Advertisement

Acta Biotheoretica

, Volume 42, Issue 2–3, pp 147–166 | Cite as

Organizing centres and symbolic dynamic in the study of mixed-mode oscillations generated by models of biological autocatalytic processes

  • P. Tracqui
Article

Abstract

The organization of the complex mixed-mode oscillations generated, in a three-dimensional variable space, by an autocatalytic process formalized as a cubic monomial is analyzed. The generation of the temporal patterns is elucidated by complementary approaches dealing with the three-variable differential continuous system itself and with successive discrete applications modelling its first return map. The extent to which the underlying bifurcation structures could constitute a fingerprint of autocatalytic processes is discussed in connection with the modelling of biological systems.

Keywords

Biological System Temporal Pattern Variable Space Complementary Approach Continuous System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aguda, B.D., L.L. Hofmann Frisch and L.F. Olsen L.F. (1990). Experimental evidence for the coexistence of oscillatory and steady states in the peroxidase-oxidase reaction. J. Am. Chem. Soc. 112: 6652–6656.Google Scholar
  2. Albahadily, F.N. and M. Schell (1988). An experimental investigation of periodic and chaotic electrochemical oscillations in the anodic dissolution of copper in phosphoric acid. J. Chem. Phys. 88: 4312–4319.Google Scholar
  3. Alexander, J.C. and D.Y. Cai (1991). On the dynamics of bursting systems. J. Math. Biol. 29: 405–423.Google Scholar
  4. Alexandre, A. and H.B. Dunford (1991). A new model for oscillations in the peroxydase-oxydase reaction. Biophys. Chem. 40: 189–195.Google Scholar
  5. Barkley, D. (1988). Near-critical behavior for one-parameter families of circle maps. Phys. Lett. A 129: 219–222.Google Scholar
  6. Cook, G.B., P. Gray, D.G. Knapp and S.K. Scott (1989). Bimolecular routes to cubic autocatalysis. J. Phys. Chem. 93: 2749–2755.Google Scholar
  7. Decroly, O. (1987). Du comportement periodique simple aux oscillations complexs dans les syst mes biochimiques: bursting, chaos et attracteurs etranges. Ph D Thesis, Universit Libre de Bruxelles.Google Scholar
  8. Decroly, O. and A. Goldbeter (1985). Selection between multiple periodic regimes in a biochemical system: Complex dynamic behaviour resolved by use of one dimensional maps. J. Theor. Biol. 113: 649–671.Google Scholar
  9. Decroly, O. and A. Goldbeter (1987). From simple to complex oscillatory behaviour: analysis of bursting in a multiply regulated biochemical system. J. Theor. Biol. 124: 219–250.Google Scholar
  10. Degn, H. (1968). Bistability caused by substrate inhibition of peroxydase in an open reaction system. Nature 217: 1047–1050.Google Scholar
  11. Escher, C. and J. Ross (1979). Multiple ranges of flow rate with bistability and limit cycles for Schlögl's mechanism in a CSTR. J. Chem. Phys. 79: 3373–3777.Google Scholar
  12. Feigenbaum, M.J. (1979). The Universal metric properties of nonlinear transformations, J. Stat. Phys. 21: 669–706.Google Scholar
  13. Feigenbaum, M.J. (1983). Universal behavior in nonlinear systems. Physica D, 7: 16–39.Google Scholar
  14. Gaspard, P. and X.J. Wang (1987). Homoclinic orbits and mixed-mode oscillations in far-from-equilibrium systems. J. Stat. Phys. 48: 151–199.Google Scholar
  15. Goldbeter, A. (eds.) (1989). Cell to Cell Signalling: From Experiments to Theoretical Models. London, Academic Press.Google Scholar
  16. Gray, P. and S.K. Scott (2983). Autocatalytic reactions in the isothermal, continuous stirred tank reactor. Chem. Eng. Sci. 38: 29–43.Google Scholar
  17. Gray, P. and S.K. Scott (1986). A new model for oscillatory behaviour in closed systems: The autocatalator. Ber. Bunsenges. Phys. Chem. 90: 985–996.Google Scholar
  18. Harms, H.M., U. Kaptaina, W.R. Külpmann, G. Brabant and R.D. Hesch (1989). Pulse amplitude and frequency modulation of parathyroid hormone in plasma. J. Clin. End. Metab. 69: 843–850.Google Scholar
  19. Knobil, E. (1981). Patterns of hormonal signals and hormone action. N. Engl. J. Med. 305: 1582–1583.Google Scholar
  20. Larter R., C.L. Bush, T.R. Lonis and B.D. Aguda (1987). Multiple steady states, complex oscillations, and the devil's staircase in the peroxidase-oxidase reaction. J. Chem. Phys. 87: 5765–5771.Google Scholar
  21. Mann, S. (1990). Molecular recognition in biomineralization. Nature 332: 119–124.Google Scholar
  22. Mundy, G.R. (1990). Calcium Homeostasis: Hypercalcemia and Hypocalcemia. London, Martin Dunitz.Google Scholar
  23. Nicolis, G. and I. Prigogine (1977). Self Organization in Nonequilibrium Systems. New York, J. Wiley & Sons.Google Scholar
  24. Nitzan, A., P. Ortoleva, J. Deutch and J. Ross (1974). Fluctuations and transitions at chemical instabilities; The analogy to phase transitions. J. Chem. Phys. 61: 1056–1074.Google Scholar
  25. Noszticzius, Z., W.D. McCormick and H.L. Swinney (1989). Use of bifurcation diagrams as fingerprints of chemical mechanisms. J. Phys. Chem. 93: 2796–2800.Google Scholar
  26. Olsen, L.F. (1983). An enzyme reaction with a strange attractor. Phys. Lett. A 94: 454–457.Google Scholar
  27. Olsen, L.F. and H. Degn (1977). Chaos in an enzyme reaction. Nature 267: 177–178.Google Scholar
  28. Olsen, L.F. and H. Degn (1985). Chaos in biological systems. Quart. Rev. Biophys. 18: 165–225.Google Scholar
  29. Otwinowski, M., W.G. Laidlaw and R. Paul (1990). Structural instability and sustained oscillations in an extended Brusselator model. Can. J. Phys. 68: 743–750.Google Scholar
  30. Peng, B., S.K. Scott and K. Showalter (1990). Period-doubling and chaos in a three-variable autocatalator. J. Phys. Chem. 94: 5243–5246.Google Scholar
  31. Perault-Staub, A.M., J.F. Staub and G. Milhaud (1990). Extracellular calcium homeostasis. In: Heersche J.N.M. & Kanis J.A., eds. Bone and Mineral Research, Vol. 7, pp. 1–102. Amsterdam New York, Elsevier.Google Scholar
  32. Procaccia, I. (1988). Universal properties of dynamically complex systems: the organization of chaos. Nature 333: 618–623.Google Scholar
  33. Ringland, J. and M. Schell (1989). The Farey tree embodied in bimodal maps of the interval. Phys. Lett. A 136: 379–386.Google Scholar
  34. Ringland, J., N. Issa and M. Schell (1990). From U sequence to Farey sequence: A unification of one-paramefer scenarios. Phys. Rev. A 41: 4223–4235.Google Scholar
  35. Ringland, J. and M. Schell (1991). Genealogy and bifurcation skeleton for cycles of the iterated two-extremum map of the interval. SIAM J. Math. Anal. 22: 1354–1371.Google Scholar
  36. Schell, M. and F.N. Albahadily (1989). Mixed-mode oscillations in an electrochemical system. II. A periodic-chaotic sequence. J. Chem. Phys. 90: 822–828.Google Scholar
  37. Schellenberger, W., M. Kretschmer, K. Eschrich and E. Hofmann (1988). Dynamic structures in the fructose-6-phosphate/fructose-1,6-Bisphosphatase cycle. In: Lamprecht I. & A.I. Zotin, eds. Thermodynamics and Pattern Formation in Biology, pp. 205–222. Walter de Gruyter & Cop.Google Scholar
  38. Schlögl, F. (1971). On thermodynamics near a steady State. Z. Physik 248: 446–458.Google Scholar
  39. Schlögl, F. (1972). Chemical reaction models for non-equilibrium phase transitions. Z. Physik 253: 147–161.Google Scholar
  40. Sel'kov, E.E. (1968). Self-oscillations in glycolysis. Europ. J. Biochem. 4: 79–86.Google Scholar
  41. Staub, J.F., P. Tracqui, P. Brezillon, G. Milhaud and A.M. Perault-Staub (1988). Calcium metabolism in the rat: a temporal self-organized model. Am. J. Physiol. 254: R134-R139.Google Scholar
  42. Staub, J.F., P. Tracqui, S. Lausson, G. Milhaud and A.M. Perault-Staub (1989). A physiological view of in vivo calcium dynamics: The regulation of a nonlinear self-organized system. Bone 10: 77–86.Google Scholar
  43. Steinmetz, C.G. and R. Larter (1991). The quasiperiodic route to chaos in a model of the peroxidase-oxidase reaction. J. Chem. Phys. 94: 1388–1396.Google Scholar
  44. Tracqui, P., J.F. Staub and A.M. Perault-Staub (1989). Analysis of degenerate Hopf bifurcations for a nonlinear model of rat metabolism. Nonlin. Anal.: Th. Meth. Appl. 13: 429–457.Google Scholar
  45. Tracqui, P. (1993). Homoclinic tangencies in an autocatalytic model of interfacial processes at the bone surface. Physica D 62: 275–289.Google Scholar
  46. Tyson, J. and S. Kauffman (1975). Control of mitosis by a continuous biochemical oscillation: synchronization; spatially inhomogeneous oscillations. J. Math. Biol. 1: 289–310.Google Scholar
  47. Tyson, J.J. (1991). Modeling the cell division cycle: cdc2 and cyclin interactions. Proc. Natl. Acad. Sci. USA 88: 7328–7332.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • P. Tracqui
    • 1
  1. 1.C.N.R.S. URA 163, Service de BiophysiqueC.H.U. St AntoineParis Cedex 12

Personalised recommendations