Influence of Rigidity and Knot Complexity on the Knotting of Confined Polymers

We employ computer simulations and thermodynamic integration to analyze the effects of bending rigidity and slit confinement on the free energy cost of tying knots, ΔF_knotting, on polymer chains under tension. A tension-dependent, nonzero optimal stiffness κ_min exists, for which ΔF_knotting is minimal. For a polymer chain with several stiffness domains, each containing a large amount of monomers, the domain with stiffness κ_min will be preferred by the knot. A local analysis of the bending in the interior of the knot reveals that local stretching of chains at the braid region is responsible for the fact that the tension-dependent optimal stiffness has a nonzero value. The reduction in ΔF_knotting for a chain with optimal stiffness relative to the flexible chain can be enhanced by tuning the slit width of the 2D confinement and increasing the knot complexity. The optimal stiffness itself is independent of the knot types we considered, while confinement shifts it toward lower values.