Parabolic Variational Problems of 1-Laplacian Type and Finite Element Models of Protein Diffusion and Homogenized Cone Photoreceptor Visual Transduction

Abstract

Two projects in classical analysis and two projects in computational mathematics applied to biology are considered. First a necessary and sufficient condition for the continuity at a point of minima for parabolic variational integrals with linear growth is obtained. Second a topology of convergence by which parabolic variational solutions of the p-Laplacian with time independent Dirichlet data tend to those for the variational 1-Laplacian is quantified. The third project numerically models Laplace-Beltrami driven diffusion of proteins along tube shaped cell membranes to understand the influence of tubular curvature on protein membrane surface diffusion. The Finite Element code for this project was built using B-splines. The fourth project applies homogenization and concentrated capacity to a PDE system, with nonlinear couple in the Neumann data, for diffusing, biological 2nd messengers Ca2+ and cGMP in photoreceptors of vertebrate retina. A standard diffusion model and homogenized model have been implemented through Matlab based Finite Element code. Convergence of the homogenized model towards the standard diffusion model is reported as well as a performance comparison. Cones are fragile and hard to isolate in wetbench settings. A collection of numerical experiments are reported that virtually modify the biochemical kinetics and morphologies of the cone photoreceptor cells.