Understanding the reversible looping kinetics of a long chain polymer molecule in solution with Dirac Delta coupling. An exact analytical perspective

In this article, we present an exact analytically solvable two state model for calculating the rate of reversible looping of a long chain polymer molecule in solution. In our two state model, one of the two states represent an open chain polymer molecule and the other state represents a closed chain.. One can quantitatively measure the kinetic parameters such as the rate constants, the long term rate constant and the average term rate constant of the reaction from macroscopic measurements like the length of the polymer, the bond length and the relaxation time. The dynamics of the end-to-end distance of both open chain and the closed chain polymer are represented by a Smoluchowski-like equation for a single particle doing random walk under the realm of two different harmonic potentials. The coupling between these two potentials are assumed to be a Dirac Delta function in our model. We have also incorporated the effect of all other chemical reactions involving at least one end of the open chain polymer molecule on the rate of end-to-end loop formation. The closed chain polymer molecule can be converted to an open-chain molecule by breaking any bond - the effect of these reactions on the rate of end-to-end loop formation as well as the rate involving one of the ends are also considered in our model.