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Theory in Biosciences

, Volume 126, Issue 1, pp 15–21 | Cite as

Spectral plots and the representation and interpretation of biological data

  • Anirban BanerjeeEmail author
  • Jürgen Jost
Original Paper

Abstract

It is basic question in biology and other fields to identify the characteristic properties that on one hand are shared by structures from a particular realm, like gene regulation, protein–protein interaction or neural networks or foodwebs, and that on the other hand distinguish them from other structures. We introduce and apply a general method, based on the spectrum of the normalized graph Laplacian, that yields representations, the spectral plots, that allow us to find and visualize such properties systematically. We present such visualizations for a wide range of biological networks and compare them with those for networks derived from theoretical schemes. The differences that we find are quite striking and suggest that the search for universal properties of biological networks should be complemented by an understanding of more specific features of biological organization principles at different scales.

Keywords

Biological network  Laplacian spectrum Graph spectrum Spectral plot Metabolic network Transcription network Protein–protein interaction network Neuronal network Foodweb 

References

  1. Albert R, Barabási A-L (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47–97CrossRefGoogle Scholar
  2. Atay FM, Jost J, Wende A (2004) Delays, connection topology, and synchronization of coupled chaotic maps. Phys Rev Lett 92(14):144101PubMedCrossRefGoogle Scholar
  3. Atay FM, Bıyıkoğlu T, Jost J (2006) Synchronization of networks with prescribed degree distributions. IEEE Trans Circuits Syst I 53(1):92–98CrossRefGoogle Scholar
  4. Atay FM, Bıyıkoğlu T, Jost J (2007) Network synchronization: spectral versus statistical properties. Phys D (to appear)Google Scholar
  5. Banerjee A, Jost J (2007a) Laplacian spectrum and protein–protein interaction networks (preprint)Google Scholar
  6. Banerjee A, Jost J (2007b) On the spectrum of the normalized graph Laplacian (preprint)Google Scholar
  7. Banerjee A, Jost J (2007c) Graph spectra as a systematic tool in computational biology. Discrete Appl Math (submitted)Google Scholar
  8. Barabási A-L, Albert RA (1999) Emergence of scaling in random networks. Science 286:509–512PubMedCrossRefGoogle Scholar
  9. Bolobás B (1998) Modern graph theory. Springer, BerlinGoogle Scholar
  10. Bolobás B (2001) Random graphs. Cambridge University Press, LondonGoogle Scholar
  11. Chung F (1997) Spectral graph theory. AMS, New YorkGoogle Scholar
  12. Dorogovtsev SN, Mendes JFF (2003) Evolution of Networks. Oxford University Press, OxfordGoogle Scholar
  13. Erdős P, Rényi A (1959) On random graphs. Public Math Debrecen 6:290–297Google Scholar
  14. Godsil C, Royle G (2001) Algebraic graph theory. Springer, BerlinGoogle Scholar
  15. Ipsen M, Mikhailov AS (2002) Evolutionary reconstruction of networks. Phys Rev E 66(4):046109Google Scholar
  16. Jeong H, Tombor B, Albert R, Oltval ZN, Barabási AL (2000) The large-scale organization of metabolic networks. Nature 407(6804):651–654PubMedCrossRefGoogle Scholar
  17. Jost J (2007a) Mathematical methods in biology and neurobiology, monograph (to appear)Google Scholar
  18. Jost J (2007b) Dynamical networks. In: Feng JF, Jost J, Qian MP (eds) Networks: from biology to theory. Springer, BerlinGoogle Scholar
  19. Jost J, Joy MP (2001) Spectral properties and synchronization in coupled map lattices. Phys Rev E 65(1):16201–16209CrossRefGoogle Scholar
  20. Jost J, Joy MP (2002) Evolving networks with distance preferences. Phys Rev E 66:36126–36132CrossRefGoogle Scholar
  21. Merris R (1994) Laplacian matrices of graphs—a survey. Lin Alg Appl 198:143–176CrossRefGoogle Scholar
  22. Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii D, Alon U (2002) Network Motifs: simple building blocks of complex networks. Science 298(5594):824–827PubMedCrossRefGoogle Scholar
  23. Mohar B (1997) Some applications of Laplace eigenvalues of graphs. In: Hahn G, Sabidussi G (eds) Graph symmetry: algebraic methods and applications. Springer, Berlin, pp 227–277Google Scholar
  24. Newman M (2003) The structure and function of complex networks. SIAM Rev 45:167–256CrossRefGoogle Scholar
  25. Ohno S (1970) Evolution by gene duplication. Springer, BerlinGoogle Scholar
  26. Shen-Orr SS, Milo R, Mangan S, Alon U (2002) Network Motifs in the transcriptional regulation network of Escherichia coli. Nat Genet 31(1):64–68PubMedCrossRefGoogle Scholar
  27. Simon H (1955) On a class of skew distribution functions. Biometrika 42:425–440Google Scholar
  28. Vázquez A (2002) Growing networks with local rules: Preferential attachment, clustering hierarchy and degree correlations. cond-mat/0211528Google Scholar
  29. Wagner A (1994) Evolution of gene networks by gene duplications—a mathematical model and its implications on genome organization. Proc Nat Acad Sci USA 91(10):4387–4391PubMedCrossRefGoogle Scholar
  30. Watts DJ, Strogatz SH (1998) Collective dynamics of ‘Small-World’ networks. Nature 393(6684):440–442PubMedCrossRefGoogle Scholar
  31. White JG, Southgate E, Thomson JN, Brenner S (1986) The structure of the nervous-system of the Nematode Caenorhabditis-Elegans. Philos Trans R Soc Lond Ser B Biol Sci 314(1165):1–340Google Scholar
  32. Wolfe KH, Shields DC (1997) Molecular evidence for an ancient duplication of the entire yeast genome. Nature 387(6634):708–713PubMedCrossRefGoogle Scholar
  33. Zhu P, Wilson R (2005) A study of graph spectra for comparing graphs. In: British machine vision conference 2005, BMVA, pp 679–688Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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