Memoire Online - La mécanique statistique des membranes biologiques confinées

Consider now a polymeric fractal of arbitrary topology confined inside a tubular vesicle of equilibrium diameter H. The host solvent is assumed to be a good solvent. The polymer is confined if only if its three-dimensional Flory radius _RF3 is much larger than the diameter H, that is H << RF3. At fixed parameters ? and p, this condition implies that the polymer mass M must be greater than some typical value M^* that scales as M^* (?/p)^5D/3(D+2). Now, if the parameter M and p are fixed, the polymer is

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

confined is only if ? << ?* ^, ^pM3(D+2)/5D. This behavior clearly indicates that the confinement is more favorable when the tubular vesicle is of small bending rigidity constant. We note that the confinement condition depends on the nature of the polymeric fractal through its spectral dimension D. Also, the solvent quality is another factor that influences this confinement.

The confinement of polymers of arbitrary topology inside a rigid tube is largely investigated in Ref. [8]. In this paper, to achieve the conformational study of a confined polymer, we use a generalized FD theory based on the following free energy [8]

kBT =

_R2

+ v R²₀

_M2

(13)

R?H²

where R? denotes the polymer parallel extension to the tube-axis, and v is the excluded volume parameter.

There, R0 ^, ^aM1/d°, is the ideal radius and R?H² represents the volume occupied by the fractal. Minimizing the above free energies with respect to R? yields the desired result

Firstly, notice that, in any case, the parallel radius naturally depends on polymer and tubular vesicle

Secondly, at fixed polymer mass M, the parallel extension is important for those tubular vesicles of small

Finally, the above behavior is valid as long as the parallel extension R? remains below the maximal

extension of the polymer, that is we must have R? < aM¹/^D (maximal extension). This gives a minimal tube diameter

Therefore, the confinement of the polymeric fractal implies that the tube diameter is in the interval Hmin < H < RF3. For instance, for linear polymers, _Hmin ^, a, and _Hmin ^, aM¹/⁸, for branched ones. The minimum pore size for linear polymers is then independent on the molecular-weight. This means that the linear polymers find their way even through a very small pore. It is not the case for branched polymers, for which the minimum pore size increases with the total mass. As pointed out in Ref. [8], it should be possible to construct a porous medium that may separate a mixture of branched and linear molecules on a basis of their connectivity. Indeed, one can choose an appropriate minimum pore size through which linear polymers can pass, whereas branched polymers cannot. Now, combining relations (12) and (15) yields a minimal value for the bending rigidity constant (at fixed pressure difference)

Thus, a tubular membrane confines a polymeric fractal only if its bending modulus is in the interval ?min < ? < ?*, where the typical value ?* is that given above.

The following paragraph is devoted to the confinement of a polymer of arbitrary topology to two parallel fluctuating fluid membranes.

Consider a lamellar phase formed by two parallel bilayer membranes. The cohesion between these lipid bilayers is provided by long-ranged van der Waals forces [54]. But these attractive interactions are balanced, at short membrane separation, by strong repulsion coming from hydration forces for uncharged bilayers [55]. In addition, for bilayers carrying electric charges, the attractive interaction is reduced by

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

For two parallel bilayer membranes that are a finite distance l apart, the total interaction energy (per unit area) is the following sum

The first part represents the hydration energy. The hydration forces that act at small separation of the order of 1 nm, have been discovered for multilayers under external stress [55]. The adopted form for the hydration energy is an empirical exponential decay

The typical values of amplitude AH and potential-range ?H are AH ? 0.2 J/m² and ?H ? 0.3 nm. The electrostatic energy between two charged membranes also decays exponentially, that is

provided that the separation l is greater than the Debye-H·uckel length. The potential amplitude and its range depend on ionic concentration in the aqueous solution, ?, and the surface charge density at membranes, ?0. More precisely, ?E and AE scale as : ?E ^?-1/2 and AE ?²₀?-¹/². The last part is the attractive van der Waals energy that results from polarizabilities of lipid molecules and water molecules. This interaction energy has the standard form

The Hamaker constant is in the range W ? 10-²² -10-²¹ J. In the above expression, the bilayer thickness ? is of the order of ? ? 4 mm.

In principle, one must add a steric interaction energy that originates from membranes undulations. According to Helfrich [45], this energy (per unit area) is

with kB the Boltzmann's constant, T the absolute temperature, and ? the common bending rigidity constant of the two membranes. But, for two bilayers of different bending rigidity constants ?1 and ?2, we have ? = ?1?2/ (?1 + ?2). The precise value of coefficient cH remains an open debate. Indeed, Helfrich gave for this coefficient the value cH ? 0.23 [45], but computer simulations predict smaller prefactors, namely, cH ? 0.16 [58], cH ? 0.1 [59], cH ? 0.07 [60] and cH ? 0.08 [61]. Therefore, the steric interaction energy is significant only for those membranes of small bending modulus. Of course, this energy vanishes for rigid interfaces (? ? 8).

We note that the lamellar phase remains stable at the minimum of the potential, provided that the potential-depth is comparable to the thermal energy kBT. This depends, in particular, on the value of W-amplitude.

From a theoretical point of view, Lipowsky and Leibler [62] had predicted a phase transition that drives the system from a state where the membranes are bound to a state where they completely separated. Such a phase transition is first-order if the steric repulsive energy is taken into account. But, if this energy is ignored (for relatively rigid membranes), the transition is rather second-order. We restrict ourselves to second-order phase transitions, only. The authors have shown that there exists a certain threshold W_c beyond which the van der attractive interactions are sufficient to bind the membranes together, while below this characteristic amplitude, the membrane undulations dominate the attractive forces, and then, the membranes separate completely. In fact, the critical value W_c depends on the parameter of the problem, which are temperature T, and parameters AH, ?H, ? and ?. For room temperatures and AH ? 0.2 J/m², ?H ? 0.3 nm, and ? ? 4 nm, one has W_c ? (6.3 - 0.61) × 10-²¹ J, when the bending rigidity constant is in the range ? ? (1 - 20) × 10-¹⁹ J. For instance, for egg lecithin, one has [63] ? ? (1 - 2) × 10-¹⁹ J, and the corresponding threshold W_c is in the interval W_c ? (6.3 - 4.1) × 10-²¹ J. We note that the typical value W_c corresponds to some temperature, T_c, called unbinding critical temperature [62, 64].

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

Let us first consider uncharged membranes, and notice that the Hamaker constant may be varied changing the polarizability of the aqueous medium. It was found [62] that, when the critical amplitude is approached from above, the mean-separation between the two membranes diverges according to

From an experimental point of view, critical fluctuations in membranes were considered in some experiment [65], and in particular, the mean-separation was measured.

We have now all ingredients to study the conformation of a single polymeric fractal of arbitrary topology, which is confined to two parallel fluid membranes forming a lamellar phase.

La mécanique statistique des membranes biologiques confinées

B. Parallel extension to the cylinder-axis