The Michaelis–Menten Reaction at Low Substrate Concentrations: Pseudo-First-Order Kinetics and Conditions for Timescale Separation

Bulletin of Mathematical Biology, vol. 86, pp. 68

Abstract

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 $$s_0 \ll K$$ s 0 ≪ K , where $$K=k_2/k_1$$ 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, USA

2. Santiago D. Schnell corresponding

   University of Notre Dame

   Affiliation as printed

   Department of Biological Sciences and Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA. santiago.schnell@nd.edu

   Department of Biological Sciences and Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA

3. Sebastian Walcher Aachen Mathematik A

   RWTH Aachen University

   Affiliation as printed

   Mathematik A, RWTH Aachen, 52056, Aachen, Germany