Tag Archives: strange attractor

Instability and invariant measure in the mathematical model for oxidative phosphorylation and ATP synthesis in the cell

11.12.2023   Uncategorized   No comments

V. I. Grytsay

Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kyiv;
e-mail: vigrytsay@gmail.com

Received: 13 October 2023; Revised: 28 November 2023;
Accepted: 01 December 2023; Available on-line: 18 December 2023

The aim of this work was to analyze the process of oxidative phosphorylation and ATP synthesis in a cell using a mathematical model. The scenario of occurrence of the autoperiodic and chaotic modes depending on the ATP dissipation values was determined. The invariant measure of the strange attractor was calculated, and histograms of its projections on the phase plane were plotted. Some recommendations were made on how to eliminate biochemically the chaotic mode and restore the stability of the self-organization of the cell biosystem.

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Self-organization and fractality in a metabolic processes of the Krebs cycle

28.10.2015   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

The metabolic processes of the Krebs cycle is studied with the help of a mathematical model. The autocatalytic processes resulting in both the formation of the self-organization in the Krebs cycle and the appearance of a cyclicity of its dynamics are determined. Some structural-functional connections creating the synchronism of an autoperiodic functioning at the transport in the respiratory chain and the oxidative phosphorylation are investigated. The conditions for breaking the synchronization of processes, increasing the multiplicity of cyclicity, and for the appearance of chaotic modes are analyzed. The phase-parametric diagram of a cascade of bifurcations showing the transition to a chaotic mode by the Feigenbaum scenario is obtained. The fractal nature of the revealed cascade of bifurcations is demonstrated. The strange attractors formed as a result of the folding are obtained. The results obtained give the idea of structural-functional connections, due to which the self-organization appears in the metabolism running in a cell. The constructed mathematical model can be applied to the study of the toxic and allergic effects of drugs and various substances on cell metabolism.

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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.

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