Mathematical analysis of a spatially coupled reaction-diffusion system for signaling networks in biological cells

Abstract

The main purpose of this thesis was the mathematical analysis of certain spatially coupled Reaction-Diffusion Systems arising in the description of Signaling Networks in biological cells. The spatial coupling between a diffusion in the cytosolic bulk and reaction and diffusion processes on the boundary surface is given by a Robin-type boundary condition that introduces a source term in the boundary equations. This thesis provides existence and well-posedness results for prototypes of such models. The main results were the following: Considering regular data and classical diffusion operators (in case of the surface equations expressed by the Laplace-Beltrami operator) we find existence of classical solutions and well-posedness based on an operator splitting approach that decouples bulk and surface equations. Complementary to the classical setting we have also considered possibly nonsmooth diffusion operators that for example can model heterogeneous domains with specific properties on the membrane. The existence and well-posedness result was based on an implicit discretization in time which reduces the given nonlinear parabolic system to a sequence of nonlinear recursive elliptic problems. This was then solved by an application of the theory of monotone perturbed operators. Furthermore, the results were applied to rigorously justify an asymptotic model reduction to a nonlocal two-variable system on the membrane.