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Reduction of Multi-Scale Systems of Chemical Langevin Equations

Vassilios Sotiropoulos1, Marie-Nathalie Contou-Carrere, Prodromos Daoutidis2, and Yiannis N. Kaznessis3. (1) Chemical Engineering and Material Science, University of Minnesota, 151 Amundson Hall, 421 Washington Ave SE, Minneapolis, MN 55455, (2) Chemical Engineering and Materials Science, University of Minnesota, 151 Amundson Hall, Minneapolis, MN 55455, (3) Chemical Engineering. and Materials. Science, University of Minnesota, 499, Walter Library, 117, Pleasant St. SE, Minneapolis, MN 55455

Trajectories that sample the probability distribution of continuous Markov processes, governed by Fokker-Plank equations, can be obtained as solutions of the Chemical Langevin equation (CLE) or systems of CLEs. CLEs are stochastic differential equations (SDEs) and like ordinary or partially differential equations, their solution can be a stiff numerical problem whenever the underlying physical system exhibits multiple time scales. The problem of multiple time scales can be addressed through model reduction of the system of CLEs. Although extensive work has been done in this field concerning deterministic deferential equations, the extension to a system of SDEs is not straightforward.

In this work we intend to develop a systematic framework that appropriately reduces the initial system of CLEs to subsystems with similar time scales. Specifically, by applying an appropriate transformation to the original system, the initial state vector of the CLE system is decomposed to fast and slow subvariables. This allows for a non-stiff description by treating each set differently. A set of sufficient and necessary conditions arises, which has to be met to ensure the existence of the appropriate transformation. The second step is to treat each of the two subsets independently. We extend and apply the method of adiabatic elimination to the systems under consideration. Fast variables are assumed to relax to a pseudo-stationary density under the assumption that the slow variables remain constant. Slow variables are approximated through a Fokker-Plank equation which governs the probability density of only the slow variables. Ultimately the distribution of the slow variables can be sampled through the solutions of a system of CLEs that correspond only to the slow subspace. The final step is to compute the approximated solution of the initial CLEs system by simply multiplying the two independent probability densities. The rigorousness of the proposed framework will be examined through illustrative biological examples that lie in the continuous Markov process regime.