Spectral Representation and Reduced Order Modeling of the Dynamics of Stochastic Reaction Networks via Adaptive Data Partitioning

Dynamical analysis tools are well established for deterministic models. However, for many biochemical phenomena in cells the molecule count is low, leading to stochastic behavior that causes deterministic macroscale reaction models to fail. The main mathematical framework representing these phenomena is based on jump Markov processes that model the underlying stochastic reaction network. Conventional dynamical analysis tools do not readily generalize to the stochastic setting due to nondifferentiability and absence of explicit state evolution equations. We developed a reduced order methodology for dynamical analysis that relies on the Karhunen-Loeve decomposition and polynomial chaos expansions. The methodology relies on adaptive data partitioning to obtain an accurate representation of the stochastic process, especially in the case of multimodal behavior. As a result, a mixture model is obtained that represents the reduced order dynamics of the system. The Schlogl model is used as a prototype bistable process that exhibits time scale separation and leads to multimodality in the reduced order model.

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