"Dynamics of biochemical, genetic and neuronal networks"

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The recent development of technology for simultaneously measuring expression of many genes has led to a profusion of data, but also a great need for a 'systems theory' of gene regulation to provide a framework for understanding dynamics of expression patterns. Parameters (such as reaction rates and degradation rates) are typically not known with any precision, and experiments to pin them down can be difficult and expensive, so it has proven useful to develop qualitative mathematical models that do not include all physiological details, but retain interaction structure. One such approach uses piecewise-linear differential equations in which interaction terms are modelled crudely as step functions. These 'Glass networks' allow a considerable amount of analysis (by a division of phase space into 'boxes'), but lead to additional mathematical difficulties because of the discontinuities in the equations. This model framework is also the basis for computational tools to carry out automated qualitative analysis of gene networks, which are already being used, but which suffer from an inadequate handling of some of these mathematical difficulties. One main thrust of my research is to improve our understanding of these issues, and the computational tools that depend on them. Gene regulation is carried out by proteins in cells, which are themselves influenced by external signals via 'pathways' of biochemical reactions. Detailed mathematical models of these signalling pathways can help to make quantitative predictions about effects of interventions, such as the use of 'chimeric proteins' to link cell proliferation pathways to cell death pathways, with the aim of killing cancer cells. Yet another context where I use network theory to understand cellular processes is in networks of neurons with particular architectures and how they respond to various types of electrical stimuli. The equations are different but principles are similar. This is relevant, for example, to the effects of stimulators used to treat Parkinson's disease. Mathematical techniques sometimes bridge problems or even fields, and I also propose to explore applications of the piecewise-linear approach to problems in Physics relating to sustained oscillations called 'breathers' in chains of particles.