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，以近似求解均相和非均相生化系统的数学模型。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对重要的生物过程进行建模，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。以前，我开发了有效的策略来解决齐次生化动力学的随机连续模型的数值初值问题(IVP)。此外，我还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的生物过程提供了不可或缺的工具。特别是，生化动力学的随机模型在今天引起了人们的高度兴趣，因为它们在生命科学中的许多重要实际应用在许多方面影响了加拿大人，包括改善人们的健康。这些模型大多相当复杂。它们不能用解析数学技术来求解，因此需要用数值方法来近似它们的解。通常，这些问题是非线性的，并表现出数学上的僵硬，这是由于在时间和分子种群数量上存在多个相互作用的尺度。因此，这些模型的快速模拟是一项艰巨的任务。本方案的目标是：(1)通过建立自适应的时间步长格式，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)开发有效且准确的策略，用于生化系统离散随机模型的灵敏度分析；(3)构造更高效、更可靠的格式来逼近非均相生化动力学的随机离散模型的解，从而避免出现负数。(4)设计策略以克服这些模型在数值积分中的僵硬。

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.**