to pearl- membrane models where due to hard sphere repulsion P (cos 0 ) = 0 for ..... BINDER. K., J. Chem. Phys. 75 (1981) 2994. [19]. BAUMGARTNER. A.,.
Phys.
J.
(1991)
1France1
NOVEMBRE1991,
1549-1556
1549
PAGE
ciassificaiion
Physics
Absiracts
68.10C
87.20C
82.70K
Does
polymerized
a
A.
crumple ?
membrane
Baumghrtner
Institut
Festk6rperforschung,
fur
(Received16
Evidence Monte
on
gyration
scales
R
membranes,
in
with
with
respect
conformations
simulations
of
the
surfaces
Jiilich,
D-5170
Germany
of polymerized membranes is reported. This is self-avoiding plaquette surfaces. The radius of according to R~ A ° ~. It is shown that the correlation (n(0).n(x)) for decays rapidly to plaquette zero
flexible A
area
normals
hard-sphere membranes exhibiting long ranged strong P(cos 0 ) of enclosed angles between adjacent plaquettes between plaquette and hard-sphere membranes.
to
distribution
differences
to
Jfilich,
July 1991)
29
crumpled
plaquette contrast
probability
The
for
Carlo
between
function
accepted
July 1991,
Abstract.
based
Forschungszentrum
correlations. discussed
is
Introduction.
I.
Polymerized (Or tethered)
membranes
ill and are realized, biomembranes [2]. Similar as collapse in poor solvent
polymers to
can
e-g-, in in the
and
be
considered
two-dimensional
as
polymer sheets of polymers, polymerized
cross-linked case
exhibit
to
swollen
Or
in
analogs Of linear Of protein networks
membranes
conformations
expected
are
good
under
solvent
conditions.
question important but still controversial the polymerized membranes under against bending. At low temperatures resistance
is
An
avoiding
conformations
deformations,
which
fluctuations. dimension
a
fractal
dimension
the
to
leads
to
conformations
anomalous
an
»
stiffening
the temperature [4, 5] which
of the
membrane
is
self-similar
are
conformations
the of
is
membrane due
2
D
constraint
=
increasing
With
crumpled
to
with
»
with
concerned
local
the
expected [3] resistance
surface
in the
expected
to
exhibit
to
to
«
in-plane
undergo
flat shear
thermal
of
presence
transition
a
by
characterized
and
of self-
dependent
temperature
fractal
a
2.5.
D =
However,
avoiding membranes
direction with
mean
the
surfaces are
parallel field
expected crumpled phase has evidence [6-12], but instead flat to
at
the
estimates
all
temperatures
average
[5,13]
surface and
and normal
been
never
has
been
relevant («
observed
given
that
fluctuations
rough phase »).
in
simulations
based
self-
This
disagreement
is in
experimental results [14]. recent evidence against a « crumpling phenomena » is current particular membrane model, the « pearl-membrane »
important to point out, that our solely on simulations of a consisting of tethered between hard spheres. Since in this model the ratio extended length i~~~ of the tether has i~,~ of the hard sphere and the It is
of
self-avoiding tethered found only in the are
diameter
the to
be
chosen
1550
JOURNAL
PHYSIQUE
DE
lit
I
II
approximately i~,~li~~~ »0.5 in order to maintain the self-avoidance of the surface am during the simulations of this model [5-12], we that the spheres induce long suspect range correlations between the triangles, which prevents the surface to crumple. In fact, it has been pointed out [12], that second neighbor repulsions only between spheres are sufficient to crumpled conformations. suppress Therefore it obvious look for a model where spheres as a tool to maintain selfto seems avoidance This is achieved by tethered triangulated surface where the not are necessary. a triangles are flexible and impenetrable («flexible impenetrable plaquettes»). This model provides for the first time some evidence that flexible polymerized membranes crumpled. are from simulations of the plaquette The results model reported and discussed below. are
2.
Model
and
boundaries
and
of cell
models be
plaquette spherically
of
types
Two
technique.
simulation
have
closed
which
membranes
by changing
induced
models
the
been
simulated:
membranes
exhibit
called
[15], the
pressure
membranes «open» vesicles ». Vesicles of are
shapes [2]. Shape
different
many
osmotic
«
temperature
with interest
transformation
as can
composition
the
or
free
of
lipids. The initial configuration of a molecular model of a vesicle consists of a spherically closed triangular mesh. Starting from an icosahedron, one adds new points on each triangle foflowed by a subsequent rescaling of all bonds to the desired length [16]. This procedure insures that of the grid points have 6 neighbors and each bond has approximately the same length. most The vesicles consist of N 10 x 3~ + X vertices (or ») with km I. The number monomers ) while the number of bonds (or edges ») is of triangles (or « plaquettes ») is N~ 2(N x quantities satisfy Euler's theorem where ). These 3 (N N N~ N~ + N~ X X =
=
=
=
Euler characteristic, and g is the number of handles of the 2 g is the genus or manifold, which is g 0 for a sphere and g I for a torus. The initial configuration of membrane with free boundary consists of an open vertices (k 2) triangular mesh, where the shape of the mesh is 3 k(k I N I + on a a m 2
x
=
=
=
=
hexagon. Accordingly, or
tethers
«
Monte
location
which
free
to
Carlo
In
