PNNL

Abstract

In order to investigate analytically the looping time of a free-draining polymer (Rouse model), we revisit the celebrated Wilemski-Fixman (WF) theory. The WF theory introduces a sink term in the Fokker-Planck equation to account for the complicated boundary condition satisfied by the looping effect. We use perturbation methods to make theoretical predictions of the looping time for two popular choices for the sink, namely the Delta and Heaviside sinks. For both types of sink, we show that under the condition of small capture radius (compared to the Kuhn length), WF can produce all known analytical and asymptotic results obtained by other means. This includes the mixed scaling regime which combines Doi's N2 scaling and Szabo, Schulten & Schulten's NvN/? scaling. In addition, again for the case of small capture radius, we find an extra term in the analytical expression for the looping time which has not appeared previously in the literature. Numerical results obtained through Monte Carlo simulations corroborate the theoretical findings. The mathematical constructions developed here can be applied to other diffusion limited catalytically activated chemical reactions.