Contributions to the Analysis of Biochemical Reaction-Diffusion Networks Stability, Analysis, and Numerical Solutions

In this thesis we address dynamic systems problems that arise from the study of biochemical networks. Here we prefer a rigorous treatment of the differential equations that govern their spatio-temporal dynamics, at the cost of studying simplified scenarios of the biological systems under study. Although these simplified scenarios do not model all aspects of the complex interplay in the biological system, they are derived to study the relationship between specific causes and effects. However, by abstracting the systems under study, we obtain the benefit of having models that represent a large variety of processes. For instance, a simple activation mechanism studied here may be used to model the autoactivation of the effector caspase in the apoptosis pathway, the activation of the Akt/mTOR complex implicated in muscular growth, and twospecies population dynamics. In particular, we derive analytical expressions for the equilibrium points of a circular protein activation mechanism with an arbitrary number of intermediate steps and characterise its local stability. Later we analyse the signalling progression due to a protein autoactivation in a long cell. Furthermore, we avail of a projection method for partial differential equations to obtain associated ordinary differential equations that will assist on the reduction of the computational load for the numerical solution of a class of reaction diffusion networks. This projection method will also be used to compute the time-integral of some species concentration in a class of reaction-diffusion networks. Since we chose a theoretical approach, our results provide analytical expressions that link the kinetic parameters and topology of the reaction network with its dynamical behaviour. These formulas can be further studied to analyse the sensitivity of the systems characteristic with respect to variation of parameters as well as explicitly unveiling the main processes that affect the features of interest. We believe that these theoretical approaches provide a deeper insight in selected biochemical pathways such as: the Akt/mTOR activation pathway, mediated by the IGF receptor; the core apoptosis pathway; and Ca2+ homeostasis in non-excitable cells