Biopolym. Cell. 2014; 30(5):403-409.
Self-organization and chaos in the metabolism of a cell
1Grytsay V. I., 2Musatenko I. V.
  1. Bogolyubov Institute for Theoretical Physics, NAS of Ukraine
    14 b Metrologichna Str., Kyiv, Ukraine, 03680
  2. Taras Shevchenko National University of Kyiv
    64, Volodymyrska Str., Kyiv, Ukraine, 01601


Aim. To study the dynamics of auto-oscillations arising at the level of enzyme-substrate interaction in a cell and to find the conditions for the self-organization and the formation of chaos in the metabolic process. Methods. A mathematical model of the metabolic process of steroids transformation in Arthrobacter globiformis. The mathematical apparatus of nonlinear dynamics. Results. The bifurcations resulting in the appearance of strange attractors in the metabolic process are determined. The projections of the phase portraits of attractors are constructed for some chosen modes. The total spectra of Lyapunov's indices are calculated. The structural stability of the attractors obtained is studied. By the general scenario of formation of regular and strange attractors, the structural-functional connections in the metabolic process in the cell are found. Their physical nature is investigated. Conclusions. The presented model explains the mechanism of formation of auto-oscillations observed in the A. globiformis cells and demonstrates a possibility of the mathematical modeling of metabolic processes for the physical explanation of the self-organization of a cell and its vital activity.
Keywords: metabolic process, mathematical model, self-organization, bifurcation, strange attractor, Lyapunov indices


[1] Akhrem AA, Titov YuA. Steroids and microorganisms. Moscow: Nauka. 1970; 526 p.
[2] Gachok VP, Grytsay VI. Kinetic model of macroporous granule with the regulation of biochemical processes. Dokl Akad Nauk SSSR. 1985; 282:51–3.
[3] 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;33(6):661-7.
[4] 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;33(6):668-80.
[5] Grytsai VI. Self-organization in a macroporous gel with immobilized cells. A kinetic model of the bioselective membrane of biosensor. Dopovidi Nats Akad Nauk Ukriny. 2000; (2):175–9.
[6] Grytsai VI. Self-organization in a reactionary-diffusive porous medium. Dopovidi Nats Akad Nauk Ukrainy. 2000; (3):201–6.
[7] Grytsay VI. Ordered structure in a mathematical model biosensor. Dopov Nats Akad Nauk Ukr. 2000; 11:112–6.
[8] Grytsay VI. Self-organization in the biochemical process in immobilized cells of the bioselective membrane of biosensor. Ukr J Phys. 2001; 46(1):124–7.
[9] Andreev VV, Grytsay VI. Modeling of inactive zones in porous granules katalizatora and biosensor. Matem Modelir. 2005; 17(2):57–64.
[10] 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.
[11] Grytsay VI, Andreev VV. The role of diffusion in the active structures formation in porous reaction-diffusion media. Matem Modelir. 2006; 18(12):88–94.
[12] Grytsay VI. Unsteady conditions in porous reaction-diffusion. Medium Romanian J Biophys. 2007; 17(1):55–62.
[13] Grytsay VI. The uncertainty in the evolution structure of reaction-diffusion medium bioreactor. Biofiz Visn. 2007; 2:92–7.
[14] Grytsay VI. Formation and stability of morphogenetic fields of immobilized cell in bioreactor. Biofiz Visn. 2008; 2:25–34.
[15] Grytsay VI. Structural instability of biochemical process. Ukr J Phys. 2010; 55(5):599–606.
[16] Dorofeyev AG, Glagolev MV, Bondarenko TF, Panikov NS. The unusual growth kinetics of Arthrobacter globiformis and its explanation. Mikrobiologiia. 1992; 61(1):33–42.
[17] Skichko AS, Koltsova EM. A mathematical model to describe the fluctuations biomass of bacteria. Teor Osnov Khim Tekhn. 2006; 40(5):540–50.
[18] Sel'kov EE. Self-oscillations in glycolysis. 1. A simple kinetic model. Eur J Biochem. 1968;4(1):79-86.
[19] Hess B, Boiteux A. Oscillatory phenomena in biochemistry. Annu Rev Biochem. 1971;40:237-58.
[20] Goldbeter A, Lefever R. Dissipative structures for an allosteric model. Application to glycolytic oscillations. Biophys J. 1972;12(10):1302-15.
[21] Goldbeter A, Caplan SR. Oscillatory enzymes. Annu Rev Biophys Bioeng. 1976;5:449-76.
[22] Chaos in chemistry and biochemistry. Eds. RJ. Field, L. Gyorgyi. World Scientific Press. Singapore. 1993; 289 p.
[23] Kordium VA, Irodov DM, Maslova OO, Ruban TA, Sukhorada EM, Andrienko VI, Shuvalova NS, Likhachova LI, Shpilova SP. Fundamental biology reached a plateau – development of ideas. Biopolym Cell. 2011;27(6):480-98.
[24] Turing AM. The chemical basis of morphogenesis. Philos Trans R Soc Lond B Biol Sci. 1952; 237(641):37–72.
[25] Nicolis G, Prigogine I. Self-organization in nonequilibrium systems: from dissipative structures to order through fluctuations. New York: Wiley. 1977; 491 p.
[26] Romanovskii YuM, Stepanova NV, Chernavskii DS. Mathematical biophysics. Moskow: Nauka, 1975. 305 p.
[27] Akhromeyeva TS, Kurdyumov SP, Malinetskii GG, Samarskii AA. Nonstationary dissipative structures and diffusion-induced chaos in nonlinear media. Phys Rep. 1989; 176(5–6):189–370.
[28] Kuznetsov SP. Dynamical chaos. Moskow: Fiz. Mat. Nauka., 2001; 296p.
[29] Varfolomeev SD, Lukovenkov AV. Stability in chemical and biological systems: Multistage polyenzymatic reactions. Russ J Phys Chem A. 2010; 84(8):1315–23.
[30] Anishchenko VS. Complex oscillations in simple systems. Moscow: Nauka. 1990; 312 p.