Stoichiometric network analysis of a reaction system with conservation constraints.

Stoichiometric Network Analysis (SNA) is a powerful method that can be used to examine instabilities in modelling a broad range of reaction systems without knowing the explicit values of reaction rate constants. Due to a lack of understanding, SNA is rarely used and its full potential is not yet fulfilled. Using the oscillatory carbonylation of a polymeric substrate [poly(ethylene glycol)methyl ether acetylene] as a case study, in this work, we consider two mathematical methods for the application of SNA to the reaction models when conservation constraints between species have an important role. The first method takes conservation constraints into account and uses only independent intermediate species, while the second method applies to the full set of intermediate species, without the separation of independent and dependent variables. Both methods are used for examination of steady state stability by means of a characteristic polynomial and related Jacobian matrix. It was shown that both methods give the same results. Therefore, as the second method is simpler, we suggest it as a more straightforward method for the applications.

The HPA axis and ethanol: a synthesis of mathematical modelling and experimental observations.

Stress and alcohol use are interrelated-stress contributes to the initiation and upholding of alcohol use and alcohol use alters the way we perceive and respond to stress. Intricate mechanisms through which ethanol alters the organism's response to stress remain elusive. We have developed a stoichiometric network model to succinctly describe neurochemical transformations underlying the stress response axis and use numerical simulations to model ethanol effects on complex daily changes of blood levels of cholesterol, 6 peptide and 8 steroid hormones. Modelling suggests that ethanol alters the dynamical regulation of hypothalamic-pituitary-adrenal (HPA) axis activity by affecting the amplitude of ultradian oscillations of HPA axis hormones, which defines the threshold with respect to which the response to stress is being set. These effects are complex-low/moderate acute ethanol challenge (<8 mM) may reduce, leave unaltered or increase the amplitude of ultradian cortisol (CORT) oscillations, giving rise to an intricate response at the organism level, offering also a potential explanation as to why apparently discordant results were observed in experimental studies. In contrast, high-dose acute ethanol challenge (>8 mM) increases instantaneous CORT levels and the amplitude of ultradian CORT oscillations in a dose-dependent manner, affecting the HPA axis activity also during the following day(s). Chronic exposure to ethanol qualitatively changes the HPA axis dynamics, whereas ethanol at intoxicating levels shuts down this dynamic regulation mechanism. Mathematical modelling gives a quantitative biology-based framework that can be used for predicting how the integral HPA axis response is perturbed by alcohol.

Dynamic transitions in a model of the hypothalamic-pituitary-adrenal axis.

Dynamic properties of a nonlinear five-dimensional stoichiometric model of the hypothalamic-pituitary-adrenal (HPA) axis were systematically investigated. Conditions under which qualitative transitions between dynamic states occur are determined by independently varying the rate constants of all reactions that constitute the model. Bifurcation types were further characterized using continuation algorithms and scale factor methods. Regions of bistability and transitions through supercritical Andronov-Hopf and saddle loop bifurcations were identified. Dynamic state analysis predicts that the HPA axis operates under basal (healthy) physiological conditions close to an Andronov-Hopf bifurcation. Dynamic properties of the stress-control axis have not been characterized experimentally, but modelling suggests that the proximity to a supercritical Andronov-Hopf bifurcation can give the HPA axis both, flexibility to respond to external stimuli and adjust to new conditions and stability, i.e., the capacity to return to the original dynamic state afterwards, which is essential for maintaining homeostasis. The analysis presented here reflects the properties of a low-dimensional model that succinctly describes neurochemical transformations underlying the HPA axis. However, the model accounts correctly for a number of experimentally observed properties of the stress-response axis. We therefore regard that the presented analysis is meaningful, showing how in silico investigations can be used to guide the experimentalists in understanding how the HPA axis activity changes under chronic disease and/or specific pharmacological manipulations.

Modelling of the thyroid hormone synthesis as a part of nonlinear reaction mechanism with feedback.

The synthesis of thyroid hormones in the hypothalamic-pituitary-thyroid (HPT) axis was studied. For this purpose, a reaction model for HPT axis with stoichiometric relations between the main reaction species was postulated. Using the law of mass action, this model has been transformed into a set of nonlinear ordinary differential equations. This new model has been examined by stoichiometric network analysis (SNA) with the aim to see if it possesses the ability to reproduce oscillatory ultradian dynamics founded on the internal feedback mechanism. In particular, a feedback regulation of TSH production based on the interplay between TRH, TSH, somatostatin and thyroid hormones was proposed. Besides, the ten times larger amount of produced T4 with respect to T3 in the thyroid gland was successfully simulated. The properties of SNA in combination with experimental results, were used to determine the unknown parameters (19 rate constants of particular reaction steps) necessary for numerical investigations. The steady-state concentrations of 15 reactive species were tuned to be consistent with the experimental data. The predictive potential of the proposed model was illustrated on numerical simulations of somatostatin influence on TSH dynamics investigated experimentally by Weeke et al. in 1975. In addition, all programs for SNA analysis were adapted for this kind of a large model. The procedure of calculating rate constants from steady-state reaction rates and very limited available experimental data was developed. For this purpose, a unique numerical method was developed to fine-tune model parameters while preserving the fixed rate ratios and using the magnitude of the experimentally known oscillation period as the only target value. The postulated model was numerically validated by perturbation simulations with somatostatin infusion and the results were compared with experiments available in literature. Finally, as far as we know, this reaction model with 15 variables is the most dimensional one that have been analysed mathematically to obtain instability region and oscillatory dynamic states. Among the existing models of thyroid homeostasis this theory represents a new class that may improve our understanding of basic physiological processes and helps to develop new therapeutic approaches. Additionally, it may pave the way to improved diagnostic methods for pituitary and thyroid disorders.

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.

Predictive modeling of the hypothalamic-pituitary-adrenal (HPA) axis response to acute and chronic stress.

Detailed dynamics of the hypothalamic-pituitary-adrenal (HPA) axis is complex, depending on the individual metabolic load of an organism, its current status (healthy/ill, circadian phase (day/night), ultradian phase) and environmental impact. Therefore, it is difficult to compare the HPA axis activity between different individuals or draw unequivocal conclusions about the overall status of the HPA axis in an individual using single time-point measurements of cortisol levels. The aim of this study is to identify parameters that enable us to compare different dynamic states of the HPA axis and use them to investigate self-regulation mechanisms in the HPA axis under acute and chronic stress. In this regard, a four-dimensional stoichiometric model of the HPA axis was used. Acute stress was modeled by inducing an abrupt change in cortisol level during the course of numerical integration, whereas chronic stress was modeled by changing the mean stationary state concentrations of CRH. Effects of acute stress intensity, duration and time of onset with respect to the ultradian amplitude, ultradian phase and the circadian phase of the perturbed oscillation were studied in detail. Bifurcation analysis was used to predict the response of the HPA axis to chronic stress. Model predictions were compared with experimental findings reported in the literature and relevance for pharmacotherapy with glucocorticoids was discussed.

Intermittent Chaos in the CSTR Bray-Liebhafsky Oscillator-Specific Flow Rate Dependence.

Dynamic states with intermittent oscillations consist of a chaotic mixture of large amplitude relaxation oscillations grouped in bursts, and between them, small-amplitude sinusoidal oscillations, or even the quiescent parts, known as gaps. In this study, intermittent dynamic states were generated in Bray-Liebhafsky (BL) oscillatory reaction in an isothermal continuously-fed, well-stirred tank reactor (CSTR) controled by changes of specific flow rate. The intermittent states were found between two regular periodic states and obtained for specific flow rate values from 0.020 to 0.082 min-1. Phenomenological analysis based on the quantitative characteristics of intermittent oscillations, as well as, the largest Lyapunov exponents calculated from experimentally obtained time series, both indicated the same type of behavior. Namely, fully developed chaos arises when approaching to the vertical asymptote which is somewhere between two bifurcations. Hence, this study proposes described route to fully developed chaos in the Bray-Liebhafsky oscillatory reaction as an explanation for experimentally observed intermittent dynamics. This is in correlation with our previously obtained results where the most chaotic intermittent chaos was achieved between the periodic oscillatory dynamic state and stable steady state, generated in BL under CSTR conditions by varying temperature and inflow potassium iodate concentration. Moreover, it was shown that, besides the largest Lyapunov exponent, analysis of chaos in experimentally obtained intermittent states can be achieved by a simpler approach which involves using the quantitative characteristics of the BL reaction evolution, that is, the number and length of gaps and bursts obtained for the various values of specific flow rates.

Stoichiometric network analysis and associated dimensionless kinetic equations. Application to a model of the Bray-Liebhafsky reaction.

The stoichiometric network analysis (SNA) introduced by B. L. Clarke is applied to a simplified model of the complex oscillating Bray-Liebhafsky reaction under batch conditions, which was not examined by this method earlier. This powerful method for the analysis of steady-states stability is also used to transform the classical differential equations into dimensionless equations. This transformation is easy and leads to a form of the equations combining the advantages of classical dimensionless equations with the advantages of the SNA. The used dimensionless parameters have orders of magnitude given by the experimental information about concentrations and currents. This simplifies greatly the study of the slow manifold and shows which parameters are essential for controlling its shape and consequently have an important influence on the trajectories. The effectiveness of these equations is illustrated on two examples: the study of the bifurcations points and a simple sensitivity analysis, different from the classical one, more based on the chemistry of the studied system.

Corticosterone oscillations during mania induction in the lateral hypothalamic kindled rat-Experimental observations and mathematical modeling.

Changes in the hypothalamic-pituitary-adrenal (HPA) axis activity constitute a key component of bipolar mania, but the extent and nature of these alterations are not fully understood. We use here the lateral hypothalamic-kindled (LHK) rat model to deliberately induce an acute manic-like episode and measure serum corticosterone concentrations to assess changes in HPA axis activity. A mathematical model is developed to succinctly describe the entwined biochemical transformations that underlay the HPA axis and emulate by numerical simulations the considerable increase in serum corticosterone concentration induced by LHK. Synergistic combination of the LHK rat model and dynamical systems theory allows us to quantitatively characterize changes in HPA axis activity under controlled induction of acute manic-like states and provides a framework to study in silico how the dynamic integration of neurochemical transformations underlying the HPA axis is disrupted in these states.

Determination of paracetamol in pure and pharmaceutical dosage forms by pulse perturbation technique.

A new procedure for kinetic determination of paracetamol in pharmaceuticals is proposed. The method is based on potentiometric monitoring of the concentration perturbations of the matrix reaction system being in a stable non-equilibrium stationary state close to the bifurcation point. In the case considered as the matrix system, the Bray-Liebhafsky oscillatory reaction is used. The response of the matrix system to the perturbations by different concentrations of paracetamol is followed by a Pt-electrode. Proposed method relies on the linear relationship between maximal potential shift, DeltaEm, and the logarithm of added paracetamol amounts. It is obtained in optimized experimental conditions for variable amounts of paracetamol in the range 0.0085 and 1.5 micromol. The sensitivity and precision of proposed method were quite good (0.0027 micromol as the limit of detection and 2.4% as R.S.D.). Some aspects of possible chemical interactions between paracetamol and matrix are discussed. Applicability of the proposed method to the direct determination of paracetamol in pharmaceutical formulations was demonstrated.

Modelling cholesterol effects on the dynamics of the hypothalamic-pituitary-adrenal (HPA) axis.

A mathematical model of the hypothalamic-pituitary-adrenal (HPA) axis with cholesterol as a dynamical variable was derived to investigate the effects of cholesterol, the primary precursor of all steroid hormones, on the ultradian and circadian HPA axis activity. To develop the model, the parameter space was systematically examined by stoichiometric network analysis to identify conditions for ultradian oscillations, determine conditions under which dynamic transitions, i.e. bifurcations occur and identify bifurcation types. The bifurcations were further characterized using numerical simulations. Model predictions agree well with empirical findings reported in the literature, indicating that cholesterol levels may critically affect the global dynamics of the HPA axis. The proposed model provides a base for better understanding of experimental observations, it may be used as a tool for designing experiments and offers useful insights into the characteristics of basic dynamic regulatory mechanisms that, when impaired, may lead to the development of some modern-lifestyle-associated diseases.

Numerically simulated pH-induced reactivation of catalytic activity of horseradish peroxidase.

A model mechanism that accounts for the experimentally observed pH-dependent enhancement of enzymatic activity of horseradish peroxidase through the catalase-like pathway is proposed. Predictions of the model are tested against a number of experimental results to confirm that kinetic constants used in the numerical simulation are correctly chosen and that the model can be used to emulate the reaction between horseradish peroxidase and hydrogen peroxide in a wide range of conditions.

Mathematical modeling of the hypothalamic-pituitary-adrenal system activity.

Mathematical modeling has proven to be valuable in understanding of the complex biological systems dynamics. In the present report we have developed an initial model of the hypothalamic-pituitary-adrenal system self-regulatory activity. A four-dimensional non-linear differential equation model of the hormone secretion was formulated and used to analyze plasma cortisol levels in humans. The aim of this work was to explore in greater detail the role of this system in normal, homeostatic, conditions, since it is the first and unavoidable step in further understanding of the role of this complex neuroendocrine system in pathophysiological conditions. Neither the underlying mechanisms nor the physiological significance of this system are fully understood yet.

Malonic acid concentration as a control parameter in the kinetic analysis of the Belousov-Zhabotinsky reaction under batch conditions.

The influence of the initial malonic acid concentration [MA]0 (8.00 x 10(-3) < or = [MA]0 < or = 4.30 x 10(-2) mol dm(-3)) in the presence of bromate (6.20 x 10(-2) mol dm(-3)), bromide (1.50 x 10(-5) mol dm(-3)), sulfuric acid (1.00 mol dm(-3)) and cerium sulfate (2.50 x 10(-3) mol dm(-3)) on the dynamics and the kinetics of the Belousov-Zhabotinsky (BZ) reactions was examined under batch conditions at 30.0 degrees C. The kinetics of the BZ reaction was analyzed by the earlier proposed method convenient for the examinations of the oscillatory reactions. In the defined region of parameters where oscillograms with only large-amplitude relaxation oscillations appeared, the pseudo-first order of the overall malonic acid decomposition with a corresponding rate constant of 2.14 x 10(-2) min(-1) was established. The numerical results on the dynamics and kinetics of the BZ reaction, carried out by the known skeleton model including the Br2O species, were in good agreement with the experimental ones. The already found saddle node infinite period (SNIPER) bifurcation point in transition from a stable quasi-steady state to periodic orbits and vice versa is confirmed by both experimental and numerical investigations of the system under consideration. Namely, the large-amplitude relaxation oscillations with increasing periods between oscillations in approaching the bifurcation points at the beginning and the end of the oscillatory domain, together with excitability of the stable quasi-steady states in their vicinity are obtained.

Determination of ascorbic acid in pharmaceutical dosage forms and urine by means of an oscillatory reaction system using the pulse perturbation technique.

A simple and reliable method for the determination of ascorbic acid (AA) is proposed and validated. It is based on potentiometric monitoring of the concentration perturbations of an oscillatory reaction system in a stable nonequilibrium stationary state close to the bifurcation point. The response of the Bray-Liebhafsky (BL) oscillatory reaction as a matrix, to the perturbation by different concentrations of AA, is followed by a Pt electrode. The linear relationship between maximal potential shift and the logarithm of the amount of AA is obtained between 0.01 and 1.0 micromol. The sensitivity of the proposed method (as the limit of detection) is 0.009 micromol and the method has excellent sample throughput (30 samples per hour). The procedure was used for AA determination in pharmaceutical formulations and urine. The results are in agreement with those obtained using the official method. Some aspects of the possible mechanism of AA action on the BL oscillating chemical system are discussed.

Kinetic determination of morphine by means of Bray-Liebhafsky oscillatory reaction system using analyte pulse perturbation technique.

A novel kinetic method for micro-quantitative determinations of morphine (MH) is proposed and validated. The method is based on the potentiometric monitoring of the concentration perturbations of the oscillatory reaction system being in a stable non-equilibrium stationary state close to the bifurcation point between stable and oscillatory state. The response of the Bray-Liebhafsky (BL) oscillatory reaction as a matrix system, to the perturbations by different concentrations of morphine, is followed by a Pt-electrode. The proposed method relies on the linear relationship between maximal potential shift, DeltaE(m), and the logarithm of the added morphine amounts in the range of 0.004-0.18 micromol. Under optimum conditions, the sensitivity of the proposed method (as the limit of detection) is 0.001 micromol and the method is featured by good precision (R.S.D.=1.6%) as well as the excellent sample throughput (45 samples h(-1)). In addition to standard solution analysis, this approach was successfully applied for quantitative determination of morphine in a typical pharmaceutical dosage form. Some aspects of the possible mechanism of morphine action on the BL oscillating chemical system are discussed in detail.

Complex and chaotic oscillations in a model for the catalytic hydrogen peroxide decomposition under open reactor conditions.

Numerous periodic and aperiodic dynamic states obtained in a model for hydrogen peroxide decomposition in the presence of iodate and hydrogen ions (the Bray-Liebhafsky reaction) realized in an open reactor (CSTR), where the flow rate was the control parameter, have been investigated numerically. Between two Hopf bifurcation points, different simple and complex oscillations and different routes to chaos were observed. In the region of the mixed-mode evolution of the system, the transitions between two successive mixed-mode simple states are realized by period-doubling of the initial state leading to a chaotic window in which the next dynamic state emerges mixed with the initial one. It appears in increasing proportions in concatenated patterns until total domination. Thus, with increasing flow rate the period-doubling route to chaos was obtained, whereas with decreasing flow rate the peak-adding route to chaos was obtained. Moreover, in very narrow regions of flow rates, chaotic mixtures of mixed-mode patterns were observed. This evolution of patterns repeats until the end of the mixed-mode region at high flow rates that corresponds to chaotic mixtures of one large and many small amplitude oscillations. Starting from the reverse Hopf bifurcation point and decreasing the flow rate, simple small amplitude sinusoidal oscillations were encountered and then the period-doubling route to chaos. With a further decreasing flow rate, the mixed-mode oscillations emerge inside the chaotic window.

Microquantitative determination of hesperidin by pulse perturbation of the oscillatory reaction system.

A simple and reliable kinetic method for the determination of hesperidin (Hesp) is developed. It is based on potentiometric monitoring of the concentration perturbations of the matrix reaction system which is in a stable non-equilibrium stationary state close to the bifurcation point. The Bray-Liebhafsky oscillatory reaction is used as the matrix system. The response of the matrix to perturbations by different concentrations of Hesp is followed by using a Pt electrode. A linear relationship between maximal potential shift and the logarithm of Hesp concentrations is obtained between 7.5 and 599.4 microg mL-1. The limit of detection is 0.65 microg mL-1. The described procedure has been successfully applied to the determination of Hesp from different sources (capsules, industrial and hand-squeezed orange juice, and white wine).