ON THE QUASI-STEADY-STATE APPROXIMATION IN AN OPEN MICHAELIS–MENTEN REACTION MECHANISM

The candidate solution reproduces the paper’s key steps and results: (i) the L^2 energy estimate showing exponential attraction to the c=w(s) QSS variety with time constant (k_-1+k_2)^{-1} and asymptotic bound using v ≤ k0 on the positively invariant wedge W1 (matching Proposition 1 and eqs. (42)–(43)) ; (ii) the dimensionless qualifier ε_o = k2 eT/[K_M(k_-1+k_2)] as a uniform smallness condition justifying the open sQSSA on W1 when k2 eT > k0 (eq. (45)) , consistent with the model’s bound |c−w(s)| ≲ ε_o eT; and (iii) the error indicator δ(s) from the invariance-equation expansion, its local maximum δ_m = (27/256) (k2 eT)/(k1 K_M^2) (1−r)^4 with r=k0/(k2 eT), the endpoint value δ(0)=k0/(k1 K_M^2), and the sharp switching threshold r<0.0767 determining whether δ_m or δ(0) dominates (eqs. (51)–(54)) . The open sQSSA ṡ = k0 − (k2 eT s)/(K_M+s) is exactly the paper’s reduction (eq. (26)) . Minor presentational differences (e.g., choice of inequality constants, writing δ(s) via u=s/(s+K_M)) do not affect correctness.