Numerical Methods with Applications to Biochemical Networks

我的研究方向是数值分析和科学计算。本建议的主要目标是发展和分析数值方法近似的数学模型的解决方案，搅拌均匀和空间分布的生化网络。这项研究将有助于现有软件的改进，并将允许生命科学关键领域的研究人员解决比以前可能的更具挑战性的问题，有效和可靠地近似解决他们的模型，并分析和预测复杂系统的行为。在过去，我的研究重点是开发有效的策略来解决均匀搅拌和空间分布的生化网络随机模型的数值初值问题和微分代数方程的初值问题。应用数学建模和模拟来研究关键的生物过程是一个令人兴奋的新研究领域。生物化学网络的随机模型由于其广泛的重要实际应用而引起了人们的高度兴趣。通常，这些数学模型是非常复杂的。它们可能是非线性的，并且由于在时间和分子种群数上存在多个尺度而表现出数学上的刚度。通常，这种模型不能用分析数学工具来求解。然后，需要数值策略来逼近它们的解。设计快速仿真技术来研究生物化学系统的复杂随机模型是一项具有挑战性的任务。本提案旨在(1)扩展我之前在均匀搅拌和空间分布的生化系统随机模型方面的工作，使用自适应时间步进方案和高阶技术改进生物学相关模型的数值方法，(2)设计混合方法来近似反应扩散系统离散随机模型的解。(3)为生物化学系统离散随机模型的敏感性分析提供有效和准确的策略；(4)为离散随机生化动力学模型的可识别性分析构建可靠和有效的数值技术。

My research is in the area of numerical analysis and scientific computing. The main objectives of this proposal are to develop and analyze numerical methods for approximating the solution of mathematical models of well-stirred and spatially distributed biochemical networks. This research will contribute to the improvement of existing software and will allow researchers in a crucial area of life sciences to solve more challenging problems than was previously possible, to effectively and reliably approximate the solution to their models and to analyze and predict the behaviour of complex systems. In the past, my research focused on the development of efficient strategies for solving numerically initial value problems for stochastic models of well-stirred and spatially distributed biochemical networks and initial value problems for differential algebraic equations. The application of mathematical modelling and simulations to study critical biological processes is an exciting and new area of research. Stochastic models of biochemical networks are of high interest today, due to their wide variety of important practical applications. Often, these mathematical models are highly complex. They may be non-linear and exhibit mathematical stiffness, due to the presence of multiple scales in time and molecular population numbers. Typically, such models cannot be solved by analytic mathematical tools. Then, numerical strategies are necessary to approximate their solution. Designing fast simulation techniques for studying complex stochastic models of biochemical systems which are also stiff is a challenging task. This proposal aims to (1) extend my previous work on stochastic models of well-stirred and spatially distributed biochemical systems, to improve the numerical methods for biologically relevant models, using adaptive time-stepping schemes and higher-order techniques, (2) design hybrid methods for approximating the solution of discrete stochastic models of reaction-diffusion systems, (3) develop effective and accurate strategies for sensitivity analysis of discrete stochastic models of biochemical systems, and (4) construct reliable and effective numerical techniques for identifiability analysis of discrete stochastic biochemical kinetic models.