  There is a vast amount of literature concerning the appropriateness of
various perturbation parameters for the standard quasi-steady state
approximation in the Michaelis-Menten reaction mechanism, and also concerning
the relevance of these parameters for the accuracy of the approximation by the
familiar Michaelis-Menten equation. Typically, the arguments in the literature
are based on (heuristic) timescale estimates, from which one cannot obtain
reliable quantitative estimates for the error of the quasi-steady state
approximation. We take a different approach. By combining phase plane analysis
with differential inequalities, we derive sharp explicit upper and lower
estimates for the duration of the initial transient and substrate depletion
during this transitory phase. In addition, we obtain rigorous bounds on the
accuracy of the standard quasi-steady state approximation in the slow dynamics
regime. Notably, under the assumption that the quasi-steady state approximation
is valid over the entire time course of the reaction, our error estimate is of
order one in the Segel-Slemrod parameter.
