# Topological Indices of Proteins

## TL;DR

This paper introduces new topological indices for proteins based on discrete curve analysis, accounting for singularities and structural motifs, to better classify and understand protein folding dynamics.

## Contribution

It extends the standard topological classification of proteins by incorporating indices related to singularities in the Frenet framing, applicable to discrete protein models.

## Key findings

- Computed integer topological indices for large protein datasets.
- Demonstrated the relation between singularities and protein structural motifs.
- Proposed topological indices as tools for analyzing protein folding dynamics.

## Abstract

Protein molecules can be approximated by discrete polygonal chains of amino acids. Standard topological tools can be applied to the smoothening of the polygons to introduce a topological classification of proteins, for example, using the self-linking number of the corresponding framed curves. In this paper we add new details to the standard classification. Known definitions of the self-linking number apply to non-singular framings: for example, the Frenet framing cannot be used if the curve has inflection points. Meanwhile in the discrete proteins the special points are naturally resolved. Consequently, a separate integer topological characteristics can be introduced, which takes into account the intrinsic features of the special points. For large number of proteins we compute integer topological indices associated with the singularities of the Frenet framing. We show how a version of the Calugareanu's theorem is satisfied for the associated self-linking number of a discrete curve. Since the singularities of the Frenet framing correspond to the structural motifs of proteins, we propose topological indices as a technical tool for the description of the folding dynamics of proteins.