Interplay between writhe and knotting for swollen and compact polymers

The role of the topology and its relation with the geometry of biopolymers under different physical conditions is a nontrivial and interesting problem. Aiming at understanding this issue for a related simpler system, we use Monte Carlo methods to investigate the interplay between writhe and knotting of ring polymers in good and poor solvents. The model that we consider is interacting self-avoiding polygons on the simple cubic lattice. For polygons with fixed knot type we find a writhe distribution whose average depends on the knot type but is insensitive to the length $N$ of the polygon and to solvent conditions. This “topological contribution” to the writhe distribution has a value that is consistent with that of ideal knots. The standard deviation of the writhe increases approximately as $\sqrt{N}$ in both regimes and this constitutes a geometrical contribution to the writhe. If the sum over all knot types is considered, the scaling of the standard deviation changes, for compact polygons, to $\sim N^{0.6}$. We argue that this difference between the two regimes can be ascribed to the topological contribution to the writhe that, for compact chains, overwhelms the geometrical one thanks to the presence of a large population of complex knots at relatively small values of $N$. For polygons with fixed writhe we find that the knot distribution depends on the chosen writhe, with the occurrence of achiral knots being considerably suppressed for large writhe. In general, the occurrence of a given knot thus depends on a nontrivial interplay between writhe, chain length, and solvent conditions.

💡 Research Summary

The paper investigates how the topological property of knotting and the geometric property of writhe interact in ring polymers under different solvent conditions, using a lattice model of interacting self‑avoiding polygons (ISAP) on the simple cubic lattice. Monte‑Carlo simulations were performed for a wide range of chain lengths (from a few thousand up to tens of thousands of monomers) and for many knot types, including the unknot, trefoils, figure‑eight knots, and more complex composite knots. Two thermodynamic regimes were explored: a good‑solvent (swollen) regime where the polymer is expanded, and a poor‑solvent (compact) regime where the polymer collapses into a dense globule.

For a fixed knot type the average writhe ⟨Wr⟩ was found to depend only on the knot itself; it is essentially independent of chain length N and of the solvent quality. This “topological contribution” to the writhe matches the values predicted for ideal knots, confirming that each knot carries a characteristic writhe offset that does not change with the surrounding geometry. In contrast, the standard deviation σ(Wr) grows as √N in both regimes, reflecting a purely geometric fluctuation that scales like a random walk. When the sum over all knot types is taken, the compact regime shows a different scaling, σ(Wr)∼N^0.6, whereas the swollen regime retains the √N law. The authors attribute the altered exponent in the compact case to the high prevalence of complex knots at relatively small N; these knots contribute a large topological writhe component that dominates the geometric one.

The study also examined the inverse problem: fixing the writhe and asking how the knot distribution changes. For large absolute values of writhe, achiral knots (e.g., the 4₁ knot) become strongly suppressed, while chiral knots with the same handedness as the writhe are enhanced. This demonstrates that the probability of a given knot is not a simple function of chain length alone but results from a non‑trivial interplay among writhe, length, and solvent condition.

Overall, the work provides a quantitative picture of how topological constraints (knot type) and geometric fluctuations (writhe) combine in polymers. It shows that the topological contribution to writhe is a robust, length‑independent offset, whereas the geometric contribution follows universal scaling laws that can be overridden by the presence of many complex knots in dense, compact states. These findings have implications for understanding DNA supercoiling, protein folding, and other biopolymer phenomena where both knotting and writhe play functional roles.