Multi-scale stochastic systems motivated by biological models

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The challenge of systems biology is identifying how interactions of the proteome and genome produce the function and behaviour of a cell. Cells are inherently "noisy" systems, and how complex biochemical pathways are affected by intrinsic and extrinsic stochasticity is not well understood. A cellular network can be modelled mathematically using the principles of chemical kinetics. At the cellular level, chemical dynamics are often dominated by the action of regulatory molecules present at levels of only a few copies per cell. Intrinsic noise due to random fluctuations of these components can have significant consequences, and stochastic modelling is necessary in order to fully describe the set of possible outcomes. In many cases reaction rates can also vary over several orders of magnitude, which implies that fluctuations of effective changes in abundances occur on multiple time-scales. Many important intracellular processes, from biochemical signalling to gene regulation, are also greatly affected by the location and motility of the molecular types involved. Stochastic models of these systems use density dependent continuous time Markov chains to represent the evolution of different types of species connected by the prescribed set of interactions between them. For prediction and simulation purposes it is useful to reduce the modelling and computational complexity of the problem, while still capturing the essential stochastic behaviour of the system. The multiscale nature of the models may be exploited in order to reduce some of the complexity. A rigorous approach poses a number of interesting and challenging mathematical questions. The primary goal of my research program is to develop model reduction techniques for such stochastic models and identify the possible effects due to noise. This involves deriving novel results combining nonlinear dynamics with stochastic averaging, diffusion approximations and large deviations for density dependent Markov chains on multiple time scales. Specific short and long term goals will allow us to better understand: (i) the combined effect of stochastic reactions and molecular movement in spatial propagation of outcomes; as well as (ii) the effect of rare events in multi-scale reaction dynamics, and their role in enhancing and spatially spreading bistable dynamics and specific initial outcomes. In the long run, they will help us identify a set of spatial phenomena arising in heterogeneous reaction systems with multiple time-scales, and conditions under which they arise. The mathematical tools developed along the way will allow us to give a probabilistic analysis of the long term behaviour of heterogeneous reaction diffusion systems. This is an ongoing research program, for which I have already developed an appropriate mathematical framework for multi-scale density dependent continuous time Markov chains and derived a number of relevant results.