Numerical Methods with Applications to Biochemical Systems

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My research lies in the realm of numerical analysis and scientific computing. In the proposed research program, I plan to design and analyze numerical methods for approximating the solution of mathematical models of homogeneous and heterogeneous biochemical systems. This research will lead to improvement of existing software and will enable researchers in a key area of life sciences to model important biological processes much more productively, to effectively and reliably approximate the solution to their problems and to analyze and predict the behaviour of complex models. Previously, I developed efficient strategies for solving numerically initial value problems (IVP) for stochastic continuous models of homogeneous biochemical kinetics. In addition, I improved the computational cost of numerical algorithms for solving initial value problems for differential algebraic equations from existing exponential to polynomial cost. Mathematical modelling and simulations provide indispensable tools for studying critical biological processes. In particular, stochastic models of biochemical kinetics are of high interest today due to their many important practical applications in life sciences that affect Canadians in many ways, including improving peoples' health. Most of these models are quite complex. They cannot be solved by analytic mathematical techniques and therefore numerical methods are required to approximate their solution. Often, these problems are nonlinear and exhibit mathematical stiffness, due to the presence of multiple interacting scales in time and molecular population amounts. Consequently, fast simulation of these models constitutes a difficult task.  The objectives of this proposal are: (1) improving the speed of computation for solving numerically stochastic continuous and stochastic discrete models of homogeneous biochemical kinetics, by building adaptive time-stepping schemes (2) developing effective and accurate strategies for sensitivity analysis of discrete stochastic models of biochemical systems, (3) constructing more efficient and reliable schemes for approximating the solution of a stochastic discrete model of heterogeneous biochemical kinetics, that of the Reaction-Diffusion Master Equation, which prevent negative population numbers (4) designing strategies to overcome stiffness in the numerical integration of these models.