AMC-SS: Analysis and Computation of Multi-Scale Stochastic Chemical Kinetic Systems with Application to Genetic Regulatory Networks

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Proteins play an essential role in the development and functioning of organisms. The synthesis of cellular proteins is a multi-step process inside living cells. The genetic information stored in DNA directs the production of proteins through a bio-chemical process called gene expression. When a specific gene is expressed, its DNA is first transcribed into a single stranded sequence of mRNA. The mRNA sequence is then translated into a sequence of amino acids as the protein is formed. These steps form the three levels of gene expression: transcription, translation, and post-translational modification. Genetic Regulatory Networks (GRNs), consisting of genes, proteins, small molecules within cells in relatively low concentrations, and their interactions, function to regulate gene expression process for the production of proteins in response to certain physical, chemical, and biological stimuli. GRNs can only be effectively modeled as stochastic systems with random molecular fluctuations instead of deterministic systems. Very often GRNs are multi-scale in terms of multiple steady states, reaction rates, and molecular concentrations, making standard simulation algorithms inefficient. Simplifying and reducing complex chemical and biological models enables easier conceptualization and interpretation and efficient simulations of inherently stochastic systems. The main objective of the proposed research project is to provide simplified dynamics and to design efficient numerical schemes for complex stochastic chemical kinetic systems exhibiting multiple steady states and multiple time and concentration scales. Applications will be emphasized on Genetic Regulatory Networks, for which only stochastic modeling incorporating random molecular fluctuations has proved to be successful. On the mathematical side, using asymptotic analysis and probability theory, the transition rates and transition pathways of chemical kinetic systems with multiple steady states will be investigated, and the effective dynamics for stochastic chemical kinetic systems with multiple well-separated time and concentration scales will be studied. On the numerical side, computational methods will be developed to simulate the reduced dynamics obtained from the mathematical work. Computer-based optimization methods will be adopted to find the transition rates and pathways of multi-stable systems. Convergence and error estimates for the proposed numerical schemes will be proved mathematically. Issues like efficiency, robustness, adaptivity, and parallelism of the computing schemes will also be studied. The proposed research will advance the frontiers of Applied Mathematics through significant generalizations and developments of the modeling, analytical, and computational techniques for stochastic and multi-scale systems. It will help to understand functional issues of chemically reacting networks at the system level, which is becoming the new focus of genomic research. The ideas from the stochastic and multi-scale analysis can be applied to a spectrum of biological, chemical, physical, and other scientific problems involving multi-scale modeling and can broaden the scope of research topics in the related fields. The research projects will also promote interdisciplinary interactions between applied mathematicians, chemists, and biologists.