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Current issue   Ukr. J. Phys. 2015, Vol. 60, N 6, p.561-574
https://doi.org/10.15407/ujpe60.06.0561   Paper

Grytsay V.I.

Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
(14b, Metrolohichna Str., Kyiv 03680, Ukraine; e-mail: vgrytsay@bitp.kiev.ua)

Lyapunov Indices and the Poincaré Mapping in a Study of the Stability of the Krebs Cycle

Section: Nonlinear processes
Language: English

Abstract: On the basis of a mathematical model, we continue the study of the metabolic Krebs cycle (or the tricarboxylic acid cycle). For the first time, we consider its consistency and stability, which depend on the dissipation of a transmembrane potential formed by the respiratory chain in the plasmatic membrane of a cell. The phaseparametric characteristic of the dynamics of the ATP level depending on a given parameter is constructed. The scenario of formation of multiple autoperiodic and chaotic modes is presented. Poincar´e sections and mappings are constructed. The stability of modes and the fractality of the obtained bifurcations are studied. The full spectra of Lyapunov indices, divergences, KS-entropies, horizons of predictability, and Lyapunov dimensionalities of strange attractors are calculated. Some conclusions about the structural-functional connections determining the dependence of the cell respiration cyclicity on the synchronization of the functioning of the tricarboxylic acid cycle and the electron transport chain are presented.

Key words: Krebs cycle, metabolic process, self-organization, strange attractor, bifurcation, Feigenbaum scenario.

References:

  1. B.P. Belousov, in: Autowave Processes in Systems with Diffusion (Gor'kii State Univ., Gor'kii, 1951), p. 76 (in Russian).
  2. H.A. Krebs and W.A. Johnson, Enzymologia, No. 4, 148 (1937).
  3. R. Bohnensack and E.E. Sel'kov, Studia biophys. 66, 47 (1977).
  4. A.E. Lyubarev and B.I. Kurganov, Molekul. Biol. 21, 1286 (1987).
  5. E.M.T. El-Mansi, G.C. Dawson, and C.F.A. Bryce, Comput. Appl. Biosci. 10, 295 (1994).
  6. R. Ramakrishna, J.S. Edwards, A. McCulloch, and B.O. Palsson, Am. J. Physiol. Regul. Integr. Comp. Physiol. 280, R695 (2001).
  7. S. Cortassa, M.A. Aon, E. Marban, R.L. Winslow, and B. O'Rourke, Biophys. J. 84, 2734 (2003). CrossRef
  8. K. Yugi and M. Tomita, Bioinform. 20, 1795 (2004). CrossRef
  9. V.K. Singh and I. Ghosh, Theor. Biol. Med. Model. 3, 27 (2006). CrossRef
  10. E. Mogilevskaya, O. Demin, and I. Goryanin, J. of Biol. Phys. 32(3-4), 245 (2006). CrossRef
  11. D.L. Nelson and M.M. Cox, Lehninger Principles of Biochemistry (Freeman, New York, 2008).
  12. V.P. Gachok, Kinetics of Biochemical Processes (Naukova Dumka, Kiev, 1988) (in Russian).
  13. V.P. Gachok, Strange Attractors in Biosystems (Naukova Dumka, Kiev, 1989) (in Russian).
  14. J. Monod, Recherches sur la Croissanse des Cultures Bacteriennes (Hermann, Paris, 1942).
  15. V.S. Podgorskii, Physiology and Metabolism of MethanolConsuming Yeast (Naukova Dumka, Kiev, 1982) (in Russian).
  16. L.N. Drozdov-Tikhomirov and N.T. Rakhimova, Mikrobiol. 55, 775 (1986).
  17. G.Yu. Riznichenko, Mathematical Models in Biophysics and Ecology (Inst. of Computer Studies, Moscow–Izhevsk, 2003) (in Russian).
  18. C.M. Watteeuw, W.B. Armiger, D.L. Ristroph, and A.E. Humphrey, Biotechnol. Bioeng. 21, 1221 (1979). CrossRef
  19. W.B. Armiger, A.R. Moreira, J.A. Phillips, and A.E. Humphrey, in: Utilization of Cellulose Materials in Inconventional Food Production (Plenum, New York, 1979), p. 111.
  20. V.I. Grytsay and I.V. Musatenko, Ukr. Biokhim. Zh. 85, 191 (2013).
  21. V. Grytsay and I. Musatenko, in: Chaotic Modeling and Simulation (CMSIM) (2014), Vol. 3, p. 207.
  22. E.E. Selkov, Europ. J. Biochem. 4, 79 (1968). CrossRef
  23. B. Hess and A. Boiteux, Annu. Rev. Biochem. 40, 237 (1971). CrossRef
  24. A. Goldbeter and R. Lefer, Biophys. J. 12, 1302 (1972). CrossRef
  25. A. Godlbeter and R. Caplan, Annu. Rev. Biophys. Bioeng. 5, 449 (1976). CrossRef
  26. Chaos in Chemical and Biochemical Systems, edited by R. Field and L. Gy¨orgyi (World Scientific, Singapore, 1993).
  27. P. Mitchell, FEBS Lett. 43, 189 (1974). CrossRef
  28. V.S. Anishchenko, Complex Oscillations in Simple Systems(Nauka, Moscow, 1990) (in Russian).
  29. S.P. Kuznetsov, Dynamical Chaos (Fiz.-Mat. Nauka, Moscow, 2001) (in Russian).
  30. V.P. Gachok and V.I. Grytsay, Dokl. Akad. Nauk SSSR 282, 51 (1985).
  31. V.P. Gachok, V.I. Grytsay, A.Yu. Arinbasarova, A.G. Medentsev, K.A. Koshcheyenko, and V.K. Akimenko, Biotechn. Bioengin. 33, 661 (1989). CrossRef
  32. V.P. Gachok, V.I. Grytsay, A.Yu. Arinbasarova, A.G. Medentsev, K.A. Koshcheyenko, and V.K. Akimenko, Biotechn. Bioengin. 33, 668 (1989). CrossRef
  33. V.I. Grytsay, Dopov. Nats. Akad. Nauk Ukr., No. 2, 175 (2000).
  34. V.I. Grytsay, Dopov. Nats. Akad. Nauk Ukr., No. 3, 201 (2000).
  35. V.I. Grytsay, Dopov. Nats. Akad. Nauk Ukr., No. 11, 112 (2000).
  36. V.I. Grytsay, Ukr. J. Phys. 46, 124 (2001).
  37. V.V. Andreev and V.I. Grytsay, Matem. Modelir. 17, No. 2, 57 (2005).
  38. V.V. Andreev and V.I. Grytsay, Matem. Modelir. 17, No. 6, 3 (2005).
  39. V.I. Grytsay and V.V. Andreev, Matem. Modelir. 18, No. 12, 88 (2006).
  40. V.I. Grytsay, Medium. Romanian J. Biophys. 17, 55 (2007).
  41. V.I. Grytsay, Biofiz. Visn., No. 2, 92 (2007).
  42. V.I. Grytsay, Biofiz. Visn., No. 2, 25 (2008).
  43. V.I. Grytsay, Ukr. J. Phys. 55, 599 (2010).
  44. V.I. Grytsay and I.V. Musatenko, Ukr. Biochem. J. 85, 93 (2013). CrossRef
  45. V.I. Grytsay and I.V. Musatenko, Ukr. J. Phys. 58, 677 (2013). CrossRef
  46. V.I. Grytsay and I.V. Musatenko, in: Self-Organization and Chaos, Chaotic Modeling and Simulation (CMSIM), No. 4, 539 (2013).
  47. V.I. Grytsay and I.V. Musatenko, Biopolym. Cell 30, 404 (2014). CrossRef
  48. M.J. Feigenbaum, J. Stat. Phys. 19, 25 (1978). CrossRef
  49. A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 154, 754 (1959).
  50. Ya.B. Pesin, Usp. Mat. Nauk. 32, No. 4, 55 (1977).
  51. J.L. Kaplan and J.A. Yorke, Ann. N. Y. Acad. Sci. 316, 400 (1979). CrossRef
  52. J.L. Kaplan and J.A. Yorke, in: Functional Differential Equations of Fixed Points, edited by H.O. Peitgen, H.O. Walther (Springer, Berlin, 1979), Vol. 730, p. 204. CrossRef