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

TL;DR

This paper investigates the conditions under which the stochastic quasi-steady-state approximation remains valid in open enzyme reactions, revealing that timescale separation alone does not guarantee its applicability, unlike in deterministic models.

Contribution

It provides a geometric singular perturbation theory perspective explaining why stochastic quasi-steady-state approximation validity differs from deterministic cases.

Findings

Timescale separation does not imply singular perturbation.

The stochastic quasi-steady-state approximation has more restrictive validity conditions.

Geometric singular perturbation theory clarifies the discrepancy in approximation validity.

Abstract

The quasi-steady-state approximation is widely used to develop simplified deterministic or stochastic models of enzyme catalyzed reactions. In deterministic models, the quasi-steady-state approximation can be mathematically justified from singular perturbation theory. For several closed enzymatic reactions, the homologous extension of the quasi-steady-state approximation to the stochastic regime, known as the stochastic quasi-steady-state approximation, has been shown to be accurate under the analogous conditions that permit the quasi-steady-state reduction of the deterministic counterpart. However, it was recently demonstrated that the extension of the stochastic quasi-steady-state approximation to an open Michaelis--Menten reaction mechanism is only valid under a condition that is far more restrictive than the qualifier that ensures the validity of its corresponding deterministic quasi-steady-state approximation. In this paper, we suggest a possible explanation for this discrepancy from the lens of geometric singular perturbation theory. In so doing, we illustrate a misconception in the application of the quasi-steady-state approximation: timescale separation does not imply singular perturbation.