Proteins analysed as virtual knots

TL;DR

This paper introduces a novel method using virtual knots to analyze and identify topological knotting in open protein chains, overcoming limitations of traditional closure methods and revealing new insights into protein topology.

Contribution

The authors develop a new approach employing virtual knots to study open curves, providing a more accurate topological analysis of protein chains and extending previous knotting results.

Findings

Virtual knot analysis reveals additional knotted proteins.

Comparison shows virtual knots better capture topological ambiguity.

Identification of new topologically interesting protein cases.

Abstract

Long, flexible physical filaments are naturally tangled and knotted, from macroscopic string down to long-chain molecules. The existence of knotting in a filament naturally affects its configuration and properties, and may be very stable or disappear rapidly under manipulation and interaction. Knotting has been previously identified in protein backbone chains, for which these mechanical constraints are of fundamental importance to their molecular functionality, despite their being open curves in which the knots are not mathematically well defined; knotting can only be identified by closing the termini of the chain somehow. We introduce a new method for resolving knotting in open curves using virtual knots, a wider class of topological objects that do not require a classical closure and so naturally capture the topological ambiguity inherent in open curves. We describe the results of analysing proteins in the Protein Data Bank by this new scheme, recovering and extending previous knotting results, and identifying topological interest in some new cases. The statistics of virtual knots in protein chains are compared with those of open random walks and Hamiltonian subchains on cubic lattices, identifying a regime of open curves in which the virtual knotting description is likely to be important.