Numerical Methods with Applications to Biochemical Networks

我的研究领域是数值分析和科学计算。该提案的主要目标是开发和分析数值方法，用于近似求解搅拌良好和空间分布的生化网络的数学模型。这项研究将有助于改进现有的软件，并将使生命科学的一个关键领域的研究人员能够解决比以前更具有挑战性的问题，有效和可靠地近似解决其模型，并分析和预测复杂系统的行为。在过去，我的研究重点是发展有效的策略，解决数值初值问题的随机模型搅拌和空间分布的生化网络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的研究领域。生物化学网络的随机模型由于其广泛的重要的实际应用而受到高度关注。通常，这些数学模型非常复杂。它们可能是非线性的，并表现出数学刚度，由于存在多个尺度的时间和分子群体数量。通常，这种模型不能通过分析数学工具来解决。然后，数值策略是必要的，以近似他们的解决方案。设计快速模拟技术研究复杂的随机模型的生化系统，也是刚性的是一项具有挑战性的任务。 本论文的目的是：（1）扩展我以前在充分搅拌和空间分布的生化系统的随机模型方面的工作，改进生物相关模型的数值计算方法，使用自适应时间步进格式和高阶技术，（2）设计混合方法来近似反应扩散系统的离散随机模型的解，（3）为生化系统的离散随机模型的灵敏度分析开发有效和精确的策略;（4）为离散随机生化动力学模型的可辨识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.