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Center for Nonlinear Studies |
Many processes in science, society and technology evolve according discrete state (jump) Markov process. The master equation (ME) for such processes is typically infinite dimensional and is unlikely to be computationally tractable without further reduction. My recently proposed Finite State Projection (FSP) technique allows for a bulk reduction of the ME while explicitly keeping track of its own approximation error. In my previous talk, I showed how this error can be been reduced in order to obtain more accurate ME solutions. In this talk, we show that this ``error" has far more significance than simply the distance between the approximate and exact solutions of the ME. In particular, I will show that apart from its use as a measure for the quality of the FSP approximation, this error term serves as an exact measure of the rate of first transition from one system region to another. I will demonstrate how this term may be used to (i) directly determine the statistical distributions for stochastic switch rates, escape times, trajectory periods, and trajectory bifurcations, and (ii) evaluate how likely its is that a system will express certain behaviors during certain intervals of time. I will also present two additional systems-theory based FSP model reduction approaches that are particularly useful in such studies. I will illustrate the benefits of the original FSP and these new approaches with in depth analysis two stochastic switches in biology: The Pap Pili Epigenetic switch in E. coli and Gardner's genetic toggle switch.
Host: Nick Hengartner (nickh@lanl.gov)