# Understanding reversible looping kinetics of a long polymer molecule in solution. Exact solution for two state model with delta function coupling

## TL;DR

This paper presents an exact analytical solution for the looping kinetics of a long polymer in solution using a two-state model with delta function coupling, accounting for various chemical reactions affecting loop formation.

## Contribution

It introduces an exact solution for the two-state polymer looping model with delta function coupling, including effects of additional chemical reactions on looping rates.

## Key findings

- Derived expressions for long-term and average rate constants.
- Quantified the impact of chemical reactions on looping kinetics.
- Provided a comprehensive analytical framework for polymer looping dynamics.

## Abstract

In this paper, the looping kinetics of a long polymer chain in solution has been investigated by using two state model, where one state represents open chain polymer molecule and the other represents closed chain polymer molecule. The dynamics of end-to end distance of both open chain and closed chain polymer is represented by a Smoluchowski-like equation for a single particle under two different harmonic potentials. The coupling between these two potentials are assumed to be a Dirac Delta function in our model. We evaluate two rate constants, the long term and the average rate constant. We have also incorporated the effect of all other chemical reactions involving at least one end of the open chain polymer molecule - on rate of end-to-end loop formation. The closed chain polymer molecule can be converted to a open chain molecule by breaking of any bond - the effect of this reaction on the rate of end-to-end loop formation is also considered in our model.