Biological membranes are important structural units in the cell. Composed of a lipid bilayer with embedded proteins, most exploration of membranes has focused on the proteins. While proteins play a vital role in membrane function, the lipids themselves can behave in dynamic ways which affect membrane structure and function. Furthermore, the dynamic behavior of the lipids can affect and be affected by membrane geometry. A novel fluid membrane model is developed in which two different types of lipids flow in a deforming membrane, modelled as a two-dimensional Riemannian manifold that resists bending. The two lipids behave like viscous Newtonian fluids whose motion is determined by realistic physical forces. By examining the stability of various shapes, it is shown that instability may result if the two lipids forming the membrane possess biophysical qualities, which cause them to respond differently to membrane curvature. By means of numerical simulation of a simplified model, it is shown that this instability results in curvature induced phase separation. Applying the simplified model to the Golgi apparatus, it is hypothesized that curvature induced phase separation may occur in a Golgi cisterna, aiding in the process of protein sorting.
In addition to flowing tangentially in the membrane, lipids also flip back and forth between the two leaflets in the bilayer. While traditionally assumed to occur very slowly, recent experiments have indicated that lipid flip-flop may occur rapidly. Two models are developed that explore the effect of rapid flip-flop on membrane geometry and the effect of a pH gradient on the distribution of charged lipids in the leaflets of the bilayer. By means of a stochastic model, it is shown that even the rapid flip-flop rates observed are unlikely to be significant inducers of membrane curvature. By means of a nonlinear Poisson- Boltzmann model, it is shown that pH gradients are unlikely to be significant inducers of bilayer asymmetry under physiological conditions.