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La mécanique statistique des membranes biologiques confinées

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par Khalid EL HASNAOUI
Faculté des sciences Ben M'Sik Casablanca - Thèse de doctorat  2011
  

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V. CONCLUSIONS

In this work, we have reexamined the computation of the Casimir force between two parallel walls delimitating a fluctuating fluid membrane that is immersed in some liquid. This force is caused by the thermal fluctuations of the membrane. We have studied the problem from both static and dynamic point of view.

We were first interested in the time variation of the roughening, L1 (t), starting with a membrane that is inially in a flat state, at a certain temperature. Of course, this length grows with time, and we found that : L1 (t) t°? (91 = 1/4), provided that the hydrodynamic interactions are ignored. For real systems, however, these interactions are important, and we have shown that the roughness increases more rapidly

as : ?L1 (t) t? (91 = 1/3). The dynamic process is then stopped at a final r (or rh) that represents the required time over which the biomembrane reaches its final equilibrium state. The final time behaves as : r D4 (or rh ? D3), with D the film thickness.

Now, assume that the system is explored at scales of the order of the wavelength q-1, where q = (4ð/À) sin (9/2) is the wave vector modulus, with À the wavelength of the incident radiation and 9 the

scattering-angle. In these conditions, the relaxation rate, r (q), scales with q as : r-1 (q) q1/°? = q4

( h (q) q1/?°? = q3)

or r-1 . Physically speaking, the relaxation rate characterizes the local growth of the

height fluctuations.

Afterwards, the question was addressed to the computation of the Casimir force, II. At equilibrium, using an appropriate field theory, we found that this force decays with separation D as : II D-3, with a known amplitude scaling as #c-1, where #c is the membrane bending rigidity constant. Such a force is then very small in comparison with the Coulombian one. In addition, this force disappears when the

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

temperature of the medium is sufficiently lowered.

The dynamic Casimir force, II (t), was computed using a non-dissipative Langevin equation (with noise), solved by the time height-field. We have shown that : II (t) t°f (èf = 2è1 = 1/2). When the hydrodynamic interactions effects are important, we found that the dynamic force increases more rapidly as :

( )

IIh (t) '-- t?°f ?èf = 2?è1 = 2/3 .

Notice that we have ignored some details such as the role of inclusions (proteins, cholesterol, glycolipids, other macromolecules) and chemical mismatch on the force expression. It is well-established that these details simply lead to an additive renormalization of the bending rigidity constant. Indeed, we write êeffective = ê + äê, where ê is the bending rigidity constant of the membrane free from inclusions, and äê is the contribution of the incorporated entities. Generally, the shift äê is a function of the inclusion concentration and compositions of species of different chemical nature (various phospholipids forming the bilayer). Hence, to take into account the presence of inclusions and chemical mismatch, it would be sufficient to replace ê by êeffective, in the above established relations.

As last word, we emphasize that the results derived in this paper may be extended to bilayer surfactants, although the two systems are not of the same structure and composition. One of the differences is the magnitude order of the bending rigidity constant.

APPENDIX

To show formula (17), we start from the partition function that we rewrite on the following form

J I j

-H [h]

Z = Dh exp =

kBT

D/2

f

-D/2

dz (z) . (5.0)

113

Also, it is easy to see that the membrane mean-roughness is given by

L2 1 =

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