Vesicle deformations by clusters of transmembrane proteins

0.46 MB·English

by  Amir Houshang Bahrami

Vesicle deformations by clusters of transmembrane proteins
Amir Houshang Bahrami1 and Mir Abbas Jalali1,a)
Department of Mechanical Engineering, Computational Mechanics Laboratory
Sharif University of Technology, Azadi Avenue, Tehran, Iran
(Dated: 1 February 2011)
We carry out a coarse-grained molecular dynamics simulation of phospholipid vesicles with transmembrane
proteins. We measure the mean and Gaussian curvatures of our protein-embedded vesicles and quantita-
tively show how protein clusters change the shapes of their host vesicles. The effects of depletion force and
vesiculation on protein clustering are also investigated. By increasing the protein concentration, clusters are
fragmentedtosmallerbundles,whicharethenredistributedtoformmoresymmetricstructurescorresponding
1
to lower bending energies. Big clusters and highly aspherical vesicles cannot be formed when the fraction of
1
protein to lipid molecules is large.
0
2

n
I. INTRODUCTION II and model the vesicle surface using sphericalharmon-
a
ics. In section III, we present a method for computing
J
9 Biological membranes are found in various com- the mean and Gaussian curvatures at the locations of
2 plex shapes1,2 which are closely related to membrane hydrophilic heads of lipid chains and proteins. The lo-
functions such as biconcave shape of erythrocytes or cal curvature information together with the sizes of pro-

] corkscrew shape of spirochetes. Shape variety depends tein clusters help us to investigate the deformations of
t
f mainly on protein concentration, operation of external host vesicles in section IV. We also study the size distri-
o forces on membranes in cellular environments3, mem- bution, formation and fragmentation of protein clusters.
s
We summarize our fundamental results in section V.
. brane movement, fusion and budding processes, varia-
t
a tionsinlipidcomposition,andvesicletrafficking4. Mem-
m
brane local curvature is representative for its shape,
- and proteins are believed to play an important role II. MODEL DESCRIPTION
d
in membrane conformation. Proteins behave as both
n
generating3,4 and sensing5 elements for the membrane
o Mesoscopicmodelshavebeenwidelyusedtostudythe
curvature4,6. While protein aggregation induces shape
c physics of membranes16. Lipids can spontaneously ag-
[ transformation, membrane curvature may also generate
feedback on protein aggregation and yield attractive6,7 gregate and form various membranes19,20. When lipid
1 chains are assembled in the form of a closed 3D surface
or repulsive6,8 curvature-mediated interactions between
v and trap water molecules, a vesicle is generated. En-
8 them. It has also been shown that bending rigidity and tropy is the main driving mechanism of this process18.
8 membrane thickness affect protein functioning9,10.
To construct vesicles, we insert initially flat rectangu-
6 Proteins affect membrane curvature through various
lar lipid bilayers (which may contain proteins) inside a
5 ways such as scaffold and local curvature mechanisms,
. box of water molecules. Such a configurationis unstable
1 and by their integration (as transmembrane proteins)
because the tails of boundary lipids are repelled by wa-
0 with the membrane3,4,6,11. However, the experimental
ter molecules and the bilayer is compressed by in-plane
1 measurements of variations in the membrane curvature
1 forces. Consequently,thebilayerbucklesandclosesitself
are not easy. The role of proteins on the membrane
: to acquire a minimum potential energy state. Vesicles
v deformations has been studied using continuum elastic
formedthroughthis bilayer vesicle transitionprocess,
i modeling6,8,12, particle-based13 and mesoscopic14 simu- →
X with the progenitor bilayer being surrounded by solvent
lations, and hybrid elastic-discrete particle models15. In
r particles(withouttouchingsimulationboundaries),have
all these studies, lipid bilayers represent a liquid envi-
a amorerelaxedpressuredistribution. Furthermore,using
ronment with freely diffusing lipid chains that dissolve
bilayer vesicle transition process we obtain vesicles of
topological deformations. The aggregation of somewhat →
different sizes in a more controllable process.
rigid proteins remarkably reduces the diffusion of lipid
Our simulation box has dimensions of L L L
chains and causes shape variations. x × y × z
with the x-axis being normal to the initial bilayer mid
In this study, we use a coarse-grained model16 and
plane. We use periodic boundary conditions, and choose
generalize the method of Markvoort et al.18 to generate
asufficiently largeboxso thatthe bilayerdoesnotreach
phospholipid vesicles from initially rectangular and flat
to the boundaries before vesiculation. The number of
bilayers with different concentrations of transmembrane
protein and lipid molecules are denoted by N and N ,
proteins. We discuss our simulation method in section p l
respectively. The particles that constitute the elements
of our setups either are water particles (type 1), or have
hydrophobic (type 2) and hydrophilic (type 3) natures.
We follow reference [19], and model each lipid chain by
a)Electronicmail: [email protected] one hydrophilic head and four hydrophobic tail particles
2
(a) (b) tweenneighboringstrands(inaproteinmolecule)provide
enough bending rigidity for our relatively short protein
molecules.
We implement equilibrium molecular dynamics simu-
lation of an NVT ensemble17 with velocity Verlet algo-
rithm for integration in the time domain. We use the
integration time step δt=0.005t with
0
mσ2 1
t = = , k =1, T =1.35, (3)
0 B
s48kBT √64.8
FIG. 1. (a) A flexible lipid chain with one hydrophilic head
(blue) and four hydrophobic tail particles (red). (b) Protein andsetthenumber densityofparticles19 to ρ=2/3. All
molecules consisting of seven strands. Each strand has two
Particleshaveequalmassesofm=1,andalllengthsare
hydrophilicends(yellow particles) and fivehydrophobicpar-
scaledbyσ. Theparametersusedinourmodelsproduce
ticles in the middle (green). Particles of neighboring strands
correct physical properties of bilayers, including diffu-
are connected byelastic rods (black lines).
sion coefficients, density profiles and mechanical proper-
ties like surface tension and stress distribution18,19. Af-
tervesicleformation,thepositionvectorsofparticlesare
(Fig. 1a). All particles interact through Lennard-Jones measuredwithrespecttothevesiclecenter,andthelipid
(LJ) potential17 with a cut off radius r =2.5σ. or protein heads exposed to water molecules outside the
c
We use the integer subscripts i and j for particle vesicle are tagged as surface particles. We assign an in-
types, and index a particle in a molecule by the sub- teger number n to each surface particle, and denote the
script s. For the pairs (i,j) = (1,2) and (2,3) the inter- total number of surface particles by N. It is remarked
actionpotential is V =4ǫ (σ /r)9, andfor other pairs that there is not a meaningful correlation between the
ij ij ij
we use U = 4ǫ [(σ /r)12 (σ /r)6] where σ ,ǫ =1 physical location of each particle and its number n. The
ij ij ij ij ij ij
(i,j =1,2,3)19. A harmonic−bond potential of the form identifiernis usedonlyforstatisticalpurposes. Further-
Ubond = K (r σ)2 is applied between neigh- more, a single number n is assigned to each surface par-
s,s+1 b | s,s+1| − ticle, i.e., there are, respectively, one and seven particle
boring particles inside a lipid chain, and we have set
K = 5000ǫ σ−2 where ǫ = 1. With these as- identifiers corresponding to each lipid chain and protein
b s,s+1 s,s+1
sumptions,lessthan10%ofbondlengthesfluctuatemore molecule.
than 2% around σ [19]. In some case studies, to include
bending rigidity in our lipid and protein chains, we use
the potential III. LOCAL MEAN AND GAUSSIAN CURVATURES
Ubend =K 1 rs−1,s.rs,s+1 , (1) To measure the local curvature of vesicles, with and
s−1,s,s+1 bend(cid:18) −|rs−1,s||rs,s+1|(cid:19) without proteins, we express the radial distance of sur-
=Kbend(1 cosφs), (2) faceparticlesfromthevesiclecenterintermsofspherical
−
harmonics21 as
where K is the bending coefficient of non-flexible
bend
chains19. φs defines the bending angle between adjacent lmax l
bonds in a single chain. Protein molecules (Fig. 1b) are r(θ,φ)= [amAm(φ,θ)+bmBm(φ,θ)], (4)
l l l l
composedofsevenstrandsinahexagonalarrangement14. l=0 m=0
X X
A particle in each strand interacts with all of its neigh-
where
bors, within the same protein, through the potentials
Usb,osn+d1 and Usb−en1d,s,s+1 defined above. In this way, lipid- Am =Re[Ym], Bm =Im[Ym], (5)
protein interactions are the same as lipid-lipid interac- l l l l
tions. (2l+1)(l m)!
Ym =( 1)meimφ − Pm(cosθ). (6)
In this study, we work with simple proteins whose l − s 4π(l+m)! l
lengths are adjusted in a way that the hydrophobic mis-
match effect is minimum, and such that a single protein HerePmareassociatedLegendrefunctionsandi=√ 1.
doesnotaffectthemembraneconformationconsiderably. l −
We find the coordinates of surface particles (r ,φ ,θ ),
n n n
Shape variationsthat we report,are thus causedonly by
define
clustering. Both flexible and rigid chains can be used
in lipid and protein molecules. The experiments of sec- R= r (φ ,θ ) r (φ ,θ ) ... r (φ ,θ ) T,(7)
1 1 1 2 2 2 N N N
tion IV show that adding bending rigidity does not in-
duce remarkable qualitative or quantitative changes in and colle(cid:2)ct all constants coefficients am and bm in(cid:3)single
l l
the protein tilting angle or in the aggregation products. column vectors a and b, respectively. The superscript T
Infact,thestrongharmonicbondpotentialsassumedbe- means transpose. We also define the matrices A and B
3
whose elements are Am(φ ,θ ) and Bm(φ ,θ ), respec-
l n n l n n TABLE I. The protein and lipid content, and averaged cur-
tively. The discrete form of equation (4) thus becomes
vatures of the simulated vesicles V –V . The quantity p¯ has
1 4 b
R=Y x, Y= A B , x= a , (8) been computed for thebilayers B2–B4.
· b
(cid:26) (cid:27) V1 V2 V3 V4
which in practice, has mo(cid:2)re equat(cid:3)ions than unknowns. Nl 2300 2120 1940 1760
We calculate x using the singularvalue decomposition Np 0 20 40 60
ofY,andobtainthepositionofanysurfaceparticlefrom H¯p - 0.0460 0.0381 0.0418
H¯ 0.0564 0.0549 0.0582 0.0567
(4) and l
H¯ 0.0564 0.0541 0.0544 0.0527
r=r sin(θ)cos(φ)ˆi+sin(θ)sin(φ)ˆj+cos(θ)kˆ , (9) H˜ 0.0068 0.0135 0.0155 0.0158
K¯ 0.0030 0.0029 0.0029 0.0027
where (ˆi,ˆjh,kˆ) are the unit vectors in Cartesian cioordi- K˜ 0.0014 0.0015 0.0015 0.0015
nates. Letusdefiner andr asthefirstorder,andr , p¯ - 1.66 5 4.61
φ θ φφ
r , and r as the second order partial derivatives of r p¯b - 1.53 2.66 3.52
φθ θθ
in(9)withrespecttoφandθ. Thecoefficientsofthefirst Qc 0 3 5 8
Q 0 9 3 5
fundamental form of the surface are thus determined as f
p(1) - 6 14 13
E =r r , F =r r , G=r r . (10) p(2) - 3 11 11
φ φ φ θ θ θ
· · ·
p(3) - 2 5 11
Defining the unit vector normal to the vesicle surface as
p(4) - - 5 6
nˆ = (r r )/r r , one finds the coefficients of the
φ× θ | φ× θ| p(5) - - 2 6
second fundamental form:
p(6) - - - 4
L=r nˆ, M =r nˆ, N =r nˆ. (11) p(7) - - - 2
φφ φθ θθ
· · · p(8) - - - 2
ThemeancurvatureH andtheGaussiancurvatureK at
thelocationofeachsurfaceparticlecanthusbecomputed
using22
LN M2 EN 2FM +GL A. Protein-embedded vesicles
H = − , K = − . (12)
EG F2 2(EG F2)
− − We replace some lipid chains of our initially flat bi-
The principal curvatures (C ,C ) are related to H and
1 2 layerby proteinmolecules while keepingthe areaofgen-
K through H = (C +C )/2 and K = C C 22. In our
1 2 1 2 eratingbilayer almostconstant. We randomly distribute
numerical experiments we have truncated the series (4)
proteins in the bilayer sheet, but observe that they form
at l = 5. Including l > 5 terms had 3% im-
max max ≈ smallclustersafterthe bending ofmembraneandduring
provementinfractionalerrors. Toperformaglobalshape
vesicle formation. The vesiculation process takes from
classification of vesicles, we compute the average curva-
5 105δt for model V to 2 106δt for protein-
tures ≈ × 1 ≈ ×
embedded vesicles. That is because proteins (or clus-
H¯ 1 N H(n) tersofproteins)resistagainstbucklingbydecreasingthe
= , (13)
K¯ N K(n) fluidity of the progenitor membrane. After vesicle for-
(cid:26) (cid:27) nX=1(cid:26) (cid:27) mation, we have waited for about 5 105 time steps to
and their corresponding standard deviations H˜ and K˜ ensure that vesicles have reached to ×an equilibrium con-
for particles living on the surface of model vesicles. dition so that the number and area of protein clusters
remain constant.
Wehaveconstructedthreevesicleswithinitialbilayers
IV. SIMULATION RESULTS
of approximately equal surface areas and different pro-
tein concentrations. We name these vesicles V , V and
2 3
We first consider the case of lipid and protein chains V , which have been formed inside N 161000 water
4 w
≈
withnobendingrigidity. Ourfirstmodel,whichiscalled molecules. The numbers of lipid and protein molecules
V1, is a vesicle without proteins and it is composed of in our models have been givenin Table I. Fig. 2 demon-
Nl = 2300 similar lipid chains. The initial flat bilayer stratesthreedimensionalviewsofthesevesiclesandtheir
is bent inside the water(solvent) particles andgradually cross sections with the largest diameter so that the ma-
formsanalmostsphericalvesicleshownintheleftpanels jorityofclustersarevisualized. Wehavealsoshownsome
ofFig. 2. To measurethe sphericity ofthis vesiclequan- waterparticlesinsideandoutsidevesicles. Itisseenthat
titatively, we use a spherical harmonics expansion with larger protein clusters have induced lower curvatures in
lmax = 5, and fit a 3D surface to the surface particles. their neighborhood, and consequently, prominent defor-
The mean local curvature H(n) is computed from (12) mations in their host vesicles.
and used in (13) to find H¯ = 0.0564 and H˜ = 0.0068. We also investigate several protein-embedded flat bi-
ThesmallvalueofH˜ comparedtoH¯ confirmsthespher- layerswheretheaggregationofproteinsiscausedmainly
ical nature of vesicle V . by the depletion force. Comparing the sizes and popula-
1
4
FIG. 2. Three dimensional views (top row) and the most informative cross sections (bottom row) of our simulated vesicles
without (model V ) and with transmembrane proteins. From left to right: models V , V , V and V .
1 1 2 3 4
dinates (r ,φ ,θ ) for n=1,2,...,N. We have plotted
n n n
H and K (µ l,p) in Fig. 4 for models V –V . The
µ µ 1 4
≡
curvatures at different surface points have been marked
bydifferentsymbolsandcolors. Thescatteredplotspro-
videaquantitativeinsighttotheeffectofproteinsonthe
vesicle shape. It is seen that the fluctuations of H are
higherinmodelswithproteins. Aswe mentionedbefore,
there is no correlation between the location of particles
and their corresponding identifier n. Therefore, in Fig.
4, points with close values of n are not close physically.
FIG. 3. Top views of ourflat bilayers B3 (left panel) and B4 The oscillatory behavior of the graphs could change by
(right panel) with transmembrane proteins. The bilayers B3 renumbering the surface points but the major minima
and B have the same number of proteins and areas of the
4 alwayscoincide with protein clusters,which flatten their
vesicles V and V , respectively.
3 4 host vesicles locally.
We define H¯ and H¯ as the averages of mean curva-
l p
tures (taken over the particles of the same kind) corre-
sponding to lipid and protein heads, respectively. The
tion of clusters formed in bilayers and vesicles helps us average mean curvature H¯ for all surface particles (pro-
tobetterunderstandtherolesofentropy-andcurvature- tein and lipid heads) and its corresponding standardde-
driven aggregation of proteins during vesiculation. We viation H˜ have been given in Table I together with H¯
l
simulate three flat bilayers, which extend to the sides of and H¯ . Lower values of H¯ for models V –V confirm
p 2 4
the simulation box. The surface areas of these bilayers
that transmembrane proteins reduce the averaged mean
and the population of their proteins match those of the
curvature through creating low-curvature clusters. Dur-
vesicles V –V . We label these bilayers as B , B and
2 4 2 3 ing vesiculation, protein molecules aggregate and form
B . They remain in a flat equilibrium state because of
4 clusters. The number of clusters and the population of
periodic boundary conditions that keep them in touch
proteins in each cluster are determined by (i) the ran-
with the sides of the simulation box. Fig. 3 displays the
dom arrangement of proteins in the generating bilayer
snapshots of B and B .
3 4 membrane (ii) the interplay between depletion force and
We split the molecules of the outer layer of vesicles curvature induced interactions (iii) system temperature
into two groups of lipids and proteins, and denote by and the concentration of protein molecules. If protein
H (n) and H (n) the mean curvatures at the locations molecules are scattered in a vesicle, the membrane will
l p
of lipid and protein heads, respectively. We track only maintain its sphericity. However, if the same number of
initially tagged particles living on the outer surface of proteins build a single cluster, the highest variation in
vesicle (because the probability of flip-flop motions is vesicle shape will be observed. In our simulations we
low) and compute the local curvature having their coor- have not seen these two extreme cases. Several clusters
5
FIG. 4. Scattered circles and squares mark, respectively, the mean curvature H (left panels) and the Gaussian curvature K
(rightpanels)atthelocation ofproteins(filledredcircles) andlipidheads(bluesquares). TheaveragedcurvaturesH¯p andH¯l
as well as the average Gaussian curvature K¯ and its standard deviation K˜ have been given in Table I. From top to bottom:
models V , V , V and V . The horizontal axis variable is the normalized real numberq=n/N for thenth surface particle.
1 2 3 4
with different populations ofproteins areusually formed H˜ is 10%of H¯ for the near-sphericalvesicle V , but it
1
≈
in our vesicles. increases remarkably for protein-embedded vesicles. As
clustersgrow,thecurvatureassociatedwiththeensemble
of lipids increases,and the vesicle becomes more aspher-
ical. This can be understood from the larger value of
B. Curvature-mediated clustering
H¯ in vesicle V . A reverse phenomenon is also possible:
l 3
ifduringthe vesicleformationthe curvaturedecreasesin
Let us denote the number of scattered free proteins of
certainregions,proteinswillmigratethereandformlow-
a model by Q , and the number of its clusters by Q .
f c curvature clusters. To show the correlation between the
Moreover, we indicate by p(m) the number of proteins numberofproteinsinclustersandH¯,wehavecomputed
in the mth cluster. For models V , V and V , we have
2 3 4 the quantity
reported the values of Q , Q and p(m) in Table I. It is
f c
seen that the biggest cluster have been formed in vesicle
1 Qc
VH¯3painsdlonwoetriinnVv4e,swichleicVh3htahsamnVor4e.pTrhoetesitnasn.dCaorndsdeeqvuieanttiolyn, p¯= Qf +Qc "Qf +m=1p(m)#, (14)
X
6
TABLEII.Thenumberandsizes of protein clustersfor vesi-
clesV4–V6 withNp =60anddifferentinitialrandomarrange-
mentsof proteins.
V V V
4 5 6
p¯ 4.61 4.61 4.61
Qc 8 9 9
Q 5 4 4
f
p(1) 13 14 14
p(2) 11 11 9
p(3) 11 8 8
FIG. 5. Three dimensional views of dissociated clusters in
p(4) 6 7 7
p(5) 6 6 6 vesicle V7 with Np =20 protein molecules initially located in
a single big cluster.
p(6) 4 3 5
p(7) 2 3 3
p(8) 2 2 2
p(9) - 2 2
C. Convergence tests
and given its magnitude in Table I. We also define p¯
b
using a formula similar to (14), but for our model bilay- We have continued our simulations until the size and
ers B that correspond to vesicles V (i = 2,3,4). The number of clusters become constantin a relaxedequilib-
i i
computed values of p¯ are given in Table I. riumstate. Toassurethatsimulatedvesiclesareinequi-
b
librium, we have carriedout various experiments. In the
Since bilayers remain flat during simulation, the ef- firstexperimentwegeneratedtwovesiclesfromthe same
fect of membrane curvature on the aggregation process progenitor bilayer of V , but using two different sets of
4
is ignorable and p¯ indicates the contribution of the de- randomly distributed proteins. We have given the prop-
b
pletion force to cluster formation. Comparingthe values ertiesofnewvesiclesV andV inTableII. Althoughthe
5 6
of p¯ and p¯ clearly shows that curvature induced inter- vesicles V , V and V start from different initial condi-
b 4 5 6
actions, during vesiculation,facilitate the cluster growth tions, there are minor differences between the properties
and we get p¯>p¯ . Moreover,p¯ is a monotonic function of their clusters. Notably, they posses the same shape
b b
ofN whilep¯isnot. Thereasonisthelackofaneffective indicator p¯=4.61.
p
fragmentation mechanism in flat bilayers: larger protein
concentrationsalwaysleadtobiggerclusters. Invesicles, To demonstrate that we usually reacha physical equi-
however,thetendencytoformastructurewithminimum librium and not a kinetically trapped state, we designed
bendingenergyleadstothefragmentationofbigclusters a second experiment and produced a vesicle from an ini-
at the turning (maximum) point of the function p¯(Np). tial bilayer where Np=20 proteins (similar to vesicle V2)
had formed aninitial big cluster. In the resulting vesicle
Forallsurfacepointsincludinglipidandproteinheads, V , the proteins of the initial single cluster are dissoci-
7
we have also calculated (see Table I) the average Gaus- atedintothreesmallerseparateclustersasshowninFig.
sian curvature K¯ and its standard deviation K˜ using 5. This result is, again, consistent with the general fea-
(12) and (13). Right panels of Fig. 4 show the dis- tures of V , which had been obtained from a completely
2
tribution of K for lipid and protein heads. Based on different initial condition. It is worth noting that we
the Gauss-Bonnet theorem, any compact manifold have observed a transient interplay between clustering
M
withoutboundary,istopologicallyequivalenttoasphere and fragmentation well before reaching the equilibrium
and the surface integral K dA ofGaussiancurvature state.
M
will be invariant. Since our vesicles become aspherical
R
through the shape transformations induced by protein We have repeated our simulations with ρ = 0.8 and
clustering, Gauss-Bonnet theorem applies and all mod- withnon-flexiblelipid andproteinchains. Usinga bend-
els V –V must be topologically equivalent. Given that ing stiffness K = 5 in lipids and K = 80 in pro-
1 4 bend bend
the head groups of proteins and lipids are identical in tein strands, the bilayer vesicle transition is slowed
→
our simulations and the areas of generating bilayers are down but we do not observe considerable change, either
almost the same, the surface integral will be approxi- quantitativeorqualitative,intheclusteringphenomenon
mately equal to AK¯. Data in Table I show that both and the shape transformation of vesicles: the numbers
K¯ and K˜ remain invariant (from one vesicle to another) and sizes of final clusters and the shape parameter p¯are
within a reasonable error threshold. This confirms the similar in allmodels. By making stiffer molecular chains
self-consistency of our models and the results obtained and increasing the density, lipid diffusion is decreased,
from spherical harmonic expansions. which in turn, yields a longer relaxation time.
7
V. CONCLUSIONS 2U.Seifert,Advances inPhysics,46(1),13(1997).
3J. Zimmerberg, and M.M. Kozlov, Nature Rev. Mol. Cell Biol.,
7,9(2006).
We have studied the phenomena of protein clustering 4H.T.McMahon,andJ.L.Gallop,Nature,438,590(2005).
and membrane shape transformation during membrane 5V.Vogel,M.Sheetz, NatureRev.Mol.CellBiol.7,265(2006).
vesiculation and afterwards. Comparing relaxed vesicles 6R.Phillips,T.Ursell,P.Wiggins,andP.Sens,Nature,459,379
and bilayersshows that proteinclustering during vesicu- (2009).
7B.J.Reynwar,G.Illya,V.A.Harmandaris,M.M.Mu¨ller,K.Kre-
lation occurs due to both entropy-driven depletion force
mer,andM.Deserno,Nature,447,461(2007).
and the curvature-mediated interactions. The latter ef- 8K.S.Kim,J.Neu,andG.Oster,Biophys.J.,75,2274(1998).
fect enhances the generation of larger protein clusters 9O.S.Andersen,andR.E.Koeppe, Annu. Rev. Biophys.Biomol.
anddetermines bilayer’sbending rigidity. Once the vesi- Struct.,36,107(2007).
cle is formed, protein clusters locally flatten their host 10M.O. Jensen, and O.G. Mouritsen, Biochim. Biophys. Acta,
vesicles and increase the bending energy as H¯ and H¯ 1666,205(2004).
l 11M.Kozlov,Nature,463,439(2010).
increase. The system, however, cannot tolerate the in- 12T.R. Weikl, M.M. Kozlov, and W. Helfrich, Phys. Rev. E, 57,
crease in the bending energy for protein concentrations 6988(1998).
beyond a critical value. By increasing the protein con- 13D.E.Chandler,et.al.,Biophys.J.,95,2822(2008).
centration, bigger protein clusters are not formed in our 14M. Venturoli, B. Smit, M.M. Sperotto, Biophys. J., 88, 1778
(2005).
simulations, or they break apart. We anticipate a uni-
15A. Naji, P.J. Atzberger, and F.L.H. Brown, Phys. Rev. Lett.,
form distribution of fragmented clusters, like the shape 102,138102(2009).
of a soccer ball. Our observations show that low protein 16M. Venturoli, M.M. Sperotto,M. Kranenburg, B. Smit, Phys.
concentrations do not lead to efficient cluster formation. Rep.,437,1(2006).
17M.P.Allenand D.J. Tildesley, Computer Simulation of Liquids
(ClarendonPress,Oxford,1987).
18A.J. Markvoort, K. Pieterse, M.N. Steijaert, P. Spijker, and
ACKNOWLEDGMENTS
P.A.J.Hilbers,J.Phys.Chem.B,,109,22649(2005).
19R.Goetz, R.Lipowsky,J.Chem.Phys.,108(17),7397(1998).
20R. Goetz, G. Gompper, R. Lipowsky, Phys. Rev. Lett., 82(1),
Thisworkwaspartiallysupportedbytheresearchvice-
221(1999).
presidencyatSharifUniversityofTechnology. We thank 21G.B. Arfken, Mathematical Methods for Physicists (Academic
the referees for their constructive comments, which sub- Press,NewYork,1985).
stantially improved the paper. 22S.A.Safran,StatisticalThermodynamicsofSurfaces,Interfaces,
and Membranes(Addison-WesleyPublishingCompany, 1994).
1R.Lipowsky,Curr.Opin.Struct.Biol.,5,531(1995).