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Characterisation of the Idiotypic Immune Network Through Persistent Entropy

  • Matteo RuccoEmail author
  • Filippo Castiglione
  • Emanuela Merelli
  • Marco Pettini
Conference paper
Part of the Springer Proceedings in Complexity book series (SPCOM)

Abstract

In the present work we intend to investigate how to detect the behaviour of the immune system reaction to an external stimulus in terms of phase transitions. The immune model considered follows Jerne’s idiotypic network theory. We considered two graph complexity measures—the connectivity entropy and the approximate von Neumann entropy—and one entropy for topological spaces, the so-called persistent entropy. The simplicial complex is obtained enriching the graph structure of the weighted idiotypic network, and it is formally analyzed by persistent homology and persistent entropy. We obtained numerical evidences that approximate von Neumann entropy and persistent entropy detect the activation of the immune system. In addition, persistent entropy allows also to identify the antibodies involved in the immune memory.

Keywords

Topological data analysis Idiotypic network Complex networks Graph entropy Persistent entropy Approximate von Neumann entropy Information theory 

Notes

Acknowledgments

We acknowledge the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme (FP7) for Research of the European Commission, under the FP7 FET-Proactive Call 8—DyMCS, Grant Agreement TOPDRIM, number FP7-ICT-318121.

References

  1. 1.
    Binchi, J., Merelli, E., Rucco, M., Petri, G., Vaccarino, F.: jHoles: A tool for understanding biological complex networks via clique weight rank persistent homology. Elect. Notes Theoret. Comput. Sci. 306, 5–18 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chan, J.M., Carlsson, G., Rabadan, R.: Topology of viral evolution. Proc. Natl. Acad. Sci. 110(46), 18566–18571 (2013)Google Scholar
  3. 3.
    Chintakunta, H., Gentimis, T., Gonzalez-Diaz, R., Jimenez, M.-J., Krim, H.: An entropy-based persistence barcode. Pattern Recogn. 48(2), 391–401 (2015)CrossRefGoogle Scholar
  4. 4.
    de Silva, V., Ghrist, R.: Coverage in sensor networks via persistent homology. Algebraic Geom. Topol. 7(339–358), 24 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Soc. (2010)Google Scholar
  6. 6.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28(4), 511–533 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Felice, D., Mancini, S., Pettini, M.: Quantifying networks complexity from information geometry viewpoint. arXiv preprint arXiv:1310.7825 (2013)
  8. 8.
    Han, L., Escolano, F., Hancock, E.R., Wilson, R.C.: Graph characterizations from von neumann entropy. Pattern Recogn. Lett. 33(15), 1958–1967 (2012)CrossRefGoogle Scholar
  9. 9.
    Hoffmann, G.W.: A theory of regulation and self-nonself discrimination in an immune network. Eur. J. Immunol. 5(9), 638–647 (1975)Google Scholar
  10. 10.
    Horak, D., Maletić, S., Rajković, M.: Persistent homology of complex networks. J. Stat. Mech.: Theory Exp. 2009(03), P03034 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ibekwe, A.M., Ma, J., Crowley D.E., Yang, C.-H., Johnson, A.M., Petrossian, T.C., Lum, P.Y.: Topological data analysis of Escherichia coli O157: H7 and non-O157 survival in soils. Frontiers Cell. Infect. Microbiol. 4 (2014)Google Scholar
  12. 12.
    Jankowski, A., Skowron, A.: Practical Issues of Complex Systems Engineering: Wisdom Technology Approach (2014)Google Scholar
  13. 13.
    Jonsson, J.: Simplicial Complexes of Graphs, vol. 1928. Springer (2008)Google Scholar
  14. 14.
    Merelli, E., Pettini, M., Rasetti, M.: Topology driven modeling: the IS metaphor. Nat. Comput. 1–10 (2014)Google Scholar
  15. 15.
    Mortveit, H., Reidys, C.: An Introduction to Sequential Dynamical Systems. Springer Science & Business Media (2007)Google Scholar
  16. 16.
    Ortiz-Arroyo, D., Akbar Hussain, D.M.: An information theory approach to identify sets of key players. In: Intelligence and Security Informatics, pp. 15–26. Springer (2008)Google Scholar
  17. 17.
    Passerini, F., Severini, S.: The von Neumann entropy of networks. arXiv preprint arXiv:0812.2597 (2008)
  18. 18.
    Petri, G., Expert, P., Turkheimer, F., Carhart-Harris, R., Nutt, D., Hellyer, P.J., Vaccarino, F.: Homological scaffolds of brain functional networks. J. R. Soc. Interface 11(101), 20140873 (2014)CrossRefGoogle Scholar
  19. 19.
    Petri, G., Scolamiero, M., Donato, I., Vaccarino, F.: Topological strata of weighted complex networks. PloS ONE 8(6), e66506 (2013)ADSCrossRefGoogle Scholar
  20. 20.
    Rapin, N., Lund, O., Castiglione, F.: Immune system simulation online. Bioinformatics 27(14), 2013–2014 (2011)CrossRefGoogle Scholar
  21. 21.
    Rucco, M., Falsetti, L., Herman, D., Petrossian, T., Merelli, E., Nitti, C., Salvi, A.: Using Topological Data Analysis for diagnosis pulmonary embolism. arXiv preprint arXiv:1409.5020 (2014)
  22. 22.
    Stein, D.L., Newman, C.M.: Nature versus nurture in complex and not-so-complex systems. In: ISCS 2013: Interdisciplinary Symposium on Complex Systems, pp. 57–63. Springer (2014)Google Scholar
  23. 23.
    Tomita, E., Tanaka, A., Takahashi, H.: The worst-case time complexity for generating all maximal cliques and computational experiments. Theoret. Comput. Sci. 363(1), 28–42 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Matteo Rucco
    • 1
    Email author
  • Filippo Castiglione
    • 2
  • Emanuela Merelli
    • 1
  • Marco Pettini
    • 3
  1. 1.School of Science and TechnologyUniversity of CamerinoCamerinoItaly
  2. 2.Institute for Applied Mathematics (IAC) CNRRomeItaly
  3. 3.Centre de Physique ThéoriqueAix-Marseille UniversityMarseilleFrance

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