Allosteric oscillatory enzymes: influence of the number of protomers on metabolic periodicities.

The study of a concerted allosteric model for an enzyme activated by the reaction product shows that this system can generate sustained metabolic oscillations regardless of the number of protomers constituting the enzyme. The analysis extends the results previously obtained in a dimeric model for the phosphofructokinase reaction which produces glycolytic periodicities. When the substrate and product concentrations evolve on comparable time scales, the amplitude of oscillations significantly drops as the number of enzyme subunits evolves from 2 to 8. The width of the domain of substrate injection rates which produce oscillations and the periodic variation in enzyme activity also depend on the number of protomers and on the time scale structure of the system. Theoretical predictions are compared with the experiments on glycolytic oscillations in yeast and muscle, and with the structural characteristics of phosphofructokinase. The results are also discussed in relation with the mechanism of cyclic AMP oscillations in the slime mold Dictyostelium discoideum.     

Allosteric oscillatory enzymes : Influence of the number of protomers on metabolic periodicities

The study of a concerted allosteric model for an enzyme activated by the reaction product shows that this system can generate sustained metabolic oscillations regardless of the number of protomers constituting the enzyme. The analysis extends the results previously obtained in a dimeric model for the phosphofructokinase reaction which produces glycolytic periodicities. When the substrate and product concentrations evolve on comparable time scales, the amplitude of oscillations significantly drops as the number of enzyme subunits evolves from 2 to 8. The width of the domain of substrate injection rates which produce oscillations and the periodic variation in enzyme activity also depend on the number of protomers and on the time scale structure of the system. Theoretical predictions are compared with the experiments on glycolytic oscillations in yeast and muscle, and with the structural characteristics of phosphofructokinase. The results are also discussed in relation with the mechanism of cyclic AMP oscillations in the slime mold Dictyostelium discoideum.     

On the role of enzyme cooperativity in metabolic oscillations: analysis of the Hill coefficient in a model for glycolytic periodicities.

The role of enzyme cooperativity in the mechanism of metabolic oscillations is analyzed in a concerted allosteric model for the phosphofructokinase reaction. This model of a dimer enzyme activated by the reaction product accounts quantitatively for glycolytic periodicities observed in yeast and muscle. The Hill coefficient characteristic of enzyme-substrate interactions is determined in the model, both at the steady state and in the course of sustained oscillations. Positive cooperativity is a prerequisite for periodic behavior. A necessary condition for oscillation in a dimer K system is a Hill coefficient larger than 1.6 at the unstable stationary state. The analysis suggests that positive as well as negative effectors of phosphofructokinase inhibit glycolytic oscillations by inducing a decrease in enzyme cooperativity. The results are discussed with respect to glycolytic and other metabolic periodicities.     

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.     

The regulatory design of an allosteric feedback loop: the effect of saturation by pathway substrate

Aspects of metabolic regulation can be fruitfully studied with a combination of generic modelling, control analysis and graphical analysis using rate characteristics. This paper analyses a prototypical supply-demand system consisting of a biosynthetic subsystem subject to allosteric inhibition by its product and a demand process that consumes this product. The effect of changes in affinity of the committing supply enzyme for the pathway substrate on the regulatory properties of the supply subsystem is compared for the Monod-Wyman-Changeux and the reversible Hill allosteric enzyme models. We found that the Hill model has a distinct advantage in that the steady-state concentration at which it maintains the product is set by the half-saturating product concentration and is independent of changes in the degree of saturation for substrate. In contrast, with the Monod-Wyman-Changeux model this set point varies with affinity for substrate. Explicitly incorporating reversibility in all rate equations made it possible to distinguish between kinetic and thermodynamic aspects of regulation. Combining the supply and demand rate characteristics allows us to explore both the control distribution at steady state and the regulatory performance of the system over a wide range of demand activities.     

Simple kinetic models explaining critical phenomena in enzymatic reactions with isomerization of the enzyme and substrate

Kinetic models for enzyme reactions are considered which take into account enzyme and substrate isomerization. Application of graph-theoretic methods allows to reveal fragments in schemes which may induce multiple stead-states or concentrational selfoscillations. The role of substrate isomers in the inhibition of enzyme isomers to produce critical phenomena is considered. The boundaries of parameter domains for critical phenomena are estimated. It is shown that the controlled change in concentrations of substrate and enzyme isomers may be important in regulation of enzyme systems, if different enzyme isomers are inhibited mainly by different substrate isomers. The models are used for interpretation of possible critical phenomena in the open reaction catalyzed by lactate dehydrogenase. It is shown that lactate dehydrogenase may act as a trigger in carbohydrate metabolism by changing "critically" its activity in relation to changes in pH and pyruvate fluxes. Slow enzyme inhibition by enolpyruvate is suggested as a possible reason for glycolytic oscillations.     

Simple and complex spatiotemporal structures in a glycolytic allosteric enzyme model

Pattern formation in glycolysis is studied with a classical reaction-diffusion allosteric enzyme model. It is found that, similar to recent experimental reports in the yeast extracts, a small magnitude local perturbation can induce transient target waves in a two dimensional oscillatory medium. An above threshold stimulation generates target waves which eventually evolve into spatiotemporal chaos upon collisions with the boundary or other wave activities. Detailed simulation studies show that the studied simple glycolytic reaction-diffusion model can support three types of spatiotemporal behaviors which are independent of the boundary conditions: (1) a spatially uniform stable steady state, (2) periodic global oscillations and (3) spatiotemporal chaos.     

Self-organization and dynamics of an open futile cycle

The dynamic behaviour of an open futile cycle composed of two enzymes has been investigated in the vicinity of a steady-state. A necessary condition required for damped or sustained oscillations of the system is that enzyme E₂, which controls recycling of the substrate S₂, be inhibited by an excess of this substrate. In order for the system to be neutrally stable and therefore to exhibit sustained oscillations, it is not necessary for antagonist enzyme E₁ to be activated by its product S₂. If it is enzyme E₁ which is inhibited by an excess of its substrate S₁, the system has a saddle point. Other conditions for stability or instability of the system have been determined. If the enzyme E₁, which is not inhibited by the substrate, exhibits a slow conformational transition of the mnemonical type, this transition dramatically alters the stability behavior of the system. If the mnemonical enzyme E₁ were exhibiting a positive kinetic co-operativity, decreasing the rate of the conformational transition of the mnemonical enzyme will increase the stability of the whole system and will tend to damp the oscillations in the vicinity of the steady-state. If conversely the mnemonical enzyme E₁ were exhibiting a negative kinetic co-operativity, decreasing the rate of the enzyme conformational transition will decrease the stability of the system and will tend to create or amplify oscillations of the system taken as a whole. If these results may be extended to more complex metabolic cycles, involving more than two enzymes, it may be tentatively considered that positive co-operativity associated with slow transition has emerged in the course of evolution in order to limit temporal instabilities of metabolic cycles. Alternatively one may speculate that the “biological function” of negative co-operativity is to create or amplify these temporal instabilities.     

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.     

Role of substrate inhibition kinetics in enzymatic chemical oscillations.

Two chemical kinetic models are investigated using standard nonlinear dynamics techniques to determine the conditions under which substrate inhibition kinetics can lead to oscillations. The first model is a classical substrate inhibition scheme based on Michaelis-Menten kinetics and involves a single substrate. Only when this reaction takes place in a flow reactor (i.e., both substrate and product are taken to follow reversible flow terms) are oscillations observed; however, the range of parameter values over which such oscillations occur is so narrow it is experimentally unobservable. A second model based on a general mechanism applied to the kinetics of many pH-dependent enzymes is also studied. This second model includes both substrate inhibition kinetics as well as autocatalysis through the activation of the enzyme by hydrogen ion. We find that it is the autocatalysis that is always responsible for oscillatory behavior in this scheme. The substrate inhibition terms affect the steady-state behavior but do not lead to oscillations unless product inhibition or multiple substrates are present; this is a general conclusion we can draw from our studies of both the classical substrate inhibition scheme and the pH-dependent enzyme mechanism. Finally, an analysis of the nullclines for these two models allows us to prove that the nullcline slopes must have a negative value for oscillatory behavior to exist; this proof can explain our results. From our analysis, we conclude with a brief discussion of other enzymes that might be expected to produce oscillatory behavior based on a pH-dependent substrate inhibition mechanism.     

Yeast glycolytic oscillations that are not controlled by a single oscillophore: a new definition of oscillophore strength

Biochemical oscillations, such as glycolytic oscillations, are often believed to be caused by a single so-called ‘oscillophore’. The main characteristics of yeast glycolytic oscillations, such as frequency and amplitude, are however controlled by several enzymes. In this paper, we develop a method to quantify to which extent any enzyme determines the occurrence of oscillations. Principles extrapolated from metabolic control analysis are applied to calculate the control exerted by individual enzymes on the real and imaginary parts of the eigenvalues of the Jacobian matrix. We propose that the control exerted by an enzyme on the real part of the smallest eigenvalue, in terms of absolute value, quantifies to which extent that enzyme contributes to the emergence of instability. Likewise the control exerted by an enzyme on the imaginary part of complex eigenvalues may serve to quantify the extent to which that enzyme contributes to the tendency of the system to oscillate. The method was applied both to a core model and to a realistic model of yeast glycolytic oscillations. Both the control over stability and the control over oscillatory tendency were distributed among several enzymes, of which glucose transport, pyruvate decarboxylase and ATP utilization were the most important. The distributions of control were different for stability and oscillatory tendency, showing that control of instability does not imply control of oscillatory tendency nor vice versa. The control coefficients summed up to 1, suggesting the existence of a new summation theorem. These results constitute proof that glycolytic oscillations in yeast are not caused by a single oscillophore and provide a new, subtle, definition for the oscillophore strength of an enzyme.     

The steady-state kinetics of isotope exchange for one substrate–one product enzymic reactions

Steady-state kinetic equations for isotope exchange are derived for a number of one substrate-one product enzymic mechanisms in which two molecules of substrate or product can be combined with an enzyme molecule at the one time (e.g. allosteric mechanisms). The usual assumption, that the radioactive material is distributed among the substrate and product components according to a first-order law, is not valid. One can recognize whether isotope-exchange kinetics of an enzyme reaction follows first-order behaviour by using various initial concentrations of the labelled substance added to a mixture.     

Conditions for effective allosteric feedforward and feedback in metabolic pathways

Whether an allosteric feedback or feedforward modifier actually has an effect on the steady-state properties of a metabolic pathway depends not only on the allosteric modifier effect itself, but also on the control properties of the affected allosteric enzyme in the pathway of which it is part. Different modification mechanisms are analysed: mixed inhibition, allosteric inhibition and activation of the reversible Monod-Wyman-Changeux and reversible Hill models. In conclusion, it is shown that, whereas a modifier effect on substrate and product binding (specific effects) can be an effective negative feedback mechanism, it is much less effective as a positive feedforward mechanism. The prediction is that catalytic effects that change the apparent limiting velocity would be more effective in feedforward activation.     

Age dependent changes of a biochemical rhythm--the glycolytic oscillator of the blowfly Phormia terraenovae

1. By monitoring changes of fluorescence of NADH the frequencies, amplitudes and maximum slopes of the glycolytic oscillator of Phormia were analyzed in 5, 9, 15 and 21-day-old male flies. 2. In order to evaluate the possible existence of circadian rhythms within the oscillatory system, all determinations were repeated eight times/day. 3. In addition, the activities of three key enzymes of glycolysis, PFK, GAPDH and PK, which are central to the glycolytic oscillator were measured with respect to age and day time. 4. With increasing age the amplitudes of oscillations increased together with the maximum slopes of the oscillatory waves. The frequency appeared to be independent of age. 5. Variations of enzyme activities over the day indicated an age dependent circadian rhythm which, due to the simultaneous activity changes of the three measured enzymes, was not reflected in the whole oscillatory system. 6. The results suggest that modifications in the allosteric regulation of enzymes are responsible for the age dependent changes of the glycolytic oscillator.     

Oscillations and trigger phenomena in fructose-2,6-bis-phosphate metabolism. A mathematical model

A mathematical model describing metabolism of fructose-2,6-bisphosphate (F2, 6P2), which is a powerful mediator in glycolysis, is investigated. The model takes into account inhibitory effect of F2, 6P2 and fructose-6-phosphate (F6P) on protein kinase, which phosphorylates the bifunctional enzyme fructose-6-phosphate-2-kinase/fructose-2,6-bisphosphatase. Such a mechanism of enzyme chemical modification in the presence of F2, 6P outflow from the F6P in equilibrium with F2, 6P2 cycle, caused by nonspecific phosphatases, can display trigger phenomena and sustained oscillations in F2, 6P2 metabolism and in the whole glycolytic system. The results obtained suggest that earlier models of the generation of glycolytic oscillations should be revised.     

Identification and biochemical characterization of two novel UDP-2,3-diacetamido-2,3-dideoxy-alpha-D-glucuronic acid 2-epimerases from respiratory pathogens

For a long period lactate was considered as a dead-end product of glycolysis in many cells and its accumulation correlated with acidosis and cellular and tissue damage. At present, the role of lactate in several physiological processes has been investigated based on its properties as an energy source, a signalling molecule and as essential for tissue repair. It is noteworthy that lactate accumulation alters glycolytic flux independently from medium acidification, thereby this compound can regulate glucose metabolism within cells. PFK (6-phosphofructo-1-kinase) is the key regulatory glycolytic enzyme which is regulated by diverse molecules and signals. PFK activity is directly correlated with cellular glucose consumption. The present study shows the property of lactate to down-regulate PFK activity in a specific manner which is not dependent on acidification of the medium. Lactate reduces the affinity of the enzyme for its substrates, ATP and fructose 6-phosphate, as well as reducing the affinity for ATP at its allosteric inhibitory site at the enzyme. Moreover, we demonstrated that lactate inhibits PFK favouring the dissociation of enzyme active tetramers into less active dimers. This effect can be prevented by tetramer-stabilizing conditions such as the presence of fructose 2,6-bisphosphate, the binding of PFK to f-actin and phosphorylation of the enzyme by protein kinase A. In conclusion, our results support evidence that lactate regulates the glycolytic flux through modulating PFK due to its effects on the enzyme quaternary structure.     

Effects of primary sequence differences on the global structure and function of an enzyme: a study of pyruvate kinase isozymes

Pyruvate kinase is an important glycolytic enzyme which is expressed differentially as four distinct isozymes whose catalytic activity is regulated in a tissue-specific manner. The kidney isozyme is known to exhibit sigmoidal kinetics, whereas the muscle isozyme exhibits hyperbolic kinetic properties. By integration of the crystallographic [Stuart, D. I., Levine, M., Muirhead, H., & Stammers, D.K. (1979) J. Mol. Biol. 134, 109-142] and primary sequence data [Noguchi, T., Inoue, H., & Tanaka, T. (1986) J. Biol. Chem. 261, 13807], it was shown that the primary sequence for the C alpha 1 and C alpha 2 regions may constitute the allosteric switching site. To provide insights into the effects of the localized sequence change on the global structural and functional behavior of the enzyme, kinetic studies under a wide spectrum of conditions were conducted for both the muscle and kidney isozymes. These conditions include measurements of enzyme activity as a function of substrate concentrations with different concentrations of allosteric inhibitors or activators. These results showed that both isozymes exhibit the same regulatory properties although quantitatively the distribution of active and inactive forms and the various dissociation constants which govern the binding of substrate and allosteric effectors with the enzyme are different. For such a majority of equilibrium constants to be altered, the localized primary sequence change must confer global perturbations which are manifested as differences in the various equilibrium constants. Structural information about these two isozymes was provided by phase-modulation measurement of the fluorescence lifetime of tryptophan residues under a variety of experimental conditions.(ABSTRACT TRUNCATED AT 250 WORDS)     

Influence of phosphate on the allosteric behavior of yeast phosphofructokinase.

Initial rate data obtained with purified yeast phosphofructokinase (PFK) show an ATP dependent kinetic cooperativity with respect to fructose-6-phosphate. In the presence of 25 mM phosphate, the cooperativity index (Hill number) is related to the half saturation concentration of fructose-6-phosphate as predicted by the concerted allosteric model in the case of a “K-system”. In the absence of phosphate, however, the kinetic behavior of yeast PFK is more complex and the cooperativity index is invariant with respect to the half saturation concentration of fructose-6-phosphate which is increased by ATP. In both cases, 5′AMP behaves as a strong activator of the enzyme. These kinetic data suggest that the two distinct functions of ATP as phosphate donnor and as allosteric inhibitor, respectively, are supported by different binding sites. These regulatory properties of yeast PFK are discussed in relation to glycolytic oscillations.     

Exploring the genetic control of glycolytic oscillations in Saccharomyces Cerevisiae

ABSTRACT: BACKGROUND: A well known example of oscillatory phenomena is the transient oscillations of glycolytic intermediates in Saccharomyces cerevisiae, their regulation being predominantly investigated by mathematical modeling. To our knowledge there has not been a genetic approach to elucidate the regulatory role of the different enzymes of the glycolytic pathway. RESULTS: We report that the laboratory strain BY4743 could also be used to investigate this oscillatory phenomenon, which traditionally has been studied using S. cerevisiae X2180. This has enabled us to employ existing isogenic deletion mutants and dissect the roles of isoforms, or subunits of key glycolytic enzymes in glycolytic oscillations. We demonstrate that deletion of TDH3 but not TDH2 and TDH1 (encoding glyceraldehyde-3-phosphate dehydrogenase: GAPDH) abolishes NADH oscillations. While deletion of each of the hexokinase (HK) encoding genes (HXK1 and HXK2) leads to oscillations that are longer lasting with lower amplitude, the effect of HXK2 deletion on the duration of the oscillations is stronger than that of HXK1. Most importantly our results show that the presence of beta (Pfk2) but not that of alpha subunits (Pfk1) of the hetero-octameric enzyme phosphofructokinase (PFK) is necessary to achieve these oscillations. Furthermore, we report that the cAMP-mediated PKA pathway (via some of its components responsible for feedback down-regulation) modulates the activity of glycoytic enzymes thus affecting oscillations. Deletion of both PDE2 (encoding a high affinity cAMP-phosphodiesterase) and IRA2 (encoding a GTPase activating protein- Ras-GAP, responsible for inactivating Ras-GTP) abolished glycolytic oscillations. CONCLUSIONS: The genetic approach to characterising the glycolytic oscillations in yeast has demonstrated differential roles of the two types of subunits of PFK, and the isoforms of GAPDH and HK. Furthermore, it has shown that PDE2 and IRA2, encoding components of the cAMP pathway responsible for negative feedback regulation of PKA, are required for glycolytic oscillations, suggesting an enticing link between these cAMP pathway components and the glycolysis pathway enzymes shown to have the greatest role in glycolytic oscillation. This study suggests that a systematic genetic approach combined with mathematical modelling can advance the study of oscillatory phenomena.     

Allosteric transition and substrate binding are entropy-driven in glucosamine-6-phosphate deaminase from Escherichia coli

Glucosamine-6P-deaminase (EC 3.5.99.6, formerly glucosamine-6-phosphate isomerase, EC 5.3.1.10) from Escherichia coli is an attractive experimental model for the study of allosteric transitions because it is both kinetically and structurally well-known, and follows rapid equilibrium random kinetics, so that the kinetic K(m) values are true thermodynamic equilibrium constants. The enzyme is a typical allosteric K-system activated by N-acetylglucosamine 6-P and displays an allosteric behavior that can be well described by the Monod-Wyman-Changeux model. This thermodynamic study based on the temperature dependence of allosteric parameters derived from this model shows that substrate binding and allosteric transition are both entropy-driven processes in E. coli GlcN6P deaminase. The analysis of this result in the light of the crystallographic structure of the enzyme implicates the active-site lid as the structural motif that could contribute significantly to this entropic component of the allosteric transition because of the remarkable change in its crystallographic B factors.