Skip to main content

Advertisement

SpringerLink
Log in

Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

Delay-differential equations are commonly used to model genetic regulatory systems with the delays representing transcription and translation times. Equations with delayed terms can also be used to represent other types of chemical processes. Here we analyze delayed mass-action systems, i.e. systems in which the rates of reaction are given by mass-action kinetics, but where the appearance of products may be delayed. Necessary conditions for delay-induced instability are presented in terms both of a directed graph (digraph) constructed from the Jacobian matrix of the corresponding ODE model and of a species-reaction bipartite graph which directly represents a chemical mechanism. Methods based on the bipartite graph are particularly convenient and powerful. The condition for a delay-induced instability in this case is the existence of a subgraph of the bipartite graph containing an odd number of cycles of which an odd number are negative.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bélair J., Campbell S.A. and van den Driessche P. (1996). Frustration, stability, and delay-induced oscillations in a neural network model. SIAM J. Appl. Math. 56: 245–255

    Article  MATH  MathSciNet  Google Scholar 

  2. Bellman R.E. and Cooke K.L. (1963). Differential Difference Equations. Academic Press, New York

    MATH  Google Scholar 

  3. Biggs N. (1989). Discrete Mathematics. Oxford Science, Oxford

    MATH  Google Scholar 

  4. Bliss R.D., Painter P.R. and Marr A.G. (1982). Role of feedback inhibition in stabilizing the classical operon. J. Theor. Biol. 97: 177–193

    Article  Google Scholar 

  5. Buchholtz F. and Schneider F.W. (1987). Computer simulation of T3/T7 phage infection using lag times. Biophys. Chem. 26: 171–179

    Article  Google Scholar 

  6. Campbell S.A. (2001). Delay independent stability for additive neural networks. Diff. Eqs. Dyn. Syst. 9: 115–138

    MATH  Google Scholar 

  7. Elowitz M.B. and Leibler S. (2000). A synthetic oscillatory network of transcriptional regulators. Nature 403: 335–338

    Article  Google Scholar 

  8. Epstein I.R. (1990). Differential delay equations in chemical kinetics: some simple linear model systems. J. Phys. Chem. 92: 1702–1712

    Article  Google Scholar 

  9. Epstein I.R. and Luo Y. (1991). Differential delay equations in chemical kinetics. Nonlinear models: The cross-shaped phase diagram and the Oregonator. J. Chem. Phys. 95: 244–254

    Article  Google Scholar 

  10. Fiedler M. (1986). Special Matrices and their Applications in Numerical Mathematics. Kluwer, Dordrecht

    MATH  Google Scholar 

  11. Gantmakher F.R. (1959). Applications of the Theory of Matrices. Interscience Publishing, New York

    Google Scholar 

  12. Goldbeter A. (1995). A model for circadian oscillations in the Drosophila period protein (PER). Proc. R. Soc. Lond. B 261: 319–324

    Article  Google Scholar 

  13. Gupta A. and Markworth A.J. (2000). The Portevin-Le Châtelier effect: A description based on time delay. Phys. Status Solidi B 217: 759–768

    Article  Google Scholar 

  14. Hale J. and Verduyn Lunel S.M. (1993). Introduction to Functional Differential Equations. Springer, New York

    MATH  Google Scholar 

  15. Hofbauer J. and So J.W.-H. (2000). Diagonal dominance and harmless off-diagonal delays. Proc. Amer. Math. Soc. 128: 2675–2682

    Article  MATH  MathSciNet  Google Scholar 

  16. Jacquez J.A. and Simon C.P. (1993). Qualitative theory of compartmental systems. SIAM Rev. 35: 43–79

    Article  MATH  MathSciNet  Google Scholar 

  17. Laidler K.J. (1987). Chemical Kinetics, 3rd edn. Harper & Row, New York

    Google Scholar 

  18. Lancaster P. and Tismenetsky M. (1985). The Theory of Matrices. Academic, Orlando

    MATH  Google Scholar 

  19. MacDonald N. (1989). Biological Delay Systems: Linear Stability Theory. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  20. Mahaffy J.M. (1984). Cellular control models with linked positive and negative feedback and delays. I. The models. J. Theor. Biol. 106: 89–102

    Article  Google Scholar 

  21. Mahaffy J.M. and Pao C.V. (1984). Models of genetic control by repression with time delays and spatial effects. J. Math. Biol. 20: 39–57

    Article  MATH  MathSciNet  Google Scholar 

  22. Mincheva, M., Roussel, M.R.: Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models doi:10.1007/s00285-007-0099-1

  23. Monk N.A.M. (2003). Oscillatory expression of Hes1, p53 and NF-κB driven by transcriptional time delays. Curr. Biol. 13: 1409–1413

    Article  Google Scholar 

  24. Nazarenko V.G. and Reich J.G. (1984). Theoretical study of oscillatory and resonance phenomena in an open system with induction of enzyme by substrate. Biomed. Biochim. Acta 43: 821–828

    Google Scholar 

  25. Pieroux D. and Mandel P. (2003). Bifurcation diagram of a complex delay-differential equation with cubic nonlinearity. Phys. Rev. E 67: 056213

    Article  MathSciNet  Google Scholar 

  26. Pinney J.W., Westhead D.R. and McConkey G.A. (2003). Petri net representations in systems biology. Biochem. Soc. Trans. 31: 1513–1515

    Article  Google Scholar 

  27. Pomerening J.R., Kim S.Y. and Ferrell Jr. J.E. (2005). Systems-level dissection of the cell-cycle oscillator: Bypassing positive feedback produces damped oscillations. Cell 122: 565–578

    Article  Google Scholar 

  28. Poole R.K. (2005). Nitric oxide and nitrosative stress tolerance in bacteria. Biochem. Soc. Trans. 33: 176–180

    Article  Google Scholar 

  29. Roussel C.J. and Roussel M.R. (2001). Delay differential equations and the model equivalence problem in chemical kinetics. Phys. Can. 57: 114–120

    Google Scholar 

  30. Roussel M.R. (1996). The use of delay differential equations in chemical kinetics. J. Phys. Chem. 100: 8323–8330

    Article  Google Scholar 

  31. Roussel M.R. (1998). Approximating state-space manifolds which attract solutions of systems of delay-differential equations. J. Chem. Phys. 109: 8154–8160

    Article  Google Scholar 

  32. Rubinow S.I. (1975). Some mathematical problems in biology. Bull. Amer. Math. Soc. 81: 782–794

    Article  MATH  MathSciNet  Google Scholar 

  33. Saaty T.L. (1981). Modern Nonlinear Equations. Dover, New York

    MATH  Google Scholar 

  34. olde Scheper T., Klinkenberg D., Pennartz C. and van Pelt J. (1999). A mathematical model for the intracellular circadian rhythm generator. J. Neurosci. 19: 40–47

    Google Scholar 

  35. Smolen P., Baxter D.A. and Byrne J.H. (2001). Modeling circadian oscillations with interlocking positive and negative feedback loops. J. Neurosci. 21: 6644–6656

    Google Scholar 

  36. Stépán G. (1989). Retarded Dynamical Systems: Stability and Characteristic Functions. Longman, New York

    MATH  Google Scholar 

  37. Wang R., Jing Z. and Chen L. (2005). Modelling periodic oscillations in gene regulatory networks by cyclic feedback systems. Bull. Math. Biol. 67: 339–367

    Article  MathSciNet  Google Scholar 

  38. Zevedei-Oancea I. and Schuster S. (2005). A theoretical framework for detecting signal transfer routes in signalling networks. Comput. Chem. Eng. 29: 597–617

    Article  Google Scholar 

Download references

Author information

Author notes

  1. Maya Mincheva

    Present address: Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706-1388, USA

Authors and Affiliations

  1. Department of Chemistry and Biochemistry, University of Lethbridge, Lethbridge, AB T1K 3M4, Canada

    Maya Mincheva & Marc R. Roussel

Authors
  1. Maya Mincheva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marc R. Roussel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marc R. Roussel.

Additional information

This work was funded by the Natural Sciences and Engineering Research Council of Canada.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mincheva, M., Roussel, M.R. Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays. J. Math. Biol. 55, 87–104 (2007). https://doi.org/10.1007/s00285-007-0098-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-007-0098-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation