Interactions of Fluid Membranes

Following Helfrich we assume that the net result of these collisions is that each membrane experiences an effective interaction with its nearest neighbors which has as its lowest energy state, the periodic configuration, where 

This interaction is represented as a quadratic form in the deviation of (Z +i — Zn) — d from zero, or equivalently in y) - hn x, y). In addition, there 

Here k is the bending modulus, and the subscripts of xx and yy represent two derivatives of the membrane position variable — i.e., h + h yy) is the mean curvature of the nth membrane. This expression is correct for membranes with gentle undulations (Vh 1) otherwise the simple expression for the curvature is incorrect and the area constraints must be reconsidered as well. The compressional elastic constant, B, represents an effective repulsion between the membranes and will be computed self-consistently. Note that this Hamiltonian is unchanged if the positions of all the membranes are uniformly shifted, representing a trivial translation of the system. Fourier transforming in both the z direction (Fourier wavevector Q with an upper cutoff of nlD due to the periodicity) and the a — y plane (Fourier wavevector q — qy)) we have 

The Boltzmann factor corresponding to the Hamiltonian of Eq. (6.56) is a Gaussian, so the free energy, F is easily evaluated from 

Performing the integral, we find that the difference in free energy per unit volume, Af, between the many membrane system and that of a single membrane (where d 00 and we anticipate that B 0) is given by 

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Helfrich, W. (1978). Steric interaction of fluid membranes in multilayer systems. Z. Natur-forsch., A33, 305-315. 

Helfrich, W., Steric Interactions of Fluid Membranes in Multilayer Systems., Z. Naturforsch., 33a, 305-315 (1978). 

The only feasible procedure at the moment is molecular dynamics computer simulation, which can be used since most systems are currently essentially controlled by classical dynamics even though the intermolecular potentials are often quantum mechanical in origin. There are indeed many intermolecular potentials available which are remarkably reliable for most liquids, and even for liquid mixtures, of scientific and technical importance. However potentials for the design of membranes and of the interaction of fluid molecules with membranes on the atomic scale are less well developed. 

From J. T. Segrest and L. D. Kohn, Protein-lipid interactions of the membrane penetrating MN-glycoprotein from the human erythrocyte, Protides of the Biological Fluids, 21st colloquium, ed. by J. Peeters, Pergamon Press, New York, 1973. 

W. Helfrich and R.-M. Servuss, Nuotfo Cimento D, 3, 137 (1984). Undulations, Steric Interaction, and Cohesion of Fluid Membranes. 

In some sense the physical properties of fluid membranes are unique because they have negligible surface tension. Consequently, their free energy is governed by their geometrical shape and its fluctuations. The rigidity k associated with the restoring force to layer bending is then the important modulus which in many cases will determine the physical state of the membrane. From a biophysical view point the physical nature of a fluid membrane surface may in some cases have a profound influence on the precise mechanism of membrane-membrane interactions which influence processes such as cell-cell contact. 

II. FREE ENERGY OF FLUID MEMBRANES AND INTERMEMBRANE INTERACTIONS II.1. Flat Interfaces with kc kgT... 

The indirect interaction of fluid amphiphilic bilayers is generally assumed to arise from their undulations, i.e. from out-of-plane fluctuations controlled by the bending rigidity. Pure steric interaction is characterized by a hard core potential for the direct interaction. Its energy per unit area in a stack of membranes was calculated to be ... 

Whereas the main challenge for the first bilayer simulations has been to obtain stable bilayers with properties (e.g., densities) which compare well with experiments, more and more complex problems can be tackled nowadays. For example, lipid bilayers were set up and compared in different phases (the fluid, the gel, the ripple phase) [67,68,76,81]. The formation of large pores and the structure of water in these water channels have been studied [80,81], and the forces acting on lipids which are pulled out of a membrane have been measured [82]. The bilayer systems themselves are also becoming more complex. Bilayers made of complicated amphiphiles such as unsaturated lipids have been considered [83,84]. The effect of adding cholesterol has been investigated [85,86]. An increasing number of studies are concerned with the important complex of hpid/protein interactions [87-89] and, in particular, with the structure of ion channels [90-92]. 

The preceding accomplishments are applied in nature, but required tremendous amounts of basic research on mass transfer, interactions of materials with biological components, fluid dynamics, separation processes (especially chromatography and membrane separations), and biochemical kinetics. 

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