The two-state reversible kinetics of a long polymer molecule in solution with a delocalized coupling term. An exact analytical model.

We develop a mathematical recipe for calculating the exact delocalized rate constant for the end loop in the motion of a long polymer undergoing reversible reaction between two disjunct states, the open and the closed state due to the effect of the surrounding solvent. The Smoluchowski-like equation mathematically represents the diffusion of the end monomers of the open polymer chain and the closed chain in terms of a free particle performing random motion under different harmonic potentials. The coupling term between the two potentials is assumed to be represented by a delocalized coupling term. It consists of a collection of several Dirac Delta functions that take care of the physical representation of the multi-state looping phenomenon. For a favorable disposition towards the more realistic experimental behavior, the rate constant have effect incorporated from all the bond making and bond breaking phenomenon of the chemical reactions different from the end loop formation. It not only involves the end looping but can evaluate for reactions where only one of the ends of the open-chain is involved.