The Michaelis--Menten reaction at low substrate concentrations: Pseudo-first-order kinetics and conditions for timescale separation

arXiv (Cornell University)

Abstract

We demonstrate that the Michaelis-Menten reaction mechanism can be accurately approximated by a linear system when the initial substrate concentration is low. This leads to pseudo-first-order kinetics, simplifying mathematical calculations and experimental analysis. Our proof utilizes a monotonicity property of the system and Kamke's comparison theorem. This linear approximation yields a closed-form solution, enabling accurate modeling and estimation of reaction rate constants even without timescale separation. Building on prior work, we establish that the sufficient condition for the validity of this approximation is $s_0 \ll K$, where $K=k_2/k_1$ is the Van Slyke-Cullen constant. This condition is independent of the initial enzyme concentration. Further, we investigate timescale separation within the linear system, identifying necessary and sufficient conditions and deriving the corresponding reduced one-dimensional equations.

Authors 3

1. Justin Eilertsen

   American Mathematical Society

   Affiliation as printed

   Mathematical Reviews American Mathematical Society 416 4th Street Ann Arbor , MI , 48103

2. Santiago D. Schnell
3. Sebastian Walcher Aachen Mathematik A

   RWTH Aachen University

   Affiliation as printed

   Mathematik A , RWTH Aachen D-52056 Aachen , Germany