  The quasi-steady-state approximation is widely used to develop simplified
deterministic or stochastic models of enzyme catalyzed reactions. In
deterministic models, the quasi-steady-state approximation can be
mathematically justified from singular perturbation theory. For several closed
enzymatic reactions, the homologous extension of the quasi-steady-state
approximation to the stochastic regime, known as the stochastic
quasi-steady-state approximation, has been shown to be accurate under the
analogous conditions that permit the quasi-steady-state reduction of the
deterministic counterpart. However, it was recently demonstrated that the
extension of the stochastic quasi-steady-state approximation to an open
Michaelis-Menten reaction mechanism is only valid under a condition that is far
more restrictive than the qualifier that ensures the validity of its
corresponding deterministic quasi-steady-state approximation. In this paper, we
explain this discrepancy through the rigorous application of geometric singular
perturbation theory. In so doing, we illustrate a common misconception in the
application of the quasi-steady-state approximation: a problem which involves
timescale separation is not always a singular perturbation problem.
