Time-dependent corrections to effective rate and event statistics in Michaelis-Menten kinetics

We generalize the concept of the geometric phase in stochastic kinetics to a noncyclic evolution. Its application is demonstrated on kinetics of the Michaelis-Menten reaction. It is shown that the nonperiodic geometric phase is responsible for the correction to the Michaelis-Menten law when parameters, such as a substrate concentration, are changing with time. We apply these ideas to a model of chemical reactions in a bacterial culture of a growing size, where the geometric correction qualitatively changes the outcome of the reaction kinetics.

💡 Research Summary

The paper extends the concept of a geometric phase—originally identified in stochastic kinetics for cyclic parameter variations—to non‑cyclic, time‑dependent evolutions and demonstrates its impact on the classic Michaelis–Menten (MM) description of enzyme catalysis.
The authors start from the master equation for a three‑state Markov model of the MM reaction (free enzyme E, enzyme‑substrate complex ES, and product P). By introducing a counting field χ for the number of product molecules, they construct a “tilted” transition‑rate matrix H(χ,t) whose smallest eigenvalue λ₀(χ,t) governs the long‑time behavior of the generating function. In the adiabatic limit (slow parameter changes) the eigenvalue can be split into a dynamical part λ₀ and a geometric contribution A(R)·Ṙ, where R(t) denotes the set of time‑dependent parameters (e.g., substrate concentration