Topological Analysis of Chaos in a Three-Variable Biochemical Model

Abstract

A three-variable biochemical prototype involving two enzymes with autocatalytic regulation proposed by Decroly and Goldbeter (1987) is analyzed using a topological approach. A two-branched manifold, a so-called template, is thus identified. For certain control parameter values, this template is a horseshoe template with a global torsion of two half-turns. This implies that the bifurcation diagram can be described using the usual sequences associated with a unimodal map with a differentiable maximum as well as exemplified by the logistic map. Moreover, a type-I intermittency associated with a saddle-node bifurcation is exhibited. The dynamics from a single time series are also investigated to determine whether it is possible to investigate the dynamics of this biochemical model from the measure of a single concentration.

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REFERENCES

  1. Abarbanel, H. D. I, R. Brown, J. J. Sidorowich and L. Sh. Tsimring (1993). The analysis of observed chaotic data in physical systems. Review of Modern Physics 65(4): 1331-1388.

    Google Scholar 

  2. Broomhead, D. S. and G. P. King (1986). Extracting qualitative dynamics from experimental data. Physica D 20: 217-236.

    Google Scholar 

  3. Cao, L. (1997). Practical method for determining the minimum embedding dimension of a scalar time series. Physica D 110: 43-52.

    Google Scholar 

  4. Collet, P. and J. P. Eckmann (1980). Iterated maps on the interval as dynamical systems. Progress in Physics. Ed. A. Jaffe and D. Ruelle. Birkhäuser, Boston.

    Google Scholar 

  5. Coullet, P. and C. Tresser (1978). Itéerations d'endomorphismes et groupe de renormalisation. Journal de Physique 8(39): C5-C25.

    Google Scholar 

  6. Decroly, O. and A. Goldbeter (1982). Birhythmicity, chaos and other patterns of temporal self-organization in a multiply regulated biochemical system. Proceedings of the Natural Academy of Science (USA) 79: 6917-6921.

    Google Scholar 

  7. Decroly, O. and A. Goldbeter (1987). From simple to complex oscillatory behaviour: analysis of bursting in a multiply regulated biochemical system. Journal of Theoretical Biology 124: 219-250.

    Google Scholar 

  8. Duysens, L. N. M. and J. Amesz (1957). Fluorescence spectrophotometry of reduced phosphopyridine nucleotide in intact cells in the near-ultraviolet and visible region. Biochem. Biophysica Acta 24: 19-26.

    Google Scholar 

  9. Feigenbaum, M. J. (1978). Quantitative Universality for a class of nonlinear transformation. Journal of Statistical Physics 19(1): 25-52.

    Google Scholar 

  10. Ghosh, A. K. and B. Chance (1964). Oscillations of glycolytic intermediates in yeasts cells. Biochemical Biophysical Research Communications 16: 174-181.

    Google Scholar 

  11. Gibson, J.F., J. D. Farmer, M. Casdagli and S. Eubank (1992). An analytic approach to practical state space reconstruction. Physica D 57: 1-30.

    Google Scholar 

  12. Gilmore, R. (1998). Topological analysis of chaotic dynamical systems. Review of Modern Physics 70(4): 1455-1529.

    Google Scholar 

  13. Goldbeter, A. (1996). Biochemical oscillations and regular rhythms. Cambridge University Press, Cambridge. Chapter 2.

    Google Scholar 

  14. Goldbeter, A. and R. Lefever (1972). Dissipative structures for an allosteric model: applications to glycolytic oscillations. Biophysical Journal 12: 1302-1315.

    Google Scholar 

  15. Grassberger, P. and I. Procaccia (1983). Measuring the strangeness of strange attractors. Physica D 9: 189-208.

    Google Scholar 

  16. Letellier, C., P. Dutertre and B. Maheu (1995). Unstable periodic orbits and templates of the Rössler system: toward a systematic topological characterization. Chaos 5(1): 271-282.

    Google Scholar 

  17. Letellier, C., J. Maquet, L. Le Sceller, G. Gouesbet and L. A. Aguirre (1998a). On the non-equivalence of observables in phase space reconstructions from recorded time series. Journal of Physics A 31: 7913-7927.

    Google Scholar 

  18. Letellier, C., J. Maquet, H. Labro, L. Le Sceller, G. Gouesbet, F. Argoul and A. Arnéodo (1998b). Analyzing chaotic behaviour in a Belousov-Zhabotinskii reaction by using a global vector field reconstruction. Journal of Physical Chemistry A 102: 10265-10273.

    Google Scholar 

  19. Nielsen, K., P. G. Sorensen and F. Hynne (1997). Chaos in glycolysis. Journal of Theoretical Biology 186: 303-306.

    Google Scholar 

  20. Pomeau, Y. and P. Manneville (1980). Intermittent transition to turbulence in dissipative dynamical systems. Communications in Mathematical Physics 74: 189-197.

    Google Scholar 

  21. Sel'kov, E. E. (1968). Self-oscillations in glycolysis: a simple kinetic model. European Journal of Biochemistry 4: 79-86.

    Google Scholar 

  22. Takens, F. (1981). Detecting strange attractors in turbulence. In: D. A. Rand and L. S. Young (Eds.). Dynamical Systems and Turbulence, Warwick, 1980. Lecture Notes in Mathematics, Vol. 898. Springer-Verlag, New York pp. 366-381.

    Google Scholar 

  23. Thomas, R. (1999). Deterministic chaos seen in terms of feedback circuits: analysis, synthesis, “labyrinth chaos”. International Journal of Bifurcation and Chaos 9(10): 1889-1905.

    Google Scholar 

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Letellier, C. Topological Analysis of Chaos in a Three-Variable Biochemical Model. Acta Biotheor 50, 1–13 (2002). https://doi.org/10.1023/A:1014737424752

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Keywords

  • Time Series
  • Control Parameter
  • Bifurcation Diagram
  • Single Time
  • Topological Analysis