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?
In short, yes.

To obtain the relation between τ and v, we need the actual time evolution of the concentration of the product by solving the differential equations (DEs) and then some way to get steady-state concentration. The DEs are non-linear. So we have to find approximate solutions. This paper develops a solution. It's not nice..

I will probably come back to add more. For the time being, I highly recommend those interested to browse through the paper.

And what do you mean by τ≈1 leading to failure of steady-state approximation please?

Edit: The tittle of the paper is "Analytical solution of coupled nonlinear rate equations : I. Michaelis-menten kinetics" by Phillipson