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

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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.