The Michaelis–Menten reaction at low substrate concentrations: Pseudo-first-order kinetics and conditions for timescale separation

The paper proves global upper/lower comparisons for the Michaelis–Menten system against linear surrogates G and H and a uniform-in-time O(s0^2) error for the linearization (Proposition 2(d,e)), using Kamke’s comparison theorem and an eigen-expansion/mean-value bound; all assumptions and steps check out (notably, s0 < K is imposed to keep H Hurwitz and the upper bounds viable) . The model’s solution derives the same comparisons and the O(s0^2) error via a different route: positivity of Metzler semigroups and a variation-of-constants convolution estimate yielding an explicit bound |x−x*| ≤ (k1 s0^2/4)(−A^{-1})1, uniform in t. This method is correct and complements the paper’s approach; it also shows the inequalities sup ≥ s ≥ slow and sup ≥ s* ≥ slow hold without needing s0 < K (though the paper restricts to s0 < K to ensure the upper linear system H is stable and practically useful) .