🍩 Database of Original & Non-Theoretical Uses of Topology

1. Evolutionary Homology on Coupled Dynamical Systems With Applications to Protein Flexibility Analysis (2020)

   Zixuan Cang, Elizabeth Munch, Guo-Wei Wei

   Abstract

   While the spatial topological persistence is naturally constructed from a radius-based filtration, it has hardly been derived from a temporal filtration. Most topological models are designed for the global topology of a given object as a whole. There is no method reported in the literature for the topology of an individual component in an object to the best of our knowledge. For many problems in science and engineering, the topology of an individual component is important for describing its properties. We propose evolutionary homology (EH) constructed via a time evolution-based filtration and topological persistence. Our approach couples a set of dynamical systems or chaotic oscillators by the interactions of a physical system, such as a macromolecule. The interactions are approximated by weighted graph Laplacians. Simplices, simplicial complexes, algebraic groups and topological persistence are defined on the coupled trajectories of the chaotic oscillators. The resulting EH gives rise to time-dependent topological invariants or evolutionary barcodes for an individual component of the physical system, revealing its topology-function relationship. In conjunction with Wasserstein metrics, the proposed EH is applied to protein flexibility analysis, an important problem in computational biophysics. Numerical results for the B-factor prediction of a benchmark set of 364 proteins indicate that the proposed EH outperforms all the other state-of-the-art methods in the field.

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   • Biology
   • Protein
   • Protein folding
   • Dynamical system
   • Residues alpha carbon
   • Innovate
   • Evolutionary homology

2. Representability of Algebraic Topology for Biomolecules in Machine Learning Based Scoring and Virtual Screening (2018)

   Zixuan Cang, Lin Mu, Guo-Wei Wei

   Abstract

   This work introduces a number of algebraic topology approaches, including multi-component persistent homology, multi-level persistent homology, and electrostatic persistence for the representation, characterization, and description of small molecules and biomolecular complexes. In contrast to the conventional persistent homology, multi-component persistent homology retains critical chemical and biological information during the topological simplification of biomolecular geometric complexity. Multi-level persistent homology enables a tailored topological description of inter- and/or intra-molecular interactions of interest. Electrostatic persistence incorporates partial charge information into topological invariants. These topological methods are paired with Wasserstein distance to characterize similarities between molecules and are further integrated with a variety of machine learning algorithms, including k-nearest neighbors, ensemble of trees, and deep convolutional neural networks, to manifest their descriptive and predictive powers for protein-ligand binding analysis and virtual screening of small molecules. Extensive numerical experiments involving 4,414 protein-ligand complexes from the PDBBind database and 128,374 ligand-target and decoy-target pairs in the DUD database are performed to test respectively the scoring power and the discriminatory power of the proposed topological learning strategies. It is demonstrated that the present topological learning outperforms other existing methods in protein-ligand binding affinity prediction and ligand-decoy discrimination.

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   • Chemistry
   • Electrostatics
   • Machine learning
   • Molecular biology
   • Molecule classification
   • Neural network
   • Neural Networks
   • Protein structures
   • Electrostatic persistence
   • Machine learning
   • Multi-component persistent homology
   • Multi-level persistent homology
   • Multilevel persistence
   • Persistent homology