UC Berkeley

The purpose of this thesis is to examine the mechanical aspects of biological lipid bilayers. Constituting the cell membranes of nearly every organism, lipid bilayers are composed of amphiphilic lipid molecules which self-assemble into bilayers in aqueous solution. They exhibit a fascinating combination of flexural resistance as in shell structures and lateral flow as in two-dimensional fluid sheets, giving rise to a host of complex phenomena. Lipid bilayers represent a promising object of inquiry for medical innovation, as dysfunctional lipid bilayers have been linked to disease formation. This work seeks to mathematically depict previously unexplored lipid bilayer phenomena.A simple model of lipid tilt and distension inspired by parallel research in molecular dynamics is outlined and demonstrated numerically. This simple model assumes reflection symmetry of lipid molecule orientation about the bilayer midsurface. Equilibrium configurations for a membrane of this type are presented for domains containing many closely packed voids representing transmembrane proteins. Lipid tilt and distension patterns arise due to the amphipathic nature of these proteins. Then, a model for a lipid bilayer involving independent tilt fields for top and bottom leaflets of the bilayer is developed. This loosened restriction leads to a complex but more general theory. A model for a lipid bilayer with a conforming elastic cytoskeleton is then proposed and equilibrium equations are established. The spontaneous curvature of the conventional Helfrich theory is shown to arise naturally as a mechanical aspect of this system.