Methods of Membrane Domain Investigation in Noisy Environments
Dr. Maria Kiskowski Byrne, Department of Mathematics and Statistics, University of South Alabama.
Dr. Anne Kenworthy, Depts. of Molecular Physiology & Biophysics and Cell & Developmental Biology, Vanderbilt University School of Medicine.
While protein clustering in the plasma cell membrane has
been well-established, determining the size, longevity and
importance of lipid domains within the bilayer has proved elusive1.
A promising experimental technique for studying lipid organization is FRET. FRET provides nanoscale
resolution and reports on lipid positions without altering the membrane. The model of Berney and Danuser 2
solves the forward problem: given a distribution of fluorophores, all relevant aspects of membrane FRET
can be simulated with an individual-based model. However, of most interest experimentally is the inverse
problem: given a specific FRET signature, what is the underlying distribution of fluorophores? We have
answered this question in the context of lipids randomly distributed in idealized disk-shaped domains3 but
further research is required to address these questions in the context of irregular, biologically relevant
domains. For this, we have developed a stochastic model of domain formation based on simplified
individual lipid-lipid interactions. We investigate the potential of FRET is identify domain properties in this
context.
Fluorescence Resonance Energy Transfer
Motivation
A 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.
The 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
Drawn by P. Kinnunen, CEO of Kibron
into different membrane domains.
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..
Time
The Fluid-Mosaic Model With Microdomains
Domain Size
Noisy Membranes
Raft Fraction 
Lipid distributions are generated for varying
raft fractions (blue) and varying domain size.
To simulate FRET, fluorophores are
randomly assigned to lipids.
Modeling FRET
Donors excite with
constant rate kE,
which models
constant illumination.
0
1
Un-excited
Excited
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.
Image: Heetderks and Weiss
assigned an index 
and each node of a
square lattice is
occupied by exactly
one lipid species.
Random Initial Conditions
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.
Sorting After 100 Timesteps
Lipid Diffusion
Each lipid species is
Lattice Energy
Lipid Species
Lipid Partitioning Model
For all distributions, FRET
robustly depends upon the
local acceptor density
δ=(N-1) fA/N
N = total # of lipids in a domain
fA = fraction of lipids labeled
with acceptors
The ratio of the global density
to the local density provides
the domain fraction.
Excited fluorophores
decay with constant
rate kD, which models
exponential decay.
Y = Y0 e
-kDt
The lifetime of the
fluorophore
Is 1/kD=.
FRET Efficiency = (# Actual Transfers) / (# Possible Transfers)
= (Acceptor Fluorescence) / (Acceptor + Donor Fluorescence)
Results: Intra & Inter-Domain FRET
Intra-domain FRET:
acceptors and donors are
located within domains
0→1
Excitation
1 → 0 Decay or Transfer
Transfer occurs between every unexcited acceptor and every
excited donor at rate kT, which depends upon their molecular
separation r :
kt = kD * (R0/r)6
The Lipid Raft Hypothesis
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.
Due to the sensitive dependence
of FRET on inter-molecular
separation, FRET has been
used as an amazingly accurate
“spectroscopic ruler”5.
Inter-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
Comparison: Ripley’s K
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 .
Ripley’s K reports on the domain radius.
For a random distribution,
K(r) is r2. K(r) can be
normalized so that its
expected value is r :
L(r)= [K(r)/]7.
For idealized
distributions, the radius
r that minimizes L'(r) is
independent of the
domain separation and
corrsponds to 2R.8
L(r) for points randomly
distributed (dotted line)
and clustered within
domains of radius R,
separation S.
And the domain separation.
The ratio of the radius that
maximizes H(r)=L(r)-r and
minimizes H’(r)=L’(r) scales
with the domain separation.
Lipids diffuse by stochastic random walk
in a way which decreases system energy by
the Metropolis algorithm: Neighboring lipids
Summary
switch locations if switching decreases the
energy of the system. Otherwise, the switch
is permitted depending on the temperature
of the system.
Lattice Energy Over Time
Rule: “like” lipids have lower
coupling energies than unlike.
These rules cause lipid species
to sort from a random
distribution into clusters.
A simple model
based on lipidlipid interactions
generates
stochastic
distributions that
vary
systematically in
domain fraction
and domain size.
FRET simulations indicate
that intra-domain FRET is a
robust measure of the intradomain probe density (that
can be used to estimate the
domain fraction) and interdomain FRET provides a
measure of the domain
width before and after the
percolation transition.
Domain interaction and
variable domain shape
make interpretation of
the Ripley K statistic
more complex. In
particular the statistic
measures the length of
the domain along its
widest dimension so
that it would be most
useful before the
percolation transition.
References
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
92: 3040--3051.
4. Forster, T. (1948) Intramolecular energy migration and fluorescence. Ann
Phys. (Leipzig) 2: 55--75.
5. Stryer L, Haugland R.P. (1967) Energy transfer: a spectroscopic ruler. Proc
Natl Acad Sci U S A. 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. Besag, J.E. (1977) Comments on Ripley's paper: J. R. Stat. Soc. B. 39:
193–195.
8. Kiskowski, M., Hancock, J.F., Kenworthy, A. On the use of Ripley’s K
function and its derivatives to determine domain size, submitted.