On the validity of the stochastic quasi-steady-state approximation in open enzyme catalyzed reactions: Timescale separation or singular perturbation?

The paper identifies the Tikhonov–Fenichel parameter values π1, π2, π3 for the open Michaelis–Menten system and uses Fenichel projection to recover the sQSSA and QEA (π1 → ṡ = k0 − v s/(KM + s); π3 → ṡ = k0 − v s/(KS + s)), exactly as the model does for π1 and π3. See the listed TFPVs and the projected sQSSA/QEA formulas, including equations (19a–c), (23), and (25) . The paper also states the LNA validity indicator from Thomas–Straube–Grima (equation (14)) and argues that εS ≪ 1 alone is insufficient; one also needs small k0 (α ≪ 1), or alternatively small k2 (β ≪ 1), aligning the parameters with neighborhoods of π1 or π3, respectively . The model reproduces the same geometric reductions and goes further by providing an explicit quantitative equivalence: for any η there is δ(η) such that F(α,β) < δ and εS < δ imply proximity to π1 or π3, and conversely, proximity to π1 or π3 with εS ≪ 1 forces F = O(η). This strengthens the paper’s qualitative equivalence claim with a clean inequality argument. Minor issues: the paper’s decomposition vector for the π1 perturbation appears to have a typographical error in the ċ-component (k2eT s instead of k1eT s) in (20)/(22)/(A.3), though the projected dynamics are still stated correctly . Overall, both are correct and substantially the same proof line (TFPVs + Fenichel projection + LNA criterion), with the model adding explicit quantitative bounds.