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

Abstract

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.

Classical quasi-steady state reduction—A mathematical characterization

“…Among the numerous follow-up publications to [35] we only mention some recent papers, viz. the extensive discussion by Goussis [18], a definition of QSS in Kollar and Siskova [22] which includes exponential attraction to some manifold, and the work by Radulescu et al [31], Samal et al [32], Samal et al [33] who formalized the slow-fast arguments by employing methods from tropical geometry. The approach by Segel and Slemrod (as well as the publications based on it) requires an a priori designation of "slow" and "fast" variables.…”

“…Eventually, as we have seen, both requirements lead to the same conditions. (In contrast, in their definition of validity for QSS, Kollar and Siskova [22] require a less restrictive invariance condition but a more restrictive convergence condition. Expressed in the terminology used in the present paper, they do not require invariance of U π * but stability and exponential attractivity for all initial values on U π * .)…”

Michaelis-Menten dynamics in protein subnetworks

“…From the criterion of [12] the Michaelis-Menten description will then become exact for γ → ∞, irrespective of the values ofk − c,ue andk − c,pe [13]. The same exactness statement applies to the more general case where there is also a back reaction from product and enzyme to form an enzyme complex [14,15].…”

“…The final term in (15), r(t), is what is known as the random force. It accounts for the fact that because of the interaction between subnetwork and bulk, the time evolution of the subnetwork observables cannot be closed.…”

“…As a result the memory function should in general have both slow contributions that decay on O(1) timescales, and fast contributions that decay for time differences of O(1/γ). As the memory function appears as a weight in an integral over the past (15), the fast contributions only matter for γ → ∞ if their amplitude is proportional to γ so that the integral over all time differences remains finite. Accounting also for subleading terms in the amplitude dependence then suggests the following decomposition of the memory function:…”