Oscillatory kinetics of the peroxidase-oxidase reaction in an open system. Experimental and theoretical studies

Complex dynamics in an unexplored simple model of the peroxidase-oxidase reaction

A previously overlooked version of the so-called Olsen model of the peroxidase-oxidase reaction has been studied numerically using 2D isospike stability and maximum Lyapunov exponent diagrams and reveals a rich variety of dynamic behaviors not observed before. The model has a complex bifurcation structure involving mixed-mode and bursting oscillations as well as quasiperiodic and chaotic dynamics. In addition, multiple periodic and non-periodic attractors coexist for the same parameters. For some parameter values, the model also reveals formation of mosaic patterns of complex dynamic states. The complex dynamic behaviors exhibited by this model are compared to those of another version of the same model, which has been studied in more detail. The two models show similarities, but also notable differences between them, e.g., the organization of mixed-mode oscillations in parameter space and the relative abundance of quasiperiodic and chaotic oscillations. In both models, domains with chaotic dynamics contain apparently disorganized subdomains of periodic attractors with dinoflagellate-like structures, while the domains with mainly quasiperiodic behavior contain subdomains with periodic attractors organized as regular filamentous structures. These periodic attractors seem to be organized according to Stern-Brocot arithmetics. Finally, it appears that toroidal (quasiperiodic) attractors develop into first wrinkled and then fractal tori before they break down to chaotic attractors.

Turing Instabilities and Rotating Spiral Waves in Glycolytic Processes

We study single-frequency oscillations and pattern formation in the glycolytic process modeled by a reduction in the well-known Sel'kov's equations (Sel'kov in Eur J Biochem 4:79, 1968), which describe, in the whole cell, the phosphofructokinase enzyme reaction. By using averaging theory, we establish the existence conditions for limit cycles and their limiting average radius in the kinetic reaction equations. We analytically establish conditions on the model parameters for the appearance of unstable nonlinear modes seeding the formation of two-dimensional patterns in the form of classical spots and stripes. We also establish the existence of a Hopf bifurcation, which characterizes the reaction dynamics, producing glycolytic rotating spiral waves. We numerically establish parameter regions for the existence of these spiral waves and address their linear stability. We show that as the model tends toward a suppression of the relative source rate, the spiral wave solution loses stability. All our findings are validated by full numerical simulations of the model equations. Finally, we discuss in vitro evidence of spatiotemporal activity patterns found in glycolytic experiments, and propose plausible biological implications of our model results.

Complexity in subnetworks of a peroxidase-oxidase reaction model

The peroxidase-oxidase (PO) reaction is a paradigmatic (bio)chemical system well suited to study the organization and stability of self-sustained oscillatory phases typically present in nonlinear systems. The PO reaction can be simulated by the state-of-the-art Bronnikova-Fedkina-Schaffer-Olsen model involving ten coupled ordinary differential equations. The complex and dynamically rich distribution of self-sustained oscillatory stability phases of this model was recently investigated in detail. However, would it be possible to understand aspects of such a complex model using much simpler models? Here, we investigate stability phases predicted by three simple four-variable subnetworks derived from the complete model. While stability diagrams for such subnetworks are found to be distorted compared to those of the complete model, we find them to surprisingly preserve significant features of the original model as well as from the experimental system, e.g., period-doubling and period-adding scenarios. In addition, return maps obtained from the subnetworks look very similar to maps obtained in the experimental system under different conditions. Finally, two of the three subnetwork models are found to exhibit quint points, i.e., recently reported singular points where five distinct stability phases coalesce. We also provide experimental evidence that such quint points are present in the PO reaction.

Complexity of a peroxidase-oxidase reaction model

The peroxidase-oxidase oscillating reaction was the first (bio)chemical reaction to show chaotic behaviour. The reaction is rich in bifurcation scenarios, from period-doubling to peak-adding mixed mode oscillations. Here, we study a state-of-the-art model of the peroxidase-oxidase reaction. Using the model, we report systematic numerical experiments exploring the impact of changing the enzyme concentration on the dynamics of the reaction. Specifically, we report high-resolution phase diagrams predicting and describing how the reaction unfolds over a quite extended range of enzyme concentrations. Surprisingly, such diagrams reveal that the enzyme concentration has a huge impact on the reaction evolution. The highly intricate dynamical behaviours predicted here are difficult to establish theoretically due to the total absence of an adequate framework to solve nonlinearly coupled differential equations. But such behaviours may be validated experimentally.

Dynamics of Rössler Prototype-4 System: Analytical and Numerical Investigation

In this paper, the dynamics of a 3D autonomous dissipative nonlinear system of ODEs-Rössler prototype-4 system, was investigated. Using Lyapunov-Andronov theory, we obtain a new analytical formula for the first Lyapunov’s (focal) value at the boundary of stability of the corresponding equilibrium state. On the other hand, the global analysis reveals that the system may exhibit the phenomena of Shilnikov chaos. Further, it is shown via analytical calculations that the considered system can be presented in the form of a linear oscillator with one nonlinear automatic regulator. Finally, it is found that for some new combinations of parameters, the system demonstrates chaotic behavior and transition from chaos to regular behavior is realized through inverse period-doubling bifurcations.

Chaos in the peroxidase-oxidase oscillator

The peroxidase-oxidase (PO) reaction involves the oxidation of reduced nicotinamide adenine dinucleotide by molecular oxygen. When both reactants are supplied continuously to a reaction mixture containing the enzyme and a phenolic compound, the reaction will exhibit oscillatory behavior. In fact, the reaction exhibits a zoo of dynamical behaviors ranging from simple periodic oscillations to period-doubled and mixed mode oscillations to quasiperiodicity and chaos. The routes to chaos involve period-doubling, period-adding, and torus bifurcations. The dynamic behaviors in the experimental system can be simulated by detailed semiquantitative models. Previous models of the reaction have omitted the phenolic compound from the reaction scheme. In the current paper, we present new experimental results with the oscillating PO reaction that add to our understanding of its rich dynamics, and we describe a new variant of a previous model, which includes the chemistry of the phenol in the reaction mechanism. This new model can simulate most of the experimental behaviors of the experimental system including the new observations presented here. For example, the model reproduces the two main routes to chaos observed in experiments: (i) a period-doubling scenario, which takes place at low pH, and a period-adding scenario involving mixed mode oscillations (MMOs), which occurs at high pH. Our simulations suggest alternative explanations for the pH-sensitivity of the dynamics. We show that the MMO domains are separated by narrow parameter regions of chaotic behavior or quasiperiodicity. These regions start as tongues of secondary quasiperiodicity and develop into strange attractors through torus breakdown.

An appreciation of the prescience of Don Gilbert (1930-2011): master of the theory and experimental unravelling of biochemical and cellular oscillatory dynamics

We review Don Gilbert's pioneering seminal contributions that both detailed the mathematical principles and the experimental demonstration of several of the key dynamic characteristics of life. Long before it became evident to the wider biochemical community, Gilbert proposed that cellular growth and replication necessitate autodynamic occurrence of cycles of oscillations that initiate, coordinate and terminate the processes of growth, during which all components are duplicated and become spatially re-organised in the progeny. Initiation and suppression of replication exhibit switch-like characteristics, that is, bifurcations in the values of parameters that separate static and autodynamic behaviour. His limit cycle solutions present models developed in a series of papers reported between 1974 and 1984, and these showed that most or even all of the major facets of the cell division cycle could be accommodated. That the cell division cycle may be timed by a multiple of shorter period (ultradian) rhythms, gave further credence to the central importance of oscillatory phenomena and homeodynamics as evident on multiple time scales (seconds to hours). Further application of the concepts inherent in limit cycle operation as hypothesised by Gilbert more than 50 years ago are now validated as being applicable to oscillatory transcript, metabolite and enzyme levels, cellular differentiation, senescence, cancerous states and cell death. Now, we reiterate especially for students and young colleagues, that these early achievements were even more exceptional, as his own lifetime's work on modelling was continued with experimental work in parallel with his predictions of the major current enterprises of biological research.

Systems and synthetic biology approaches in understanding biological oscillators

Background

Self-sustained oscillations are a ubiquitous and vital phenomenon in living systems. From primitive single-cellular bacteria to the most sophisticated organisms, periodicities have been observed in a broad spectrum of biological processes such as neuron firing, heart beats, cell cycles, circadian rhythms, etc. Defects in these oscillators can cause diseases from insomnia to cancer. Elucidating their fundamental mechanisms is of great significance to diseases, and yet challenging, due to the complexity and diversity of these oscillators.

Results

Approaches in quantitative systems biology and synthetic biology have been most effective by simplifying the systems to contain only the most essential regulators. Here, we will review major progress that has been made in understanding biological oscillators using these approaches. The quantitative systems biology approach allows for identification of the essential components of an oscillator in an endogenous system. The synthetic biology approach makes use of the knowledge to design the simplest, de novo oscillators in both live cells and cell-free systems. These synthetic oscillators are tractable to further detailed analysis and manipulations.

Conclusion

With the recent development of biological and computational tools, both approaches have made significant achievements.

Build to understand: synthetic approaches to biology

In this review we discuss how synthetic biology facilitates the task of investigating genetic circuits that are observed in naturally occurring biological systems. Specifically, we give examples where experimentation with synthetic gene circuits has been used to understand four fundamental mechanisms intrinsic to development and disease: multistability, stochastic gene expression, oscillations, and cell-cell communication. Within each area, we also discuss how mathematical modeling has been employed as an essential tool to guide the design of novel gene circuits and as a theoretical basis for exploring circuit topologies exhibiting robust behaviors in the presence of noise.

Effects of tyramine and 4-aminophenol on the oscillating peroxidase-oxidase reaction

The peroxidase-oxidase oscillator, a model of biological oscillations, is usually studied in conjunction with the effector molecule, 2,4-dichlorophenol. In this account, we present evidence of the effects of a naturally occurring phenol, tyramine, on the reaction, and also those of the structurally similar 4-aminophenol. Whereas 2,4-dichlorophenol gives rise to sustained oscillations at 40 μM, it was discovered that tyramine promotes damped oscillations at a concentration of 120 μM. Oxidation of NADH was completely inhibited by 4-aminophenol and ascorbate. In separate experiments, the peroxidase-catalyzed ring coupling of tyramine and 4-aminophenol was observed, which in the case of tyramine, may provide an explanation for the damping of oscillations.

PPARs, Cardiovascular Metabolism, and Function: Near- or Far-from-Equilibrium Pathways

Peroxisome proliferator-activated receptors (PPAR α, β/δ and γ) play a key role in metabolic regulatory processes and gene regulation of cellular metabolism, particularly in the cardiovascular system. Moreover, PPARs have various extra metabolic roles, in circadian rhythms, inflammation and oxidative stress. In this review, we focus mainly on the effects of PPARs on some thermodynamic processes, which can behave either near equilibrium, or far-from-equilibrium. New functions of PPARs are reported in the arrhythmogenic right ventricular cardiomyopathy, a human genetic heart disease. It is now possible to link the genetic desmosomal abnormalitiy to the presence of fat in the right ventricle, partly due to an overexpression of PPARγ. Moreover, PPARs are directly or indirectly involved in cellular oscillatory processes such as the Wnt-b-catenin pathway, circadian rhythms of arterial blood pressure and cardiac frequency and glycolysis metabolic pathway. Dysfunction of clock genes and PPARγ may lead to hyperphagia, obesity, metabolic syndrome, myocardial infarction and sudden cardiac death, In pathological conditions, regulatory processes of the cardiovascular system may bifurcate towards new states, such as those encountered in hypertension, type 2 diabetes, and heart failure. Numerous of these oscillatory mechanisms, organized in time and space, behave far from equilibrium and are “dissipative structures”.

Design principles of biochemical oscillators

Cellular rhythms are generated by complex interactions among genes, proteins and metabolites. They are used to control every aspect of cell physiology, from signalling, motility and development to growth, division and death. We consider specific examples of oscillatory processes and discuss four general requirements for biochemical oscillations: negative feedback, time delay, sufficient 'nonlinearity' of the reaction kinetics and proper balancing of the timescales of opposing chemical reactions. Positive feedback is one mechanism to delay the negative-feedback signal. Biological oscillators can be classified according to the topology of the positive- and negative-feedback loops in the underlying regulatory mechanism.

Biofilm effects on the peroxidase-oxidase reaction

In experiments on the kinetics of the peroxidase-oxidase oscillatory reaction in pH 5.l acetate buffer, biofilms form in less than 48 h on the quartz reactor surface. The nominally homogeneous peroxidase system shows dynamical changes in response to this biofilm growth, partially explaining subtle differences among dynamics observed over time and between laboratories. Kinetics data and model computations are correlated with micrographs of biofilm formation. It is evident that bare quartz also interacts with reaction species, so that the surface area-to-volume ratio is an important parameter on which observed dynamics depend.

Oscillations in peroxidase-catalyzed reactions and their potential function in vivo

The peroxidase-oxidase reaction has become a model system for the study of oscillations and complex dynamics in biochemical systems. In the present paper we give an overview of previous experimental and theoretical studies of the peroxidase-oxidase reaction. Recent in vitro experiments have raised the question whether the reaction also exhibits oscillations and complex dynamics in vivo. To investigate this possibility further we have undertaken new experimental studies of the reaction, using horseradish extracts and phenols which are widely distributed in plants. The results are discussed in light of the occurrence and a possible functional role of oscillations and complex dynamics of the peroxidase-oxidase reaction in vivo.

Oscillatory reaction of catalase wrapped by liposome

To date we have been studying the enzymatic oscillatory reaction caused by gradual entry of substrate via semi-permeable membrane. It has been found that many enzymes cause oscillatory reaction. Here we present an oscillatory reaction of enzyme wrapped by liposome. We used catalase as an enzyme, since its oscillatory reaction has been already investigated in detail in the absence of liposome. Distinct oscillation with shorter period than without liposome was obtained. On the other hand, it was shown that the presence of liposome facilitated the permeation rate of hydrogen peroxide through semi-permeable membrane. This is thought to be factor of shortening the oscillation period compared with that in the absence of liposome. Oscillation was also temperature dependent. Our finding may provide an important insight into the study of enzyme reaction taking part in rhythms in living systems.

Human myeloperoxidase catalyzes an oscillating peroxidase-oxidase reaction

We have studied the peroxidase-oxidase reaction catalyzed by human myeloperoxidase in an open system where both substrates-molecular oxygen and NADH-are supplied continuously to the reaction mixture. The reaction shows oscillatory kinetics at pH values around 5, provided that the reaction medium in addition to the enzyme and the substrates also contains an aromatic electron mediator such as melatonin or 4-hydroxybenzoic acid and chloride ions at concentrations >1mM. The experimental findings can be simulated by a detailed model of the reaction. The results are important for our understanding of oxidant production in neutrophils.

Mechanism of protection of peroxidase activity by oscillatory dynamics

The peroxidase-oxidase reaction is known to involve reactive oxygen species as intermediates. These intermediates inactivate many types of biomolecules, including peroxidase itself. Previously, we have shown that oscillatory dynamics in the peroxidase-oxidase reaction seem to protect the enzyme from inactivation. It was suggested that this is due to a lower average concentration of reactive oxygen species in the oscillatory state compared to the steady state. Here, we studied the peroxidase-oxidase reaction with either 4-hydroxybenzoic acid or melatonin as cofactors. We show that the protective effect of oscillatory dynamics is present in both cases. We also found that the enzyme degradation depends on the concentration of the cofactor and on the pH of the reaction mixture. We simulated the oscillatory behaviour, including the oscillation/steady state bistability observed experimentally, using a detailed reaction scheme. The computational results confirm the hypothesis that protection is due to lower average concentrations of superoxide radical during oscillations. They also show that the shape of the oscillations changes with increasing cofactor concentration resulting in a further decrease in the average concentration of radicals. We therefore hypothesize that the protective effect of oscillatory dynamics is a general effect in this system.

Mechanism of melatonin-induced oscillations in the peroxidase-oxidase reaction

Melatonin induces oscillations in the peroxidase-oxidase (PO) reaction catalyzed by horseradish peroxidase. We present here studies of the effect of pH, enzyme concentration, and concentration of melatonin on the oscillation frequency. We also present a mechanistic model to explain the experimentally observed changes in oscillation frequency. Using the data obtained here we are able to predict that oscillations will also occur in the PO reaction catalyzed by myeloperoxidase. Myeloperoxidase is an important protein in activated neutrophils and we provide evidence that the oscillations of NAD(P)H, superoxide and hydrogen peroxide in these cells may involve this enzyme. Thus, our experimental system can be considered a model system for the nonrespiratory oxygen metabolism in activated neutrophils and other similar cells participating in the defence against invading pathogens.

Oscillations in the peroxidase-oxidase reaction: a comparison of different peroxidases

The nonlinear behavior of the peroxidase-oxidase reaction was studied using structurally different peroxidases. For the first time sustained oscillations with peroxidases other than horseradish peroxidase in a single-enzyme system were observed. All peroxidases that showed significant oxidase activity were able to generate sustained oscillations. When adjusting the overall reaction rate, either of the two modifiers 2,4-dichlorophenol or Methylene blue could be omitted from the reaction. Due to the observation of different enzyme intermediates when using different peroxidases, we conclude that the mechanisms responsible for oscillatory kinetics may vary from one peroxidase to the other.

A hydrogen peroxide assay based on the peroxidase-oxidase reaction. Numerical simulation of the reaction mechanism

In order to explain an assay method for H2O2, based on the horseradish peroxidase/NADH/H2O2 system, we performed a numerical simulation of this complex reaction mechanism. For this purpose, we applied a simplified version of a model designed by Yokota and Yamazaki to our experimental conditions [Yokota, K. & Yamazaki, I. (1977) Biochemistry 16, 1913-1918]. This modification leads to a kinetic behaviour of the system very similar to our experimental results when the presence of superoxide dismutase is simulated. These simulations confirm that hydrogen peroxide concentrations are proportional to the total amount of NADH oxidized. We also showed that superoxide dismutase activity can be determined by this assay method. Moreover, in samples containing superoxide dismutase and hydrogen peroxide, the simulations showed that both superoxide dismutase and hydrogen peroxide can be determined using our assay method.

Free radicals in iron-containing systems

All oxidative damage in biological systems arises ultimately from molecular oxygen. Molecular oxygen can scavenge carbon-centered free radicals to form organic peroxyl radicals and hence organic hydroperoxides. Molecular oxygen can also be reduced in two one-electron steps to hydrogen peroxide in which case superoxide anion is an intermediate; or it can be reduced enzymatically so that no superoxide is released. Organic hydroperoxides or hydrogen peroxide can diffuse through membranes whereas hydroxyl radicals or superoxide anion cannot. Chain reactions, initiated by chelated iron and peroxides, can cause tremendous damage. Chain carriers are chelated ferrous ion; hydroxyl radical .OH, or alkoxyl radical .OR, and superoxide anion O2-. or organic peroxyl radical RO2.. Of these free radicals .OH and RO2. appear to be most harmful. All of the biological molecules containing iron are potential donors of iron as a chain initiator and propagator. An attacking role for superoxide dismutase is proposed in the phagocytic process in which it may serve as an intermediate enzyme between NADPH oxidase and myeloperoxidase. The sequence of reactants is O2----O2-.----H2O2----HOCl.

Computer simulation of sustained oscillations in peroxidase-oxidase reaction

A system of differential equations of second order exhibiting transitional behaviour and sustained oscillations has been obtained for a complete scheme of the peroxidase-oxidase reaction. The concentrations of hydrogen peroxide and of hydrogen donor radicals are slow variables of the system. The most essential reactions responsible for oscillations have been selected. Analysis of the system in phase plane and in parameter space has been carried out. The dependence of oscillation period and amplitude on the parameter values has been investigated.