Numerical Methods with Applications to Biochemical Networks

我的研究方向是数值分析和科学计算。这项建议的主要目标是开发和分析数值方法来近似求解搅拌良好且空间分布的生化网络的数学模型的解。这项研究将有助于改进现有的软件，并将使生命科学关键领域的研究人员能够解决比以前更具挑战性的问题，有效和可靠地近似他们的模型的解决方案，并分析和预测复杂系统的行为。过去，我的研究主要集中在发展有效的策略来解决搅拌良好且空间分布的生化网络的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。生化网络的随机模型因其广泛的实际应用而备受关注。通常，这些数学模型非常复杂。它们可能是非线性的，并表现出数学上的刚性，这是由于时间和分子种群数量存在多个尺度。通常，这样的模型不能用分析数学工具求解。然后，需要数值策略来逼近它们的解。设计快速的模拟技术来研究生化系统的复杂随机模型是一项具有挑战性的任务。这一建议的目的是：(1)扩展我以前在搅拌良好和空间分布的生化系统随机模型方面的工作，利用自适应时间步长格式和高阶技术，改进生物相关模型的数值方法；(2)设计用于逼近反应扩散系统离散随机模型解的混合方法；(3)发展有效和准确的生化系统离散随机模型灵敏度分析的策略；(4)构建可靠和有效的离散随机生化动力学模型可辨识性分析的数值方法。

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.