Pacific Institute for the Mathematical Sciences - PIMS

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.

PIMS Distinguished Lecture 2007