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.
Highlights
Folded linear molecular chains are ubiquitous in biology
Despite being interesting and innovative, these studies have had a limited impact on protein science as the vast majority of identified proteins fall into one topology class, i.e., the unknot [22]
We introduce singquandles, which are an extension of quandles that have been used to distinguish knots with singularities. We further extend this idea to the idea of a bondle, which is a quandle that can be applied in the presence of bonds
Summary
Folded linear molecular chains are ubiquitous in biology. Proteins and nucleic acids are linear polymers responsible for most cellular functions, for the inheritance of biological information, and are subject to changes during evolution and pathologies [8, 25]. This paper presents a new knot theory for folded linear molecular chains and looks to classify the topological structure of proteins through the application of certain aspects of knot theory. We define singular crossings to exist where a protein has intra-chain interactions ( called contacts). These contacts take one of two forms. 5, we introduce singquandles, which are an extension of quandles that have been used to distinguish knots with singularities We further extend this idea to the idea of a bondle, which is a quandle that can be applied in the presence of bonds.
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