Bulletin of Mathematical Biology

, Volume 73, Issue 10, pp 2277–2304 | Cite as

Oscillations in Biochemical Reaction Networks Arising from Pairs of Subnetworks

Original Article


Biochemical reaction models show a variety of dynamical behaviors, such as stable steady states, multistability, and oscillations. Biochemical reaction networks with generalized mass action kinetics are represented as directed bipartite graphs with nodes for species and reactions. The bipartite graph of a biochemical reaction network usually contains at least one cycle, i.e., a sequence of nodes and directed edges which starts and ends at the same species node. Cycles can be positive or negative, and it has been shown that oscillations can arise as a result of either a positive cycle or a negative cycle. In earlier work it was shown that oscillations associated with a positive cycle can arise from subnetworks with an odd number of positive cycles. In this article we formulate a similar graph-theoretic condition, which generalizes the negative cycle condition for oscillations. This new graph-theoretic condition for oscillations involves pairs of subnetworks with an even number of positive cycles. An example of a calcium reaction network with generalized mass action kinetics is discussed in detail.


Biochemical and chemical reaction networks Bipartite graph Generalized mass action kinetics Oscillations Negative feedback cycle 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alon, U. (2007). Network motifs: theory and experimental approaches. Nat. Rev. Genet., 8, 450–461. CrossRefGoogle Scholar
  2. Angeli, D., De Leenheer, P., & Sontag, E. D. (2006). On the structural monotonicity of chemical reaction networks. In 45th IEEE conference on decision and control, pp. 7–12. CrossRefGoogle Scholar
  3. Angeli, D., Ferrell, J. E., & Sontag, E. D. (2004). Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems. Proc. Natl. Acad. Sci. USA, 101, 1822–1827. CrossRefGoogle Scholar
  4. Asner, B. (1970). On the total nonnegativity of the Hurwitz matrix. SIAM J. Appl. Math., 18, 407–418. MathSciNetMATHCrossRefGoogle Scholar
  5. Chickarmane, V., Kholodenko, B., & Sauro, H. (2007). Oscillatory dynamics arising from competitive inhibition and multisite phosphorylation. J. Theor. Biol., 244, 68–76. MathSciNetCrossRefGoogle Scholar
  6. Clarke, B. (1980). Stability of complex reaction networks. Adv. Chem. Phys., 43, 1–213. CrossRefGoogle Scholar
  7. Clarke, B., & Jiang, W. (1993). Method for deriving Hopf and saddle-node bifurcation hypersurfaces and application to a model of the Belousov–Zhabotinskii system. J. Chem. Phys., 99, 4464–4478. CrossRefGoogle Scholar
  8. Craciun, G., & Feinberg, M. (2005). Multiple equilibria in complex chemical reaction networks, I: the injectivity property. SIAM J. Appl. Math., 65, 1526–1546. MathSciNetMATHCrossRefGoogle Scholar
  9. Craciun, G., & Feinberg, M. (2006). Multiple equilibria in complex chemical reaction networks, II: the species-reactions graph. SIAM J. Appl. Math., 66, 1321–1338. MathSciNetMATHCrossRefGoogle Scholar
  10. Craciun, G., Tang, Y., & Feinberg, M. (2006). Understanding bistability in complex enzyme-driven reaction networks. Proc. Natl. Acad. Sci. USA, 103(23), 8697–8702. CrossRefGoogle Scholar
  11. Elowitz, M., & Leibler, S. (2000). A synthetic oscillatory network of transcriptional regulators. Nature, 403, 335–338. CrossRefGoogle Scholar
  12. Fallat, S. (2001). Bidiagonal factorization of totally nonnegative matrices. Am. Math. Mon., 108, 697–712. MathSciNetMATHCrossRefGoogle Scholar
  13. Forger, D., & Peskin, C. (2003). A detailed predictive model of the mammalian circadian clock. Proc. Natl. Acad. Sci. USA, 100, 14806–14811. CrossRefGoogle Scholar
  14. Gantmacher, F. R. (1959). Applications of the theory of matrices. New York: Interscience. MATHGoogle Scholar
  15. Gatermann, K., Eiswirth, M., & Sensse, A. (2005). Toric ideals and graph theory to analyze Hopf bifurcation in mass action systems. J. Symb. Comput., 40, 1361–1382. MathSciNetMATHCrossRefGoogle Scholar
  16. Goldbeter, A. (1995). A model for circadian oscillations in the Drosophila period protein (PER). Proc. R. Soc. Lond. B, Biol. Sci., 261, 319–324. CrossRefGoogle Scholar
  17. Goldbeter, A. (2007). Biological rhythms as temporal structures. Adv. Chem. Phys., 135, 253–295. CrossRefGoogle Scholar
  18. Goodwin, B. C. (1965). Oscillator behavior in enzymatic control processes. Adv. Enzyme Regul., 3, 425–438. CrossRefGoogle Scholar
  19. Goryachev, A., & Pokliho, A. (2008). Dynamics of Cdc42 network embodies a Turing-type mechanism of yeast cell polarity. FEBS Lett., 582, 1437–1443. CrossRefGoogle Scholar
  20. Griffith, J. (1968). Mathematics of cellular control processes, I: negative feedback to one gene. J. Theor. Biol., 20, 202–208. CrossRefGoogle Scholar
  21. Harary, F. (1969). Graph theory. Reading: Addison-Wesley. Google Scholar
  22. Horn, F., & Jackson, R. (1972). General mass action kinetics. Arch. Ration. Mech. Anal., 47, 81–116. MathSciNetCrossRefGoogle Scholar
  23. Kitano, H. (2002). Systems biology: a brief review. Science, 295, 1662–1664. CrossRefGoogle Scholar
  24. Kitano, H. (2004). Biological robustness. Nat. Rev. Genet., 5, 826–837. CrossRefGoogle Scholar
  25. Kruse, K., & Julicher, Fr. (2005). Oscillations in cell biology. Curr. Opin. Cell Biol., 17, 20–26. CrossRefGoogle Scholar
  26. Kuznetsov, Y. (2004). Elements of applied bifurcation theory. New York: Springer. MATHGoogle Scholar
  27. Liu, W. M. (1994). Criterion of Hopf bifurcations without using eigenvalues. J. Math. Anal. Appl., 182, 250–256. MathSciNetMATHCrossRefGoogle Scholar
  28. Lancaster, P., & Tismenetsky, M. (1985). The theory of matrices. Orlando: Academic Press. MATHGoogle Scholar
  29. Locke, J. et al. (2006). Experimental validation of a predicted feedback loop in the multi-oscillator clock of Arabidopsis thaliana. Mol. Syst. Biol., 2, 59. CrossRefGoogle Scholar
  30. Milo, R. et al. (2002). Network motifs: Simple building blocks of complex networks. Science, 298, 824–827. CrossRefGoogle Scholar
  31. Mincheva, M., & Craciun, G. (2008). Multigraph conditions for multistability, oscillations and pattern formation in biochemical reaction networks. Proc. IEEE, 96, 1281–1291. CrossRefGoogle Scholar
  32. Mincheva, M., & Roussel, M. R. (2006). A graph-theoretic method for detecting Turing bifurcations. J. Chem. Phys., 125, 204102. CrossRefGoogle Scholar
  33. Mincheva, M., & Roussel, M. R. (2007a). Graph-theoretic methods for the analysis of chemical and biochemical networks, I: multistability and oscillations in ordinary differential equation models. J. Math. Biol., 55, 61–86. MathSciNetMATHCrossRefGoogle Scholar
  34. Mincheva, M., & Roussel, M. R. (2007b). Graph-theoretic methods for the analysis of chemical and biochemical networks, II: oscillations in networks with delays. J. Math. Biol., 55, 87–104. MathSciNetMATHCrossRefGoogle Scholar
  35. Reidl, J. et al. (2006). Model of calcium oscillations due to negative feedback in olfactory cilia. Biophys. J., 90, 1147–1155. CrossRefGoogle Scholar
  36. Rockafellar, R. T. (1970). Convex analysis. Princeton: Princeton University Press. MATHGoogle Scholar
  37. Schuster, S. (1999). Studies on the stoichiometric structure of enzymatic reaction systems. Theory Biosci. 118, 125–139. Google Scholar
  38. 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
  39. Smolen, P., Baxter, D. A., & Byrne, J. H. (2000). Modeling transcriptional control in gene networks. Methods, recent results, and future directions. Bull. Math. Biol., 62, 247–292. CrossRefGoogle Scholar
  40. Tsai, T. et al. (2008). Robust, tunable biological oscillators from interlinked positive and negative feedback loops. Science, 321, 126–129. CrossRefGoogle Scholar
  41. Tyson, J. J. (1975). Classification of instabilities in chemical reaction systems. J. Chem. Phys., 62, 1010–1015. CrossRefGoogle Scholar
  42. Tyson, J. J. (1991). Modeling the cell division cycle: cdc2 and cyclin interactions. Proc. Natl. Acad. Sci. USA, 88, 7328–7332. CrossRefGoogle Scholar
  43. Tyson, J. J., & Othmer, H. (1978). The dynamics of feedback control circuits in biochemical pathways. Prog. Theor. Biol., 5, 1–62. Google Scholar
  44. Tyson, J. J., Chen, K. C., & Novak, B. (2003). Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Curr. Opin. Cell Biol., 15, 221–231. CrossRefGoogle Scholar
  45. Volpert, A., & Hudyaev, S. (1985). Analyses in classes of discontinuous functions and equations of mathematical physics. Dordrecht: Martinus Nijhoff (Chap. 12). Google Scholar
  46. Volpert, A., & Ivanova, A. (1987). Mathematical models in chemical kinetics. In Mathematical modeling (pp. 57–102). Moscow: Nauka (in Russian). Google Scholar

Copyright information

© Society for Mathematical Biology 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorthern Illinois UniversityDeKalbUSA

Personalised recommendations