Autooscillatory Dynamics in a Mathematical Model of the Metabolic Process in Aerobic Bacteria. Influence of the Krebs Cycle on the Self-Organization of a Biosystem
aa r X i v : . [ q - b i o . O T ] M a y GENERAL PHYSICS
ISSN 2071-0186. Ukr. J. Phys. ZZZZ. Vol. YY, No. XX doi: V.I. GRYTSAY, A.G. MEDENTSEV, A.YU. ARINBASAROVA Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine (14b, Metrolohichna Str., Kyiv 03680, Ukraine; e-mail: [email protected]) G.K. Skryabin Institute of Biochemistry and Physiology of Microorganisms of the RAS (5, Prosp. Nauki, Pushchino, Moscow region, Russian Federation; e-mail:[email protected])
AUTOOSCILLATORY DYNAMICSIN A MATHEMATICAL MODEL OF THE METABOLICPROCESS IN AEROBIC BACTERIA. INFLUENCEOF THE KREBS CYCLE ON THE SELF-ORGANIZATIONOF A BIOSYSTEM
We have modeled the metabolic process running in aerobic cells as open nonlinear dissipativesystems. The map of metabolic paths and the general scheme of a dissipative system partic-ipating in the transformation of steroids are constructed. We have studied the influence ofthe Krebs cycle on the dynamics of the whole metabolic process and constructed projectionsof the phase portrait in the strange attractor mode. The total spectra of Lyapunov exponents,divergences, Lyapunov’s dimensions of the fractality, Kolmogorov–Sinai entropies, and pre-dictability horizons for the given modes are calculated. We have determined the bifurcationdiagram presenting the dependence of the dynamics on a small parameter, which defines sys-tem’s physical state.K e y w o r d s : mathematical model, metabolic process, self-organization, deterministic chaos,Fourier series, strange attractor, bifurcation.
1. Introduction
We will study the causes for the appearance of an au-tooscillatory dynamics in the metabolic process run-ning in aerobic bacteria which was observed experi-mentally in a bioreactor with
Arthrobacter globiformis cells [1–5]. The consideration is carried on with thehelp of the synergetic method of modeling of themetabolic processes developed in works V.I. Grytsay[6–33]. The application of this method will allow us toconsider the self-organization and the dynamic chaosin the metabolic processes in cells and in organism asa whole and to establish the physical laws for theirvital processes. c (cid:13) V.I. GRYTSAY, A.G. MEDENTSEV,A.YU. ARINBASAROVA, 2019
The mentioned microorganism is referred to oxy-gen-breathing bacteria arisen 2.48 bln years ago. Dueto the evolution of the metabolic processes of pro-tobionts, the microorganisms not consuming oxygentransited to oxygen-breathing bacteria with their sub-sequent evolution to eukaryotes. The considerationof a specific biochemical process of transformation ofsteroids allows us to use the experimentally deter-mined parameters in the construction of a model andto conclude about structural-functional connectionsunder conditions of the self-organization in the vitalactivity of cells. The relative simplicity of the givenmetabolic process of bacteria under study gives pos-sibility to model the whole metabolic process in acell as an open dissipative system, in which two mainsystems (namely, the system of transformation of a .I. Grytsay, A.G. Medentsev, A.Yu. Arinbasarova
Fig. 1.
Map of metabolic paths of a cell
Arthrobacter globiformis substrate and the respiratory chain) necessary for itslife are self-organized.
2. Mathematical Model
In order to develop a mathematical model of the pro-cess of transformation of steroids by aerobic bacte-ria
Arthrobacter globiformis , we use the map of theirmetabolic paths at the transformation of steroids (Fig. 1) obtained as a result of the processing of ex-perimental data [2, 3].Since we consider only the process of transforma-tion of steroids, we separate the localized dissipativesystem of transformation of steroids, which interactswith other metabolic processes in a cell. Its generalscheme is presented in Fig. 2.Based on the scheme in Fig. 2, we developed amathematical model of the metabolic process in a cell ISSN 2071-0186. Ukr. J. Phys. ZZZZ. Vol. YY, No. XX utooscillatory Dynamics in a Mathematical Model of the Metabolic Process in the following form: dGdt = G N + G + γ ψ − l V ( E ) V ( G ) − α G, (1) dPdt = l V ( E ) V ( G ) − l V ( E ) V ( N ) V ( P ) − α P, (2) dBdt = l V ( E ) V ( N ) V ( P ) − k V ( ψ ) V ( B ) − α B, (3) E dt = E G β + G (cid:18) − P + mNN + P + mN (cid:19) −− l V ( E ) V ( G ) + l V ( e ) V ( Q ) − α E , (4) de dt = − l V ( e ) V ( Q ) + l V ( E ) V ( G ) − α e , (5) dQdt = 6 lV (2 − Q ) V ( O ) V (1) ( ψ ) −− l V ( e ) V ( Q ) − l V ( Q ) V ( N ) , (6) dO dt = O N + O − lV (2 − Q ) V ( O ) V (1) ( ψ ) − α O , (7) dE dt = E P β + p Nβ + N (cid:18) − BN + B (cid:19) −− l V ( E ) V ( N ) V ( P ) − α E , (8) dNdt = − l V ( E ) V ( N ) V ( P ) − l V ( Q ) V ( N )) ++ k V ( B ) ψK + ψ + N N + N − α N, (9) dψdt = l V ( E ) V ( G ) + l V ( N ) V ( Q ) − αψ, (10)where V ( X ) = X/ (1 + X ) ; V (1) ( ψ ) = 1 / (1 + ψ ) ; V ( X ) is a function describing the adsorption of theenzyme in the region of a local bond; and V (1) ( ψ ) is a function characterizing the influence of a kineticmembrane potential on the respiratory chain.In the modeling, it is convenient to introduce thefollowing dimensionless parameters: l = l = k =0 . ; l = l = 0 . l = 0 . ; l = l = 0 . ; l = 1 . ; l = 2 . ; k = 1 . ; E = 3 ; β = 2 ; N = 0 . ; m =2 . ; α = 0 . ; a = 0 . ; α = 0 . ; E = 1 . ; β = 0 . ; β = 1 ; N = 0 . ; α = 0 . ; G = 0 . ; N = 2 ; γ = 0 . ; α = 0 . ; α = α = α = α =0 . ; O = 0 . ; N = 0 . ; N = 0 . ; N = 1 ; K = 0 . .Equations (1)–(9) describe variations in the con-centrations of hydrocortisone ( G ) (1); prednisolone Fig. 2.
General scheme of the process of transformation ofsteroids by a cell
Arthrobacter globiformis
Fig. 3.
Kinetic curves for some variables of the stationary ( N = 0 ) and autooscillatory n ( N = 0 . ) modes ( P ) (2); 20 β -oxyderivative of prednisolone ( B ) (3);oxidized form of 3-ketosteroid- ∆ ′ -dehydrogenase ( E ) (4); reduced form of 3-ketosteroid- ∆ ′ -dehydrogenase( e ) (5); oxidized form of the respiratory chain ( Q ) (6); oxygen ( O ) (7); β -oxysteroid-dehydrogenase( E ) (8); and (9) N AD · H ( reduced form of nicoti-namide adenin dinucleotide) ( N ) . Equation (10)shows a change in the kinetic membrane poten-tial ( ψ ) . ISSN 2071-0186. Ukr. J. Phys. ZZZZ. Vol. YY, No. XX .I. Grytsay, A.G. Medentsev, A.Yu. Arinbasarova Fig. 4.
Projections of the phase portrait of a strange attractor arising at N = 0 . The reduction of parameters of the system to thedimensionless form was presented in [2, 3].The calculations according to the given mathemat-ical model (1)–(10) were carried out with the applica-tion of the theory of nonlinear differential equations[34].The analogous modeling of bioprocesses was real-ized in a lot of works (see, e.g., [35–40]).
3. Results of the Study
Within the constructed mathematical model (1)–(10), we performed the computational experiments,by studying the dependence of the kinetics of the metabolic process in a cell on the Krebs cycle [26].These both processes are coupled with each other bythe level of NADH (N). The variation in the amountof this metabolite during the Krebs cycle affects therespiratory chain and the activity of enzyme E (seeFig. 2). In Fig. 3, we present the kinetics ofsome components in two modes: for N = 0 and N = 0 . . The change in this parameter causesthe transition from the stationary mode to the au-tooscillatory one n . Such modes were observed inthe experiment with a bioreactor [4, 5]. However, thenature of such oscillations was not clarified, thoughsome hypotheses were advanced [5]. ISSN 2071-0186. Ukr. J. Phys. ZZZZ. Vol. YY, No. XX utooscillatory Dynamics in a Mathematical Model of the Metabolic Process
Fig. 5.
Projections of the phase portrait of a strange attractor arising at N = 0 . Fig. 6.
Bifurcation diagram of the dependence of the form of attractors of the dynamic process on the parameter N ISSN 2071-0186. Ukr. J. Phys. ZZZZ. Vol. YY, No. XX .I. Grytsay, A.G. Medentsev, A.Yu. Arinbasarova Table 1.
Total spectra of Lyapunov exponents ( λ − λ ) , divergences ( Λ) , Kolmogorov–Sinai entropies( h ) , foresight horizons ( t min ) , and Lyapunov’s dimen-sions of strange attractors ( D F r ) for the dynamicmodes at different N Lyapu-nov’sindices N n x λ –0.0000056 0.0000280 0.0005342 0.000607 λ –0.004731 –0.000586 0.000019 0.000019 λ –0.0049847 –0.0046497 –0.0047941 –0.00478731 λ –0.0083491 –0.0082351 –0.0079388 –0.00776693 λ –0.0230134 –0.0237059 –0.0230698 –0.023414386 λ –0.0302575 –0.0290098 –0.0292015 –0.0289861 λ –0.079155 –0.0796078 –0.80446039 –0.080484059 λ –0.0870153 –0.0860939 –0.0833784 –083210056101 λ –0.1807851 –1.7927866 –0.1787882 –0.178729 λ –0.5214070 –0.5136290 –0.5147199 0.5145829 Λ –0.9354453 –0.9245716 –0.9217877 –0.921326 h t min D F r In the investigation of the physical dynamical pat-tern of the mentioned oscillations in cells, we testedthe observed autooscillations for the stability by Lya-punov. For different values of N , we calculated thetotal spectra of Lyapunov exponents (see Table 1),which enabled us to establish the dynamics of the pro-cess. The form of constructed attractors characterizesthe mode of the self-organization of the metabolicprocess in a cell or the mode of dynamic chaos asthe transient mode describing the adaptation of themetabolism to a change in the nutrition of a cell fromthe external medium. By the determined values ofLyapunov exponents for strange attractors, we calcu-lated the Lyapunov dimensions of their fractalities,Kolmogorov–Sinai entropies, and foresight horizons[35]. On the basis of those data, we may judge aboutthe structure of strange attractors. Some their pro-jections are shown in Fig. 4, 5 ( N = 0 . ).Then we calculated the bifurcation diagram pre-senting the dynamics of the metabolic process as a function of N (see Fig. 6). There, the transitionsfrom the 1-fold mode to multiple modes, as well asstrange attractors, are clearly seen.
4. Conclusions
A mathematical model of the open dissipative sys-tem with localized metabolic process involving aer-obic bacteria is presented. The general map of itsmetabolic paths is constructed.The synergetic method to study the self-organization and dynamical chaos in metabolic pro-cesses in a cell and the whole organism is developed.In adreement with experimental data, we have de-termined the map of paths of the metabolic processrunning in aerobic bacteria and the general schemeof a dissipative system of transformation of steroids.Using the constructed mathematical model, we havestudied the dependence of the dynamics on a changein the small parameter of the Krebs cycle and foundthe modes of autooscillations and strange attrac-tors. The total spectra of Lyapunov exponents, di-vergences, Lyapunov’s dimensions of the fractality,Kolmogorov–Sinai entropies, and predictability hori-zons are calculated. The bifurcation diagram pre-senting the dependence of the dynamics of the pro-cess on a small parameter determining the modeof self-organization or dynamical chaos in the cellmetabolism is constructed.The obtained scientific results have also practicalmeaning, by presenting a physical interpretation ofthe causes for the appearance of destructive autooscil-latory modes observed in biotechnological processesrunning in bioreactors (Section 2 and [4, 5]). Thevariables of the mathematical model which dependon a small parameter will allow the bioengineers tocompetently control the course of a biotechnologicalprocess.
The present work was partially supported bythe Program of Fundamental Research of the De-partment of Physics and Astronomy of the Na-tional Academy of Sciences of Ukraine “Mathemat-ics models of non-equilibrium process in open system”No. 0120U100857.
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АВТОКОЛИВНА ДИНАМIКАВ МЕТАБОЛIЧНОМУ ПРОЦЕСI МАТЕМАТИЧНОЇМОДЕЛI АЕРОБНОЇ БАКТЕРIЇ. ВПЛИВ ЦИКЛУ КРЕБСА НА САМООРГАНIЗАЦIЮ БIОСИСТЕМИР е з ю м еПроведено моделювання метаболiчного процесу аеробноїклiтини як вiдкритої нелiнiйної дисипативної системи. По-будована карта її метаболiчних шляхiв i загальна схема ди-сипативної системи, яка приймає участь у трансформацiїстероїдiв. Дослiджено вплив циклу Кребса на динамiку вцiлому метаболiчного процесу, побудовано проекцiї фазово-го портрету в режимi дивного атрактора. Розрахованi по-внi спектри показникiв Ляпунова, дивергенцiй, ляпуновсь-кi розмiрностi фрактальностi, ентропiї Колмогорова–Сiнаята горизонтипередбачування в даних режимах. Побудова-на бiфуркацiйна дiаграма залежностi динамiки вiд малогопараметра, що впливає на фiзичний стан системи.8