Looping of a long chain polymer in solution: Simple derivation for exact solution for a delta function sink

TL;DR

This paper introduces a straightforward analytical approach to model polymer looping in dilute solutions using a Smoluchowski-like equation with a delta function sink, providing exact solutions expressed via Laplace transforms.

Contribution

It presents a simple, exact analytical method for solving intra-molecular polymer reactions modeled by a delta function sink, extending previous complex approaches.

Findings

Solution expressed in terms of Laplace transform of Green's function.

Both long-term and average rate constants depend on polymer parameters.

Results agree with more detailed existing methods.

Abstract

A simple analytical method for solving intra-molecular reactions of polymer chain in dilute solution is formulated. The physical problem of looping can be modeled mathematically with the use of a Smoluchowski-like equation with a Dirac delta function sink of finite strength. Here we have proposed a very simple method of solution. The solution is expressed in terms of Laplace Transform of the Green's function for end-to-end motion of the polymer in absence of the sink. We have used two different rate constants, the long term rate constant and the average rate constant. The average rate constant and long term rate constant varies with several parameters such as length of the polymer (N), bond length (b) and the relaxation time ${\tau_R}$. Our results are in agreement with that produced by more general and detailed method.