Knot Theory for Proteins: Gauss Codes, Quandles and Bondles

TL;DR

This paper extends knot theory to include intra-chain interactions in proteins, using Gauss codes and algebraic structures like quandles and bondles to classify protein topology beyond traditional methods.

Contribution

It introduces a novel framework incorporating intra-chain bonds into knot theory for proteins, including the development of bondles for topological distinction.

Findings

Extended knot theory to account for intra-chain bonds.

Represented protein topology using Gauss codes.

Defined bondles as a new algebraic tool for protein classification.

Abstract

Proteins are linear molecular chains that often fold to function. The topology of folding is widely believed to define its properties and function, and knot theory has been applied to study protein structure and its implications. More that 97% of proteins are, however, classified as unknots when intra-chain interactions are ignored. This raises the question as to whether knot theory can be extended to include intra-chain interactions and thus be able to categorize topology of the proteins that are otherwise classified as unknotted. Here, we develop knot theory for folded linear molecular chains and apply it to proteins. For this purpose, proteins will be thought of as an embedding of a linear segment into three dimensions, with additional structure coming from self-bonding. We then project to a two-dimensional diagram and consider the basic rules of equivalence between two diagrams. We further consider the representation of projections of proteins using Gauss codes, or strings of numbers and letters, and how we can equate these codes with changes allowed in the diagrams. Finally, we explore the possibility of applying the algebraic structure of quandles to distinguish the topologies of proteins. Because of the presence of bonds, we extend the theory to define bondles, a type of quandle particularly adapted to distinguishing the topological types of proteins.