Bilayer membranes undulation forces

The interest in lipid bilayers is due to their relevance to biological membranes [1], They exhibit a richness of structures due to the interplay between many different inter- and intrabilayer forces. Among all the multilamellar bilayer structures, probably the most pertinent to biological membranes are the lamellar ones. Their equilibrium spacing is considered to be the result of a balance between attractive and repulsive forces. While the former forces are just the usual van der Waals interactions, the latter are composed of double layer forces (for charged bilayers) [2], hydration forces (due to the structuring of water near interfaces) [3] and repulsive forces generated by the thermal undulation of the membranes [4]. 

The undulation force arises from the configurational confinement related to the bending mode of deformation of two fiuid bilayers, like surfactant lamellas or lipid membranes. This mode consists in undulation of the bilayer thickness and constant area of the bilayer midsurface (Fig. 30b). Helfrich et al. [396,397] established that two such bilayers, at a mean surface-to-surface distance of h, experience a repulsive disjoining pressure given by... 

Different components contribute to the force between lipid bilayers van der Waals attraction, hydration repulsion, steric repulsion, electrostatic forces, and undulation forces. Electrostatic double-layer forces arise if charged lipids are present in the membrane. Surface charges also occur in the presence of divalent ions. Ca and adsorb to lipids, which leads to net positive charge even at concentration of 1 mM [1257]. Double-layer forces are suppressed by adding monovalent salt into the solution. 

Undulation forces not only act between lipid bilayers in the L phase but also prevent vesicles from coagulation and act between membranes and other surfaces [1266]. Emulsions are stabilized by undulation forces [1267]. 

The dependence of the interaction force between two undulating phospholipid bilayers and of the root-mean-square fluctuation of their separation distances on the average separation can be determined once the distribution of the intermembrane separation is known as a function of the applied pressure. However, most of the present theories for interacting membranes start by assuming that the distance distribution is symmetric, a hypothesis invalidated by Monte Carlo simulations. Here we present an approach to calculate the distribution of the intermembrane separation for any arbitrary interaction potential and applied pressure. The procedure is applied to a realistic interaction potential between neutral lipid bilayers in water, involving the hydration repulsion and van der Waals attraction. A comparison with existing experiments is provided. 

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