Dynamics of genetic, biochemical and analog electronic networks

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The development of technology for simultaneously measuring expression of many genes and concentrations of many proteins in cells (genomics and proteomics) has led to a profusion of data, but also a great need for a 'systems theory' of gene regulation and cell biochemistry to provide a framework for understanding dynamics of expression and protein activity. 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, and simplify things too much to catch some important biochemical details. One main thrust of my research is to improve our understanding of these networks, but also to expand their scope, to bridge the gap between mathematical tractability and biochemical realism. Gene regulation is carried out by proteins in cells, which are themselves influenced by external signals via 'pathways' of biochemical reactions. I am working on a much-expanded qualitative framework that should allow fundamental insights into the behaviour of a wide range of biochemical reaction networks, including but not limited to gene regulation processes. On the other hand, more detailed mathematical models of these signalling pathways can help to make quantitative predictions about effects of interventions, and I am investigating: the use of 'chimeric proteins' to link cell proliferation pathways to cell death pathways, with the aim of killing cancer cells; the diffusion of drug cocktails through tumours and their ability to kill tumour cells by triggering appropriate signalling pathways; and the growth of neural tissue on 'scaffolds,' ultimately for use in repairing spinal chord injuries. Mathematical techniques sometimes apply in very different fields, and the equations used in qualitative models of gene networks turn out to be of the same type as the equations that describe analog electronic circuits used as 'true random number generators' (TRNGs), crucial in encryption systems for cyber security. The possibility of hacking these TRNGs necessitates development of more resilient designs, and the chaotic behaviour possible in the gene network systems, combined with intrinsic noise in all electronic circuits, provides an excellent means to produce effective new circuit designs. The mathematical tractability of these piecewise-linear equations allows chaotic behaviour to be achieved (and proved) by clever construction.