Ukr.Biochem.J. 2013; Volume 85, Issue 2, Mar-Apr, pp. 93-104

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

Self-oscillatory dynamics of the metabolic process in a cell

07.10.2015   Uncategorized   No comments

V. I. Grytsay1, I. V. Musatenko2

1Bogolyubov Institute for Theoretical Physics,
National Academy of Sciences of Ukraine, Kyiv;
e-mail: vgrytsay@bitp.kiev.ua;
2Taras Shevchenko National University of Kyiv, Ukraine;
e-mail: ivmusatenko@gmail.com

In this work, a mathematical model of self-oscillatory dynamics of the metabolism in a cell is studied. The full phase-parametric characteristics of variations of the form of attractors depending on the dissipation of a kinetic membrane potential are calculated. The bifurcations and the scenarios of the transitions “order-chaos”, “chaos-order” and “order-order” are found. We constructed the projections of the multidimensional phase portraits of attractors, Poincaré sections, and Poincaré maps. The process of self-organization of regular attractors through the formation torus was investigated. The total spectra of Lyapunov exponents and the divergences characterizing a structural stability of the determined attractors are calculated. The results obtained demonstrate the possibility of the application of classical tools of nonlinear dynamics to the study of the self-organization and the appearance of a chaos in the metabolic process in a cells.

Keywords: , , , , ,

References:

  1. Turing AM. The chemical basis of morphogenesis.  Phil Trans Roy Soc Lond Series B. 1952 Aug;237(641):37-72.
  2. Nicolis G., Prigogine I. Self-Organization in Nonequilibrium Systems. From Dissipative Structures to Order through Fluctuations – Wiley, New York. 1977.
  3. Romanovskii YuM, Stepanova NV, Chernavskii DS. Mathematical Biophysics. Nauka, Moskow, 1984. 304 p. (in Rissian).
  4. Akhromeyeva TS, Kurdyumov SP, Malinetskii GG, Samarskii AA. Nonstationary dissipative structures and diffusion-induced chaos in nonlinear media. Physics Reports. 1989 May; 176(5-6):189-370. CrossRef
  5. Gachok VP. Kinetics of Biochemical Processes. Naukova Dumka, Kiev, 1988. 218 p. (in Russian).
  6. Gachok VP. Strange Attractors in Biosystems. Naukova Dumka, Kiev, 1989. 238 p. (in Russian).
  7. Akhromeyeva TS, Kurdyumov SP, Malinets­kii GG, Samarskii AA. Nonstationary Structures and Diffusion Chaos. Nauka, Moscow, 1992. 541 p. (in Russian).
  8. Strizhak PE. Deterministic Chaos in Chemistry. Akademperiodika, Kiev, 2002. 288 p. (in Russian).
  9. Akhrem AA, Titov YuA. Steroids and Microorganisms. Nauka, Moscow, 1970. 525 p. (in Russian).
  10. Dorofeev AG, Glagolev MV, Bondarenko TF, Panikov NS. The unusual growth kinetics of Arthrobacter globiformis and its explanation. Mikrobiologiia. 1992;61(1):33-42.
  11. Skichko AS, Kol’tsova EM.  Mathematical model for describing oscillations of bacterial biomass. Theor Foundat Chem Eng. 2006;40(5):503-513. CrossRef
  12. Gachok VP, Grytsay VI. A kinetic-model of macroporous granule with the regulations of biochemical processes. Doklady Akad Nauk USSR. 1985;282(1):51-53.
  13. Selkov EE. Self-Oscillations in Glycolysis. Eur J Biochem. 1968 Mar;4(1);79-86. PubMed, CrossRef
  14. Hess B, Boiteux A. Oscillatory phenomena in biochemistry. Annu Rev Biochem. 1971;40(1):237-58. Review. PubMed, CrossRef
  15. Goldbeter A, Lefer R. Dissipative structures for an allosteric model. Application to glycolytic oscillations. Biophys J. 1972 Oct;12(10):1302-15. PubMed, PubMedCentral, CrossRef
  16. Goldbeter A, Caplan SR. Oscillatory enzymes. Annu Rev Biophys Bioeng. 1976;5(1):449-76. Review. PubMed, CrossRef
  17. Gachok VP, Grytsay VI, Arinbasarova AY, Medentsev AG, Koshcheyenko KA, Akimenko VK. Kinetic model of hydrocortisone 1-en-dehydrogenation by Arthrobacter globiformis. Biotechnol Bioeng. 1989 Feb 5;33(6):661-7. PubMed, CrossRef
  18. Gachok VP, Grytsay VI, Arinbasarova AY, Medentsev AG, Koshcheyenko KA, Akimenko VK. Kinetic model for the regulation of redox reaction in steroid transformation by Arthrobacter globiformis cells. Biotechnol Bioeng. 1989 Feb 5;33(6):668-80. PubMed,  CrossRef
  19. Grytsay VI. Self-organization in macroporosity gel structure with immobilization cells. Kinetic model of bioselective membrane of biosensor. Dopovidi Nats Akad Nauk Ukrainy. 2000;(2):175-179.
  20. Grytsay VI. Self-organization in reaction-diffusion environment. Dopovidi Nats Akad Nauk Ukrainy. 2000;(3):201-206.
  21. Grytsay VI. Ordered structure in a mathematical model biosensor. Dopovidi Nats Akad Nauk Ukrainy. 2000;(11):112-116.
  22. Grytsay VI, Gachok VP, Prylutskyy Yu, Kucherenko ME, Matyshevska OP, Prylutska SV, Scharff P. Modeling of the Biochemical Processes in the Bioselective Membrane of Biosensor Using the Colloidal Fullerene Particles. EUROCARBON 2000, 1-st World Conference on Carbon. Germany. Berlin. 2000. P. 969-970.
  23. Grytsay VI. Self-Organization in the Biochemical Process in Immobilized Cells of the Bioselective Membrane of a Biosensor. Ukr Fizych Zhurn. 2001;46(1):124-127.
  24. Andreev VV, Grytsay VI. Modeling of inactive zones in porous catalyst granules and in a biosensor. Matem Modelir. 2005;17(2):57–64.
  25. Andreev VV, Grytsay VI. Influence of heterogeneity of diffusion-reaction process for the formation of structures in the porous medium. Matem Modelir. 2005;17(6):3-12.
  26. Grytsay VI, Andreev VV. The diffusion role on non-active structures formation in porous reaction-diffusion medium. Matem Modelir. 2006;18(12):88-94.
  27. Grytsay VI. Unsteady Conditions in Porous Reaction-Diffusion Medium. Romanian J Biophys. 2007;17(1):55-62.
  28. Grytsay VI. The uncertainty in the evolution structure of reaction-diffusion medium bioreactor. Biofiz Visn. 2007;2(19):92-97.
  29. Grytsay VI. Formation and stability of morphogenetic fields of immobilized cell in bioreactor. Biofiz Visn. 2008;2:25-34.
  30. Grytsay VI. Structural instability of a biochemical process. Ukr J Phys. 2010;55(2):599-606.
  31. Kuznetsov SP. Dynamical Chaos. Nauka, Moscow, 2001. 296 p. (in Russian).
  32. Anishchenko V., Astakhov V., Neiman A. et al. Nonlinear Dynamics of Chaotic and Stochastic System. Tutorial and Modern Developments – Springer, Berlin, 2007.
  33. Chaos in Chemical and Biochemical System / Ed. By R. Field, L. Györgyi. – World Scientific Press, Singapore, 1993.
  34. Varfolomeev SD, Lukovenkov AV. Stability in chemical and biological systems: Multistage polyenzymatic reactions. Rus J Phys Chem A. 2010;84(8):1315-1323. CrossRef
  35. Shimada I., Nagashima T.  A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems. Progr Theor Phys. 1979;61(6):1605-1616. CrossRef
  36. Holodniok M, Klic A, Kubicek M, Marek M. Methods of Analysis of Nonlinear Dynamical Models. Academia, Praha, 1986 (in Czech).
Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License.