Solutions of Equations for the Regulation of Kinase Activity in a Finite Cylindrical Cell

In this work, inspired by the reality in organisms and particularly the shape of axon of the neuron, new mathematical models of regulation of kinase activity are presented. In the view of mathematician, those models are diffusion equations defined in finite cylinders but with mixed Robin boundary and Dirichlet boundary conditions. The first part of the thesis focuses mainly on the model with the linear mixed Robin boundary and Dirichlet boundary conditions. By use of variational principle and eigenvalue problem, the results are provided on the existence, uniqueness and boundedness of the weak solution ( kinase concentration ) of an abstract elliptical equation related to the kinase activity model. For the kinase activity model itself, the bound can be expressed as the function of relevant parameters. Furthermore, this work also obtains the existence of the time-dependent solution to the reaction diffusion equation generalized from this kinase activity model. Base on those results, the time-dependent solutions are presented in integral form. This work has shown the exponential convergence of the time-dependent solution to the solution of its corresponding steady state equation. The second part has demonstrated the existence and boundedness of the weak solution by use of variational principle for the kinase model with mixed nonlinear boundary conditions. Then the series representation of the nonzero solution is shown. Moreover, a critical equality in bifurcation analysis is obtained. Based on this equality, when a parameter varies, bifurcation analysis is demonstrated from the special case to the more general case. Particularly, the critical (bifurcation) value of this biological parameter has been determined mathematically as a function of other biological parameters. By use of those theoretical results, some corresponding biological explanations are also provided. All of those have significance when considering biology signalling and biology control.