Figure 2: The graphical illustration of the characteristic timescale for the Michaelis–Menten reaction mechanism (12).

When σ₁ ≪ 1, the timescale t_s1 is the characteristic time of the substrate species. The solid black curve is the numerical solution to the mass action equations (13a)–(13b) and the vertical dashed/dotted line corresponds to t = t_s1. The dotted horizontal line corresponds to the scaled characteristic value $(1-\ell )s_1^0$. The constants (without units) used in the numerical simulation are: $e_1^0=1$, k₁ = 0.01, k₂ = 10, k₋₁ = 1 and $s_1^0=100$. Time has been mapped to the t_∞ scale: t_∞(t) = 1 – 1/ln[t + exp(1)].