Deterministic-like model reduction for a class of multi-scale stochastic differential equations with application to biomolecular systems

We consider a class of singularly perturbed stochastic differential equations with linear drift terms, and present a reduced-order model that approximates both slow and fast variable dynamics when the time-scale separation is large. We show that, on a finite time interval, moments of all orders of the slow variables for the original system become closer to those of the reduced-order model as time-scale separation is increased. A similar result holds for the first and second moments of the fast variable approximation. Biomolecular systems with linear propensity functions, modeled by the chemical Langevin equation fit the class of systems considered in this work. Thus, as an application example, we analyze the trade-offs between noise and information transmission in a typical gene regulatory network motif, for which both slow and fast variables are required. 1