Effects of Intrachain Interactions on the Knot Size of a Polymer

Knots occur frequently in our daily life and also in long polymers. Studies of spontaneously occurring knots in a linear polymer have focused on their size and probabilities. In this work, we report counterintuitive phenomena about knot sizes and explain them by an analytic theory. Our simulations show that short-range intrachain attractions (repulsions) lead to shrinking (swelling) of knot sizes, trends which follow intuition. However, long-range attractions (repulsions) surprisingly lead to swelling (shrinking) of knot sizes. Such counterintuitive trends are because larger knots contain more interacting monomer pairs if the interaction range is larger than a critical value. The critical interaction range varies from a fraction of to multiple persistence lengths, and so the long-range interaction regime can be reached for DNA, peptide chains, or synthetic polymers under depletion attractions induced by colloids (e.g., proteins) or via Coulomb repulsions. Our results suggest that probabilities and sizes of knots can be controlled independently through adjusting the range and the strength of intrachain interactions.