Homology of homologous knotted proteins

TL;DR

This paper introduces a topological framework using persistent homology to analyze and classify knotted protein structures, effectively capturing their geometric features and homology even in noisy data.

Contribution

It develops a mathematical pipeline that applies persistent homology to quantify and cluster trefoil knotted proteins, demonstrating robustness to noise and providing geometric insights.

Findings

Persistent homology faithfully represents protein homology.

The method clusters trefoil proteins by entanglement depth.

Topological differences are localized and geometrically interpretable.

Abstract

Quantification and classification of protein structures, such as knotted proteins, often requires noise-free and complete data. Here we develop a mathematical pipeline that systematically analyzes protein structures. We showcase this geometric framework on proteins forming open-ended trefoil knots, and we demonstrate that the mathematical tool, persistent homology, faithfully represents their structural homology. This topological pipeline identifies important geometric features of protein entanglement and clusters the space of trefoil proteins according to their depth. Persistence landscapes quantify the topological difference between a family of knotted and unknotted proteins in the same structural homology class. This difference is localized and interpreted geometrically with recent advancements in systematic computation of homology generators. The topological and geometric quantification we find is robust to noisy input data, which demonstrates the potential of this approach in contexts where standard knot theoretic tools fail.