Mathematical Modeling of Biomembranes

This proposal aims at a comprehensive understanding of the dynamics and stability of biomembranes under applied fields. Both applied flow fields and electric fields will be considered. Asymptotic and computational methods will be used to determine the predictions of the model equations. The objective of the proposed research is the development an effective zero-thickness model for a fluid-embedded lipid bilayer membrane. The predictions of the model equations will be applied to explain observed features of membrane behavior such as the appearance of sharp edges, vesicle bursting, and stability of the thin liquid film between membranes undergoing electrofusion. The dynamics of a compound vesicle will also be considered. Generalizations of the lipid bilayer model are proposed to account for micro and macropores. Mathematically, these are challenging free boundary problems exhibiting complex dynamics. Continuum theory will be used to model the motion of the lipid membrane interface and the surrounding liquids, e.g., the Stokes equations for the flow in the bulk and a coupled system of partial differential equations describing the interface evolution and membrane charging. The interface conditions are related to the jump in stress and electric potential across the interface and they will depend on the geometry of the interface and several physical constants, e.g., the bending rigidity and membrane capacitance. A computational method is proposed to solve these complicated transient three-dimensional free-boundary problems. Recent experimental work will be used as a guide in selecting our mathematical models. An investigation beyond the range of the experimental situations will be done to fully understand the models and as a guide to the development of more realistic models of cells.