(13f) Biased Sampling of 3D Polymer Conformations in an External Field Using Brownian Bridges

The equilibrium conformation of polymer molecules in an external field is of fundamental interest to the calculation of macroscopic polymer properties. Here, we build upon a mathematical method [Krishnaswami et al. (1997). J. Chem. Phys., 107(15), 5929-5944.] for efficient sampling of continuous polymer chains described by a stochastic differential equation along the backbone’s contour length (e.g., Gaussian chains, worm-like chains, etc.). This method is based on the concept of a Brownian bridge, which is a stochastic process whose endpoint is constrained through the addition of a biasing drift derived from Bayes theorem. We note for a polymer system, one can formulate a Brownian bridge to exclude small Boltzmann weights during sampling, and hence only sample regions of high probability in phase space. In this talk, we formulate numerical schemes to generate such bridges in applications where one wants to sample polymer chains in an arbitrary, external potential. The biasing drift needed for the Brownian bridge is a hitting probability (which can be thought of as an entropic potential), which we obtain by solving a Backwards Kolmogorov equation (or equivalently, a Backwards Fokker Planck equation). Thus, through an additional cost of solving a PDE once at the beginning of a simulation, we can guarantee our stochastic differential equations will be able to generate efficient configurations of polymer chains. We show examples for a wide range of potentials where polymer conformations are shown to satisfy exact energy constraints either at a fixed value or within a stipulated range. We conclude by discussing how to extend this technique to include segment-segment interactions (e.g., excluded volume) and multiple chains. We also discuss how to obtain approximate biasing, entropic potentials through approximations to the Backwards Kolmogorov equation for more complicated situations.