Geometric Predictors of Knotted and Linked Arcs

Inspired by how certain proteins sense knots and entanglements in DNA molecules, here, we ask if local geometric features that may be used as a readout of the underlying topology of generic polymers exist. We perform molecular simulations of knotted and linked semiflexible polymers and study four geometric measures to predict topological entanglements: local curvature, local density, local 1D writhe, and nonlocal 3D writhe. We discover that local curvature is a poor predictor of entanglements. In contrast, segments with maximum local density or writhe correlate as much as 90% of the time with the shortest knotted and linked arcs. We find that this accuracy is preserved across different knot types and also under significant spherical confinement, which is known to delocalize essential crossings in knotted polymers. We further discover that nonlocal 3D writhe is the best geometric readout of the knot location. Finally, we discuss how these geometric features may be used to computationally analyze entanglements in generic polymer melts and gels.

Automated summary

Inspired by proteins, researchers find that local geometric features can predict knots and entanglements in polymers. Segments with maximum local density or writhe correlate up to 90% of the time with the shortest knotted and linked arcs. Nonlocal 3D writhe is the best geometric readout of the knot location.