🍩 Database of Original & Non-Theoretical Uses of Topology

1. Multidimensional Persistence in Biomolecular Data (2015)

   Kelin Xia, Guo-Wei Wei

   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.

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   • Anisotropic filtration
   • Protein flexibility
   • Protein folding
   • Point cloud
   • proteins
   • Confirm
   • Betti numbers
   • Multidimensional persistence
   • Persistent homology
   • Topological denoising

2. Atom-Specific Persistent Homology and Its Application to Protein Flexibility Analysis (2020)

   David Bramer, Guo-Wei Wei

   Abstract

   Recently, persistent homology has had tremendous success in biomolecular data analysis. It works by examining the topological relationship or connectivity of a group of atoms in a molecule at a variety of scales, then rendering a family of topological representations of the molecule. However, persistent homology is rarely employed for the analysis of atomic properties, such as biomolecular flexibility analysis or B-factor prediction. This work introduces atom-specific persistent homology to provide a local atomic level representation of a molecule via a global topological tool. This is achieved through the construction of a pair of conjugated sets of atoms and corresponding conjugated simplicial complexes, as well as conjugated topological spaces. The difference between the topological invariants of the pair of conjugated sets is measured by Bottleneck and Wasserstein metrics and leads to an atom-specific topological representation of individual atomic properties in a molecule. Atom-specific topological features are integrated with various machine learning algorithms, including gradient boosting trees and convolutional neural network for protein thermal fluctuation analysis and B-factor prediction. Extensive numerical results indicate the proposed method provides a powerful topological tool for analyzing and predicting localized information in complex macromolecules.

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   • Atomic analysis
   • Atomic properties
   • Protein flexibility
   • atomic configurations
   • proteins
   • Bottleneck distance
   • Convolutional neural network
   • Gradient boosting tree
   • Persistence diagrams
   • Persistent homology
   • Wasserstein distance