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Every cell contains several millions of diffusing and reacting biological molecules. The interactions between these molecules ultimately manifest themselves in all aspects of life, from the smallest bacterium to the largest whale. One of the greatest open scientific challenges is to understand how the microscopic chemistry determines the macroscopic biology. Key to this challenge is the development of mathematical and computational models of biochemistry with molecule-level detail, but which are sufficiently coarse to enable the study of large systems at the cell or organism scale. Two such models are in common usage: the reaction-diffusion master equation, and Brownian dynamics. These models are utterly different in both their history and in their approaches to chemical reactions and diffusion, but they both seek to address the same reaction-diffusion question. Here we make an in-depth study into the physical validity of these models under various biological conditions, determining when they can reliably be used. Taking each model in turn, we propose modifications to the models to better model the realities of the cellular environment, and to enable more efficient computational implementations. We use the models to make predictions about how and why cells behave the way they do, from mechanisms of self-organisation to noise reduction. We conclude that both models are extremely powerful tools for clarifying the details of the mysterious relationship between chemistry and biology.

Dans cette thèse, on s'intéresse à la compréhension qualitative de systèmes d'EDP de type FitzHugh Nagumo. Elle est basée sur les propriétés excitable et oscillante du système d'EDO de type FitzHugh Nagumo lorsqu'on varie la valeur d'un paramètre. Après avoir analysé les propriétés du système d'EDO, on contruit des systèmes d'EDP par couplage de Réaction Diffusion ou opérateur des ondes. La simulation numérique des systèmes montre l'émergence de patterns complexes pertinents en biologie et physiologie. D'un point de vue mathématique, cela correspond à des attracteurs non triviaux, et divers théorèmes y sont montrés.

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (leaves 93-99).

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 50-51). Also available in electronic version. Access restricted to campus users.

Thesis (M. Sc.)--University of Stellenbosch, 2008. / Includes bibliographical references. Also available via the Internet.

Thesis (Ph. D.)--Ohio State University, 2006. / Available online via OhioLINK's ETD Center; full text release delayed at author's request until 2007 Mar 21

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 122-126).

Thesis (Ph. D.)--University of Sydney, 2008. / Includes graphs. Title from title screen (viewed November 28, 2008) Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science. Includes bibliographical references. Also available in print form.

Dissipative spots are found in physical experiments of many branches of natural science. In this thesis we use three-component reaction-diffusion systems on two-dimensional domains in order to generate these patterns. Using a dynamical system approach we proceed with a Fourier analysis on a linearized reaction-diffusion system in order to provide the bifurcation conditions for a given homogeneous state. We validate our results and establish it's limitations through numerical experiments. We report very interesting behavior during these simulations, notably hysteresis and multi-stability. We will then turn our attention to the relatively unexplored phenomenon of rotating spots. Based on previous work done for spiral waves, we investigate the effect of translational symmetry-breaking on a rotating spot mainly through careful numerical analysis.