Reactive conformations and non-Markovian reaction kinetics of a Rouse polymer searching for a target in confinement

We investigate theoretically a diffusion-limited reaction between a reactant attached to a Rouse polymer and an external fixed reactive site in confinement. The present work completes and goes beyond a previous study that showed that the distribution of the polymer conformations at the very instant of reaction plays a key role in the reaction kinetics and its determination enables the inclusion of non-Markovian effects in the theory. Here we describe in detail this non-Markovian theory and compare it with numerical stochastic simulations and a Markovian approach, in which the reactive conformations are approximated by equilibrium ones. We establish the following results. Our analysis reveals a strongly non-Markovian regime in one dimension, where the Markovian and non-Markovian dependences of the reaction time on the initial distance are different. In this regime, the reactive conformations are so different from equilibrium conformations that the Markovian expressions of the reaction time can be overestimated by several orders of magnitudes for long chains. We also show how to derive qualitative scaling laws for the reaction time in a systematic way that takes into account the different behaviors of monomer motion at all time and length scales. Finally, we also give an analytical description of the average elongated shape of the polymer at the instant of the reaction and show that its spectrum behaves as a slow power law for large wave numbers.