  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.
