On boundary value problems for intracellular subdiffusion and signaling pathways, and for geometric flows

Doctoral thesis, 2006

The main part of this thesis concerns mathematical models for diffusion of proteins inside cells, including reactions between the proteins. Initially, such models are applied to describe signaling pathways in yeast cells, and the properties of the model are studied, especially in relation to models that do not include diffusion. The results show that it is sometimes necessary to include diffusion in the model to capture important aspects of the biological system. The thesis also contains work on the numerical methods used to compute solutions to the reaction-diffusion equations inside domains. Specifically, the Immersed Interface Method, which allows efficient numerical solution inside arbitrary domains using uniform rectangular grids, is applied on Boolean grids, which give the same accuracy as uniform grids while using fewer grid points. The third part of the thesis also concerns models for protein diffusion inside cells, but now describing the phenomenon subdiffusion (or anomalous diffusion), which has been observed inside cells and manifests itself as a qualitatively different, and slower, diffusion behaviour of proteins. The cause of this phenomenon is the crowdedness of the interior of the cell, where other proteins and larger structures interfere with the motion of the proteins. In the thesis, a new mathematical model for anomalous diffusion in the form of a parabolic pseudo-differential equation is proposed, and a proof of existence of solutions for boundary value problems representing anomalous diffusion inside a cell is given. Experiments using Fluorescence Correlation Spectroscopy which support the model have also been performed. Finally, the thesis contains a convergence result for a computational scheme for approximation of mean curvature flows inside a domain, that is the description of the motion of surfaces which move at each point with a velocity depending on the mean curvature at that point. The scheme allows a quite general dependence on the curvature and concerns the case when the moving surface is inside a domain and intersects the domain boundary at a right angle.