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In this thesis we want to investigate the use of delay and diffusion in modelling prostate cancer cell dynamics while undergoing treatment; to study any limitations in their use; and to provide insights into optimal therapeutic strategies. We first analyse a discrete delay system to represent the phenomenon of neuroendocrine trans differentiation, which is the ability of cells to change their intrinsic nature in response to a hostile environment. For this system, we prove some basic properties of its solutions, which ensure the biological significance of the model, and the local and global stability properties of the tumour free equilibrium. We perform sensitive and bifurcation analyses, and produce numerical simulations that reflect the discovered behaviour. Based on the results of this analysis, we propose a new distributed delay system, and as in the previous case we analytically derive basic properties of the solutions, and prove the existence of a tumour free and a tumour present equilibrium. Local stability properties of the equilibria are also analysed, and numerical simulations using the uniform and exponential distributions are produced that show the stability switch of the tumour present equilibrium with the uniform distribution. The final model we consider is a hybrid cellular automata system to incorporate the diffusion of cells and chemicals in the in vivo representation introduced in the second system. We explore the effect of different drug therapies on the prostate cancer tumour, which we consider to have two phenotypes, one which is sensitive to one of the three considered drugs, and one resistant to the same drug.

Two-component regulatory systems allow bacteria to respond to and survive the environments in which they are situated. Specifically focusing on the sensor kinase element of these systems, much is left unknown regarding the overall functions of many of these protein types. This work centres on the EnvZ/OmpR system within Escherichia coli and more specifically how the osmosensing kinase EnvZ transduces signal across the membrane. Many studies have detailed the mechanisms of cytoplasmic and periplasmic sections of these proteins, yet the transmembrane helices have remained relatively untouched due to difficulties in probing within the membrane in vivo. Transmembrane helix movements have been detailed within similar kinase proteins, which are described for comparison. The experimental approach centred around three mutant libraries of EnvZ – two singlecysteine libraries (transmembrane helix 1 or 2) and one double-cysteine library (both transmembrane domains). The cysteines replaced residues within the transmembrane helices of EnvZ and were predicted to form a sulphydryl crosslink between two dimerising proteins under oxidising conditions. This was proposed to occur only if the positions were within an appropriate proximity meaning that some area of the helix would crosslink and others would not, revealing an interaction profile. The signal output of the system was also tested for tolerance to each mutation. These experiments were performed in the presence and absence of a hyperosmotic stimulus in order to suggest differences in the interaction and signal output profiles of the active versus inactive forms of EnvZ. The single-cysteine libraries revealed no differences in the interaction profile of the first transmembrane helix and a non-piston type movement within the second transmembrane helix. The double-cysteine library generally corroborated these results, adding a TM1/2 interaction profile. This profile included residues within the periplasmic and cytoplasmic ends of the helices but not within the membrane core. The dynamic range of the system was also calculated for the double-cysteine library and compared to “absence of stimulus” signal output results. This revealed an exponential decay relationship across the 89-member library, with only five outliers to this trend. While the overall movement of the EnvZ transmembrane helices cannot be conclusively stated from this work, several interesting features of this mechanism have been described. In particular, the lack of interaction between TM1 and TM2’ helices may suggest an inner membrane space is created that contributes to the sensing mechanism of the system. Ultimately, the results collected do not support the piston-type displacements seen in similar kinases, leading to the proposal of a new model. Further work is required to test the proposed model as well as understanding the role of EnvZ within a wider context.