Detecting Structural Invariants in Biological Reaction Networks

  • Jörn Behre
  • Luís Filipe de Figueiredo
  • Stefan Schuster
  • Christoph KaletaEmail author
Part of the Methods in Molecular Biology book series (MIMB, volume 804)


The detection and analysis of structural invariants in cellular reaction networks is of central importance to achieve a more comprehensive understanding of metabolism. In this work, we review different kinds of structural invariants in reaction networks and their Petri net-based representation. In particular, we discuss invariants that can be obtained from the left and right null spaces of the stoichiometric matrix which correspond to conserved moieties (P-invariants) and elementary flux modes (EFMs, minimal T-invariants). While conserved moieties can be used to detect stoichiometric inconsistencies in reaction networks, EFMs correspond to a mathematically rigorous definition of the concept of a biochemical pathway. As outlined here, EFMs allow to devise strategies for strain improvement, to assess the robustness of metabolic networks subject to perturbations, and to analyze the information flow in regulatory and signaling networks. Another important aspect addressed by this review is the limitation of metabolic pathway analysis using EFMs to small or medium-scale reaction networks. We discuss two recently introduced approaches to circumvent these limitations. The first is an algorithm to enumerate a subset of EFMs in genome-scale metabolic networks starting from the EFM with the least number of reactions. The second approach, elementary flux pattern analysis, allows to analyze pathways through specific subsystems of genome-scale metabolic networks. In contrast to EFMs, elementary flux patterns much more accurately reflect the metabolic capabilities of a subsystem of metabolism as well as its integration into the entire system.

Key words

Elementary flux modes Elementary flux patterns K-Shortest elementary flux modes Petri nets Signal transduction Stoichiometric network analysis Structural robustness 



We thank Ines Heiland for the reconstruction of the medium-scale model of arginine metabolism in E. coli and for valuable discussions. Financial support from the German Ministry for Research and Education (BMBF) to C.K. and J.B. within the frameworks of the Forsys Partner initiative and the HepatoSys project, respectively, and from the Fundação Calouste Gulbenkian, Fundação para a Ciência e a Tecnologia (FCT) and Siemens SA Portugal (PhD grant SFRH/BD/32961/2006) to L.F.F. is gratefully acknowledged.


  1. 1.
    Schuster S, Fell DA, Dandekar T, (2000) A general definition of metabolic pathways useful for systematic organization and analysis of complex metabolic networks. Nat Biotechnol, 18(3):326–332.PubMedCrossRefGoogle Scholar
  2. 2.
    Palsson BØ. (2006) Systems BiologyProperties of Reconstructed Networks. Cambridge University Press, New York.Google Scholar
  3. 3.
    Price ND, Reed JL, Palsson BØ. (2004) Genome-scale models of microbial cells: evaluating the consequences of constraints. Nat Rev Microbiol, 2(11):886–897.Google Scholar
  4. 4.
    Feist AM, Palsson BØ. (2008) The growing scope of applications of genomescale metabolic reconstructions using Escherichia coli. Nat Biotechnol, 26(6):659–667.Google Scholar
  5. 5.
    Paley SM, Karp PD. (2002) Evaluation of computational metabolic-pathway predictions for Helicobacter pylori. Bioinformatics, 18(5):715–724.PubMedCrossRefGoogle Scholar
  6. 6.
    Notebaart RA, Teusink B, Siezen RJ, Papp B. (2008) Co-regulation of metabolic genes is better explained by flux coupling than by network distance. PLoS Comput Biol, 4:e26.PubMedCrossRefGoogle Scholar
  7. 7.
    Lautenbach K. (1973) Exakte Bedingungen der Lebendigkeit für eineKlasse von Petri-Netzen (in German). GMD Report, 82.Google Scholar
  8. 8.
    Murata T. (1989) Petri nets: properties, analysis and applications. Proc IEEE, 77(4):541–580.CrossRefGoogle Scholar
  9. 9.
    Starke PH. (1990) Analyse von Petri-Netz-Modellen. Teubner Verlag, Leipzig.Google Scholar
  10. 10.
    Matsuno H, Doi A, Nagasaki M, Miyano S. (2000) Hybrid Petri net representation of gene regulatory network. Pac Symp Biocomput, 5:341–352.Google Scholar
  11. 11.
    Wu J, Voit E. (2009) Hybrid modeling in biochemical systems theory by means of functional Petri nets. J Bioinform Comput Biol, 7:107–134.PubMedCrossRefGoogle Scholar
  12. 12.
    Hardy S, Robillard PN. (2004) Modeling and simulation of molecular biology systems using Petri nets: modeling goals of various approaches. J Bioinform Comput Biol, 2(4):595–613.PubMedCrossRefGoogle Scholar
  13. 13.
    Chaouiya C. (2007) Petri net modelling of biological networks. Brief Bioinform, 8(4):210–219.CrossRefGoogle Scholar
  14. 14.
    Pfeiffer T, Sánchez-Valdenebro I, Nuño JC, Montero F, Schuster S. (1999) METATOOL: for studying metabolic networks. Bioinformatics, 15(3):251–257.PubMedCrossRefGoogle Scholar
  15. 15.
    Schuster S, Dandekar T, Fell DA. (1999) Detection of elementary flux modes in biochemical networks: a promising tool for pathway analysis and metabolic engineering. Trends Biotechnol, 17(2):53–60.PubMedCrossRefGoogle Scholar
  16. 16.
    Schilling CH, Letscher D, Palsson BØ. (2000) Theory for the systemic definition of metabolic pathways and their use in interpreting metabolic function from a pathway-oriented perspective. J Theor Biol, 203(3):229–248.Google Scholar
  17. 17.
    Larhlimi A, Bockmayr A. (2009) A new constraint-based description of the steady-state flux cone of metabolic networks. Discrete Appl Math, 157(10):2257–2266.CrossRefGoogle Scholar
  18. 18.
    Papin JA, Stelling J, Price ND, Klamt S, Schuster S, Palsson BØ. (2004) Comparison of network-based pathway analysis methods. Trends Biotechnol, 22(8):400–405.Google Scholar
  19. 19.
    Trinh CT, Wlaschin A, Srienc F. (2009) Elementary mode analysis: a useful metabolic pathway analysis tool for characterizing cellular metabolism. Appl Microbiol Biotechnol, 81(5):813–826.PubMedCrossRefGoogle Scholar
  20. 20.
    Carlson R, Fell D, Srienc F. (2002) Metabolic pathway analysis of a recombinant yeast for rational strain development. Biotechnol Bioeng, 79(2):121–134.PubMedCrossRefGoogle Scholar
  21. 21.
    Trinh CT, Carlson R, Wlaschin A, Srienc F. (2006) Design, construction and performance of the most efficient biomass producing E. coli bacterium. Metab Eng, 8(6):628–638.Google Scholar
  22. 22.
    Schuster S. (2004) Metabolic Pathway Analysis in Biotechnology. In Metabolic Engineering in the Post Genomic Era. Edited by Kholodenko BN, Westerhoff HV, Horizon Scientific, Wymondham, 181–208.Google Scholar
  23. 23.
    de Graaf AA. (2000) Metabolic Flux Analysis of Corynebacterium glutamicum. In Bioreaction Engineering, Modelling and Control. Edited by Schügerl K, Springer, New York, 506–555.Google Scholar
  24. 24.
    Gayen K, Venkatesh KV. (2006) Analysis of optimal phenotypic space using elementary modes as applied to Corynebacterium glutamicum. BMC Bioinformatics, 7:445.PubMedCrossRefGoogle Scholar
  25. 25.
    Schuster S, von Kamp A, Pachkov M. (2007) Understanding the roadmap of metabolism by pathway analysis. Methods Mol Biol, 358:199–226.PubMedCrossRefGoogle Scholar
  26. 26.
    Wittmann C, Weber J, Betiku E, Krömer J, Böhm D, Rinas U. (2007) Response of fluxome and metabolome to temperature-induced recombinantprotein synthesis in Escherichia coli. J Biotechnol, 132(4):375–384.PubMedCrossRefGoogle Scholar
  27. 27.
    Kröomer JO, Wittmann C, Schröder H, Heinzle E. (2006) Metabolic pathway analysis for rational design of l-methionine production by Escherichia coli and Corynebacterium glutamicum. Metab Eng, 8(4):353–369.CrossRefGoogle Scholar
  28. 28.
    Çakir T, Tacer CS, Ülgen KO. (2004) Metabolic pathway analysis of enzymedeficient human red blood cells. Biosystems, 78(1–3):49–67.PubMedCrossRefGoogle Scholar
  29. 29.
    Schuster S, Kenanov D. (2005) Adenine and adenosine salvage pathways in erythrocytes and the role of S-adenosylhomocysteine hydrolase. A theoretical study using elementary flux modes. FEBS J, 272(20):5278–5290.Google Scholar
  30. 30.
    Förster J, Gombert AK, Nielsen J. (2002) A functional genomics approach using metabolomics and in silico pathway analysis. Biotechnol Bioeng, 79(7):703–712.PubMedCrossRefGoogle Scholar
  31. 31.
    Pachkov M, Dandekar T, Korbel J, Bork P, Schuster S. (2007) Use of pathway analysis and genome context methods for functional genomics of Mycoplasma pneumoniae nucleotide metabolism. Gene, 396(2):215–225.PubMedCrossRefGoogle Scholar
  32. 32.
    Liao JC, Hou SY, Chao YP. (1996) Pathway analysis, engineering, and physiological considerations for redirecting central metabolism. Biotechnol Bioeng, 52:129–140.CrossRefGoogle Scholar
  33. 33.
    Fischer E, Sauer U. (2003) A novel metabolic cycle catalyzes glucose oxidation and anaplerosis in hungry Escherichia coli. J Biol Chem, 278(47):46446–46451.PubMedCrossRefGoogle Scholar
  34. 34.
    Schuster S, Höfer T. (1991) Determining all extreme semi-positive conservation relations in chemical reaction systems: a test criterion for conservativity. J Chem Soc Faraday Trans, 87:2561–2566.CrossRefGoogle Scholar
  35. 35.
    Schuster S, Pfeiffer T, Moldenhauer F, Koch I, Dandekar T. (2000) Structural analysis of metabolic networks: elementary flux Modes, analogy to Petri nets, and application to Mycoplasma pneumoniae. German Conference on Bioinformatics, 115–120.Google Scholar
  36. 36.
    Gevorgyan A, Poolman MG, Fell DA. (2008) Detection of stoichiometric inconsistencies in biomolecular models. Bioinformatics, 24(19):2245–2251.PubMedCrossRefGoogle Scholar
  37. 37.
    Schuster S, Hilgetag C,Woods JH, Fell DA. (2002) Reaction routes in biochemical reaction systems: algebraic properties, validated calculation procedure and example from nucleotide metabolism. J Math Biol, 45(2):153–181.PubMedCrossRefGoogle Scholar
  38. 38.
    Clarke BL. (1981) Complete set of steady states for the general stoichiometric dynamical system. J Chem Phys, 75(10):4970–4979.CrossRefGoogle Scholar
  39. 39.
    Hofestädt R. (1996) A Petri net application to model metabolic processes. Syst Anal Model Sim, 16(2):122.Google Scholar
  40. 40.
    Reddy VN, Liebman MN, Mavrovouniotis ML. (1996) Qualitative analysis of biochemical reaction systems. Comput Biol Med, 26:9–24.PubMedCrossRefGoogle Scholar
  41. 41.
    Voss K, Heiner M, Koch I. (2003) Steady-state analysis of metabolic pathways using Petri nets. In Silico Biol, 3(3):367–387.PubMedGoogle Scholar
  42. 42.
    Colom JM, Silva M. (1991) Convex Geometry and Semiflows in P/T Nets: A Comparative Study of Algorithms for Computation of Minimal P-semiflows. In APN 90: Proceedings on Advances in Petri nets 1990. Springer-Verlag, New York, 79–112.Google Scholar
  43. 43.
    Schuster S, Pfeiffer T, Moldenhauer F, Koch I, Dandekar T. (2002) Exploring the pathway structure of metabolism: decomposition into subnetworks and application to Mycoplasma pneumoniae. Bioinformatics, 18(2):351–361.PubMedCrossRefGoogle Scholar
  44. 44.
    Sackmann A, Heiner M, Koch I. (2006) Application of Petri net based analysis techniques to signal transduction pathways. BMC Bioinformatics, 7:482.PubMedCrossRefGoogle Scholar
  45. 45.
    Burgard AP, Nikolaev EV, Schilling CH, Maranas CD. (2004) Flux coupling analysis of genome-scale metabolic network reconstructions. Genome Res, 14(2):301–312.PubMedCrossRefGoogle Scholar
  46. 46.
    Sackmann A, Formanowicz D, Formanowicz P, Koch I, Blazewicz J. (2007) An analysis of the Petri net based model of the human body iron homeostasis process. Comput Biol Chem, 31:1–10.PubMedCrossRefGoogle Scholar
  47. 47.
    Kielbassa J, Bortfeldt R, Schuster S, Koch I. (2009) Modeling of the U1 snRNP assembly pathway in alternative splicing in human cells using Petri nets. Comput Biol Chem, 33:46–61.PubMedCrossRefGoogle Scholar
  48. 48.
    Klamt S, Saez-Rodriguez J, Gilles ED. (2007) Structural and functional analysis of cellular networks with CellNetAnalyzer. BMC Syst Biol, 1:2.PubMedCrossRefGoogle Scholar
  49. 49.
    Klamt S, Gilles ED. (2004) Minimal cut sets in biochemical reaction networks. Bioinformatics, 20(2):226–234.PubMedCrossRefGoogle Scholar
  50. 50.
    Klamt S. (2006) Generalized concept of minimal cut sets in biochemical networks. Biosystems, 83(2–3):233–247.PubMedCrossRefGoogle Scholar
  51. 51.
    Terzer M, Stelling J. (2008) Large-scale computation of elementary flux modes with bit pattern trees. Bioinformatics, 24(19):2229–2235.PubMedCrossRefGoogle Scholar
  52. 52.
    Hucka M, Finney A, Sauro HM, Bolouri H, Doyle JC, Kitano H, Arkin AP, Bornstein BJ, Bray D, Cornish-Bowden A, Cuellar AA, Dronov S, Gilles ED, Ginkel M, Gor V, Goryanin II, Hedley WJ, Hodgman TC, Hofmeyr JH, Hunter PJ, Juty NS, Kasberger JL, Kremling A, Kummer U, Le Novère N, Loew LM, Lucio D, Mendes P, Minch E, Mjolsness ED, Nakayama Y, Nelson MR, Nielsen PF, Sakurada T, Schaff JC, Shapiro BE, Shimizu TS, Spence HD, Stelling J, Takahashi K, Tomita M, Wagner J, Wang J. (2003) The Systems Biology Markup Language (SBML): A medium for representation and exchange of biochemical network models. Bioinformatics, 19(4):524–531.PubMedCrossRefGoogle Scholar
  53. 53.
    von Kamp A, Schuster S. (2006) Metatool 5.0: fast and exible elementary modes analysis. Bioinformatics, 22(15):1930–1931.Google Scholar
  54. 54.
    Sauro HM, Hucka M, Finney A, Wellock C, Bolouri H, Doyle J, Kitano H. (2003) Next generation simulation tools: the Systems Biology Workbench and BioSPICE integration. OMICS, 7(4):355–372.PubMedCrossRefGoogle Scholar
  55. 55.
    Keating SM, Bornstein BJ, Finney A, Hucka M. (2006) SBMLToolbox: an SBML toolbox for MATLAB users. Bioinformatics, 22(10):1275–1277.PubMedCrossRefGoogle Scholar
  56. 56.
    Poolman MG. (2006): ScrumPy: metabolic modelling with Python. IEE Proc Syst Biol, 153(5):375–378.PubMedCrossRefGoogle Scholar
  57. 57.
    Schwarz R, Liang C, Kaleta C, Kühnel M, Hoffmann E, Kuznetsov S, Hecker M, Griffiths G, Schuster S, Dandekar T. (2007) Integrated network reconstruction, visualization and analysis using YANAsquare. BMC Bioinformatics, 8:313.CrossRefGoogle Scholar
  58. 58.
    Kanehisa M, Araki M, Goto S, Hattori M, Hirakawa M, Itoh M, Katayama T, Kawashima S, Okuda S, Tokimatsu T, Yamanishi Y. (2008) KEGG for linking genomes to life and the environment. Nucleic Acids Res, 36(Database issue):D480–D484.Google Scholar
  59. 59.
    Heiner M, Richter R, Schwarick M, Rohr C. (2008) Snoopy – a tool to design and execute graph-based formalisms. Petri Net Newsletter, 74:8–22.Google Scholar
  60. 60.
    Rohr C, Marwan W, Heiner M. (2010) Snoopy – a unifying Petri net framework to investigate biomolecular networks. Bioinformatics, 26(7):974–975.Google Scholar
  61. 61.
    Kitano H. (2004) Biological robustness. Nat Rev Genet, 5(11):826–837.PubMedCrossRefGoogle Scholar
  62. 62.
    Wolf J, Becker-Weimann S, Heinrich R. (2005) Analysing the robustness of cellular rhythms. IEE Proc Syst Biol, 2:35–41.Google Scholar
  63. 63.
    Jacobsen EW, Cedersund G. (2008) Structural robustness of biochemical network models-with application to the oscillatory metabolism of activated neutrophils. IET Syst Biol, 2:39–47.PubMedCrossRefGoogle Scholar
  64. 64.
    Stelling J, Klamt S, Bettenbrock K, Schuster S, Gilles ED. (2002) Metabolic network structure determines key aspects of functionality and regulation. Nature, 420(6912):190–193.PubMedCrossRefGoogle Scholar
  65. 65.
    Wilhelm T, Behre J, Schuster S. (2004) Analysis of structural robustness of metabolic networks. IEE Proc Syst Biol, 1:114–120.Google Scholar
  66. 66.
    Behre J, Wilhelm T, von Kamp A, Ruppin E, Schuster S. (2008) Structural robustness of metabolic networks with respect to multiple knockouts. J Theor Biol, 252(3):433–441.PubMedCrossRefGoogle Scholar
  67. 67.
    Feist AM, Henry CS, Reed JL, Krummenacker M, Joyce AR, Karp PD, Broadbelt LJ, Hatzimanikatis V, Palsson BØ. (2007) A genome-scale metabolic reconstruction for Escherichia coli K-12 MG1655 that accountsfor 1260 ORFs and thermodynamic information. Mol Syst Biol, 3:121.PubMedCrossRefGoogle Scholar
  68. 68.
    Papin JA, Palsson BØ. (2004) Topological analysis of mass-balanced signaling networks: a framework to obtain network properties including crosstalk. J Theor Biol, 227(2):283–297.Google Scholar
  69. 69.
    Zevedei-Oancea I, Schuster S. (2005) A theoretical framework for detecting signal transfer routes in signalling networks. Comput Chem Eng, 29(3):597–617.CrossRefGoogle Scholar
  70. 70.
    Klamt S, Saez-Rodriguez J, Lindquist JA, Simeoni L, Gilles ED. (2006) A methodology for the structural and functional analysis of signaling and regulatory networks. BMC Bioinformatics, 7:56.PubMedCrossRefGoogle Scholar
  71. 71.
    Xiong M, Zhao J, Xiong H. (2004) Network-based regulatory pathways analysis. Bioinformatics, 20(13):2056–2066.PubMedCrossRefGoogle Scholar
  72. 72.
    Behre J, Schuster S. (2009) Modeling signal transduction in enzyme cascades with the concept of elementary flux modes. J Comput Biol, 16(6):829–844.PubMedCrossRefGoogle Scholar
  73. 73.
    Gianchandani EP, Papin JA, Price ND, Joyce AR, Palsson BØ. (2006) Matrix formalism to describe functional states of transcriptional regulatory systems. PLoS Comput Biol, 2(8 e101):0902–0917.Google Scholar
  74. 74.
    Varma A, Palsson BØ. (1994) Metabolic flux balancing: basic concepts, scientific and practical use. Bio/Technology, 12(10):994–998.Google Scholar
  75. 75.
    Acuña V, Marchetti-Spaccamela A, Sagot MF, Stougie L. (2010) A note on the complexity of finding and enumerating elementary modes. Biosystems, 99(3):210–214.CrossRefGoogle Scholar
  76. 76.
    de Figueiredo LF, Podhorski A, Rubio A, Beasley JE, Schuster S, Planes FJ. (2009) Calculating the K-Shortest Elementary Flux Modes in Metabolic Networks. In Proceedings of the MATHMOD 2009 in Vienna. Edited by Troch I, Breitenecker F, 736–747.Google Scholar
  77. 77.
    Dandekar T, Moldenhauer F, Bulik S, Bertram H, Schuster S. (2003) A method for classifying metabolites in topological pathway analyses based on minimization of pathway number. Biosystems, 70(3):255–270.PubMedCrossRefGoogle Scholar
  78. 78.
    Schwartz JM, Gaugain C, Nacher J, de Daruvar A, Kanehisa M. (2007) Observing metabolic functions at the genome scale. Genome Biol, 8(6):R123.PubMedCrossRefGoogle Scholar
  79. 79.
    Kaleta C, de Figueiredo LF, Schuster S. (2009) Can the whole be less than the sum of its parts? Pathway analysis in genome-scale metabolic networks using elementary flux patterns. Genome Res, 19(10):1872–1883.PubMedCrossRefGoogle Scholar
  80. 80.
    Oh MK, Rohlin L, Kao KC, Liao JC. (2002) Global expression profiling of acetate-grown Escherichia coli. J Biol Chem, 277(15):13175–13183.PubMedCrossRefGoogle Scholar
  81. 81.
    Croes D, Couche F, Wodak SJ, van Helden J. (2006) Inferring meaningful pathways in weighted metabolic networks. J Mol Biol, 356:222–236.PubMedCrossRefGoogle Scholar
  82. 82.
    Planes FJ, Beasley JE. (2009) Path finding approaches and metabolic pathways. Discrete Appl Math, 157(10):2244–2256.CrossRefGoogle Scholar
  83. 83.
    de Figueiredo LF, Schuster S, Kaleta C, Fell DA. (2009) Can sugars be produced from fatty acids? A test case for pathway analysis tools. Bioinformatics, 25:152–158.PubMedCrossRefGoogle Scholar
  84. 84.
    Kacser H, Acerenza L. (1993) A universal method for achieving increases in metabolite production. Eur J Biochem, 216(2):361–367.PubMedCrossRefGoogle Scholar
  85. 85.
    Niederberger P, Prasad R, Miozzari G, Kacser H. (1992) A strategy for increasing an in vivo flux by genetic manipulations. The tryptophan system of yeast. Biochem J, 287(Pt 2):473–479.Google Scholar
  86. 86.
    Meléndez-Hevia E, Waddell TG, Montero F. (1994) Optimization of metabolism: the evolution of metabolic pathways toward simplicity through the game of the pentose phosphate cycle. J Theor Biol, 166(2):201–220.CrossRefGoogle Scholar
  87. 87.
    Wu WH, Morris DR. (1973) Biosynthetic arginine decarboxylase from Escherichia coli. Purification and properties. J Biol Chem, 248(5):1687–1695.Google Scholar
  88. 88.
    Blethen SL, Boeker EA, Snell EE. (1968) Argenine decarboxylase from Escherichia coli. I. Purification and specificity for substrates and coenzyme. J Biol Chem, 243(8):1671–1677.Google Scholar
  89. 89.
    Schuster S, Klamt S, Weckwerth W, Pfeiffer T. (2002) Use of network analysis of metabolic systems in bioengineering. Bioproc Biosyst Eng, 24:363–372.CrossRefGoogle Scholar
  90. 90.
    Reed JL, Palsson BØ. (2004) Genome-scale in silico models of E. coli have multiple equivalent phenotypic states: assessment of correlated reaction subsets that comprise network states. Genome Res, 14(9):1797–1805.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Jörn Behre
    • 1
  • Luís Filipe de Figueiredo
    • 2
  • Stefan Schuster
    • 3
  • Christoph Kaleta
    • 3
    Email author
  1. 1.Theoretical Systems BiologyInstitute of Food Research (IFR), Norwich Research ParkNorwichUK
  2. 2.Chemoinformatics and Metabolism TeamEuropean Bioinformatics Institute (EBI)CambridgeshireUK
  3. 3.Department of BioinformaticsFriedrich Schiller University JenaJenaGermany

Personalised recommendations