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For cellular biochemical reaction systems where the numbers of molecules is small, or in small tumours with few cells, significant fluctuation is associated with the events that drive the time evolution of the system. This intrinsic, molecular noise can give rise to behaviour that is very different from the predictions of deterministic rate equation models.

Unfortunately, there are few analytic methods for examining the qualitative behaviour of stochastic systems. An important focus of my research is on methods that extend deterministic analysis to include leading-order corrections due to the molecular noise. These methods allow the steady-state behaviour of the stochastic model to be easily computed, facilitate the mapping of stability phase diagrams that include stochastic effects, and reveal how model parameters affect noise susceptibility in a manner not accessible to numerical simulation.

This work is being done in collaboration with Brian Ingalls, Francis Poulin, Siv Sivaloganathan, and Mohammad Kohandel.

Background reading:

Here a general method is derived to treat intrinsic fluctuations in a spatially-varying system, whose deterministic counterpart is a system of nonlinear partial differential equations.

An article introducing the effective stability approximation that was developed to modify traditional linear stability analysis in order to include the effect of internal noise in a straightforward manner.

In the past, I have taught a graduate course in stochastic processes based upon these lecture notes.