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Proteins analysed as virtual knots

View Article: PubMed Central - PubMed

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.

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Related in: MedlinePlus

Classical and virtual knot types found amongst different projection/closure directions for a protein backbone chain.The protein backbone shown has PDB ID: 4K0B, chain A (Sulfolobus solfataricus S-adenosylmethionine synthetase)49. Each point is coloured according to the knot type (classical or virtual) found by closure/projection in that direction. Classical and virtual knot types are coloured according to the legend. (a) Classical knots resulting from 3-dimensional sphere closure in each direction. (b) Virtual knot types resulting from virtual closure of the diagram obtained from projection in each direction. (c) and (d) are Mollweide projections of (a) and (b). These images are constructed from sampling 10,000 directions in each case. Antipodal points on the sphere are always associated with the same knot type under virtual closure (up to possibly distinct mirrors for certain virtual knot types), but may produce different classical knots on sphere closure. This protein is considered strongly trefoil (31) knotted under sphere closure, and strongly v21 virtually knotted under virtual closure; it is an unusually strong exemplar of this behaviour, described in the following Section.
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f3: Classical and virtual knot types found amongst different projection/closure directions for a protein backbone chain.The protein backbone shown has PDB ID: 4K0B, chain A (Sulfolobus solfataricus S-adenosylmethionine synthetase)49. Each point is coloured according to the knot type (classical or virtual) found by closure/projection in that direction. Classical and virtual knot types are coloured according to the legend. (a) Classical knots resulting from 3-dimensional sphere closure in each direction. (b) Virtual knot types resulting from virtual closure of the diagram obtained from projection in each direction. (c) and (d) are Mollweide projections of (a) and (b). These images are constructed from sampling 10,000 directions in each case. Antipodal points on the sphere are always associated with the same knot type under virtual closure (up to possibly distinct mirrors for certain virtual knot types), but may produce different classical knots on sphere closure. This protein is considered strongly trefoil (31) knotted under sphere closure, and strongly v21 virtually knotted under virtual closure; it is an unusually strong exemplar of this behaviour, described in the following Section.

Mentions: We analyse open curves in terms of the fractions of directions giving different knot types under sphere or virtual closure. Figure 3(a–d) demonstrates this for an example protein chain, for both closure methods: directions are coloured according to the knot types both on a sphere and in (area-preserving) Mollweide projection. In the sphere closure maps (b), (c), 59% of directions give a trefoil knot 31, which therefore dominates and so this backbone was determined by ref. 5 to be 31 knotted (alongside 34% unknots and 7% more complex knots shown by the smaller islands). Much of the area identified as 01 or 31 under sphere closure in (c), becomes, in the corresponding virtual closure map (d), the virtual knot v21 in 54% of different projections. This curve therefore has strong virtual character, and its virtual knot type reflects the ambiguity of the open curve between the unknot and trefoil knot.


Proteins analysed as virtual knots
Classical and virtual knot types found amongst different projection/closure directions for a protein backbone chain.The protein backbone shown has PDB ID: 4K0B, chain A (Sulfolobus solfataricus S-adenosylmethionine synthetase)49. Each point is coloured according to the knot type (classical or virtual) found by closure/projection in that direction. Classical and virtual knot types are coloured according to the legend. (a) Classical knots resulting from 3-dimensional sphere closure in each direction. (b) Virtual knot types resulting from virtual closure of the diagram obtained from projection in each direction. (c) and (d) are Mollweide projections of (a) and (b). These images are constructed from sampling 10,000 directions in each case. Antipodal points on the sphere are always associated with the same knot type under virtual closure (up to possibly distinct mirrors for certain virtual knot types), but may produce different classical knots on sphere closure. This protein is considered strongly trefoil (31) knotted under sphere closure, and strongly v21 virtually knotted under virtual closure; it is an unusually strong exemplar of this behaviour, described in the following Section.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5304221&req=5

f3: Classical and virtual knot types found amongst different projection/closure directions for a protein backbone chain.The protein backbone shown has PDB ID: 4K0B, chain A (Sulfolobus solfataricus S-adenosylmethionine synthetase)49. Each point is coloured according to the knot type (classical or virtual) found by closure/projection in that direction. Classical and virtual knot types are coloured according to the legend. (a) Classical knots resulting from 3-dimensional sphere closure in each direction. (b) Virtual knot types resulting from virtual closure of the diagram obtained from projection in each direction. (c) and (d) are Mollweide projections of (a) and (b). These images are constructed from sampling 10,000 directions in each case. Antipodal points on the sphere are always associated with the same knot type under virtual closure (up to possibly distinct mirrors for certain virtual knot types), but may produce different classical knots on sphere closure. This protein is considered strongly trefoil (31) knotted under sphere closure, and strongly v21 virtually knotted under virtual closure; it is an unusually strong exemplar of this behaviour, described in the following Section.
Mentions: We analyse open curves in terms of the fractions of directions giving different knot types under sphere or virtual closure. Figure 3(a–d) demonstrates this for an example protein chain, for both closure methods: directions are coloured according to the knot types both on a sphere and in (area-preserving) Mollweide projection. In the sphere closure maps (b), (c), 59% of directions give a trefoil knot 31, which therefore dominates and so this backbone was determined by ref. 5 to be 31 knotted (alongside 34% unknots and 7% more complex knots shown by the smaller islands). Much of the area identified as 01 or 31 under sphere closure in (c), becomes, in the corresponding virtual closure map (d), the virtual knot v21 in 54% of different projections. This curve therefore has strong virtual character, and its virtual knot type reflects the ambiguity of the open curve between the unknot and trefoil knot.

View Article: PubMed Central - PubMed

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.

No MeSH data available.


Related in: MedlinePlus