In chemistry and biology, we often come across chemical reaction networks (CRNs) where one or more of the species exhibit a different intrinsic time scale and tend to reach the equilibrium state faster than others. The quasi-steady-state approximation (QSSA) is a common tool to simplify the description of the dynamics of such systems. Traditionally, the QSSAs have been derived from deterministic ordinary differential equation (ODE) models using perturbation theory. In this talk, we will consider a stochastic multiscale approximation technique for Continuous Time Markov Chain (CTMC) models of CRNs and apply it to the important class of chemical reaction networks known as the Michaelis—Menten (MM) models of enzyme kinetics to derive several QSSAs. In particular, we derive three different QSSAs, namely the standard QSSA (sQSSA), the total QSSA (tQSSA) and the reverse QSSA (rQSSA), from the stochastic model of MM enzyme kinetics.

In the second part of the talk, we will consider the issue of delays in chemical reactions. Some biophysical processes such as gene transcription and translation are known to have a significant gap between the initiation and the completion of the processes, which renders the usual assumption of exponentially distributed inter-event times in CTMC models untenable. We relax this assumption by incorporating age-dependent random time delays into the system dynamics. We do so by constructing a measure-valued Markov process on a more abstract state space, which allows us to keep track of the "ages" of molecules participating in a chemical reaction. We study the large-volume limit of such age-structured systems. We show that, when appropriately scaled, the stochastic system can be approximated by a system of Partial Differential Equations (PDEs) in the large-volume limit, as opposed to Ordinary Differential Equations (ODEs) in the classical theory. We show how the limiting PDE system can be used for the purpose of further model reductions and for devising efficient simulation algorithms.