Intermediate time scale in the first product formation time distribution of Michaelis-Menten kinetics with inhibitors

Michaelis-Menten kinetics is one of the most recognized models in enzyme kinetics, crucial for understanding biochemical reactions in various metabolic processes. In this study, we perform a stochastic analysis of the Michaelis-Menten kinetics with inhibitory mechanisms, which significantly enriches the description of the reaction. Using the Fock space formalism, we reformulate the master equation into a Schrödinger-type form. We examine reversible inhibitions and analyze the averaged number of participating substances, identifying a stiffness behavior in all scenarios. For partial inhibition, we show that the formalism correctly captures the phenomenon of inhibitor-activator duality, where the inhibitor transition from pure inhibition role to functionally favors product formation. We calculate the first product formation time distribution, which characterizes the time statistic of the first product formation. An intermediate timescale emerges in addition to the two known regimes typically observed in first-passage problems. This timescale is associated with the introduction of new pathways by inhibitory mechanisms. Altogether, the results offer a perspective on inhibited enzymatic reactions and illustrate how the Fock space formalism can be applied to the analysis of low-copy-number chemical reactions.