Gated reactions in discrete time and space

Abstract

How much time does it take two molecules to react? If a reaction occurs upon contact, the answer to this question boils down to the classic first-passage time problem: find the random time it takes the two molecules to meet. However, this is not always the case as molecules switch stochastically between reactive and nonreactive states. In such cases, the reaction is said to be "gated" by the internal states of the molecules involved which could have a dramatic influence on kinetics. A unified, continuous-time, approach to gated reactions on networks was presented in [Phys. Rev. Lett. 127, 018301, 2021]. Here, we build on this recent advancement and develop an analogous discrete-time version of the theory. Similar to continuous-time, we employ a renewal approach to show that the gated reaction time can always be expressed in terms of the corresponding ungated first-passage and return times; which yields formulas for the generating function of the gated reaction-time distribution and its corresponding mean and variance. In cases where the mean reaction time diverges, we show that the long-time asymptotics of the gated problem is inherited from its ungated counterpart, where only the pre-factor of the power law tail changes. The discretization of time also gives rise to new phenomena that do not exist in the continuous-time analogue. Crucially, when the internal gating dynamics is in, or out of, phase with the spatial process governing molecular encounters resonance and anti-resonance phenomena emerge. These phenomena are illustrated using two case studies which also serve to show how the general approach presented herein greatly simplifies the analysis of gated reactions.