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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
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.
Broomhead, D. S. and G. P. King (1986). Extracting qualitative dynamics from experimental data. Physica D 20: 217-236.
Cao, L. (1997). Practical method for determining the minimum embedding dimension of a scalar time series. Physica D 110: 43-52.
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.
Coullet, P. and C. Tresser (1978). Itéerations d'endomorphismes et groupe de renormalisation. Journal de Physique 8(39): C5-C25.
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.
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.
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.
Feigenbaum, M. J. (1978). Quantitative Universality for a class of nonlinear transformation. Journal of Statistical Physics 19(1): 25-52.
Ghosh, A. K. and B. Chance (1964). Oscillations of glycolytic intermediates in yeasts cells. Biochemical Biophysical Research Communications 16: 174-181.
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.
Gilmore, R. (1998). Topological analysis of chaotic dynamical systems. Review of Modern Physics 70(4): 1455-1529.
Goldbeter, A. (1996). Biochemical oscillations and regular rhythms. Cambridge University Press, Cambridge. Chapter 2.
Goldbeter, A. and R. Lefever (1972). Dissipative structures for an allosteric model: applications to glycolytic oscillations. Biophysical Journal 12: 1302-1315.
Grassberger, P. and I. Procaccia (1983). Measuring the strangeness of strange attractors. Physica D 9: 189-208.
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.
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.
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.
Nielsen, K., P. G. Sorensen and F. Hynne (1997). Chaos in glycolysis. Journal of Theoretical Biology 186: 303-306.
Pomeau, Y. and P. Manneville (1980). Intermittent transition to turbulence in dissipative dynamical systems. Communications in Mathematical Physics 74: 189-197.
Sel'kov, E. E. (1968). Self-oscillations in glycolysis: a simple kinetic model. European Journal of Biochemistry 4: 79-86.
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.
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.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
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
Issue Date:
DOI: https://doi.org/10.1023/A:1014737424752