The investigator Rathinam aims to develop efficient and robust leaping methods and their theory. Leaping methods provide an efficient and approximate way to do computer simulations of continuous-time discrete-state Markov processes. These are dynamical systems in which quantities change in time by discrete jumps, and the size and time of the next jump occurs according to probabilities that depend on the current value. Motivating applications are in intracellular gene regulatory models which involve small numbers of large molecules and are essentially discrete and stochastic in nature. Applications from other fields include financial market models as well as internet traffic of data.
The leaping methods work in the same spirit as time-stepping methods for differential equations which describe the time evolution of deterministic and continuous dynamical systems such as a spacecraft. However, the state-of-the-art time-stepping methods for differential equations are far more sophisticated than any of the leaping methods which apply to stochastic and discrete systems. This project aims to improve the existing leaping methods further as well as to design new leaping methods so that they could be an effective simulation tool. A particular focus of application will be to intracellular gene regulatory models that involve several chemical species, most of which are present in small molecular numbers, several reaction channels, and multiple time scales. The project will develop methods as well as their quantitative analysis.
Broader impacts of this project include an increased collaboration between systems biologists and mathematicians, enhancing cross-fertilization of ideas. The participation of graduate and undergraduate students in this project will provide them with the opportunity to develop skills in the research and development of numerical simulation tools for stochastic dynamics, a valuable expertise of future relevance in the financial, pharmaceutical, and biotechnology industries. As an educational component the tools and techniques developed will be incorporated into existing and new graduate courses on stochastic methods.
