Hard rod and frustum model of two-dimensional vesicles

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Hard rod and frustum model of two-dimensional vesicles

Ryota Morikawa, Yukio Saito

To cite this version:

Ryota Morikawa, Yukio Saito. Hard rod and frustum model of two-dimensional vesicles. Journal de

Physique II, EDP Sciences, 1994, 4 (1), pp.145-161. �10.1051/jp2:1994120�. �jpa-00247945�

Classification

Physics

Abstracts

87.20 36.20C 05.20

Hard rod and IFustum model of two-dimensional vesicles

Ryota

Morikawa and Yukio Saito

Department

of

Physics,

Keio

University, 3-14-1Hiyoshi,

Kohoku-ku, Yokohama 223,

Japan

(Received

12

July

1993,

accepted

in final form 13 October

1993)

Abstract. A two-dimensional vesicle of hard rods

subject

to _a

shadowing

interaction is

analy2ed by

Monte Carlo simulations. From the mode

analysis

of the

shape

fluctuation of the vesicle, the

bending rigidity

is estimated

microscopically

^and the relation between the rod

length

and the

bending rigidity

is studied. A

scaling

relation of _{a mean} square radius of

gyration

is

investigated

and compared with that

proposed by

Leibler,

Singh

and Fisher.

Increasing

osmotic pressure a vesicle

performs

_a

shape

transition from

a circular to

bi-lobocyte shape.

A vesicle with _a spontaneous curvature is also studied

by using

molecules of _a frustum

shape.

1. Introduction.

It is

widely

known that natural and artificial membrane vesicles _can

change

their

shapes

under various conditions

[1-3].

^A

typical example

is the red blood cell of _a closed membrane _or vesicle which deforms

by varying temperature, pH

of the

solution,

osmotic pressure, and so on. In order to understand _a mechanism and the relevant

physical quantities governing

these

shape transformations,

a continuous elastic model is

widely

in use

[4-6].

^{The so-called} spontaneous-

curvature model [7]

proposed by

Helfrich [5] has the curvature elastic energy per unit _area of

a fluid membrane in the form _as:

g~ =

~~c(ci

_{+ c2}

co)~

+ k~cic2

(1)

2

where _~c and kc are the curvature elastic modulus

(bending rigidity)

and Gaussian _curvature elastic

modulus,

_cl and _{c2 are} the two

principal curvatures,

^and _co is the

spontaneous

curvature.

Under _an external field such _{as an} osmotic pressure difference AP between the outside and the inside of the

vesicle,

the total free energy of the vesicle is

represented

_as

Fc

=

/ _gods

+

/ _{APdV, (2)}

where the first and the second

integrals

are taken _over the surface and the volume of the

vesicle,

respectively. By minimizing Fc

for _a

given

surface _area

S,

various

shapes

of the

vesicle,

which

have been observed in

experiments,

_are realized.

Though

this and the related models

[8,

9]

explain shapes

of the vesicle in real

systems,

^{effects of the thermal} noise and

shape

fluctuation

are not taken into consideration.

Thermal noise has been taken into account

by

_means of Monte Carlo simulation of micro-

scopic

^{models such} as tethered hard

spheres

in three dimensions

[10, 11]. Scaling

^{laws for the}

mean radius of

gyration

under various conditions and

crumpling phenomenon

have been found.

Since the simulation of the tethered membrane in three dimensions needs _a

long

CPU

time, Leibler, Singh,

Fisher

(LSF)

have studied _a tethered

ring

in two dimensions

[12-16]

in detail.

Albeit with all these _success of continuum and tethered

ring models, they

_are

stillinadequate

when _we want to understand

microscopic

effects such _as the individual

shapes

of the molecules

or interactions between them. For

example,

in real

system, phospholipid

molecules which _con- struct the

biological

membranes

together

with

protein

molecules [17] are

anisotropic

and have

polar hydrophilic

heads and

oily hydrophobic

tails. The

origin

of the

lipid bilayer

formation is believed to be _a

hydrophobic

interaction

[18]. Hydrophobic

chains of

phospholipids

lead

configurational ordering

in the

surrounding

water molecules and _{cause an} entropy ^reduction and the free energy increase. For detailed

understanding

of the

microscopic properties

of the

membrane,

the

hydrophobic

effect is

indispensable.

The

bending elasticity

is believed to be in- duced

by

such _an

anisotropy

of the molecule

[19],

^{and the}

spherical

molecule _as in the tethered

membrane model is insufficient. In

addition,

in

describing shape

transformations

accompanied by topological changes

such _as

endocytosis

_or

exocytosis,

tethered membrane model needs _an

artificial bond recombination

[20].

Taking

account of _an

anisotropy

of

phospholipid

molecules and

microscopic

interactions between

them,

_we have

recently proposed

_a

simple

model

[21],

^{in which} a

pair

of

phospholipid

molecules in both

layers

is

regarded

_{as a}

single

hard rod. These hard rods cohere and form lamellars _or vesicles

by

the attractive

interaction,

which mimics the

hydrophobic

interaction.

Here _we describe the extended

investigation

_on various

shapes

of _a

ring

vesicle in _a two- dimensional system. ^{We estimate the}

bending rigidity

from

microscopic

interactions between rod molecules instead of

treating

it _{as an a}

priori given

parameter as in LSF model. We also

study topological changes

of the membrane induced

by

the spontaneous ^{curvature in} two dimensions. Since in LSF model there is _an

assumption

of the

connectivity

between hard

spheres, topological changes

of the vesicle _are not allowed and effects of the spontaneous curvature for the

thermodynamic properties

of the model

only

leads _to trivial _energy shift

[I].

^In our

model,

_on the

contrary,

the cohesion of rod molecules _are caused

by

a caricature

hydrophobic

interaction _or

"shadowing interaction",

and the

topological changes

of the vesicle

are

permitted.

We introduce _a spontaneous curvature into _our model

by changing

the

shape

of _a molecule from _a rod to _a frustum.

In section 2 the model is

explained

in detail and is studied

by

the Monte Carlo simulation at finite temperatures. From the simulation

results,

_a

bending rigidity

of _a continuum elastic

model is estimated. We also

investigate scaling

relations of _{a mean square} radius of

gyration

and of _{a mean area} of

vesicles,

and compare the results with those

proposed by

LSF. When _an osmotic _pressure is

applied

_on the

vesicle,

_a

shape

transition of _a vesicle from circular to bi-

lobocyte

is

expected according

to LSF. This effect of the osmotic pressure on vesicles is studied in section 3. In section 4 the model with _a spontaneous ^{curvature is studied}

by using

molecules of _a frustum

shape

and observe

topological change

^{of vesicular}

shape.

Final _summary and

conclusion is

given

in section 5.

2. A hard-rod model of _a vesicle.

2.I A _{MODEL wiTH SHADOWING} INTERACTION. We consider _a hard-rod

system

in two

dimensions _as is shown in

figure

I. Each rod has _a hard _{core to} prevent rods from

crossing

each

other,

and the minimum distance between _two rods is _{set a.} To stabilize the membrane

structure, we introduce _an attractive

shadowing

interaction to mimic the

hydrophobic

inter- action

[18].

^When a rod I is

isolated,

both of its sides _are naked and face _water such that the

ordering

is induced in the

surrounding

water molecules with _an energy ^cost J _per

length.

When another rod

j stays

close to the rod

I, part

^{of the rod I is covered}

by

the shadow of the rod

j projected

_on rod I and is shielded from the water

(Fig. I).

We _assume that

only

the

remaining

naked part of the rod I _causes the _energy increase. Therefore _the

"hydrophobic"

energy is written _as

Eh

₌ J

£

_In~,

₍₃₎

1

where In~ is a total

length

of the naked part ⁱⁿ rod

I,

and the summation is taken _over all rods.

Compared

to the hard-rod diameter _a, the

length

of the rod is set

long enough

to

yield

sufficient cohesion.

The

shielding

is most effective when the

neighboring

I and

j

rods _are

parallel

with each

other,

and

perpendicular

to the vector r~~,

connecting

the _mass centers of the _two rods. Since

we do not know

precisely

how far the

ordering

of water molecules extends _or how it

decays,

_we

simply

_assume here that the effect of the

screening

extends

uniformly

in the

shielding region,

and vanishes outside of _a cutoff. The

shielding region

is rather

arbitrary

chosen _{as a}

rectangle

with _a

height

21 and _a width

2rc

around the rod _as is shown in

figure

I. When two

widely separated

rods _are tilted

against

each

other,

the

projected length

is short and the

shielding

2rc

i

, i

i

, ,

i ,

i

21

' '

~----

Fig.

I.

Lipid bilayer

is

regarded

_{as a} hard rod. Hard rod interact with

shadowing

attraction,

modeling

the

hydrophobic

interaction.

becomes less effective. When shadows from different rods

overlap, they

cannot contribute to the

shielding additively,

and thus the

present

interaction is not _a

simple twc-body

force but _a

many-body

_one.

When N rods form _a

ring,

the

ground

state

configuration

is _a circle with _an allowed minimum radius such that every rod

points

out

radially

with the

neighboring

inner ends

being

in contact.

The radius in the

ground

state is calculated to be

Ro

=

Ii

+

al sin(7rN~~)) /2

with _arJ energy

Eo

=

2NJ(Ro +1/2)(1- cos(27rN~~)).

At _a finite

temperature

^the

ring

deforms _to

gain

the orientational

entropy

of each rod and the

configurational entropy

associated with the

global shape

variation. To allow for the

deformation,

the radius of the

ring

increases at finite

temperatures.

^The

shape

of _a vesicle is described

by

_a contour line

connecting

^the _mass centers of rod molecules.

By assuming

that the

shape

variation of the vesicle takes

place

_very

slowly

compared

to the

adjustment

of the rod orientation to the normal direction of the

shape,

the energy for _a

given

vesicle

shape

is

represented

_as

~~

_" ~

~j ⁽ sin~~~/2)(

^~

~

~~

~°~~~~~'

_~~~

where

As~

is the

separation

between the _{mass centers} of the rods I and _{I +}

I,

^and

A#~

is _a difference of the orientations of rods I and I +1. With the

assumption

that the

separations

between

neighboring

rods _are all

equal

_as

(As)

=

SIN

for _a total

perimeter length

_s, and that the

angle

difference

A#

is

small,

the _energy

(4)

is written in _a continuum limit _as

Eh

*

~~/~~ / _H~ds

+

J(As) / _{(H(ds, (5)}

where H

=

limAs-oA#/As

is the curvature. For _a

singly

connected

ring

with _a

positive

curvature, ^I-e- an

oval,

the

integral

of curvature

(H( along

the closed contour is _constant to be 27r and the second term

gives

_a trivial contribution. Then the _energy

Eh

reduces to the elastic

energy of _a continuum

theory

in two

dimensions;

Ec

=

/ _{H~ds, (6)}

with the curvature elastic modulus _or the

bending rigidity

~ estimated _as

~ =

Jl(As). (7)

At

higher

temperatures, ^where fluctuations _cannot be

neglected

_{any more,} the estimation

(7)

_may be

insufficient,

and _we estimate _~ from the undulation of the contour of the vesicle.

For _a

shape

described

by

_a

polar

coordinate _{as r}

=

r(R),

the undulations of the vesicle

shape

are

decomposed

into Fourier modes

[22]

as

When the

perimeter length

_s is fixed and the

equipartition

of energy among different modes is

adopted,

the

amplitude

is obtained to be [21]

~~'~"~~~

87r2~~~ ₁₎₂

^~~~

By using equation (9),

_{we may} estimate

accurately

the

bending rigidity

_~

by obtaining

mode

amplitudes ((un(~)

^{and the}

perimeter length

s from the simulation.

2.2 SIMULATION. In order to understand the

thermodynamic

behaviors in

detail,

_as in the

case when the

shape

deviates

largely

from the circle at

high temperatures,

we have to

rely

on

Monte Carlo method because of the

complexity

of _our

microscopic system.

In _a Monte Carlo

trial,

_a rod is chosen

randomly.

Its center of _mass and orientaion _are tried to be shifted

randomly.

These trials _are

accepted

_or

rejected according

to the Boltzmann

weight exp(-AE/kBT),

which is calculated from the associated _energy

change

AE and the

temperature

^{of the}

system kBT

in order to

satisfy

the detailed balance. The maximum shift and rotation sizes _{are so} chosen that about

fifty percent

of the trials _are

accepted.

In the

simulation,

the rod

length

is taken _to be 10 times of the hard _core radius _a, and the attraction range rc is set 3a. As _an initial

configuration

we prepare N=100 rods in _a

circular

ring

of the

ground

state

configuration

in _an

infinitely

extended space. The radius in the

ground-state

is

Ro

" 20.9a and the energy per rod is

Eo /NJa

=0.102. This _energy is

quite

low

compared

to the value of 20 when the rod is isolated. Simulations _{are run up} to I x

10~

Monte Carlo steps

(MCS)

for every rod at various

temperatures.

At low

temperatures, kBT/Ja <I.I,

^the

ring

vesicle is stable

(Fig. 2a).

At

kBT/Ja

m 1.3

~w

3 the

ring

breaks apart at _one or few

points

to form

fragments (Fig. 2b).

At still

higher

temperatures ^the system ^breaks down to _a collection of isolated

rods,

as shown in

figure

2c. At

a temperature

kBTc /Ja

=

1.2,

_we obtain _a precursory behavior of

phase

transition such that _a rod leaves and _{returns to} the vesicle. At low

temperatures,

T <

Tc,

where the vesicle is

stable,

the _energy

Eh/NJa

is found to be

proportional

to the

temperature kBT/Ja,

and the

specific

heat

GIN

=

((E)) (Eh)~)/NkBT~

remains constant about

unity.

The

specific

heat obtained

by

the energy fluctuation is

compatible

with that obtained

by

the temperature derivative of

the energy,

indicating

that _an

equilibrium

^is reached in the simulation. Since _our Monte Carlo simulation does not contain the kinetic energy

contribution,

the value

unity

of the

specific

heat

can be

interpreted

that there _are

essentially

two vibrational

degrees

of freedom in _our

ring

vesicle. These two

degrees

of freedom _may be the vibration of the rod orientation and that of the _mass center of the rod normal to the

ring,

_or of the

shape

fluctuation.

20

,jfAW'>

~

)

iV

5 z

fi

)

j

(

Z ,,

&

/

_(b)

£j

^~

~~~~~~~

(C)

~

(

~-~

~,'~

%llnJld

Z

$'

~ ~

#~~~~ $

~'

~~~

' ~'

t '

/

/ ,

0 2 3 4 5

k~T

/ Ja

Fig.

2. Energy _per rod

(Eh)/JaN

_versus temperature ^and snapshot of the

configuration

of 100 rods at

(a) kBT/Ja

₌ 0.5~

(b)

1.6 and

(c)

3,o.

2.3 ESTIMATE _{OF BENDING} RIGIDITY.

Bending rigidity

_~ in the

microscopic

^model can be estimated

by using

^the

shape fluctuation, equation (9).

The inverse root square of the mode

amplitude I/ ((un(~)

is obtained from the

simulation,

and the result is

plotted

in

figure

3

against

the m6de number

n~

_{I at various}

temperatures

^{for the}

system

^{with N} ₌ loo. At low

temperatures

the de,>iation of the vesicle

shape

from the circle is small and relation

(9)

is found _to be satisfied. From the

slope

of

figure 3,

^the

bending rigidity

_~ is estimated

by

the least

square fit to

equation (9).

_~ is found to increase at

relatively

low

temperatures, kBT/Ja

<

0.2,

but remains almost constant at

~/Ja~

_m 32 at

kBT/Ja

_m 0.2

~w o-G _as is shown in

figure

4.

Estimation

by equation (7) yields ~/Ja~

_m 24 in

agreement

^of the order. The difference may be due to the

neglect

of the

separation

fluctuation in the estimation

(7).

For

kBT/Ja

_{> o-G,} the

linearity

of

((un(~)~~/~

to

n~

_I becomes _poor, and the estimation of the

bending rigidity

_~ is not

fully

confidential. Here the fluctuation

((un(~)

of the mode

with small _n is

expected

to become

large,

but the

neighboring

inner ends of hard rods _may

contact each

other,

and the hard _core

repulsion

between them reduces the fluctuation from the

expected

value. Also the contribution of the total curvature in

equation (5)

_may stiffen the vesicle when the fluctuation becomes

large.

According

to the

rough

estimate

equation (7)

^of the

bending rigidity,

_~ should be insensitive to the

system

size for

large N,

since the average

separation

^{of the} two

neighboring

rods

(As

is determined

by

the local interaction of the hard _core

repulsion

and the

shadowing

attraction.

Simulation shows the

systematic

increase of

~/Ja~

when N increases from 40 to

loo,

but it

seems to converge ^to an

asymptotic

constant value at

kBT/Ja

< 0.6 for

large

N _as shown in

figure

4. For the

system

with 40 rods _~ deviates

largely

from the

asymptotics.

Since the effect of the molecular size limits the

shape

fluctuation of the

vesicle,

the smallness of the

system

soo

. kBT/Ja=0.I _.

o----~- k~T/Ja=0.3 . .

400 _. kBT/Ja=0.5

. .

i kBT/Ja=0.7 ^° O

j~ +-k~T/Ja=0.9 .

*

o o o

ji

300 ^°

e

m ;.'o

[

_. . ^" . . .

R :.' ^~

..

i~

₂₀₀

;-"I'

_ll~

.

~ ~ ~

° ~

° 2

u

.." I-I'

, ^u

_:~'

. N=100

loo

(~"~~

u N=40

0 0 20 30 40 50 °

n2 k~T

/ Ja

Fig.

3

Fig.

4

Fig.

3. Mode

amplitude ((un(~)~~~~

versus mode number n~ I, at various temperatures. The lines represent ^least square fits and from the

slopes

^the

bending

rigidity is estimated.

Fig.

4

Bending rigidity

_~ estimated from

equation (9)

_versus temperature for systems

containing

various numbers of rods N.

. kBT/Ja=0.5

o k~T/Ja=0.I

.

.---- keT/Ja=0.05

kBT/Ja=0

:"j~

._."/ ^o

_.£I' ,--II o_/*

~

."/~,l'"

:$~ .l'~

,:$~ l'~

:7~ ^.I'

o

/ _a

Fig.

5.

Bending rigidity

_{~ versus} rod

length lla~

at various temperatures ^{for N} ₌ 100. The three

curves are obtained from equation

(11)

with

fitting

parameters

e(0.5)

₌ 1.42,

e(0,I)

₌ 1.46 and

e(0.05)

= 1.23. A solid _curve represents the _case of

2ero temperature; _~ =

Jl(a

+ lsin

7r/N).

size affects

strongly

in the estimation of the

bending rigidity

_even at low

temperatures.

Another characteristics of _~

expected

from

equation (7)

is its

dependence

_on the

length

of the rod I. At _zero

temperature

As is

proportional

to the rod

length

_as

Aso

= a + sin _7r

IN.

Therefore,

the

bending rigidity

_~ is _a

quadratic

function of I at

kBT/Ja

= 0. At finite

temperatures, kBT/Ja

=

0.05,

0.I and

0.5,

_we obtained _~ from simulations of

systems

^with various rod

lengths

_as is shown in

figure

5. _~ is well fitted _to be _a

quadratic

function of

I,

as is drawn

by

the lines in

figure

5. The result is consistent with that of the _mean

separation (As),

which _can be well fitted

by

the linear law in _as

(As(T, I))

=

kjT)

₊

lijT). (lo)

~ and

(As)

satisfies the relation

~ =

Ji(As(T, i))e(T), (11)

with _a

parameter BIT) being independent

of the rod

length

I.

We also find that the

stability

of _a vesicular

shape depends strongly

_on the rod

length

I. At

kBT/Ja

= 0.5 the vesicle is stable when >

5a,

but _torn off when < 4a. Destabilization of

a vesicle for _a small

aspect

^ratio I

la

is well

expected,

since for _a

system

^with = 0 the model reduces to a

simple

hard-core

system

and no cohesion is

expected

at _zero pressure. Even at the lowest

temperature

as

kBT/Ja

=

0.01,

_a

system

with

= a can not

keep

a

ring

form

stably

but breaks _up at one

point.

At

kBT/Ja

= 0.2 all the rods become isolated. This

represents

the

importance

of _a

large

aspect ratio

la

for the formation of _a stable

layer

structure.

2.4 SCALING _FORMS. For _a

polymer

chain connected with N _monomers, many universal

laws have been derived

by applying

the

theory

of _a

self-avoiding

walk

[23].

^{One of them is the}

scaling

relation between the

Flory

^radius

RF

and the

degree

of

polymerization

N _as

RF

+~

N~.

The

Flory exponent

u is

independent

of the details of the _monomer structure. For _a

membrane,

scaling

laws for the _square radius of

gyration (RI

and the _{mean area}

(A)

have been studied in detail

by using

^LSF

[12-14],

or

Ostrowsky

and

Peyraud

_[15] model in two dimensions. We

expect

^{that the} same

scaling

laws _are satisfied in _our model of _a vesicle constructed

by

hard rods.

Due to the

bending

_{energy cost,} thermal fluctuation is restricted to modes with

wavelength longer

than the

rigidity length:

L

₌

~/kBT. (12)

In _our

model,

with N

=

100,

the

rigidity length L

decrease from 480a at

kBT/Ja

= 0.05 to

40a at

kBT/Ja

= 1.1. If the

system

size _s is smaller than

L,

_or the dimensionless

quantity

§ "

8/lK l13)

is less than

unity,

_we

expect

a circular

vesicle,

and square radius of

gyration (RI

and _mean

area

(A)

_are then

proportional

to

s~.

_{On the other}

hand,

at

high temperatures

^when

L

becomes smaller than _{s, we may} expect a fractal

shape

of _a vesicle. Then

(RI

_and

_(A)

_{may have the}

scaling

relation [13]

(RI)

=

~

)~ ^{U(y), (14)}

27r

IA)

~2

=

pvlv), lls)

with

scaling

forms

U(y)

and

V(y)

with

U(0)

=

V(0)

=

1,

and

U(y)

~w

V(y)

~w

y~~(~~~)

=

y~~/~

for _{y »} I. Here _u is

expected

to be

3/4

in two dimension.

By

linear mode

analysis,

_we calculate

U(y)

and

V(y)

_up to the first order of _y for small _y;

~~~~ 327r~ 4~

~ ~~~~

Vlv)

= i

~v. _Ii?)

Both

analyses

_are valid for _{y <} I where the

higher

order terms of _{un are}

negligible.

The

scaling (14)

and

(15)

_are

investigated

for

systems

^with N ₌

40,

_Go,

80,

and 100. We observe _a stiff

region

where

(A)

_~w

(RI

~w

s~ for _{y <}

I,

^{and the linear mode}

analysis (16, 17) yields satisfactory

fit to the data for _y < I.

However,

the soft

region

with behaviors

(A) (RI

~w

~w

s~~

with the

Flory

_{exponent u}

=

3/4

is not obtained.

Instead,

at

high temperatures

with y »

I,

thermal fluctuation is _too wild _to

destroy

the vesicular conformation itself

(Fig. I).

To observe the soft

region

with

large

_{y, we} have to simulate _a

large system,

^which is out of _our

present capability.

3. Osmotic _pressure effect.

3.I TRANSITION _To Bi-LOBOCYTE. In _a real

system,

osmotic pressure of the vesicle

plays

an

important

^role to the deformation of the

shape.

With the curvature elastic model

(2),

the

bending rigidity

_~c, the

spontaneous

curvature co and the osmotic pressure AP control the vesicle

shapes

^such _as

discocyte, stomatocyte,

dumbbell and _{so on}

[6].

^{In LSF model}

[12, 13],

the circular vesicle is transformed into

bi-lobocyte by

osmotic pressure, and _a

phenomenon

of nonlinear

flickering

is also observed. We

expect

a similar

shape

transition in _our model caused

by

external _or osmotic pressure

applied

to the vesicle.

.o

_=

,41 h

( <I

>,

$ _,j

(<

cQ

~$,,~J'

~~~

~ )

~

(~ ) _~

r, j

~ j lj

~'~

~

'4fl4~

16)

. 4n<A>/s~

O y~

o ooi

APa/J

Fig.

6. Reduced _area

4x(A)/s~

and compressibility _{~iT versus} osmotic _pressure

/hPa/J.

Center

of mass of about 40 statistically

independent

vesicle

configurations

have been

superimposed

with

alignment

of the

principal

_axes of inertia at

(a) kBT/Ja

= 0.003,

(b)

0.006 and

(c)

0.009.

The osmotic pressure is defined

by

_{a pressure} difference between the interior and the exterior of the vesicle:

AP _e

P~x tin. (18)

In Monte Carlo

simulation,

the _energy of the

system

is chosen _as

Ehp

=

Eh

+

APA, (19)

where A is _{an area} enclosed

by

the

connecting

line of the _mass centers of the consecutive rods.

Therefore,

when the vesicle breaks

apart

or some rods

change

their

position

in the

vesicle,

_our simulation loses the

validity.

First _we carry ^out the simulation with N

= 100 rods at

kBT/Ja

₌ 0.5.

Figure

6

depicts

the variation of vesicular

shape

_{as a} function of _an external _{over pressure,} AP _> 0. It shows the

superposition

of the center of _mass of each rod at various time

steps

at various pressures.

When AP is

small,

the

shape

of the vesicle is circular and does not differ much from that at zero pressure

(Fig. 6a).

At

APa/J

=

0.006,

a fluctuation of the vesicle

shape

becomes _very

large (Fig. 6b),

and for

higher

_pressure

APa/J

>

0.008,

the vesicle

collapses

into

bi-lobocyte (Fig. 6c).

The

compressibility

for the vesicle is obtained from the _area fluctuation _as

~~~

~~~~

~T =

(A)k~T

^~~°~

~T has _a maximum at

APa/J

= 0.006

(in Fig. 6)

associated with the

shape

transition. A reduced _area

47r(A) /s~

also reflects the

shape transition,

_as is shown in

figure

6. It is almost

constant for

APa/J

<

0.005,

but decreases

sharply

when the vesicle takes _a

bi-lobocyte shape

for

large

hp.

For

large

_pressure

difference, APa/J

_>

0.01,

two _concave sides of

bi-lobocyte

contact with each

other,

and _some rod molecules

change

their _concave sides. Thus with _{a very}

large AP,

the _area of the vesicle _cannot be defined

appropriately

with _our

algorithm

and the simulation has to be terminated.

JOURNAL DE PHYS'QUE II -T 4 N'' J~NU~R~ lQQ4 6

3.2 LINEAR STABILITY ANALYSIS OF THE SHAPE. Here _we

analyze

the

stability

of the vesicle

shape by

_means of _a mode

analysis.

The _energy of the

system

is written in the continuum

theory

_as

Ecp

=

Ec

+ AP

/

_dA.

₍₂₁₎

By decomposing

the vesicle

shape

in the Fourier modes _as in

equation (8),

and

expanding

the energy up to the second order of mode

amplitudes,

_we obtain the thermal _average of the mode

amplitude

_as

~~'~"~~~

87r2~(n2

)(n2~~~~-

APs3

/(27r)3~)

^~~~~

From

equation (22),

the

amplitude

_u2 of the mode number _{n =} 2 is found to

diverges

at

APO

e

247r~~/s~.

^{For the}

present

case at

kBT/Ja

=

o-S, parameters

are estimated

roughly

as

sla

= 220 and

~/Ja~

= 30. Therefore _we expect that at

APca/J

_m

_0.002,

the vesicle should become unstable and

collapse

into

bi-lobocyte.

The order of this estimation _agrees to the result of

simulations, showing

that the

shape

transition _occurs at

APa/J

_m 0.006. The

discrepancy

_may be attributed _to the _term

J(As) f (H(ds

in

equation (5),

since the

shape

of the vesicle is _no

longer

oval _near the

instability point

and the total curvature

f (H(ds

varies

over 27r. For

large

_pressure

difference,

the effect of the total curvature in

equation (5)

increases and restrains the

shape

from

transforming

into non-oval

shape,

I.e.

bi-lobocyte.

Therefore

larger

_pressure difference

APca/J

m 0.006 is

required

in the simulation than the theoretical

expectation APca/J

_m 0.002 to the transformation into

bi-lobocyte.

The

reciprocal

of the mode

amplitude Rn

=

1/[((un(~)(n~ _1)]

is

plotted against

the mode number

n~

_{I for} various pressures as shown in

figure

7. For

APa/J

<

0.006, Rn

is

linearly proportional

to

n~

_{I and the}

slopes

do not vary much _{as pressure} varies. Hence from

equation (22),

the

bending rigidity

is almost

independent

of the _{osmotic pressure} hp. In

figure 7,

the

o APa/J=0.001

~~~~

u APa/J=0.003

800 _O APa/J=0.005

%~

. APa/J=0.007

~l . APa/J=0.009

)600

fl

j

(400

~

200

j

4

6

,

~

0 20 40

n~ -1

Fig.

7. Reciprocal mode amplitude

(((un(~)(n~ -1)]~~

versus mode number n~ _{I, at various}

pres-

sures. The solid line represents _a theoretical

expectation (23)

^{when the}

divergence

of mode

amplitude

occurs at _{n =} 2.

data of

Rn

for

APa/J

< 0.006

stay

above the critical

line;

(un(2))n2

1) ~~~

~'~~ ~~ ~~~~

where the

instability

of the second

mode,

_n

=

2,

takes

place.

For

APa/J

>

0.006,

the vesicular

shape

deviates

largely

^from _a

simple

circle and the value of

Rn

becomes small

except

_{n =} 3,

indicating

that the

relation(22) completely

breaks down.

The linear mode

analysis

of the reduced _area

47r(A) /s~, equation (17),

_can be extended for the

present

case with _pressure difference. It is

expected

to

satisfy

the relation

~)/~

₌ l

)Z(W), 124)

where

Z(w)

is a

scaling

form

~~~°~

j~(1~~~2)

/~

~°~ ~~°' ~~~~

with _a dimensionless parameter w defined _as

fi(26)

" _" ~

APO

The relation

(25)

is shown

by

_a solid _curve in

figure

8. When _{w -} 1 _or AP

-

0, Z(1)

=

3/4

and

equation (24)

reduces to

equation (17).

When _w

- wc = 2 _or AP _-

APO,

the

scaling

form

Z(w) diverges.

However since the reduced _area

47r(A) /s~

should be

positive,

the

scaling

form

Z(w)

should be less than 47r~

Iv

which is about 12 for _n

= 100 and

= 10a for instance.

Linear

analysis

becomes

meaningless

around _w

= 2.

In the

present system,

we

expect

^the

universality

of the

scaling

form

Z(w)

=

47r~(1- 47r(A) /s~) Iv against scaling parameter

w. The

parameter

w

depends

not

only

_on AP but also

. N=100 lla=10

o N=100,lla=8

. N=100 lla= 6 *

u N= 80 lla=10 °

h4 O

,a o

. "° ^oQ.

~ ~p. 2.

f~ a C~

3

on the

bending rigidity

~ and the

perimeter length

_s. These

parameters

_~ and _s

depends

_on

the rod

length

I.

Therefore,

_{we vary}

_parameter

_w either

by varying

the _{over pressure} AP _or the rod

length

I. The

system investigated

have the combination of the number of rods N and the rod

length

I

la

_as

(N, la)

₌

(loo, lo), (100, 8), (100, 6)

and

(80,10)

at

kBT/Ja

= 0.5. The

results show that

Z(w)

increases very

slowly

at _w <

2.9,

but

rapidly

at _w > 2.9 _as is shown in

figure

10. Hence the critical

parameter

_{wc may} take the value around 2.9.

The simulation results of the LSF model

[12,

13] can be

interpreted

with the

present parametrization

that the fluctuations become

anomalously large

at _w

= 2.0

~w 2,4 and then _a

new

type

of

shape

_as

bi-lobocyte

_appears when _w > 3.4. The critical

parameter

wc m 2.9 in

our model is not much different from _w

= 2.0

~w 2.4 in LSF model. We also observe the critical behavior similar to

"flickering"

found in LSF model around _wc.

4. A frustum

system

^{for vesicles.}

4.I SPONTANEOUS _{CURVATURE IN Two} DimENsioNs. To

investigate shapes

of the vesicle in three

dimensions,

the

spontaneous

curvature in

equation (1) plays

_an

important

role in _a

continuum

theory

_[6]. It is connected to the

asymmetry

of the inner and the exterior of the

lipid bilayer

membrane. The difference of _two

layers

_may be due to the chemical

difference,

to the different number of

lipids,

or to the difference in the environment. In _two

dimensions,

the energy of _a vesicle with _a

spontaneous

curvature

Ho

without the osmotic pressure is written _as

Ec~

=

f /(H Ho )~ds. (27)

2

As

long

_{as we} consider _a

single vesicle, Ho gives

_a trivial contribution [1]. ^{But when the}

topology

^of a vesicle

changes, Ho Plays

_a crucial role. When there _{are m} vesicles in the

system,

the

geometrical

relation

yields

the

integral:

/Hds

₌ ^27rm.

(28)

Then

equation (27)

is written _as

Ec~

=

~

H~ds

₊

~~°

ds

27r~Hom, (29)

2 2

~

where the

integral

is _over total

perimeter

of vesicles. The second term in

equation (29)

has the

same form of the interfacial tension of vesicles. The

topological

contribution _to the _energy

Ec~

is realized

by

the third

term,

and it

produces

_a discrete variation in the energy

by varying

the number of vesicles _m.

4.2 A FRUSTUM MODEL. The effect of spontaneous ^curvature can be introduced into the

microscopic

model

by changing

a

shape

of the molecule from _a rod into _a

frustum,

which is shown in

figure

9. The

physical

or

biological meanings

of such _a model _{are as} follows. In _a

binary

mixture of _an

amphiphilic

molecule with

cosurfactant,

membrane _may be constructed from _a

triple

of two main

lipids

and _one cosurfactant [1]. ^We may

regarded

this

triplet

_{as a}

frustum

shaped particle.

Also _one of the

pair

^of anionic

phospholipids

which form

membrane,

is

kept

in contact with divalent

cations,

the latter reduce _a

repulsive

radius of the

polar

head

effectively [18],

^{and thus the}

pair

of anionic

phospholipids

becomes

asymmetric

and _can be modeled

by

_a frustum.

71

1-_

--- ----

'3~~ _~

,

i ,

, , , i

, i

i ,'

i

i

j j '

i 21

'

Fig.

9. Scheme of the

frustum-shaped

molecule.

In

figure 9,

both sides of the

frustum,

each

having length

I, make _an

angle

_R~ with each other.

When

they

are

naked,

each side _{cost an energy} Jl _as before. The attractive and

repulsive

interaction between the frustum

shaped

molecules _are

essentially

the _{same as} between the rod

shaped

molecules. In order to prevent ^the inversion of _a frustum _or, in _a real

bilayer system,

to

prevent

^the

exchange

of molecules between _two

monolayers,

we introduced additional restriction

on the interaction such that when the direction _n~ of the i-th frustum is

opposite

to

that,

_n~,

of the

neighboring j-th frustum,

I-e- _n~ n~ ^<

0, they

_are ineffective to shield each other. This restriction

keeps

the

bigger

end of the frustum exterior of the membrane. In the Monte Carlo simulation _{we never} encounter this inversion _so far.

An energy of _a vesicle constructed

by

N frustums is estimated _as

following

at low

tempera-

ture. The

shape

of _a vesicle is described

by

the contour lines

connecting

the centers of frustum molecules with _a total

perimeter

s.

By assuming

that the average orientation of the frustum molecule is in normal direction of the

shape,

the _energy between _a frustum I and I ₊ I is

represented

_as

ohs,1 ~

J(I

₊

ii (I cos(R~ A#~ ) (30)

~~~~

~

As~

i

sin(R~/2) cos(Aj~/2)

"

(Sin i(Rs A41)/211

^{' ~~~~}

where

As~

is the

separation

between the centers of the frustums I and I +

I,

and

A#~

is the

angle

difference of the normals of the frustums I and I ₊ I. Inner ends of the

neighboring

frustums _are closer than the outer ends for

A#~

> R~, while outer ends _are closer than the inner ends for

A#~

< Rs. At

A#~

= R~, a vesicle is the _most stable.

The total energy of the vesicle

Ehs

is derived from

equation (30)

_as

Ehs

₌

~j _{Jl(I cos(R~ A#~))}

+

~j

₂

^As~

_sin

(

cos

@~

^sin

~ /~~ (32)

~

Assuming

that the frustums _are

separated equidistantly

_as

(As)

=

SIN,

and that the

angle A#~

and _{R~ are}

small,

the average energy for _a

given

vesicle

shape

in _a continuum limit is

approximated

_as

Ehs

=

~~/~~ /(H Ho)~ds

+

J(As) (1- _~°~) / _{(H Holds (33)}

2

with _a

spontaneous

curvature

Hoi

Ho

₌

R~/(As). ₍₃₄₎

The first term in

equation (33) corresponds

to

equation (27)

and

representation

of the

bending rigidity

is the _{same as} that of the rod model shown in

equation (7).

The

spontaneous

curvature

Ho

is found _to be related to the

asymmetry

_R~ ^{of the exterior and interior size of} a molecule. The second term in

equation (33)

is similar to the total curvature term

J(As) f (H(ds

in

equation (5). Except

H _m

Hoi

^the

sign

of H

Ho

remains definite for small

configurational fluctuation,

and the second _{term may} be

negligible. However,

when the

topology

of the vesicle

changes,

this term also

gives

a discrete variation additional _to the contribution

given

_as

equation (29).

4.3 SIMULATION. To realize the

topological change

of the

membrane,

_we simulate the

frustum model _at finite temperature

by

the Monte Carlo method.

First,

_{we prepare} the

ring- shaped

vesicle constructed

by

N

= loo frustums and carry ^out the simulation for various _R~

at

kBT/Ja

= 0.5. On

increasing

_R~ from 0 to

0.047r,

the fluctuation of the _area of the vesicle is enhanced and the

compressibility

_~T in

equation (20)

has maximum around _R~

= 0.047r.

The enhancement _may be caused

by

the fact that the vesicle with N

= 100 is metastable at _R~

= 0.047r

=

27r/(N/2)

and

trys

to

split

into two. On further

increasing

_R~, the average curvature

(H)

= 27r

IN

differs

largely

from

Hoi

and the energy increase associated with _a small deformation _{or curvature}

change

bH becomes _very

large

to suppress the

shape

fluctuation.

This fluctuation

suppression happens

for 0.067r < R~ <

0.147r,

and at _R~

= 0.147r fluctuation vanishes

eventually.

For _R~

larger

than

0.147r,

the vesicle

"ruptures"

and breaks into many smaller vesicles. The

stiffening

for 0.047r < _R~ is also observable in the reduced _area

47r(A) /s~.

It

approaches unity

for

large

_R~,

indicating

that the

shape

of the vesicle becomes circular at

large

_R~ like _a stiff vesicle at y ^< I

(in

Sect.

2.4).

The energy per frustum

Ehs/N

for various sizes of vesicle

(N

=

20, 25, loo)

is obtained

by

simulation at

kBT/Ja

₌ 0.5 and is shown in

figure

lo. It looks _a

quadratic

function of _R~ with

a minimum

depending

_on the number N of frustums.

Accordingly,

_a vesicle constructed

by

N = loo frustums with _R~

=

0.17r,

for

example,

has _a

high

_energy, and

gains

_energy

by splitting

into several smaller vesicles. The minimum energy ^state is _a collection of 5

vesicles,

each of which is constructed

by

20 frustums.

We simulate this relaxation _process of vesicle

splitting by using

Monte Carlo simulation.

Initially

_{we prepare} ^N ₌ ^{100 frustums in} a circular

ring

and then

gradually

increase the

2

. N=100

1 N=25

+ N=20

z 1.5

7

m

~f

A

~ l

~'~0

_{0.04 0.08 0,12}

0.16 6~ In

Fig.

10. Energy _per frustum

(Ehs) /JaN

_versus

angle

of frustum es

lx,

at

kBT/Ja

= 0.5 for various sizes of vesicles; N

= 20, 25 and 100.

"'l'vz&

"li'll'j $ 4 4 ,< ll j,,'"

3 M

~i>li/,

~llllll'~~V< _d~~ ,>1'/ § ~

~ ) ii,A~

"/W~ fl,~

~"l/~ i$

z

(b)

Ii (h

&

~

~

W

/~

fij

v

~~~

~il>~/,

,,lV1'/ ~i G

4 ) ₊

j ^4h

~j if

§ ,@

~j,

jjill?

~ ~ ~ ix1

0~j

Fig.

11. Variation of the energy per frustum

(Ehs)/JaN against

Monte Carlo step

(MCS)

at

kBT/Ja

= 0.9 for _a system ^with es

=

x/10,

N

= 100. Inserts show the

topological changes

of _a vesicle;

(a) "heart"-shape

of the _torn vesicle which is the final

configuration

at

kBT/Ja

= 0.8, and

(b)-(c)

the process of

"furling".

temperature. ^A connected vesicle is overheated above

kBT/Ja

= 0.5 to remain in _a metastable

state. At

kBT/Ja

= 0.8 the closed vesicle tears off at a

point

and the broken ends curl inside to form like _a '~heart"

(Fig. lla).

This state is _very

unstable,

and at _a

higher

temperature

kBT/Ja

=

0.9,

two

edges

^of the torn vesicle ~~furl"

(Fig. 11b)

and _new smaller vesicles _are born

one

by

_one

(Fig. 11c). Finally,

there _appear 4 vesicles.

During

this rupture process, the energy per frustum is lowered from

Ehs/NJa

Gt 1.535

(at kBT/Ja

=

0.8)

to

Ehs/NJa

Gt

1.153(at kBT/Ja

=

o-g),

as shown in

figure

11. The number of the

newly

born

vesicles, four,

in the