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.