Mathematical Modelling of the Dynamics of Cellular Processes

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This project develops and uses mathematical tools to study the dynamics of a variety of cellular physiological processes. The specific projects include a study to better understand how the electrical signal that coordinates cardiac activity is transmitted from cell to cell, including the effects of ephaptic coupling; how random fluctuations of ion channels can lead to spontaneous electrical activity in nerves; how dividing cells coordinate the separation of their duplicated chromosomes; how proteins are made, sorted into vesicles, transported and processed, (recycled or destroyed) in a well-controlled manner, and how bacteria build their flagellar motors to well-defined specifications. All of these projects employ nonlinear partial differential equations and/or stochastic differential equations while the tools of mathematical modeling, dynamical systems theory, bifurcation theory, asymptotic analysis, stochastic processes, and numerical simulation are all used in this work. While the biological problems to be studied are quite diverse, they are unified by their many common mathematical features. Thus, this work has the two-fold goal of using mathematical modeling approaches to understand specific complex cellular processes, and to discover common principles and transferable concepts that are in operation in many different areas of biology. In this way, this work contributes to the development of a fundamental and unified theory of how biological processes work, and in doing so, will contribute to our understanding of many issues relating to human health and medicine.