Understanding the stability of lipid bilayers

Lipid bilayers demonstrate an intriguing combination of properties: at a microscopic scale they consist of individual molecules without covalent bonding, but at larger scales they show behaviour reminiscent of elastic solids. Energy-based models, going back to Helfrich, are often used to describe this macroscopic behaviour; in these models the bilayer is represented as a smooth surface, and the energy functional penalizes the curvature of this surface.

In this talk I describe our work to understand, mathematically, the connection between a microscopic, diffusion-based model of the lipids on one hand and the macroscopic Helfrich-type surface model on the other. This is too complex a task to do in full generality: instead we consider a simplified continuum lipid model that is derived from Density Functional Theory, in which volume exclusion effects are represented by a mean field. For this simplified model we can investigate the connection in detail, and rigorously, resulting in an expansion of the microscopic energy in terms of the layer thickness. The terms of this expansion represent macroscopic quantities, such as the preferred thickness, the penalty to breakage, and the bending stiffness. This gives an interesting insight into the relative importances of these quantities.