Reactive conformations and non-Markovian cyclization kinetics of a Rouse polymer

We investigate theoretically the physics of diffusion-limited intramolecular polymer reactions. The present work completes and goes beyond a previous study [T. Guérin, O. Bénichou, and R. Voituriez, Nat. Chem. 4, 568 (2012) https://doi.org/10.1038/nchem.1378] that showed that the distribution of the polymer conformations at the very instant of reaction plays a key role in the cyclization kinetics, and takes explicitly into account the non-Markovian nature of the reactant motion. Here, we present in detail this non-Markovian theory, and compare it explicitly with existing Markovian theories and with numerical stochastic simulations. A large focus is made on the description of the non-equilibrium reactive conformations, with both numerical and analytical tools. We show that the reactive conformations are elongated and are characterized by a spectrum with a slowly decreasing tail, implying that the monomers that neighbor the reactive monomers are significantly shifted at the instant of reaction. We complete the study by deriving explicit formulas for the reaction rates in the Markovian Wilemski-Fixman theory when the reactants are located in arbitrary positions in the chain. We also give a simple scaling argument to understand the existence of two regimes in the reaction time, that come from two possible behaviors of monomer motion, which can be either diffusive or subdiffusive.