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

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.

Figures (1)

• Figure 1: Illustration of Proposition \ref{['baseprop']}. The solution to the Michaelis--Menten system (\ref{['eqmmirrev']}) converges to the solution of the linear Michaelis--Menten system (\ref{['eqmmlin']}) as $s_0\to 0$. In all panels, the solid black curve is the numerical solution to the Michaelis--Menten system (\ref{['eqmmirrev']}). The thick yellow curve is the numerical solution to linear Michaelis--Menten system (\ref{['eqmmlin']}). The dashed/dotted curve is the numerical solution to the linear system defined by matrix $G$. The dotted curve is the numerical solution to the linear system defined by matrix $H$. All numerical simulations where carried out with the following parameters (in arbitrary units): $k_1=k_2=k_{-1}=e_0=1$. In all panels, the solutions have been numerically-integrated over the domain $t\in[0,T],$ where $T$ is selected to be long enough to ensure that the long-time dynamics are sufficiently captured. For illustrative purposes, the horizontal axis (in all four panels) has been scaled by $T$ so that the scaled time, $t/T$, assumes values in the unit interval: $\frac{t}{T}\in[0,1]$. Top Left: The numerically-obtained time course of $s$ with $s_0=0.5$ and $c(0)=0.0$. Top Right: The numerically-obtained time course of $c$ with $s_0=0.5$ and $c(0)=0.0$. Bottom Left: The numerically-obtained time course of $s$ with $s_0=0.1$ and $c(0)=0.0$. Bottom Right: The numerically-obtained time course of $c$ with $s_0=0.1$ and $c(0)=0.0$. Observe that the solution components of (\ref{['eqmmlin']}) become increasingly accurate approximations to the solution components of (\ref{['eqmmirrev']}) as $s_0$ decreases.

Theorems & Definitions (13)

• Proposition 1
• Lemma 1
• Remark 1
• Lemma 2
• Proposition 2
• proof
• Remark 2
• Remark 3
• Remark 4
• Remark 5