Memoire Online - La mécanique statistique des membranes biologiques confinées

Consider a fluctuating fluid membrane that is confined to two interacting parallel walls 1 and 2. We denote by D their finite separation. Naturally, the separation D must be compared to the bulk membrane roughness, L⁰₁, when the system is unconfined (free membrane). The membrane is confined only when the condition L << L⁰₁ is fulfilled. For the opposite condition, that is L >> L⁰₁, we expect finite size corrections.

We assume that these walls are located at z = -D/2 and z = D/2, respectively. Here, z means the perpendicular distance. For simplicity, we suppose that the two surfaces are physically equivalent. We design by V (z) the interaction potential exerted by one wall on the fluid membrane, in the absence of the other. Usually, V (z) is the sum of a repulsive and an attractive potentials. A typical example is provided by the following potential [22]

Vh (z) = Ahe-^z/^Ah (2.0)
represents the repulsive hydration potential due to the water molecules inserted between hydrophilic lipid heads [22]. The amplitude Ah and the potential-range ëh are of the order of : Ah ^, 0.2 J/m² and ëh ^, 0.2-0.3 nm. In fact, the amplitude Ah is Ah = Ph xëh, with the hydration pressure Ph ^, 10⁸-10⁹ Pa. There, _VvdW (z) accounts for the van der Waals potential between one wall and biomembrane, which are a distance z apart. Its form is as follows

with the Hamaker constant H ^, 10-²² - 10-²¹ J, and ä ^, 4 nm denotes the membrane thickness. For large distance z, this implies

VvdW (z) ? z4 . (2.0)
Generally, in addition to the distance z, the interaction potential V (z) depends on certain length-scales, (î1, ...,î_n), which are the interactions ranges. The fluid membrane then experiences the following total potential

U (z) = V 2 - z + V ₂ + z 2 < z < D 2 . (2.0)
In the Monge representation, a point on the membrane can be described by the three-dimensional position vector r = (x, y, z = h (x, y)), where h (x, y) E [-D/2, D/2] is the height-field. The latter then fluctuates around the mid-plane located at z = 0.

The Statistical Mechanics for the description of such a (tensionless) fluid membrane is based on the standard Canham-Helfrich Hamiltonian [9, 23]

1-l [h] = dxdy ₂ (?h)² + W (h) , (2.0)
with the membrane bending rigidity constant ê. The latter is comparable to the thermal energy kBT, where T is the absolute temperature and kB is the Boltzmann's constant. There, W (h) is the interaction potential per unit area, that is

W (h) = L2 , (2.0)
where the potential U (h) is defined in Eq. (2), and L is the lateral linear size of the biomembrane. Let us discuss the pair-potential W (h).

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

Firstly, Eq. (2) suggests that this total potential is an even function of the perpendicular distance h, that is

Secondly, when they exist, the zeros h0's of the potential function U (h) are such that

This equality indicates that, if h0 is a zero of the potential function, then, -h0 is a zero too. The number of zeros is then an even number. In addition, the zero h0's are different from 0, in all cases. Indeed, the quantity V (D/2) does not vanish, since it represents the potential created by one wall at the middle of the film. We emphasize that, when the potential processes no zero, it is either repulsive or attractive. When this same potential vanishes at some points, then, it is either repulsive of attractive between two consecutive zeros.

Thirdly, we first note that, from relation (2), we deduce that the first derivative of the potential function, with respect to distance h, is an odd function, that is W^' (-h) = -W^' (h). Applying this relation to the midpoint h = 0 yields : W^' (0) = 0. Therefore, the potential W exhibits an extremum at h = 0, whatever the form of the function V (h). We find that this extremum is a maximum, if V ^?(D/2) < 0, and a minimum, if V ^?(D/2) > 0. The potential U presents an horizontal tangent at h = 0, if only if V ^?(D/2) = 0. On the other hand, the general condition giving the extrema {h_m} is

dV dh

????h= D 2 -hm

dV =

????h= D 2 +hm

. (2.0)

Since the first derivative W^' (h) is an odd function of distance h, it must have an odd number of extremum points. The point h = h_m is a maximum, if

and a minimum, if

d²V dh²

????h= D 2 -hm

d²V dh²

????h= D 2 +hm

(2.0)

d²V dh²

????h= D 2 -hm

d²V dh²

????h= D 2 +hm

. (2.0)

The above deductions depends, of course, on the form of the interaction potential V (h).

Fourthly, a simple dimensional analysis shows that the total interaction potential can be rewritten on the following scaling form

where (î1, ...,î_n) are the ranges of various interactions experienced by the membrane, and Ö(x1, ..., xn+1) is a (n + 1)-factor scaling-function.

Finally, we note that the pair-potential W (h) cannot be singular at h = 0. It is rather an analytic function in the h variable. Therefore, at fixed ratios îi/D, an expansion of the scaling-function Ö, around the value h = 0, yields

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

We restrict ourselves to the class of potentials that exhibit a minimum at the mid-plane h = 0. This assumption implies that the coefficient ã is positive definite, i.e. ã > 0. Of course, such a coefficient depends on the ratios of the scale-lengths îi to the separation D.

In confinement regime where the distance h is small enough, we can approximate the total interaction potential by its quadratic part. In these conditions, the Canham-Helfrich Hamiltonian becomes

The prefactor ã will be computed below. The above expression for the elastic constant u gives an idea on its dependance on the film thickness D. In addition, we state that this coefficient may be regarded as a Lagrange multiplier that fixes the value of the membrane roughness.

Thanks to the above Hamiltonian, we calculate the mean-expectation value of the physical quantities, like the height-correlation function (propagator or Green function), defined by

G (x - x^', y - y^') = (h (x, y) h (x^',y^')) - (h (x, y)) (h (x^',y^')) . (2.0)

where ä (x) denotes the one-dimensional Dirac function, and ? = ?²/?x² + ?²/?y² represents the two-dimensional Laplacian operator. We have used the notations : = ê/kBT and u = u/kBT, to mean the reduced membrane elastic constants.

Such a quantity measures the fluctuations of the height-function (fluctuations amplitude) around the equilibrium plane located at z = 0. We show in Appendix that the membrane roughness is exactly given by

provided that one is in the confinement-regime, i.e. D << L⁰_?. Notice that the above equality indicates that the roughness is independent on the geometrical properties of the membrane (through ê). We emphasize that this relation can be recovered using the argument that each point of the membrane has equal probability to be found anywhere between the walls [24].

This formula clearly shows that this elastic constant decays with separation D as D-⁴. The term uh²/2 then describes a confinement potential that ensures the localization of the membrane around the mid-plane. Integral over the hole plane R² of this term represents the loss entropy due to the confinement of the membrane. The value (19) of the elastic constant is compatible with the constraint (17).

Therefore, the elaborated model is based on the Hamiltonian (13), with a quadratic confinement potential. We can say that the presence of the walls simply leads to a confinement of the membrane in a region

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

We define now another length-scale that is the in-plane correlation length, L?. The latter measures the correlations extent along the parallel directions to the walls. More precisely, the propagator G (x - x^', y - y^') fails exponentially beyond L?, that is for distances d such that d =

In contrary to L?, the length-scale L? depends on the geometrical characteristics of the membrane (through ê).

The next steps consist in the computation of the Casimir force at and out equilibrium.

La mécanique statistique des membranes biologiques confinées

II. THEORETICAL FORMULATION