This journal is©The Royal Society of Chemistry 2016 Soft Matter, 2016, 12, 1791--1800 | 1791

Cite this: SoftMatter, 2016,

12, 1791

Hydrodynamics of bilayer membranes withdiffusing transmembrane proteins

Andrew Callan-Jones, Marc Durand and Jean-Baptiste Fournier

We consider the hydrodynamics of lipid bilayers containing transmembrane proteins of arbitrary shape. This

biologically-motivated problem is relevant to the cell membrane, whose fluctuating dynamics play a key role

in phenomena ranging from cell migration, intercellular transport, and cell communication. Using Onsager’s

variational principle, we derive the equations that govern the relaxation dynamics of the membrane shape,

of the mass densities of the bilayer leaflets, and of the diffusing proteins’ concentration. With our generic

formalism, we obtain several results on membrane dynamics. We find that proteins that span the bilayer

increase the intermonolayer friction coefficient. The renormalization, which can be significant, is in inverse

proportion to the protein’s mobility. Second, we find that asymmetric proteins couple to the membrane

curvature and to the difference in monolayer densities. For practically all accessible membrane tensions

(s 4 10�8 N m�1) we show that the protein density is the slowest relaxing variable. Furthermore, its

relaxation rate decreases at small wavelengths due to the coupling to curvature. We apply our formalism to

the large-scale diffusion of a concentrated protein patch. We find that the diffusion profile is not self-similar,

owing to the wavevector dependence of the effective diffusion coefficient.

1 Introduction

Biological membranes are lipid bilayers forming the envelopesof plasma membranes, nuclei, organelles, tubules and transportvesicles within a cell.1 They are versatile structures, both fluidand elastic, that can change shape or topology in order toaccomplish the cell functions. From the dynamical point ofview, membranes can be viewed as a system of four fluid phasesin contact: a pair of two-dimensional (2D) lipid phases and twothree-dimensional (3D) aqueous phases. These phases, separatedbut strongly coupled,2,3 exhibit nontrivial multiphase flowbehaviors.3–5 The dynamics of bilayer membranes containingtransmembrane proteins at a high concentration are especiallychallenging because the proteins form a fifth phase thateffectively interdigitates two components of the multiphaseflow. This situation corresponds to the actual biological one,where macromolecular crowding effects are ubiquitous andwhich are known to make molecules in cells behave in radicallydifferent ways than in artificial lipid vesicles.1

The study of the diffusive behaviour of proteins embeddedin a membrane, taking into account the hydrodynamics of themembrane and that of the surrounding solvent, originated withthe seminal work of Saffman and Delbruck on cylindricalinclusions in flat membranes.6 Since then, investigations intothe effects on single protein diffusion both of membrane height

fluctuations and of the coupling between membrane curvatureand protein shape have been carried out. While earlier workfound that membrane fluctuations tend to enhance the diffusioncoefficient D of curvature-inducing proteins,7,8 more recentstudies that take greater account of the surrounding membranedeformation caused by a protein, showed that D is actuallyreduced.9,10 This predicted lowering of D has been recentlyverified experimentally.11

Despite these advances, there has been relatively little workdone on the collective diffusive behavior of many, interactingtransmembrane proteins, taking into account the bilayer struc-ture of the membrane and in turn the influence of the proteinson the in-plane and out-of-plane membrane dynamics. In fact,almost thirty years ago, it was proposed that the energeticcoupling between a curved membrane and asymmetric proteinswould effectively result in greater attraction between proteins,12,13

suggesting a reduction in the cooperative diffusion coefficient.However, the membrane was treated as a single, mathematicalsurface, thus neglecting the influence of transmembrane proteinson the coupling between the two monolayers. The influence of thebilayer structure of the membrane on single protein diffusion hasrecently been studied, in particular for proteins located in one ofthe two monolayers,14 yet no comparable work at the many proteinlevel has been proposed. We note that though a very generaltheoretical framework was developed some time ago,15 a handytheory describing the multiphase dynamics of a deformablemembrane bilayer with diffusing transmembrane proteins isstill lacking.

Universite Paris Diderot, Sorbonne Paris Cite, Laboratoire Matiere et Systemes

Complexes (MSC), UMR 7057 CNRS, F-75205 Paris, France

Received 7th October 2015,Accepted 14th December 2015

DOI: 10.1039/c5sm02507a

www.rsc.org/softmatter

Soft Matter

PAPER

Publ

ishe

d on

18

Dec

embe

r 20

15. D

ownl

oade

d by

SC

D U

nive

rsité

Par

is 7

on

03/0

2/20

16 1

4:48

:02.

View Article OnlineView Journal | View Issue

1792 | Soft Matter, 2016, 12, 1791--1800 This journal is©The Royal Society of Chemistry 2016

Simple, single-phase bilayer membranes already have a complexhydrodynamic behavior16,17 due to their soft out-of-plane elasticity.18

The complete equations describing the hydrodynamics of bilayermembranes, including curvature and in-plane elasticities, inter-monolayer friction, monolayer 2D viscosity and solvent 3Dviscosity, were first derived by Seifert and Langer for almostplanar membranes.3 The method employed was a careful balance ofin-plane and out-of-plane elastic and viscous stresses. Generalizationto non-linear membrane deformations was achieved by Arroyoet al.19–21 using covariant elasticity and Onsager’s variationalprinciple;22,23 see also ref. 24. Importantly, the monolayers mustbe treated as compressible fluids. Indeed, while the 3D density ofthe lipid region remains almost constant, the 2D lipid densitycan significantly change because the membrane thickness is freeto adapt.25 Moreover, any lipid density difference between thetwo monolayers couples with the membrane curvature.26,27 Inthe early studies of membrane hydrodynamics the bilayer struc-ture of the membrane was neglected.16,17 While this is a goodapproximation24 for tensionless membranes at length-scalesmuch larger than microns, experiments and theoretical studieshave shown that taking the bilayer structure into account isessential at shorter length-scales.2–4,28,29 This is chiefly due tothe importance of the dissipation due to the intermonolayerfriction, caused by the relative motion of the lipid tails, occurringwhen the two monolayers have different velocities.2,30

In this paper, we derive the coupled equations describingthe multi-phase flow of a deformable bilayer membrane host-ing transmembrane proteins that diffuse collectively. For a one-component, or even a two-component lipid bilayer membrane,the hydrodynamical equations for each monolayer can be writtenin a relatively simple manner by balancing standard 2D and 3Dhydrodynamic stresses with the in-plane pressure gradient and theintermonolayer stress.3,31 This is no longer possible for a bilayermembrane with spanning proteins. Indeed, the two monolayers arenot only coupled through the intermonolayer stress, they are alsocoupled through the protein mass conservation law, manifested bythe equality of the protein fluid velocity in the two monolayers,which connects the diffusive flows of both monolayers. Thisconstraint produces additional in-plane stresses that cannot beunderstood simply. Using Onsager’s variational principle, becauseit is based on minimizing the total energy dissipation under all therelevant constraints, we are able to determine the constitutiveequations consistently and to uncover these nontrivial stresses.

One simple consequence, that is readily understood, is thatwith respect to the hypothetical situation where the proteins wouldbe broken into independent halves living in each monolayer, theintermonolayer friction coefficient b increases in inverse proportionto the proteins’ mobility G. Indeed, imagine the simple case ofup-down symmetric proteins within a bilayer flowing in such away that the two monolayers have exactly opposite velocities. Bysymmetry, the proteins remain immobile, and friction arises notonly from the lipid tails of the contacting monolayers, but alsofrom the dissipation (p 1/G) due to the flow within the mono-layers between the lipids and the proteins.

From our coupled dynamical equations, we obtain thecollective diffusion coefficient Deff of up-down asymmetric

proteins and its wavevector dependence. Relaxation of smallwavevector (q) disturbances of the density of asymmetric trans-membrane proteins couples to the relative motion of themonolayers, which involves intermonolayer friction. This leadsto a b-proportional reduction of the effective diffusion coefficientDeff(q), as compared with equivalent but symmetric proteins. Incontrast, for large wavevectors, the energetic coupling betweenprotein density and membrane curvature further reduces theeffective diffusion coefficient Deff(q), in agreement with ref. 12and 13. The crossover between the two regimes occurs at

wavevector qc �ffiffiffiffiffiffiffiffis=k

p, with s the membrane tension and k

the membrane bending modulus. Below qc, tension effectivelyflattens the membrane, thus only the protein concentration andthe difference in monolayer densities couple; above qc, shortscale membrane deformations favor spatially inhomogeneousprotein concentration, thus lowering Deff. Finally, we confirm thewavevector dependence of the protein diffusion coefficient byexamining, with our formalism, the dynamics of an initiallylocalized excess concentration of proteins.

The present article is organized as follows. In Section 2,using Onsager’s variational principle,23 we derive the generalequations describing the multiphase flow of the system. InSection 3, we calculate the relaxation modes of the dynamicalequations coupling the membrane shape, the lipid density-difference and the protein density. We discuss the role ofmembrane tension and protein asymmetry, and we derive thewavevector-dependent effective diffusion coefficient of the proteins.In Section 4, we analyse the spreading of a concentrated spot ofproteins and we discuss its anomalous diffusion. In Section 5, wegive our conclusions and discuss the perspectives of our approach.

2 Hydrodynamic description

To understand the coupling between the protein diffusion ina bilayer membrane and the membrane curvature, we firstestablish the dynamical equations governing the relaxation ofsmall fluctuations in membrane shape, and densities of lipidand protein relative to a flat, homogeneous configuration. Letus consider a one lipid species bilayer membrane hosting anon-uniform mass fraction c of (identical) asymmetric trans-membrane proteins. The space is parametrized in Cartesiancoordinates by R = (x>,z), with x> = (x,y). The membrane shapeis described, in the Monge gauge, by the height h(x>,t) of thebilayer’s midsurface M above the reference plane (x,y), with tthe time. Quantities with the superscript + and � will refer tothe upper and lower monolayer, respectively (Fig. 1). Consider-ing one protein, we call 1/s+ (resp. 1/s�) the fraction of its masslying in the upper (resp. lower) monolayer, with

1/s+ + 1/s� = 1. (1)

Let r� be the total (lipid + protein) mass density withineach monolayer and c� the protein mass fraction within eachmonolayer. Then, r�c� is the protein mass density in eachmonolayer, and c� is thus related to the protein mass fraction cthrough

Paper Soft Matter

Publ

ishe

d on

18

Dec

embe

r 20

15. D

ownl

oade

d by

SC

D U

nive

rsité

Par

is 7

on

03/0

2/20

16 1

4:48

:02.

View Article Online

This journal is©The Royal Society of Chemistry 2016 Soft Matter, 2016, 12, 1791--1800 | 1793

s�r�c� = (r+ + r�)c. (2)

Note that r�, c� and c vary along the membrane, while s� areconstant coefficients. Like in ref. 3, all densities are defined onM.

We consider small deviations with respect to a flat membraneground state with uniform protein mass fraction c0 and massdensities r�0 . Thus, we write

r� = r�0 (1 + r�), (3)

c� = c�0 + f�, (4)

c = c0 + f. (5)

The variables h(x>,t), r�(x>,t) and f(x>,t) are our mainvariables and they will be considered as first-order quantities.Note that c�0 , as well as f�, follow from eqn (2):

c�0 ¼rþ0 þ r�0r�0 s� c0; (6)

and, to linear order in the densities,

f� ¼ � r�0 c0r�0 s� rþ � r�ð Þ þ rþ0 þ r�0

r�0 s� fþOð2Þ: (7)

That is, the monolayer protein mass fractions couple not onlyto f, but also to the difference in monolayer densities r+ � r�.

Since the monolayers have little interactions apart frombeing pushed into contact by hydrophobic forces, the Hamiltonianof the bilayer can be written as

H ¼ðMdS gþ H; rþ; cþð Þ þ g� H; r�; c�ð Þ½ � (8)

’ðd2x?

s2rhð Þ2þ

Xe¼�

f e r2h; re;fe� �" #; (9)

where dS is the elementary area and H Cr2h/2 the mean curvatureofM. Eqn (9) is obtained by expanding eqn (8) at quadratic order.The membrane tension, s, arises from the zeroth-order terms in g�

multiplied by dS/d2x> C 1 + (rh)2/2. The most general quadraticform of f� can be written as32

f � ¼ k�

4r2h� �2 þ k�

2r� � e�r2hþ b�f�� �2

� l�k�e� r2h� �

f� þ a�k�

2f�

2

;

(10)

which generalizes the form proposed by Seifert and Langer forprotein-free membranes.3 Note that, although the lipids are thesame in both monolayers, the elastic constants k�, k�, e�, etc.,are monolayer dependent if the proteins are up-down asym-metric. The Gaussian curvature elasticity term can be neglectedthanks to the Gauss–Bonnet theorem despite the inhomogenityof the corresponding stiffness, �k�, arising from the pro-teins.33,34 Indeed, taking into account the dependence of �k�

in f� would result in a higher-order, cubic term, since theGaussian curvature is already a second-order quantity. We shallassume throughout that the system is far from protein phaseseparation. Here k� is the monolayer bending rigidity,18 k� thestretching modulus, e� a density–curvature coupling constantthat can be interpreted as the distance between M and themonolayer neutral surface, and b�, l� and a� are dimension-less coupling constants arising from the presence of theproteins, normalized by k�. All these constants depend inprinciple on the background protein mass fraction c0 and onthe bare mass densities r�0 , hence on the tension s,35,36

although these dependences should be moderate.37 Identifyingthem would require a specific microscopic model, which isbeyond the scope of this work.

Let us call V� the solvent velocity on both sides of themembrane, v� the in-plane barycentric velocity of the lipid +protein binary fluid, in each monolayer, and v+

p = v�p thebarycentric velocity of the transmembrane proteins. Thesequantities are obtained from the total mass flux g� and theprotein mass flux g�p using the relations g� = r�v� and g�p =r�c�v�p . Our model is therefore valid only on length scalesgreater than the inter-protein spacing. We assume that thevelocities are generated by the relaxation of the membraneonly, hence they are also small quantities that can be consid-ered parallel to (x,y) at first-order. In the following, we use theconvention that Latin indices represent either x or y whileGreek indices represent either x, y or z. Assuming no slip atthe membrane surface and no permeation, we have on themembrane, at z = 0:

V�i = v�i , Vz =:h, (11)

where a dot indicates partial time derivative. The mass con-tinuity equations read:38

:r� + qi(r

�v�i ) = 0, (12)

Fig. 1 Membrane with asymmetric, spanning proteins. (a) Geometricdescription of a membrane with height h and solvent velocities V�.(b) Monolayers have total mass densities r� and barycentric velocities v�.Proteins have total mass fraction c and velocity vp.

Soft Matter Paper

Publ

ishe

d on

18

Dec

embe

r 20

15. D

ownl

oade

d by

SC

D U

nive

rsité

Par

is 7

on

03/0

2/20

16 1

4:48

:02.

View Article Online

1794 | Soft Matter, 2016, 12, 1791--1800 This journal is©The Royal Society of Chemistry 2016

r�:c� + r�v�i qic

� + qi J�i = 0, (13)

where the protein diffusion currents are defined by J� =r�c�(v�p � v�) and the last equation follows from qt(r

�c�) +qig�p,i = 0. We note that, with the help of eqn (2), (12), and (13),

the protein continuity equation can be rewritten in terms of thebilayer protein mass fraction:

r _cþ rvi@ic ¼ �1

2@i s

þJþi þ s�J�i� �

þ 1

2@i cDrDvið Þ; (14)

where r = r+ + r� is the total mass density, vi = (r+v+i + r�v�i )/r is

the barycentric velocity, Dr = r+ � r� and Dvi = v+i � v�i . Thus, the

total protein density is convected by the bilayer barycentricvelocity, as expected. A more striking observation is thatthe protein mass current relative to vi depends not only onthe diffusive fluxes, but also on the relative monolayervelocity Dvi.

A crucial element of this problem is the following. Since theproteins span the bilayer, v+

p = v�p , and the monolayer proteincurrents must obey the following constraint:

s+J +i � s�J�i = (r+ + r�)c(v�i � v+

i ), (15)

as a consequence of eqn (2) and the definition of the currents.As we show below, this relation implies that transmembraneproteins slow down the relative motion between monolayers.

In the Stokes approximation, i.e., neglecting all inertialeffects, the dynamical equations of the whole system can beobtained by minimizing the total Rayleighian of the system,

R ¼ 1

2Ps þ Pmð Þ þ _H, equal to half the total dissipated power

plus the time derivative of the Hamiltonian.22,23 The dissipa-tion in the solvent reads

Ps ¼ðz4 0

d3RZDþabDþab þ

ðzo 0

d3RZD�abD�ab; (16)

where D�ab ¼1

2@aV

�b þ @bV�a

� �. The dissipation within the

membrane reads

Pm ¼Xe¼�

ðd2x? Z2d

eijd

eij þ

l22deiid

ejj þ

1

2GJei J

ei

� �

þðd2x?

b

2vþi � v�i� �2

;

(17)

where d�ij ¼1

2@iv�j þ @jv�i

� �, Z2 and l2 are the in-plane mono-

layer viscosities (compressible fluid), G is the coefficient asso-ciated with the dissipation due to the diffusion currents,38 andb is the intermonolayer friction coefficient.2,3 As in the case ofthe elastic constants, these dissipative coefficients depend on c0

and r�0 .14,39–41 Note that for the sake of simplicity we haveassumed that Z2, l2, and G are the same in both monolayers.

The time derivative of the Hamiltonian is

_H ¼ðd2x? �s r2h

� �_h

n

þXe¼�

f er _r� þ f ef_f� þ r2f eh

� �_h

h i);

(18)

where f �h = qf �/q(r2h), f �r = qf �/qr�, and f �f = qf �/qf�,The Rayleighan R must be extremalized with respect to all

the variables expressing the rate of change of the system::h, v�i ,

V �z , V �i , _r�, _f�, J�i ,22,23 and Lagrange multiplier fields mustbe introduced in order to implement the various constraints.Let o(x>) be the Lagrange field associated with (2),{m(x>),g(x>),z(x>),w(x>)} associated with (11–13), and oi(x>)associated with (15). The bulk incompressibility conditionqaV�a = 0 will be implemented by the Lagrange fields p�(R)corresponding to the solvent’s pressure. Performing the con-strained extremalization of R yields the bulk Stokes equations:

�2ZqbD�ab + qap� = 0, (19)

qaV�a = 0. (20)

and the surface equations in the plane z = 0 of the unperturbedmembrane (in the order of the dynamical variables listedabove):

r2( f +h + f�h ) � sr2h + g+ + g� = 0, (21)

�2Z2qjd�ij � l2qid

�jj � b(v+

i � v�i ) � r�qiz� + r�w�qic

�

� oi(r+ + r�)c + m�i = 0, (22)

�p� 8 2ZD�zz � g� = 0, (23)

82ZD�zi � m�i = 0, (24)

f �r + r�0 z� = 0, (25)

f �f + r�w� = 0, (26)

G�1J�i � qiw� � s�oi = 0, (27)

:r� + qi(r

�v�i ) = 0, (28)

r�:c� + r�v�qic

� + qi J�i = 0, (29)

v�i � V�i = 0, (30):h � V�z = 0, (31)

s+r+c+ = s�r�c� = (r+ + r�)c, (32)

s+J +i � s�J�i = (r+ + r�)c(v�i � v+

i ). (33)

Note that in addition to h, r� and f�, the velocities, thediffusion currents and, therefore, all the Lagrange multipliersare first-order quantities. Linearizing these surface equations,eliminating the Lagrange multipliers, the diffusion currentsand f� (thanks to eqn (7)) is straightforward and yields thefollowing reduced set of surface equations:

r2fh � sr2h + p+ � p� � 2Z(D+zz � D�zz) = 0, (34)

Paper Soft Matter

Publ

ishe

d on

18

Dec

embe

r 20

15. D

ownl

oade

d by

SC

D U

nive

rsité

Par

is 7

on

03/0

2/20

16 1

4:48

:02.

View Article Online

This journal is©The Royal Society of Chemistry 2016 Soft Matter, 2016, 12, 1791--1800 | 1795

�2Z2qjd�ij � l2qid

�jj � (b + db)(v+

i � v�i ) + qi f �r

8 Q(s+r�0 qi f +f � s�r+

0qi f �f) 8 2ZD�zi = 0, (35)

_r� + qiv�i = 0, (36)

_f � GS(s�r�0r2f +f + s+r+

0r2f �f) � Tqi(v+i � v�i ) = 0, (37)

in which fh = f +h + f �h , s2 ¼ 1

2sþð Þ2þ s�ð Þ2

h i� 4, r0 ¼

1

2rþ0 þ r�0� �

,

Q = c0r0/(s2r+0r�0 ), S = s+s�/(4s2r+

0r�0 r0), T = c0[(s+)2r+0 � (s�)2r�0 ]/

(4s2r0), and

db ¼ 2

Gr0c0

s

� �2: (38)

These equations, eqn (34–37), resemble the ones obtained by Seifertand Langer for a one-component lipid bilayer membrane,3 yetthere are fundamental differences. These arise from the termsproportional to db and Q in eqn (35), and, of course, fromeqn (37), which describes the protein mass conservation. Beforecommenting on these differences, we explain the physicalmeaning of these four equations. The first one, eqn (34), isthe balance of stresses normal to the membrane, with r2fh and�sr2h the elastic stresses and 2ZD�zz � p� the stresses arisingfrom the solvent. Eqn (35) is the balance of stresses parallel tothe membrane in each monolayer: the first two terms are theviscous stresses, the third term is the intermonolayer friction,the fourth term the gradient of two-dimensional pressure, thefifth term (p Q) is an extra intermonolayer stress and the lastterm is the solvent shear stress. Eqn (36) is the mass continuityequation and eqn (37) is the protein conservation equation.

In order to grasp the differences with the dynamics ofmembranes deprived of spanning objects, we consider the limitwhere the proteins are located only in the upper monolayer,which corresponds to s+ = 1 and s� B s - N. In this casedb - 0, Qs+ B s�2 - 0, Qs� B s�1 - 0, and Ss+ - 0, and allthe extra terms mentioned above vanish. These terms arise thusspecifically from the spanning character of the proteins. Asexplained in the introduction, the increase of the intermono-layer friction coefficient b arises from the dissipation p 1/Gcaused by the motion of the lipids relative to the proteins. Notethat a similar conclusion was found in a simulation study.42

More precisely, in the process of obtaining eqn (34)–(37), theprotein currents J�i are found to bear a term proportional tothe relative barycentric velocity v+

i � v�i . This comes in partfrom eqn (15). Thus the dissipation associated with diffusion,J�i J�i /(2G), contains a term proportional to (v+

i � v�i )2 whichincreases b. Note that in eqn (38) the dependence on c0 ishidden, since G depends on c0. Hence in the dilute limit db Bc0 since G B c0. As for the stresses p Q, which are proportionalto the gradient of the energy density f�f , they are elastic stressestransmitted by the spanning character of the proteins.

3 Relaxation modes

The equations being linear, we may decompose allquantities into independent in-plane Fourier modes:h x?; tð Þ ¼

Ðd2qð2pÞ�2hqðq; tÞ exp iq � x?ð Þ, etc. Let us define the

average density �r = r+ + r� and the density-difference

r = r+ � r�. (39)

Taking the Fourier transform of the membrane eqn (34–36) andeliminating the bulk and surface velocities yields a system offour equations that give

:hq, _rq, _�rq and _fq as linear combinations

of hq, rq, �rq and fq. This procedure, involving heavy calculations,is detailed in the Appendix.

3.1 Reduction to three dynamical variables

We shall assume throughout typical parameter values: k� C10�19 J,43 k� C 0.1 J m�2,43 e� C 1 nm, Z C 10�3 J s m�3, Zs =10�9 J s m�2,44 b C 109 J s m�4,45–47 and D0 C 10�12 m2 s�1 forthe diffusion coefficient of transmembrane proteins.48 Thecoefficients a�, b�, and l� are expected to be of order unity.49

As in the case of protein-free membranes,3 it turns out, giventhe parameter values given above, that the variable �rq is alwaysmuch more rapid than hq, rq and fq. It can therefore beeliminated by setting the right-hand-side of eqn (68) equal tozero and solving for �rq. One then obtains a reduced dynamicalsystem of the form

_hq

_rq

_fq

0BBBB@

1CCCCA ¼ A

hq

rq

fq

0BBB@

1CCCA: (40)

In the general case, the coefficients of the matrix A are toocumbersome to be given explicitly. Fig. 2a and b show examplesof the three relaxation rates g1 4 g2 4 g3 given by the negativeof the eigenvalues of A, for different membrane tensions anddegree of protein asymmetry. Note that they never cross.

To simplify the expression for the elements of A, we considerthe case of a weak asymmetry which we assume to be presentonly in the curvature coupling constants. Without loss ofgenerality, it can be expressed in terms of a small parameter

n, by setting 1/s� = 1/2 � n, e� = e � ne, b� = b � nb, l� = l � nl,and k� = k, k� = k, r�0 = r0, a� = a. We then obtain

A = A(0) + nA(1) + O(n2) (41)

with

Að0Þ ¼

�~kq3 þ sq4Z

ke 1� c00� �q

4Z0

ke 1� c00� �q4

Bþ Zqþ Zsq2�

k 1� c000� �q2

2 Bþ Zqþ Zsq2ð Þ 0

0 0 �Gkaq2

r02

0BBBBBBBBBB@

1CCCCCCCCCCA:

(42)

where B = b + db, ~k ¼ kþ 2ke2, Zs ¼ Z2 þ1

2l2, c00 ¼ c0ðbþ lÞ,

c000 ¼ 2bc0 � c0

2 aþ b2� �

, and

Að1Þ13 ¼

ke1

2Zq; A

ð1Þ23 ¼

2~bkBþ Zqþ Zsq2

q2; (43)

Soft Matter Paper

Publ

ishe

d on

18

Dec

embe

r 20

15. D

ownl

oade

d by

SC

D U

nive

rsité

Par

is 7

on

03/0

2/20

16 1

4:48

:02.

View Article Online

1796 | Soft Matter, 2016, 12, 1791--1800 This journal is©The Royal Society of Chemistry 2016

Að1Þ31 ¼

Ge1r02þ

2ec0 1� c00� �

Bþ Zqþ Zsq2

kq4; (44)

Að1Þ32 ¼

2c0aþ ~b� �

G

r02�

c0 1� c000� �

Bþ Zqþ Zsq2

0@

1Akq2; (45)

with e1 = (2b + b + 2l + l)e + le and ~b ¼ bþ 1

2b

c0b� 1ð Þ, the

other elements of A(1) being zero.It can be first noticed that, for symmetrical proteins (n = 0),

the matrix A = A(0) is block-diagonal and h and r are coupledexactly as in protein-free membranes, but with renormalizedcoefficients.50

Next we compare the basic relaxation rates in this problem,given by the diagonal elements of the matrix A. Although theydo not coincide in general with the actual rates gi, they provideuseful informations. First, the h-like relaxation rate, associatedwith A(0)

11, is ~kq3 þ sq� ��

ð4ZÞ (green dashed line in Fig. 2a and b).Second, the r-like rate, associated with A(0)

22 and originatingfrom the relative monolayer movement, is Ckq2/B (red dashedline in Fig. 2a and b). Here, we have assumed c0

00 = O(1) and B c

Zq + Zsq2, which we shall assume throughout since it holds

typically for q�1\ 1 nm, given the orders of magnitude listed

above. The ratio of the h-like rate to the r-like rate is minimumwhere the green dashed line of Fig. 2a (s = 0) or Fig. 2b (s a 0)attains its minimum, i.e., at

qc ¼ffiffiffiffiffiffiffiffis=~k

p: (46)

Comparing the h-like rate and the r-like rate yields the cross-over tension

sc ¼4

~kkZB

2

t10�8N m�1 (47)

that separates the vanishing tension regime (Fig. 2a) from thefinite tension regime (Fig. 2b); it is defined by the conditionthat for s4 sc the h-like rate is always larger that the r-like rate(as in Fig. 2b).

Finally, the f-like rate, associated with A(0)33, is D0q2 (blue

dashed line in Fig. 2a and b), where

D0 ¼kGar02

: (48)

The ratio of the f-like rate to the r-like rate (that both have adiffusive character p q2) is D0B/k; this leads us to introduce thesmall dimensionless parameter:

z ¼ D0b

k 10�2: (49)

We thus expect in general fq to be the slowest variable.With the above considerations, we can write the renormalized

intermonolayer friction coefficient [eqn (38)], for symmetricmembranes, as

B ¼ b 1þ ac02

2z

: (50)

Thus, the slower the diffusion of the proteins, the larger therenormalization of the intermonolayer friction coefficient. Sincewe expect aE 1 in the absence of specific interactions among theproteins,49 and assuming typically c0 E 0.3, the renormalization is

Fig. 2 Relaxation rates and effective diffusion coefficient for asymmetricproteins. All rates are normalized by q2. (a) Zero tension case. The blacklines show the three relaxation rates g1 4 g2 4 g3 versus wavevector q.The dashed lines show the h-like rate A11 in green, the r-like rate A22 inred and the f-like rate A33 in blue. Besides the parameters given in thetext, b = 5 108 J s m�4, n = 0.25, e = 0.1 nm, a = 1, b = 1, l = 1. (b) Sameplots in the finite tension case, s Z sc. Four values of s are shown:s = 10�5 N m�1 (black), s = 10�6 N m�1 (dark gray or dark green),s = 10�7 N m�1 (gray or green), and s = 10�8 N m�1 (light gray or lightgreen). The dots indicate the effective diffusion coefficient Deff; itis indistinguishable from the slowest normalized rate. (c) Asymptoticeffective diffusion coefficients D0

eff for q - 0 (three upper blue curves)and DN

eff for q -N (three lower red curves), versus asymmetry parametern. In each group, from top to bottom: b = 109 J s m�4, b = 5 109 J s m�4

and b = 1010 J s m�4. The other specific parameters are e = a = b = 0 ands = 10�6 N m�1. (For interpretation of the references to color, the readeris referred to the web version of this paper.)

Paper Soft Matter

Publ

ishe

d on

18

Dec

embe

r 20

15. D

ownl

oade

d by

SC

D U

nive

rsité

Par

is 7

on

03/0

2/20

16 1

4:48

:02.

View Article Online

This journal is©The Royal Society of Chemistry 2016 Soft Matter, 2016, 12, 1791--1800 | 1797

large, owing to the smallness of z. Thus, transmembraneproteins are expected to have a significant influence on theintermonolayer movement, and hence, on the relaxation ofmembrane bending fluctuations.3,46,47,51

The actual relaxation rates gi can be interpreted as follows.The quickest rate, g1, always coincides (more or less perfectly)with one of the three basic relaxation rates (Fig. 2a and b).For instance, at high q, the quickest rate (upper black line)coincides with the h-like rate (green dashed line). This meansthat hq relaxes very quickly with rq and fq ‘‘frozen’’. Themedium rate (intermediate black line), g2, is mainly parallelbut below another one of the three basic rates, indicating thesecond-quickest variable. For instance, at high q, this is ther-like rate, which implies that after hq has relaxed, rq is relaxingwith both hq ‘‘slaved’’ and fq still ‘‘frozen’’. Then, the slowestrate, g3, is parallel and below the slowest basic rate, indicatingthe slowest variable. For instance, at high q, fq is the slowestvariable and it relaxes with hq and rq ‘‘slaved’’. This analysisapplies wherever the three rates are well separated, outside thecrossover regions. For instance, at zero tension and low q(Fig. 2a) the sequence from quickest to slowest relaxing variablesis rq, fq and hq. Actually, hq is the slowest variable only for s = 0and low q. At finite tensions, s 4 sc, the slowest variable isalways fq (Fig. 2b) and its relaxation rate steps downwards forq 4 qc; the slowing down of protein diffusion at large q has longbeen recognized as resulting from the coupling to membranecurvature.12

3.2 Effective protein dynamics at finite membrane tension

From Fig. 2b and the discussion just above, we see that forordinary membranes, for which s \ 10�8 N m�1

c sc, therelaxation rate g1 coincides with the relaxation rate of hq, and isat least an order of magnitude greater than the other two.Therefore, hq adiabatically follows the dynamics of rq and fq,

and by setting:hq = 0 in eqn (40) we are able to reduce the

dynamical problem to two variables, governed by the matrix

B ¼A22 �

A21A12

A11A23 �

A21A13

A11

A32 �A31A12

A11A33 �

A31A13

A11

0BBB@

1CCCA: (51)

The two relaxation rates (negative of eigenvalues) G1 4 G2 of B arenot always well separated, and one cannot eliminate adiabaticallyrq in order to get the dynamics of fq. Instead, keeping only theslower rate, one has fq(t) B exp[�q2Deff(q)t], where Deff = G2/q2 and

G2 ¼ �1

2B11 þ B22 þ B11 � B22ð Þ2þ4B12B21

h i1=2� . The precise

form of Deff(q), the effective diffusion coefficient of the proteins,is quite complicated for arbitrary q, but can be calculated numeri-cally, as shown in Fig. 2c. Its coincidence with q�2g3 justifies theadiabatic approximation for hq.

We consider now two limiting forms of Deff(q). First, for q -

0, the asymptotic effective diffusion coefficient, D0eff, is given at

first order in the small parameter z and to O(n2) by

D0eff

D0’ 1� 2n2z

2bþ b� �2a bc0 � 1ð Þ2

: (52)

Since z p b, we find that the protein diffusion coefficient atlong wavelengths is, for asymmetric bilayers, reduced by theintermonolayer friction b (see Fig. 2c): this effect arises fromthe coupling between f and r in the free energy terms b�f�

2

through eqn (7).Second, for q c qc, we write

DN

eff = D0eff + DD, (53)

where in general DD is very complicated. However, DD vanishesfor symmetric membranes (n = 0) and no protein–membranecurvature coupling (l = l = 0), in agreement with ref. 12. For thesimple asymmetric case e = l = 0 and c0 - 0, expanding againto O(n2) and to first order in z yields

DDD0¼ �8n2ke

2

kl2

a1þ 2z

l2bþ b� 2

ke2

k

� �: (54)

Thus, the effect of intermonolayer friction, b p z, is a second-ary effect compared to the dominant role played by the protein–membrane curvature coupling l (see Fig. 2c).

4 Diffusion of a concentrated proteinspot

As an application of the above formalism, we consider thediffusion of a concentrated spot of asymmetric transmembraneproteins. The relaxation of the protein density field, c(x>,t), iscoupled to the membrane height, h(x>,t), and to the density-difference r(x>,t). Defining the state of the protein–membranesystem by the column matrix X(x>,t) = (h(x>,t),r(x>,t),f(x>,t))T,we ask how does X(x>,t) evolve given that proteins are initiallyconcentrated in a circular region of radius w in a flat, equili-brium membrane:

X(x>,0) = f0e�r 2/(2w2)(0,0,1)T, (55)

To find out, we work in Fourier space. The time evolution of theFourier transform is determined by solving

:X(q,t) = AX(q,t) for

each q, subject to the initial condition:

Xðq; 0Þ ¼ðd2rX x?; 0ð Þe�iq�x? ¼ 2pw2f0e

�12w2q2ð0; 0; 1ÞT : (56)

We denote S(q) the matrix whose columns are the threeeigenvectors of A; therefore Ad = S�1AS is a diagonal matrixwhose elements are the negative of the relaxation rates gi(q).With a change of basis, we define X0(q,t) = S�1X(q,t), whichsatisfies

:X0(q,t) = AdX0(q,t), with solution X0(q,t) = eAdtX0(q,0).

Therefore, we obtain X(q,t) = SeAdtS�1X(q,0). As a result, the

Soft Matter Paper

Publ

ishe

d on

18

Dec

embe

r 20

15. D

ownl

oade

d by

SC

D U

nive

rsité

Par

is 7

on

03/0

2/20

16 1

4:48

:02.

View Article Online

1798 | Soft Matter, 2016, 12, 1791--1800 This journal is©The Royal Society of Chemistry 2016

solution in real space is reconstructed, yielding

X x?; tð Þ ¼ðd2q

ð2pÞ2SeAdtS�1Xðq; 0Þeiq�x?

¼ð10

dq

2pqJ0ðqrÞSeAdtS�1Xðq; 0Þ;

(57)

and thus,

fðr; tÞ ¼ f0w2Xi

ð10

dqqJ0ðqrÞ

S3iðqÞe�12w2q2�giðqÞtS�1i3 ðqÞ: (58)

We have used the symmetry of revolution of the problem tocarry out the angular integration, which yields the Besselfunction of zeroth order J0(qr), where r = |x>|. We thereforesee that the diffusion of the protein spot, viewed in real space,is governed not only by the bare diffusion constant D0, butmore generally by all the three rates gi.

The time evolution of the protein concentration spot isdescribed in Fig. 3. The q-dependence of the effective diffusioncoefficient Deff(q), as seen in Fig. 2b, is reflected in the evolution

of the real space protein mass fraction f(r,t). As shown inFig. 3a, at short times (but still D0t 4 w2) and for r2 t 2D0t, theprotein diffusion is controlled by DN

eff. Indeed, for times suchthat qc

ffiffiffiffiffiffiffiffiD0tp

� 1,

fðr; tÞ ’ f0w2a13

2D1eff te� r2

4D1eff

t; (59)

where aN

3 � limq-NS33S33�1. Note that, for large r, f oscillates

slightly about zero, due to the coupling with membrane curvaturefor large q; in Fig. 3a this is seen as a divergence in the logarithm oftf(r,t).

In contrast, at long times qcffiffiffiffiffiffiffiffiD0tp

1� �

, the protein diffusionis controlled by D0

eff, and

fðr; tÞ ’ f0w2a03

2D0eff t

e� r2

4D0efft; (60)

where a03 � limq-0S33S33

�1. These behaviors are confirmed byFig. 3b, showing the half width at half maximum of f, denoted

r1/2, versus t. For short times, r1/2 tends to 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD1eff t ln 2

p, while for

long times, r1/2 tends to 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD0

eff t ln 2q

. Thus, the coupling of

protein diffusion to membrane curvature is felt at short times(through DN

eff), whereas the coupling to in-plane motion is felt atlong times (through D0

eff).

5 Conclusion and perspectives

To summarize, we developed in this paper a generic formalism,based on Onsager’s variational principle, that allowed us todescribe the dynamics of lipid bilayers containing transmembraneproteins of arbitrary shape. This approach, valid on length scaleslarger than the mean inter-protein distance, reveals that thepresence of proteins that span the bilayer opposes monolayersliding, and hence significantly increases the intermonolayerfriction coefficient. The correction to this coefficient is shown tovary in inverse proportion to the protein’s mobility. Experimen-tally, transmembrane protein-enhanced intermonolayer frictionhas recently been invoked to explain observations of slow forcerelaxation on membrane tubes extracted from Giant UnilamellarVesicles.52 Applying our formalism to the simple situation whereprotein enrichment is localized initially in a circular regionwithin a flat membrane, we show that for practically all acces-sible membrane tensions, protein density is the slowest relaxingvariable, and the spreading of the protein spot follows ananomalous diffusion behavior. Precisely, at short and long times,the spreading passes from a normal diffusion regime at shorttimes to another normal diffusion regime (i.e. with anotherdiffusive coefficient) at long times. To our knowledge, diffusionof transmembrane proteins has been probed only at very lowconcentrations.11 The anomalous diffusion behavior expected atlarger protein concentrations could be tested experimentally: theinitial patch of proteins could be realized either by applying amagnetic field on proteins with grafted magnetic particles, or bycoalescing a small vesicle with high concentration of proteinsand a large vesicle with moderate concentration of proteins.53

Fig. 3 Diffusion of a concentrated protein spot in a tensed membrane.(a) Logarithmic plot of tf(r,t) normalized by 2D0/(f0w2) versus r2/t for timesranging from t = 1 s to t = 1000 s, with two profiles per time decade (red tofuchsia, from bottom to top). Short and long time behaviors are given bydotted and dashed lines, controlled by DN

eff and D0eff (see text). (b) Half width

at half maximum, r1/2, as a function of t. Short and long time behaviors aregiven by dotted and dashed lines, controlled by DN

eff and D0eff (see text).

In addition to the parameter values given in text, s = 10�8 N m�1,b = 5 109 J s m�4, e = 0.1 nm, a = 1, b = 1, l = 1, n = 0.25, and w = 10�8 m.

Paper Soft Matter

Publ

ishe

d on

18

Dec

embe

r 20

15. D

ownl

oade

d by

SC

D U

nive

rsité

Par

is 7

on

03/0

2/20

16 1

4:48

:02.

View Article Online

This journal is©The Royal Society of Chemistry 2016 Soft Matter, 2016, 12, 1791--1800 | 1799

Some proteins, like Gramicidin, can exist in the form of mono-mers located separately in the two monolayers, or in the form ofa dimer spanning the whole bilayer.54 In light of our model, itwould be interesting to test experimentally the effect of thisdimerization on the effective intermonolayer friction coefficient.

AppendixA Dynamical equations for the membrane in Fourier space

In reciprocal space, eqn (34–37) take the form:

�q2Xe¼�

f ehrreq þ f ehff

eq þ f ehhhq

� �þ sq2hq þ 4Zq _hq ¼ 0; (61)

2Zsq2v�q �B vþq � v�q

� �þ iq f �rrr

�q þ f �rff

�q þ f �rhhq

� �

� iqQXe¼�

eser�e0 f efrreq þ f efff

eq þ f efhhq

� �þ 2Zqv�q ¼ 0;

(62)

_r�q + iqv�q = 0, (63)

_fþ Sq2Xe¼�

s�er�e0 f efrreq þ f efff

eq þ f efhhq

� �� iqT vþq � v�q

� �¼ 0;

(64)

where v�q is the component of the velocity parallel to q. Theabove coefficients f �ij , where (i, j) A {r,f,h}2, are definedthrough the reciprocal space relations: f �r = f �rrr

�q + f �rff

�q +

f �rhhq, f �f = f �frr�q + f �fff

�q + f �fhhq, and f �h = f �hrr

�q + f �hff

�q + f �hh

hq. Explicitly, they read

f �rr ¼ k�; f �rf ¼ k�b�; f �rh ¼ �e�k�q2;

f �fr ¼ k�b�; f �ff ¼ k�a� þ k�b�2

; f �fh ¼ �e�k� l� þ b�� �

q2;

f �hr ¼ � e�k�; f �hf ¼ �e�k� l� þ b�� �

;

f �hh ¼�1

2k� þ 2k�e�

2� �

q2:

(65)

The term 4Zq:hq in eqn (61) is the Fourier transform of the bulk

stress p+ � p� � 2Z(D+zz � D�zz) evaluated at z = 0; the term

2Zsq2v�q in eqn (62), where Zs ¼ Z2 þ

1

2l2, is the Fourier transform

of �2Z2qjd�ij � l2qid

�jj ; finally, the term 2Zqv�q in eqn (62) is the

Fourier transform of the bulk stress 82ZD�zi evaluated in z = 0.For a detailed derivation of those contributions, see, e.g., Section8 of ref. 24.

Expressing f�q in terms fq, rq and �rq using eqn (7) andeliminating the velocities v�q yields the system of equationsgiven in the body of the text:

4Z _h ¼ f þhh þ f�hh � s� �

qhþ 1

2f þhr � f �hr þ 2Y�f �hf � 2Yþf þhf

� �qr

þ 1

2f þhr þ f �hr

� �q�rþ Xþf þhf þ X�f �hf

� �qf;

(66)

2 Bþ Zqþ Zsq2

� �_r ¼ f �rh � f þrh � 2Qs�rþ0 f

�fh þ 2Qsþr�0 f

þfh

� �q2h

� 1

2f �rr þ f þrr � 2Y�f �rf � 2Yþf þrf

h

� 2Qrþ0 s� f �fr � 2Y�f �ff

� �

� 2Qr�0 sþ f þfr � 2Yþf þff

� �iq2r

þ 1

2f �rr � f þrr � 2Qrþ0 s

�f �fr

�

þ 2Qr�0 sþf þfr

�q2�r

þ q2 X�f �rf � Xþf þrf � 2X�Qrþ0 s�f �ff

�

þ 2XþQr�0 sþf þff

�f;

(67)

2 Zþ Zsqð Þ _�r ¼� f þrh þ f �rh

� �qh

þ 1

2f �rr � f þrr � 2Y�f �rf þ 2Yþf þrf

� �qr

� 1

2f þrr þ f �rr

� �q�r� Xþf þrf þ X�f �rf

� �qf;

(68)

_f ¼ 1

2�2SM41 þ

TN41

Bþ Zqþ Zsq2

q2h

þ 1

4�2SM42 þ

TN42

Bþ Zqþ Zsq2

q2r

þ 1

4�2SM43 þ

TN43

Bþ Zqþ Zsq2

q2�r

þ 1

2�2SM44 þ

TN44

Bþ Zqþ Zsq2

q2f;

(69)

where X� = (r+0 + r�0 )/(r�0 s�) and Y� = (r80 c0)/(r�0 s�) are the

coefficients appearing in eqn (7), and the coefficients M4i andN4i are given by

M41 = r�0 s�f +fh + r+

0s+f �fh, (70)

M42 = 2r+0s+( f �fr � 2Y�f�ff) � 2r�0 s�( f+

fr � 2Y+f +ff), (71)

M43 = r+0s+f �fr + r�0 s�f +

fr, (72)

M44 = r�0 s�X+f +ff + r+

0s+X�f �ff, (73)

N41 = f +rh � f �rh + 2Q(r+

0s�f �fh � r�0 s+f +fh), (74)

N42 = f �rr + f +rr� 2Y�( f �rf� 2Qr+

0s�f �ff)� 2Y +( f +rf� 2Qr�0 s+f +

ff)

� 2Q(r+0s�f �fr + r�0 s+f +

fr), (75)

N43 = f +rr � f �rr + 2Q(r+

0s�f �fr � r�0 s+f +fr), (76)

N44 = X+f +rf � X�f �rf + 2Q(r+

0s�X�f �ff � r�0 s+X +f +ff). (77)

Soft Matter Paper

Publ

ishe

d on

18

Dec

embe

r 20

15. D

ownl

oade

d by

SC

D U

nive

rsité

Par

is 7

on

03/0

2/20

16 1

4:48

:02.

View Article Online

1800 | Soft Matter, 2016, 12, 1791--1800 This journal is©The Royal Society of Chemistry 2016

Acknowledgements

We thank S. Mangenot and P. Bassereau for useful discussions.J.-B. Fournier acknowledges financial support from the FrenchAgence Nationale de la Recherche (Contract No. ANR-12-BS04-0023-MEMINT).

References

1 H. Lodish, et al., Molecular Cell Biology, W. H. Freeman &Co, New York, 7th edn, 2012.

2 E. Evans and A. Yeung, Chem. Phys. Lipids, 1994, 73, 39.3 U. Seifert and S. A. Langer, Europhys. Lett., 1993, 23, 71.4 J.-B. Fournier, N. Khalifat, N. Puff and M. I. Angelova, Phys.

Rev. Lett., 2009, 102, 018102.5 C. E. Brennen, Fundamentals of Multiphase Flow, Cambridge

University Press, New York, 2005.6 P. G. Saffman and M. Delbruck, Proc. Natl. Acad. Sci. U. S. A.,

1975, 72, 3111–3113.7 E. Reister and U. Seifert, EPL, 2005, 71, 859.8 S. M. Leitenberger, E. Reister-Gottfried and U. Seifert, Lang-

muir, 2008, 24, 1254–1261.9 A. Naji, P. J. Atzberger and F. L. H. Brown, Phys. Rev. Lett.,

2009, 102, 138102.10 E. Reister-Gottfried, S. M. Leitenberger and U. Seifert, Phys.

Rev. E: Stat., Nonlinear, Soft Matter Phys., 2010, 81, 031903.11 F. Quemeneur, et al., Proc. Natl. Acad. Sci. U. S. A., 2014,

111, 5083.12 S. Leibler, J. Phys., 1986, 47, 507–516.13 S. Leibler and D. Andelman, J. Phys., 1987, 48, 2013.14 B. A. Camley and F. L. H. Brown, Soft Matter, 2013, 9, 4767–4779.15 M. A. Lomholt, P. L. Hansen and L. Miao, Eur. Phys. J. E: Soft

Matter Biol. Phys., 2005, 16, 439–461.16 L. Kramer, J. Chem. Phys., 1971, 55, 2097.17 F. Brochard and J.-F. Lennon, J. Phys., 1975, 36, 1035.18 W. Helfrich, Z. Naturforsch., 1973, C28, 693.19 M. Arroyo and A. DeSimone, Phys. Rev. E: Stat., Nonlinear,

Soft Matter Phys., 2009, 79, 031915.20 M. Rahimi and M. Arroyo, Phys. Rev. E: Stat., Nonlinear, Soft

Matter Phys., 2012, 86, 011932.21 M. Rahimi, A. DeSimone and M. Arroyo, Soft Matter, 2013,

9, 11033.22 H. Goldstein, C. Poole and J. Safko, Classical Mechanics,

Addison Welsley, New York, 2000.23 M. Doi, J. Phys.: Condens. Matter, 2011, 23, 284118.24 J.-B. Fournier, Int. J. Non Linear Mech., 2015, 75, 67.25 S. A. Safran, Statistical thermodynamics of surfaces, interfaces,

and membranes, Westview Press, 2003.26 S. Svetina, M. Brumen and B. Zeks, Stud. Biophys., 1985,

110, 77.27 L. Miao, U. Seifert, M. Wortis and H.-G. Dobereiner, Phys.

Rev. E: Stat., Nonlinear, Soft Matter Phys., 1994, 49, 5389.28 P. Sens, Phys. Rev. Lett., 2004, 93, 108103.

29 A.-F. Bitbol, et al., Soft Matter, 2012, 8, 6073.30 W. K. den Otter and S. A. Shkulipa, Biophys. J., 2007, 93,

423–433.31 R. Okamoto, Y. Kanemori, S. Komura and J.-B. Fournier,

2015, arXiv:1511.01324 [cond-mat.soft].32 The term Br2h is discarded as a boundary term. The terms

Br� and Bf� are discarded as bothÐd2x?r� andÐ

d2x? 1þ r�ð Þf� are constant due to the mass conservationof lipids and proteins.

33 G. Brannigan and F. L. H. Brown, Biophys. J., 2007, 92,864–876.

34 B. West, F. L. Brown and F. Schmid, Biophys. J., 2009, 96,101–115.

35 A. F. Bitbol, D. Constantin and J. B. Fournier, PLoS One,2012, 7, e48306.

36 M. C. Watson, A. Morriss-Andrews, P. M. Welch andF. L. H. Brown, J. Chem. Phys., 2013, 139, 084706.

37 There are lipids in between the rigid proteins, thereforebending involves essentially splaying the lipids and stretch-ing involves essentially changing the lipid density. Thepresence of the proteins should thus not change drasticallythe associated energy penalty per unit area.

38 L. D. Landau and E. M. Lifshitz, Fluid Mechanics,Butterworth-Heinemann, Oxford, 2000.

39 P. S. Niemela, M. S. Miettinen, L. Monticelli, H. Hammaren,P. Bjelkmar, T. Murtola, E. Lindahl and I. Vattulainen, J. Am.Chem. Soc., 2010, 132, 7574–7575.

40 N. Oppenheimer and H. Diamant, Biophys. J., 2009, 96,3041–3049.

41 B. A. Camley and F. L. H. Brown, J. Chem. Phys., 2014,141, 075103.

42 A. Khoshnood, H. Noguchi and G. Gompper, J. Chem. Phys.,2010, 132, 025101.

43 W. Rawicz, et al., Biophys. J., 2000, 79, 328.44 A. R. Honerkamp-Smith, F. G. Woodhouse, V. Kantsler and

R. E. Goldstein, Phys. Rev. Lett., 2013, 111, 038103.45 R. Merkel, E. Sackmann and E. Evans, J. Phys., 1989, 50, 1535.46 T. Pott and P. Meleard, Europhys. Lett., 2002, 59, 87.47 R. Rodrguez-Garca, L. R. Arriaga, M. Mell, L. H. Moleiro,

I. Lopez-Montero and F. Monroy, Phys. Rev. Lett., 2009,102, 128101.

48 S. Ramadurai, et al., J. Am. Chem. Soc., 2009, 131, 12650.49 The mixing entropy for proteins much larger than lipids of

size a yields a = kBT/[ka2(1 � c0)] E 1, for c0 C 0.3.50 For c0 = 0 we recover the dynamical matrix of ref. 3, except

that Zs includes dilational viscosity.51 S. A. Shkulipa, W. K. den Otter and W. J. Briels, Phys. Rev.

Lett., 2006, 96, 178302.52 C. Campillo, P. Sens, D. Koster, L.-L. Pontani, D. Levy,

P. Bassereau, P. Nassoy and C. Sykes, Biophys. J., 2013,104, 1248–1256.

53 P. Bassereau and S. Mangenot, personal communication.54 M. Goulian, et al., Biophys. J., 1998, 74, 328.

Paper Soft Matter

Publ

ishe

d on

18

Dec

embe

r 20

15. D

ownl

oade

d by

SC

D U

nive

rsité

Par

is 7

on

03/0

2/20

16 1

4:48

:02.

View Article Online