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Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis

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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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Notes

  1. Here, traditional means the Vietoris–Rips filtration on the point cloud induced by the embedding.

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  1. Zixuan Cang

    Present address: Department of Mathematics, University of California Irvine, Irvine, USA

Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, USA

    Zixuan Cang, Elizabeth Munch & Guo-Wei Wei

  2. Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, USA

    Elizabeth Munch

  3. Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, USA

    Guo-Wei Wei

  4. Department of Electrical and Computer Engineering, Michigan State University, East Lansing, USA

    Guo-Wei Wei

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  1. Zixuan Cang
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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.

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Cang, Z., Munch, E. & Wei, GW. Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis. J Appl. and Comput. Topology 4, 481–507 (2020). https://doi.org/10.1007/s41468-020-00057-9

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