Proteins analysed as virtual knots
Keith Alexander, Alexander J. Taylor & Mark R. Dennis
Scientific Reports 7, Article number: 42300 (2017)
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Applied mathematics Biophysics Computational science Protein analysis
Received: 26 September 2016 Accepted: 05 January 2017 Published online: 13 February 2017
Figure 1: Protein backbone structures as open knotted space curves.
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, which are 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.
Acknowledgements
The authors are grateful to Benjamin Bode, Paula Booth, Neslihan Gügümcü, Lou Kauffman, Annela Seddon, Joanna Sulkowska and Stu Whittington for valuable discussions. This research was funded by the Leverhulme Trust Research Programme Grant No. RP2013-K-009, SPOCK: Scientific Properties of Complex Knots. Keith Alexander was funded by the Engineering and Physical Sciences Research Council. This work was carried out using the computational facilities of the Advanced Computing Research Centre, University of Bristol.
Author information
Affiliations
H H Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK
Keith Alexander, Alexander J. Taylor & Mark R. Dennis
Contributions
K.A. carried out the protein analysis and virtual knotting routines. A.J.T. carried out the classical knot identification and random chain analysis, and suggested the original problem. M.R.D. directed the study and drafted the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Keith Alexander or Alexander J. Taylor or Mark R. Dennis.
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