On the kratky-porod model for semi-flexible polymers in an external force field

Abstract

We prove, by means of matrix-valued stochastic processes, the convergence, in a suitable scaling limit, of the position vectors along a polymer in the discrete freely rotating chain model to that of the continuous Kratky Porod model, building on an earlier result for the original model without a force, and showing that it holds when an external force field is added to the system. In doing so, we also prove that the process of tangent vectors satisfies a stochastic differential equation, showing that it is the sum of a spherical Brownian motion and a projective drift term, and we analyze this equation to derive properties about the polymer in the regimes of high and low values of the force parameter and persistence length, or stiffness parameter of the molecule. If the force is much stronger than the persistence length, the polymer tends toward the rigid rod limit, aligned in the direction of the force; while if the force is much weaker, the polymer is aligned in its original direction, as in the case of the unforced model. The most interesting case occurs when both force and stiffness parameters are very large, in which case the polymer takes on the shape of a deterministic curve that satisfies an ordinary differential equation.