Abstract
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, ΔFknotting, on polymer chains under tension. A tension-dependent, nonzero optimal stiffness κmin exists, for which ΔFknotting is minimal. For a polymer chain with several stiffness domains, each containing a large amount of monomers, the domain with stiffness κmin 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 ΔFknotting 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.
Highlights
While in the macroscopic world it is clear that the effort needed to tie a knot in wire or string will always increase if it is made more rigid, for equivalent microscopic objects, polymers, the same does not hold
The effect of confinement on κmin as well as the amount by which ΔFknotting is reduced for the optimal rigidity κmin depends sensitively on the tension applied to the polymer chain
We have demonstrated that the local stretching at the braiding region and close to the crossing points is the physical mechanism responsible for the minimization of the free energy penalty of knotting of a linear polymer for nonvanishing values of the bending rigidity
Summary
While in the macroscopic world it is clear that the effort needed to tie a knot in wire or string will always increase if it is made more rigid, for equivalent microscopic objects, polymers, the same does not hold. In contrast to three dimensions where they are weakly localized, knots in polymers adsorbed on a surface are strongly localized.[14,15] Considering the properties of polymers confined in a slit, simulations of DNA found a nonmonotonic dependence of the knotting probability on the slit width,[16] and for flexible polymers evidence was found that the particular topology is important.[17] While previous work on knotting in confinement has focused on polymers that have one specific stiffness, here we apply a simple model for a polymer chain under tension to investigate the dependence of ΔFknotting on rigidity for various widths of the geometrical confinement.
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