Model Reduction of Genetic-Metabolic Networks via Time Scale Separation

  • Juan Kuntz
  • Diego Oyarzún
  • Guy-Bart Stan


Model reduction techniques often prove indispensable in the analysis of physical and biological phenomena. A succesful reduction technique can substantially simplify a model while retaining all of its pertinent features. In metabolic networks, metabolites evolve on much shorter time scales than the catalytic enzymes. In this chapter, we exploit this discrepancy to justify the reduction via time scale separation of a class of models of metabolic networks under genetic regulation . We formalise the concept of a metabolic network and employ Tikhonov’s Theorem for singularly perturbed systems. We demonstrate the applicability of our result by using it to address a problem in metabolic engineering: the genetic control of branched metabolic pathways. We conclude by providing guidelines on how to generalise our result to larger classes of networks.


Time scale separation Model reduction Genetic-metabolic networks 



We thank Aivar Sootla for very useful discussions about various topics described in this chapter and Alexandros Houssein and Keshava Murthy for their valuable advice regarding how to improve this script.


  1. 1.
    Alon U (2006) An introduction to systems biology: design principles of biological circuits. Chapman and Hall/CRC, LondonGoogle Scholar
  2. 2.
    Baldazzi V, Ropers D, Geiselmann J, Kahn D, de Jong H (2012) Importance of metabolic coupling for the dynamics of gene expression following a diauxic shift in Escherichia coli. J Theor Biol 295:100–115CrossRefPubMedGoogle Scholar
  3. 3.
    Cornish-Bowden A (2004) Fundamentals of enzyme kinetics, 3rd edn. Portland Press, LondonGoogle Scholar
  4. 4.
    Craciun G, Pantea C, Sontag ED (2011) Graph theoretical analysis of multistability and monotonicity for biochemical reaction networks, vol 4. Springer, Berlin, pp 63–72Google Scholar
  5. 5.
    Feinberg M (1987) Chemical reaction network structure and the stability of complex isothermal reactors I. The deficiency zero and deficiency one theorems. Chem Eng Sci 42(10):2229–2268Google Scholar
  6. 6.
    Flach EH, Schnell S (2006) Use and abuse of the quasi-steady-state approximation. Syst Biol 153(4):187–191CrossRefGoogle Scholar
  7. 7.
    Heinrich R, Schuster S (1996) The regulation of cellular systems. Springer, BerlinGoogle Scholar
  8. 8.
    Khalil HK (2002) Nonlinear systems, 2nd edn. Prentice Hall, Englewood CliffsGoogle Scholar
  9. 9.
    Kokotovic P, Khalil HK, O’Reilly J (1986) Singular perturbation methods in control: analysis and design. Academic Press, New YorkGoogle Scholar
  10. 10.
    Kumar A, Josić K (2011) Reduced models of networks of coupled enzymatic reactions. J Theor Biol 278(1):87–106, 1101.1104Google Scholar
  11. 11.
    Lin CC, Segel LA (1988) Mathematics applied to deterministic problems in the natural sciences. SIAM, PhiladelphiaGoogle Scholar
  12. 12.
    Madigan MT, Martinko JM, Stahl DA, Clark DP (2011) Brock biology of microorganisms, 13th edn. Pearson Education, New JerseyGoogle Scholar
  13. 13.
    Nielsen J, Keasling JD (2011) Synergies between synthetic biology and metabolic engineering. Nat Biotechnol 29(8):693–695CrossRefPubMedGoogle Scholar
  14. 14.
    Oyarzún DA, Stan GB (2012) Design tradeoffs in a synthetic gene control circuit for metabolic networks. In: Proceedings of the 31st American control conference, MontrealGoogle Scholar
  15. 15.
    Oyarzún DA, Stan GB (2013) Synthetic gene circuits for metabolic control: design tradeoffs and constraints. J Royal Soc Interface 10(78):20120671Google Scholar
  16. 16.
    Oyarzún DA, Chaves M, Hoff-Hoffmeyer-Zlotnik M (2012) Multistability and oscillations in genetic control of metabolism. J Theor Biol 295:139–153CrossRefPubMedGoogle Scholar
  17. 17.
    Radde N, Bar NS, Banaji M (2010) Graphical methods for analysing feedback in biological networks. A survey. Int J Syst Sci 41(1):35–46Google Scholar
  18. 18.
    Scheer M, Grote A, Chang A, Schomburg I, Munaretto C, Rother M, Sohngen C, Stelzer M, Thiele J, Schomburg D (2011) Brenda, the enzyme information system in 2011. Nucleic Acids Res 39:D670–D676Google Scholar
  19. 19.
    Segel LA, Slemrod M (1989) The quasi-steady-state assumption: a case study in perturbation. SIAM Rev 31:446–477CrossRefGoogle Scholar
  20. 20.
    Sontag ED (1989) Remarks on stabilization and input-to-state stability. In: Proceedings of the 28th IEEE conference on decision and control. IEEE, pp 1376–1378Google Scholar
  21. 21.
    Tikhonov AN (1948) On dependence of the solutions of differential equations on a small parameter. Mat Sb 22:193–204 (in Russian)Google Scholar
  22. 22.
    Tikhonov AN (1952) Systems of differential equations containing a small parameter multiplying the derivative. Mat Sb 31:575–586 (in Russian)Google Scholar
  23. 23.
    Vasil’eva AB (1963) Asymptotic behaviour of solutions to certain problems involving non-linear differential equations containing a small parameter multiplying the highest derivatives. Uspekhi Mat Nauk 18:15–86Google Scholar
  24. 24.
    Zaslaver A, Mayo AE, Rosenberg R, Bashkin P, Sberro H, Tsalyuk M, Surette MG, Alon U (2004) Just-in-time transcription program in metabolic pathways. Nat Genet 36(5):486–491CrossRefPubMedGoogle Scholar
  25. 25.
    Zhang F, Keasling J (2011) Biosensors and their applications in microbial metabolic engineering. Trends Microbiol 19(7):323–329CrossRefPubMedGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Bioengineering, Centre for Synthetic Biology and InnovationImperial College LondonLondonUK
  2. 2.Departments of MathematicsImperial College LondonLondonUK

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