Timescale disparity and the reduction of the chemical master equation: A geometric approach via the linear noise approximation
The chemical master equation dictates the stochastic behavior of chemical reaction networks in small volumes that are well-mixed and homogeneous. Although the Gillespie algorithm is capable of generating exact realizations of the master equation, it is often computationally expensive, especially when the network exhibits disparate reaction rates and timescales. Consequently, model reduction techniques are frequently employed to reduce the computational demands of the stochastic simulation algorithm while preserving accurate timecourse statistics. Often, the reduction of the master equation is facilitated through the adaptation of deterministic techniques from singular perturbation theory, resulting in homologous -- yet fundamentally heuristic -- reduced master equations. Such heuristic reductions are often accurate when applied to linear reaction networks but inaccurate when applied to nonlinear reaction networks. The failure of heuristic reductions is not fully understood, and it remains unclear when and why the heuristic reduction of the master equation will succeed. In this work, we take a significant step towards bridging this divide by proving that every first-order reaction network admits an accurate heuristic reduction of its associated master equation provided a specific geometric criterion holds. We also discuss the implications of this result as it pertains to nonlinear reaction networks. Specifically, we explain, through the lens of geometric perturbation theory, why the heuristically reduced CME of the nonlinear Michaelis-Menten reaction network loses precision when substrate concentrations are moderate.