Multidimensional persistence in biomolecular data

• Published in: Journal of computational chemistry Vol. 36; no. 20; pp. 1502 - 1520
• Main Authors: Xia, Kelin, Wei, Guo-Wei
• Format: Journal Article
• Language: English
• Published: United States Blackwell Publishing Ltd 30.07.2015; Wiley Subscription Services, Inc
• Subjects: Algorithms; anisotropic filtration; Anisotropy; Molecular Dynamics Simulation; multidimensional persistence; multifiltration; multiscale persistence; Nanoparticles; Partial differential equations; protein flexibility; Protein Folding; Proteins; Proteins - chemistry; topological denoising; anisotropic filtration; multiscale persistence; multidimensional persistence; protein flexibility; protein folding; multifiltration; topological denoising
• ISSN: 0192-8651; 1096-987X
• DOI: 10.1002/jcc.23953

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 pseudomultidimensional 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 cryoelectron microscopy data, and the scale dependence of nanoparticles. 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. © 2015 Wiley Periodicals, Inc. Persistent homology has emerged as a popular technique for the topological simplification of massive biomolecular data. Resolution‐based multidimensional persistent homology is introduced to bridge the gap between traditional topology and geometry. The utility, robustness, and efficiency of the proposed topological methods for protein folding, protein flexibility analysis, the topological denoising of cryo‐electron microscopy data, and the scale dependence of nanoparticles are demonstrated.