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A topological measurement of protein compressibility

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Abstract

In this paper we partially clarify the relation between the compressibility of a protein and its molecular geometric structure. To identify and understand the relevant topological features within a given protein, we model its molecule as an alpha filtration and hence obtain multi-scale insight into the structure of its tunnels and cavities. The persistence diagrams of this alpha filtration capture the sizes and robustness of such tunnels and cavities in a compact and meaningful manner. From these persistence diagrams, we extract a measure of compressibility derived from those topological features whose relevance is suggested by physical and chemical properties. Due to recent advances in combinatorial topology, this measure is efficiently and directly computable from information found in the Protein Data Bank (PDB). Our main result establishes a clear linear correlation between the topological measure and the experimentally-determined compressibility of most proteins for which both PDB information and experimental compressibility data are available. Finally, we establish that both the topological measurement and the linear correlation are stable with respect to small perturbations in the input data, such as those arising from experimental errors in compressibility and X-ray crystallography experiments.

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Notes

  1. From the finiteness of \({\mathsf {K}}\), we have a natural choice of \(S_{\mathcal F}\) for any filtration \({\mathcal F}\) of \({\mathsf {K}}\) since there are only finitely many indices in \([0,A]\) where new simplices get introduced.

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Acknowledgments

The authors thank Fumihide Nouno for valuable discussions. M. G. was partially supported by FAPESP Grants 2013/07460-7 and 2010/00875-9 and by CNPq Grant 306453/2009-6. Y. H. and S. I. were partially supported by JSPS Grant-in-Aid for Challenging Exploratory Research. M. K., K. M., and V. N. were partially supported by NSF Grants DMS-0915019, DMS-1125174, and CBI-0835621 and by contracts from DARPA and AFOSR.

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Authors and Affiliations

  1. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, São Carlos, SP, 13560-970, Brazil

    Marcio Gameiro

  2. Institute of Mathematics for Industry, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Yasuaki Hiraoka

  3. Department of Mathematical and Life Sciences, Hiroshima University, 1-3-1, Kagamiyama, Higashi-Hiroshima, 739-8526, Japan

    Shunsuke Izumi

  4. Department of Mathematics, Hill Center-Busch Campus Rutgers, The State University of New Jersey, 110 Frelinghusen Rd, Piscataway, NJ, 08854-8019, USA

    Miroslav Kramar

  5. Department of Mathematics and BioMaPS, Hill Center-Busch Campus Rutgers, The State University of New Jersey, 110 Frelinghusen Rd, Piscataway, NJ, 08854-8019, USA

    Konstantin Mischaikow

  6. Department of Mathematics, The University of Pennsylvania, DRL, 209 South 33rd Street, Philadelphia, PA, 19104, USA

    Vidit Nanda

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  1. Marcio Gameiro
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  2. Yasuaki Hiraoka
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  3. Shunsuke Izumi
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  5. Konstantin Mischaikow
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Correspondence to Yasuaki Hiraoka.

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Gameiro, M., Hiraoka, Y., Izumi, S. et al. A topological measurement of protein compressibility. Japan J. Indust. Appl. Math. 32, 1–17 (2015). https://doi.org/10.1007/s13160-014-0153-5

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  • DOI: https://doi.org/10.1007/s13160-014-0153-5

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