Fast Flux Module Detection Using Matroid Theory

  • Arne C. Müller
  • Frank J. Bruggeman
  • Brett G. Olivier
  • Leen Stougie
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8394)


Flux balance analysis (FBA) is one of the most often applied methods on genome-scale metabolic networks. Although FBA uniquely determines the optimal yield, the pathway that achieves this is usually not unique. The analysis of the optimal-yield flux space has been an open challenge. Flux variability analysis is only capturing some properties of the flux space, while elementary mode analysis is intractable due to the enormous number of elementary modes. However, it has been found by Kelk et al. 2012, that the space of optimal-yield fluxes decomposes into flux modules. These decompositions allow a much easier but still comprehensive analysis of the optimal-yield flux space.

Using the mathematical definition of module introduced by Müller and Bockmayr 2013, we discovered that flux modularity is rather a local than a global property which opened connections to matroid theory. Specifically, we show that our modules correspond one-to-one to so-called separators of an appropriate matroid. Employing efficient algorithms developed in matroid theory we are now able to compute the decomposition into modules in a few seconds for genome-scale networks. Using that every module can be represented by one reaction that represents its function, in this paper, we also present a method that uses this decomposition to visualize the interplay of modules. We expect the new method to replace flux variability analysis in the pipelines for metabolic networks.


metabolic networks FBA flux modules matroid theory 


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  1. 1.
    Avis, D., Fukuda, K.: A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete & Computational Geometry 8(1), 295–313 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Beard, D.A., Babson, E., Curtis, E., Qian, H.: Thermodynamic constraints for biochemical networks. Journal of Theoretical Biology 228, 327–333 (2004)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Burgard, A.P., Vaidyaraman, S., Maranas, C.D.: Minimal reaction sets for escherichia coli metabolism under different growth requirements and uptake environments. Biotechnology Progress 17, 791–797 (2001)CrossRefGoogle Scholar
  4. 4.
    Cunningham, W.H.: A combinatorial decomposition theory. PhD thesis, University of Waterloo, Ontario, Canada (1973)Google Scholar
  5. 5.
    Feist, A.M., Palsson, B.Ø.: The biomass objective function. Current Opinion in Microbiology 13, 344–349 (2010)CrossRefGoogle Scholar
  6. 6.
    Fukuda, K., Prodon, A.: Double description method revisited. In: Deza, M., Manoussakis, I., Euler, R. (eds.) CCS 1995. LNCS, vol. 1120, pp. 91–111. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  7. 7.
    Gansner, E.R., Koutsofios, E., North, S.C., Vo, K.-P.: A technique for drawing directed graphs. IEEE Transactions on Software Engineering 19(3), 214–230 (1993)CrossRefGoogle Scholar
  8. 8.
    Gansner, E.R., North, S.C.: An open graph visualization system and its applications to software engineering. Software - Practice and Experience 30(11), 1203–1233 (2000)CrossRefzbMATHGoogle Scholar
  9. 9.
    Kelk, S.M., Olivier, B.G., Stougie, L., Bruggeman, F.J.: Optimal flux spaces of genome-scale stoichiometric models are determined by a few subnetworks. Scientific Reports 2, 580 (2012)CrossRefGoogle Scholar
  10. 10.
    Khachiyan, L., Boros, E., Borys, K., Elbassioni, K., Gurvich, V.: Generating all vertices of a polyhedron is hard. Discrete Computational Geometry 39, 174–190 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Khannapho, C., Zhao, H., Bonde, B.L., Kierzek, A.M., Avignone-Rossa, C.A., Bushell, M.E.: Selection of objective function in genome scale flux balance analysis for process feed development in antibiotic production. Metabolic Engineering 10(5), 227–233 (2008)CrossRefGoogle Scholar
  12. 12.
    Krogdahl, S.: The dependence graph for bases in matroids. Discrete Mathematics 19, 47–59 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Mahadevan, R., Schilling, C.H.: The effects of alternate optimal solutions in constraint-based genome-scale metabolic models. Metabolic Engineering 5, 264–276 (2003)CrossRefGoogle Scholar
  14. 14.
    Müller, A.C., Bockmayr, A.: Fast thermodynamically constrainted flux variability analysis. Bioinformatics 29(7), 903–909 (2013)CrossRefGoogle Scholar
  15. 15.
    Müller, A.C., Bockmayr, A.: Flux modules in metabolic networks. Journal of Mathematical Biology (2013) (in press, preprint),
  16. 16.
    Oliveira, J.S., Bailey, C.G., Jones-Oliveira, J.B., Dixon, D.A.: An algebraic-combinatorial model for the identification and mapping of biochemical pathways. Bulletin of Mathematical Biology 63, 1163–1196 (2001)CrossRefGoogle Scholar
  17. 17.
    Orth, J.D., Thiele, I., Palsson, B.Ø.: What is flux balance analysis. Nature Biotechnology 28, 245–248 (2010)CrossRefGoogle Scholar
  18. 18.
    Oxley, J.: Matroid Theory, 2nd edn. Oxford Graduate Texts in Mathematics. Oxford University Press, New York (2011)CrossRefzbMATHGoogle Scholar
  19. 19.
    Jason Papin, A., Stelling, J., Price, N.D., Klamt, S., Schuster, S., Palsson, B.Ø.: Palsson. Comparison of network-based pathway analysis methods. Trends in Biotechnology 22(8), 400–405 (2004)CrossRefGoogle Scholar
  20. 20.
    Price, N.D., Reed, J.L., Palsson, B.Ø.: Genome-scale models of microbial cells: evaluating the consequences of constraints. Nature Reviews Microbiology 2, 886–897 (2004)CrossRefGoogle Scholar
  21. 21.
    Santos, F., Boele, J., Teusink, B.: A practical guide to genome-scale metabolic models and their analysis. Methods in Enzymology 500, 509 (2011)CrossRefGoogle Scholar
  22. 22.
    Schellenberger, J., Que, R., Fleming, R.M.T., Thiele, I., Orth, J.D., Feist, A.M., Zielinski, D.C., Bordbar, A., Lewis, N.E., Rahmanian, S., Kang, J., Hyduke, D.R., Palsson, B.Ø.: Quantitative prediction of cellular metabolism with constraint-based models: the COBRA toolbox v2.0. Nature Protocols 6(9), 1290–1307 (2011)CrossRefGoogle Scholar
  23. 23.
    Schuster, S., Hilgetag, C.: On elementary flux modes in biochemical systems at steady state. J. Biol. Systems 2, 165–182 (1994)CrossRefGoogle Scholar
  24. 24.
    Schuster, S., Schuster, R.: Detecting strictly detailed balanced subnetworks in open chemical reaction networks. Journal of Mathematical Chemistry 6, 17–40 (1991)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Terzer, M.: Large scale methods to enumerate extreme rays and elementary modes. PhD thesis, Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 18538, 2009 (2009)Google Scholar
  26. 26.
    Terzer, M., Stelling, J.: Large-scale computation of elementary flux modes with bit pattern trees. Bioinformatics 24(19), 2229–2235 (2008)CrossRefGoogle Scholar
  27. 27.
    Truemper, K.: Partial matroid representations. European Journal of Combinatorics (1984)Google Scholar
  28. 28.
    Varma, A., Palsson, B.Ø.: Metabolic flux balancing: Basic concepts, scientific and practical use. Nature Biotechnology 12, 994–998 (1994)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Arne C. Müller
    • 1
    • 2
    • 3
  • Frank J. Bruggeman
    • 8
  • Brett G. Olivier
    • 4
    • 5
    • 7
  • Leen Stougie
    • 4
    • 6
  1. 1.Department of Mathematics and Computer ScienceFreie Universität BerlinBerlinGermany
  2. 2.International Max Planck Research School for Computational Biology and Scientific Computing (IMPRS-CBSC)Max Planck Institute for Molecular GeneticsBerlinGermany
  3. 3.Berlin Mathematical School (BMS)BerlinGermany
  4. 4.Centre for Mathematics and Computer Science (CWI)AmsterdamThe Netherlands
  5. 5.Molecular Cell PhysiologyVU UniversityAmsterdamThe Netherlands
  6. 6.Operations ResearchVU UniversityAmsterdamThe Netherlands
  7. 7.Netherlands Institute for Systems BiologyAmsterdamThe Netherlands
  8. 8.Systems BioinformaticsVU UniversityAmsterdamThe Netherlands

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