Self-organization and chaos in the metabolism of a cell

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.


INTRODUCTION
One of the basic problems of natural science is the development of a unified theory of the self-organization of the matter. The idea of a cell as a nonlinear open biochemical system is the basis for the studies in this trend. It can help to study physical laws of formation and life activity of cells.
It is impossible to consider the biochemical evolution realized in the Nature on the whole. Therefore, the following assertion is taken as a basis of our study: "During the biochemical evolution including the self-organization from a primary "broth" on the Earth, only those biochemical processes have conserved and evolved, which are the predecessors of contemporary metabolic processes running in cells."Thus, it is of primary importance to study the internal dynamics of a vital cell itself. This will allows one to clarify the laws of biochemical evolution on the Earth and the laws of selforganization of a cell as the open nonlinear system.
For this aim, authors investigate biochemical process of steroids transformation of the Arthrobacter globiformis cell [1]. Earlier jointly with experimenters, the mathematical model of this process was constructed. Within this model, the various biotechnological modes arising in a bioreactor under a transformation of steroids were described [2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The study of the mathematical model has demonstrated the possibility of the appearance of various auto-oscillatory modes in this biotechnological process. In experiments, this phenomenon of oscillatory dynamics was revealed later for the cells that consume other substrata as well [16,17].
Analogous oscillatory modes were observed in the processes of photosynthesis, glycolysis, variations of the concentration of calcium in cells, oscillations in heart muscle, and other biochemical processes [18][19][20][21][22][23]. Though the nature of the appearance of such auto-oscillations can be different, their study allows one to investigate step-bystep the laws of self-organization in biosystems.
In the present work, we state that the reason for the external auto-oscillations observed in biosystems containing Arthrobacter globiformis cells is the internal autooscillations in cells. Depends on external conditions, diffusion, etc., the autooscillations can change and appear as auto-oscillations, or as chaos in the biochemical process of the bioreactor. We will consider auto-oscillations arising on the level of enzyme-substrate interactions and in the respiratory chain. The parameters of such autooscillations can hardly be measured in experiments. The auto-oscillations organize themselves in the total metabolic process of cells at autocatalysis.
It is clear that the construction of the most general universal mathematical model of a cell, which would describe the maximally possible number of metabolic processes running in it is the problem of great importance. But since it is very difficult to attain the full correspondence of the results obtained on the basis of a model and experimental 4 data, we will restrict ourselves by the modeling of the most significant experimentally measurable parts of the metabolic process, which was obtained from experiments.
Though we consider the specific metabolic process, its basic elements such as the consumption of a substrate, respiratory chain, and positive feedback reflect the general mechanisms guiding any metabolic process in any cell.
The development of such models for other metabolic processes in a cell will allow one to study step-by-step the process of self-organization on the whole.
In addition, the metabolic processes are described by various nonlinear interactions, which give possibility to investigate a lot of nonlinearities that are inherent in open systems of any nature and eventually form the universal types of selforganization observed in the Nature [24][25][26][27].

MATHEMATICAL MODEL
The general scheme of metabolic processes running in Arthrobacter globiformis cells under the transformation of steroids is presented in Fig. 1. In the article, the part of the metabolic process [3,4] is discussed, where autocatalysis regime is appearing. The mathematical model of the metabolic process in a cell is constructed in accordance with the given general scheme: where The main parameters of the system (1)-(10) were defined from stationary regimes, which appropriate experimental characteristics [3,4]. Using dimensionless, the parameters are: Solutions of the mathematical model were investigated using the nonlinear differential equations theory [28][29][30].
The precision of calculations is 8 10 − . The time of the entrance on the attractor is 6 10 .
The bifurcation diagrams were built on the plane The full spectrum of Lyapunov indices were obtained, using Benettine algorithm, with Gram-Schmidt orthogonalization [28]. The types of the attractors were defined, using the first and the second Lyapunov indices. For the regular attractor are

RESULTS OF STUDIES
Here, we continue study the dynamics of modes of the mathematical model (1)-(10) under a variation of the dissipation of the kinetic membrane potential α . We analyze the various types of auto-oscillatory modes, as well as the scenarios of the appearance of bifurcations under the transition of the dynamical process from one type of attractors to another one, with the help of the construction of bifurcation diagrams.
As a result of the numerical experiment, we obtain the bifurcation diagrams and 8 calculated the total spectra of Lyapunov indices for the most typical typical modes (see Table 1).
For the parametr α =0.032180 (Table 1) Table 1) arises. It smoothly 9 transits via the intermittence into the strange attractor     Table 1). The "chaos-order" transition arises. Let us consider the figure from the right to the left. It is very similar to Fig. 3,a. As earlier, a strange attractor is formed via the intermittence from the regular attractor through two bifurcations with the doubling of a period and the intermittence. The "order-chaos" transition  suddenly arises as a result of the intermittence (Table 1). We observe the transition "order-chaos". As α increases to 0.03254 and more (Fig. 5,b), the laminar part of the trajectory of the attractor The further decrease of the dissipation due to the self-organization forms a strange attractor with multiplicity, which is larger than the previous value by 1. The subsequent decrease of α causes the destruction of auto-oscillations arisen in the respiratory chain. As a result, only the laminar part of oscillations in the nutrient chain is conserved in the metabolic process, and a new regular attractor arises.
In the subsequent work, we will present the results of the further study of the dynamics of the given metabolic process. We will construct the scenario of the formation of other types of regular and strange attractors, calculate the Lyapunov dimensions of their fractality and KS-entropies, and discuss the "predictability horizons". We will carry out the spectral analysis of the obtained attractors and will study the structure of the chaos of attractors, the hierarchy of its forms, and the influence of the chaos on the stability of the metabolic process and the adaptation and the functioning of a cell. Table 1. Total spectra of Lyapunov exponents for attractors of the system under study ( 4 λ - 9 λ are not important for our investigation).