Hysteresis,oscillations, and pattern formation in realistic immobilized enzyme systems

Summary Hysteresis, oscillations, and pattern formation in realistic biochemical systems governed by P.D.E.s are considered from both numerical and mathematical points of view. Analysis of multiple steady states in the case of hysteresis, and bifurcation theory in the cases of oscillations and pattern formation, account for the observed numerical results. The possibility to realize these systems experimentally is their main interest, thus bringing further arguments in favor of theories explaining basic biological phenomena by diffusion and reaction.     

A theoretical treatment of damped oscillations in the transient state kinetics of single-enzyme reactions

An extension of the available kinetic theory for reactions in the transient state is presented which establishes that single-enzyme reactions may exhibit damped oscillations under the conditions of standard kinetic experiments performed by stopped-flow techniques. Such oscillations may occur for reasonable magnitudes of rate constants in the enzymic reaction mechanism and at physiological concentrations of enzyme and substrate. In the simplest reaction systems, the oscillations will be strongly damped and lead to progress curves resembling those of a reaction governed by standard exponential transients; statistical regression methods may then have to be applied for their detection and characterization. The observation that single-enzyme reactions may exhibit oscillatory behaviour points to a previously unrecognized possible source of the damped oscillations observed in metabolic systems such as the pathways of glycolysis or photosynthesis.     

Equivalence of branched and unbranched Michaelian pathways concerning periodic signal transmission

Sinusoidal oscillation transmission through branched metabolic pathways is studied. Two systems are analyzed, which are composed of two convergent reaction branches and differ in the length of one of them. Linear kinetics is assumed first. Michaelis-Menten enzymes are then considered by using previous results that suggest their behavior with respect to propagation of oscillations is close to linearity around the mean input flux. As a result, there exist ways to modulate the activity of the enzymes so that propagation is equivalent for branched and specific unbranched pathways. Cells may have taken advantage of such a possibility in cases where oscillations have a biological role.     

Stability analysis of an artificial biomolecular oscillator with non-cooperative regulatory interactions

Oscillators are essential to fuel autonomous behaviours in molecular systems. Artificial oscillators built with programmable biological molecules such as DNA and RNA are generally easy to build and tune, and can serve as timers for biological computation and regulation. We describe a new artificial nucleic acid biochemical reaction network, and we demonstrate its capacity to exhibit oscillatory solutions. This network can be built in vitro using nucleic acids and three bacteriophage enzymes, and has the potential to be implemented in cells. Numerical simulations suggest that oscillations occur in a realistic range of reaction rates and concentrations.     

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.     

Mathematical models of endocrine systems

There is proposed a generalized mathematical model of endocrine systems, consisting of a set of differential equations which describe a chain of chemical reactions. The product of each reaction stimulates or inhibits some other reaction in the chain except possibly the last, which may or may not influence the system. At least one reaction must be independent and able to proceed without stimulation or inhibition by the products of other reactions. If only two reactions of the type assumed constitute a closed chain, sustained periodic variations in the concentrations of the reaction products cannot occur. If the chain consists of three or more reactions forming a closed loop, sustained oscillations, such as are observed in the menstrual cycle or in the mental disorder called periodic catatonia, can occur under suitable conditions. In this case, the concentrations of the system components exhibit relaxation oscillations characterized by periodic degeneration of the system when an independent reaction becomes completely inhibited by other reaction products. A set of conditions sufficient to produce periodicities in component concentrations is presented. Application of the model to the normally periodic system of the menstrual cycle and to the abnormal endocrine system which causes periodic catatonia is discussed.     

Oscillatory enzyme reactions and Michaelis–Menten kinetics

Oscillations occur in a number of enzymatic systems as a result of feedback regulation. How Michaelis–Menten kinetics influences oscillatory behavior in enzyme systems is investigated in models for oscillations in the activity of phosphofructokinase (PFK) in glycolysis and of cyclin-dependent kinases in the cell cycle. The model for the PFK reaction is based on a product-activated allosteric enzyme reaction coupled to enzymatic degradation of the reaction product. The Michaelian nature of the product decay term markedly influences the period, amplitude and waveform of the oscillations. Likewise, a model for oscillations of Cdc2 kinase in embryonic cell cycles based on Michaelis–Menten phosphorylation–dephosphorylation kinetics shows that the occurrence and amplitude of the oscillations strongly depend on the ultrasensitivity of the enzymatic cascade that controls the activity of the cyclin-dependent kinase.     

Oscillations and efficiency in glycolysis

We suggest that temporal oscillations of concentrations of intermediates in biochemical reaction systems may enhance the efficiency of free energy conversion (reduce dissipation) in those reactions. Experiments on glycolysis are used to estimate the Gibbs free energy changes along the glycolysis mechanism, and to postulate a construct for the glycolysis "machine" which involves: the PFK reaction as the primary oscillophor; the GAPDH reaction as a phase-shifting device; and the PK reaction with the property of intrinsic oscillatory response at resonance with the driving frequency. Analysis of a simple reaction mechanism with these postulated properties shows that the conversion of free energy from reactants to products is more efficient in an oscillatory than a steady state operation. The efficiency of free energy conversion in glycolysis from glucose + ADP to products + ATP is estimated to be increased by 5--10% due to oscillations. This may have been relevant for the evolutionary development of oscillations such as in glycolysis, especially in anaerobic cells.     

Mean Lifetime and First-Passage Time of the Enzyme Species Involved in an Enzyme Reaction. Application to Unstable Enzyme Systems

Taking as starting point the complete analysis of mean residence times in linear compartmental systems performed by Garcia-Meseguer et al. (Bull. Math. Biol. 65:279–308, 2003) as well as the fact that enzyme systems, in which the interconversions between the different enzyme species involved are of first or pseudofirst order, act as linear compartmental systems, we hereby carry out a complete analysis of the mean lifetime that the enzyme molecules spend as part of the enzyme species, forms, or groups involved in an enzyme reaction mechanism. The formulas to evaluate these times are given as a function of the individual rate constants and the initial concentrations of the involved species at the onset of the reaction. We apply the results to unstable enzyme systems and support the results by using a concrete example of such systems. The practicality of obtaining the mean times and their possible application in a kinetic data analysis is discussed.     

Oscillations in Biochemical Reaction Networks Arising from Pairs of Subnetworks

Biochemical reaction models show a variety of dynamical behaviors, such as stable steady states, multistability, and oscillations. Biochemical reaction networks with generalized mass action kinetics are represented as directed bipartite graphs with nodes for species and reactions. The bipartite graph of a biochemical reaction network usually contains at least one cycle, i.e., a sequence of nodes and directed edges which starts and ends at the same species node. Cycles can be positive or negative, and it has been shown that oscillations can arise as a result of either a positive cycle or a negative cycle. In earlier work it was shown that oscillations associated with a positive cycle can arise from subnetworks with an odd number of positive cycles. In this article we formulate a similar graph-theoretic condition, which generalizes the negative cycle condition for oscillations. This new graph-theoretic condition for oscillations involves pairs of subnetworks with an even number of positive cycles. An example of a calcium reaction network with generalized mass action kinetics is discussed in detail.     

Finding complex oscillatory phenomena in biochemical systems. An empirical approach

Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmicity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the cAMP signalling system of Dictyostelium discoideum amoebae.     

From bistability to oscillations in a model for the isocitrate dehydrogenase reaction

Considered is a bienzymatic system consisting of isocitrate dehydrogenase (IDH, EC 1.1.1.42), which transforms NADP(+) into NADPH, and of diaphorase (DIA, EC 1.8.1.4), which catalyzes the reverse reaction. Experimental evidence as well as a theoretical model show the possibility of a coexistence between two stable steady states in this reaction system. The phenomenon originates from the regulatory properties of IDH. We extend the analysis of a theoretical model proposed for the IDH-DIA bienzymatic system and investigate the occurrence of different modes of bistability, with or without hysteresis, i.e. in the presence of two or only one limit point bounding the domain of multiple steady states. The analysis indicates that the two types of bistability may sometimes be observed sequentially as a given control parameter is progressively increased. We further obtain conditions in which sustained oscillations develop in the model. These results establish the isocitrate dehydrogenase reaction coupled to diaphorase as a suitable candidate for further experimental and theoretical studies of bistability and oscillations in biochemical systems.     

Diffusive coupling can discriminate between similar reaction mechanisms in an allosteric enzyme system

Background  

A central question for the understanding of biological reaction networks is how a particular dynamic behavior, such as bistability or oscillations, is realized at the molecular level. So far this question has been mainly addressed in well-mixed reaction systems which are conveniently described by ordinary differential equations. However, much less is known about how molecular details of a reaction mechanism can affect the dynamics in diffusively coupled systems because the resulting partial differential equations are much more difficult to analyze.     

Spatiotemporal behaviors in immobilized enzyme systems

The immobilization of enzymes within an artificial membrane, with a homogeneous distribution of the active sites, allows a simple modelling in a well defined context. The systems are described by non-linear PDE'S, taking into account enzyme reaction and metabolite diffusion. These equations can exhibit several types of behaviors, qualitatively different from those observed in solution, such as hysteresis, oscillations and pattern formations. Preliminary experimental results have shown the existence of sustained oscillations and instabilities with immobilized acetylcholinesterase and phosphofructokinase.     

Influence of parameter values on the oscillation sensitivities of two p53–Mdm2 models

Biomolecular networks that present oscillatory behavior are ubiquitous in nature. While some design principles for robust oscillations have been identified, it is not well understood how these oscillations are affected when the kinetic parameters are constantly changing or are not precisely known, as often occurs in cellular environments. Many models of diverse complexity level, for systems such as circadian rhythms, cell cycle or the p53 network, have been proposed. Here we assess the influence of hundreds of different parameter sets on the sensitivities of two configurations of a well-known oscillatory system, the p53 core network. We show that, for both models and all parameter sets, the parameter related to the p53 positive feedback, i.e. self-promotion, is the only one that presents sizeable sensitivities on extrema, periods and delay. Moreover, varying the parameter set values to change the dynamical characteristics of the response is more restricted in the simple model, whereas the complex model shows greater tunability. These results highlight the importance of the presence of specific network patterns, in addition to the role of parameter values, when we want to characterize oscillatory biochemical systems.

Electronic supplementary material

The online version of this article (doi:10.1007/s11693-015-9173-y) contains supplementary material, which is available to authorized users.     

Simulation of chemical oscillations in membranes

A computer simulation model has been developed to follow chemical oscillations in a membrane for immobilized enzyme systems. It is a discrete particle type model which follows the spatial and temporal fluctuations of the concentrations in a reaction involving two substrates. The parameters can be readily varied to allow dissipative structures to result from the sustained nonlinear reaction kinetics and to determine which parameters cause damping of the oscillations. The nature of the diffusion mechanism allows extension to more than one dimension.     

Dancing between the devil and deep blue sea: the stabilizing effect of enemy-free and victimless sinks

Theoretical and empirical studies have shown that enemy–victim interactions in spatially homogenous environments can exhibit diverging oscillations which result in the extinction of one or both species. For enemy–victim models with overlapping generations, we investigate the dynamical implications of spatial heterogeneity created by enemy-free sinks or victimless sinks. An enemy-free sink is a behavioral, physiological or ecological state that reduces or eliminates the victim's vulnerability to the enemy but cannot sustain the victim population. For victims that move in an ideal-free manner, we prove that the inclusion of an enemy-free sink shifts the population dynamics from diverging oscillations to stable oscillations. During these stable oscillations, the victim disperses in an oscillatory manner between the enemy-free sink and the enemy-occupied patch. Enemy-free sinks with lower mortality rates exhibit oscillations with smaller amplitudes and longer periods. A victimless sink, on the other hand, is a behavioral, physiological or ecological state in which the enemy has limited (or no) access to its victims. For enemies that move in an ideal-free manner, we prove that victimless sinks also stabilize diverging oscillations. Simulations suggest that suboptimal behavior due to information gathering or learning limitations amplify oscillations for systems with enemy-free sinks and dampen oscillations for systems with victimless sinks. These results illustrate that the coupling of a sink created by unstable enemy–victim interactions and a sink created by unsuitable environmental conditions can result in population persistence at the landscape level.     

Simplest Kinetic Schemes for Biochemical Oscillators

The topological structure of the simplest critical fragments in biochemical systems has been characterized. The conditions are considered where the critical fragments induce oscillations of the concentrations of the system participants. To illustrate, three biochemical systems (transport of ions through a membrane, protein phosphorylation, and two-substrate reaction) are discussed. The kinetic schemes of these systems contain one of the discovered critical fragments. Relaxation oscillations of the concentrations of the system participants were demonstrated using the numerical integration method.     

Allosteric regulation, cooperativity, and biochemical oscillations

Allosteric regulation is associated with a number of periodic phenomena in biochemical systems. The cooperative nature of such regulatory interactions provides a source of nonlinearity that favors oscillatory behavior. We assess the role of cooperativity in the onset of biochemical oscillations by analyzing two specific examples. First, we consider a model for a product-activated allosteric enzyme which has previously been proposed to account for glycolytic oscillations. While enzyme cooperativity plays an important role in the occurrence of oscillations, we show that these may nevertheless occur in the absence of cooperativity when the reaction product is removed in a Michaelian rather than linear manner. The second model considered was recently proposed to account for signal-induced oscillations of intracellular calcium. This phenomenon originates from a nonlinear process of calcium-induced calcium release. Here also, the cooperative nature of that positive feedback favors the occurrence of oscillations but is not absolutely required for periodic behavior. Besides underlining the importance of cooperativity, the results highlight the role of diffuse nonlinearities distributed over several steps within a regulated system: even in the absence of cooperativity, such mild nonlinearities (e.g., of the Michaelian type) may combine to raise the overall degree of nonlinearity up to the level required for oscillations.