Mechanics of biomembranes

The focus of this work is to model the physical behavior of biomembranes. These membranes are composed of oppositely oriented lipid molecules which are free to diffuse on the surface. They constitute an integral part of the cell, and various intracellular organelles in eukaryotic cells. In addition to providing a basic structure, the membranes play a significant role in the trafficking of macromolecules and transduction of signals across them. A knowledge of the underlying mechanics of biomembranes is thus essential to gain insight into the physiology of cells. To this end, we extend the existing mathematical theory of biomembranes and apply it to model various phenomena associated with them.;We develop the framework to model the mechanics of membranes with non-uniform properties. Constant spontaneous curvature, which accounts for asymmetry in the bending response of the membrane, is replaced by a variable distribution representing the influence of attached proteins. The associated mathematical model is used to predict the equilibrium shapes of membranes in the vicinity of nuclear pores and in the process of protein-assisted endocytosis. Further, we develop the theory of coexistent phases in lipid bilayers organized by surface curvatures. The theory of curvature elasticity of membranes is extended with a classical framework for treating phase equilibria using non-convex energy density. A simplified version of the theory is used to study the necking and budding of closed vesicles. Finally, we present a generalized variational treatment of the theory of lipid bilayer films with free edges. In particular, wetting of rigid substrates and the various anchoring conditions known from the theory of liquid crystals are discussed. We formulate the theory to model the adhesion of potentially non-uniform bilayers to planar and curved substrates. The theory is applied to investigate the force deflection response of vesicles adhered to substrates.