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

  • V. I. Grytsay Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
Keywords: Krebs cycle, metabolic process, self-organization, strange attractor, bifurcation, Feigenbaum scenario

Abstract

On the basis of a mathematical model, we continue the study of the metabolic Krebs cycle (or the tricarboxylic acid cycle). For the rst 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 phase-parametric 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. Poincare 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.

References

B.P. Belousov, in: Autowave Processes in Systems with Diffusion (Gor'kii State Univ., Gor'kii, 1951), p. 76 (in Russian).

H.A. Krebs and W.A. Johnson, Enzymologia, No. 4, 148 (1937).

R. Bohnensack and E.E. Sel'kov, Studia biophys. 66, 47 (1977).

A.E. Lyubarev and B.I. Kurganov, Molekul. Biol. 21, 1286 (1987).

E.M.T. El-Mansi, G.C. Dawson, and C.F.A. Bryce, Comput. Appl. Biosci. 10, 295 (1994).

R. Ramakrishna, J.S. Edwards, A. McCulloch, and B.O. Palsson, Am. J. Physiol. Regul. Integr. Comp. Physiol. 280, R695 (2001).

S. Cortassa, M.A. Aon, E. Marban, R.L. Winslow, and B. O'Rourke, Biophys. J. 84, 2734 (2003).

http://dx.doi.org/10.1016/S0006-3495(03)75079-6

K. Yugi and M. Tomita, Bioinform. 20, 1795 (2004).

http://dx.doi.org/10.1093/bioinformatics/bth125

V.K. Singh and I. Ghosh, Theor. Biol. Med. Model. 3, 27 (2006).

http://dx.doi.org/10.1186/1742-4682-3-27

E. Mogilevskaya, O. Demin, and I. Goryanin, J. of Biol. Phys. 32(3-4), 245 (2006).

http://dx.doi.org/10.1007/s10867-006-9015-y

D.L. Nelson and M.M. Cox, Lehninger Principles of Biochemistry (Freeman, New York, 2008).

V.P. Gachok, Kinetics of Biochemical Processes (Naukova Dumka, Kiev, 1988) (in Russian).

V.P. Gachok, Strange Attractors in Biosystems (Naukova Dumka, Kiev, 1989) (in Russian).

J. Monod, Recherches sur la Croissanse des Cultures Bacteriennes (Hermann, Paris, 1942).

V.S. Podgorskii, Physiology and Metabolism of MethanolConsuming Yeast (Naukova Dumka, Kiev, 1982) (in Russian).

L.N. Drozdov-Tikhomirov and N.T. Rakhimova, Mikrobiol. 55, 775 (1986).

G.Yu. Riznichenko, Mathematical Models in Biophysics and Ecology (Inst. of Computer Studies, Moscow–Izhevsk, 2003) (in Russian).

C.M. Watteeuw, W.B. Armiger, D.L. Ristroph, and A.E. Humphrey, Biotechnol. Bioeng. 21, 1221 (1979).

http://dx.doi.org/10.1002/bit.260210711

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.

V.I. Grytsay and I.V. Musatenko, Ukr. Biokhim. Zh. 85, 191 (2013).

V. Grytsay and I. Musatenko, in: Chaotic Modeling and Simulation (CMSIM) (2014), Vol. 3, p. 207.

E.E. Selkov, Europ. J. Biochem. 4, 79 (1968).

http://dx.doi.org/10.1111/j.1432-1033.1968.tb00175.x

B. Hess and A. Boiteux, Annu. Rev. Biochem. 40, 237 (1971).

http://dx.doi.org/10.1146/annurev.bi.40.070171.001321

A. Goldbeter and R. Lefer, Biophys. J. 12, 1302 (1972).

http://dx.doi.org/10.1016/S0006-3495(72)86164-2

A. Godlbeter and R. Caplan, Annu. Rev. Biophys. Bioeng. 5, 449 (1976).

http://dx.doi.org/10.1146/annurev.bb.05.060176.002313

Chaos in Chemical and Biochemical Systems, edited by R. Field and L. Gy¨orgyi (World Scientific, Singapore, 1993).

P. Mitchell, FEBS Lett. 43, 189 (1974).

http://dx.doi.org/10.1016/0014-5793(74)80997-X

V.S. Anishchenko, Complex Oscillations in Simple Systems(Nauka, Moscow, 1990) (in Russian).

S.P. Kuznetsov, Dynamical Chaos (Fiz.-Mat. Nauka, Moscow, 2001) (in Russian).

V.P. Gachok and V.I. Grytsay, Dokl. Akad. Nauk SSSR 282, 51 (1985).

V.P. Gachok, V.I. Grytsay, A.Yu. Arinbasarova, A.G. Medentsev, K.A. Koshcheyenko, and V.K. Akimenko, Biotechn. Bioengin. 33, 661 (1989).

http://dx.doi.org/10.1002/bit.260330602

V.P. Gachok, V.I. Grytsay, A.Yu. Arinbasarova, A.G. Medentsev, K.A. Koshcheyenko, and V.K. Akimenko, Biotechn. Bioengin. 33, 668 (1989).

http://dx.doi.org/10.1002/bit.260330603

V.I. Grytsay, Dopov. Nats. Akad. Nauk Ukr., No. 2, 175 (2000).

V.I. Grytsay, Dopov. Nats. Akad. Nauk Ukr., No. 3, 201 (2000).

V.I. Grytsay, Dopov. Nats. Akad. Nauk Ukr., No. 11, 112 (2000).

V.I. Grytsay, Ukr. J. Phys. 46, 124 (2001).

V.V. Andreev and V.I. Grytsay, Matem. Modelir. 17, No. 2, 57 (2005).

V.V. Andreev and V.I. Grytsay, Matem. Modelir. 17, No. 6, 3 (2005).

V.I. Grytsay and V.V. Andreev, Matem. Modelir. 18, No. 12, 88 (2006).

V.I. Grytsay, Medium. Romanian J. Biophys. 17, 55 (2007).

V.I. Grytsay, Biofiz. Visn., No. 2, 92 (2007).

V.I. Grytsay, Biofiz. Visn., No. 2, 25 (2008).

V.I. Grytsay, Ukr. J. Phys. 55, 599 (2010).

V.I. Grytsay and I.V. Musatenko, Ukr. Biochem. J. 85, 93 (2013).

http://dx.doi.org/10.15407/ubj85.02.093

V.I. Grytsay and I.V. Musatenko, Ukr. J. Phys. 58, 677 (2013).

http://dx.doi.org/10.15407/ujpe58.07.0677

V.I. Grytsay and I.V. Musatenko, in: Self-Organization and Chaos, Chaotic Modeling and Simulation (CMSIM), No. 4, 539 (2013).

V.I. Grytsay and I.V. Musatenko, Biopolym. Cell 30, 404 (2014).

http://dx.doi.org/10.7124/bc.0008B9

M.J. Feigenbaum, J. Stat. Phys. 19, 25 (1978).

http://dx.doi.org/10.1007/BF01020332

A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 154, 754 (1959).

Ya.B. Pesin, Usp. Mat. Nauk. 32, No. 4, 55 (1977).

J.L. Kaplan and J.A. Yorke, Ann. N. Y. Acad. Sci. 316, 400 (1979).

http://dx.doi.org/10.1111/j.1749-6632.1979.tb29484.x

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.

http://dx.doi.org/10.1007/BFb0064319

Published
2019-01-17
How to Cite
Grytsay, V. (2019). Lyapunov Indices and the Poincare Mapping in a Study of the Stability of the Krebs Cycle. Ukrainian Journal of Physics, 60(6), 561. https://doi.org/10.15407/ujpe60.06.0561
Section
Nonlinear processes