Development of high-order numerical methods for the dynamics of suspended lipid membranes with internal structure

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The shape and rigidity of cell membranes are dynamic features related to cell function. An idealized model of the cell membrane that includes these features is a lipid bilayer vesicle, comprised of two or more chemical components capable of phase separation and endowed with different inherent rigidities. Examples include several lipid-cholesterol systems. In general, compositional fluxes and elastic moduli may be derived from a common energy potential, resulting in a tightly coupled system. These vesicles have characteristic thicknesses that are much less than characteristic diameters, suggesting their treatment as co-dimension 1 surfaces. This proposal links three research topics to develop a continuum model of such vesicles in fluid media. First is the high-order interface tracking. Embedded boundary methods will be developed to couple the motion of a membrane suspended in, and containing, viscous fluid. Second is the dynamics -- chemical and mechanical -- of a membrane with time dependent shape. An approach to this problem is to employ cartesian grid finite volume methods with a phase field approach to compute dynamics (Cahn-Hilliard for chemistry, and solid mechanics for motion) in an annulus, with the desired behavior obtained in the limit of zero thickness. The problems of front tracking and interface dynamics must be tightly coupled, e.g., through predictor-corrector and relaxation strategies. The third problem concerns constitutive modeling - the development of energy potentials, constrained by experimental phase diagrams, elasticity measurements, and the symmetry requirements of frame invariance, to couple the effects of chemistry with shape.This proposed work will advance the understanding of numerical partial differential equations of mixed hyperbolic-elliptic type on time-dependent domains. A numerical model of vesicle dynamics, linking the dynamics of shape and chemistry, will be developed for the purpose of better understanding experiments on heterogeneous vesicles, both natural and biomimetic, and ultimately to aid in understanding the connections between cell membrane chemistry, shape, and function.