La mécanique statistique des membranes biologiques confinées

II. UNCONFINED POLYMERIC FRACTAL

Consider a single polymeric fractal of arbitrary topology (linear polymers, branched polymers, polymer networks, ...). We assume that the considered polymer is trapped in a good solvent. We denote by

RF aM1/dF (1)

its gyration (or Flory) radius, where dF is the Hausdorff fractal dimension, M is the molecular-weight (total mass) of the considered polymer, and a denotes the monomer size. The mass M is related to the linear dimension N by : M = N^D, where D is the spectral dimension [9]. The latter is defined as the Hausdorff dimension corresponding to the maximal extension of the fractal.

Naturally, the Hausdorff dimension depends on the Euclidean dimensionality d, the spectral dimension D and the solvent quality. When the polymer is ideal (without excluded volume forces), its Hausdorff dimension, d⁰_F, is a known simpler function of D [9]

d0 F = 2D 2 - D .

(2)

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

For linear polymers (D = 1), d⁰_F = 2 [2], for ideal branched ones (D = 4/3), d⁰_F = 4 [9], and for crumpled membranes (D = 2), d⁰_F = 8.

Because of the positivity of the Hausdorff dimension, the above expression makes sense only for D < 2. Indeed, this condition is fulfilled for any complex polymer with spectral dimension in the interval 1 = D < 2 [9].

A polymeric fractal in good solvent is swollen, because of the presence of the excluded volume forces. The polymer size increases with increasing total mass M according to the power law (1). The first implication of the polymer swelling is that, the actual Hausdorff dimension dF is quite different from the Gaussian one, defined in Eq. (2). However, there exists a special value of the Euclidean dimensionality called upper critical dimension _duc [39,40], beyond which the polymeric fractal becomes ideal. This upper dimension is naturally a D-dependent function, which can be determined [39] using a criterion of Ginzburg type, usually encountered in critical phenomena [41,42]. According to Ref. [39], _duc is given by

4D

duc =

2 - D . (3)

For instance, the upper critical dimension is 4 for linear polymers [2], and 8 for branched ones [39]. We emphasize that, in general, the Hausdorff fractal dimension dF cannot be exactly computed. Many techniques have been used to determine its approximate value, in particular, the Flory-de Gennes (FD) theory [2]. Using a generalized FD approach, it was found that the fractal dimension is given by [39]

dF = D d + 2

D + 2 , (4)

below the critical dimension, and it equals the Gaussian fractal dimension d⁰_F described above. For dimension 3, we have

5D

dF (3) =

D + 2 . (5)

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For instance, for linear polymers, dF (3) = 5/3, and dF (3) = 2, for branched ones (animals).

In the following paragraph, we shall focus our attention on the conformational study of a single polymeric fractal confined to a long tubular vesicle.

III. CONFINED POLYMERIC FRACTAL IN GEOMETRY I
A. Useful backgrounds

Before studying the conformation of a single polymeric fractal, we recall some basic backgrounds concerning the equilibrium shape of tubular vesicles. This can be done using Differential Geometry machineries.

The tubular vesicle is essentially formed by two adjacent leaflets (inner and outer) that are composed of amphiphile lipid molecules. These permanently diffuse with the molecules of the surrounded aqueous medium. Such a diffusion then provokes thermal fluctuations (undulations) of the membrane. This means that the latter experiences fluctuations around an equilibrium plane we are interested in.

Consider a biomembrane of arbitrary topology. A point of this membrane can be described by two local coordinates (u1, u2). From surfaces theory point of view, at each point, there exists two particular curvatures (minimal and maximal), called principal curvatures, denoted C1 = 1/R1 and C2 = 1/R2. The quantities R1 and R2 are the principal curvature radii. With the help of the principal curvatures, one constructs two invariants that are the mean-curvature

1 C = 2 (C1 + C2) ,

(6)

and the Gauss curvature

K = C1C2 . (7)

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We recall that C1 and C2 are nothing else but the eigenvalues of the curvature tensor [43].

To comprehend the geometrical and physical properties of the biomembranes, one needs a good model. The widely accepted one is the fluid mosaic model proposed by Singer and Nicholson in 1972 [44]. This model consists to regard the cell membrane as a lipid bilayer, where the lipid molecules can move freely in the membrane surface like a fluid, while the proteins and other amphiphile molecules (cholesterol, sugar molecules, ...) are simply embedded in the lipid bilayer. We note that the elasticity of cell membranes crucially depends on the bilayers in this model. The elastic properties of bilayer biomembranes were first studied, in 1973, by Helfrich [45]. The author recognized that the lipid bilayer could be regarded as smectic-A liquid crystals at room temperature, and proposed the following curvature free energy

? ? ? ?

?

F = (2C + 2C0)² dA + ?G KdA + ?dA + p dV . (8)
2

where dA denotes the area element, and V is the volume enclosed within the lipid bilayer. In the above definition, ? accounts for the bending rigidity constant, C0 for the spontaneous curvature, ?G for the Gaussian curvature, ? for the surface tension, and p for the pressure difference between the outer and inner sides of the vesicles. The first order variation gives the shape equation of lipid vesicles [46]

p - 2?C + ? (2C + C0) (2C² - C0C - 2K) + ??² (2C) = 0 , (9)

with the surface Laplace-Bertlami operator ?2 = 1 vg

? (vggij ? ) , (10) ?u^i ?uj

where gⁱj is the metric tensor on the surface and g = det ^(gij). For open or tension-line vesicles, local differential equation (9) must be supplemented by additional boundary conditions we do not write [47]. The above equation have known three analytic solutions corresponding to sphere [46], v2-torus [48 - 51] and biconcave disk [52].

For cylindrical (or tubular) vesicles, one of the principal curvature is zero, and we have

1

C = - R , K = 0 , (11)

where R is the radius of the cylinder. If we ignore the boundary conditions (assumption valid for very long tubes), the uniform solution to equation (9) is

_(4?)1/3

H = 2 , (12)

p

where H is the equilibrium diameter. We have neglected the surface tension and spontaneous curvature contributions, in order to have a simplified expression for the equilibrium diameter.

The above relation makes sense as long as the pressure difference is smaller than a critical value p_c that scales as [53] : p_c ? ?/R³. The latter is in the range 1 to 2 Pa. The meaning of the critical pressure is that, beyond p_c, the vesicle is unstable. This implies that the equilibrium diameter must be greater than the critical one H_c = 2 (4?/p_c)^1/3.

In what follows, we shall use the idea that consists to regard the tubular vesicle as a rigid cylinder of effective diameter H that depends on the characteristics of the bilayer through the parameters ? and p.