Understanding looping kinetics of a long polymer molecule in solution. Exact solution for delta function sink model

TL;DR

This paper develops an exact analytical diffusion model for polymer looping kinetics in dilute solutions, providing explicit formulas for rate constants and their dependence on polymer parameters.

Contribution

It introduces a precise solution to the Smoluchowski equation with a delta function sink, detailing how rate constants depend on polymer length, bond length, and relaxation time.

Findings

Derived explicit expressions for short and long term rate constants.

Long term rate constant is independent of initial conditions.

Rate constants vary with polymer length, bond length, and relaxation time.

Abstract

A diffusion theory for intramolecular reactions of polymer chain in dilute solution is formulated. We give a detailed analytical expression for calculation of rate of polymer looping in solution. 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. The solution of this equation is expressed in terms of Laplace Transform of the Greens function for end to end motion of the polymer in absence of the sink. We have defined two different rate constants, the long term rate constant and the short term rate constant. The short term rate constant and long term rate constant varies with several parameters such as length of the polymer, bond length and the relaxation time. The long term rate constant is independent of the initial probability distribution.