• Українська
  • English

< | Next >

Current issue   Ukr. J. Phys. 2017, Vol. 62, N 5, p.448-458
https://doi.org/10.15407/ujpe62.05.0448    Paper

Grytsay V.I.

Bogolyubov Institute for Theoretical Physics
(14b, Metrologichna Str., Kyiv 03680, Ukraine)

Spectral Analysis and Invariant Measure in the Study of a Nonlinear Dynamics of the Metabolic Process in Cells

Section: Nonlinear Processes
Original Author's Text:  Ukrainian/English

Abstract: The metabolic process in a cell is modeled with the use of the Fourier transformation. The histograms of the invariant measures of chaotic attractors are constructed. In particular, a scenario of adaptation of the metabolic process under a change in the dissipation of a kinetic membrane potential and the sequence of the modes of self-organization and deterministic chaos are determined. Respectively, the spectral mapping of the attractors of these modes is considered. The structural-functional connections of the metabolic process in a cell as an integral dissipative system are analyzed.

Key words: mathematical model, metabolic process, self-organization, deterministic chaos, Fourier series, strange attractor, invariant measure, bifurcation.

References:

  1. V.P. Gachok, V.I. Grytsay. Kinetic model of macroporous granule with the regulation of biochemical processes. Dokl. Akad. Nauk SSSR 282, No. 1, 51 (1985).
  2. V.P. Gachok, V.I. Grytsay, A.Yu. Arinbasarova, A.G. Medentsev, K.A. Koshcheyenko, V.K. Akimenko. Kinetic model of hydrocortisone 1-en-dehydrogenation by Arthrobacter globiformis. Biotechn. Bioengin. 33, 661 (1989).
    https://doi.org/10.1002/bit.260330602
  3. V.P. Gachok, V.I. Grytsay, A.Yu. Arinbasarova, A.G. Medentsev, K.A. Koshcheyenko, V.K. Akimenko. Kinetic model for the regulation of redox reaction in steroid transformation by Arthrobacter globiformis cells. Biotechn. Bioengin. 33, 668 (1989). Gachok, V. P., Grytsay, V. I., Arinbasarova, A. Yu., Medentsev, A. G., Koshcheyenko, K. A. and Akimenko, V. K. (1989), Kinetic model for the regulation of redox reaction in steroid transformation by Arthrobacter globiformis cells. Biotechnol. Bioeng., 33: 668–680.
    https://doi.org/10.1002/bit.260330603
  4. V.I. Grytsay. Self-organization in the macroporous structure of the gel with immobilized cells. Kinetic model of the bioselective membrane of a biosensor. Dopov. Nats. Akad. Nauk Ukr. No. 2, 175 (2000).
  5. V.I. Grytsay. Self-organization in a reaction-diffusion porous media. Dopov. Nats. Akad. Nauk Ukr. No. 3, 201 (2000).
  6. V.I. Grytsay. Ordered structure in a mathematical model of biosensor. Dopov. Nats. Akad. Nauk Ukr. No. 11, 112 (2000).
  7. V.I. Grytsay. Self-organization of the biochemical process in immobilized cells of the bioselective membrane of a biosensor. Ukr. J. Phys. 46, 124 (2001).
  8. V.V. Andreev, V.I. Grytsay. Modeling of inactive zones in porous granules of a catalyst and in a biosensor. Matem. Modelir. 17, No. 2, 57 (2005).
  9. V.V. Andreev, V.I. Grytsay. Influence of heterogeneity of the diffusion-reaction process for the formation of structures in the porous medium. Matem. Modelir. 17, No. 6, 3 (2005).
  10. V.I. Grytsay, V.V. Andreev. The role of diffusion in the active structures formation in porous reaction-diffusion media. Matem. Modelir. 18, No. 12, 88 (2006).
  11. V.I. Grytsay. Unsteady conditions in porous reactiondiffusion. Roman. J. Biophys. 17, No. 1, 55 (2007).
  12. V.I. Grytsay. The uncertainty in the evolution structure of a reaction-diffusion medium bioreactor. Biofiz. Visn. 19, No. 2, 92 (2007).
  13. V.I. Grytsay. Morphogenetic field forming and stability of bioreactor immobilization cells. Biofiz. Visn. 20, No. 1, 48 (2008).
  14. V.I. Grytsay. Prediction structural instability and type of the attractor of a biochemical process. Biofiz. Visn. 23, No. 2, 77 (2009).
  15. V.I. Grytsay. Structural instability of a biochemical process. Ukr. J. Phys. 55, No. 2, 599 (2010).
  16. V.I. Grytsay, I.V. Musatenko. Self-oscillatory dynamics of the metabolic process in a cell. Ukr. Biochem. J. 85, No. 2, 93 (2013).
    https://doi.org/10.15407/ubj85.02.093
  17. V.I. Grytsay, I.V. Musatenko. The structure of a chaos of strange attractors within a mathematical model of the metabolism of a cell. Ukr. J. Phys. 58, No. 7, 677 (2013).
    https://doi.org/10.15407/ujpe58.07.0677
  18. V. Grytsay, I. Musatenko. A mathematical model of the metabolism of a cell. Self-organization and chaos. Chaotic Modeling and Simulation (CMSIM) No. 4, 539 (2013).
  19. V.I. Grytsay, I.V. Musatenko. Self-organization and chaos in the metabolism of a cell. Biopolym. Cell 30 No. 5, 403 (2014).
    https://doi.org/10.7124/bc.0008B9
  20. A.A. Akhrem, Yu.A. Titov. Steroids and Microorganisms (Nauka, 1970) (in Russian).
  21. S.P. Kuznetsov. Dynamical Chaos (Fizmatlit, 2001) (in Russian).
  22. V.S. Anishchenko. Complex Oscillations in Simple Systems (Nauka, 1990) (in Russian).
  23. Yu.M. Romanovskii, N.V. Stepanova, D.S. Chernavskii. Mathematical Biophysics (Nauka, 1984) (in Rissian).
  24. G.G. Malinetskii, A.B. Potapov. Modern Problems of Nonlinear Dynamics (Editorial URSS, 2002) (in Russian).
  25. E.E. Sel'kov. Self-Oscillations in Glycolysis. Europ. J. Biochem. 4, 79 (1968).
    https://doi.org/10.1111/j.1432-1033.1968.tb00175.x
  26. M. Holodniok, A. Klic, M. Kubicek, M. Marek. Methods of Analysis of Nonlinear Dynamical Models (Academia, 1986) (in Czech).
  27. G.Yu. Riznichenko. Mathematical Models in Biophysics and Ecology (Inst. of Computer. Studies, 2003) (in Russian).
  28. V.S. Podgorskij. Physiology and Metabolism of MethanolAssimilating Yeast (Naukova Dumka, 1982) (in Russian).
  29. V. Anishchenko, V. Astakhov, A. Neiman, T. Vadicasova, L. Schimansky-Geir. Nonlinear Dynamics of Chaotic and Stochastic System. Tutorial and Modern Developments (Springer, 2007) [ISBN: 978-3-540-38168-6].
  30. Chaos in Chemical and Biochemical Systems. Ed. by R. Field, L. Gyorgyi (World Scientific, 1993).
    https://doi.org/10.1142/1706
  31. V.A. Kordium, D.M. Irodov, O.O. Maslova, T.A. Ruban, E.M. Sukhorada, V.I. Andrienko, N.S. Shuvalova, L.I. Likhachova, S.P. Shpilova. Fundamental biology reached a plateau – development of ideas. Biopolymers & Cells 27 (6), 480 (2011).
    https://doi.org/10.7124/bc.00011B
  32. V.I. Grytsay. The conditions of the self-organization in the multienzyme prostacyclin-thromboxane system. Visn. Kyiv. Univ. No. 3, 372 (2002).
  33. V.I. Grytsay, V.P. Gachok. The modes of self-organization в prostacyclin-thromboxane system. Visn. Kyiv. Univ. No. 4, 365 (2002).
  34. V.I. Grytsay, V.P. Gachok. Ordered structures in the mathematical system of prostacyclin and thromboxane model. Visn. Kyiv. Univ., Ser. Fiz.-Mat. Nauk. No. 1, 338 (2003).
  35. V.I. Grytsay. Modeling of processes in the multienzyme prostacyclin and thromboxane system. Visn. Kyiv. Univ. No. 4, 379 (2003).
  36. V.P. Gachok, Kinetics of Biochemical Processes. (Naukova Dumka, Kiev, 1988) (in Russian).
  37. V.P. Gachok. Strange Attractors in Biosystems (Naukova Dumka, 1989) (in Russian).
  38. S.D. Varfolomeev, A.T. Mevkh, V.P. Gachok. Kinetic model of the multienzyme system of blood prostanoid synthesis. 1. Mechanism of stabilization of the levels of thromboxane and prostacyclin. Molek. Biol. 20, No. 4, 957 (1986).
  39. S.D. Varfolomeev, V.P. Gachok, A.T. Mevkh. Kinetic behavior of the multienzyme system of blood prostanoid synthesis. BioSystems 19, 45 (1986).
    https://doi.org/10.1016/0303-2647(86)90033-X
  40. V.I. Grytsay, I.V. Musatenko. Self-organization and fractality in metabolic processes of the Krebs cycle. Ukr. Biokhim. Zh. 85, No. 5, 191 (2013).
  41. V. Grytsay, I. Musatenko. Nonlinear self-organization dynamics of a metabolic process of the Krebs cycle. Chaotic Modeling and Simulation (CMSIM) 3, 207 (2014).
  42. V. Grytsay. Lyapunov indices and the Poincare mapping in a study of the stability of the Krebs cycle. Ukr. J. Phys. 60, No. 6, 561 (2015).
    https://doi.org/10.15407/ujpe60.06.0561
  43. V.I. Grytsay. Self-organization and fractality in the metabolic process of glycolysis. Ukr. J. Phys. 60, No. 12, 1243 (2015).
    https://doi.org/10.15407/ujpe60.12.1251
  44. V.I.Grytsay. Self-organization and chaos in the metabolism of hemostasis in a blood vessel. Ukr. J. Phys. 61, No. 7, 648 (2016).
    https://doi.org/10.15407/ujpe61.07.0648
  45. V.I. Grytsay. A mathematical model of the metabolic process of atherosclerosis. Ukr. Biochem. J. 88, No. 4, 75 (2016).
    https://doi.org/10.15407/ubj88.04.075
  46. V. Grytsay. Self-organization and fractality created by gluconeogenesis in the metabolic process. Chaotic Modeling and Simulation (CMSIM) 2, 113 (2016).
  47. A. Golub, O. Matyshevska, S. Prylutska, V. Sysoyev, L. Ped, V. Kudrenko, E. Radchenko, Yu. Prylutskyy, P. Scharff, T. Braun. Fullerenes immobilized at silica surface: topology, structure and bioactivity. J. Mol. Liq. 105, No. 2–3, 141 (2003).
    https://doi.org/10.1016/S0167-7322(03)00044-8
  48. Yu.I. Prylutskyy, V.M. Yashchuk, K.M. Kushnir, A.A. Golub, V.A. Kudrenko, S.V. Prylutska, I.I. Grynyuk, E.V. Buzaneva, P. Scharff, T. Braun, O.P. Matyshevska. Biophysical studies of fullerene-based composite for bionanotechnology. Mater. Sci. Engineer. C 23, Nos. 1–2, 109 (2003).
    https://doi.org/10.1016/S0928-4931(02)00244-8
  49. A.D. Suprun, Yu.I. Prylutskyy, A.M. Shut, M.S. Miroshnichenko. Towards a dynamical model of skeletal muscle. Ukr. J. Phys. 48, No. 7, 704 (2003).
  50. Yu.I. Prylutskyy, A.M. Shut, M.S. Miroshnychenko, A.D. Suprun. Thermodynamic and mechanical properties of skeletal muscle contraction. Int. J. Thermophys. 26, No. 3, 827 (2005).
    https://doi.org/10.1007/s10765-005-5580-8
  51. A.D. Suprun, A.M. Shut, Yu.I. Prylutskyy. Simulation of the Hill equation for fiber skeletal muscle contraction. Ukr. J. Phys. 52, No. 10, 997 (2007).
  52. M. Zabolotnyy, Yu. Barabash, Yu. Sklyarov, Yu. Prylutskyy. The model of photoinduced changes in the pigmentprotein complex of reaction center. Ukr. Bioorg. Acta No. 1, 27 (2010).
  53. N.S. Piskunov. Differential and Integral Calculi (Nauka, 1978) (in Russian).
  54. J.L. Kaplan, J.A. Yorke. The onset of chaos in a fluid flow model of Lorenz. Ann. N. Y. Acad. Sci. 316, 400 (1979).
    https://doi.org/10.1111/j.1749-6632.1979.tb29484.x
  55. J.L. Kaplan, J.A. Yorke. Chaotic behavior of multidimensional difference equations. In Functional Differential Equations and Approximations of Fixed Points, edited by H.O. Peitgen, H.O. Walther (Springer, 1979), p. 204.
    https://doi.org/10.1007/BFb0064319
  56. A.G. Dorofeev, M.V. Glagolev, T.F. Bondarenko, N.S. Panikov. Unusual growth kinetics of Arthrobacter globiformis and its explanation. Mikrobiol. 61, 33 (1992).
  57. Ya.B. Pesin. Characteristic Lyapunov indices and the ergodic theory. Usp. Mat. Nauk 32, No. 4, 55 (1977).