Instability region in models of nonlinear reaction systems. The Stoichiometric Network Analysis

Stability analysis of reaction systems is described by the application of the Stoichiometric Network Analysis to the three-variable-autocatalator. Although simple, this model is complex enough to describe complex forms of nonlinear dynamics phenomena, like mixed-mode oscillations and chaos. Therefore, stability analysis of such model is not a trivial task. Using the Stoichiometric Network Analysis for this purpose makes the process clear and leads to the reliable result. The method is described briefly in few general steps and all of them are further clarified through the application to the chosen example. First, the reaction rates in steady state are decomposed to contributions of independent pathways, called extreme currents. Then, linearized operator is constructed. Finally, through the analysis of the principal minors of the essential part of this operator, simple stability criterion is identified.Organized on the occasion of the 110th anniversary since the birth of Nikolay Nikolaevich Bogolyubov, October 11–12, 2019, Belgrade, Serbia, published in 2020, Editors: B. Dragovich, Ž. Čupić

Various dinamical states in the Bray-Liebhafsky oscillatory reaction- from periodicity to intermittent chaos

The Bray-Liebhafsky (BL) is one of the most analyzed oscillatory reaction both experimentally and numerically. Most of the experimentally obtained dynamical states of this reaction realized in a continuously fed well stirred tank reactor (CSTR) are successfully simulated. Beside others, numerous structured chaotic dynamical states were obtained between each two periodic states in the period doubling rout to chaos with respect to specific flow rate as the control parameter. It was an universal scenario throughout the whole mixed-mode region, as well as throughout other mixed-mode regions obtained under different initial conditions. However, the intermittent oscillations consistent of chaotic mixture of large-amplitude relaxation oscillations, grouped in bursts and small-amplitude sinusoidal ones or even quiescent parts between them known as gaps were also generated experimentally in the Bray–Liebhafsky reaction by varying different parameters such as temperature, flow rate or reactant concentrations under CSTR conditions. Nevertheless, it will be shown here that intermittent oscillations can be simulated by already published model of the BL reaction network

Ultradian oscillations of corticotropin-releasing hormone (crh) and arginine vasopressin (avp) in modelling of hypothalamic-pituitary-adrenal axis: influence of feedback loop between crh and cortisol

The previously proposed stoichiometric model of the Hypothalamic-Pituitary-Adrenal (HPA) axis activity that took into account arginine vasopressin (AVP), has been further developed to emulate ultradian oscillations of corticotropin-releasing hormone (CRH) and AVP. With this aim, additional coupling of HPA consisting hormones was introduced into this model by reaction between CRH and cortisol (CORT). How additional coupling of hormones affects HPA axis ultradian dynamics and reflects on ultradian oscillations of AVP and CRH concentrations was examined by using numerical simulations and bifurcation analysis. Results show that the rate constant of newly incorporated reaction alone is sufficient to be adjusted only for CRH to exhibit oscillations with optimally prominent amplitudes. Also, oscillation frequencies of CRH were found to be in accordance with findings in the literature under all investigated condition

Kinetic modelling of testosterone-related differences in the hypothalamic–pituitary–adrenal axis response to stress

The sex hormone testosterone (TTS) and the hypothalamic–pituitary–adrenal (HPA) axis mutually control one another’s activity, wherein TTS suppresses corticotrophin releasing hormone (CRH) stimulated HPA axis activity, whereas the activation of HPA axis has an inhibitory effect on TTS secretion. With an intention to explain these phenomena, a network reaction model is developed from the previously postulated stoichiometric models for HPA activity where main dynamic behaviors are controlled by two catalytic steps (one autocatalytic and one autoinhibitory) with respect to cortisol, both found experimentally. The capacity of the model to emulate TTS effects on HPA axis dynamics and its response to acute CRH-induced stress is examined using numerical simulations. Model predictions are compared with empirically obtained results reported in the literature. Thus, the reaction kinetic examinations of nonlinear biochemical transformations that constitute the HPA axis, including the negative feedback effect of TTS on HPA axis activity, recapitulates the well-established fact that TTS dampens HPA axis basal activity, decreasing both cortisol level and the amplitude of ultradian cortisol oscillations. The model also replicates TTS inhibitory action on the HPA axis response to acute environmental challenges, particularly CRH-induced stress. In addition, kinetic modelling revealed that TTS induced reduction in ultradian cortisol amplitude arises because the system moves towards a supercritical Hopf bifurcation as TTS is being increased. © 2017, The Author(s)

L-tyrosine influence on the reaction kinetics of iodatehydrogen peroxide oscillatory reaction

The impact of L-tyrosine amino acid on the kinetics of the BL oscillatory reaction was investigated under closed reactor conditions. The study was focused on examining the sensitivity of the BL reaction matrix to tyrosine perturbations. A high sensitivity of the BL matrix to very low tyrosine concentrations was observed

Applicability of Bray-Liebhafsky reaction for chemical computing

The first discovered homogeneous oscillatory reaction was the Bray-Liebhafsky (BL) one, described in a paper published exactly 100 years ago. However, the applicability of oscillatory reactions in chemical computing was recently discovered. Here we intend to expose the native computing concept applied to intermittent states of the BL reaction, because we believe that this particular state may have some advantages. For this purpose, numerical simulations will be used based on the known model. Sequences of perturbations will be introduced by adding iodate (IO3 - ) and hydrogen peroxide (H2O2), separately, as well as in various combinations with one another. It will be shown that dynamic states obtained after perturbations with same species depend very much on the sequence in which these species were used in perturbations. Additionally, it will be shown that obtained dynamic states shift the system from chaotic intermittent dynamic state to different complex periodic states. Hence, the applicability of the BL reaction system in chemical computing was demonstrated

Numerical simulations of the oscillatory dynamics in the Bray-Liebhafsky reaction perturbed by L-tyrosine

It is well known that almost all living or biological systems are naturally in the oscillatory dynamic states and can be considered as biochemical reaction systems. These oscillatory dynamic states can be caused by internal self-organized phenomena, but also by external periodic variations of temperature, light, food, or seasonal changes. The hypothalamic-pituitary-thyroid (HPT) axis is one such nonlinear system with feedback that is always in an oscillatory dynamic state and L-tyrosine is its main representative. The biological importance of L-tyrosine interactions with iodine species was the motivation for modelling of the oscillatory dynamics in the Bray-Liebhafsky (BL) reaction perturbed by L- tyrosine. Also, direct experimental investigation of metabolic processes in the human body is extremely complex to be done, and therefore any alternative approach is of great importance. Therefore, the BL reaction has the potential to be used as a model of the biological system due to certain characteristics shared with the considered processes; it is characterized by its oscillatory dynamics and based on the chemistry of hydrogen peroxide and iodine compounds commonly present in the thyroid gland, where L-tyrosine is iodinated. The impact of L-tyrosine on the dynamics of the Bray-Lienbhafsky oscillatory reaction was investigated numerically using the proposed model. The study was focused on the examination of the sensitivity of the BL reaction to L-tyrosine perturbation. The obtained results indicated possible pathways of influenc

Influence of arginine vasopressin on the ultradian dynamics of Hypothalamic-Pituitary-Adrenal axis

Numerous studies on humans and animals have indicated that the corticotrophin-releasing hormone (CRH) and arginine vasopressin (AVP) stimulate both individually and synergistically secretion of adrenocorticotropic hormone (ACTH) by corticotropic cells in anterior pituitary. With aim to characterize and better comprehend the mechanisms underlying the effects of AVP on Hypothalamic-Pituitary-Adrenal (HPA) axis ultradian dynamics, AVP is here incorporated into our previously proposed stoichiometric model of HPA axis in humans. This extended nonlinear network reaction model took into account AVP by: reaction steps associated with two separate inflows of AVP into pituitary portal system, that is synthesized and released from hypothalamic parvocellular and magnocellular neuronal populations, as well as summarized reaction steps related to its individual and synergistic action with CRH on corticotropic cells. To explore the properties of extended model and its capacity to emulate the effects of AVP, nonlinear dynamical systems theory and bifurcation analyses based on numerical simulations were utilized to determine the dependence of ultradian oscillations on rate constants of the inflows of CRH and AVP from parvocellular neuronal populations, the conditions under which dynamical transitions occur due to their synergistic action and, moreover, the types of these transitions. The results show that under certain conditions, HPA system could enter into oscillatory dynamic states from stable steady state and vice versa under the influence of synergy reaction rate constant. Transitions between these dynamical states were always through supercritical Andronov-Hopf bifurcation point. Also, results revealed the conditions under which amplitudes of ultradian oscillations could increase several-fold due to CRH and AVP synergistic stimulation of ACTH secretion in accordance with results reported in the literature. Moreover, results showed experimentally observed superiority of CRH as a stimulator of ACTH secretion compared to AVP in humans. The proposed model can be very useful in studies related to the role of AVP and its synergistic action with CRH in life-threatening circumstances such as acute homeostasis dynamic crisis, autoimmune inflammations or severe hypovolemia requiring instant or several-days sustained corticosteroid excess levels. Moreover, the model can be helpful for investigations of indirect AVP-induced HPA activity by exogenously administered AVP used in therapeutic treatment

Mathematical Modeling of the Hypothalamic-Pituitary-Adrenal Axis Dynamics in Rats

The hypothalamic-pituitary-adrenal (HPA) axis is a dynamic regulatory network of biochemical reactions that integrates and synchronizes the nervous and the endocrine systems functions at the organism level. In order to describe how this vast network of biochemical interactions operates, we have developed a nonlinear eleven-dimensional stoichiometric model that concisely describes key biochemical transformations that comprise the HPA axis in rats. In a stoichiometric model of a biochemical system, the outcomes of complex biochemical pathways are succinctly described by stoichiometric relations. In this representation, substances that initiate, i.e. enter a pathway are regarded to behave as reactants; substances that are generated in a pathway are regarded to behave as products; and the rates at which products of a pathway appear are jointly proportional to the concentrations of the reactants. In order to derive rate constants for specific biochemical reaction pathways, we have resorted to our recently developed nonlinear reaction model that concisely describes biochemical transformations in the HPA axis in humans. In this way, a mathematical framework is developed to describe in the form of a system of ordinary differential equations (ODEs) the integration of biochemical pathways that constitute the HPA axis on chemical kinetics basis. This, in turn, allows us to use numerical simulations to investigate how the underlying biochemical pathways are intertwined to give an integral HPA axis response at the organism level to a variety of external or internal perturbators of the HPA dynamics. Given that the HPA axis is a nonlinear dynamical network, its response is complex and often cannot be intuitively predicted, stoichiometric modeling can be harnessed for gaining additional insights into dynamical functioning of this complex neuroendocrine system.Belgrade, Serbia, June 20-24, 2016 [http://alas.matf.bg.ac.rs/~websites/bioinfo/

Development of methods for stability examination of non-equilibrium steady states of complex reaction systems

U okviru ovog rada urađena je detaljna analiza, prilagođavanje i dopuna različitih metoda za ispitivanje stabilnosti stacionarnih stanja sa ciljem njihove efikasnije primene u istraživanju dinamičkih stanja složenih reakcionih sistema. Klasična analiza stabilnosti, analiza stehiometrijskih mreža, metode numeričke kontinuacije i analiza dinamičkih stanja primenjene su na različite probleme koji se sreću prilikom izvođenja analize stabilnosti, sa ciljem da se pokaže kako se dobijeni rezultati što bolje mogu iskoristiti radi dobijanja korisnih informacije o dinamičkim stanjima i načinu funkcionisanja pomenutih sistema. U zavisnosti od problema koji je obrađivan korišćeni su model autokatalatora, modeli oscilatornih reakcija Belousov- Zhabotinsky i Bray-Liebhafsky, kao i model HPA sistema. Primenom klasične analize stabilnosti, pristupa koji se zasniva na određivanju analitičkih izraza za svojstvene vrednosti nalaženjem nula karakterističnog polinoma, na modelu autokatalatora pokazano je da je primena ovog pristupa ograničena na one reakcione sisteme koji imaju najviše dve intermedijerne vrste. U slučaju analize stehiometrijskih mreža (SNA), obrađen je niz problema koji se javljaju pri primeni ove metode u ispitivanju stabilnosti neravnotežnih stacionarnih stanja modela složenih reakcionih sistema i predložena su rešenja za njihovo prevezilaženje. Za potrebe analize napisani su programi u MATLAB programskom paketu, koji omogućavaju brzo izračunavanje matrice ekstremnih struja E primenom opisanih algoritama, kao i efikasno određivanje negativnih dijagonalnih minora matrice brzine struja V(j). Pored navedenog, objašnjeno je kako izabrati intermedijerne vrste bitne za izvođenje analize stabilnosti, odrediti i pojednostaviti dobijene uslove nestabilnosti. Pokazano je i kako odrediti funkcionalne delove modela odgovorne za nastanak bifurkacija sedlasti-čvor i Andronov-Hopf, što je urađeno na modelima HPA sistema i BL reakcije. Takođe je ispitana i važnost brzina reakcija vss i brzina struja j za...In this doctoral thesis detailed analysis of the various methods for stability examinations of stationary states was carried out with the main goal to improve their efficiency in examination of the dynamical states of complex reaction systems. Different approaches such as: classical stability analysis, a stoichiometric networks analysis, methods of numerical continuation and analysis of dynamical states were applied to the different problems encountered when performing stability analysis with aim to show how obtained results can be utilized in the most useful way in order to acquire informations about dynamical states and the way analyzed systems functioning. Depending on the problem which was processed different models such as: autocatalator, models of the Belousov-Zhabotinsky and Bray-Liebhafsky oscillating reactions and the model of the hypothalamic-pituitary-adrenal (HPA) axis were analyzed. Classical stability analysis, an approach that is based on the determination of the analytical expression for the eigenvalues through finding the zeros of the characteristic polynomial, was applied to the model of autocatalator and it was demonstrated that the application of this approach is limited to those reaction systems that have at most two intermediate species. In the case of stoichiometric network analysis (SNA) a number of problems that arise when this method is applied in examination of the stability of non-equilibrium steady state of the models of complex reaction systems were analyzed and their solutions were proposed. For the purpose of the analysis, programs which perform fast calculation of the matrix of extreme current E by applying the algorithms described in this thesis and also allow efficient determination of the negative diagonal minors of the matrix V(j) were written in MATLAB program package. In addition, it was explained how to choose the intermediate species essential for performing stability analysis, determine and simplify obtained instability conditions. It was also shown on the models of HPA system and BL reactions how to determine the functional parts of the analyzed..