Multidimensional persistence in biomolecular data

TL;DR

This paper introduces two new types of multidimensional persistence methods to analyze biomolecular data, demonstrating their robustness and utility in applications like protein folding, cryo-EM data denoising, and nanoparticle analysis.

Contribution

The paper presents pseudo-multidimensional and multiscale multidimensional persistence, advancing topological data analysis for complex biomolecular datasets.

Findings

Topological transitions observed in protein folding.

Effective separation of noise and molecular signatures.

Multiscale analysis reveals local and global topological features.

Abstract

Persistent homology has emerged as a popular technique for the topological simplification of big data, including biomolecular data. Multidimensional persistence bears considerable promise to bridge the gap between geometry and topology. However, its practical and robust construction has been a challenge. We introduce two families of multidimensional persistence, namely pseudo-multidimensional persistence and multiscale multidimensional persistence. The former is generated via the repeated applications of persistent homology filtration to high dimensional data, such as results from molecular dynamics or partial differential equations. The latter is constructed via isotropic and anisotropic scales that create new simiplicial complexes and associated topological spaces. The utility, robustness and efficiency of the proposed topological methods are demonstrated via protein folding, protein flexibility analysis, the topological denoising of cryo-electron microscopy data, and the scale dependence of nano particles. Topological transition between partial folded and unfolded proteins has been observed in multidimensional persistence. The separation between noise topological signatures and molecular topological fingerprints is achieved by the Laplace-Beltrami flow. The multiscale multidimensional persistent homology reveals relative local features in Betti-0 invariants and the relatively global characteristics of Betti-1 and Betti-2 invariants.