Journal of Statistical Physics

, Volume 144, Issue 2, pp 429–442 | Cite as

Whose Entropy: A Maximal Entropy Analysis of Phosphorylation Signaling

  • F. Remacle
  • T. G. Graeber
  • R. D. Levine


High throughput experiments, characteristic of studies in systems biology, produce large output data sets often at different time points or under a variety of related conditions or for different patients. In several recent papers the data is modeled by using a distribution of maximal information-theoretic entropy. We pose the question: ‘whose entropy’ meaning how do we select the variables whose distribution should be compared to that of maximal entropy. The point is that different choices can lead to different answers. Due to the technological advances that allow for the system-wide measurement of hundreds to thousands of events from biological samples, addressing this question is now part of the analysis of systems biology datasets. The analysis of the extent of phosphorylation in reference to the transformation potency of Bcr-Abl fusion oncogene mutants is used as a biological example. The approach taken seeks to use entropy not simply as a statistical measure of dispersion but as a physical, thermodynamic, state function. This highlights the dilemma of what are the variables that describe the state of the signaling network. Is what matters Boolean, spin-like, variables that specify whether a particular phosphorylation site is or is not actually phosphorylated. Or does the actual extent of phosphorylation matter. Last but not least is the possibility that in a signaling network some few specific phosphorylation sites are the key to the signal transduction even though these sites are not at any time abundantly phosphorylated in an absolute sense.


Information theory Prior distribution Systems biology Signal transduction High throughput experiments Phosphoproteomics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Levine, R.D.: Molecular Reaction Dynamics. Cambridge University Press, Cambridge (2005) CrossRefGoogle Scholar
  2. 2.
    Levine, R.D., Bernstein, R.B.: Energy disposal and energy consumption in elementary chemical-reactions—information theoretic approach. Acc. Chem. Res. 7, 393–400 (1974) CrossRefGoogle Scholar
  3. 3.
    Remacle, F., et al.: Information-theoretic analysis of phenotype changes in early stages of carcinogenesis. Proc. Natl. Acad. Sci. USA 107(22), 10324–10329 (2010) ADSCrossRefGoogle Scholar
  4. 4.
    Graeber, T.G., et al.: Maximal entropy inference of oncogenicity from phosphorylation signaling. Proc. Natl. Acad. Sci. USA 107(13), 6112–6117 (2010) ADSCrossRefGoogle Scholar
  5. 5.
    Levine, R.D.: Invariance and the distribution of maximal entropy. Kinam 3, 403 (1981) Google Scholar
  6. 6.
    Levine, R.D.: Dynamical symmetries. J. Phys. Chem. 89, 2122 (1985) CrossRefGoogle Scholar
  7. 7.
    Levine, R.D.: Information theoretical approach to inversion problems. J. Phys. A 13, 91–108 (1980) MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Callen, H.B.: Thermodynamics and an Introduction to Thermostatics. Wiley, New York (1985) Google Scholar
  9. 9.
    Remacle, F., Levine, R.D.: The elimination of redundant constraints in surprisal analysis of unimolecular dissociation and other endothermic processes. J. Phys. Chem. A 113(16), 4658–4664 (2009) CrossRefGoogle Scholar
  10. 10.
    Mayer, J.E., Mayer, M.G.: Statistical Mechanics. Wiley, New York (1966) Google Scholar
  11. 11.
    Margolin, A.A., Califano, A.: Theory and limitations of genetic network inference from microarray data. Ann. N.Y. Acad. Sci. 1115, 51–72 (2007) ADSCrossRefGoogle Scholar
  12. 12.
    Ziv, E., Nemenman, I., Wiggins, C.H.: Optimal signal processing in small stochastic biochemical networks. PLoS ONE 2(10), e1077 (2007) ADSCrossRefGoogle Scholar
  13. 13.
    Banavar, J.R., Maritan, A., Volkov, I.: Applications of the principle of maximum entropy: from physics to ecology. J. Phys., Condens. Matter 22(6) (2010) Google Scholar
  14. 14.
    Krawitz, P., Shmulevich, I.: Entropy of complex relevant components of Boolean networks. Phys. Rev. E 76 (2007) Google Scholar
  15. 15.
    Lezon, T.R., et al.: Using the principle of entropy maximization to infer genetic interaction networks from gene expression patterns. Proc. Natl. Acad. Sci. USA 103(50), 19033–19038 (2006) ADSCrossRefGoogle Scholar
  16. 16.
    Locasale, J.W., Wolf-Yadlin, A.: Maximum entropy reconstructions of dynamic signaling networks from quantitative proteomics data. PLoS ONE 4(8) (2009) Google Scholar
  17. 17.
    Mora, T., et al.: Maximum entropy models for antibody diversity. Proc. Natl. Acad. Sci. USA 107(12), 5405–5410 (2010) ADSCrossRefGoogle Scholar
  18. 18.
    Roudi, Y., Nirenberg, S., Latham, P.E.: Pairwise maximum entropy models for studying large biological systems: when they can work and when they can’t. PLoS Comput. Biol. 5(5) (2009) Google Scholar
  19. 19.
    Theis, F.J., Bauer, C., Lang, E.W.: Comparison of maximum entropy and minimal mutual information in a nonlinear setting. Signal Process. 82(7), 971–980 (2002) zbMATHCrossRefGoogle Scholar
  20. 20.
    Schneidman, E., et al.: Network information and connected correlations. Phys. Rev. Lett. 91, 238701 (2003) ADSCrossRefGoogle Scholar
  21. 21.
    Tkacik, G., Calan, C.G., Jr., Bialek, W.: Information flow and optimization in transcriptional regulation. Proc. Natl. Acad. Sci. USA 105, 12265–12270 (2008) ADSCrossRefGoogle Scholar
  22. 22.
    Skaggs, B.J., et al.: Phosphorylation of the ATP-binding loop directs oncogenicity of drug-resistant BCR-ABL mutants. Proc. Natl. Acad. Sci. USA 103(51), 19466–19471 (2006) ADSCrossRefGoogle Scholar
  23. 23.
    Alhassid, Y., Levine, R.D.: Experimental and inherent uncertainties in the information theoretic approach. Chem. Phys. Lett. 73(1), 16–20 (1980) MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Kinsey, J.L., Levine, R.D.: Performance criterion for information theoretic data-analysis. Chem. Phys. Lett. 65(3), 413–416 (1979) ADSCrossRefGoogle Scholar
  25. 25.
    Agmon, N., Alhassid, Y., Levine, R.D.: Algorithm for finding the distribution of maximal entropy. J. Comput. Phys. 30(2), 250–258 (1979) ADSzbMATHCrossRefGoogle Scholar
  26. 26.
    Janes, K.A., Lauffenburger, D.A.: A biological approach to computational models of proteomic networks. Curr. Opin. Chem. Biol. 10(1), 73–80 (2006) CrossRefGoogle Scholar
  27. 27.
    van den Berg, R.A., et al.: Centering, scaling, and transformations: improving the biological information content of metabolomics data. BMC Genomics 7, 142 (2006) CrossRefGoogle Scholar
  28. 28.
    Bar-Even, A., et al.: Noise in protein expression scales with natural protein abundance. Nat. Genet. 38(6), 636–643 (2006) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Département de Chimie, B6cUniversité de LiègeLiègeBelgium
  2. 2.Crump Institute for Molecular Imaging and Department of Molecular and Medical Pharmacology, David Geffen School of MedicineUniversity of CaliforniaLos AngelesUSA
  3. 3.Department of Chemistry and BiochemistryUniversity of CaliforniaLos AngelesUSA
  4. 4.Institute of ChemistryThe Hebrew University of JerusalemJerusalemIsrael

Personalised recommendations