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.     

Dissipative Structures for an Allosteric Model: Application to Glycolytic Oscillations

An allosteric model of an open monosubstrate enzyme reaction is analyzed for the case where the enzyme, containing two protomers, is activated by the product. It is shown that this system can lead to instabilities beyond which a new state organized in time or in space (dissipative structure) can be reached. The conditions for both types of instabilities are presented and the occurrence of a temporal structure, consisting of a limit cycle behavior, is determined numerically as a function of the important parameters involved in the system. Sustained oscillations in the product and substrate concentrations are shown to occur for acceptable values of the allosteric and kinetic constants; moreover, they seem to be favored by substrate activation. The model is applied to phosphofructokinase, which is the enzyme chiefly responsible for glycolytic oscillations and which presents the same pattern of regulation as the allosteric enzyme appearing in the model. A qualitative and quantitative agreement is obtained with the experimental observations concerning glycolytic self-oscillations.     

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.     

Control of phosphofructokinase from rat skeletal muscle. Effects of fructose diphosphate, AMP, ATP, and citrate.

Under conditions used previously for demonstrating glycolytic oscillations in muscle extracts (pH 6.65, 0.1 to 0.5 mM ATP), phosphofructokinase from rat skeletal muscle is strongly activated by micromolar concentrations of fructose diphosphate. The activation is dependent on the presence of AMP. Activation by fructose diphosphate and AMP, and inhibition by ATP, is primarily due to large changes in the apparent affinity of the enzyme for the substrate fructose 6-phosphate. These control properties can account for the generation of glycolytic oscillations. The enzyme was also studied under conditions approximating the metabolite contents of skeletal muscle in vivo (pH 7.0, 10mM ATP, 0.1 mM fructose 6-phosphate). Under these more inhibitory conditions, phosphofructokinase is strongly activated by low concentrations of fructose diphosphate, with half-maximal activation at about 10 muM. Citrate is a potent inhibitor at physiological concentrations, whereas AMP is a strong activator. Both AMP and citrate affect the maximum velocity and have little effect on affinity of the enzyme for fructose diphosphate.     

Interaction of D-fructose and fructose 1-phosphate with yeast phosphofructokinase and its influence on glycolytic oscillations.

Fermentation of D-fructose- and D-glucose induced glycolytic oscillations of different period lengths in Saccharomyces carlsbergensis. Recent studies suggested, that D-fructose or one of its metabolites interacted with phosphofructokinase (ATP:D-fructo-6-phosphate 1-phosphofructokinase, EC 2.7.1.11), the core of the glycolytic 'oscillator'. In order to explore the kinetics of interaction, the influence of D-fructose and fructose 1-phosphate on purified yeast phosphofructokinase was studied. D-fructose concentrations up to 0.3 mM stimulated the enzyme, while a further increase led to competitive inhibition. The Hill coefficient for fructose 6-phosphate decreased from 2.8 to 1.0. Fructose 1-phosphate acted in a similar way, up to 1 mM activation and inhibition competitive to fructose 6-phosphate at higher concentration (2.0--3.5 mM) with the same effect on the Hill coefficient. The inhibition patterns obtained with D-fructose or fructose 1-phosphate suggest a sequential random reaction mechanism of yeast phosphofructokinase with fructose 6-phosphate and MgATP2-. The mode of interaction of phosphofructokinase with D-fructose and fructose 1-phosphate is discussed. The influence of both effectors resulted in altered enzyme kinetics, which may cause the different period lengths of glycolytic oscillations.     

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.     

Influence of fructose 2,6-bisphosphate on the phosphofructokinase/fructose 1,6-bisphosphatase cycle

In a reconstituted enzyme system multiple stationary states and oscillatory motions of the substrate cycle catalyzed by phosphofructokinase and fructose 1,6-bisphosphatase are significantly influenced by fructose 2,6-bisphosphate. Depending on the initial conditions, fructose 2,6-bisphosphate was found either to generate or to extinguish oscillatory motions between glycolytic and gluconeogenic states. In general, stable glycolytic modes are favored because of the efficient activation of phosphofructokinase by this effector. The complex effect of fructose 2,6-bisphosphate on the rate of substrate cycling correlates with its synergistic cooperation with AMP in the activation of phosphofructokinase and inhibition of fructose 1,6-bisphosphatase.     

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.     

On the mechanisms of glycolytic oscillations in yeast

This work concerns the cause of glycolytic oscillations in yeast. We analyse experimental data as well as models in two distinct cases: the relaxation-like oscillations seen in yeast extracts, and the sinusoidal Hopf oscillations seen in intact yeast cells. In the case of yeast extracts, we use flux-change plots and model analyses to establish that the oscillations are driven by on/off switching of phosphofructokinase. In the case of intact yeast cells, we find that the instability leading to the appearance of oscillations is caused by the stoichiometry of the ATP-ADP-AMP system and the allosteric regulation of phosphofructokinase, whereas frequency control is distributed over the reaction network. Notably, the NAD+/NADH ratio modulates the frequency of the oscillations without affecting the instability. This is important for understanding the mutual synchronization of oscillations in the individual yeast cells, as synchronization is believed to occur via acetaldehyde, which in turn affects the frequency of oscillations by changing this ratio.     

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.     

Turing patterns inside cells

Concentration gradients inside cells are involved in key processes such as cell division and morphogenesis. Here we show that a model of the enzymatic step catalized by phosphofructokinase (PFK), a step which is responsible for the appearance of homogeneous oscillations in the glycolytic pathway, displays Turing patterns with an intrinsic length-scale that is smaller than a typical cell size. All the parameter values are fully consistent with classic experiments on glycolytic oscillations and equal diffusion coefficients are assumed for ATP and ADP. We identify the enzyme concentration and the glycolytic flux as the possible regulators of the pattern. To the best of our knowledge, this is the first closed example of Turing pattern formation in a model of a vital step of the cell metabolism, with a built-in mechanism for changing the diffusion length of the reactants, and with parameter values that are compatible with experiments. Turing patterns inside cells could provide a check-point that combines mechanical and biochemical information to trigger events during the cell division process.     

Modeling of glycolytic wave propagation in an open spatial reactor with inhomogeneous substrate influx

The spatio-temporal dynamics of traveling waves in glycolysis as it occurs in yeast extract have been studied, both theoretically and experimentally. We describe this phenomenon with the distributed Selkov model that accounts for the reactions of phosphofructokinase, which is a key enzyme of the glycolytic reaction cascade. To describe the experimentally observed phase waves in an open spatial reactor we introduce a non-homogeneous flux of substrate in the model. The experimental observation that waves can change their direction of propagation during the experiment is considered in the model. The mechanism for such a change in wave direction is discussed.     

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.     

Metabolite Concentrations and Phase Relations in d-Fructose Induced Glycolytic Oscillations

The frequency of glycolytic oscillations in yeast cells is only slightly modified by the consumption rate of either glucose or fructose, but it is reduced to 1 /2 to 2/3 if the ketohexose is the only substrate. Phase relations between NADH, adenine nucleotides and sugar phosphates (G-6-P, F-6-P, FDP, DAP) are independent of the hexose fermented. With fructose as a substrate the amplitudes of adenylates and hexose phosphates are distinctly smaller than with glucose. The maximum of G-6-P is higher with glucose and the minimum concentration of FDP is higher with fructose. In the transition to anaerobiosis FDP, adenosine 5′-diphosphate and adenosine 5′-phosphate are extremely high, whereas G-6-P, F-6-P and ATP concentrations are lower if fructose is the substrate fermented. The results are indicative for different control characteristics of the phosphofructokinase step, but on their basis it cannot be distinguished between a direct interaction of fructose (or one of its derivatives) on the phosphofructokinase kinetics or the existence of a bypass for fructose which might be able to withdraw a part of the substrate from the control point at the phosphofructokinase.     

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.     

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.     

Allosteric activation and competitive inhibition of yeast phosphofructokinase by d-fructose.

Purified phosphofructokinase from bakers yeast is activated by D-fructose in low concentrations (up to 1 mM) and inhibited by high concentrations. The stimulatory effect of D-fructose is similar, but smaller than that of AMP. In the presence of AMP (0.4 mM or higher) D-fructose does no longer stimulate, but its inhibitory effect persists (KI = 8 mM). Its dualistic action on phosphofructokinase activity indicates that D-fructose might induce low frequency in glycolytic oscillations by direct interaction with the enzyme.     

Metabolic adaptation of the renal carbohydrate metabolism. II

Summary The effects of a high carbohydrate diet on the renal gluconeogenic and glycolytic capacities and on the activities of the main enzymes of the carbohydrate metabolism, fructose 1,6-bisphosphatase, phosphofructokinase and pyruvate kinase have been studied. These parameters have been analysed in two separate and isolated fractions of the renal tubule, the proximal convoluted (PCT) and the distal convoluted (DCT) zones. The results presented in this study show a rapid adaptation capacity of the kidney in response to the high amount of dietary carbohydrate, which are characterized by a decrease in the glucose production and fructose 1,6-bisphosphatase activity in the proximal tubules, and an increase in the glycolytic flux and phosphofructokinase and pyruvate kinase activities in the distal tubules. The changes in these enzyme activities took place only at subsaturating substrate concentrations and not at maximum velocity which suggest that they are probably due to an allosteric and/or covalent modifications and so, they are independent of variations in the cellular levels of the enzymes.     

Dynamics of a biochemical system with multiple oscillatory domains as a clue for multiple modes of neuronal oscillations

We analyze the behavior of a two-variable biochemical model in conditions where it admits multiple oscillatory domains in parameter space. The model represents an autocatalytic enzyme reaction with input of substrate both from a constant source and from non-linear recycling of product into substrate. This system was previously studied for birhythmicity, i.e. the coexistence between two stable periodic regimes (Moran and Goldbeter 1984), and for multithreshold excitability (Moran and Goldbeter 1985). When two distinct oscillatory domains obtain as a function of the substrate injection rate, the system is capable of exhibiting two markedly different modes of oscillations for slightly different values of this control parameter. Phase plane analysis shows how the multiplicity of oscillatory domains depends on the parameters that govern the underlying biochemical mechanism of product recycling. We analyze the response of the model to various kinds of transient perturbations and to periodic changes in the substrate input that bring the system through the two ranges of oscillatory behavior. The results provide a qualitative explanation for experimental observations (Jahnsen and Llinas 1984b) related to the occurrence of two different modes of oscillations in thalamic neurones.