With the completion of numerous genome projects for bacteria, yeast, and humans, there is an increasing interest in understanding how molecules encoded within the genomes interact to define various functional networks of the cell. Network of integrated molecular reactions tend to involve many different molecular species, thus posing complex analytical problems. For prediction and simulation purposes it is essential to reduce both the model and computational complexity of the problem, while still capturing all the essential characteristics and potential behavior of the network. This project will systematically develop stochastic models for chemical reaction networks, beginning with classical Markov chain models and developing new models that take into account the stepwise development of reactions involving RNA and DNA molecules. Specific issues to be addressed include scaling limits based on the wide range of time and other quantitative scales in the system, model reduction through scaling limit approximations and other approaches, the implications of the combinatorial restrictions the reaction structure places on the system, sensitivity analysis for the parameters of the stochastic models, and statistical methods for model validation based on data that is frequently obtained through indirect and/or aggregated measurements.
At the level of the cell, the chemical dynamics may well be dominated by the action of regulatory molecules that are present at levels of only a few copies per cell. Therefore, the molecular fluctuations of these components may determine the dynamics of the reaction network. These molecular fluctuations appear to have significant consequences; the observed large variation in rates of development, morphology and concentration of molecular species in a cell often lead to a randomization of phenotypic outcomes and non-genetic population heterogeneity. Since these fluctuations may have profound effects on the physiology of the cell, stochastic models for the intra-cellular reaction networks and careful statistical analysis appear to be essential if the system is to be well understood. The project will also provide a fertile training ground for graduate students and postdoctoral researchers. 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 cell and molecular biology.
