# 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 - SourceAvailable from: sciencedirect.com[Show abstract] [Hide abstract]

**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 - [Show abstract] [Hide abstract]

**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

Page 1

Phase Transitions of the Coupled Membrane-Cytoskeleton Modify

Cellular Shape

Alex Veksler and Nir S. Gov

Department of Chemical Physics, The Weizmann Institute of Science, Rehovot, Israel

ABSTRACT

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

INTRODUCTION

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,

neuronaldendrites,etc.),andforthecellmotility(lamellipodia

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

network (7).

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

curvature.

doi: 10.1529/biophysj.107.113282

Submitted May 22, 2007, and accepted for publication July 18, 2007.

Address reprint requests to Alex Veksler, Tel.: 972-8-934-6031; E-mail:

alexander.veksler@weizmann.ac.il.

Editor: Alexander Mogilner.

? 2007 by the Biophysical Society

0006-3495/07/12/3798/13$2.00

3798 Biophysical JournalVolume 93 December 20073798–3810

Page 2

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

separation diagrams.

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

and theoretically

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.

THE MODEL

Model description

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,

FIGURE 1

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

Page 3

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)

terms.

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-

merization (3).

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.

Model equations

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)

1T

a2ðflnf1ð1 ? fÞlnð1 ? fÞÞ

J

2a2fð1 ? fÞ1J

F ¼

Z

S

1

2ðs ? afÞð=hÞ21k

2

=2h1f

R

??2

1

4ð=fÞ2

?

d2r;

(1)

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

composition.

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)

?

@hðr ~;tÞ

@t

¼

Z

S

?

dF

dhðr ~9;tÞ1fðfðr ~9;tÞ?f0Þ

?

Oðjr ~?r ~9jÞd2r9;

(2)

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

Page 4

Oðjr ~?r ~9jÞ¼

ð8pdhÞ?1; jr ~?r ~9j#d

0;

jr ~?r ~9j$d:

?

(3)

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

@hðr ~;tÞ

@t

¼d

8h

?

dF

dhðr ~;tÞ1fðfðr ~;tÞ?f0Þ

??

:

(4)

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

(56,57)

@fðr ~;tÞ

@t

¼Da2

T= fðr ~;tÞ=

dF

dfðr ~;tÞ

??

;

(5)

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

rescaled parameters

?

?k=2ðf1=2hÞ1fðf?f0Þ

@f

@t¼D=

2

1e2 k

@h

@t¼e

8h

?s=2h?a=f11

2

??

=h

?

??

;

(6)

f11

?

2?a

?

= ð4T?J1e2kÞf116T

2ð=hÞ21k=2h?J

3f3

??

?

2=2f

???

:

(7)

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

linearized equations

where two new parameters are defined

S [s?aðf011=2Þ;

m[4Tð114f2

0Þ?J:

(9)

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

following cases:

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

@

@t

dhðq;tÞ

dfðq;tÞ

??

¼ L

dhðq;tÞ

dfðq;tÞ

??

L[

?

D

T0

e

8hT0ðSq21kq4Þ

f011

2

e

8hT0ðf 1kq2Þ

f011

2

??

e2kq4

?D

T0

??

ðm1e2kÞq21J

2e2q4

??

0

B

B

@

1

C

C

A;

(8)

Membrane-Cytoskeleton Phase Separation3801

Biophysical Journal 93(11) 3798–3810

Page 5

conditions that give rise to this instability are given in the

Appendix.

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?

?ð2km1e2JSÞ=ð2e2JkÞ

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

c¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip

(Fig. 2 b).

FIGURE 2

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

Page 6

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);

q2

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

p/ ?2ð4Tð114f2

0Þ ? JÞ=ðe2JÞ (for T=T0? 1). In this

p/ðaðf01

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

all temperatures.

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

FIGURE 3

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

Page 7

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

model parameters.

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.

FIGURE 4

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

p)

0)). (d) Same graph as panel c,

phas the same linear

FIGURE 5

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

Page 8

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

(48).

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

FIGURE 6

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

Page 9

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.

Itisinterestingtothinkofa mappingbetweenthestructural

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

FIGURE 7

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

Page 10

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

v1;2ðqÞ¼1

2

?

TrðLÞ6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

TrðLÞ2?4DetðLÞ

q

?

:

(10)

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

q2¼0; q2

p;n¼s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

Jðf1?f2Þ

r

6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

Jðf ?f2Þ

r

;

(11)

where we set q2

n# q2

p. The f-values are given by

f1[ 11m

e2k

??

S

f2[f1?ðm12?mÞ2

2e4J

;

(12)

where m12[ – e2JS/2k, and

s¼

11; m,m12

?1; m.m12:

?

(13)

The roots of Tr (L) are given by

q2¼0; q2

tr¼?8Dhðf011=2Þðm1e2kÞ1eS

4e2DhJðf011=2Þ1ek

:

(14)

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

q2

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

stable.

The physical parameters of the system may be divided into two sets:

por q2

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

parameters.

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

Page 11

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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a0¼ s=f0;

acr¼s ? 2k2=J

f011=2;T10¼

J ? e2k

4ð114f2Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

T12¼ ?Jð2k ? e2SÞ

8kð114f2Þ;T20¼J ? e2ð

2Jðs ? aðf011=2ÞÞ

p

? Jðs ? aðf011=2ÞÞ=2kÞ

4ð114f2

0Þ

;

ð15Þ

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