Molecular dynamics simulations of the structure of closed tethered membranes

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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

(~)

and

Gary

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 data

on red blood cell skeletons.

1 Introduction.

Recently,

there has been _an extensive amount of

work,

both theoretical and

numerical,

in-

vestigating

the

properties

of D dimensional

self-avoiding

tethered surfaces embedded into d dimensions _11, 2]. ^{While linear}

polymers (D

=

I)

is a well-known

example

^of this type ^of sys-

tem, recent interest has focused _on 2-dimensional tethered surfaces

(D

₌

2).

Kantor _et al. [3]

suggested,

based _{on a}

simple Flory-level theory,

^that in the _presence of

only

excluded volume interactions, _a tethered membrane would

crumple

and that its size

Rg

^{would scale} as

R(f

+~

N,

where N is the number of _monomers in the membrane and

df

is the fractal dimension. For

D = 2 and d

= 3,

Flory theory predicts

that df

= 2.5 [3]. This conclusion _was

supported by

renormalization _group calculations [3-6] ^which

suggested

that the flat

phase

_was unstable in d = 3 and

by early

Monte Carlo simulations [3]. However, more extensive computer ^simula-

tions [7-15] on

large

system ^{has shown}

convincingly

that in fact tethered membranes do not

crumple

but remain flat

(R(

+~

N)

if _one includes

only

excluded volume interactions. This lack of _a

crumpling

transition in d

= 3 has been

explained ii Ii

in terms of _an

implicit bending

rigidity

which is induced by the self-avoidance requirement _even when _no such _term is present in the

microscopic

Hamiltonian.

Then,

if

bending rigidity

is

relevant,

_one cannot expect ^the

Flory theory

to work. An alternative

explanation by

Goulian [16] ^based on a

generalization

of Edward's model which _{uses a} Gaussian variational

approximation

^and

by

Guitter and Palmeri

ii?]

and Le Doussal [18] ^based on

expansion

in

large embedding

_space dimension d suggests that the flat

phase

is stable for d

= 3 and tethered membranes

crumple only

for d > 4. This is in agreement ^{with the} recent molecular

dynamics

simulations of Grest

[15]. However,

in the presence of attractive

interactions,

Liu and Plischke [19] ^{find that the membranes}

undergo

_a

phase

transition from the

high

temperature flat

phase

to a low temperature

crumpled phase.

This

crumpled phase

_seems to exist _{over a} range of temperatures and is characterized

by

_a fractal dimension df _+~ 2.5.

While _some

understanding

of

why self-avoiding

membranes do not

crumple

is

beginning

to emerge, there remain _a number of unsettled issues.

Important

unresolved issues include the precise value for the exponent

(

that describes the size of the

height

fluctuations

(h~

_+~ ^{N~ and}

the

interpretation

of several _recent

scattering experiments

_on tethered surfaces.

Lipowsky

and Girardet [20] ^used a

scaling analysis

to show that

(

>

1/2

and

suggested

from _a Monte Carlo simulation for _a continuum model of _a solid-like elastic sheet that

(

₌

1/2.

This result

agreed

with _a

simple one-loop

self-consistent

theory

of Nelson and Peliti [21] which

predicted (

=

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 _on

bead-spring

models which _gave

(

= 0.64 ~ 0.04.

However,

Abraham [22] ^has

pointed

out that this

larger

value for

(

_may be due to finite size effects since all these bead-

spring

type simulations _were

performed

_{on open} membranes with _a free perimeter. ^Abraham

replaced

the

free-perimeter

with _a

periodic boundary

condition

(pbc) by using

_a

computational

box which _was allowed to vary in size

using

_a constant pressure molecular

dynamics technique

and found

(

₌ 0.53 ~

0.03,

consistent with

Lipowsky

and Girardet [20].

Recently

Le Doussal and

Radzihovsky

_[23] carried out _a self-consistent

screening approximation

which

improved

_on the Nelson-Peliti

theory 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 molecular

dynamics

simulations _on closed vesicles in which _we find

(

₌ 0.55 ~

0.02, slightly larger than,

but consistent with Abraham's result [22]. ^This

finding gives

further

evidence _to the

suggestion

that the

larger

^values of

(

obtained earlier for _open membranes _are

probably

due to finite size effects due to the free

perimeter. (See

note added in

proof.

A

second, important,

unresolved issue _concerns the

interpretation

of _recent

scattering

_ex-

periments

_on

experimental

realizations of

polymerized

membranes. A laser

scattering

mea-

surement [24] ^{of the static} structure function

S(q)

for

graphite

oxide

crystalline

membranes suggests that

they

_are

crumpled

and not flat _as indicated

by

computer simulations. For _a flat

membrane,

_one expects ^that for

large enough

systems,

S(q)

+~

q~~

while for _a

crumpled

membrane

S(q)

+~

q~~.~.

While Wen _et al. [24] ^{do observe} a power law

decay

with _a

slope

of 2.54, one should not

interpret

this _as

convincing

evidence for _a

crumpled phase,

since these

experiments

_{are on}

randomly

oriented membranes. Abraham and Goulian [25] ^{have shown} that for _a wide range of _q, the

isotropically averaged S(q)

~w q~~.~~ ^for open, tethered _mem- branes

containing

_{as many as} 29 420 _monomers.

They

found _no evidence for _a

q~~ regime

_even

though

these membranes _are

definitely

flat. The

expected q~~ regime

which must be present for

large enough samples

is

apparently suppressed

and

only

the

rough regime

[26] ^{in which}

S(q)

+~

q~~+'

is observed. This

interpretation

is consistent with the computer simulations of Abraham and Goulian [25] on open membranes where

(

_cs 0.64. It also _agrees with _recent

X-ray

^and

light scattering experiments [27,

28] on isolated red blood cell

(MC)

skeletons in

high

salt in which a q~~

regime

at low _{q was} followed

by

a q~~.~~

regime

for

larger

_q, consistent

with the behavior

expected

for _a flat membrane. While it is

fairly

certain that the interaction in the red blood cell skeletons is

purely repulsive,

it is

possible

that attractive interactions [19] are

important

for the

graphite

oxide

crystalline

membranes and

they

_are in fact

crumpled.

Additional

experiments

_are needed _to

clarify

whether _attractive interactions

play

_a role in this

case.

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 in

an expanded

state. In viva, the _average node distance is about

1/3

of the contour length of spectrin tetramers

(2000 h).

While the

graphite

oxide

crystalline

membranes [24] are open membranes with _a

large

local

bending rigidity,

induced

by

the close

packing

of _monomers, the RBC skeleton is _a closed

vesicle, roughly spherical,

made _up of flexible worm-like chains in _a

triangulated

network. The ABC skeleton

(Fig. I)

is attached to the

cytoplasmic

side of the

liquid lipid

cell membrane and

can be isolated

by detergent

treatment. The

spectrin

tetramers which made _up the chains have

a contour

length

of _cz

20001

_and

a

persistence length

of100 200

1

_in

high

ionic

strength

buffer. Mammalian ABC skeletons contain

+~ 70 000

triangular

meshes.

Obviously,

this system is too

complex

to simulate

directly

_on the computer.

However,

the essential

properties

of the membrane _on intermediate

length

scales _can be obtained from _a

study

of _a

coarse-grained

^bead-

spring

model which has the _same

topology. Following

Abraham's

study

of _open membranes of

linear flexible chains connected _at tri-functional vertices [12], we

study

_a

triangular

_array of linear chains

containing

_{n monomers} connected to form

a closed vesicle _as shown in

figure

2. All but 12 of the vertices _are 6-fold coordinated. The

remaining

12 vertices _are S-fold

coordinated,

as is necessary to close the vesicle. In

high salt,

the actin

oligomers

have _{a mean}

spacing

of

400 500

1,

or about I

IS 1/4

of their

fully

extended

length. Using

this _{as a means} of

mapping

to _our bead

spring model,

we estimate that each linear chain should contain about 50-60 _monomers for the model studied here. Since the

largest

system we can

equilibrate

and

study

in _a few hundred hours of

Gray

time is about 25 000 monomers, our initial studies _were

limited _{to n} < 16 in order to increase the number of

triangles

in the system.

Larger

systems will have to wait for the next

generation

^of supercomputers.

However,

in

spite

of the fact that

our

largest samples

_are about 50 times smaller than the

experimental

_systems and contain

only

excluded volume

interactions,

the _cross sections of the simulated membranes _compare very

favorably

with the

shape

of ABC skeletons. Results for

S(q)

_are also in agreement with

recent

X-ray

and

light 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 that

the 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 section

3,

_we present our results for the

eigenvalues

of the _mo- ment of inertia tensor, the radius of

gyration

^{and the} static structure function for membranes

containing

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 the

membranes and then

briefly

summarize our results in section 5.

2. Model and method.

Each membrane consists of N _monomers of _{mass m} connected

by

anharmonic

springs.

^The

monomers interact

through

a shifted Lennard-Jones

potential given by

U°(r)

=

~~ ~~~~~ ~~~

~

~

~~ ~ ~ ~~'

(l)

0 _r > r~,

with _{r~ =}

2~/~~.

_This

purely repulsive potential

represents ^the excluded volume interactions that _are dominant in the _case of _monomers immersed in a

good

solvent. For _monomers which

are tethered

(nearest neighbors)

there is _an additional attractive interaction

potential

_as de- scribed elsewhere [14,

15].

^No

explicit bending

terms are included.

Denoting

the total

potential

^of _monomer I

by U;,

the

equation

of _motion for _monomer I is

given by

~

~

~~'

^{~' ~ ~~}

~

~

~'~~~'

_~~~

Here r is the bead friction which acts to

couple

the _monomers to the heat bath.

W;(t)

describes the random force

acting

_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/~.

The

equations

of motion _are then solved

using

_a

velocity-

Verlet

algorithm

_[29] with _a time step At

= 0.012r [15]. ^{With this choice of} parameters, the average bond

length

between nearest

neighbors

that _are tethered is 0.97~. The program

was vectorized for _a supercomputer

following

the

procedure

described

by

Grest et al. [30]

except that the

periodic boundary

conditions _were removed and additional interactions between

monomers 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

XMP

14/s.

Further details of the

method _can be found elsewhere [31].

The simulations _were

performed

_on two-dimensional

triangular

_arrays of _{n monomers con-} nected to form _a closed vesicle. This _was done

by

construction of _an icosahedron with _nv

monomers per

edge.

An additional _{n monomers} connected

linearly

_were then added between

each node. The total number of _monomers N ₌

10n((1

₊

3n)

₊ 2 with

30n( edges

of

length

Lo ₌

nv(n

+

I)

and

20n(

faces. As _seen in table

I,

for _n

= 0 _we studied systems _up to 8 412 _monomers

corresponding

to nv "

29,

about twice _as

large

_as that considered

by

Komura

and

Baumg£rtner [32], though

^their closed vesicles did not have the icosahedron symmetry.

Our

largest

system, ^N ₌ 25002, n = 8 is

comparable

to the

largest

_open membrane with free perimeter ^studied

by

Abraham [12].

Typically,

0.5 -1.0 _x 10~ _time steps were needed to

equilibrate

the system. ^This

equilibration

_was checked

by monitoring

the autocorrelation

function 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 about

1/4 -1IS

of its

fully

extended

length.

With

only

excluded volume interactions _as

studied

here,

the _average distance between nodes for _n > 8

approximately

^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)

_cz

1/4.

An alternative

mapping

^based on

comparing

the persistence

length

for the spectrin tetramers with that of the

bead-spring

model would suggest a small value of _n.

However,

from _our

experience

_on

polymer melts, comparing

^the

persistence length

is not _as

meaningful

since there is _some arbitrariness in the definition of _a statistical segment. Due to the slow

equilibration

of these

membranes,

the

largest sample

_we

can 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 of

triangles

in _our system.

To

analyze

_our results _we calculated several

quantities. Every

500 steps the inertia matrix, its

eigenvalues

I,

(li

_<

12

<

13)

and the radius of

gyration squared, R(, given by

their

sum were calculated. These quantities were also used to check the

equilibration

of the system

through

their autocorrelation. For closed membranes in the flat

phase,

all three

eigenvalues

should scale _as N. Results for total duration of the _run,

(R()

and

(h~)

are

given

in table I.

If _we treat the data saved _every 500 time steps as

statistically independent,

then the _error in

Table I. The number of _{monomers per}

edge

_nv in the

original icosahedron,

number of

monomers n between each

node,

total number of _monomers N and total duration of the

run, Tt

IT,

after

equilibration

for the closed vesicles studied here. Also shown is the _average

mean square radius of

gyration (R()

and the _{mean square}

height

fluctuation

(h~)

^{for each}

membrane.

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 this

surely

underestimates the _{error as} the

configurations

_are

surely

not

independent. 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 the

spherically averaged

structure

factor, S(q), given by

S(q)

=

( ~e'~'~~'~~'~ ). (4)

;,j

The

angle

brackets represent a

configurational

_average

typically

taken _every 5000 time steps.

For each _q

= q (, ^we

averaged

_over 20 random orientations.

3 Results.

Figure

3 shows

typical

results for three vesicles with

approximately

the _same number of

monomers for _n

= 0,8 ^and 16. For _n

=

0,

the membrane retains the

shape

characteristic of _an icosahedra. This system has _a

relatively large

local

bending rigidity

due to the

high

connectivity

of each _monomer. The

magnitude

of the

height

fluctuations is

relatively

small _as

seen in table I.

Only

^{for much}

larger

systems would _one expect ^this system to look less faceted.

For _n

= 8 and

16, however,

the local

bending rigidity

is

significantly

reduced and there is little evidence of the icosahedra

topology

_even ^for systems as small _{as a} few thousand _monomers. In

figure

4 _we show a

typical configuration

for 16002 and _compare its cross-sections

through

the center with RBC skeletons in

high

salt. Note the

similarity

between the RBC skeletons and the simulated membranes _even

though

the simulated membranes _are

significantly

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 of

gyration (R()

_versus N is shown in

figure

5 for the membranes studied. Least

squared

fits _to the data for each _n show that

(R()

+~

n~~

with ug = 0.99 ~ 0.02

for _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 ^from

previous

simulations _{on open} membranes _as well _as

Komura and

Baumgfirtner's

simulations [32] ^{for closed membranes with} n = 0. One

interesting

feature of this data is that _to first

approximation (R() depends

_{on n} and _nv

only through

N

+~

n[n.

This _can be understood if _one considers the

elementary triangles

that make _up the membrane. Since the distance between nodes

(Re)

+~

(n

+

I)",

the surface _{area per}

triangle

scales _as n~" Since the number of

triangles

is

20n[

and the membrane is

approximately spherical,

_one would expect ^that

(R()

^would scale _as the surface _area, which _is

proportional

to

n(R(

+~

n[nl.l~

+~

Nn°.l~ Therefore,

the dominant contribution _to the size of the membrane is due _to

N,

the total number of _monomers. The number of _monomers between each node

enters

only weakly

via _a power n°.l~ for fixed N. Similar argument at the theta

point

where

R(

+~ N suggests ^that

(R()

would

depend only

_on N and would be

essentially independent

of the number of _monomers between each node for membranes of the

topology

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 the

eigenvalues

of the _moment of inertia tensor I, versus N _are

presented

for _n

= 0 and 8. Least _square fits to the data suggest ^{that all of these}

eigenvalues satisfy

_a

scaling

relaxation I;

+~ N"~. For _n

= 0, the three values of _{u, are}

essentially

I, with the ratio 11

/13 increasing slightly

with

increasing

N. However, for _n

= 8, ^the effective exponents u; are very different and the'ratio

11 /13

increases

slightly

as N increases. Best fits

to 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 of

large N,

all three

eigenvalues

must be

the _same. Differences in the I,'s for finite N _{are a} result of fluctuations in the

height

h which

vanish in

comparison

to the radius of the membrane _as N _{- co} since

(

< I. This _can be _seen very

simply by considering

the energy ^{cost to}

produce

_a fluctuation in the membrane for

large

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 the

Legendre 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

form

E ₌

K(h/R)~. (7)

Here _K is the renormalized

bending rigidity

which scales _as

R~,

where _~

= 2

2(

for D

= 2.

Therefore,

for E of order

kBT, (h~ /R~)

+~ R~~ which vanishes _as R _{- co.} The

disparity

in the values of _u; is due to finite size corrections. Thus the

ratio11 /13

must increase with

N,

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 8.0}

9.0 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 similar

argument

also

applies

to the

largest

2

eigenvalues

for _open membranes which also must be

equal

in the

asymptotic

limit. This

point

has

apparently

been overlooked in _a

number of

previous simulations, apparently

because for membranes with _no holes

(n

=

0),

the

dependence of11 /13

_on N is _very weak.

Only

for

perforated

membranes _as those studied here is this correction to

scaling

_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 free

perimeter

_can

give

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 free

perimeter.

His simulations gave ^a

significantly

lower estimate for

(.

An alternative method for

eliminating

the free

perimeter

is to

study

closed vesicles _{as we} have done here. In this case, the

height

fluctuations

(h~)

can be

easily

measured from the fluctuations of the _mean

squared

radius of

gyration,

j~2 ~

(~j~2)2 jl/2

₌ j~4 j~2

2jl/2 (~)

g

g) g)

In

figure

7, we present results for

(h~)

_versus N for _n

= 0 and 8. We find

(

₌ 0.55 ~ 0.02

for _n

= 0 and

(

₌ 0.56 ~ 0.02 for _n

= 8,

only slightly larger

than Abraharns [22] ^{results for} membranes with

pbc (

₌ 0.53 ~ 0.03. Our results _are

significantly

lower than those observed for _open membranes with free

perimeters, (

₌ 0.64 ~

0.03,

_as well _as earlier results of Komura

and

Baumgfirtner

_[32] for _a closed vesicle with _n

=

0, (

₌ o.645 ~ o.08. Our results _are close _to those obtained from _a self-consistent

screening approximation by

Le Doussal and

Radzihovsky

[23] ^{who find}

(

= o.59.

One of the

original

motivations for

studying large

membranes with holes _was to

investigate

whether

reducing

the local

binding rigidity

^would cause the membranes to

crumple.

Abraharn [12] ^showed very

conclusively

in his

study

of

perforated

_open ^membranes that the order _param- eter

[33],

^defined as the ratio of the the mean square radius of

gyration

_over the membrane size,

vanishes

only

as the _mass fraction of the

perforated

membrane vanishes. In

figure

8, _we present similar evidence for _our closed membrane. Here _we

plot (R()/L(,

where

Lo

₌

nv(n

₊

I)

is the

length

^of an

edge

in the initial

configuration

of the membrane and is _{a measure} of the

membrane's size. In

agreement

with

Abraham,

_we find that the membranes remain flat and do not

crumple,

even when the holes _are

quite large

and the _mass fraction is

quite

low.

Experimentally

_x-ray and

light scattering

have been used to

study randomly

oriented _mem- branes

[24, 27, 28].

^In

figure

9, _we present _our ^{results for the scaled}

S(q) IN

for _n

= 8. Unlike open

membranes,

there is extra structure in

S(q)

because the membrane is closed. Since the vesicles _are to first

approximation spherical

shells of diameter 2R

+~

Nl/~,

it is not

surpris- ing

that the first few minima scale _as

qNl/~

These oscillations _{are more}

pronounced

than

one finds in the

experimental

data due both to the ideal

monodispersity

and

relatively

^low

value of _n. For _{n =}

o,

these oscillations

persist

_over the entire _{q range.}

However,

for _n

=

8,

there is also _a

high

_q

regime

^{which scales} as

qN~/~

with b _cz o.85 ~

o.05,

consistent with the

isotropically averaged S(q)

for _open membranes

[25, 26].

^{Because of the} oscillations due to the

monodispersity

of the

sample

at small _q, this

scaling regime

is

relatively

small and extends

only

from 3 <

lnqN~/~

_< 5. This value of b

corresponding

to

roughness

exponent

(

= 3

2/b

_cz o.65 ~ o.05. While this value is

slightly larger,

^it is not

particulary

reliable

compared

to that determined

directly

^{from the}

height fluctuations,

due to the limited

scaling regime

in

S(q).

4.

Dynamics.

To determine if the membranes studied have

equilibrated,

it is useful to measure the time

dependent

correlation function for the _{mean square} radius of

gyration, 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 membranes

are

weakly coupled

to _a heat

bath,

^{the membranes should have} a Rouse-like relaxation. Kan- tor et al. [3]

suggested, following

the

analysis

^for _a ^linear

polymer

chain

[34],

^{that the}

longest

relaxation time _rR

+~

Nl+"

For _a flat

membrane,

_u

= I and the

analogy

^would

predict

that _rR

+~

N~.

However _{as seen} in

figure lo,

_our data scale _as

t/N,

not _as

t/N~.

This is in agreement ^{with earlier data of Komura and}

Baumg£rtner

^{for closed vesicles with} n = o.

They suggested

_one

possible explanation

for this faster relaxation observed is that for flat membranes, the relaxation has both _an

in-plane

and _an

out-of-plane

component. ^The

in-plane longitudinal

relaxation time scales _as

N,

similar to the _case of permanent crosslinked networks

[35],

whereas the

out~of-plane

relaxation time scales _as

Nl+'.

_This

explanation

is consistent with both _our data and that of Komura and

Baumgfirtner

if _{one assumes} that the

amplitude

of the

in-plane

contribution to

#(t)

is much

larger

than the

out-of-plane amplitude. Unfortunately,

neither

our data _nor theirs is accurate

enough

to tell whether this second relaxation mode is present.

We find that the

longest

relaxation time _rR

depends strongly

_on both the system size N and the number of _{monomers n} between each

node,

unlike

(R().

This result _can be understood

by recalling

^{that for} _a ^linear

polymer

ⁱⁿ _a

good solvent,

the

longest

relaxation time scales _as

n~+~",

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

"~

/

0.4

i~

~

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

~-4

m

~ -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 with

n = 8.

o-o

-o.5

+J

~lo0 .

~ -

~'~0.0

~~

Note added in

proof

_:

Zhang,

^{Davies and Kroll} [37] ^have

recently

shown that for closed

membranes,

there is _a linear

coupling

between the

out-of-plane

undulation modes and the

in-plane phonon modes,

which _suppresses the

out-of-plane

fluctuations _at

long length

scales.

This leads to _{a new}

scaling

^relation for what _we identified _as the

height

fluctuations _< h~ _>

in

equation (8).

As defined in

equation (8), Zhang

et al.

point

out that _< h~ _>+~

N~i,

where (1 ₌

(2 ~/2)/(2

+

~),

instead of

N~,

with

(

₌ (1

~/2). Reinterpreting

_our results in terms of

(i gives

_~ = o.85 ~ 0.04 and

(

₌ o.58 ~

o.02,

^which _agrees with the theoretical

prediction

^of

Le Doussal and

Radzihovsky

_[23].

Acknowledgements.

We thank D. C.

Morse,

S. T.

Milner,

D. R.

Nelson,

T. C.

Lubensky,

and T. A. Witten for

helpful

discussions.

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