Journal of Mathematical Chemistry

, Volume 49, Issue 2, pp 531–545 | Cite as

Globally attractive oscillations in open monosubstrate allosteric enzyme reactions

Original Paper


Here we study the dynamical properties of glycolytic and other similar biochemical oscillation-generating processes by means of the analysis of a model proposed by Golbdeter and Lefever (Bioph J 13:1302–1315, 1972) in a reduced form proposed by Keener and Sneyd (Mathematical physiology, chap 1, Springer Verlag, Berlin, 2009). After showing that the orbits of the system are bounded, we give some conditions for the existence of oscillations and for the global arrest of them. Then, after deriving an equivalent Lienard-Newton’s equation we assess uniqueness and the global stability of the arising limit cycle. Finally, we shortly investigate the possibility of breaking of the spatial symmetry. Some biological remarks end the work.


Limit cycles Glycolysis Global stability Lienard’s equation Turing’s bifurcations 


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  1. 1.
    Alberts B., Bray D., Lewis J., Raff M., Roberts K., Watson J.D.: Molecular Biology of the Cell. Garland, NewYork (1994)Google Scholar
  2. 2.
    Decroly O., Goldbeter A.: Birhythmicity, chaos, and other patterns of temporal selforganization in a multiply regulated biochemical system. Proc. Natl Acad. Sci. USA 79, 6917–6921 (1982)CrossRefGoogle Scholar
  3. 3.
    D’Onofrio A.: Mathematical analysis of the Tyson model of the regulation of the cell division cycle. Nonlinear Anal. 62, 817–831 (2005)CrossRefGoogle Scholar
  4. 4.
    A. D’Onofrio, Uniqueness of glycolytic oscillations suggested by Selkovć6s model. J. Math. Chem. (2010) doi: 10.1007/s10210-010-9674-6
  5. 5.
    Duysens L.N.M., Amesz J.: Fluorescence spectrophotometry of reduced phosphopyridine nucleotide in intact cells in the near-ultraviolet and visible region. Biochim. Biophys. Acta 24, 19–26 (1957)CrossRefGoogle Scholar
  6. 6.
    Edelstein-Keshet L.: Mathematical Models in Biology. SIAM publishing, Philadelphia (2004)Google Scholar
  7. 7.
    Erle D., Mayer K.H., Plesser T.: The existence of stable limit cycles for enzymatic reactions with positive feedbacks. Math. Biosc. 44, 191–208 (1978)CrossRefGoogle Scholar
  8. 8.
    Ghosh A., Chance B.: Oscillations of glycolytic intermediates in yeast cells. Biochem. Biophys. Res. Commun. 16, 174–181 (1964)CrossRefGoogle Scholar
  9. 9.
    Golbdeter A., Lefever R.: Dissipative structures for an allosteric model. Bioph. J. 13, 1302–1315 (1972)Google Scholar
  10. 10.
    Goldbeter A., Moran F.: Dynamics of a biochemical system with multiple oscillatory domains as a clue for multiple modes of neuronal oscillations. Eur. Biophys. J. 15, 277–287 (1988)CrossRefGoogle Scholar
  11. 11.
    Goldbeter A.: Biochemical oscillations and cellular rhythms: the molecular bases of periodic and chaotic behaviour. CambridgeUniversity Press, Cambridge (1996)CrossRefGoogle Scholar
  12. 12.
    Goldbeter A.: Biological Rythms as Temporally Dissipative Structures. In: Rice, S.A. (eds) Special Volume in Memory of Ilya Prigogine Advances in Chemical Physics, Volume 135, Wiley, London (2007)Google Scholar
  13. 13.
    Hale J.K., Kocack H.: Dynamics and Bifurcations. Springer, Heidelberg (2003)Google Scholar
  14. 14.
    Hess B., Boiteux A., Kruger J.: Cooperation of glycolytic enzymes. Adv. Enzyme Regul. 7, 149–167 (1969)CrossRefGoogle Scholar
  15. 15.
    Hwang T.-W., Tsai H.-J.: Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions. J. Phys. A: Math. Gen. 38, 821–18223 (2005)CrossRefGoogle Scholar
  16. 16.
    Ibsen K.H., Schiller K.W.: Oscillations of nucleotides and glycolytic intermediates in aerobic suspensions of Ehrlich Ascites tumor cells. Biochim. Biophys. Acta 131, 405–407 (1967)CrossRefGoogle Scholar
  17. 17.
    Kar S., Ray D.S.: Nonlinear dynamics of glycolysis. Mod. Phys. Lett. B 18, 653–678 (2004)CrossRefGoogle Scholar
  18. 18.
    Keener J., Sneyd J.: Mathematical Physiology. Springer, Heidelberg (2009)Google Scholar
  19. 19.
    Kuang Y., Freedman H.I.: Uniqueness of limit cycles in gause-type models of predator-prey systems. Math. Biosc. 88, 67–84 (1988)CrossRefGoogle Scholar
  20. 20.
    Lotka A.J.: Undamped oscillations derived from the law of mass action. J. Am. Chem. Soc. 42, 1595–1599 (1920)CrossRefGoogle Scholar
  21. 21.
    Monod J., Wyman J.J., Changeaux J.P.: On the nature of allosteric transitions: a plausible model. J. Mol. Biol. 12, 88–118 (1965)CrossRefGoogle Scholar
  22. 22.
    Murray J.D.: Mathematical Biology. Springer, Heidelberg (2003)Google Scholar
  23. 23.
    Nicolis G., Prigogine I.: Symmetry breaking and pattern selection in far–from–equilibrium systems. PNAS 78, 659–663 (1981)CrossRefGoogle Scholar
  24. 24.
    Othmer H.G., Aldridge J.A.: The effects of cell densities and metabolite flux on cellular dynamics. J. Math. Biol. 5, 169–200 (1978)Google Scholar
  25. 25.
    Poulsen A.K., Petersen M.O., Olsen L.F.: Single cell studies and simulation of cell-cell interactions using oscillating glycolysis in yeast cells. Biophys. Chem. 125, 275–280 (2007)CrossRefGoogle Scholar
  26. 26.
    Prigogine I., Lefever R., Goldbeter A., Herschowitz–Kaufman M.: Symmetry breaking instabilities in biological systems. Nature 223, 913–916 (1981)CrossRefGoogle Scholar
  27. 27.
    Prigogine I., Nicolis G.: Self-Organization in Non-Equilibrium Systems. Wiley, new York (1977)Google Scholar
  28. 28.
    Pye E.K., Chance B.: Sustained sinusoidal oscillations of reduced pyridine nucleotide in a cell-free extract of S. carlsbergiensis. PNAS 55, 888–894 (1981)CrossRefGoogle Scholar
  29. 29.
    Sel’kov E.E.: Self-oscillations in glycolysis. Eur. J. Biochem. 4, 79–86 (1968)CrossRefGoogle Scholar
  30. 30.
    D.E. Strier, S.P. Dawson, Turing patterns inside cells. PLOS One, e1053 (2007)Google Scholar
  31. 31.
    Winfree A.: The Geometry of Biological Time. 2nd edn. Springer, Heidelberg (2001)Google Scholar
  32. 33.
    Yang J.H., Yang L., Qu Z.L. et al.: Glycolytic oscillations in isolated rabbit ventricular myocytes. J. Biol. Chem. 283, 36321–36327 (2008)CrossRefGoogle Scholar
  33. 34.
    Zhifen Z.: Proof of the uniqueness theorem of generalized Lienard’s equations. App. Anal. 23, 63–76 (1986)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Experimental OncologyEuropean Institute of OncologyMilanoItaly

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