DYNAMICS OF BILAYER MEMBRANE

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Submitted on 1 Jan 1979

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DYNAMICS OF BILAYER MEMBRANE

A. Massih, J. Naghizadeh

To cite this version:

A. Massih, J. Naghizadeh. DYNAMICS OF BILAYER MEMBRANE. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-451-C3-454. �10.1051/jphyscol:1979390�. �jpa-00218786�

DYNAMICS OF BILAYER MEMBRANE

(*)

A. R. MASSIH and J. NAGHIZADEH

Institute of Biochemistry and Biophysics, University of Tehran, P.O. Box 314-1700 Tehran, Iran

RBsumC.

-

La thCorie prCsentke fait appel aux modbles de Rouse, pour dkcrire la chaine isolke, et de Marcelja, pour dkrire la double couche. Ainsi les forces agissant sur un groupe mtthylkne de la chaine sont celles de rappel, browniennes, visqueuses, qui apparaissent dans le modhle de Rouse, plus, les forces de pression latbrales du modhle de Marcelja. On calcule la viscositC dynamique.

Abstract. - The theory combines the Rouse model for a single chain with the liquid crystal theory for bilayer membrane developed by Marcelja. Thus the forces acting on a bead within a chain are spring forces, Brownian force and dissipative or viscous force which appear in Rouse- theory plus the liquid crystal and lateral pressure forces.

The dynamic viscosity of the system is calculated.

1 . Introduction. - Extensive experimental study of both natural and synthetic bilayer membranes has been made in recent years [l, 2, 31. Whereas interpretation of measured data in naturally occurring membranes is complicated, the properties of synthetic bilayers are somewhat simpler to interpret. Due to the fact that the structure of these bilayers is well established, theoretical models can be worked out to further elucidate the nature of physical phenomena.

The model of bilayer membrane presented by Marcelja [4, 51 treats the problem of thermodynamics and phase transitions in these systems by introducing the concept of liquid crystal theory. Marcelja was able to calculate the transition temperature from isotropic to nematic phase. The transition tempe- ratures predicted by Marcelja model are in good agreement with experiments. Other experiments indicate that in bilayer systems the liquid chain is in constant motion parallel to the plane of the bilayer. This motion has been studied by spin labelled experiments by McConnel and coworkers [6] and Sackman and Trauble [7,8]. The information obtained from these measurements gives the lateral diffusion coefficient of the chain in the isotropic phase for which both experimental group report a value of the order of 2 X 10-* cm2/s. More recent experiments by Webb and coworkers [g] extended these extended studies using florescence spectroscopy and photo- bleaching techniques. They have measured the lateral diffusion coefficient in both isotropic and nematic state for bilayers and have found that the diffusion

(*) This research has been supported by a grant from the ministry of Science and higher education of Iran.

coefficient has a larger value by a factor of three in the nematic phase.

These experiments and similar ones require a dyna- mical theory of membranes for interpretation.

Although the increase in the lateral diffusion coeffi- cient in the nematic phase would involue a more complete theory in the phase transition region, a preliminary theory away from the critical region is here developed assuming that the chain behaves as a Rouse chain [10].

2. The model. -We consider a single polymer chain of N links, each link with effective length b embedded among similar chains in the membrane. One end of the chain is constrained to move only in the X-y plane, the other end being free. Actually a bilayer consist of two such systems where chains are constrained between parallel planes. However for this purpose it is sufficient to consider one of the chains in the bilayer.

Consider now the link i in this single chain. This link consist of the carbon (bead) number i

-

^{1 at}

one end and carbon number i at the other. One is interested in writing the equation of motion for the system of N beads. Now one must characterize the forces acting on each bead, this has been done by Rouse and Zimm [10, 111 for a system of beads connected together by harmonic springs.

In our problem there is an additional force arising from the surrounding polymers in the membrane, the latter forces also has been studied in the equili- brium case by Marcelja [5]. Our object is to combine these two types of forces. The Langevin equation

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979390

C3-452 A. R. MASSIH AND J. NAGHIZADEH

for the jth bead for a free polymer in solution is given by

d 2 ~ , -

m - _dt2 - - [Rj - KB T Vj log

+

^{- VjU (1)}

d .

where m

-

is the inertial force,

[Rj

is the fric- dt2

tional force,

5

being the friction coefficient, KB T Vj log $ is the random force and VjU is the spring forces given in more detail

with Z , = 0 and K being the spring constant. In order to incorporate the interpolymer interactions we apply the method employed in theory of liquid crystals. One begins by writing the Ising like Hamil- tonian [12],

the contribution of repulsive forces. Marcelja [5]

has discussed these forces in detail and has called them lateral pressure forces. The dependence of the lateral pressure on order parameter V is given by an equation of the form PA, L,/L with L being the total projection of the chain in the z direction and A, and L, are the area and the length respectively, of the chain in the frozen nematic phase. Using this concept and relating it to the order parameter the final equation becomes nonlinear in V. In order to avoid this difficulty one notes that, as far as small oscillations are concerned, the forces exerted by neighbouring polymers are of elastic type and the lateral pressure force should appear as proportional to the order parameter or the square of the projection along the z axis.

~ C r t h e r scrutiny reveals that this result can also be obtained by the expansion of non-linear Marcelja form of the lateral pressure. It has also been used by Marcelja in his earlier paper [4] as a logical analogy of the Ising type Hamiltonian eq. (4).

We can write the interpolymer forces on the ith - -

X = - C A i j q i q j - H C q i (3) bead (along z axis) as

i j i

axi ax, aei

where qi is the order parameter associated with a

F . =

- = ^{- p - =} czi

82,

aei azi

⁽⁸⁾

bead i on one polymer and qj is that associated with where bead j on a neighbouring polymer, Aij is the coupling

constant. The second term in eq. (3) represents the 3

c = - ( v o ~ < ~ j ) b2 ^j + H ) . (9) contribution of external forces which in this case

are the elastic (repulsive) interpolymer forces. We now

apply the mean field approximation to eq. (3) we Combining eq. (8) with eq. (1) we obtain the ~ a n ~ e v i n

obtain the form equation of motion of the ith bead for the system under

consideration

xi

⁼ - v0

C <

^{V j ) V i} - HVi

j

(4) m d2Ri --- =

dt2 ^-

mi

^{- K, T Vi log}

+

^{- ViU}

+

^{czi (10)}

where v, is given by

j

3. Reduction of the equation of motion. - Follow- Here the mean order Parameter ( V j ) is obtained ing the standard procedure given in Yamakawa [13], as a solution of the self-consistent equation we neglect the inertial term in eq. (10) and utilize the continuity equation for the distribution function

+

[dQVjexP[~vo ( V j ) rti] given by

J

1 ( 6 )

So

In this manner we obtain the so called generalized

dx($x2

- 4)

exp[/3vO(~x2 -

$)l

diffusion equation for our system which in the matrix

- 1 notation can be written as

So

^dx exp[pv0(t x2 -

B]

- a*

- - ^{[ - l} VT[K+AR

+

^KB TV* - C@] -

A t

V C

For the case of a single chain embedded in membrane

the order parameter is given explicity as - VTvO

\1/

(12)

qi = (3 cos2 Oi - *)

.

(7) where superscript T indicates the transpose Moreover the second term in eq. (4) as stated, is VT ^E (VO V1

.. .

^{VN) ,}

v0 being the medium velocity

R =

and

We now transform eq. (12) into normal coordinates.

The resulting equation is

- VTU,O 1C/ (13) where

5

being the normal coordinates V = Q - I T V I and

VT =

l7tQ-l

V, being the differential operator with respect to

6.

4 . Model calculations. - It is instructive to compute the mean square projection of link j along the Z axis, ( ZjZ ). This may be done in the standard way by multiplying eq. (13) by Zf and integrating over all appropriate coordinates. As usual one arrives at a differential equation for ( 2; ), in this case given by :

The steady state solution of eq. (14) can be obtained easily, which is :

Comparing eq. (15) with the similar expression obtained by Rouse [l31 for a single chain in solution, one notes that the average bead length projection increases due to presence of a factor C in the deno- minator. This increase is however limited by the average bead length b, thus at the frozen state one could replace ( Z? ) by b2 and obtain the limiting value of C.

Further quantity of interest is the viscosity of bilayer. This quantity is indirectly measured in the experiments of lateral diffusion coefficient cited in the introduction. It is also of interest in determining the general flow behaviour of a fluid system with a membrane. The latter system has been studied by Saffman and Delbriick. We solve for the viscosity by introducing a fluid flow term into eq. (13). This flow has a time periodic function associated with it and has a linear velocity gradient in the Z direction.

It is explicitly given by

with

g = g o ^eiwt

Such a flow can be realized experimentally when the bilayer is immersed in a fluid which itself has a Z velocity gradient. The situation which arises has been described by Saffman and Delbriick [14].

It is possible to compute the membrane viscosity under these conditions by standard methods described by Zimm [l l]. The viscosity is given by the expression

MO is the local viscosity associated with friction constant

5

and p is the chain density ;

( Xj Zj ) may be computed similar to computation of ( Zf ) described above.

The long time result for the viscosity is as follows 2 K, T(Kilj - C)

M - M o = p C

i (2 K l j - C)

(7

^-

^{$ +}

^iw)

5. Discussion. - The model introduced in this paper has chosen the friction constant experienced by any bead to be isotropic in all directions. This situation is true for a polymer dissolved in solution.

But for a membrane with liquid crystal characteristics the friction coefficient

5

should in general include the anisotropy of the system. However since our paper considers the transition region between isotropic and nematic phase, the anisotropy of friction coeffi- cient is indirectly introduced through a liquid crystal characteristic in constant CZ.

C3-454 A. R. MASSIH AND J. NAGHIZADEH

Thus as the system goes from the disordered to the nematic phase the increase in the anisotropy of the system is exhibited by the increase in para- meter C and the anisotropy of the effective friction coefficient can be calculated from the equations.

The C may be calculated from measurements of the mean order parameter via eq. (15). The measured viscosity of the membrane would depend on the mea- suring instrument. For those experiments that measure local relaxation of labelled elements, the local vis- cosity described by the friction constant

5

is significant.

However an experiment can be conceived whereby the bilayer is inserted in a flowing fluid parallel to the direction-of flow. If such a Auid has a velocity gradient along the Z-axis of the membrane, it would

be two-dimensional flow with a boundary condition along the interface described by sudden change of viscosity. This situation as pointed above has been investigated by Saffman and Delbriick [14]. It is possible to design an experiment in this manner to measure the viscosity of the membrane. This viscosity would be different from the local viscosity characterized by the friction coefficient

l.

We have attempted in the foregoing section to study the latter viscosity via Rouse's model. The lateral diffusion coefficient could be obtained in this scheme, however the experimental effect whereby this quantity changes from the disordered to nematic phase, does not come out of this theory.

References

[l] METCALFE, J. C., BIRDSALL, N. J. M,, FEENEY, J., LEE, A. G., LEVINE, Y. K. and PARTINGTON, P., Nature 233 (1971) 199.

[2] LEVINE, Y. K., BIRDSALL, N. J. M,, LEE, A. G. and MET- CALFE, J. C., Biochemistry 11 (1972) 1416.

[3] LEE, A. G., BIRDSALL, N. J. M. and METCALFE, J. C., Bioche- mistry 12 (1973) 1650.

[4] MARCELJA, S., Nature 241, 451-453.

[5] MARCELJA, S., Biochimica et Biophysica Acta 367 (1974) 165- 176.

[6] MCCONNELL, H. M. in Spin Labelling, edited by L. J. Berliner (Academic Press, New York) 1976.

[7] SACKMANN, E. and TRAUBLE, H., J. Amer. Chem. Soc. 94 (1972) 4482-449 1.

[8] TRAUBLE, H. and SACKMANN, E., J. Amer. Chem. Soc. 94 (1972) 4499-4510.

[9] WEBB, W. W., in Electrical Phenomena At The Biological Membrane Level (Edited b y E. Roux (Elsevier Scientific Publishing Company, Amsterdam)) 1977.

[l01 ROUSE, P. E. Jr., J. Chem. Phys. 21 (1953) 1272.

[11] Z ~ M , B. H., J. Chem. Phys. 24 (1956) 269.

j12J BLINC, R., in Local Properties at Phase Transition, edited by K. A. Muller (North Holland, Amsterdam) 1976.

[l31 YAMAKAWA, H., Modern Theory of Polymer Solutions, chap- ter V1 (Harper and Row, New York) 1971.

[l41 SAEEMAN, P. G. and DELBRUCK, M., Proc. Nat. Acad. Sci.

USA 72 (1975) 3113.