Geometric Predictors of Knotted and Linked Arcs

Abstract

Inspired by how certain proteins "sense" knots and entanglements in DNA molecules, here we ask if there exist local geometric features that may be used as a read-out of the underlying topology of generic polymers. We perform molecular simulations of knotted and linked semiflexbile polymers and study four geometric measures to predict topological entanglements: local curvature, local density, local 1D writhe and non-local 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 delocalise essential crossings in knotted polymers. We further discover that

Machine Learning Understands Knotted Polymers

“…However, one can consider other rotational invariant features. For instance, a recent work 63 suggested that the local density along the ring (i.e., the average number of neighboring beads) could be a set of rotational and translational features that can faithfully capture the underlying topology of the rings.…”

Geometric learning of knot topology

“…In this work, we set out to test the use of a different type of geometric feature that our group recently utilised to identify essential crossings of a knot and plectoneme-like double folding of ring polymers. 25,26 More specifically, we focused on a generalisation of the Gauss linking integral applied to a single closed curve, often associated with its writhe 27 and average crossing number. 25,28 This choice is inspired by the intuition that writhe captures the geometrical entanglement of a curve with itself, and we thus define a generalised local segment-to-segment (StS) writhe aswhere r ( x ) and t ( x ) are the 3D position of, and the tangent at, segment x , respectively.…”

“… 23 since we use a smaller training dataset and simpler NNs. We then trained the same NNs using a range of other geometric features, such as local curvature, density and 1D writhe 26 (see ESI † for details), and found that most of them performed more poorly, or at best equally, with respect to the XYZ representation ( Fig. 1(E) ).…”

Machine Learning in Soft Matter: From Simulations to Experiments

“…Equally, these methods have been used to study phase and structural transitions in polymer nanocomposites, [224] to respectively identify the two-state and three-state phase transitions of polymers adsorbed on homogeneous patterned and striped-patterned surfaces, [225] and to identify various knot types in polymers. [226][227][228][229] Furthermore, some researchers have combined ML with traditional simulation methods to enhance the efficiency of relevant processes. Bhattacharya et al [230] developed a universal artificial intelligence tool, called dPOLY, for analyzing MD trajectories and predicting phase and phase transitions in polymers.…”