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 the understanding of biochemical reactions in several metabolic processes. In this study, we perform a stochastic analysis of the Michaelis-Menten kinetics with the introduction of inhibitory mechanisms, which significantly diversifies the study of the reaction. We apply the Fock space formalism to reformulate the master equation, transforming it into a Schrödinger-type equation. We investigate reversible inhibitions and analyze the behavior of the averaged number of substances involved, identifying a stiffness behavior in all scenarios. In a specific case of partial inhibition, we observe that the inhibitor can act as an activator that favors product formation. We calculate the first product formation time (FPFT), which characterizes the time statistic of the first product formation. We observe the emergence of an intermediate time scale in addition to the two known time scales typical in first-passage problems. This intermediate time scale is closely aligned with the slow-binding kinetics observed in experimentally observed enzymatic reactions that involve inhibitors. This intermediate time scale is related to the new pathways introduced by the presence of inhibitors. This study offers a new perspective on inhibited enzymatic reactions, and demonstrates the usefulness of the Fock space formalism in the analysis of complex chemical systems.