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 root 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 similar to 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.