# Non-Markovian closure kinetics of flexible polymers with hydrodynamic interactions

## TL;DR

This paper develops a non-Markovian theoretical model for polymer closure kinetics with hydrodynamic interactions, accurately matching simulations and revealing significant memory effects that influence cyclization times.

## Contribution

It introduces a non-Markovian analysis based on preaveraged Zimm dynamics, improving over Markovian approximations and deriving new scaling laws for cyclization times.

## Key findings

- Markovian models overestimate cyclization times by up to a factor of 2.
- Scaling law for large N: T ~ N^{3/2} log(N/b^2).
- Non-Markovian approach aligns closely with numerical simulations.

## Abstract

This paper presents a theoretical analysis of the closure kinetics of a polymer with hydrodynamic interactions. This analysis, which takes into account the non-Markovian dynamics of the end-to-end vector and relies on the preaveraging of the mobility tensor (Zimm dynamics), is shown to reproduce very accurately the results of numerical simulations of the complete non linear dynamics. It is found that Markovian treatments, based on a Wilemski-Fixman approximation, significantly overestimate cyclization times (up to a factor 2), showing the importance of memory effects in the dynamics. In addition, this analysis provides scaling laws of the mean first cyclization time (MFCT) with the polymer size $N$ and capture radius $b$, which are identical in both Markovian and non-Markovian approaches. In particular, it is found that the scaling of the MFCT for large $N$ is given by $T\sim N^{3/2}\ln (N/b^2)$, which differs from the case of the Rouse dynamics where $T\sim N^{2}$. The extension to the case of the reaction kinetics of a monomer of a Zimm polymer with an external target in a confined volume is also presented.