest to probe for simplified, idealized distributions of lipids3 but distributions in vivo are complex and have many important sources of noise. We develop a stochastic Monte Carlo model based on simplified individual lipid-lipid interactions to study the inverse problem in the context of noise. [DETAILS BELOW AND BELOW RIGHT ]

Lipid Partitioning Model

Motivation

The Fluid-Mosaic Model With MicrodomainsThe fluid-mosaic model of Singer and Nicholson, in which the plasma membrane is a phospholipid bilayer embedded with proteins, includes lipid membrane domains within the “mosaic”. Protein clustering occurs as proteins form complexes or preferentially partition into different membrane domains. Drawn by P. Kinnunen, CEO of Kibron

Image: Heetderks and Weiss

In simple lipid mixtures, phospholipids with long, ordered chains sort into gel domains and those with short, disordered chains sort into fluid domains. The addition of cholesterol to gel domains forms a liquid ordered phase, the proposed state of lipid rafts.

It is hypothesized that this may also occur in vivo so that the cell membrane phase separates into liquid-ordered domains (lipid rafts) and liquid-disordered domains that compartmentalize membrane proteins.

The Lipid Raft Hypothesis

Caveolae: flask-shaped structures enriched with caveolin.

GPI-anchored proteins cluster in the apical ends of epithelial cells.

Lipid rafts??? .. small, enriched in sphingolipid and cholesterol, involved in signal transduction, protein sorting and membrane transport..

The coupling energy (e.g., partition

coefficient) between any two lipid

species i and j is i i. The total

energy of the system is defined as

the sum of the coupling energies of

all adjacent nodes on the lattice. Lipi

d S

peci

es

Each lipid species is

assigned an index

and each node of a

square lattice is

occupied by exactly

one lipid species.

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 depending on the temperature

of the system. Latt

ice

Ene

rgy

Lipi

d D

iffus

ion

These rules cause lipid species

to sort from a random

distribution into clusters.

Rule: “like” lipids have lower coupling energies than unlike.

Random Initial Conditions Sorting After 100 Timesteps Lattice Energy Over Time

FRET: Fluorescence Resonance Energy TransferA fluorophore with an excited electron may transfer its electronic energy to another fluorophore (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.

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 separation4.

Due to the sensitive dependence of FRET on inter-molecular separation, FRET has been used as an amazingly accurate “spectroscopic ruler”5.

Dr. Maria Audi Kiskowski Byrne, Department of Mathematics & Statistics, University of South Alabama.Dr. Anne Kenworthy, Depts. of Molecular Physiology & Biophysics and Cell & Developmental Biology, Vanderbilt University School of Medicine.

Modeling FRET

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

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

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

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

Y = Y0 e -kDt

kt = kD * (R0/r)6

The lifetime of the fluorophore

Is 1/kD=.

0 Un-excited 0 → 1 Excitation

1 Excited 1 → 0 Decay or Transfer Intra-Domain FRETIntra-domain FRET: acceptors and donors are located together within domainsFor disk distributions, FRET robustly depends upon the local acceptor density δ:

N = total # of lipids in a domain

fA = fraction of lipids labeled with acceptors

FIND DOMAIN FRACTION: The ratio of the global density to the local density provides the domain fraction.

fAN

N 1

Inter-Domain FRETInter-domain FRET: donors are located within domains and acceptors are located outside domains

FRET functionally depends upon the domain width.

Blue: distributions before the percolation transition

Red: during the percolation transition

Black: distributions after the percolation transition

Lipid distributions are generated for varying raft fractions (blue) and varying domain size. To simulate FRET, fluorophores are randomly assigned to raft and/or non-raft lipids.

Noisy Membranes

Raft Fraction

D

omai

n S

ize

T

ime

Results3

Results for Noisy Environments

1. Edidin, M. 2003. The state of lipid rafts: from model membranes to cells. Annu. Rev. Biophys. Biomol. Struct. 32:257–283.

2. Berney C, Danuser G. 2003. FRET or no FRET: A quantitative comparison. Biophysical Journal 84(6): 3992-4010.

3. Kiskowski, M., Kenworthy, A. 2007. In silico characterization of resonance energy transfer for disk-shaped membrane domains, Biophysical Journal, in press. Biophysical Journal, 92: 3040-3051.

4. Forster, T. (1948). Intramolecular energy migration and fluorescence. Ann Phys. (Leipzig) 2: 55-75.

5. Stryer L, Haugland RP. Energy transfer: a spectroscopic ruler. Proc Natl Acad Sci U S A. 1967 Aug;58(2):719–726.

6. Ripley, B. D. 1978. Spectral analyses and the analysis of pattern in plant communities. Journal of Ecology 66: 965–981.

7. Kiskowski, M, and A. Kenworthy, 2009. On the use of Ripley’s K function and its derivatives to analyze domain size, Biophysical Journal, doi:10.1016/j.bpj.2009.05.039.

8. Besag, J.E. 1977, Comments on Ripley's paper: J. R. Stat. Soc. B. 39 193–195.

References

Ripley’s K6 is the second moment property of a spatial point pattern (the expected # of points within a distance r of another point) normalized by the # of points per area .

For a random distribution, K(r) is r2. K can be normalized so that its expected value is r (linear): L(r)= [K(r)/]7.

K(r) and L(r) for points randomly distributed (dotted lines) and distributed

within domains of radius R, separation S.

For Ideal Disks, FRET, Ripley’s KReport on Domain

Spatial Scale

FRET

For systematized departures from ideal disks, FRET (●) measures spatial scale of

the minor axis and Ripley’s K (□) measures spatial scale of the major axis.

No

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l Va

ria

tion

in

Ra

diu

s

Do

ma

in C

on

tou

r

Ecc

en

tric

ity

An Independent Statistic: Ripley’s K

Ripley’s K

Intra-domain FRET robustly predicts the

domain fraction.

Inter-domain FRET measures an

increase in domain radius as domain fraction increases and reports on the

percolation transition.

Domain width increases with domain fraction.

Black: raft lipid labeled with donors.

Red: non-raft lipid labeled with donors.