Modelling the spatial effects of the signal transduction process
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In this thesis we construct and analyze a cell signal transduction model for a biological cell which takes into account three dimensional geometry, diffusion, and spatial separation and localization of activating and deactivating enzymes. The deactivation of signalling proteins occurs throughout the cytosol and activation is localized to specific sites in the cell. The model consists of a system of linear reaction diffusion equations (PDEs) and nonlinear boundary conditions defined over a spherical domain with spherical compartments within its interior. Using asymptotic methods we obtain steady state solutions and determine their stability. We also find asymptotic time dependent solutions to the PDE model. We do this analysis for a model with and without delay. The delay is used to model the time lapse during enzyme reactions and also the recovery times associated with conformational changes during the phosphorylation process. For the model without delay, the full PDE system is approximated by a system of differential algebraic equations. The full model with delay is approximated by a system of delay differential algebraic equations. From our results we can detect complex signalling behavior such as bistability, robust switches, and sustained oscillations which all come about from different bifurcations. The simulations of the full three dimensional systems correspond well with simulations of the approximating differential algebraic systems. In this thesis we also introduce a model with cell surface receptors. We model the clustering of these receptors by introducing circular patches on the surface of the sphere. We then investigate the effect that the number of clusters has on the signalling pathway.