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.
