Polynomial Invariant of Molecular Circuit Topology

TL;DR

This paper introduces polynomial invariants for circuit topology, enabling classification of molecular chains with complex contacts, advancing the understanding of molecular folding and structure.

Contribution

It develops efficient polynomial invariants for circuit topology, capable of handling various contact types, and provides computational tools and data tables for chains with up to three contacts.

Findings

Polynomial invariants effectively classify molecular chain topologies.

The invariants distinguish chains with different contact arrangements.

Computational implementation and contact chain tables are provided.

Abstract

The topological framework of circuit topology has recently been introduced to complement knot theory and to help in understanding the physics of molecular folding. Naturally evolved linear molecular chains, such as proteins and nucleic acids, often fold into 3D conformations with critical chain entanglements and local or global structural symmetries stabilised by formation contacts between different parts of the chain. Circuit topology captures the arrangements of intra-chain contacts within a given folded linear chain and allows for the classification and comparison of chains. Contacts keep chain segments in physical proximity and can be either mechanically hard attachments or soft entanglements that constrain a physical chain. Contrary to knot theory, which offers many established knot invariants, circuit invariants are just being developed. Here, we present polynomial invariants that are both efficient and sufficiently powerful to deal with any combination of soft and hard contacts. A computer implementation and table of chains with up to three contacts is also provided.