Studies of global and local entanglements of individual protein chains using the concept of knotoids

Abstract

We study here global and local entanglements of open protein chains by implementing the concept of knotoids. Knotoids have been introduced in 2012 by Vladimir Turaev as a generalization of knots in 3-dimensional space. More precisely, knotoids are diagrams representing projections of open curves in 3D space, in contrast to knot diagrams which represent projections of closed curves in 3D space. The intrinsic difference with classical knot theory is that the generalization provided by knotoids admits non-trivial topological entanglement of the open curves provided that their geometry is frozen as it is the case for crystallized proteins. Consequently, our approach doesn't require the closure of chains into loops which implies that the geometry of analysed chains does not need to be changed by closure in order to characterize their topology. Our study revealed that the knotoid approach detects protein regions that were classified earlier as knotted and also new, topologically interesting regions that we classify as pre-knotted.

Figure 1

A diagram of the trefoil knot.

Figure 2

Reidemeister moves. Top Row: Reidemeister Moves I and II. Bottom Row: Reidemeister Move III.

Figure 3

Types of knotoids. (a) A non-trivial pure knotoid diagram with two crossings, (b) A knot-type knotoid with three crossings and (c) A knot-type knotoid with three crossings.

Figure 4

The forbidden moves. Crossing under (ΦI) and over (ΦII) an arc adjacent to an endpoint is prohibited.

Figure 5

Projection of a protein chain using two different techniques. Top Row: Knotoids technique. (a) The two black infinite lines pass through the N and C termini of the protein chain and are perpendicular to a chosen plane. (b) The two infinite lines pass through the N and C termini of the reduced protein backbone are perpendicular to a chosen plane. Bottom Row: Stochastic closure technique. (c) A choice of closing direction and the two rays extending from the termini towards that direction. The ends of the two rays are connected when they exit the sphere that contains the protein chain. (d) The resulting knot diagram.

Figure 6

Projection Maps. Top Row: The knotoid technique for the protein 3KZN. Bottom Row: The stochastic closure technique for the protein 3KZN. The maps were created using Blender 2.78 (http://www.blender.org. Used under CC BY-SA 3.0, https://creativecommons.org/licenses/by-sa/3.0/. No changes were made.) and R 3.3.2.

Figure 7

Fingerprints. (a) The knotoid fingerprint of DehI. (b) The knot fingerprint of DehI.

Figure 8

Instance of the trimming process of protein 3BJX. Notice that only one end of projected chain is progressively trimmed. The knotoid type change is indicated with the change of the color of the diagram and the used colors correspond to these used to indicate knotoid types in Fig. 7.