Protein Knots

Mathematically, knots are only well defined in closed (circular) loops. However, both the N- and C-termini of open proteins are typically located close to the surface of the protein and can be connected unambiguously: We reduce the protein to its backbone and draw two lines outward starting at the termini in the direction of the connection line between the centre of mass of the backbone and the respective ends. The two lines are joined by a big loop, and the structure is topologically classified by the determination of its Alexander polynomial. To determine an estimate for the size of the knotted core, we successively delete amino acids from the N-terminus until the protein becomes unknotted. The procedure is repeated at the C-terminus starting with the last N-terminal deletion structure that contained the original knot. For each deletion, the outward-pointing line through the new termini is parallel to the respective lines computed for the full structure. Unfortunately, the size of a knot is not always precisely determined by this procedure, so reported sizes should therefore only be treated as approximate.

To speed up calculations, the KMT reduction scheme is used. This algorithm successively deletes amino acids that are not essential to the topological structure of the protein. It is also employed to create a reduced representation of the knot. In the course of our investigations we came up with a number of stringent criteria that a structure should satisfy to be classified as knotted:

1. The Alexander polynomial should yield a knot.
2. There should not be any gaps in the polypeptide backbone. (See below.)
3. The knot should persist if two amino acids are removed from each end. (This prevents knots formed by just a few residues at the end of the chain passing through the loop - "shallow knots" and knots which only appear due to our specific loop closure procedure.)

Unfortunately, there are some structures containing regions of the backbone that were not resolved and for which coordinates are not reported in PDB (a gap in the structure). Mobile loops may not be resolved by X-ray crystallography unless they are stabilized by a ligand or by protein engineering, for example. If the polypeptide chain contains a gap, the knot is reported if a knot is present in at least one fragment of the chain and the structure that results from gaps being bridged with straight lines contains a knot. These criteria form the basis of our list of known knots. We have also included knotted structures with gaps if at least one homolog is knotted.

References

• Kolesov G, Virnau P, Kardar M, Mirny LA. Protein knot server: detection of knots in protein structures. Nucleic Acids Res. 2007 May 21 [PDF]
  
• Virnau P, Mirny LA and Kardar M. Intricate knots in proteins: function and evolution. PLoS Comput Biol. 2006 Sep 15;2(9):e122. Epub 2006 Jul 28.
  
• Mirny lab webpage

Questions about particular knots? Contact Peter Virnau.