# What is the scaling for % completion time in a Michaelis–Menten scenario as a function of substrate concentration?

Say I have a straightforward Michaelis-Menten reaction where an enzyme $E$ at some fixed concentration $[E]$ forms an interaction complex $ES$ with a substrate $S$ (with initial concentration $[S]_0$), and at a rate $k_{cat}$ can irreversibly convert the substrate $S$ to a product $P$. The rate of interaction complex formation $ES$ is $k_f$ and its dissociation rate is $k_r$, such that overall, we have the reaction:

$E + S \rightleftharpoons^{k_f}_{k_r} ES \to_{k_{cat}} E + P$

Is there an explicit expression for the time $\tau_{v}$ to convert some percentage $v$ of the substrate $S$ to product $P$? In lieu of an explicit expression for $\tau_{v}$, what scaling for this time should we expect for $\tau_{v}$ as a function $[S]$ (holding $[E]$ fixed)?

Ultimately, for $\tau_{v} \approx 1$ I suppose the steady-state approximation will begin to fail?