Interfacial Mechanics of Cell Membranes: Stochastic Exterior Calculus Approaches for Curved Fluid Lipid-Protein Bilayers

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Mathematical approaches are playing an increasingly important role in the biological sciences. This ranges from the development of new models and theories that provide insights into biological processes, to the development of new analytic tools and software that are useful in experimental design and in the analysis of experimental data. In cell biology, lipid bilayer membranes can be thought of as effectively two-dimensional materials comprising of a heterogeneous mixture of lipids, proteins, and other small molecules. The individual and collective dynamics of these molecules are fine-tuned to carry out complex cellular processes ranging from signaling to synaptic transmission to the regulation of shapes of organelles. The effective two-dimensional fluid-elastic nature of cell membranes yields interfacial phenomena and complicated geometric shapes effecting both molecular interactions and dynamics that can be very distinct from their bulk three dimensional counterparts. To gain a deeper understanding of cellular processes in curved protein-lipid bilayers, there is a need for new approaches for incorporating the important roles played by geometry. This research develops new mathematical approaches and software for handling complicated geometries in investigations conducted through laboratory experiments and computational simulations of membrane-protein systems. The educational activities of the project will further enhance the training of both undergraduate and graduate students in new quantitative approaches to the biological sciences. The research of this project will develop new mathematical approaches to capture diffusive transport of proteins within curved membranes arising from both active collective motions and passive thermal fluctuations, hydrodynamics and viscoelastic deformations of curved surfaces, and discrete heterogeneous interactions between inclusions incorporating stochastic kinetics. Mathematical approaches will be extended for both continuum mechanics and coarse-grained molecular descriptions so as to provide insights into the emergent mechanics and dynamics of inclusions within bilayer membranes. To perform computational simulations, a general class of numerical methods based on exterior calculus will be designed to address the challenges associated with solving differential equations on curved manifolds. The research of this project advances the next generation of theoretical models and analysis tools for investigating membrane-protein systems. Societal impacts of the work derive from fundamental contributions to the understanding of membrane-protein systems in cell biology, which may give insights into processes relevant to membrane-related technologies such as drug delivery through permeation or design of antimicrobials such as membrane disruptors.