Diffusion on Surfaces and the Boundary Periodic Unfolding Operator with an Application to Carcinogenesis in Human Cells

Abstract

In the context of periodic homogenization based on the periodic unfolding method, we extend the existing convergence results for the boundary periodic unfolding operator to gradients defined on manifolds. These general results are then used to homogenize a system of five coupled reaction-diffusion equations, three of which are defined on a manifold. The system describes the carcinogenesis of a human cell caused by Benzo-[a]-pyrene molecules. These molecules are activated to carcinogens in a series of chemical reactions at the surface of the endoplasmic reticulum. The diffusion on the endoplasmic reticulum, modeled as a Riemannian manifold, is described by the Laplace--Beltrami operator. The binding process to the surface of the endoplasmic reticulum is modeled in a nonlinear way taking into account the number of free receptors.