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，在真核细胞的现实三维模型中，准确地计算出数万到数十万个相互作用和扩散分子的随机行为。为了解决这一问题，本项目将开发准确、收敛和有效的数值方法，以近似解决来自亚细胞成像数据的复杂几何结构中的生化系统随机反应扩散模型。这将通过对偏积分-微分方程的高维耦合系统创建新的收敛反应-扩散主方程近似来完成。这些方程模拟了细胞内数万个分子的随机反应和扩散，其中包含高分辨率软x射线断层扫描成像数据得出的详细亚细胞结构。为了更好地理解生物体的功能、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，其中包括进行生化反应的蛋白质，这些生化反应在细胞间通讯、细胞生长和分裂、免疫系统功能以及癌症的发生和进展中起着重要作用。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方便计算机模拟细胞过程，这些过程是基于高分辨率实验成像数据得出的细胞内部的真实模型。本计画将理论工作与教育计划相结合，旨在改善计算数学生物学家的训练。这个跨学科的领域需要综合的技能，该项目将以综合的方式提供给学生。这些技能包括建立生物系统数学模型的能力；理解和选择合适的数值方法来解决这些模型的能力；以及在大规模计算平台上利用现有数字库实现这些方法的能力。因此，计划中的研究是由一项教育计划补充的，该计划旨在满足培养计算科学方面的科学家和工程师的需求，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.