Self-Organization and Fractality in a Metabolic Process of the Krebs Cycle

With the help of a mathematical model, the metabolic process of the Krebs cycle is studied. 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 a 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 metabolic process 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 the metabolism of a cell.

Among various metabolic processes, the Krebs cycle is common for all cells [1]. The cycle involving tricarboxylic acids occupies the principal place, to which practically all metabolic paths lead. It is the final point of the catabolism of a "cell fuel," the central place of a cell respiration, and the source of molecules-predecessors, from which the aminoacids, carbohydrates, fat acids, and other compounds important for the functioning of cells are then synthesized. Its study will allow one to find the general regularities of the operation of a cell.
As the main difficulty, the deficit of experimental data should be mentioned.
It is difficult to study separately this cycle without the consideration of the functioning of a cell as a whole, since it is impossible to determine the internal parameters of a metabolic process without the knowledge of the external parameters of a medium, where a cell lives. Our studies are based on the mathematical model of the unstable growth of cells Candida utilis on ethanol, which was developed by Professor V.P. Gachok [11,12] in view of the experimental data published in [13]. Analogous problems, namely the modeling of a cell growth, were considered by J. Monod, V.S. Podgorskii, L.N. Drozdov-Tikhomirov, N.T. Rakhimova, G.Yu. Riznichenko, and others [14][15][16][17][18].
Our refined mathematical model involves the formation of carbon dioxide and considers its influence on the kinetics of a metabolic process.
In the present work, we will study the unstable modes of the cultivation of cells and will find the parameters and structural-functional connections of the metabolic process running in a cell that admit the appearance of oscillations. We will consider the intracellular autooscillations characteristic of the metabolic process on the level of redox reactions of the Krebs cycle. These autooscillations are related to the cyclicity of the process and characterize the self-organization inside a cell.
We note that the description of the metabolic process within the presented model is based not on the use of conditional reagents, as was made earlier in other models, but involves the real intracellular components, which will allow us to find an actual mechanism of self-organization.

MATHEMATICAL MODEL
The general scheme of the process is presented in Fig. 1. According to it with regard for the mass balance, we have constructed the mathematical model given by Eqs. (1) - (19). ), , , , where is the function that describes the adsorption of the enzyme in the region of a local coupling. The variables of the system are dimensionless.
The internal parameters of the system are as follows: The model covers the processes of substrate-enzymatic oxidation of ethanol to acetate, cycle involving tri-and dicarboxylic acids, glyoxylate cycle, and respiratory chain. Model (1)- (19) is improved as compared with the model used in [11][12], since it involves the formation of 2 CO in of the Krebs cycle, which affects the running of the metabolic process. Some parameters of our model are taken from [11][12].
The incoming ethanol S is oxidized by the alcohol dehydrogenase enzyme 1 E to acetaldehyde  Fig. 1 as an enzymatic reaction with the consumption of 2 S and 3 S and the formation of 7 S . The parameter 3 k controls the activity of the активность glyoxylatelinked way (3), (4), (8). The yield of 7 S into cytosol is controlled by its concentration, which can increase due to 9 S , by causing the inhibition of its transport with the participation of protons of mitochondrial membrane.
The formed malate 9 S is used by a cell for its growth, namely for the biosynthesis of protein X (11). The energy consumption of the given process is supported by the process ADP ATP → . The presence of ethanol in the external solution causes the "ageing" of external membranes of cells, which leads to the inhibition of this process. The inhibition of the process also happens due to the enhanced level of the kinetic membrane potential ψ . The parameter 0 µ is related to the lysis and the washout of cells.
In the model, the respiratory chain of a cell is represented in two forms: oxidized, Q , (12) and reduced, q , ones. They obey the integral of motion A change of the concentration of oxygen in the respiratory chain is determined by Eq. (13).
The activity of the respiratory chain is affected by the level of NADH (14). In the respiratory chain, the kinetic membrane potential ψ (16) is created under the running of reducing processes q Q → . It is consumed at the substrate-linked phosphorylation ATP ADP → in the respiratory chain and the Krebs cycle. Its enhanced level inhibits the biosynthesis of protein and process of reduction of the respiratory chain.

RESULTS OF STUDIES
According to the constructed mathematical model (1) The formed ATP is consumed for the growth of cells and in the Krebs cycle itself ADP ATP → (2-nd autocatalytic cycle).
The preservation of the functioning of the Krebs cycle depends of the key parameters of the system relating to the inverse paths of the reduction of oxalocitrate. They are the parameters 15 k and 6 k . The former determines the rate of oxidation, , in the respiratory chain and depends on its parameters: . The rate of the whole autocatalytic process + ↔ NAD NADH is determined by the rate of the redox process in the respiratory chain itself q Q ↔ .
The parameter 6 k determines the influence of the concentration of ADP on the rate of the metabolic process of the Krebs cycle.
Thus, the metabolic process in the Krebs cycle is characterized by the appearance of autooscillations, whose frequency and stability depend on the synchronization of the 1-st and 2-nd autocatalytic cycles. Let us consider the kinetics of autooscillations in the metabolic process of the Krebs cycle, which arise in its cyclic mode. We will study the stability of the Krebs cycle on the whole.
During one turnover of a cycle, one molecule of acetyl-CoA is completely oxidized to malate, and a new molecule of acetyl-CoA is formed at the input. In such a way, the continuous process of operation of the Krebs cycle is running and manifests the autooscillatory character. In Fig. 2 15 k =0.23.
In Fig. 3 (Fig. 3,a) and magnify it ( Fig. 3,b). This part of the phase-parametric diagram is identical to the whole diagram in Fig. 3,a.
This indicates that, after the ( 2 + j )-th bifurcation, the next bifurcation arises, and so on to ( n j + ).
In other words, we have a cascade of bifurcations. At the following decrease in the scale of the diagram, the pattern will repeat up to the critical value of parameter 15 k , after which we observe the appearance of a deterministic хаос. For the derived sequence of bifurcations, we have the relation  (Fig. 3,a), the windows of periodicity appear. The deterministic chaos is destroyed, and periodic and quasiperiodic modes are established. Outside these windows, chaotic modes hold. The identical windows of periodicity are observed also on less scales of the diagram (Fig. 3,b). In other words, the phase-parametric diagrams on a small scale ( Fig. 3,b) and on a large one (Fig. 3,a)   As examples of the successive period doubling for the autoperiodic modes of the system by the the Feigenbaum scenario, w present the projections of the phase portraits of appropriate regular attractors in Fig. 4,a-d. In Fig. 4,e, we show a regular attractor of the 6-fold quasiperiodic mode 0 2 6 ⋅ ≈ arising in the window of periodicity at 2543 . 0 15 = k (Fig. 3,a). be strictly periodic (Fig. 2). The periodicity and the amplitudes change and become chaotic (Fig. 5).
This occurs due to the violation of the synchronism of metabolic processes in the system of transport of electrons ( ) of the redox process in the respiratory chain ( q Q ↔ ) and the substrate-linked phosphorylation ATP ADP → .