Extension and Justification of Quasi-Steady-State Approximation for Reversible Bimolecular Binding

The quasi-steady-state approximation (QSSA) is commonly applied in chemical kinetics without rigorous justification. We provide details of such a justification in the ubiquitous case of reversible two-step bimolecular binding in which molecules as an intermediate step of the reaction form a transient complex. First, we justify QSSA in the regime that agrees with the results in the literature and is characterized by max{R₀, L₀} ≪ K(m). Here, R₀ and L₀ are the initial concentrations of reacting receptor and ligand, respectively, and K(m) is the Michaelis constant. We also validate QSSA under an alternative condition that can be viewed as partially irreversible binding, and it does not require a tight bound on R₀ and L₀ but rather requires k₂ + k₋₂ ≪ k₋₁. Here, k₋₁ is the rate constant of decomposition of the transient complex to the ligand and the receptor, and k₂ and k₋₂ are the forward and the reverse rate constants of transformation of the complex to the product, respectively. Furthermore, we provide arguments that QSSA can also be accurate in a regime when max{R₀, L₀} ≈ K(m) and k₂ + k₋₂ ≈ k₋₁ if |R₀ - L₀| ≪ K(m). The derived conditions may be of practical use as they provide weaker requirements for the validity of QSSA compared to the existing results.