Mathematical Modelling of Cellular and Physiological Processes

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The main theme of my scientific research program is to understand and explain complex biological processes, in particular cellular and physiological processes, through the development and analysis of mathematical models. I will focus on models that describe spatio-temporal phenomena.******One area of investigation is to understand the spatial patterns exhibited by microtubules in the presence of motor proteins. Microtubules are protein polymers in the cytoplasm of cells that play a crucial role in providing structure to cells, cell division, cell motility, and cellular transport. Microtubules are observed to form a variety of patterns, such as asters and vortices, and parallel and anti-parallel bundles, both in vivo and in vitro. The spatial patterning is the result of interactions with motor proteins, which can walk along microtubules and crosslink adjacent microtubules, affecting the orientation of microtubules with respect to each other. Detailed mathematical modelling of the interaction of microtubules and motor proteins will lead to insights into the mechanisms underlying self-organization of molecular components into a variety of spatial structures. The model consists of nonlinear partial integro-differential equations, and this research also will advance techniques for the analysis of such equations.******Another area of investigation pertains to a relatively new class of model, namely the birth-jump model. Birth-jump models are appropriate for describing the dynamics of populations (at all levels, from populations of cells to populations of animals) where the growth and spatial spread processes cannot be decoupled, in particular in situations where newly generated cells or individuals are transported immediately to distant locations. The birth-jump model has been studied for specific limiting cases, but it remains to establish theoretical results for the general case. This work will advance the theory of nonlinear integro-differential equations, and provide a solid foundation for the use of birth-jump models in biological applications.