Abstract
Polymers can be modeled as open polygonal paths and their closure generates knots. Knotted proteins detection is currently achieved via high-throughput methods based on a common framework insensitive to the handedness of knots. Here we propose a topological framework for the computation of the HOMFLY polynomial, an handedness-sensitive invariant. Our approach couples a multi-component reduction scheme with the polynomial computation. After validation on tabulated knots and links the framework was applied to the entire Protein Data Bank along with a set of selected topological checks that allowed to discard artificially entangled structures. This led to an up-to-date table of knotted proteins that also includes two newly detected right-handed trefoil knots in recently deposited protein structures. The application range of our framework is not limited to proteins and it can be extended to the topological analysis of biological and synthetic polymers and more generally to arbitrary polygonal paths.
Highlights
The topological study of biological polymers has led to important insights into their structural properties and evolution [1,2]
We have presented a novel topological framework for the HOMFLY polynomial computation of polygonal paths based on the geometric construction of Conway skein triples
Validation on tabulated knots and links demonstrates the global method robustness and the effectiveness of the greedy selection of the crossing to be switched. These evidences have been further confirmed by the polynomial computation of protein structures, leading to an upto date table of knotted structures
Summary
The topological study of biological polymers has led to important insights into their structural properties and evolution [1,2]. From a topological point of view polymers can be naturally modeled as sequences of 3D points, i.e. open polygonal paths. Their closure generates classical objects in topology called knots. Knots investigation was initially fostered by the discovery of knotted circular single-stranded DNA [3] and has been followed by the study of the underlying enzymatic mechanisms [4,5] and more recently by the description of the topological organization and packing dynamics of bacteriophage P4 genome [6,7]
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