Free Energy of a Knotted Polymer Confined to Narrow Cylindrical and Conical Channels

Monte Carlo simulations are used to study the conformational behavior of a semiflexible polymer confined to cylindrical and conical channels in the Odijk regime. For cylindrical confinement, we examine polymers with a single knot of topology 3₁, 4₁, or 5₁, as well as unknotted polymers that are capable of forming S-loops. We measure the variation of the free energy F with the polymer extension length X and examine the effect of varying the polymer topology, persistence length P and cylinder diameter D on the free energy functions and the knot span. We find that increasing the knot complexity increases the typical size of the knot. In addition, the scaling properties of the free energy and knot span are in agreement with predictions from a theoretical model constructed using known properties of interacting polymers in the Odijk regime. We also examine the variation of F with position of a knot in conical channels for various values of the cone angle. The behavior is in agreement with predictions from a theoretical model in which the dominant contribution to the change in F is the change in the size of the hairpins as the knot moves to the wider region of the channel.