  In this paper we derive several quasi steady-state approximations (QSSAs) to
the stochastic reaction network describing the Michaelis-Menten enzyme
kinetics. We show how the different assumptions about chemical species
abundance and reaction rates lead to the standard QSSA (sQSSA), the total QSSA
(tQSSA), and the reverse QSSA (rQSSA) approximations. These three QSSAs have
been widely studied in the literature in deterministic ordinary differential
equation (ODE) settings and several sets of conditions for their validity have
been proposed. By using multiscaling techniques introduced in Kang (and Kurtz
2013) and Ball et al. (2006) we show that these conditions for deterministic
QSSAs largely agree with the ones for QSSAs in the large volume limits of the
underlying stochastic enzyme kinetic network.
