Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis
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Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis Zixuan Cang1,2 · Elizabeth Munch1,3 · Guo-Wei Wei1,4,5 Received: 16 February 2018 / Accepted: 16 July 2020 © Springer Nature Switzerland AG 2020
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. Keywords Evolutionary homology · Local property · Dynamical systems · Protein network Mathematics Subject Classification 55N31 · 62R40 · 37N25 · 92E10
This work was supported in part by NSF Grants DMS-1721024, DMS-1761320, and IIS1900473, NIH Grant GM126189, Pfizer and Bristol-Myers Squibb. The work of EM was supported in part by NSF grants DMS-1800446, CMMI-1800466, CCF-1907591, and DEB-1904267. Extended author information available on the last page of the article
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1 Introduction Homology, a tool from traditional algebraic topology, provides an algebraic structure which encodes topological structures of different dimensions in a given space, such as connected components, closed loops, and other higher dimensional analogues (Hatcher 2002). To study topological invariants in a discrete data set, one uses the structure of the data set, such as pairwise distance information, to build a simplicial complex, which can be loosely thought of as a generalization of a graph, and then compute the homology of the complex. However, conventional homology is blind to scale, and thus retains too little geometric or physical information to be practically useful. Persistent homology, a new branch of algebraic topology, embeds multiscale information into topological invariants to achieve an interp
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