Numerical Methods with Applications to Biochemical Systems

我的研究领域是数值分析和科学计算。在拟议的研究计划中，我计划设计和分析数值方法，用于近似均相和非均相生化系统的数学模型的解决方案。这项研究将导致现有软件的改进，并将使生命科学的一个关键领域的研究人员能够更有效地模拟重要的生物过程，有效和可靠地近似解决他们的问题，并分析和预测复杂模型的行为。在此之前，我制定了有效的策略，解决数值初值问题（IVP）的随机连续模型均匀生化动力学。另外，将求解微分代数方程初值问题的数值算法的计算代价从现有的指数代价改进为多项式代价。 数学建模和模拟为研究关键生物过程提供了不可或缺的工具。特别是，随机模型的生化动力学是高度兴趣的今天，由于其在生命科学中的许多重要的实际应用，影响加拿大人在许多方面，包括改善人民的健康。这些模型中的大多数都相当复杂。它们不能通过分析数学技术来解决，因此需要数值方法来近似它们的解。通常，这些问题是非线性的，并表现出数学刚度，由于存在多个相互作用的时间尺度和分子群体数量。因此，这些模型的快速模拟构成了一项艰巨的任务。 这项建议的目标是：（1）通过建立自适应时间步进格式，提高求解均相生化动力学的数值随机连续和随机离散模型的计算速度（2）开发用于生化系统的离散随机模型的灵敏度分析的有效和准确的策略，（3）构造更有效和可靠的方案来逼近非均相生化动力学的随机离散模型的解，即反应-扩散主方程的解，其防止负总体数（4）设计策略以克服这些模型的数值积分中的刚性。

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.