CAREER: Numerical Methods for Stochastic Reaction Diffusion Equations

当今数值分析、生物物理学、计算生物学和生物化学面临的一个挑战是准确计算真实的真核细胞三维模型中数万到数十万相互作用和扩散的分子的随机行为。为了帮助解决这个问题，这个项目将开发精确、收敛和高效的数值方法来逼近来自亚细胞成像数据的复杂几何结构中生化系统的随机反应扩散模型的解。这将通过创建新的收敛的反应扩散主方程近似来实现对高维耦合偏积分-微分方程组的逼近。这些方程模拟了细胞内数以万计的分子的随机反应和扩散，这些分子包含从高分辨率软X射线断层成像数据获得的详细的亚细胞结构。为了更好地理解生物体如何功能，对环境刺激的反应，以及帮助治疗疾病，有必要了解、预测和控制单个细胞的行为。每个细胞都包含许多复杂的动态过程，涉及经历生化反应的蛋白质，这些生化反应在细胞之间的交流、细胞的生长和分裂、免疫系统功能以及癌症的发生和发展中发挥着重要作用。了解蛋白质如何在细胞内移动和相互作用，对于能够预测和控制这些动态过程至关重要。该项目将开发新的数学方程和计算方法，可用于研究蛋白质如何在细胞内移动和相互作用。这些方法的设计是为了便于在从高分辨率实验成像数据中得出的细胞内部的真实模型中对细胞过程进行计算机模拟。该项目将理论工作与旨在改善计算数学生物学家培训的教育计划相结合。这个跨学科领域需要一个综合技能，该项目将以综合的方式向学生提供这些技能。这些技能包括开发生物系统的数学模型的能力；理解和选择适当的数值方法来求解这些模型的能力；以及以利用大规模计算平台上现有的数值库的方式实施这些方法的能力。因此，计划中的研究性研究得到了一个教育计划的补充，该计划旨在满足在计算科学方面培训科学家和工程师的需要，重点是计算数学生物学。

A challenge facing numerical analysis, biophysics, computational biology, and biochemistry today is to accurately compute the stochastic behavior of tens to hundreds of thousands of interacting and diffusing molecules within a realistic three dimensional model of a eukaryotic cell. In order to contribute to the solution of this problem, this project will develop accurate, convergent, and efficient numerical methods for approximating the solutions to stochastic reaction-diffusion models of biochemical systems within the complicated geometries that come from sub-cellular imaging data. This will be done by creating new convergent reaction-diffusion master equation approximations to high dimensional coupled systems of partial integro-differential equations. These equations model the stochastic reactions and diffusion of tens of thousands of molecules within a cell containing detailed sub-cellular structures derived from high resolution soft X-ray tomography imaging data.To better comprehend how organisms function, respond to environmental stimuli, and to aid in treating disease, it is necessary to understand, predict, and control the behavior of individual cells. Each cell contains numerous complex dynamical processes involving proteins undergoing biochemical reactions that play a major role in cell to cell communication, in cell growth and division, in immune system function, and in the development and progression of cancer. Understanding how proteins move about and interact within cells is critical to being able to predict and control these dynamical processes. This project will develop new mathematical equations and computational methods that can be used to study how proteins move about and interact within cells. These methods are designed to facilitate the computer simulation of cellular processes within realistic models of the interior of cells derived from high resolution experimental imaging data. This project integrates the theoretical work with an educational program designed to improve the training of computational mathematical biologists. This interdisciplinary field requires a synthesis of skills that the project will provide to students in an integrated manner. These skills include the ability to develop mathematical models of biological systems; the ability to understand, and choose, appropriate numerical methods with which to solve these models; and the ability to implement these methods in a manner that takes advantage of existing numerical libraries on large scale computing platforms. Thus the planned research studies are complemented by an educational program designed to address the need to train scientists and engineers in the computational sciences, with an emphasis on computational mathematical biology.