Numerical Methods with Applications to Biochemical Systems

我的研究方向是数值分析和科学计算。在提出的研究计划中，我计划设计和分析近似求解同质和异质生化系统数学模型的数值方法。这项研究将导致现有软件的改进，并将使生命科学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，我开发了有效的策略来解决均匀生化动力学随机连续模型的数值初值问题（IVP）。此外，我将求解微分代数方程初值问题的数值算法的计算成本从现有的指数成本提高到多项式成本。数学建模和模拟为研究关键的生物过程提供了不可或缺的工具。特别是，生物化学动力学的随机模型由于其在生命科学中的许多重要实际应用，在许多方面影响着加拿大人，包括改善人们的健康，因此今天受到高度关注。这些模型大多相当复杂。它们不能通过解析数学技术来解决，因此需要数值方法来近似解决它们。通常，这些问题是非线性的，并表现出数学上的刚度，因为在时间和分子数量上存在多个相互作用的尺度。因此，快速模拟这些模型是一项艰巨的任务。***本建议的目标是：(1)通过构建自适应时步方案，提高求解均匀生化动力学数值随机连续和随机离散模型的计算速度；(2)为生化系统离散随机模型的敏感性分析开发有效和准确的策略；(3)为非均匀生化动力学随机离散模型的近似解构建更有效和可靠的方案；在这些模型的数值积分中，设计克服刚度的策略。**

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.**