A topological framework for the computation of the HOMFLY polynomial and its application to proteins

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.

Figure 1. A knot diagram and illustration of the Conway skein triple.

(A) Three dimensional polygonal representation of the trefoil knot (in red) and its planar diagram (in black). Two red spheres on the knot mark the 3D points and projecting down to on the planar diagram along the brown arrow. (B) The Conway skein triple is composed of three oriented diagrams that are the same outside a small region, where they look like the illustrated , and . To define the oriented sign of a crossing, approach it along the underpass in the direction of the orientation: if the overpass orientation runs from left to right, the oriented sign is , otherwise.

Figure 2. Knots met in proteins.

Illustration of the knots found in proteins, labeled according to Rolfsen names. U: the simplest knot, the unknot. : the trefoil knot and its mirror image, denoted by the , has three crossings. : the figure-eight knot is the only knot with four crossings. : the three-twist knot has five crossings. : the Stevedore's knot, the most complex knot detected in proteins.

Figure 3. Example of geometric construction of the skein configurations.

(A) Figure-eight polygonal knot diagram. Knot orientation and the crossing between the edges and are shown. (B) A clean quadrilateral around is shown in red. (C) The rotated quadrilateral (solid blue lines) is obtained by rotating (dashed red lines) along the axis. (D) Triangles to be analyzed in the topological check are shaded in green. The points and are reported respectively in red and blue. (E) The configuration, with the path highlighted in black (F) The configuration. Solid lines highlight new connections (in red) and (in blue).

Figure 4. The Increase of the number of tree nodes as a function of tree levels.

Trees of both greedy (white/black) and fixed choice (gray) algorithms have been clustered according to the number of levels (). For each cluster a box plot of the nodes number has been drawn with a width proportional to the cluster size. Solid power curves fit the reported data. Dashed red and blue curves represent respectively lower and upper estimates of node numbers. Curve expressions are shown in the legend.

Figure 5. The two newly identified right-handed trefoil knots in recently deposited protein structures.

(A) On the top, the secondary structure and the accessible surface area (in transparency) of the human Carbonic Anhydrase VII, isoform 1 (3 mdz) is shown. On the bottom, a sausage view cartoon of the same enzyme is shown. In this representation, the diameter of the sausage is proportional to the B-factor. The thicker the backbone is, the more flexible it is. (B) The same representations as in (A) are shown for the knotted core of the uncharacterized ORF from Sulfolobus Islandicus rudivirus 1 (2x4i), chain A. Colors change continuously from blue (first residue) to red (last residue). The last residue of the 2x4i protein is colored in orange, since the structure presents a gap toward its true C-terminus end and results a slip-knot when the whole structure is considered, as detailed in the text.

Figure 6. MSR reduction curve of the U2 snRNP protein Rds3p.

On the middle are illustrated the 13 reduction steps (b-n) for the Rds3p protein (2k0a) (a). The last frame (n) represents the minimal structure of the protein, a left-handed trefoil knot. On the top, the residual points are plotted for each frame a-n. The corresponding move lengths are shown on the right.

Figure 7. MSR algorithm analysis.

(A) The observed distribution of move lengths, considering only density values greater than 0.2%. (B) The mean move length significantly decreases as a function of the protein length. Values for each length percentile are reported. (C) Move length distributions are shown relatively to the first and fourth quartile of the reduction process, considering only density values greater than 0.2%. (D) Frequencies of classes of move lengths as a function of protein residual length at which they occur are dotted. LOESS curves are reported. Classes cutoffs were chosen according to move length quartiles (0,4,13) and the last 5% of residual lengths were discarded to remove frequency fluctuations.