La mécanique statistique des membranes biologiques confinées

B. Polymer parallel extension

First, we note that the polymer is confined only when its three-dimensional gyration radius _RF3 ,,, aM(D+2)/5D is much greater than the mean-separation H (T_c - T)^-?, that is H << RF3. This condition implies that the polymer confinement is possible only when temperature T is below some value T* = T_c - aM-(D+2)/5D?. This temperature is then smaller than the critical unbinding one T_c, and the shift T_c - T^* essentially depends on the polymer mass M and its spectral dimension D.

The first implication of the polymer confinement is that, its behavior becomes two-dimensional. This means that the polymer can be regarded as a two-dimensional polymeric fractal formed by blobs (pancakes) of size H. To determine the parallel extension of the polymer, R?, as before, we start from a FD free energy

F= kBT

R2 ? + v R2 0

M2 R2H , (24) ?

where v denotes the excluded volume parameter. There, R0 ^aM1/d0F is the ideal radius and R²_?H

represents the volume occupied by the fractal. Minimization of the above free energy with respect to R? gives

R? ^aM(D+2)/4D (W - Wc)?/4 ^aM(D+2)/4D (T_c - T)^?/4 , (T << T^*) . (25)

We have used Eq. (22).

The above result calls the following remarks.

First, the expression of the polymer parallel extension combines two critical phenomena, one is related to the long-mass limit of the polymeric fractal and the other to the vicinity of the unbinding transition of the membranes.

Second, in this formula, naturally appears the fractal dimension (D + 2) /4D of a two-dimensional polymeric fractal.

Third, the parallel radius becomes more and more smaller as the unbinding transition is reached. In other word, this radius is important only when the two adjacent membranes are strongly bound. Finally, the parallel radius expression may be used to determine the unbinding critical exponent 0, in X-ray experiment, for instance.

V. CONCLUDING REMARKS

In this paper, we have two objectives, namely the conformational study of a polymeric fractal inside a tubular vesicle or between two parallel membranes forming an equilibrium lamellar phase. For the former, the length scale is the equilibrium diameter that depends on the characteristics of the membrane, which are the bending modulus and the pressure difference between inner and outer aqueous media. For

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62

the second geometry, the length scale is the mean-separation between the adjacent membranes.

The main quantity to consider was the parallel extension of the confined polymeric fractal. Such a quantity was computed within the framework of an extended Flory-de Gennes theory. Notice that the same result may be recovered using scaling argument or blob model [8].

Another physical quantity to consider is the confinement free energy, denoted ?F. It is the reduced free energy of the polymer, measured from the state with H = oo. Here, H is the tube diameter or the mean-separation. The confinement free energy must have the scaling form : ?F = kBT f (RF3/H), where the unknown scaling function f (x) has the following features : f (x) 0, for x << 1 and f (x) x^m, for x >> 1. The exponent m can be obtained using the fact that ?F must be extensive as a function of the total mass M. This gives m = dF (3) = 5D/ (D + 2). Therefore, ?F = kBTM (H/a)^-5D/(D+2),

with H (?/p)^1/3, for tubular vesicles, and H (W - W_c)^-?, for lamellar phases. Notice that ?F

may be measured by comparison of the two concentrations in the pore (tube or slab) and in the bulk solution. We have Cpore/Cbulk = ? exp (-?F/kBT), where ? is a certain coefficient depending on the ratio RF3/H.

We emphasize that for charged membranes forming the lamellar phase, it was demonstrated [62] that the mean-separation between two adjacent bilayers scales as : H (? - ?_c)^-?, as ? ? ?⁺_c , where ? is the ionic concentration of the aqueous medium and ?_c is its critical value. Of course, the latter depends on the nature of the lipid system. For instance, for DPPC in CaCl2 solutions, experimental measurements [66] showed that ?_c is in the range ?_c ' 84 - 10 mM. In this case, the parallel radius of the polymer

_aM(D+2)/4D (x -- x_o)~^G/4.

scales as : R11

We said above that, for T > T_c, the shape fluctuations drive the membranes forming lamellar phase apart even in the presence of the direct attractive forces. In this case, the system recovers its bound state by a simple application of an external pressure or a lateral tension.

In the presence of an external pressure P, it was found [64] that the mean-separation H scales as : H ?L ^P-1/3 (?L being the membrane mean-roughness). Such a behavior agrees with MC data [67]. In this case, the parallel extension of the polymer obeys the scaling law : R11 aM(D+2)/4DP1/12. As it should be, this extension increases with increasing external pressure.

The role of a lateral tension is to suppress the bending undulations and the fluctuation-induced repulsion. In fact, the latter becomes short-ranged, and the long-ranged van der Waals attraction then dominates

[64]. For this case, it was found [64] that the mean-separation behaves as : H ?L ^Ó-1/2, where

Ó represents the lateral tension. Then, the resulting parallel extension is : R11 aM(D+2)/4DÓ1/8. As

expected, the parallel radius of the polymer increases as the lateral tension is augmented.

For tubular vesicles, we have assumed that they are tensionless. If it is not the case, and if the pressure difference between inner and outer aqueous media can be neglected, the equilibrium diameter is : H = 2 (2?/?)^1/2, which is solution to Eq. (9). Here, ? is the interfacial tension coefficient. In this case, the

parallel extension of the polymer is : R11 ^' aM(D+2)/3D (a²?/?)^1/3. Naturally, this extension increases with increasing interfacial tension coefficient. The minimal diameter corresponds to a typical value ?min ^' a-2?M(1-D)/D of ? (? is fixed). Therefore, the polymer is confined if only if ? < ?min.

Finally, further questions in relation with the subject are under consideration.

ACKNOWLEDGMENTS

We are much indebted to Professors T. Bickel, J.-F. Joanny and C. Marques for helpful discussions, during the »First International Workshop On Soft-Condensed Matter Physics and Biological Systems», 14-17 November 2006, Marrakech, Morocco. One of us (M.B.) would like to thank the Professor C. Misbah for fruitful correspondences, and the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kinds of hospitalities during his regular visits.

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Dans cette cinquième contribution originale, nous présentons une vue large sur le phénomène de ségrégation entre les phospholipides et des chaînes de polymère greffées sur une membrane en bicouche.

En particulier, nous discutons l'influence de la qualité du solvant et la polydispersité des chaînes de polymère greffées sur le comportement de phase critique du mélange.

Nous rappelons que la qualité du solvant apparaît à travers l'expression générale de l'énergie libre du mélange, qui nous permettra la détermination du diagramme de phase relatif à l'agrégation des chaînes de polymère ancrées

Article N°5

Condensation des polymères greffés sur

une biomembrane

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Phys. Scr. 83 (2011) 065801 (6pp) doi:10.1088/0031-8949/83/06/065801

An extended study of the phase separation

between phospholipids and grafted

polymers on a bilayer biomembrane

M Benhamou, I Joudar, H Kaidi, K Elhasnaoui, H Ridouane and H Qamar

Laboratoire de Physique des Polymères et Phénomènes Critiques, Faculté des Sciences, Ben M'sik, PO Box 7955, Casablanca, Morocco