Stochastic and deterministic models of intracellular reaction networks: analysis and algorithms

这个建议集中在细胞内反应网络的随机和确定性模型。 如果反应网络的组成分子的丰度足够高，则反应网络通常被建模为耦合的常微分方程组。 然而，如果丰度低，则标准确定性模型不能很好地表示系统的行为，而使用随机模型。 最简单的随机模型将系统视为连续时间马尔可夫链。更复杂的模型在本质上往往是混合的，其中一些组件离散建模，一些组件扩散建模，一些组件绝对连续建模（即，具有来自其他组件的随机输入的常微分方程的解）。 这个项目有两个相关的目标。 第一个是制定严格的网络结构相关的结果-通常是最简单的信息，以实验获得有关网络-与定性性质的动态反应网络，并在确定性和随机设置这样做。 第二个目标是开发和分析随机建模系统的数值模拟方法，这些方法考虑了反应网络的自然缩放和基本数学结构。 随机建模的反应网络通常涉及多个自然尺度，在开发和分析数值模拟方法时，考虑这些自然尺度至关重要。 此外，所有描述反应网络的方程都具有非常特定的数学结构，可以用于在离散和连续随机环境中开发有效的算法。本提案中解决的所有问题都有来自生物科学的动机。关于细胞动力学的实验方法（如绿色荧光蛋白）的最新进展增加了人们对生物分子如何在细胞水平上随着时间的推移相互作用的兴趣。 由于布朗力在这一水平上是重要的，这些细胞内反应网络的随机模型，结合分析和计算工具，如果要很好地理解它们，是必不可少的。 该项目还将为研究生提供一个肥沃的培训基地。 特别是，有一个训练有素的数学科学家的兴趣和必要的专业知识，以促进解决生物学中出现的问题的需求很高。

This proposal concentrates on both stochastic and deterministic models of intracellular reaction networks. If the abundances of the constituent molecules of a reaction network are sufficiently high then the reaction network is typically modeled as a coupled set of ordinary differential equations. If, however, the abundances are low then the standard deterministic models do not provide a good representation of the behavior of the system and stochastic models are used. The simplest stochastic model treats the system as a continuous time Markov chain. More complicated models tend to be hybrid in nature with some components modeled discretely, some diffusively, and some absolutely continuously (that is, solutions of ordinary differential equations with stochastic inputs from the other components). This project has two related goals. The first is to develop rigorous results relating network structure---typically the simplest information to obtain experimentally about networks---with qualitative properties of the dynamics of reaction networks, and to do so in both the deterministic and stochastic settings. The second goal is to develop and analyze numerical simulation methods for stochastically modeled systems that account for both the natural scalings and basic mathematical structure of reaction networks. Stochastically modeled reaction networks often involve multiple natural scales and it is crucial that these natural scales be accounted for when developing and analyzing numerical simulation methods. Also, all of the equations that describe reaction networks are of a very specific mathematical structure that can be utilized to develop efficient algorithms in both the discrete and continuous stochastic contexts.All of the questions being addressed in this proposal have motivation from the biosciences. Recent advances in experimental methods concerning the dynamics of the cell (such as green fluorescent protein) have increased interest in how biological molecules interact at the cellular level over the course of time. Because Brownian forces are significant at this level, stochastic models for these intracellular reaction networks, combined with analytical and computational tools, are essential if they are to be well understood. This project will also provide a fertile training ground for graduate students. In particular, there is a high demand for well-trained mathematical scientists with the interest and expertise necessary to contribute to the solution of problems arising in biology.