A COMPUTER STUDY OF A SIMPLE STATISTICAL-MECHANICAL MODEL OF PHOSPHOLIPID MONOLAYERS AND BILAYERS

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A COMPUTER STUDY OF A SIMPLE STATISTICAL-MECHANICAL MODEL OF

PHOSPHOLIPID MONOLAYERS AND BILAYERS

E. Scalas, A. Levi, A. Gliozzi

To cite this version:

E. Scalas, A. Levi, A. Gliozzi. A COMPUTER STUDY OF A SIMPLE STATISTICAL-

MECHANICAL MODEL OF PHOSPHOLIPID MONOLAYERS AND BILAYERS. Journal de

Physique Colloques, 1990, 51 (C7), pp.C7-333-C7-338. �10.1051/jphyscol:1990733�. �jpa-00231132�

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COLLOQUE DE PHYSIQUE

Colloque C7, suppl6ment au n023, Tome 51, ler decembre 1990

A COMPUTER STUDY OF A SIMPLE STATISTICAL-MECHANICAL MODEL OF PHOSPHOLIPID MONOLAYERS AND BILAYERS

E. SCALAS, A.C. LEVI* and A. GLIOZZI

Universita di Genova, Dipartimento di Fisica, Via Dodecaneso 33, :-l6146 Genova, Italy

SISSA-ISAS, Strada Costiera 11, I-34014 Trieste, Italy

k b s t r n c t

l3oth the phase diagram and the phase-transition dynamics are studied for a simple model of phospliolipid ~llo~iolayers and bilayers. The phase diagram is studied using finite size scaling and pl~momenological renormalization. The model predicts a first-order phase transition with a critical point.

The phase-transition dynamics is investigated by the Monte Carlo method with Glauber dynamics. After quenching the system below the critical temperature, the average size of the growing domains l ( 1 ) is found to follow a tll' law.

Introduction

and bilayers have been proposed [1,3,4]. Only few of these models have turned out to be ex- A phospholipid molecule consists of three actly soluble (41. Among the non-exactly solu- parts: a polar head, a glycerol backbone a d ble models, one of the simplest has been pro- two hydrocarbon chains. The phospholipids posed by Doniach [S] and Caillk et al. [6]. This are amphiphilic molecules and, in an aque- model has been studied by the mean field ap- ous environment, they form lamellar bilayers. proximation [6], the Bethe approximation [7], When they are spread a t the air-water inter- Monte Carlo simulations 181, and the Kikuchi face, they form mono-molecular layers. cluster method [g]. A firdtlorder phase tran- As the temperature increases, a pure phos- sition with a critical point is predicted by the pholipid bilayer undergoes a first-order phase model.

transition from a gel phase to a liquid-crystal phase [l]. The phenomenon is characterized

by a sharp peak of the specific heat and by

The

a decrease in the bilayer thickness. This phase transition has been considered equiva- lent to the so-called liquid-condensed to hquid- expanded transition of a pure phospholipid monolayer [2]. Extensive stndies, both exper- imental and theoretical, have been made on these transitions [1,2].

Throughout the paper we shall refer to the gel to liquid-crystal transition of a bilayer and to the liquid-condensed to liquid-expanded transition of a monolayer as the main phase transitions.

An important term in the expression for the energy of an assembly of phospholipids refers to the rotational isomerism of the hydrocarbon chains. The van der Waals interaction between the chains, the excluded volume interactions, and the interaction between head groups provide other basic terms.

The rotational isomerism is the driving phenomenon of the main phase transitions, while the translational degrees of freedom may be

In the q-state models, a bilayer is made up of two non-interacting monolayers. A monolayer consists of a triangular lattice with N lattice sites which are occupied by a saturated hydrocarbon chain with M carbon atoms. Each chain may be in one out of q states. A state n is characterized by a cross-sectional area A,, a conformational energy E, and a degeneracy D,. All the q states are conformational iso- mers of the ground all-trans state. The ro- tational isomeric model justifies the assump- tion of a discrete number of states. In this model, only three rotational configurations are allowed for each C H z

-

C H 2 bond: one trans configuration of absolute minimal energy and two gauche configurations of relative minimal energy. The number of possible chain states is thus 3M-1, and in the q-state models it is assumed that only few of these states may be accessed in the condensed system.

The Hamiltonian of the q-state models is:

neglected[5]. This is because the entropy q

change due to the translational disordering a t =

E C(E;,

^S^IIAia)Li,

the main phase transitions is nearly 0.5kn per i a=l

molecule, much less than the observed entropy ^'2

change, which is e.g. 1 4 k ~ per molecule for

- ' ^C ^C

V a ~ L i a L j p - (1) dipalmitoyl-phosphatidylcholine. < i j > a,P=l

In the last twenty years, various statistical- where L;, is 1 if site i is in state a; otherwise, mechanical models o f ~ h o s ~ h o l i ~ i d monolavers it is 0; Vac$ is the interaction energy between Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990733

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a chain in state a and a chain in state

p;

^Jo

is the strength of this interaction; the symbol

C,ij>

indicates the sum on the six nearest neighbours of site i.

This Hamiltonian contains the conformational energy:

and the interaction energy between the nearest neighbour chains

T h e term:

X &

^UAiaLia

i a=l

describes the interactions between the polar

T h e average area per chain

<

A

>

^{is re-}

lated t o the Ising magnetization per site M = l/iV

xi

^siby the following formula:

In the last section we shall consider in more detail some aspects of this simple model.

3 The Methods

3.1

Finite Size Scaling

heads of the phospholipids in terms

an ef- ~h~ partition function of semi-infinite strips fective surface pressure I1 {10].

W X n can be computed by the classical trans- If the surface pressure is fixed a t some ap- fer matrix method [ l l ] . T h e dimensionless free propriate value, the q-state models represent

energy per site of semi-infinite strips is given the bilayer, while, if the surface pressure may by the formula:

be varied, they represent the monolayet.

A two-state model may be mapped [5] into a n Ising model with the following effective Hamiltonian:

where the mapping transformations are:

L ~ G = ( I

+

s i ) / 2 (3) LiE = (1 - 3;)/2 (4)

si = *l, ( 5 )

where S ; = 1 means that the chain is in the ground state G, and S; = - 1 nieans that the chain is in a effective excited state E. More- over,

where

X?)

^isthe greatest eigenvalue of the transfer matrix, for a system of size n . T h e inverse correlation length is:

where

Xg')

is the second greatest eigenvalue.

By using the finite size scaling method, it is possible t o obtain some information on a

W X W system [12,13]. Finite size scaling is a n application of renormalization group the- ory t o finite systems. . T h e basic idea of the method is to consider l l n as a scaling field, in addition t o t h e usual temperature-like field E

and to the ordering field h. The generalized

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Kadanoff relations for free energy and the inverse correlation length in d dimensions [l21 become:

~~f

(c, h, l l n ) = L ~ ~ L ( E , h)

EL,

hL, L l n ) (11) L%(&, h, 1/72) =

EL,

h ~ , L/n). (12) Under renormalization, the system size n becomes n/L, therefore, the exponent of the field 1/n is 1. Close to the critical point, E = h = 0 and the renormalization transformations can be written as:

€&(E, 0) = L ~ T E

+

0 ( c 2 ) (13) h ~ ( 0 , h ) = ~ ~ 0 ( h 2 ) . ~ h(14) + By using equations (11-14), one can obtain the following relations for the specific heat c,, the susceptibility X,, the inverse correlation length n,, and its derivatives with respect to the temperature field n z and to the ordering field squared K::

c,, nZYT-d

nzYh-d

(15) (16)

K,, n-I (17)

nY7-' (18)

,It

^,Zyh-1 ₍₁₉₎

These equations can be used to estimate YT and Yh. A l l the other critical exponents may be computed from YT and

X,.

The presence of a regular term g~ in the expression for free energy results in the presence of constant back- ground terms in equations (15) and (16). In the next section we shall use equations (15) and (17) to study the critical exponents and the phase diagram of the two-state model.

3.2

The Monte Carlo Method

The Monte Carlo method used in statistical mechanics is a computer simulation tech- nique which was introduced by Metropolis et

al. [l41 in the fifties. Since good reviews of this method and its applications [15,16] can be found in the literature, in this subsection we shall only recall the basic ideas of the algorithm.

Let us suppose we have a lattice system with N lattice sites a t a fixed temperature T. A sequence of configurations is generated by an importance sampling, according to an appro- priate probability distribution P,. One possible choice for P, is the Boltzmann probability distribution. Once an initial configuration has been chosen, at each step the algorithm at- tempts to change the system state. A transition probability W , depending on the difference in energy between the trial state and the old state is computed and a random number s is generated. If s

<

^W,the trial state becomes the new state; otherwise the old state becomes the new one. At each step, the values of the desired quantities are computed. The steps are repeated until a sufficient number of independent configurations are generated. The av- erages of the physical quantities are then computed on the independent configurations. The transition probability must satisfy the detailed balance condition:

where El

-

Ez is difference in energy between state 1 and state 2. A possible choice of W is that proposed by Glauber [17]:

w(1 -+ 2) cc - l - tanh ( E l

-

Ez) 2 l

[ [

^2kBT

l]

⁽²¹⁾

The sequence of configurations generated by the Monte Carlo method is a result of a Marko- vian stochastic process. In the next section, we shall use this fact in tile discussion of the dynamics of the first order phase transition predicted by the two-state model.

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C7-336 COLLOQUE DE PHYSIQUE

4 The Results

4.1

Phase Diagram and Critical Exponents

Pink et al. [7] proposed a slightly modified version of the simple two-state model. They considered an excited state and an effective ground state which was a thermal average over three states of lower energy. This is possible because the time for a transition between the lower-energy states (m 10-'ls) is much shorter than the time for a transition to the excited state from any of the lower-energy states (m 1 0 - ~ s ) . Therefore in this version of the two- state model, DG, AG and EG are temperature- dependent.

I t is easy to obtain the phase diagram of this two-state model in the (11, T ) plane. If II and T are less than their critical values II, and T,, when Heff (II,T) = 0 the system undergoes a first-order phase transition. When the critical values of surface pressure and temperature are reached, a continuous phase transition occurs. Pink et al. showed that this behaviour is true in the case of the Bethe approximation.

Indeed, in the case of this model the critical temperature is known exactly and is equal to the critical temperature of the triangular Ising model: kTc/J = 3.64096. The critical pressure II, is given by the solution of the equation Heff(II,Tc) = 0. The results obtained by finite size scaling are consistent with those reported by Pink et al.; they are sulnn~arized in Fig. 1. Equation (17) yields that at the transition line the scaled inverse correlation length nK, must be asymptotically indepen-

dent of the size n. The plot of the scaled inverse correlation length versus k B T / J for the sizes 2,4,6 is shown for two different values of the surface pressure, i.e. below and above the critical point. Above the critical pressure, the scaled inverse correlation lengths of the different sizes do not intersect a t any k n T / J value, therefore no phase transition occurs.

Pink et al. conjectured that the critical order parameter exponent

p

and the criti- cal magnetic field exponent 6 of the two-state model satisfy the relation:

when II -+ H,. If eq. (22) is true, then the temperature exponent

YT

must be equal to 1.875, i.e., to Yh. As we know the critical teni- perature of the two-state model, we use equation (15) plus a constant term to estimate YT [12,18]. YT is obtained by successive three- point fits. In Fig. 2 the specific heat per site of the two state model is plotted vs. k B T / J for the sizes 2-8 and the three-point fits of YT are plotted versus size for the case of averaged ground state. The results are consistent with the assumed YT value.

4.2

The Dynamics of the First- order Phase Transition

In this subsection, we describe a typical simulated quenching experiment. The initial state of the system is an unordered state in the one- phase region above the critical temperature.

Figure 1: nn, vs. k B T / J for I1

<

II, and

n > nc

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Then the temperature is suddenly lowered below the critical point. Therefore the system is no longer in equilibrium. Its evolution towards equilibrium is observed, which involves phase separation. As Ising-like variables hibit no natural dynamics, Glauber proposed a kinetic Ising model in which the interaction of the system with a heat reservoir makes the spins fiip (in our case, it makes the chains change state) [17]. The evolution of the Ising- like variables is described by a Markovian mas- ter equation. In the Glauber dynamics, the order parameter is not conserved. Indeed, our simulated experiments begin with a sudden decrease in both temperature and surface pressure below the critical point, for a constant area per molecule, in a phospholipid monolayer. Then the evolution towards equilibrium is attained a t constant surface pressure and temperature. Therefore, during this evolution, the order parameter of the system (the average area per molecule) is not conserved.

It is generally assumed that the characteris- tic length scale l of the domain growth process during phase separation obeys the following scaling law [19,20]:

where t is the time elapsed since the quenching and X is a universal exponent, the value of which depends on the universality class of the system. The results of our simulations, for two system sizes, are shown in Fig. 4. for one choice of l (see [19]). The X value obtained is 1/2, which is consistent with both previous Monte Carlo simulations [20] and theoretical computations [21,22] for non-conserved order parameters.

References

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Wiley & Sons, New York (1987)

[2] D.A. Cadenhead in "Structure and Prop- erties of Cell Membranesn, v01 3, ed. G.

Benga, CRC Press, Boca Raton (Fl) USA (1985)

[3] A. CaillB, D. Pink, F. de Verteuil, M.J.

Zuckermann, Can. J. Phys., 58, 581 (1980) [4] J. F. Nagle, Ann. Rev. Phys. Chem., 31,

157, (1980)

[5] S. Doniach, J. Chem. Phys., 68, 4912, (1978)

[6] A. Caillk, A. Rapini, M.J. Zuckermann, A.

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[l31 M.N. Barber in

"

Phase Transitions and Critical Phenomena", v01 8 (1983) ed. C.

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[l41 N. Metropolis, A.W. Rosenbluth, M.N.

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[l51 "Monte Carlo Methods in Statistical Physics", ed. K. Binder, Springer-Verlag (1979)

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Lebowitz, Academic Press, London [22] G.F. Mazenko, O.T. Valls, M. Zannetti,

Phys Rev. B, 38, 520, (1988)

Figure 3: N M ~ K 1" vs. time t measured in Monte Carlo steps per chain

I . l... l . l . . I

2 3 4 5 6

Figure 2: c, vs. k B T / J and YT vs. size n