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Large-scale simulations of fluctuating biological membranes

Andrea Pasqua∗,1 Lutz Maibaum∗,1, 2 George Oster,3 Daniel A. Fletcher,4, 5 and Phillip L. Geissler1, 2, 5

1Department of Chemistry, University of California, Berkeley, CA 947202Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720

3Department of Molecular and Cellular Biology, University of California, Berkeley, CA 947204Department of Bioengineering, University of California, Berkeley, CA 94720

5Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720

We present a simple, and physically motivated, coarse-grained model of a lipid bilayer, suitedfor micron scale computer simulations. Each ≈ 25nm2 patch of bilayer is represented by a spher-ical particle. Mimicking forces of hydrophobic association, multi-particle interactions suppress theexposure of each sphere’s equator to its implicit solvent surroundings. The requirement of highequatorial density stabilizes two-dimensional structures without necessitating crystalline order, al-lowing us to match both the elasticity and fluidity of natural lipid membranes. We illustrate themodel’s versatility and realism by characterizing a membrane’s response to a prodding nanorod.

INTRODUCTION

Lipid bilayers form the basis of biological membranes.Integral membrane proteins and a variety of smallmolecules are embedded in this two-dimensional fluid,which is stabilized by hydrophobic interactions: the bi-layer structure effectively shields the lipid’s hydropho-bic alkane chains from exposure to the aqueous solvent.Despite this complexity on the molecular level, biologi-cal membranes at large length scales are well character-ized by surprisingly few material properties. In partic-ular, their behavior on large length scales is consistentwith that of crude elastic models, which take as inputonly a bending rigidity, i.e., the membrane’s resistanceto smooth shape deformations. It is unclear below whatscale such representations become inappropriate, in partdue to computational difficulties of incorporating ther-mal fluctuations, which invariably gain importance as thescale of observation is reduced from the macroscopic. Atthe opposite extreme computer simulations of lipid bi-layers at atomistic resolution provide detailed access tothe spectrum of thermal fluctuations. They are limited,however, to length and time scales of tens of nanometersand hundreds of nanoseconds, respectively [1].

Many important biological phenomena involve mem-brane deformations over several micrometers and spanseconds or even minutes. They thus occur on lengthscales intermediate between those natural to elastic con-tinuum models and to atomistic representations. In at-tempts to bridge this gap, a large variety of simplifiedmodels for interactions between lipid molecules have beenproposed in the literature. These range from systemati-cally coarse-grained systems [2, 3] to solvent–free heuris-tic models [4, 5]; a recent review can be found in Ref. 6.Common to these models is an attempt to mimic the am-phipathic character of individual lipid molecules. While

∗These authors contributed equally to this work.

considerably less expensive than atomistic simulations intheir computational demands, these approaches are stilllimited in the scope of fluctuations and response they canfeasibly capture.

Extending computer simulations to examine large scalebehaviors such as aggregation of membrane-associatedproteins or deformations induced by growth of an actinnetwork would appear to require coarse-graining beyondthe scale of individual lipid molecules. Several such ap-proaches have been proposed [7, 8, 9]. Most representin a discrete way the fluctuations implied by Helfrich’scontinuum model of an elastic sheet [10, 11]. These re-quire estimating local curvature of a discretized mani-fold, which can be both numerically unstable and taxingto implement. Furthermore, these approaches do not at-tempt to make a close connection with the physical forcesresponsible for membrane cohesion and elasticity.

In this work we propose a new simulation model at themany-lipid scale that follows transparently from the sta-tistical mechanics of hydrophobicity and the underlyingmembrane thermodynamics. Despite its simplicity, ourmodel successfully captures several important propertiesof lipid bilayer physics, such as intrinsic fluidity and theability to spontaneously assemble into two–dimensionalsheets. It is furthermore sufficiently versatile to repro-duce a wide range of biologically relevant elastic proper-ties.

In our model, we envision the lipid bilayer as a collec-tion of small membrane patches of size d ≈ 5 nm, roughlythe thickness of a typical bilayer [12], each comprising∼ 100 phopholipid molecules. For geometric simplicity,we represent each patch as a volume-excluding spherewith an axis of rotational symmetry pointing from onepolar head group region to the other. Cohesion of suchpatches is due of course to the presence of water: Expos-ing the hydrophobic portion (i.e., the equatorial region ofour model spheres) to solvent incurs a free energetic cost,while exposing the hydrophilic portion (i.e., the polarcaps of our model spheres) is thermodynamically advan-

tageous. At length scales >∼ 1 nm both of these contri-

2

butions should be proportional to the exposed area [13].Remarkably, these considerations alone are sufficient tosuccessfully mimic the flexibility and fluidity of naturalbilayers.

The model we have developed from these simple phys-ical notions resembles in some respects one reported longago by Leibler and coworkers [14]. They similarly con-sidered association of spherical units each representing abilayer patch comparable in size to the membrane’s thick-ness. The anisotropic interactions acting among theirparticles, however, were devised not to reflect pertinentthermodynamic driving forces at this length scale, butinstead to foster formation of fluid elastic sheets. Inour view the potential energy function used in that workwould be difficult to motivate on microscopic grounds,and no attempt was made in Ref. 14 to do so. For thisreason we expect that our approach will generalize morenaturally to describe scenarios that involve membraneproperties beyond long-wavelength fluidity and elastic-ity.

As a more immediate and practical justification for ournew approach, we found that the original implementationof the model, as described in Ref. 14, does not yield stabletwo–dimensional structures. In the Appendix we detaila revised version of that model which is consistent withpreviously reported properties. But even in this case themodel membrane’s bending rigidity is atypically small forbiophysical systems.

MODEL

For a collection of N particles, each representing apatch of lipid bilayer, we adopt the energy function

U = UHC + ǫ

N∑

i=1

Aeq(n(i)eq ) − Apol(n

(i)pol), (1)

where the hard–core potential UHC enforces the con-straint that the separation between any two particles is at

least d. The quantities n(i)eq and n

(i)pol characterize, respec-

tively, the equatorial and polar coordination numbers ofparticle i. The functions Aeq(n) and Apol(n) determinesolvent exposure of these two regions based on their co-ordination. The positive constant ǫ sets the scale of thesesolvent-mediated interactions. Based on the surface ten-sion between water and oil, γ ≈ 50mJ/m

2, and the hy-

drophobic surface area of a membrane patch, A ≈ 60nm2,we expect γA ≈ 740kBT to be an appropriate value forǫ.

In detail, we define the fluctuating density of a parti-cle’s equatorial and polar neighbors as

n(i)eq =

∑

j 6=i

Geq(rij)Heq(z2ij), (2)

n(i)pol =

∑

j 6=i

Gpol(rij)Hpol(z2ij). (3)

The contributions of particle j to the coordination densi-ties of particle i are thus determined both by the distancerij = |rij | between their centers, and by the normalized

projection zij = rij · di/rij of their separation vector rij

onto the axis of particle i (which points along the unitvector di). They are attenuated by the functions

Geq(r) = Gpol(r) =

1, if r ≤ ra,r2

b−r2

r2

b−r2

a

, if ra < r ≤ rb,

0, otherwise,

(4)

and

Heq(z2) = 1 − Hpol(z

2) =

1, if z2 ≤ z2a,

z2

b−z2

z2

b−z2

a

, if z2a < z2 ≤ z2

b ,

0, otherwise,(5)

over scales determined by parameters ra, rb, za, and zb.In this way, particle j counts toward the equatorial (po-lar) coverage of particle i only if it lies near its center andnot too far off the equatorial plane (polar axis).

For a partially occluded object, such as our idealizedmembrane particles when surrounded by neighbors, cal-culating the area accessible to solvent molecules of finitesize is a nontrivial operation. Furthermore, this task isnot sensible to carry out in detail for the reduced rep-resentation we have chosen. Essential features of thefunctions Aeq(n) and Apol(n) are nonetheless straightfor-ward to ascertain: As n increases, exposed area at firstdeclines steadily. At some value n lower than the max-imum n(max) permitted by steric constraints, it shouldnearly vanish, since perfect close-packing is not neededto thoroughly exclude solvent from the bilayer’s interior(or from the polar exterior). For n > n, variation inexposed area should be very weak. We caricature thisdependence with computationally inexpensive, piecewiselinear functions:

Aeq(n) = 1 − n

neq(n < neq), 0 (n ≥ neq), (6)

Apol(n) = 1 − n

npol(n < npol), 0 (n ≥ npol). (7)

Appropriate values of neq and npol will depend on the rel-ative thicknesses of hydrophobic and polar regions, spec-ified in our model by za and zb.

The membrane free energy we have described ismarkedly multi-body in character, but in a way that isboth physically meaningful and simple to understand. Itsparameters correspond intuitively to the geometry andchemistry of constituent molecules. Below we present re-sults of Monte Carlo computer simulations for this model,which employed single particle translations and rotationsas trial moves, as well as shape deformations of the peri-odically replicated simulation box when an external ten-sion was imposed. For these calculations we selected

3

FIG. 1: (a) Illustration of our model. Each particle corre-sponds to a fragment of lipid bilayer, comprised of a cen-tral hydrophobic core and two surrounding hydrophilic layers.The unit vector di specifies the orientation of particle i. Alsoshown are the spatial regions used to compute the numbersof equatorial (enclosed by dashed line) and polar (enclosed bydotted line) neighbors of particle i. (b) Random initial con-figuration for a trajectory of N = 864 particles. As time pro-gresses, particles quickly form two–dimensional patches thatcontinue to coarsen. Shown are snapshots after 100 (c) and1000 (d) Monte Carlo sweeps.

ra/d = 1.3, rb/d = 1.7, z2a = 0.05, z2

b = 0.2, neq = 5 andnp = 1, which yield a rigidity typical of biological mem-branes. By varying these values, it is possible to tune theelasticity of the assembled bilayer, as well as more subtleproperties like internal viscosity or rupture tension.

RESULTS

We first demonstrate that a sheet–like configurationis indeed the equilibrium state of our model at finitetemperature. Fig. 1 shows snapshots from a MonteCarlo trajectory in which an initially disperse collectionof membrane particles spontaneously organizes into two–dimensional structures that then diffuse and coalesce [22].This coarsening process is expected to proceed until onlya single membrane sheet remains in the simulation box.A free boundary in the resulting structure can be avoidedeither by forming a membrane sheet that spans the pe-riodically replicated simulation box, or by adopting aboundary–free geometry such as a spherical vesicle.

We next provide evidence that the elastic propertiesof our model match quantitatively those of natural lipidbilayers. Specifically, on length scales well beyond a par-

z2

a z2

b κ/kBT

0.01 0.2 126.8±0.8

0.02 0.2 63.4±0.5

0.05 0.2 25.2±0.1

0.1 0.2 12.1±0.1

0.4 0.6 2.1±0.03

TABLE I: Computed bending rigidities for different values ofthe model parameters z2

a and z2

b .

ticle radius d (corresponding to the bilayer’s thickness),its shape fluctuations are well described by the Helfrichmodel of incompressible fluid elastic sheets [10, 11], withmacroscopic material properties in accord with experi-mental measurements. In Helfrich’s model, a nearly flatsegment of membrane exhibits a fluctuation spectrum

⟨

∣

∣

∣hq

∣

∣

∣

2⟩

=kBTA

σq2 + κq4. (8)

Here, hq =∫

Ah(x) exp (−iq · x) dx is the Fourier trans-

form of the membrane height h(x), q is a two-dimensionalwavevector conjugate to the Cartesian position x on aflat reference membrane with area A, and 〈.〉 denotesthe equilibrium average. For lipid mixtures common incell membranes, the bending rigidity κ ranges from 10to 30 kBT [15]. To the extent that the membrane’s areaper molecule is constant, the surface tension σ plays therole of minus chemical potential and increases roughlylinearly with applied lateral tension τ .

For the purpose of estimating the normal mode fluctua-tions of Eq. 8 from simulation, we wish to avoid imposinglateral tension (requiring τ = 0). As a practical matter,however, it is convenient in this calculation to prescribea fixed box geometry (and thus a fixed set of wavevec-tors). We began by performing simulations in which thebox size was allowed to fluctuate at zero lateral tension.The resulting average box size was then adopted as aconstraint for a second set of simulations, in which wecomputed the distribution of height fluctuations [23]. Atlarge length scales we indeed observe the characteristic⟨

∣

∣

∣hq

∣

∣

∣

2⟩

∝ q−4 behavior predicted by the Helfrich model,

as shown in Fig. 2. The computed proportionality coef-ficient indicates a bending rigidity of κ ≈ 25kBT , wellwithin the range of measured values [15]. By manipulat-ing the parameters z2

a and z2b , we are able to reproduce

the elastic behavior of membrane sheets over a wide rangeof bending rigidities (Table I).

The advantage of our model lies in a facile ability toaddress mesoscale response without sacrificing the mi-croscopic basis of corresponding fluctuations. As a rep-resentative biophysical example that calls for these ca-pabilities, we considered the resistance of a fluctuatingmembrane to impingement of a nanorod oriented perpen-dicular to the lipid bilayer. Experimental realizations of

4

10−2

100

102

10−1

100

<|h

q|2

>/A

d4

qd

FIG. 2: Spectrum of height fluctuations around a flat refer-ence state, equilibrated at zero lateral tension. The dashedline is a fit to the expected behavior (8) with σ = 0, yieldinga bending rigidity κ = 25.2kBT .

this situation include extension of polymerizing actin fil-aments close to a cell membrane [16] and external forcingof a carbon nanotube against a cell wall [17].

Insets of Fig. 3 depict the reversible membrane defor-mations we have studied in this context. The proddingnanorod is modeled here as a volume-excluding, rigidspherocylinder of radius R = 3d, which defines a regionin space inaccessible to the membrane particles. Its ver-tical displacement l determines the size of the membranedeformation. We set l = 0 for a nanorod that would con-tact the membrane, in a completely flat configuration, ata single point. To prevent global translation of the mem-brane when l > 0, we constrain the vertical positions of asmall number of membrane particles. This pinning couldbe viewed as a mimicry of cytoskeletal attachments thatwould suppress overall translations in a living cell [12].

To calculate membrane–induced forces on the nanorod,we treat the rod height l as a dynamical variable subjectto an external potential, V (l) = (1/2)K (l − l0)

2, in addi-

tion to the fluctuating constraints imposed by membraneparticles. For several values of l0 we computed the forceon a protrusion of length l from simulations as

f(l) = K

〈l〉 − l0 −

⟨

(δl)2⟩

− kBT/K⟨

(δl)2⟩

(

〈l〉 − l)

, (9)

which follows from an expansion of the membrane freeenergy to quadratic order in l around l.

Force–extension curves for this extrusion process areplotted in Fig. 3 for three different values of τ . Therestoring force initially increases in proportion to filamentlength, and ultimately reaches a plateau value at large l.This limiting force is in good agreement with the con-stant force 2π

√2κσ predicted by the Helfrich model for

the stretching deformation of an elastic cylinder [18, 19],provided that σ is dominated by lateral tension. Simi-lar agreement is found for the diameter of the membrane

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 10 20 30 40 50

f[ǫ

/d]

l [d]

FIG. 3: Restoring force as a function of protrusion lengthfor three different values of the lateral tension τ : 0.0002ǫ/d2

(squares), 0.0005ǫ/d2 (circles), and 0.001ǫ/d2 (triangles). Foreach value of τ and l0 we performed computer simulations attwo values of K (0.2 and 1 in units of ǫ/d2). The force wasevaluated using (9) at a length l corresponding to the arith-metic mean of the two average lengths 〈l〉 . The solid linesshow the result 2π

√2κσ expected for long cylindrical protru-

sions in the limit of zero temperature with σ = τ [18, 19].The inset shows cutaway views of typical configurations atτ = 0.0005ǫ/d2 and l0 = 17.5, 29.5, 41.5 (left to right), illus-trating the transition from a global deformation to a localized,tubular protrusion. In our calculations the nanorod is placedat the center of the periodically replicated simulation box,and a small number of membrane particles are immobilizedat the boundary to avoid overall membrane translation.

tubule (data not shown). At intermediate extensions,f exhibits a local maximum, signaling a mechanical in-stability associated with the transformation between twodifferent classes of membrane configurations. This ob-servation is consistent with recent experiments on theformation of membrane tethers from giant vesicles [20]and conclusions drawn from continuum calculations [18].

OUTLOOK

In addition to mechanical responses exemplified hereby nanorod protrusion, several other biologically relevantmembrane processes could addressed through straight-forward extensions of our model. For example, lateraldemixing of multicomponent lipid bilayers could be stud-ied by endowing each membrane particle with a degree offreedom representing the local composition. The organi-zation of embedded proteins could be simulated using adescription of the macromolecule that is compatible withour membrane model. Studying time-dependent behav-ior would require in addition a set of dynamical rulesfor updating particle arrangements. Most simply, the

5

Monte Carlo trajectories we have used here to sampleequilibrium configurations could be interpreted as an ap-proximation of physical time evolution. Integrating equa-tions of motion that involve derivatives of potential en-ergy would require smoothed versions of the interactionswe have described. Already, the mesoscale realism andcomputational economy of our approach recommends itsuse for examining a wide range of lipid bilayer fluctuationphenomena beyond the reach of molecular models.

We thank Anthony Maggs for helpful discussions of thework presented in Ref. 14. This work was supported bythe Director, Office of Science, Office of Basic Energy Sci-ences, of the U.S. Department of Energy under ContractNo. DE-AC02-05CH11231.

APPENDIX

The model proposed by Drouffe and coworkers inRef. 14 operates on the same lengthscale and focuses onthe same degrees of freedom as the model presented inthis work. It employs an energy functional of the form

U = UHC + Uan + Uden. (10)

The first term is a pairwise repulsive energy that penal-izes overlap of different particles, which we treat as hardspheres. The second term is a pairwise anisotropic inter-action energy,

Uan = ǫ∑

i<j

B(rij){

η (di · dj)α + g

(

z2ij

)

+ g(

z2ji

)

}

,

(11)where B(rij) is a positive weight function that limits therange of the interaction to ≈ 2d, η = 1 in Ref. 14, α ∈{1, 2} depending whether the particles are symmetric,and

g(x) = 0.75x3 + 0.25x + 0.8 (12)

is a monotonically increasing function over the range ofits argument x ∈ [0, 1].

The last term in (10) is a multibody potential thatfavors configurations in which each particle is surroundedby 6 nearest neighbors,

Uden = ǫ∑

i

C(ρi), (13)

where C(ρi) = (ρi − 6)2, ρi =∑

j 6=i h(rij), and h(rij)is a weight function with unit value for small particleseparations.

To implement this model one must specify the depen-dence of B(r) and h(r) on inter-particle distance, whichwas done only graphically in Ref. 14. We assign these

quantities simple functional forms that are piecewise lin-ear in r2,

B(r) =

1.5, if r ≤ 1.85,1.5(2.02−r2)2.02−1.852 , if 1.85 < r ≤ 2.0,

0, otherwise,

(14)

h(r) =

1, if r ≤ 1.6,1.952−r2

1.952−1.62 , if 1.6 < r ≤ 1.95,

0, otherwise.

(15)

Values of B and h resulting from this choice closely re-semble those plotted in Ref. 14.

We performed Monte Carlo computer simulations ofthe model defined by (10)–(15) for α = 1 at a tem-perature T = 1.6ǫ. For these conditions, Drouffe andcoworkers report the spontaneous assembly of randomlydistributed and oriented particles into a two-dimensionalsheet, in which the orientation vectors of neighboringparticles are nearly parallel. However, in our simula-tions we find that for these parameter values the systemforms many small clusters of particles without apparentinternal order. Similar configurations are obtained whena completely ordered sheet is used as the initial condi-tion (Fig. 4a), which indicates that the inability to formstable membranes is thermodynamic in origin, and notdue to dynamical artefacts caused by the Monte Carlomethod.

The lack of stable membrane-like structure in this pub-lished version of the model is straightforward both to un-derstand and to remedy. When η = 1, the first term inthe brackets of Eq. (11) actually disfavors parallel align-ment of neighboring particles’ orientational vectors; thisenergy contribution is minimum when axes of adjacentparticles are antiparallel (in the case α = 1) or perpen-dicular (α = 2). Suspecting a typographical error inRef. 14, we set η = −1 in order to instead favor the de-sired alignment. Equilibrium states resulting from thismodification, however, still feature a collection of manysmall aggregates of particles at the temperature T = 1.6reported in Ref. 14. In this case the adhesive interac-tion (13) is insufficient to overcome the loss in entropyassociated with forming a single membrane sheet. Onlyafter reducing the temperature to T = 0.5ǫ were we ableto observe stable sheet-like structures and vesicles in oursimulations (Fig. 4b). This modified set of model param-eters results in membrane sheets with a bending rigidityκ = 4.9± 0.1kBT , which is smaller than values measuredfor biophysical phospholipid bilayers [15].

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6

FIG. 4: (a) Typical configuration of the membrane modeldeveloped by Drouffe and coworkers (α = η = 1, T = 1.6ǫ).A flat sheet of 256 particles with parallel orientation vectorswas used as the initial configuration. This structure dissolvesquickly into a disordered collection of small clusters. (b) Finalconfiguration of a trajectory starting from the same initialstate, but employing an alternative parameter set (α = 1,η = −1, T = 0.5ǫ). The initially flat sheet remains stable,and closes to form a vesicle.

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nated by entropic effects, yielding a nearly temperature–independent reduced bending rigidity κ/kBT . Transla-tion and rotation of finite membrane patches, on theother hand, proceed by displacements of individual parti-cles, which are suppressed at low temperatures. To accel-erate the self–assembly process the trajectory shown inFig. 1 was generated at a higher temperature ǫ = 25kBT .

[23] To compute the fluctuation spectrum of a mem-brane that spans the simulation box in the xy–plane we define the height function h(x, y) =P

N

i=1ziΘ(xi − x)Θ (yi − y) /

P

N

i=1Θ (xi − x)Θ (yi − y)

where Θ(x) = 1 if |x| ≤ ∆/2 and zero otherwise. Eval-uating this function for points (x, y) that form a squaregrid of resolution ∆ and taking the two–dimensionaldiscrete Fourier transform, we obtain the Fourier am-plitude hq . This procedure generates a systematic biasat large wavevectors [21]. Taking this bias into accountdoes not change the value of κ for our model.