La mécanique statistique des membranes biologiques confinées

I. INTRODUCTION

The polymer confinement finds many applications in various domains, such as biological functions, filtration, gel permeation chromatography, heterogeneous catalysis, and oil recuperation.

The physics of polymer confinement is a rich and exciting problem. Recently, much attention has been paid to the structure and dynamics of polymer chains confined to two surfaces, or inside cylindrical pores [1 - 7]. Thereafter, the study has been extended to more complex polymers, called D-polymeric fractals or D-manifolds [8] that are restricted to the same geometries. Here, D is the spectral dimension that measures the degree of the connectivity of monomers inside the polymer [9]. For instance, this intrinsic dimension is 1 for linear polymers and 4/3 for branched ones. The D-manifolds may be polymerized (or

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M. Benhamou et al. African Journal Of Mathematical Physics Volume 10(2011)55-64

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crumpled) vesicles [10]. The polymer confinement between two surfaces and in cylinders with sinusoidal undulations was also investigated [11, 12].

The confining geometries may be soft-bodies, as bilayer biomembranes and surfactants, and spherical and tubular vesicles. A particular question has been addressed to polymer confinement in surfactant bilayers of a lyotropic lamellar phase [13], where the authors reported on small-angle X-ray scattering and free-fracture electron microscopy studies of a nonionic surfactant/water/polyelectrolyte system in the lamellar phase region. The fundamental remark is that, the polymer molecules cause both local deformation and softening of the bilayer.

In the same context, it was experimentally demonstrated [14] that the polymer confinement may induce a nematic transition of microemulsion droplets. More precisely, the authors showed that upon confinement, spherical droplets deform to prolate ellipsoid droplets. The origin of such a structural transition may be attributed to a loss of the conformational entropy of polymer chains due to the confinement.

In other experiments [15 - 17], a new phase has been observed adding an neutral hydrosoluble polymer (PVP) in the lyotropic lamellar phase (CPCI/hexanol/water). Also, one has studied the effect of a neutral water-soluble polymer on the lamellar phase of a zwitterionic surfactant system [18].

The polymer confinement is also relevant for living systems and governs many biological processes. As example, we can quote membrane nanotubes that play a major role in intercellular traffic, in particular for lipid and proteins exchange between various compartments in eukaryotic cells [19 - 22]. The traffic of macromolecules and vesicles in nanotubes is ensured by molecular motors [23]. Also, these are responsible for the extraction of nanotubes [24]. The formation of tubular membrane tethers or spicules [25 - 33] results from the action of localized forces that are perpendicular to the membrane. These forces may originate from the polymerization of fibers [34], as actin [35], tubulin [36], or sickle hemoglobin [37]. Inspired by biological processes, as macromolecules and vesicles transport, we aim at a conformational study of confined polymers in aqueous media delimitated by biomembranes. More precisely, the purpose is to see how these biomembranes can modify the conformational properties of the constrained polymer. To this end, we choose two geometries : a tubular vesicle (Geometry I), and two parallel biomembranes (Geometry II). To be more general, we consider polymers or arbitrary connectivity called D-polymeric fractals or D-manifolds [38]. Linear and branched polymers, and polymerized vesicles constitute typical examples. We note that this conformational study in necessary for the description of dynamic properties of polymers restricted to these geometries.

As we shall below, the polymer confinement is entirely controlled by the confining biomembranes state. Indeed, for Geometry I, the polymer may be confined only when the confining tubular vesicle is at equilibrium. For Geometry II, the confinement is possible if only if the two parallel membranes are in the binding state.

This paper is organized as follows. In Sec. II, we briefly recall the conformational study of unconfined D-polymeric fractals in good solvent. Sec. III deals with the conformational study of polymers confined inside a tubular vesicle. In Sec. IV, we extend the study to D-polymeric fractals confined to two parallel biomembranes. Some concluding remarks are drawn in the last section.