Memoire Online - La mécanique statistique des membranes biologiques confinées

L'objectif est la détermination du coefficient ç (N) apparaissant dans la formule (7.6), du Chapitre 7. Pour cela, nous commençons par la contribution de l'entropie à l'énergie libre

Fentropique kBT

b2 %N ([-b²S^'(n)] ln I _-S^' (n)1) dn , (B.1)

1 _{L J/}

où la distribution des tailles des boucles est définie par la relation (7.2), du même chapitre. L'expression ci-dessus peut être réécrite comme

Fentropique kBT

= ç (N) Ö , (B.2)

ç (N) = a-1 11 J N (nâ/(") ln fâ - _JJ 1 1 nâ/(1-âl) dn . (B.3)

ç (N) = a^-¹{[â + ln (â - 1)] (1 - N1/(")) + â - 1 JJJ NI-â) ln N } . (B.4)

Dans cette première contribution originale, nous réexaminons le calcul de la force de Casimir entre deux plaques interactives parallèles délimitant un liquide comptant une biomembrane immergée. Nous désignerons par D, la distance qui sépare les deux plaques, et l'on suppose que cette dernière est beaucoup plus petite que la rugosité en volume, afin d'assurer le confinement de cette membrane. Cette force répulsive provient des ondulations thermiques de la membrane. Nous avons traité aussi bien l'aspect statique que l'aspect dynamique

African Journal Of Mathematical Physics Volume 8(2010)101-114

Laboratoire de Physique des Polymères et Phénomènes Critiques Facultédes Sciences Ben M'sik, P.O. 7955, Casablanca, Morocco m.benhamou@univh2m.ac.ma

We reexamine the computation of the Casimir force between two parallel interacting plates delimitating a liquid with an immersed biomembrane. We denote by D their separation, which is assumed to be much smaller than the bulk roughness, in order to ensure the membrane confinement. This repulsive force originates from the thermal undulations of the membrane. To this end, we first introduce a field theory, where the field is noting else but the height-function. The field model depends on two parameters, namely the membrane bending rigidity constant, k, and some elastic constant, u - D^-4. We first compute the static Casimir force (per unit area), 11, and find that the latter decays with separation D as : 11 - D^-3, with a known amplitude scaling as k^-1. Therefore, the force has significant values only for those biomembranes of small enough k. Second, we consider a biomembrane (at temperature T) that is initially in a flat state away from thermal equilibrium, and we are interested in how the dynamic force, 11 (t), grows in time. To do calculations, use is made of a non-dissipative Langevin equation (with noise) that is solved by the time height-field. We first show that the membrane roughness, L? (t), increases with time as : L? (t) - t¹/⁴(t < r), with the final time r - D⁴(required time over which the final equilibrium state is reached). Also, we find that the force increases in time according to : 11 (t) - t¹/²(t < r). The discussion is extended to the real situation where the biomembrane is subject to hydrodynamic interactions caused by the surrounding liquid. In this case, we show that : L? (t) - t¹/³(t < rh) and 11h (t) - t²/³(t < rh), with the new final time rh - D³. Consequently, the hydrodynamic interactions lead to substantial changes of the dynamic properties of the confined membrane, because both roughness and induced force grow more rapidly. Finally, the study may be extended, in a straightforward way, to bilayer surfactants confined to the same geometry.

I. INTRODUCTION

The cell membranes are of great importance to life, because they separate the cell from the surrounding environment and act as a selective barrier for the import and export of materials. More details concerning the structural organization and basic functions of biomembranes can be found in Refs. [1 - 7]. It is well-recognized by the scientific community that the cell membranes essentially present as a phospho-lipid bilayer combined with a variety of proteins and cholesterol (mosaic fluid model). In particular, the function of the cholesterol molecules is to ensure the bilayer fluidity. A phospholipid is an amphiphile

K. El Hasnaoui et al. African Journal Of Mathematical Physics Volume 8(2010)101-114

molecule possessing a hydrophilic polar head attached to two hydrophobic (fatty acyl) chains. The phos-pholipids move freely on the membrane surface. On the other hand, the thickness of a bilayer membrane is of the order of 50 Angstroms. These two properties allow to consider it as a two-dimensional fluid membrane. The fluid membranes, self-assembled from surfactant solutions, may have a variety of shapes and topologies [8], which have been explained in terms of bending energy [9, 10].

In real situations, the biomembranes are not trapped in liquids of infinite extent, but they rather confined to geometrical boundaries, such as white and red globules or liposomes (as drugs transport agents [11 - 14]) in blood vessels. For simplicity, we consider the situation where the biomembrane is confined in a liquid domain that is finite in one spatial direction. We denote by D its size in this direction. For a tube, D being the diameter, and for a liquid domain delimitated by two parallel plates, this size is simply the separation between walls. Naturally, the length D must be compared to the bulk roughness, L0 ₁, which is the typical size of humps caused by the thermal fluctuations of the membrane. The latter depends on the nature of lipid molecules forming the bilayer. The biomembrane is confined only when D is much smaller than the bulk roughness L⁰₁. This condition is similar to that usually encountered in confined polymers context [15].

The membrane undulations give rise to repulsive effective interactions between the confining geometrical boundaries. The induced force we term Casimir force is naturally a function of the size D, and must decays as this scale is increased. In this paper, we are interested in how this force decays with distance. To simplify calculations, we assume that the membrane is confined to two parallel plates that are a finite distance D < L⁰₁apart.

The word »Casimir» is inspired from the traditional Casimir effect. Such an effect, predicted, for the first time, by Hendrick Casimir in 1948 [16], is one of the fundamental discoveries in the last century. According to Casimir, the vacuum quantum fluctuations of a confined electromagnetic field generate an attractive force between two parallel uncharged conducting plates. The Casimir effect has been confirmed in more recent experiments by Lamoreaux [17] and by Mohideen and Roy [18]. Thereafter, Fisher and de Gennes [19], in a short note, remarked that the Casimir effect also appears in the context of critical systems, such as fluids, simple liquid mixtures, polymer blends, liquid ⁴He, or liquid-crystals, confined to restricted geometries or in the presence of immersed colloidal particles. For these systems, the critical fluctuations of the order parameter play the role of the vacuum quantum fluctuations, and then, they lead to long-ranged forces between the confining walls or between immersed colloids [20, 21].

To compute the Casimir force between the confining walls, we first elaborate a more general field theory that takes into account the primitive interactions experienced by the confined membrane. As we shall see below, in confinement regime, the field model depends only on two parameters that are the membrane bending rigidity constant and a coupling constant containing all infirmation concerning the interaction potential exerting by the walls. In addition, the last parameter is a known function of the separation D. With the help of the constructed free energy, we first computed the static Casimir force (per unit area), II. The exact calculations show that the latter decays with separation D according to a power law, that is II #c-¹(kBT)²D-³, with a known amplitude. Here, kBT denotes the thermal energy, and #c the membrane bending rigidity constant. Of course, this force increases with temperature, and has significant values only for those biomembranes of small enough #c. The second problem we examined is the computation of the dynamic Casimir force, II (t). More precisely, we considered a biomembrane at temperature T that is initially in a flat state away from the thermal equilibrium, and we were interested in how the expected force grows in time, before the final state is reached. Using a scaling argument, we first showed that the membrane roughness, L1 (t), grows with time as : L1 (t) t¹/⁴(t < r), with the final time r D⁴. The latter can be interpreted as the required time over which the final equilibrium state is reached. Second, using a non-dissipative Langevin equation, we found that the force increases in time according to the power law : II (t) t¹/²(t < r). Third, the discussion is extended to the real situation where the biomembrane is subject to hydrodynamic interactions caused by the flow of the surrounding

liquid. In this case, we show that : L1 (t) t¹/³(t < rh) and II (t) t²/³(t < rh), with the new final
time rh D³. Consequently, the hydrodynamic interactions give rise to drastic changes of the dynamic properties of the confined membrane, since both roughness and induced force grow more rapidly.

This paper is organized as follows. In Sec. II, we present the field model allowing the determination of the Casimir force from a static and dynamic point of view. The Sec. III and Sec. IV are devoted to the computation of the static and dynamic induced forces, respectively. We draw our conclusions in the last section. Some technical details are presented in Appendix.

K. El Hasnaoui et al. African Journal Of Mathematical Physics Volume 8(2010)101-114

La mécanique statistique des membranes biologiques confinées

Chapitre 11

Appendice B.

I. INTRODUCTION