# Exploring the relaxation time, bond length and length of the polymer chain in the kinetics of end-to-end looping of a long polymer chain

## TL;DR

This paper develops a theoretical model to analyze how relaxation time, bond length, and chain length influence the kinetics of polymer end-to-end looping, revealing dependencies that affect loop formation rates.

## Contribution

It introduces a Smoluchowski-like equation with a Dirac delta sink to model polymer looping kinetics, highlighting the effects of physical parameters on rate constants.

## Key findings

- Long-term rate constant inversely proportional to square root of chain length
- Short-term rate constant decreases with relaxation time for short polymers
- Bond length significantly influences looping probability and end-to-end distance

## Abstract

We present a theoretical model for understanding the kinetics of a long polymer in solution. A Smoluchowski-like equation has been used to model the problem of polymer looping with a Dirac delta function sink of finite strength. The results for rate constants have been obtained by using Green's function. In this model, the rate constants for end-to-end looping of polymer exhibits simple Arrhenius behavior. The rate constants are sensitive to relaxation time, bond length and chain length dependence of the looping probability. In particular, for short term rate constant, relaxation time makes loop formation between two ends of a polymer chain slower but has a direct dependence with long term rate constant. We also show that for shorter polymer strands looping is slower as compared to polymer having longer chain length. The long term rate constant shows inverse square root dependence on the length of the chain. The bond length strongly affects the probability of loop formation as for long constructs of polymer, the end-to-end distance is found to exhibit inverse power law relationship to bond length. We also observe that the short term rate constant for flexible polymer shows stronger affinity for loop formation.