Topological knots and links in proteins

Abstract

Twenty years after their discovery, knots in proteins are now quite well understood. They are believed to be functionally advantageous and provide extra stability to protein chains. In this work, we go one step further and search for links-entangled structures, more complex than knots, which consist of several components. We derive conditions that proteins need to meet to be able to form links. We search through the entire Protein Data Bank and identify several sequentially nonhomologous chains that form a Hopf link and a Solomon link. We relate topological properties of these proteins to their function and stability and show that the link topology is characteristic of eukaryotes only. We also explain how the presence of links affects the folding pathways of proteins. Finally, we define necessary conditions to form Borromean rings in proteins and show that no structure in the Protein Data Bank forms a link of this type.

Keywords: catenanes; disulphide bridge; folding; lasso; slipknot.

Fig. 1.

Exemplary structures of proteins with links: negative Hopf link, positive Hopf link, and positive Solomon link [Protein Data Bank (PDB) codes 2LFK, 2KQA, and 4ASL, Top to Bottom]. Middle row shows the most general link type (without orientation). The orange stripes denote disulfide bridges. The N and C letters denote the N and C termini of the protein. The arrows denote the orientation from the N to the C terminus. For details in link orientation see SI Appendix. In each panel, colors in the structure match the colors in the scheme at Right; the protein topology is presented as a solid (black or colored) line.

Fig. 2.

The method of identification of links. On each covalent loop a triangulated minimal surface is spanned (Left and Right). The necessary condition for the existence of a link is that surfaces pierce one another (Center) or, equivalently, each surface is pierced by the border of the second surface.

Fig. 3.

Crystal structure of cerato-platanin (PDB code 3SUK). (A) Covalent loops forming the Hopf link are depicted in red and blue. Cysteines closing the loops and cysteine bridges are marked in orange. (B) Solvent-exposed surface. Colors correspond to the crystal structure. Only one cysteine bridge (marked with a black circle) is partially exposed to the solvent.

Fig. 4.

The temperature dependence of unfolding rate constant for five models of TdPI protein, differing in topology. Inset shows the protein structure with native “red” and “blue” loop indicated. The stripes denote the loop closing pattern: orange and green for the (native) Hopf link and the trivial (nonnative) model. The fitted function is presented in the top right corner. On the right side the fitted values of $E_A/R$ are given. In the bottom right corner, the schemes of the Hopf link and the trivial model are presented.

Fig. 5.

Possible ways of folding of TdPI. Folding can follow three different pathways, but formation of the small covalent loop as the first event blocks folding. Moreover, in the last folding step the protein can collapse to a topological trap (in red oval), characterized by trivial topology. Green oval denotes the native, Hopf-link structure.

Fig. 6.

Exemplary structures of proteins with the Solomon link and the core of Borromean rings. (Top, Left to Right) An exemplary protein with the Solomon link (PDB code 4ASL), the covalent loops after unfolding, and the scheme of the corresponding Solomon link. (Middle, Left to Right) Structure of goat lactoperoxidase (PDB code 2E9E) forming the core of Borromean rings; the structure after smoothing—the blue surface is spanned on the main chain and on the disulfide bridge, and the red surface is spanned by the chain and is delimited by the blue surface (shown in cyan) and it is pierced by the green part of the chain; the schematic view of the protein—the solid color lines form the core of Borromean rings, shown in Bottom, Left to Right with retention of colors. Bottom Left presents the Borromean rings.