  The conditions for the validity of the standard quasi-steady-state
approximation in the Michaelis--Menten mechanism in a closed reaction vessel
have been well studied, but much less so the conditions for the validity of
this approximation for the system with substrate inflow. We analyze
quasi-steady-state scenarios for the open system attributable to singular
perturbations, as well as less restrictive conditions. For both settings we
obtain distinguished invariant slow manifolds and time scale estimates, and we
highlight the special role of singular perturbation parameters in higher order
approximations of slow manifolds. We close the paper with a discussion of
distinguished invariant manifolds in the global phase portrait.
