Monte Carlo Simulation of Vesicle Self-Organisation

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Monte Carlo Simulation of Vesicle Self-Organisation

Américo Bernardes

To cite this version:

Américo Bernardes. Monte Carlo Simulation of Vesicle Self-Organisation. Journal de Physique II,

EDP Sciences, 1996, 6 (2), pp.169-174. �10.1051/jp2:1996174�. �jpa-00248287�

Short Communication

Monte

Carlo

Simulation

of

Vesicle

Self-Organisation

Am6rico T, Bernardes

(*)

Institute for Theoretical

Physics,

Cologne

University,

50923

Cologne,

Germany

(Received

22 November 1995,

accepted

27

November1995)

PACS.05.50.+q

Lattice

theory

and

statistics;

Ising problems

PACS.82.70.-y

Disperse

systems

PACS.87.15.Da

Physical

chemistry

of solutions of

biomolecules;

condensed states

Abst+act. Vesicles, which _are

droplets

of _a fluid

encapsulated

in

a membrane, represent

the basic structure of almost all life forms. The

amphiphilic

membrane that separates different

domains acts as a tunable

filter,

selectively

allowing

the passage of chemical substances. The

formation of these aggregates is

subject

of

increasing

interest, both

experimental

and

theoret-ically,

due to the fact that

they

_can be used

as

drug

carrier or even as artificial cells. In this

paper, we report the first successful computer simulation of vesicle spontaneous formation

ob-tained by

performing

extensive Monte Carlo calculations in _a

microscopic

lattice model based

on two-tailed

arnphiphilic

chains surrounded

by

solvent molecules. The

dynamical

process of

aggregation

is also shown and _some features of this _process are discussed.

Amphiphilic

molecules are constituted of two

parts

with

opposite

properties:

a

hydrophilic

water

loving

"head" and a

hydrophobic

hydrocarbon

"tail",

the

ordinary

detergent

being

one

of the most known

example

of this

type

of molecule. Another

type

of

amphiphilic

molecules

(phospholipids)

constitute the basic

compound

of the cell membrane. In _aqueous solutions

they

_can

aggregate

in several different _ways, which will

depend

on the concentration

and/or

type of

surfactant,

temperature,

pressure etc.

[1,2].

Hydrophobic

effects

produce

aggregates

with the

polar

head outwards in contact with the water molecules

hiding

the

hydrocarbon

chains inside. At low temperatures,

amphiphiles

tend to form continuous

aggregates

and the

membranes are one of the structures most

usually

formed.

However,

with the

bilayer

struc-ture one can not avoid the contact between the

hydrocarbon

tail and water at the

edges

of the membrane. Therefore the membrane

might

lower its _energy

by bending

to form closed

structures,

called vesicles or

liposomes.

Thus,

vesicles

separate

different

portions

of the so-lution and this

property

has

important

applications

[3],

such as in

drugs

delivery

systems

or even the

production

of artificial cells

(like

artificial

erythrocytes)

[4].

However,

whereas

vesi-cles are

widespread

and form

spontaneously

in nature, their artificial

production

requires

the

input

of considerable mechanical _energy and

they

_are

usually

metastable _[5].

Recently,

some

experimental

methods were

developed

to achieve spontaneous

production,

such as mixtures of

different types of

amphiphiles

[5,6].

Computer

simulations on lattice models have been used

to

study

the

aggregation

of

amphiphiles

and/or

related features: critical micelle

concentration,

transition

temperatures,

stability

of

_aggregates

etc.

[7-15]

and

important

results have been

(*)

Permanent address: Departamento de Fisica, Universidade Federal de Ouro Preto,

Campus

do

Morro do

Cruzeiro,

34500-000 Ouro

Preto/MG,

Brazil

170 JOURNAL DE

PHYSIQUE

II N°2

~

amphiphflic

mol. blifial coring. O d O O O

ptafion~

O

~O

O O o kkk

ol

Do o o bucklblg

Fig.

I. Schematic representation

(at

top)

of _a twc-tailed

amphiphile

(here

with

length

ta

= 12

and tail

length

=

4)

used in this simulation: black circle =

head,

white =

tail,

and crossed-circle

= liaison.

Following

towards the bottom is a two-dimensional representation of a sequence of the

possible

movements

(reptation,

kink and

buckling).

Grey

circles represent water molecules

(only

those

molecules influenced

by

the movements are

shown).

obtained.

However,

_up to now

only

the

spontaneous

formation of micelles has been

obtained,

although

the

stability

of membranes has been discussed

_[13],

but

starting

the simulations with

pre-formed

membranes. In this _paper,

by performing

extensive Monte Carlo simulations with

an

improved

version of a

Larson-type

model,

we show the spontaneous formation of vesicles

from the

aggregation

of two-tailed

amphiphiles

randomly

distributed in _a cubic lattice. With

extensive observation of this _{process, we are} able to show

dynamical

aspects of the vesicle

formation.

In the

present

model for the surfactant

solution,

_as in earlier works

[13,14j,

an

amphiphilic

molecule is

represented

by

t~

connected

particles

and _a water molecule

by

_a

single

particle

on a cubic lattice of L x L x

Lz

sites. Periodic

boundary

conditions were

adopted

in the x and y directions and the movement was not allowed

through

the first and last

planes

at

z direction. Each site may be

occupied

only

once, therefore

satisfying

the excluded volume

condition. The two-tail

amphiphiles

are constituted of a "water-like" head with two sites

(defined

in the middle of the

polymer

chain,

namely

in the

positions

[t~/2j

and

[ta/2

+

lj),

two liaison sites

(that

play

a noninteraction

role)

and two tails with

(la

4)/2

sites each.

Therefore _a water

particle

is

represented by

an

Ising

spin

+I and the

amphiphilic

molecule

for

la

= 12

by

a

string

of connected sites with values:

-1,

-1,

-1,

-1,

0, +1,

+1, 0,

-1,

-1,

-lj

-I. In order to simulate the

hydrophobic

effect _{we assume} water-water and tail-tail attractions and water-tail

repulsions

through

the

following

relations between interaction

energies:

Eww

A)

B)

°.O@040: oa***OO*Oo o@ ~ ee o*o OOOOOO. C) Fig. 533 esicles

A

hows entire vesicle; in part B the vesicle

hasbeen

ectioned at _the

middle the left

portion

and

its

inner sidej and in part C only the middle layer

is

shown (here

the _membrane

different domains can be better observed). Black circles represent the head particles, mall

dark

grey

the liaison particles and light grey the tail particlesj the inner water is represented

in part B by

172 JOURNAL DE

PHYSIQUE

II N°2 600 w

)

₄₀₀

~

~/

fl

1'

(

200 E

~e+00 _le+07 2e+07 3e+07 4e+07 MC steps

Fig. 3. Dynamics of the

aggregation

process,

showing

the behaviour of the

largest

cluster

(the

parameters are the same as that described in

Fig.

2).

In the

beginning

of the simulation micelles are formed and isolated

amphiphiles

aggregate in those clusters

(linear

growth

of the

largest

cluster).

These clusters become closer each other and from the contact of the clusters _a large one is formed

(the

jumps

_on the cluster size observed in the

figure).

This is done

by

changing

an initial

configuration

by

the movement of a chosen

amphiphilic

molecule. The _move is

accepted

with

probability exp[ (Eaid

En~w)/kBT

and a MC step

corresponds

to one

attempted

to move of all

amphiphile

molecules.

The results we are

going

to show were obtained with one

sample,

starting

with

amphiphilic

molecules

uniformly

distributed in the cubic lattice. Several simulations

using

other initial

configurations

have been done in order to confirm these results. We have

performed

simulations

at

temperature

t

=

2.2,

therefore below the condensation temperature for this

type

of molecule

A)

B)

Fig.

4.

Snapshots

of the dynamical process obtained at different times. Part A shows six aggregates

basically

micelles at S-S

x10~

MC steps. Parts B and C show the fusion of two aggregates

(obtained

at tililes 1.12 x lo~ and 1.13 x

10~,

respectively).

The

largest

cluster has a membrane-like structure

whereas the other is a small vesicle. The latter moves in a amoeba-like way. Parts D and E show

different views of the membrane

bending,

obtained at 1.7 x 10~ MC steps. The symbols have the same

meaning

as that in

Figure

2.

Again,

note that boxes are

provided

as

guides,

I-e-,

they

do not represent

e

c)

D)

E)

Fig.

4.

(Contm~ed)

[14j,

in which the

amphiphiles

tend to

aggregate

in one

big

cluster.

Figure

2 shows the final

configuration

for _a simulation with 533

amphiphiles

(representing

a fraction of 0.08 of the total lattice

volume)

obtained after 4 x

10~

MC steps. In a first

inspection

part

A we observe that the heads and liaison sites _are

mostly

in the external surface

enclosing

the tails.

However,

when _one looks at the inside of this

aggregate

part

B _{one can} observe an internal surface

basically

formed

by

heads and liaison

sites,

encapsulating

a volume of water that has filled this internal

portion.

This feature can be better seen in part C that represents the middle

layer

of this

figure.

The

major

length

of this

aggregate

is oriented in the _y z

direction,

but

in other simulations we have obtained different

orientations,

that means that

qualitatively

the

final results have not been influenced

by

the restriction of the movement in the _z direction. The

dynamical

process of vesicle formation is

subject

of

great

interest and some

possible

ways have been described in the literature

[3,17j.

Examples

are: membrane

bending forming

a closed

surface,

fusion of micelles with

re-arrangement

of the

amphiphilic

molecules and fusion

174 JOURNAL DE PHYSIQUE II N°2

of vesicles.

Figure

3 shows the

dynamical

behaviour of the

largest

cluster obtained in _our

simulation. First the

amphiphiles

aggregate

in small

clusters,

basically

micelle-like

aggregates,

and the isolated

amphiphiles

tend to

aggregate

in these micelles. Afterwards _{a new process} takes

place,

when these

aggregates

fuse in

larger

_ones,

producing

the

jumps

that _one observes

in the

plot.

If

initially

these

aggregates

are

micelles,

the last ones present a more

complex

aggregation

form,

dealing

with the

problem

of

avoiding

the contact between the tails and

water molecules. Membrane-like and

pre-vesicles

aggregates

can be found. Some

pictures

of

this _process described above can be seen in

Figure

4,

which shows

snapshots

of the

system

obtained at different moments.

Firstly

the

amphiphilic

molecules

aggregate

in micelles

(part

A,

but _even here we may see a membrane-like structure the vertical

aggregate

at

top-left

corner).

Parts B and C show the moment of fusion of the

largest

aggregate

(a

membrane-like

structure)

with _a

pre-vesicle,

this becomes closer to the

larger

one

(B)

and

they

touch each

other

(C), forming

the new cluster. Parts D and E show different views of the same

aggregate.

Here,

the

bending

of the membrane _can be _seen. Note that the micelle at the bottom of

parts

B _up to E becomes smaller due to the fact that it lost

amphiphiles,

which have

migrated

to the

largest

cluster. From these results some

interesting

questions

arise: the behaviour of

single-tail

amphiphiles,

the

stability

of

membranes,

the role of the tail

bending

rigidity

etc.

Acknowledgments

I would like to thank D.

Chowdhury,

D.

Staufler,

T-B-

Liverpool,

C. Shida and V. B.

Henriques

for _many

stimulating

discussions and

suggestions.

This ~vork is

partially

supported

by

the

Brazilian

Agency

Conselho Nacional para o Desenvolvimento Cientifico e

Tecno16gico

CNPq.

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M.,

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