Phase transitions of the coupled membrane-cytoskeleton modify cellular shape.
ABSTRACT Formation of protrusions and protein segregation on the membrane is of a great importance for the functioning of the living cell. This is most evident in recent experiments that show the effects of the mechanical properties of the surrounding substrate on cell morphology. We propose a mechanism for the formation of membrane protrusions and protein phase separation, which may lay behind this effect. In our model, the fluid cell membrane has a mobile but constant population of proteins with a convex spontaneous curvature. Our basic assumption is that these membrane proteins represent small adhesion complexes, and also include proteins that activate actin polymerization. Such a continuum model couples the membrane and protein dynamics, including cell-substrate adhesion and protrusive actin force. Linear stability analysis shows that sufficiently strong adhesion energy and actin polymerization force can bring about phase separation of the membrane protein and the appearance of protrusions. Specifically, this occurs when the spontaneous curvature and aggregation potential alone (passive system) do not cause phase separation. Finite-size patterns may appear in the regime where the spontaneous curvature energy is a strong factor. Different instability characteristics are calculated for the various regimes, and are compared to various types of observed protrusions and phase separations, both in living cells and in artificial model systems. A number of testable predictions are proposed.
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ABSTRACT: Actin-based cellular protrusions are a ubiquitous feature of cell morphology, e.g., filopodia and microvilli, serving a huge variety of functions. Despite this, there is still no comprehensive model for the mechanisms that determine the geometry of these protrusions. We present here a detailed computational model that addresses a combination of multiple biochemical and physical processes involved in the dynamic regulation of the shape of these protrusions. We specifically explore the role of actin polymerization in determining both the height and width of the protrusions. Furthermore, we show that our generalized model can explain multiple morphological features of these systems, and account for the effects of specific proteins and mutations.Biophysical Journal 08/2014; 107(3):576–587. · 3.83 Impact Factor
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ABSTRACT: Lipid demixing phase transition is not needed to induce lateral lipid domains.•A coupling between lipid spontaneous curvature and the extracellular matrix suffices to induce lateral lipid domains.•Explicit demonstration of the proposed mechanism for a membrane sandwiched between a support and an actin network.•An analogous effect is observed for a membrane bound to actin via proteins that induce local membrane curvature.Biochimica et Biophysica Acta (BBA) - Biomembranes 11/2014; · 3.43 Impact Factor
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ABSTRACT: Equilibrium equations and stability conditions are derived for a general class of multicomponent biological membranes. The analysis is based on a generalized Helfrich energy that accounts for geometry through the stretch and curvature, the composition, and the interaction between geometry and composition. The use of nonclassical differential operators and related integral theorems in conjunction with appropriate composition and mass conserving variations simplify the derivations. We show that instabilities of multicomponent membranes are significantly different from those in single component membranes, as well as those in systems undergoing spinodal decomposition in flat spaces. This is due to the intricate coupling between composition and shape as well as the nonuniform tension in the membrane. Specifically, critical modes have high frequencies unlike single component vesicles and stability depends on system size unlike in systems undergoing spinodal decomposition in flat space. An important implication is that small perturbations may nucleate localized but very large deformations. We show that the predictions of the analysis are in qualitative agreement with experimental observations.SIAM Journal on Applied Mathematics 01/2012; 72(2). · 1.58 Impact Factor
Phase Transitions of the Coupled Membrane-Cytoskeleton Modify
Alex Veksler and Nir S. Gov
Department of Chemical Physics, The Weizmann Institute of Science, Rehovot, Israel
the living cell. This is most evident in recent experiments that show the effects of the mechanical properties of the surrounding
substrate on cell morphology. We propose a mechanism for the formation of membrane protrusions and protein phase
separation, which may lay behind this effect. In our model, the fluid cell membrane has a mobile but constant population of
proteins with a convex spontaneous curvature. Our basic assumption is that these membrane proteins represent small
adhesion complexes, and also include proteins that activate actin polymerization. Such a continuum model couples the
membrane and protein dynamics, including cell-substrate adhesion and protrusive actin force. Linear stability analysis shows
that sufficiently strong adhesion energy and actin polymerization force can bring about phase separation of the membrane
protein and the appearance of protrusions. Specifically, this occurs when the spontaneous curvature and aggregation potential
alone (passive system) do not cause phase separation. Finite-size patterns may appear in the regime where the spontaneous
curvature energy is a strong factor. Different instability characteristics are calculated for the various regimes, and are compared
to various types of observed protrusions and phase separations, both in living cells and in artificial model systems. A number of
testable predictions are proposed.
Formation of protrusions and protein segregation on the membrane is of a great importance for the functioning of
Membrane protrusions built of actin filaments (1) are of a
great importance both for the functioning of the living cell
(microvilli in intestinal cells, stereocilia in the inner ear cells,
and filopodia). Phase separation of membrane components
such as proteins, lipids, and cholesterol, i.e., the formation of
aggregates (rafts) on the cell membrane, is also an important
process that determines cell behavior (2,3). It has been found
experimentally that these two phenomena may be closely
related in many circumstances in living cells, where the
membrane region, deformed by the protrusion, has a very
different composition compared to flat regions (see, for ex-
ample, (4,5)). This relation of composition to membrane cur-
vature has also been demonstrated recently in a simple in vitro
model system (6), and in a model system that contains actin
Additionally, interactions of the cell with the surrounding
matrix (ECM) have been shown experimentally (8,9) to
affect the shape of cells (and even their differentiation
(8,10)). In particular, the morphological features, such as the
density and length of membrane protrusions, are affected by
the rigidity of the ECM. The feature of the living cell
primarily affected by the substrate rigidity is the adhesion of
the cell to the substrate. It has also been observed that in the
tips or along the length of protrusions, adhesion molecules
(such as integrins) are concentrated (11–13). Since for these
molecules to adhere they need to be connected to the actin
cytoskeleton, their adhesion activity is also linked to the
local level of actin polymerization (14). Both actin poly-
merization and the activity of molecular motors that enable
adhesion are dependent on the metabolism of the cell.
One can therefore make the following model which
proposes that: aggregation of some membrane proteins
(MP), adhesion of the cell to the substrate, and actin poly-
merization near the membrane, must all be related in some
way. The key feature that links the adhesion and actin
polymerization to the membrane could be the spontaneous
curvature of the membrane components. We implement these
features by considering small protein complexes that have a
convex spontaneous curvature, and also activate the poly-
merization of actin (15). Such protein complexes may include
Formins, WASP, Arp2/3, and a host of proteins. Recently a
number of membrane proteins that contain domains with
specific spontaneous curvature, and are associated with actin
filaments and polymerization, have been identified (15–17).
These proteinsare exactly the kind whichweproposedtoplay
a key role in our model (18). These membrane proteins are
considered to be laterally mobile inside the membrane and
permanently activated. Our model deals with their dynamics,
which is treated as a two-dimensional gas in the plane of the
membrane. We also consider that there may be attractive
interactions between the MP, which also contribute to their
aggregation (phase separation). The local density of these pro-
teins is now coupled to the membrane shape deformation
through the induced active forces and the spontaneous cur-
vature. Another experimental example that links membrane
convex curvature with actin-driven protrusions is Bettache
et al. (19), where lipid composition is driving the spontaneous
Submitted May 22, 2007, and accepted for publication July 18, 2007.
Address reprint requests to Alex Veksler, Tel.: 972-8-934-6031; E-mail:
Editor: Alexander Mogilner.
? 2007 by the Biophysical Society
3798 Biophysical JournalVolume 93 December 20073798–3810
The proposed continuum model is based on the Helfrich
Hamiltonian (20) for the membrane elastic energy, combined
with the free energy of a gas of mobile membrane proteins,
with the protrusive actin force added to the equations of
motion. Our goal is to build a framework applicable for
investigation of the various kinds of dynamic phase transi-
tions and the formation of membrane protrusions in the
living cells. In this model we consider dynamic-instability
transitions in the coupled membrane-cortical cytoskeleton
system. The cortical cytoskeleton is intimately coupled to the
membrane and mainly composed of actin filaments. Physi-
cally, the appearance of instability meansthe phase separation
of the MP, with the simultaneous formation of protrusions,
and we therefore plot these transitions in the form of phase
We do not consider all the intricate internal organization
of the cell, such as its nucleus, microtubules, and intracellular
organelles. Furthermore, we calculate here the instability of
the membrane-cytoskeleton system within linear stability
analysis, which is therefore limited to the initiation stage of
any structural phase transitions. Such shape transitions may
be followed by large-scale internal reorganization of the cell,
over long timescales. These rearrangements are not treated
within our phase transition model.
Membrane and vesicular deformation caused by proteins
having intrinsic curvature was extensively studied during the
last years (see, for example, (21–39)). However, it has been
shown (24) that the bending energy coefficient must be very
large, for the spontaneous curvature alone to be able to bring
about phase separation and cause the appearance of mem-
brane protrusions. Nevertheless, close to the critical temper-
ature the curvature does enhance the phase separation,as was
calculated (21) and observed (40). All these works investi-
gated systems that are in thermodynamic equilibrium, where
there are no active forces that arise from cellular metabolism
(i.e., passive systems).
The effect of cell adhesion on the membrane shape was
also studied,both experimentally
(12,13,31,32,41–45), and so was the effect of coupling the
protrusive force of actin polymerization and spontaneous
curvature (18,46–48). The main innovation of our work is
the coupling of all these factors in one unified model. We
have shown that, indeed, the protein phase separation and the
formation of membrane protrusions may be drivenby the cell
adhesion and actin polymerization, rather than by the
spontaneous curvature and aggregation alone. The results
of our model introduce a new concept: dynamic-instability
transitions in the coupled membrane-cortical cytoskeleton
system can trigger shape changes in the cell, and maybe even
influence the differentiation paths.
Let us stress here that our work deals with linear stability
analysis of the system. We therefore consider adhesion and
actin-driven protrusions in their initial stages of formation.
The later stages, when mature focal adhesions and associated
stress fibers form, are outside the scope of our work. The
response of these structures to stress through their mechano-
sensitivity is also not treated here (49). We also consider that
the elastic properties of the external substrate affect the
adhesion strength, while other effects such as its elasticity
and deformation (50), are neglected here. We nevertheless
propose that the initial type of instability that initiates the
formation of these later structures therefore determines the
local or global cell shape. Our approach of linear stability
analysis is somewhat similar to previous studies of shape
instabilities in other fields of physics (51,52).
The model is developed in the next section and its linear
stability analysis is further developed in the Appendix. In
The Results: Stability Phase Diagrams, the various instability
regimes are described in detail. Possible interpretation of the
results, with comparison to experimental data, is given in
Discussion and Conclusions.
Our model is shown schematically in Fig. 1 a. We assume
here that the membrane is approximately flat, which may
apply to many cases, such as the leading edge of a cell that is
spread on a substrate, a cell in suspension (spherical in
shape), and the top and ventral sides of a spread cell. In all
these cases, the scale of the cell curvature may be negligible
as compared to that of the protrusions that we will describe.
We assume that the cell is free to deform, so that curvature
can play a role, as we show below.
In this flat membrane we assume a finite overall concen-
tration of mobile clusters f that include a variety of proteins,
brane (solid line) that is free to deform locally (h) and contains a population
of membranal clusters (f) that have a convex spontaneous curvature and
are free to diffuse in the plane of the membrane with diffusion coefficient
D. These membranal clusters (MPs) contain proteins that promote actin
polymerization and protrusive force f, and adhesion to the surrounding ECM
(inducing effective negative surface tension a). (b) The cell membrane-
cortical cytoskeleton can be in the mixed state (i), which is featureless and
uniform, or can become unstable (ii); phase-separated, with large aggrega-
tion of the MPs and protrusions/adhesions structures.
(a) A schematic picture of our model: an overall flat mem-
Membrane-Cytoskeleton Phase Separation3799
Biophysical Journal 93(11) 3798–3810
which are able to activate actin polymerization (protrusive
force f in Fig. 1 a), and adhesion to the surrounding matrix
(negative surface tension coefficient a). These clusters are
free to diffuse in the plane of the membrane with effective
diffusion coefficient D, which may be much smaller then for
small membrane proteins, since there is the additional
friction with the cytoskeleton. The effect of adhesion on the
diffusion coefficient would lead to higher order (nonlinear)
In our simple model we do not deal with the fast molecular
timescales that build the cluster. We assume that the
components of the cluster have some affinity, and that the
resulting cluster therefore has both properties: actin poly-
merization and adhesion (in addition to having a spontaneous
curvature) (4). The cluster that we describe may link
adhesion with the promotion of actin polymerization by
either a direct link or through some signaling cascade, while
both lead to the same description. Note that there are many
types of adhesions, and some, like those on the ventral side
of the cell, may not be associated with a membrane pro-
trusion, although all seem to be associated with actin poly-
In the specific system under consideration, the clusters
(MPs) of membranal proteins are assumed to be sufficiently
small to be described by a continuum model. This means that
we are dealing with MPs that are much smaller than the
lateral size of typical membrane protrusions, which are in the
range of 100–1000 nm. Our MPs are, therefore, considered
to have a lateral size of a ;10 nm. On the other hand, the
radius of the spontaneous curvature of such clusters is
assumed to be of the order of the lateral size of the
protrusion, i.e., R ;100–1000 nm. This large value can be
attributed to the fact that it is the radius of curvature of a
membrane cluster (containing tens of proteins) and not of a
single protein. Such aggregation curvatures have been
recently reviewed in the literature (36,53,54).
In Fig. 1 b, we show schematically the two states of the
membrane (or cell) that we obtain:
1. The mixed state. In this state, we have the clusters
uniformly spread on the membrane, which therefore re-
mains featureless and flat.
2. The phrase-separated state. This state occurs when the
mixed state develops unstable modes, and the clusters are
now aggregated and the membrane no longer flat.
In this work we deal with an initially flat, infinite two-
dimensional ðr ~¼ ðx;yÞÞ membrane. We assume that while
the MP freely diffuse in the membrane, any hydrodynamical
flow effects inside the highly viscous membrane fluid are
neglected. The variables of the model are:
1. hðr ~;tÞ; the local normal displacement of the membrane.
2. fðr ~;tÞ; the local phase parameter of the MP, defined as
f [ n/nS, where nðr ~;tÞ is the local concentration and nS
is the saturation value (that is, nS[ a?2).
In this work we assume that the rigid actin cytoskeleton
fully determines the membrane shape, and any small-scale
thermal fluctuations in the membrane shape can therefore be
neglected. We further assume that this cytoskeleton network
suppresses long-range hydrodynamic flows on scales larger
than the typical mesh-size d ’ 10–100 nm, hence making all
nonlocal effects negligible.
We write down the Helfrich’s Hamiltonian (20) for the
elastic energy of the membrane, including the direct effects
of the MP concentration: spontaneous curvature and adhe-
sion. The local spontaneous curvature is assumed to depend
linearly on the MP concentration (second term in Eq. 1). The
adhesion of MP to the ECM lowers the effective energy per
unit membrane area, and therefore changes the surface
tension term (first term in Eq. 1), in proportion to the
concentration. To this elastic energy we add the terms that
describe the gas of MP: its entropy (third term) and
aggregation interaction (fourth term). The final free energy
expression reads (21,55)
a2ðflnf1ð1 ? fÞlnð1 ? fÞÞ
2a2fð1 ? fÞ1J
2ðs ? afÞð=hÞ21k
where S is the membrane area. Below we replace the full
expression for the entropy by its expansion to fourth order in
f (Ginzburg-Landau approximation): 2ððf ? 1=2Þ212=3
ðf ? 1=2Þ4Þ: For simplicity, we assume in this work that the
bending rigidity is only weakly dependent on the membrane
The equations of motion for the membrane height
displacement and for the protein phase are derived by
variation of the free energy expression (Eq. 1) (56,57)
dhðr ~9;tÞ1fðfðr ~9;tÞ?f0Þ
Oðjr ~?r ~9jÞd2r9;
where the term fðfðr ~9;tÞ ? f0Þ describes the protrusive
force of actin polymerization acting on the membrane, while
a uniform state ðh[0;f[f0Þ remains stationary. In our
model, f, s, and a are assumed to be always positive.
For free membranes, Oðjr ~? r ~9jÞ ¼ ð8phjr ~? r ~9jÞ?1is the
diagonal portion of the Oseen tensor (58), and h is the
effective viscosity coefficient of the ECM. Assuming
confined hydrodynamic flows due to the cytoskeleton, of
typical mesh-size d, we can simplify Eq. 2, by approximating
the Oseen tensor by
3800 Veksler and Gov
Biophysical Journal 93(11) 3798–3810
Oðjr ~?r ~9jÞ¼
ð8pdhÞ?1; jr ~?r ~9j#d
jr ~?r ~9j$d:
Consequently the integral is limited to the domain
jr ~? r ~9j#d; and the integrand is expanded around r ~9 ¼ r ~:
To first order in d we get
dhðr ~;tÞ1fðfðr ~;tÞ?f0Þ
The dynamical equation for f is also based on the variational
derivative of the free energy, with the additional requirement
of conservation of the total number of MPs. For simplicity
we took d ’ a in the remainder of this article.
We therefore derive a diffusion equation for the density
T= fðr ~;tÞ=
where D is the diffusion coefficient.
Note that the adhesion may also affect the effective vis-
cosity of the membrane-substrate interface, namely making
it dependent on fðr ~;tÞ and a: h/h01gðfðr ~;tÞ;aÞ; where
g is some unknown increasing function of its variables. For
simplicity, this effect is neglected here.
For clarity we prefer to work with nondimensional
parameters, as follows: The two length scales in our problem
are a and R, where their ratio, e, is defined to be the small
parameter of the system. We take R to be the unit length of
the model, and set a ¼ e R. The room temperature, T0, is
defined to be the unit energy. The bending energy, k, is taken
to be k ¼ k9T0, and the temperature is T ¼ T9T0. We take a
unit time to be the typical time of protein diffusion, tD[ R2/
D. This value turns out to be very close to the typical time of
the membrane height dynamics, th[ hR3/T0. Consequently,
both dynamic coefficients can be nondimensionalized as
D ¼ D9 R2/t (D9 ¼ 1) and h ¼ h9 T0t/R3(h9 ¼ 1.25). The
dimensions of s and a are of energy density (J/m2), hence
they can be nondimensionalized by s ¼ s9T0/R2and a ¼
a9T0/R2, while f has the dimension of pressure, and can be
rewritten by f ¼ f9T0/R3. The protein concentration pa-
rameter, f, is nondimensional by definition, and it is con-
venient to shift it to the entropy and aggregation extremum,
f0– 1/2 ¼ f90. We therefore work within small deviations
from the equilibriam composition of f0¼ 1/2. The h vari-
able becomes: h ¼ h9R, and the space and time derivatives
change to =/==R; ð@=@tÞ/ð@=@tÞ=t:
The dynamical equations (Eqs. 4 and 5) can now be
written in their final form, while omitting the primes for the
We begin from a uniform initial state and look for
conditions of instability. In our model,such instability means
the onset of protein phase separation and the initiation of
protrusions formation. The standard linear stability analysis
is performed (37,38) where, for simplicity, the system is
assumed to benonuniform alongone axis only. The variables
h and f are expanded around their initial homogeneous
values, to first order: h(x, t) ¼ h01 dh(x, t), f(x, t) ¼ f01
df(x, t), and substituted into the dynamical equations (Eqs. 6
and 7). Performing a space Fourier-transform, we get the
where two new parameters are defined
The parameter S shows the competition between the positive
surface tension s and the average adhesion of the membrane,
while m shows the competition between the (temperature-
induced) entropy and the aggregation interactions of the MP.
An unstable mode has positive growth rate v1,2(q) . 0.
The conditions for instability occur in either one of the
1. Wave instability: Tr (L)2– 4 Det (L) ¼ 0 and Tr (L) . 0.
2. Turing instability: Det (L) ¼ 0 and Tr (L) , 0.
In our model the wave instability occurs only for f , 0,
which we do not consider in this work (see, for example,
(48)), so that the only possible kind of instability in our
model is the Turing instability. The calculation details of the
Membrane-Cytoskeleton Phase Separation3801
Biophysical Journal 93(11) 3798–3810
conditions that give rise to this instability are given in the
THE RESULTS: STABILITY PHASE DIAGRAMS
In this section we discuss the results of our model, in the
form of stability phase diagrams (Figs. 2, 4, and 5); namely
we plot the regions in the physical parameter space where
the uniform system is stable, and where it is unstable. The
regions where the uniform system is stable are called the
mixed phase, since they correspond to a flat membrane and a
uniform density f0of the membrane complexes MP (Fig.
1 b). The unstable regions exhibit two types of instabilities:
type-I and -II (Fig. 2), and there the uniform system breaks
up (fragments) into aggregating complexes of MP, and
growing membrane protrusions (Fig. 1 b). Since we are
limited here to a linear stability analysis, we cannot predict
the final new steady state (if one exists) of the system, as is
usually the case for thermodynamic phase transitions. In
the limit of the passive system (vanishing f and a), the
thermodynamic phase transition occurs at the temperature
where we find the first instability occurring (Fig. 3). Since we
do not calculate the usual phase transition lines, there is no
sense here to discuss the order of the transition, i.e., first or
higher order. For the detailed analysis of the linear stability,
we refer the reader to the Appendix.
The thermodynamic phase transition occurs at T0for a
passive system (f ¼ a ¼ 0) and flat membrane with zero
surface tension, and turns out to be second-order. For finite
surface tension the instability transition is shifted to a lower
temperature, such that T20, T0(Fig. 3). For consistency, we
indeed find that this is the instability temperature we derive
from our linear analysis (Eqs. 11 and 13) in this limit (for
s ¼ 0). Note that as soon as the surface tension is finite,
and inhibits the formation of membrane protrusions, the
instability temperature is suppressed (Fig. 3 b and Fig. 6).
In the stability phase diagrams below we find two
instability regions, where v(q) . 0 (Eq. 10). The physical
meaning of both the instabilities is the same: the small
protein complexes that initially were distributed homoge-
neously, start to aggregate, and protrusions start to appear on
the initially flat membrane. However, the instability patterns
are different, as shown in Fig. 2: type I is an instability band
that starts at q ¼ 0 and lasts until qp(Fig. 2 a), with the most
unstable mode at qp. q* . 0 (Eq. 11). At later times we
expect that the system, which shows the type-I instability, to
evolve toward global phase separation, where the size of the
phase-separated domains increase to infinity (q / 0), due to
line tension (59). In a finite system, this will result in one
phase-separated domain of proteins (H. Levine, private
communication, 2007), while in a living cell the smallest
mode is polar. This polarized structure of the real cell
corresponds to two phase-separated domains on opposite
sides, which allows that tensile forces are produced by
molecular motors (myosin) to maintain the adhesion. The
effects of these motors is not included in our model. Since in
our model the domains also correspond to membrane
protrusions and adhesion, they may remain separated and
will not coalesce into one domain.
On the other hand, in the type-II instability the wavevector
in which we first get v(q) . 0 is at a nonzero value, denoted
by Eq. 11, q?
The band of instability for type-II occurs for qn# q # qp
(Fig. 2 b). In these systems we therefore expect that the
unstable domains remain with typical size ;1/ q*. Note that
second-order phase transitions correspond to our type-I
transition where q* ¼ 0 on the mixed-phase-separated
transition line. There are examples of first-order phase
transitions where a finite q* appears at the transition line,
similar to our type-II (51).
In all the diagrams below we use the following param-
eters: e ¼ 0.05, f0¼ 1/2, J ¼ 4, and k ¼ 10. In Fig. 3 a, we
plot the stability phase diagram in the actin activity (f) and
reduced temperature (T/T0) plane, for a fixed value of the
adhesion strength (a), which is in the range a0. a . acr.
We find in this diagram the two kinds of instabilities we
defined above: type-I occurs above the solid red line, while
type-II instability occurs in the shaded region between the
red and green lines. One finds for a sufficiently low
temperature, T , T10, a type-I instability that occurs even
without the actin force (f ¼ 0) or adhesion (a ¼ 0). This
phenomenon means that in this region, the aggregation is the
dominant term, and is sufficient to bring about an instability,
which corresponds to phase separation at thermodynamic
equilibrium, as was found previously (24). The transition
temperature Tcof the passive system (f ¼ 0, a ¼ 0) is T10for
0 , acr; otherwise, it is given by T20. In the range of the type-
II instability we have competing contributions of comparable
size from the aggregation, entropy, and elastic energies. Above
the transition temperature, in the mixed state, the entropy is
dominant over the aggregation and elastic energies.
For acr, 0 (low surface tension) we have always a type-II
instability for the passive system, while when acr. 0 (high
surface tension), this transition disappears for acr. a (blue
solid line in Fig. 3 b). Note that acrdepends on the value of
the surface tension s (Eq. 15). We find that the slope of the
(Fig. 2 b).
instability band starts at q ¼ 0, until qp, and has a maximum at q*. This
instability leads to global phase-separation. (b) Type-II instability: the
instability band first appears at q?
c, and then grows into an unstable band.
This instability gives rise to patterns with typical length scale.
Instability patterns of the system. (a) Type-I instability: the
3802 Veksler and Gov
Biophysical Journal 93(11) 3798–3810
critical line decreases, and the passive transition temperature
increases, with increasing values of the adhesion a. At a0the
transition line becomes horizontal and the system is unstable
at all temperatures. This means that when the adhesion is
stronger than the surface tension, it is enough to bring about
instability. The curved portion of the critical line (for small
values of f) corresponds to the type-II instability transition
(green line in Fig. 3 a). In Fig. 3 c, we plot the stability phase
diagram in terms of f versus a, for a fixed temperature in the
range T10, T , T12. In Fig. 3 d, we plot the f-a phase dia-
grams for different values of T/T0. It can be easily seen that
for T/T0. T10, T12, the type-II instability region disappears.
The most outstanding feature of this diagram is that the
critical temperature increases linearly with the activity of the
actin, above the critical temperature of the passive system.
This linear relation follows from Eq. 13. Note that there is a
large difference in the width of the unstable band above and
below Tc; below this temperature, the band width increases
linearly with decreasing temperature (Fig. 4, b and d);
limit, both f and a do not affect the asymptotic behavior of
qp. Above this temperature, the width is typically much
smaller; for a . a0, it approaches a constant q2
1=2Þ ? sÞ=k for T=T0? 1 (blue lines in Fig. 4, a and b).
Note that for a , a0there is always a finite critical tem-
perature for all f (Fig. 4 c).
In the inset of Fig. 3 a, we show that a small increase in the
ratio of the protein complex size a to its curvature radius R,
i.e., in e, results in a drastic increase in the temperature range
of the transition. The critical temperatures increase above T0
and decrease below T0, as can be seen from Eq. 13 (note that
m ¼ 0 at T0). Since the actin force promotes phase separation
through the induction of curvature, increasing the curvature
bending energy (} e2k) increases the induced shift in the
0Þ ? JÞ=ðe2JÞ (for T=T0? 1). In this
transition temperature. Note that the effect of actin force f
shifts the critical temperatures by 1–10% (depending on e),
which translates to a shift of 1–10?C. For the larger e, the
shift in the transition temperature becomes measurable.
The next phase diagram that we plot is in terms of a versus
temperature (Fig. 5). We see again that above a0, the system
is phase-separated for all temperatures; while below a0, three
is a critical temperature above which the system is mixed. In
Fig. 5 b, we find that as the actin force is increased the critical
temperature increases too, and the region of type-II instabil-
ity is reduced, until it vanishes for large enough f. This is
similar to the behavior seen in Fig. 3 d.
In Fig. 6, we plot the phase diagram in terms of the surface
tension s versus temperature. In Fig. 6 a, we keep the
adhesion zero, while plotting the diagrams for different
values of the actinforce f.When f is negligible (red lines), we
have a type-I instability below T10independent of the surface
tension, while the type-II instability vanishes (approaches
T10) when acr¼ 0, i.e., at a critical surface tension scr¼
2k2/J (Eq. 15). As the actin force increases, we find that
the phase-separated region increases into higher values of s,
and the type-II region eventually vanishes altogether.
In Fig. 6 b, we plot the effects of adhesion, for negligible
actin force. When increasing the adhesion strength, we find
again that the unstable region exists for larger values of s.
Note that for s , af0(i.e., for a . a0) we have instability at
DISCUSSION AND CONCLUSIONS
To conclude, our work presented here is the first to calculate
in a quantitative manner the spontaneous phase separation of
membranal components which is driven by the forces of
actin polymerization and adhesion. This phase separation is
f-versus-T plane. The actin force leads to an increase in
the critical temperature. The type-II instability is given
bythe shadedregion.(a)Weuse s ¼10anda¼ 10.In
all the graphs in this work we use e ¼ 0.05. In the inset
of panel a, we plot the same diagram but now for e ¼
0.2, and we find that the temperature range drastically
increases. (b) The solid lines represent the phase-
separation transition (dashed lines give the type II-type
I transition), using s ¼ 50 and blue a ¼ 0, green a ¼
70, and red a ¼ 90. The passive system is represented
by the line f [ 0. (c and d) Stability phase diagrams in
the f-versus-a plane. Here we find that increase in both
f and a have a similar effect in driving the system into
the unstable regime. (c) T ¼ 0.997T0and s ¼ 10. (d)
The solid lines represent the phase-separation transi-
tion (dashed lines give the type II-type I transition),
using s ¼ 10 and blue T ¼ 1.005T0, green T ¼
0.9985T0, and red T ¼ 0.997T0. For S . Scr, only
type-I instability is possible.
(a and b) Stability phase diagrams in the
Membrane-Cytoskeleton Phase Separation3803
Biophysical Journal 93(11) 3798–3810
tightly linked with the formation of membrane curvature, and
we therefore propose that it is the driving force for many
cellular shape transformations. One limitation of our model
is that in its present form it does not describe the behavior of
cells inside a three-dimensional matrix (61,62).
We now give several experimental observations that seem
to exhibit the phase separation we calculated above.
We begin with a more simple artificial system that was
recently observed (7). In this work, synthetic vesicles,
containing a number of reconstituted proteins that are known
to activate actin polymerization in cell, were observed as a
function of temperature. It was found that the phase-
separation temperature increased by ;8?C when actin
monomers were added into the buffer and formed a network
of filaments on the outer surface of the membrane. This
behavior is in qualitative agreement with our calculation of
Fig. 3 a, where we predict that the actin force enhances the
critical temperature. Note that, alternatively, the effect of the
actin network may be to increase the effective attractive
interaction J between the membrane proteins, and therefore
increase the critical temperature independently of the actin
polymerization force f, i.e., independent of any metabolism.
Such synthetic in vitro systems offer a good platform to test
our model, since they allow good control over the different
We next give a few examples from living cells. Note first
that in a living cell there may be a number of types of
membranal proteins and complexes that have different
values of curvature (R), density (f0), and activity (f and a)
(15,17). Each such type of MP therefore has different phase
transition conditions, and in a real cell membrane there can
therefore be a numberof phase-separationtransitions. Second,
our model assumes a uniform system, which therefore cor-
responds to cases where the active part of the cell membrane
is uniform; for example, the lamellipodia of motile cells has a
rather uniform leading edge, the growth cone of axons (63),
and possibly the entire cell periphery in adhering (nonmotile)
cells. In all these cases, structural phase transitions that lead
to the formation of protrusions, membrane phase separation,
and adhesion loci, may be examples where our model applies.
various values of f and a. (a) f ¼ 0, a grows counter-
clockwise (a ¼ 0, 10, 20, 40, 60); the scale is Log (q2
versus linear (T). (b) Same graph as panel a, but on linear-
linear scale. The solid line give qpwhile the dashed lines
give qn. Both are in units of 1/R, where R ¼ 100 nm. (c)
a ¼ 0, f grows counterclockwise (f ¼ 0, 10, 30, 45, 60); the
scale is Log (q2
p) versus linear (T); the cutoff temperature is
Tcut¼ (J(s – a(f01 1/2)) 1 e2k(f 1 a (f01 1/2) – s))/
(4(s – a(f01 1/2))(1 1 4 f2
but on linear-linear scale; again, q2
asymptotic dependence on T for T ? T0.
Graphs of q2
pversus T, for s ¼ 10 and
0)). (d) Same graph as panel c,
phas the same linear
temperature plane. We find that the adhesion can drive the
instability, increasing the transition temperature. (a) Gen-
eral features of the phase transition line (solid line), for
f ¼ 4. The shaded region shows the type-II instability
(bounded by solid and dashed lines). The passive system is
represented by the x axis where a ¼ 0. At a0(horizontal
dashed line) the transition temperature diverges. (b)
Transitions lines (solid line) for various values of the actin
force f: from top to bottom, f ¼ 0, 4, 10.
Stability phase diagram in the adhesion(a)-
3804 Veksler and Gov
Biophysical Journal 93(11) 3798–3810
Furthermore, a cell may change its overall shape and
motility due to small changes in the internal expression of the
components that drive the shape transitions, i.e., in our model
the actin and adhesion mechanisms. This may explain the
observation that identical cells from a single culture display a
number of morphologies, and even individual cells change
between these different shapes (64,65). It isfurther shown that
these different morphologies are closely related to marked
changes in the cell’s actin and adhesion organization.
Another example of striking shape transformations is
given in Baas et al. (66), where expression of a protein that is
connected with the activation of the actin cytoskeleton
changes the cell shape from an adhering starlike shape to a
compact form that has the entire cortical actin phase
separated into a single domain of tightly packed microvilli
projections. Within our model this transfers the cell from a
type-II instability due to adhesion (a) to a type-I instability
due to actin force (f) (Fig. 7 a).
1. In Runge et al. (67), it was found that membrane
receptors (CD9) exclusively aggregate at high concen-
trations on the surface of microvilli, which are membrane
protrusions that are driven by actin polymerization. The
activity of these receptors inside the microvilli enhance
their length, and therefore within our model they may be
considered as part of the complex of MP that enhance the
actin protrusive force. Whether the complex containing
these receptors also has the convex spontaneous curva-
ture, remains to be experimentally tested.
2. In the growth-cone of axons (63) it was observed that, in
the absence of myosin, the filopodia are highly static and
appear to be regularly spaced. If indeed our model is
correct, then this typical spacing between the filopodia is
given by the 1/q* that we calculate. The static nature of
these filopodia, i.e., their lack of lateral motion along the
front edge of the growth cone, arises naturally in our
model, where only Turing instability arises. When
myosin is active there is a possibility for wave instability
3. Another example where a combination of spontaneous
curvature and actin polymerization may be driving phase
separation, is in the formation of the immune synapse
(2,68,69). The spontaneous curvature of the TCR micro-
complexes is due to their closer adhesion to the
neighboring cell, as compared to the surrounding mem-
brane. They are also observed to be regions of high actin
polymerization activity, and indeed they aggregate to
form the large immune synapse. We therefore propose
that the aggregation of these complexes is triggered by an
increase in f, as shown in Fig. 3 a.
4. In cell-cell junctions the cadherin molecules need to form
a large aggregate to stabilize the adhesion (70–72). It is
further shown that when the link between the cadherins
and the actin cytoskeleton is broken, this aggregation cannot
take place. We propose that this is another example where
adhesion and actin-driven protrusion combine to give
phase separation and aggregation in the membrane.
5. In motile cells the leading edge of the lamellipodia has a
relatively large convex curvature in its cross section, and
strong actin polymerization activity. It was found that in
this region there is a membranal complex that prevents
lipid diffusion through the leading edge (73). This
densely packed complex disintegrates when the actin
polymerization is inhibited, while keeping the high
curvature artificially. This may correspond to our results
of Fig. 3 a, where the actin force enables the system to
phase-separate even when the passive system is mixed.
The curvature alone may not be sufficient to cause phase
separation, as we discussed above.
6. Experiments on the effects of the external substrate on
the shape of cells have been carried out in recent years
(8,9). In one experiment (8) it was found that astrocytes
change their shape from approximately round on soft
substrate to jagged (triangular) on a hard substrate.
Within our model this shape transition corresponds to the
phase separation we calculated; the round cells corre-
spond to the mixed phase, while the polygonal shapes
correspond to the phase-separated state, since phase-
separated domains correspond to membrane protrusions.
Since these cells adhere more strongly on harder surfaces
(74,75), this phase transition may be driven by an
tension(s)-temperature plane. We find that the increase
in the surface tension decreases the instability temperature
below the thermodynamic phase transition temperature.
Increasing f or a increases again the instability tempera-
ture. (a) Phase-separation lines (solid lines) for various
values of f (and zero adhesion a ¼ 0); from top to bottom,
f ¼ 20, 4, 0.1. The dashed lines give the transition between
type-II and type-I instability. (b) Phase-separation lines
(solid lines) for various values of a (and negligible actin
force f ¼ 0.1); from top to bottom, a ¼ 20, 0. The dashed
lines give the region of type-II instability. The horizontal
Stability phase diagram in the surface-
black line indicates the value s ¼ af0, below which the system is unstable at all temperatures. The vertical black line gives the lower critical temperature T10,
below which the system is unstable (type-I) for all surface tensions.
Membrane-Cytoskeleton Phase Separation 3805
Biophysical Journal 93(11) 3798–3810
increase of a, as is shown in Fig. 2 c and Figs. 4 and 7.
Similar transitions were observed in fibroblast cells
(12,13), where the higher the adhesion (substrate rigid-
ity), the more numerous the cell protrusions (starlike
shapes). This corresponds to the increase in qp, q*, the
more unstable the system is (Fig. 4, a and b, and Fig. 7).
A similar trend of increase in the density of protrusions
with the adhesion strength was observed in Cukierman
et al. (76).
On the other hand, the changes in shapes in response to the
increase in stiffness and adhesion can be more compli-
cated (10,77). It is observed that as the substrate stiffens,
the cell’s protrusions bundle and the star-shape is
transformed into a polarized dipolar body. This transition
is indicated by the shapes drawn in Fig. 7. We interpret
these changes in the following way: the star-shape is
typical for cells that are close to the mix-type-II transition,
with protrusions at the wavelength corresponding to the
dominant mode q*. As the adhesion increases and they
move to larger values of a, larger wavelength modes
become unstable (qndecreases; see Fig. 7 c) and mix with
the dominant mode, producing aggregation of the
protrusions. Finally, when 1/qnbecomes larger than the
cell circumference, the lowest mode (a force dipole)
dominates the large-scale ordering of the protrusions.
This polarized structure of the real cell corresponds to two
phase-separated domains on opposite sides, which allows
tensile forces produced by molecular motors (myosin) to
maintain the adhesion. The effects of these motors is not
included in our model.
Adhesion to the substrate was shown to determine the
overall fate of cells, to either proliferation or apoptosis
(78). In particular it seems that when the cells are
confined to compact shapes the adhesions are weak, and
no large aggregations of adhesion and actin are observed.
When cells are allowed to deform, their adhesion clusters
are usually in highly curved membrane protrusions,
where strong aggregation of adhesion molecules and
actin is observed. It may well be that only when cell
deformation is allowed does the phase separation of
membrane components occur due to coupling of mem-
brane shape and active forces as our model predicts. This
phase separation and aggregation of proteins can trigger
the signals (79) that determine the cell’s fate.
7. It was further found that the substrate rigidity influences
not only the cell shape but also its differentiation, in the
case of stem cells (8,10). It will be interesting in the
future to investigate the interplay between the stability
phase transition that we calculated, which occurs in the
cytoskeleton and membrane, and the resulting changes in
the cell differentiation and genetic expression. Since the
cell membrane may contain a number of types of MP,
and therefore experience a number of phase transitions as
a function of adhesion strength, each such transition may
trigger another change in the differentiation path.
Through these different transitions at the membrane-
cortical cytoskeleton the cell may respond differently to
different external substrates. An interesting conjecture
may be that the round and featureless stem cells have no
stable membranal domains and segregation, due to many
competing forces. When the cell differentiates some
genes are expressed more then others and consequently
some protein-networks can become dominant and lead to
membranal phase transitions, as we described. There
may therefore be a positive feedback loop between the
cytoskeleton-membrane transitions and the expression of
genes, which leads to differentiation.
phase transitions in the cell membrane-cortical cytoskeleton
and genetic phase transitions. Such a mapping relates the cell
response to the external substrate to the differentiation path it
chooses. Our model gives also a plausible mechanism by
which cells form specific membranal domains that are well
unstable mode q* in the (a, f) stability phase diagram (T ¼
0.997T0). The heavy black dashed line shows the mixed-
type-II phase transition and the heavy solid black line
shows the type-II/type-I phase transition. In panels b and
c, we plot the wavevectors of the unstable region (q*, blue
line; qp, red line; and qn, green line) for f ¼ 10 and f ¼ 0,
respectively (the wavevectors are in units of 1/R, where
R ¼ 100 nm). Below the contour plot, we show the
expected shapes of cells along the trajectory indicated by
the heavy arrow.
(a) Contours of the wavevector of the most
3806Veksler and Gov
Biophysical Journal 93(11) 3798–3810
segregated and driven by both actin polymerization and
adhesion. These domains serve to initiate a signaling cascade
that influences the cell’s behavior (for example, the T-cell
(69)), gene expression (62), and malignancy (80,81). Inter-
fering with the ability of the cell to form these domains, by
disrupting the actin polymerization and adhesion, will allow
us to modify the cell behavior and fate.
We now wish to suggest a number of testable predictions
that may test the validity of our model:
The most direct test of our model will be the detailed char-
acterization of the clusters that are involved in promot-
ing protrusions and adhesion, and finding if indeed
these have the spontaneous curvature that we assumed
in our model.
In vitro experiments of the kind described in Liu and
Fletcher (7) may be repeated using recently discovered
proteins that are known to have convex curvature and
associate with actin filaments (16). In such a system,
the phase-separation transition with and without actin
polymerization may be measured and test our predic-
tion of Fig. 3.
For the filopodia in the growth-cone of axons (63) (with
myosin blocked), we predict that the spacing should
decrease if the actin force f is increased, or if the ad-
hesion is increased, or if the surface tension is de-
creased. This may be tested in future experiments.
In the case of type-II instability, we predict that the wave-
length of the protrusions will increase with increasing
surface tension (Fig. 4). This may be tested on cells that
exhibit filopodia and microvilli, to test that not only
their amplitude, but their density, decreases with in-
creasing induced surface tension.
Similarly, increasing the adhesion strength (by chemical
or biological means), should result in an increase in the
density of adhesion-related structures, such as podosomes.
We further expect that increasing the surface tension
suppresses the formation of phase separation and
protrusions (Fig. 6), which can be tested in both living
cells and in vitro synthetic systems.
APPENDIX: LINEAR STABILITY ANALYSIS
In this Appendix, we give the details of the linear stability analysis of The
Model, detailed above.
Equation 8 is an eigenvalue problem, with the solution
To obtain the conditions for Turing instability, we need to calculate the roots
of Det (L) and Tr (L). The solutions of Det (L) ¼ 0 are
where we set q2
p. The f-values are given by
where m12[ – e2JS/2k, and
The roots of Tr (L) are given by
Omitting the unessential technical details, we derive here the conditions
which must hold together, for q2
transition point: 1), it must be real and positive; and 2), it must be larger than
tr: Note that at q ¼ 0, the growth rate is always zero. It should be noted that
in the stability analysis, only real wavenumbers are meaningful. Hence, only
real and positive q2can be accepted. In addition, it can be easily seen from
Eq. 8 that for large q, Det (L) . 0, and Tr(L) , 0, and the system is always
The physical parameters of the system may be divided into two sets:
nto correspond to an instability
1. The temperature T, average protein concentration f0, aggregation
potential J, and surface tension s are all intrinsic parameters of fluid
membranes, both in synthetic (passive) systems and in living cells.
2. The adhesion strength a and actin force f are parameters that arise from
metabolic activity and are therefore unique to the living system.
The control parameters of our model turn out to be m, f, and S (Eq. 9).
Two of them, m and S, are a combination of physical parameters. For
example, the relative increase of entropy with respect to the aggregation
will change m, but their simultaneous increase will not. In the phase
diagrams that we plot below, the variation in m is due only to variation
in the temperature T, while keeping f0 and J constant. We further
assume that only the entropy changes with the temperature, while all the
other parameters are temperature-independent. This means that a change
of T means a change of balance between the aggregation (J) and the
entropy. Under these assumptions, the variation of the model parameters
m, S, and f corresponds to the variation of the physical parameters T, a,
s, and f. The phase diagrams will be given in terms of the physical
Moreover, for the sake of simplicity, we choose the aggregation
coefficient J to yield m ¼ 0 for T ¼ T0, which is the temperature for
which the entropy is equal to the aggregation. In most biologically relevant
systems T0is of the order of room temperature (7). Note that the adhesion
can be both stronger or weaker (on average) than the surface tension,
resulting in either positive or negative S (Eq. 9). Strong adhesion and
formation of protrusions drives an increase in the apparent membrane area,
which results in an exponential increase of the surface tension coefficient s
in a closed system of finite overall membrane area (82). This feedback
mechanism between the surface tension and the adhesion is a nonlocal
effect, so it is not included in our model. We may conclude that this effect
confines the range of possible values to a , a0(see below). Note that this
effect limits the growth of all membranal protrusions, as observed for
example in Bohil et al. (83).
For the presentation of the stability phase diagrams, the following
parameters should be defined as
Membrane-Cytoskeleton Phase Separation3807
Biophysical Journal 93(11) 3798–3810
where a0is the point where the cell-substrate adhesion is equal (on average)
to the surface tension. The value acrwill be clarified in the following:
T10and T20are the vanishing points of f1and f2, respectively. The value
T12is the touching point of f1with f2.
We thank Sam Safran and Patricia Bassereau for useful discussions and
David Andelman for useful comments.
We thank the European Union Comp NoE grant and the Alvin and Gertrude
Levine Career Development Chair for their support. This research was
supported by the Israel Science Foundation (grant No. 337-05).
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