Gated reactions in discrete time and space

Area of Science:

• Chemical Kinetics
• Statistical Mechanics
• Theoretical Chemistry

Background:

• Molecular reactions are often gated by stochastic changes in molecular states.
• Existing continuous-time theories provide a framework for understanding gated reactions.
• Analyzing gated reactions in discrete time presents unique challenges.

Purpose of the Study:

• To develop a discrete-time theory for gated reactions analogous to existing continuous-time approaches.
• To express gated reaction times in terms of ungated first-passage and return times.
• To analyze the impact of discrete time on reaction kinetics, including asymptotic behavior and novel phenomena.

Main Methods:

• Extension of continuous-time renewal theory to a discrete-time framework.
• Derivation of formulas for the generating function, mean, and variance of gated reaction times.
• Analysis of long-time asymptotics and the emergence of interim power-law regimes.

Main Results:

• Gated reaction time is expressible via ungated first-passage and return times.
• Discrete time introduces resonances and anti-resonances absent in continuous-time models.
• An interim power-law decay regime appears when molecules are predominantly non-reactive.

Conclusions:

• The discrete-time theory simplifies the analysis of gated reactions on networks.
• The study reveals new kinetic features arising from time discretization.
• The findings offer a more comprehensive understanding of molecular reaction dynamics.