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Molecular dynamics simulations of the structure of closed tethered membranes
Irena Petsche, Gaxy Grest
To cite this version:
Irena Petsche, Gaxy Grest. Molecular dynamics simulations of the structure of closed tethered mem- branes. Journal de Physique I, EDP Sciences, 1993, 3 (8), pp.1741-1754. �10.1051/jp1:1993213�.
�jpa-00246829�
Classification Physics Abstracts 64.60C 05.40 64.60A
Molecular dynamics simulations of the structure of closed tethered membranes
Irena B. Petsche
(~)
andGary
S. Grest(~)
(~) Exxon Engineering Technology Department, Exxon Research and Engineering Company, NJ 07932, Florham Park, U.S.A.
(~)
Corporate Research Science Laboratories, Exxon Research and Engineering Company,NJ 08801, Annandale, U-S-A-
(Received
5 February 1993, accepted in final form 13 April 1993)Abstract The equilibrium structure of closed self-avoiding tethered vesicles are investigated by molecular dynamics simulations. To allow for local flexibility, the vesicles are constructed by connecting linear chins of n monomers to form a closed membrane. For all n studied,
0 < n < 16, we find that the membranes remain flat. The height fluctuations
(h~)
+~N~,
where ( = 0.55 + 0.02 and N is the total number of monomers in the vesicle. This result for (
is significantly lower than earlier estimates for open membranes with a free perimeter but is in agreement with recent estimates of Abraham for membranes without
a free perimeter. Results for the static structure factor,
S(q),
are calculated and compared to recent dataon red blood cell skeletons.
1 Introduction.
Recently,
there has been an extensive amount ofwork,
both theoretical andnumerical,
in-vestigating
theproperties
of D dimensionalself-avoiding
tethered surfaces embedded into d dimensions 11, 2]. While linearpolymers (D
=
I)
is a well-knownexample
of this type of sys-tem, recent interest has focused on 2-dimensional tethered surfaces
(D
=2).
Kantor et al. [3]suggested,
based on asimple Flory-level theory,
that in the presence ofonly
excluded volume interactions, a tethered membrane wouldcrumple
and that its sizeRg
would scale asR(f
+~N,
where N is the number of monomers in the membrane anddf
is the fractal dimension. ForD = 2 and d
= 3,
Flory theory predicts
that df= 2.5 [3]. This conclusion was
supported by
renormalization group calculations [3-6] whichsuggested
that the flatphase
was unstable in d = 3 andby early
Monte Carlo simulations [3]. However, more extensive computer simula-tions [7-15] on
large
system has shownconvincingly
that in fact tethered membranes do notcrumple
but remain flat(R(
+~N)
if one includesonly
excluded volume interactions. This lack of acrumpling
transition in d= 3 has been
explained ii Ii
in terms of animplicit bending
rigidity
which is induced by the self-avoidance requirement even when no such term is present in themicroscopic
Hamiltonian.Then,
ifbending rigidity
isrelevant,
one cannot expect theFlory theory
to work. An alternativeexplanation by
Goulian [16] based on ageneralization
of Edward's model which uses a Gaussian variationalapproximation
andby
Guitter and Palmeriii?]
and Le Doussal [18] based onexpansion
inlarge embedding
space dimension d suggests that the flatphase
is stable for d= 3 and tethered membranes
crumple only
for d > 4. This is in agreement with the recent moleculardynamics
simulations of Grest[15]. However,
in the presence of attractiveinteractions,
Liu and Plischke [19] find that the membranesundergo
aphase
transition from thehigh
temperature flatphase
to a low temperaturecrumpled phase.
This
crumpled phase
seems to exist over a range of temperatures and is characterizedby
a fractal dimension df +~ 2.5.While some
understanding
ofwhy self-avoiding
membranes do notcrumple
isbeginning
to emerge, there remain a number of unsettled issues.Important
unresolved issues include the precise value for the exponent(
that describes the size of theheight
fluctuations(h~
+~ N~ andthe
interpretation
of several recentscattering experiments
on tethered surfaces.Lipowsky
and Girardet [20] used ascaling analysis
to show that(
>1/2
andsuggested
from a Monte Carlo simulation for a continuum model of a solid-like elastic sheet that(
=1/2.
This resultagreed
with a
simple one-loop
self-consistenttheory
of Nelson and Peliti [21] whichpredicted (
=
1/2.
If this were true, then the membrane would have a finite shear modulus on
large scales,
since shear modulus scales as+~
N~»/~
and~p =
4(
2. This result is in contrast with all previous simulations onbead-spring
models which gave(
= 0.64 ~ 0.04.
However,
Abraham [22] haspointed
out that thislarger
value for(
may be due to finite size effects since all these bead-spring
type simulations wereperformed
on open membranes with a free perimeter. Abrahamreplaced
thefree-perimeter
with aperiodic boundary
condition(pbc) by using
acomputational
box which was allowed to vary in sizeusing
a constant pressure moleculardynamics technique
and found(
= 0.53 ~0.03,
consistent withLipowsky
and Girardet [20].Recently
Le Doussal andRadzihovsky
[23] carried out a self-consistentscreening approximation
whichimproved
on the Nelson-Pelititheory by allowing
non~trivial renormalization of the elastic moduli.They
found
(
= 0.590, in between the
previous
estimates. In this paper, we present the results of extensive moleculardynamics
simulations on closed vesicles in which we find(
= 0.55 ~0.02, slightly larger than,
but consistent with Abraham's result [22]. Thisfinding gives
furtherevidence to the
suggestion
that thelarger
values of(
obtained earlier for open membranes areprobably
due to finite size effects due to the freeperimeter. (See
note added inproof.
A
second, important,
unresolved issue concerns theinterpretation
of recentscattering
ex-periments
onexperimental
realizations ofpolymerized
membranes. A laserscattering
mea-surement [24] of the static structure function
S(q)
forgraphite
oxidecrystalline
membranes suggests thatthey
arecrumpled
and not flat as indicatedby
computer simulations. For a flatmembrane,
one expects that forlarge enough
systems,S(q)
+~
q~~
while for acrumpled
membraneS(q)
+~
q~~.~.
While Wen et al. [24] do observe a power lawdecay
with aslope
of 2.54, one should notinterpret
this asconvincing
evidence for acrumpled phase,
since theseexperiments
are onrandomly
oriented membranes. Abraham and Goulian [25] have shown that for a wide range of q, theisotropically averaged S(q)
~w q~~.~~ for open, tethered mem- branes
containing
as many as 29 420 monomers.They
found no evidence for aq~~ regime
eventhough
these membranes aredefinitely
flat. Theexpected q~~ regime
which must be present forlarge enough samples
isapparently suppressed
andonly
therough regime
[26] in whichS(q)
+~q~~+'
is observed. Thisinterpretation
is consistent with the computer simulations of Abraham and Goulian [25] on open membranes where(
cs 0.64. It also agrees with recentX-ray
andlight scattering experiments [27,
28] on isolated red blood cell(MC)
skeletons inhigh
salt in which a q~~regime
at low q was followedby
a q~~.~~regime
forlarger
q, consistentwith the behavior
expected
for a flat membrane. While it isfairly
certain that the interaction in the red blood cell skeletons ispurely repulsive,
it ispossible
that attractive interactions [19] areimportant
for thegraphite
oxidecrystalline
membranes andthey
are in factcrumpled.
Additional
experiments
are needed toclarify
whether attractive interactionsplay
a role in thiscase.
Fig. 1. Illustration of the
spectrinlactin
membrane skeleton of ABC(from
Ref. [27]) showing the main components and the triangulated structure. For clarity, the skeleton is drawn inan expanded
state. In viva, the average node distance is about
1/3
of the contour length of spectrin tetramers(2000 h).
While the
graphite
oxidecrystalline
membranes [24] are open membranes with alarge
localbending rigidity,
inducedby
the closepacking
of monomers, the RBC skeleton is a closedvesicle, roughly spherical,
made up of flexible worm-like chains in atriangulated
network. The ABC skeleton(Fig. I)
is attached to thecytoplasmic
side of theliquid lipid
cell membrane andcan be isolated
by detergent
treatment. Thespectrin
tetramers which made up the chains havea contour
length
of cz20001
anda
persistence length
of100 2001
inhigh
ionicstrength
buffer. Mammalian ABC skeletons contain
+~ 70 000
triangular
meshes.Obviously,
this system is toocomplex
to simulatedirectly
on the computer.However,
the essentialproperties
of the membrane on intermediatelength
scales can be obtained from astudy
of acoarse-grained
bead-spring
model which has the sametopology. Following
Abraham'sstudy
of open membranes oflinear flexible chains connected at tri-functional vertices [12], we
study
atriangular
array of linear chainscontaining
n monomers connected to forma closed vesicle as shown in
figure
2. All but 12 of the vertices are 6-fold coordinated. Theremaining
12 vertices are S-foldcoordinated,
as is necessary to close the vesicle. In
high salt,
the actinoligomers
have a meanspacing
of400 500
1,
or about I
IS 1/4
of theirfully
extendedlength. Using
this as a means ofmapping
to our beadspring model,
we estimate that each linear chain should contain about 50-60 monomers for the model studied here. Since thelargest
system we canequilibrate
andstudy
in a few hundred hours ofGray
time is about 25 000 monomers, our initial studies werelimited to n < 16 in order to increase the number of
triangles
in the system.Larger
systems will have to wait for the nextgeneration
of supercomputers.However,
inspite
of the fact thatour
largest samples
are about 50 times smaller than theexperimental
systems and containonly
excluded volumeinteractions,
the cross sections of the simulated membranes compare veryfavorably
with theshape
of ABC skeletons. Results forS(q)
are also in agreement withrecent
X-ray
andlight scattering
studies of Schmidt et al. [28].Fig. 2. Illustration of the initial state for a membrane of size N
= 9002 with n
= 8 additional
monomers between each vertex. This system contains 1080 edges and 720
triangular
faces. Note thatthe bond lengths are not exactly the same as the points have been projected onto the surface of a
sphere. If all bond lengths would be equal, the membrane would have the shape of an icosahedra.
The outline of the paper is as follows. In section 2, we describe the model and the molecular
dynamics technique
used. In section3,
we present our results for theeigenvalues
of the mo- ment of inertia tensor, the radius ofgyration
and the static structure function for membranescontaining
from 252 25 002 monomers with 0 < n < 16 monomer per linear chain. All mem- branes were connected to form an icosahedra. In section 4, we discuss the relaxation of themembranes and then
briefly
summarize our results in section 5.2. Model and method.
Each membrane consists of N monomers of mass m connected
by
anharmonicsprings.
Themonomers interact
through
a shifted Lennard-Jonespotential given by
U°(r)
=
~~ ~~~~~ ~~~
~
~
~~ ~ ~ ~~'
(l)
0 r > r~,
with r~ =
2~/~~.
Thispurely repulsive potential
represents the excluded volume interactions that are dominant in the case of monomers immersed in agood
solvent. For monomers whichare tethered
(nearest neighbors)
there is an additional attractive interactionpotential
as de- scribed elsewhere [14,15].
Noexplicit bending
terms are included.Denoting
the totalpotential
of monomer Iby U;,
theequation
of motion for monomer I isgiven by
~
~
~~'
~' ~ ~~~
~
~'~~~'
~~~Here r is the bead friction which acts to
couple
the monomers to the heat bath.W;(t)
describes the random forceacting
on each bead. It can be written as a white noise term with(w,(t) w~ (t'))
= &,~&(t
1')6k~Tr, (3)
where T is the temperature and kB is the Boltzmann constant. We have used r
= r~l and
kBT = 1.2e. Here r
=
~(mle)1/~.
Theequations
of motion are then solvedusing
avelocity-
Verletalgorithm
[29] with a time step At= 0.012r [15]. With this choice of parameters, the average bond
length
between nearestneighbors
that are tethered is 0.97~. The programwas vectorized for a supercomputer
following
theprocedure
describedby
Grest et al. [30]except that the
periodic boundary
conditions were removed and additional interactions betweenmonomers which are tethered was added. For a tethered membrane of size N
= 16 002, 1 million time steps took about 80 hours of cpu time on our
Cray
XMP14/s.
Further details of themethod can be found elsewhere [31].
The simulations were
performed
on two-dimensionaltriangular
arrays of n monomers con- nected to form a closed vesicle. This was doneby
construction of an icosahedron with nvmonomers per
edge.
An additional n monomers connectedlinearly
were then added betweeneach node. The total number of monomers N =
10n((1
+3n)
+ 2 with30n( edges
oflength
Lo =nv(n
+I)
and20n(
faces. As seen in tableI,
for n= 0 we studied systems up to 8 412 monomers
corresponding
to nv "29,
about twice aslarge
as that consideredby
Komuraand
Baumg£rtner [32], though
their closed vesicles did not have the icosahedron symmetry.Our
largest
system, N = 25002, n = 8 iscomparable
to thelargest
open membrane with free perimeter studiedby
Abraham [12].Typically,
0.5 -1.0 x 10~ time steps were needed toequilibrate
the system. Thisequilibration
was checkedby monitoring
the autocorrelationfunction of the mean square radius of
gyration
as discussed in section 4.As mentioned in the
introduction,
the average distance between nodes in the ABC skeleton is about1/4 -1IS
of itsfully
extendedlength.
Withonly
excluded volume interactions asstudied
here,
the average distance between nodes for n > 8approximately
satisfies the relation(R()
=a(n +1)~",
where u= 0.59 is the excluded volume exponent and a cz 1.8,
though
the value of a for a free chain under the same conditions is somewhat
smaller, approximately
1.5. Thus n cz 50 60 for
(Re) /(n
+I)
cz1/4.
An alternativemapping
based oncomparing
the persistence
length
for the spectrin tetramers with that of thebead-spring
model would suggest a small value of n.However,
from ourexperience
onpolymer melts, comparing
thepersistence length
is not asmeaningful
since there is some arbitrariness in the definition of a statistical segment. Due to the slowequilibration
of thesemembranes,
thelargest sample
wecan handle at present is the order of 25 000 monomers.
Therefore,
we limited n to 16, for which(Re)/(n +1)
=
0.43,
in order to have a reasonable number oftriangles
in our system.To
analyze
our results we calculated severalquantities. Every
500 steps the inertia matrix, itseigenvalues
I,(li
<12
<13)
and the radius ofgyration squared, R(, given by
theirsum were calculated. These quantities were also used to check the
equilibration
of the systemthrough
their autocorrelation. For closed membranes in the flatphase,
all threeeigenvalues
should scale as N. Results for total duration of the run,
(R()
and(h~)
aregiven
in table I.If we treat the data saved every 500 time steps as
statistically independent,
then the error inTable I. The number of monomers per
edge
nv in theoriginal icosahedron,
number ofmonomers n between each
node,
total number of monomers N and total duration of therun, Tt
IT,
afterequilibration
for the closed vesicles studied here. Also shown is the averagemean square radius of
gyration (R()
and the mean squareheight
fluctuation(h~)
for eachmembrane.
nv n N
Tt/r (R() (h~)
5 0 252 12000 14.7 0.17
7 0 492 12000 28.6 0.24
10 0 1002 12000 58.3 0.36
14 0 1962 12000 l14.1 0.53
20 0 4002 12000 232.9 0.75
29 0 8412 12000 489.6 1-IS
4 4 2082 18000 106.9 2.16
3 8 2252 18000 130.5 3.80
4 8 4002 18000 221A 5.40
6 8 9002 18000 478.3 8.6
8 8 16002 20880 832.4 11.2
3 16 4412 18000 283.0 8.7
4 16 7842 12000 479.9 10.5
5 16 12252 12000 702.3 14.6
these two
quantities
as well as the moments of inertia are much less than I%. However thissurely
underestimates the error as theconfigurations
aresurely
notindependent. Averaging
subsets of the data
suggests
that a reasonable estimate of the statistical error is about I% for(R()
and about 2-3% for(h~).
We also measured thespherically averaged
structurefactor, S(q), given by
S(q)
=( ~e'~'~~'~~'~ ). (4)
;,j
The
angle
brackets represent aconfigurational
averagetypically
taken every 5000 time steps.For each q
= q (, we
averaged
over 20 random orientations.3 Results.
Figure
3 showstypical
results for three vesicles withapproximately
the same number ofmonomers for n
= 0,8 and 16. For n
=
0,
the membrane retains theshape
characteristic of an icosahedra. This system has arelatively large
localbending rigidity
due to thehigh
connectivity
of each monomer. Themagnitude
of theheight
fluctuations isrelatively
small asseen in table I.
Only
for muchlarger
systems would one expect this system to look less faceted.For n
= 8 and
16, however,
the localbending rigidity
issignificantly
reduced and there is little evidence of the icosahedratopology
even for systems as small as a few thousand monomers. Infigure
4 we show atypical configuration
for 16002 and compare its cross-sectionsthrough
the center with RBC skeletons inhigh
salt. Note thesimilarity
between the RBC skeletons and the simulated membranes eventhough
the simulated membranes aresignificantly
smaller.ia) ib)
Fig. 3. Typical configuration for closed
self-avoiding
tethered membranes for:a)
N = 4 002(n
= 0);
b)
N= 4 002
(n
=
8)
and c) N = 4 412(n
= 16).
,,
' ,
.
ic)
Fig. 4.
a)
Isolated red blood cell skeletons in 2.5nM monovalent salt viewed under video enhanced differential interference contrast microscopy. The mean diameter is 5.3 ~ 0.4 pm [28]. Typical config~uration for the membrane of size N
= 16002
(n
= 8) shown for
b)
entire network and c) three cross sections through the center of the network. Note similarity to shape of the experimental and simulated system.The mean
squared
radius ofgyration (R()
versus N is shown infigure
5 for the membranes studied. Leastsquared
fits to the data for each n show that(R()
+~
n~~
with ug = 0.99 ~ 0.02for n
=
0,
0.95 ~ 0.03 for n= 8 and 0.89 ~ 0.04 for n
= 16. All of these results are consistent with a flat
phase
as one expects fromprevious
simulations on open membranes as well asKomura and
Baumgfirtner's
simulations [32] for closed membranes with n = 0. Oneinteresting
feature of this data is that to first
approximation (R() depends
on n and nvonly through
N
+~
n[n.
This can be understood if one considers theelementary triangles
that make up the membrane. Since the distance between nodes(Re)
+~
(n
+I)",
the surface area pertriangle
scales as n~" Since the number of
triangles
is20n[
and the membrane isapproximately spherical,
one would expect that(R()
would scale as the surface area, which isproportional
ton(R(
+~
n[nl.l~
+~
Nn°.l~ Therefore,
the dominant contribution to the size of the membrane is due toN,
the total number of monomers. The number of monomers between each nodeenters
only weakly
via a power n°.l~ for fixed N. Similar argument at the thetapoint
whereR(
+~ N suggests that
(R()
woulddepend only
on N and would beessentially independent
of the number of monomers between each node for membranes of thetopology
studied here.7.5
~ a
AlJ
«I
5.5/
nS
v a
~ ~
~
o
o
~'i.5
6.5 7.5 8.5 9.5 10.5
In N
Fig. 5. Mean square radius of gyration
(R()
versus N for closed self-avoiding membranes with
n = 0
(o),
4(x),
8 (a) and16(6).
In
figure
6, results for theeigenvalues
of the moment of inertia tensor I, versus N arepresented
for n= 0 and 8. Least square fits to the data suggest that all of these
eigenvalues satisfy
ascaling
relaxation I;+~ N"~. For n
= 0, the three values of u, are
essentially
I, with the ratio 11/13 increasing slightly
withincreasing
N. However, for n= 8, the effective exponents u; are very different and the'ratio
11 /13
increasesslightly
as N increases. Best fitsto the data
give
vi "0.99,
u2 " 0.95 and u3" 0.91 for n
= 8, At first these results seem
a little
surprising
until one notes that in the limit oflarge N,
all threeeigenvalues
must bethe same. Differences in the I,'s for finite N are a result of fluctuations in the
height
h whichvanish in
comparison
to the radius of the membrane as N - co since(
< I. This can be seen verysimply by considering
the energy cost toproduce
a fluctuation in the membrane forlarge
5.o
»
4.°
(a)
n=o~ ~
$
3.0 ~fi
~
~'i.0
6.0 7.0 B-O 9.0
In N 7.0
6.0
(b)
n=8, 1
(
5.0)
fi
1 4.0
~
°
i
~'~,o
a-o g-o lo-o
In N
Fig. 6. Eigenvalues of the moment of inertia tensor AI
(a),
12(6)
and 13(x)
versus N for(a)
n = 0 and
(b)
n = 8,
N. Excursions from the mean radius Ro of the membrane can be written as
R =
R~
+hill(Q) (5)
where Pi
(Q)
is theLegendre polynomial
of order I. The lowest order excitation which dominates(h~)
is for= 2 and the energy of the excitation has the form
E
+~
/ dA(V~h)~ (6)
which has the
scaling
formE =
K(h/R)~. (7)
Here K is the renormalized
bending rigidity
which scales asR~,
where ~= 2
2(
for D= 2.
Therefore,
for E of orderkBT, (h~ /R~)
+~ R~~ which vanishes as R - co. The
disparity
in the values of u; is due to finite size corrections. Thus theratio11 /13
must increase withN,
3.0
O
Z-o °
O O
m~ I-o
Jz V
~ o-o °
~ a
D
-i.o D
~ a
~~'~5.0
6.0 7.0 8.09.0 lo-o 11,0
In N
Fig.
7.Height
fluctuations(h~)
versus N for n= 0 and 8.
giving
rise to values of I; which have different effective exponent for the range of N studied here. A similarargument
alsoapplies
to thelargest
2eigenvalues
for open membranes which also must beequal
in theasymptotic
limit. Thispoint
hasapparently
been overlooked in anumber of
previous simulations, apparently
because for membranes with no holes(n
=
0),
thedependence of11 /13
on N is very weak.Only
forperforated
membranes as those studied here is this correction toscaling
more apparent.As discussed in the
introduction,
the actual value for the exponent(
is not well determined.For open
membranes,
an issue has been raised as to whether the freeperimeter
cangive
rise to finite size corrections which lead to an overestimation of(
for the system size studied to date.Abraham [22] used
pbc
and a constant pressure simulation to eliminate the freeperimeter.
His simulations gave asignificantly
lower estimate for(.
An alternative method foreliminating
the free
perimeter
is tostudy
closed vesicles as we have done here. In this case, theheight
fluctuations
(h~)
can beeasily
measured from the fluctuations of the meansquared
radius ofgyration,
j~2 ~
(~j~2)2 jl/2
= j~4 j~22jl/2 (~)
g
g) g)
In
figure
7, we present results for(h~)
versus N for n= 0 and 8. We find
(
= 0.55 ~ 0.02for n
= 0 and
(
= 0.56 ~ 0.02 for n= 8,
only slightly larger
than Abraharns [22] results for membranes withpbc (
= 0.53 ~ 0.03. Our results aresignificantly
lower than those observed for open membranes with freeperimeters, (
= 0.64 ~0.03,
as well as earlier results of Komuraand
Baumgfirtner
[32] for a closed vesicle with n=
0, (
= o.645 ~ o.08. Our results are close to those obtained from a self-consistentscreening approximation by
Le Doussal andRadzihovsky
[23] who find(
= o.59.
One of the
original
motivations forstudying large
membranes with holes was toinvestigate
whetherreducing
the localbinding rigidity
would cause the membranes tocrumple.
Abraharn [12] showed veryconclusively
in hisstudy
ofperforated
open membranes that the order param- eter[33],
defined as the ratio of the the mean square radius ofgyration
over the membrane size,vanishes
only
as the mass fraction of theperforated
membrane vanishes. Infigure
8, we present similar evidence for our closed membrane. Here weplot (R()/L(,
whereLo
=nv(n
+I)
is thelength
of anedge
in the initialconfiguration
of the membrane and is a measure of themembrane's size. In
agreement
withAbraham,
we find that the membranes remain flat and do notcrumple,
even when the holes arequite large
and the mass fraction isquite
low.Experimentally
x-ray andlight scattering
have been used tostudy randomly
oriented mem- branes[24, 27, 28].
Infigure
9, we present our results for the scaledS(q) IN
for n= 8. Unlike open
membranes,
there is extra structure inS(q)
because the membrane is closed. Since the vesicles are to firstapproximation spherical
shells of diameter 2R+~
Nl/~,
it is notsurpris- ing
that the first few minima scale asqNl/~
These oscillations are morepronounced
thanone finds in the
experimental
data due both to the idealmonodispersity
andrelatively
lowvalue of n. For n =
o,
these oscillationspersist
over the entire q range.However,
for n=
8,
there is also a
high
qregime
which scales asqN~/~
with b cz o.85 ~o.05,
consistent with theisotropically averaged S(q)
for open membranes[25, 26].
Because of the oscillations due to themonodispersity
of thesample
at small q, thisscaling regime
isrelatively
small and extendsonly
from 3 <lnqN~/~
< 5. This value of bcorresponding
toroughness
exponent(
= 32/b
cz o.65 ~ o.05. While this value isslightly larger,
it is notparticulary
reliablecompared
to that determineddirectly
from theheight fluctuations,
due to the limitedscaling regime
inS(q).
4.
Dynamics.
To determine if the membranes studied have
equilibrated,
it is useful to measure the timedependent
correlation function for the mean square radius ofgyration,
j~2(t) j~2(~)) j~2)2
~~~~ ~
~4~
j~2)~
~ ~~~
g g
Since the present simulations do not include solvent molecules
explicitly
and the membranesare
weakly coupled
to a heatbath,
the membranes should have a Rouse-like relaxation. Kan- tor et al. [3]suggested, following
theanalysis
for a linearpolymer
chain[34],
that thelongest
relaxation time rR
+~
Nl+"
For a flatmembrane,
u= I and the
analogy
wouldpredict
that rR
+~
N~.
However as seen infigure lo,
our data scale ast/N,
not ast/N~.
This is in agreement with earlier data of Komura andBaumg£rtner
for closed vesicles with n = o.They suggested
onepossible explanation
for this faster relaxation observed is that for flat membranes, the relaxation has both anin-plane
and anout-of-plane
component. Thein-plane longitudinal
relaxation time scales as
N,
similar to the case of permanent crosslinked networks[35],
whereas theout~of-plane
relaxation time scales asNl+'.
Thisexplanation
is consistent with both our data and that of Komura andBaumgfirtner
if one assumes that theamplitude
of thein-plane
contribution to#(t)
is muchlarger
than theout-of-plane amplitude. Unfortunately,
neitherour data nor theirs is accurate
enough
to tell whether this second relaxation mode is present.We find that the
longest
relaxation time rRdepends strongly
on both the system size N and the number of monomers n between eachnode,
unlike(R().
This result can be understoodby recalling
that for a linearpolymer
in agood solvent,
thelongest
relaxation time scales asn~+~",
where u= o.59 [34].
Therefore,
if we assume that rR scales as N+~
n(n,
rR+~
Nnl.l~.
This extra factor of nl.l~ accounts, within the accuracy of our
data,
for the difference in#(t)
we observe for 4 < n < 16.
o-a
o.6 o
"~
/
0.4i~
~
o
~'~
#
~'i.0
D-Z 0.4 0.6 D-B I-D I-Z
mass fraction
Fig. 8. -Mean square radius of gyration
(R()/Lf
versus mass fraction for 0 < n < 16. Here,
Lo =
nv(n +1)
is length of each edge in the initial icosahedra configuration.o
j~~
~
m
~ -8
d=o.85
z
z
~j
~-4m
~ -8
d=i,o
~
l 0 2 3 4 5 6
In qN~~~
Fig. 9. Scaling plots for isotropically averaged
S(q)
for the 5 closed selLavoiding membranes withn = 8.
o-o
-o.5
+J
~lo0 .
~ -
~'~0.0
~~
Note added in
proof
:Zhang,
Davies and Kroll [37] haverecently
shown that for closedmembranes,
there is a linearcoupling
between theout-of-plane
undulation modes and thein-plane phonon modes,
which suppresses theout-of-plane
fluctuations atlong length
scales.This leads to a new
scaling
relation for what we identified as theheight
fluctuations < h~ >in
equation (8).
As defined inequation (8), Zhang
et al.point
out that < h~ >+~N~i,
where (1 =(2 ~/2)/(2
+~),
instead ofN~,
with(
= (1~/2). Reinterpreting
our results in terms of(i gives
~ = o.85 ~ 0.04 and(
= o.58 ~o.02,
which agrees with the theoreticalprediction
ofLe Doussal and
Radzihovsky
[23].Acknowledgements.
We thank D. C.
Morse,
S. T.Milner,
D. R.Nelson,
T. C.Lubensky,
and T. A. Witten forhelpful
discussions.References
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