Temperature dependency and temperature compensation in a model of yeast glycolytic oscillations

Temperature sensitivities and conditions for temperature compensation have been investigated in a model for yeast glycolytic oscillations. The model can quantitatively simulate the experimental observation that the period length of glycolytic oscillations decreases with increasing temperature. Temperature compensation is studied by using control coefficients describing the effect of rate constants on oscillatory frequencies. Temperature compensation of the oscillatory period is observed when the positive contributions to the sum of products between control coefficients and activation energies balance the corresponding sum of the negative contributions. The calculations suggest that by changing the activation energies for one or several of the processes, i.e. by mutations, it could be possible to obtain temperature compensation in the yeast glycolytic oscillator.     

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.     

Kontrollmechanismen der Glycolyse

Control mechanisms of enzymic reactions are generally based on interactions between activators, inhibitors, substrates and products with enzyme proteins or on induction and repression of enzyme synthesis. All main types of control can be recognized in glycolysis. They are the basis of the network which controls the over-all glycolytic function and operates according to the feed-back principle. — Enzyme profiles may not be used for a functional definition of the metabolic state. Enzyme activities are governed by a variety of control mechanisms, which can best be recognized by steady-state and transient state analysis of metabolites and by an analysis of the system's response in titration experiments with pure enzymes under conditions whereby the system displays oscillatory behaviour of its over-all flux. The important parameter for the definition of the metabolic state is the net-flux through the system, since this parameter, along with the steady-state levels of the meabolites, gives the steady-state flux pattern, reveals the kinetic state of enzymic reactions and points to control points of metabolism. Continuous glycolytic oscillations in a cell-free extract ofSaccharomyces carlsbergensis have been observed over a period of 22 hours with a constant frequency of 0.17 min^?1 and a rate of 7.2 nMol ethanol per mg protein per min. Titration of such an extract by pure yeast enzymes reveals the gain (FDP, ADP) and damping components (ATP), which are fed back to the enzymes PFK and PK, respectively, PFK, PGK and PK operating as the control units. On the basis of the titration data as well as metabolite and enzyme activity phase relationship, the mechanism of this oscillation can be understood as a crossed feed-back interaction. Furthermore, it is discussed as the biochemical model of a physiological clock mechanism.     

Long term oscillation in glycolysis

To increase the period of glycolytic oscillations in yeast extracts (Saccharomyces uvarum), the dependence of the period on pH, on concentrations of phosphate and enzymes, and on temperature has been studied. Stable oscillatory trans were obtained at a pH value of about 6.5. Increasing the phosphate and decreasing the enzyme concentrations as well as decreasing temperature lengthened the period. By dilution of the extract with buffer while maintaining the metabolite concentrations at their initial level the period could be successively prolonged from 20 min to about 6 h.     

What makes biochemical networks tick?

In view of the increasing number of reported concentration oscillations in living cells, methods are needed that can identify the causes of these oscillations. These causes always derive from the influences that concentrations have on reaction rates. The influences reach over many molecular reaction steps and are defined by the detailed molecular topology of the network. So-called 'autoinfluence paths', which quantify the influence of one molecular species upon itself through a particular path through the network, can have positive or negative values. The former bring a tendency towards instability. In this molecular context a new graphical approach is presented that enables the classification of network topologies into oscillophoretic and nonoscillophoretic, i.e. into ones that can and ones that cannot induce concentration oscillations. The network topologies are formulated in terms of a set of uni-molecular and bi-molecular reactions, organized into branched cycles of directed reactions, and presented as graphs. Subgraphs of the network topologies are then classified as negative ones (which can) and positive ones (which cannot) give rise to oscillations. A subgraph is oscillophoretic (negative) when it contains more positive than negative autoinfluence paths. Whether the former generates oscillations depends on the values of the other subgraphs, which again depend on the kinetic parameters. An example shows how this can be established. By following the rules of our new approach, various oscillatory kinetic models can be constructed and analyzed, starting from the classified simplest topologies and then working towards desirable complications. Realistic biochemical examples are analyzed with the new method, illustrating two new main classes of oscillophore topologies.     

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.     

Control Analysis for Autonomously Oscillating Biochemical Networks

It has hitherto not been possible to analyze the control of oscillatory dynamic cellular processes in other than qualitative ways. The control coefficients, used in metabolic control analyses of steady states, cannot be applied directly to dynamic systems. We here illustrate a way out of this limitation that uses Fourier transforms to convert the time domain into the stationary frequency domain, and then analyses the control of limit cycle oscillations. In addition to the already known summation theorems for frequency and amplitude, we reveal summation theorems that apply to the control of average value, waveform, and phase differences of the oscillations. The approach is made fully operational in an analysis of yeast glycolytic oscillations. It follows an experimental approach, sampling from the model output and using discrete Fourier transforms of this data set. It quantifies the control of various aspects of the oscillations by the external glucose concentration and by various internal molecular processes. We show that the control of various oscillatory properties is distributed over the system enzymes in ways that differ among those properties. The models that are described in this paper can be accessed on http://jjj.biochem.sun.ac.za.     

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.     

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.     

Modeling of Glucose-Induced cAMP Oscillations in Pancreatic β Cells: cAMP Rocks when Metabolism Rolls

Recent advances in imaging technology have revealed oscillations of cyclic adenosine monophosphate (cAMP) in insulin-secreting cells. These oscillations may be in phase with cytosolic calcium oscillations or out of phase. cAMP oscillations have previously been modeled as driven by oscillations in calcium, based on the known dependence of the enzymes that generate cAMP (adenylyl cyclase) and degrade it (phosphodiesterase). However, cAMP oscillations have also been reported to occur in the absence of calcium oscillations. Motivated by similarities between the properties of cAMP and metabolic oscillations in pancreatic β cells, we propose here that in addition to direct control by calcium, cAMP is controlled by metabolism. Specifically, we hypothesize that AMP inhibits adenylyl cyclase. We incorporate this hypothesis into the dual oscillator model for β cells, in which metabolic (glycolytic) oscillations cooperate with modulation of ion channels and metabolism by calcium. We show that the combination of oscillations in AMP and calcium in the dual oscillator model can account for the diverse oscillatory patterns that have been observed, as well as for experimental perturbations of those patterns. Predictions to further test the model are provided.     

Conservation of glycolytic oscillations in Saccharomyces cerevisiae and human pancreatic beta-cells: a study of metabolic robustness

The present study compares two computer models of the first part of glucose catabolism in different organisms in search of evolutionarily conserved characteristics of the glycolysis cycle and proposes the main parameters that define the stable steady-state or oscillatory behavior of the glycolytic system. It is suggested that in both human pancreatic beta-cells and Saccharomyces cerevisiae there are oscillations that, despite differences in wave form and period of oscillation, share the same robustness strategy: the oscillation is not controlled by only one but by at least two parameters that will have more or less control over the pathway flux depending on the initial state of the system as well as on extra-cellular conditions. This observation leads to two important interpretations: the first is that in both S. cerevisiae and human beta-cells, despite differences in enzyme kinetics and mechanism of feedback control, evolution seems to have kept an oscillatory behavior coupled to the glucose concentration outside the cytoplasm, and the second is that the development of drugs to regulate metabolic dysfunctions in more complex systems may require further study, not only determining which enzyme is controlling the flux of the system but also under which conditions and how its control is maintained by the enzyme or transferred to other enzymes in the pathway as the drug starts acting.     

The rhythm of yeast

Although yeast are unicellular and comparatively simple organisms, they have a sense of time which is not related to reproduction cycles. The glycolytic pathway exhibits oscillatory behaviour, i.e. the metabolite concentrations oscillate around phosphofructokinase. The frequency of these oscillations is about 1 min when using intact cells. Also a yeast cell extract can oscillate, though with a lower frequency. With intact cells the macroscopic oscillations can only be observed when most of the cells oscillate in concert. Transient oscillations can be observed upon simultaneous induction; sustained oscillations require an active synchronisation mechanism. Such an active synchronisation mechanism, which involves acetaldehyde as a signalling compound, operates under certain conditions. How common these oscillations are in the absence of a synchronisation mechanism is an open question. Under aerobic conditions an oscillatory metabolism can also be observed, but with a much lower frequency than the glycolytic oscillations. The frequency is between one and several hours. These oscillations are partly related to the reproductive cycle, i.e. the budding index also oscillates; however, under some conditions they are unrelated to the reproductive cycle, i.e. the budding index is constant. These oscillations also have an active synchronisation mechanism, which involves hydrogen sulfide as a synchronising agent. Oscillations with a frequency of days can be observed with yeast colonies on plates. Here the oscillations have a synchronisation mechanism which uses ammonia as a synchronising agent.     

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.     

Desynchronisation of Glycolytic Oscillations in Yeast Cell Populations

Glycolytic oscillations of intact yeast cells of the strain Saccharomyces carlsbergensis were investigated at both the levels of cell populations and of individual cells. Individual cells showed glycolytic oscillations even at very low cell densities (e.g. 1.010⁵ cells/ml). By contrast, the collective behaviour on the population level was cell density-dependent: at high cell densities it is oscillatory, but below the threshold density of 1.010⁶ cells/ml the collective dynamics becomes quiescent. We demonstrate that the transition in the collective dynamics is caused by the desynchronisation of the oscillations of individual cells. This is characteristic for a Kuramoto transition. Spatially resolved measurements at low cell densities revealed that even cells that adhere to their neighbours oscillated with their own, independent frequencies and phases.     

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.     

Design of glycolysis

The design of the glycolytic pathway resulting from the continuous refinement of evolution is discussed with regard to three aspects. 1. Functional and structural properties of individual enzymes. The catalytic constants of the glycolytic enzymes are remarkably optimized; the turnover numbers are within one order of magnitude. The same is true for the molarities of catalytic centres in the cytosol, as is noted for yeast. Functional properties of the enzymes are reflected in their tertiary and quaternary structures. 2. Regulatory mechanisms of single enzymes. A classification of the various types of enzymic control mechanisms operating in the glycolytic pathway is given. In addition to the usual Michaelis-Menten saturation kinetics and the various types of inhibition there is control by positive and negative effectors based on oligomeric structures (fast acting, fine control) as well as regulation by chemical interconversion structures (fast acting, fine control) as well as regulation by chemical based on enzymes cascades (slow acting, very effective). 3. Functional and regulatory mechanisms of the whole glycolytic reaction pathway. A prominent feature is the high enzyme:substrate ratio, which guarantees fast response times. However, a quantitative treatment of the overall kinetics is limited by an incomplete knowledge of the enzymes' dynamic and chemical compartmentation as well as some of their control properties. From an analysis of the oscillatory state, certain control points in the glycolytic chain can be located that coincide with major branching points to other metabolic pathways. These points are controlled by fast-acting cooperative enzymes that operate in a flip-flop mechanism together with the respective antagonistic enzymes, preventing futile cycles. The gating enzymes leading to the glycogen store and the citric acid cycle are of the slow-acting but very effective interconvertible type. The combination of all the complex and intricate features of design yields a glycolytic network that enables the cell to respond to its various metabolic needs quickly, effectively and economically.     

How yeast cells synchronize their glycolytic oscillations: a perturbation analytic treatment

Of all the lifeforms that obtain their energy from glycolysis, yeast cells are among the most basic. Under certain conditions the concentrations of the glycolytic intermediates in yeast cells can oscillate. Individual yeast cells in a suspension can synchronize their oscillations to get in phase with each other. Although the glycolytic oscillations originate in the upper part of the glycolytic chain, the signaling agent in this synchronization appears to be acetaldehyde, a membrane-permeating metabolite at the bottom of the anaerobic part of the glycolytic chain. Here we address the issue of how a metabolite remote from the pacemaking origin of the oscillation may nevertheless control the synchronization. We present a quantitative model for glycolytic oscillations and their synchronization in terms of chemical kinetics. We show that, in essence, the common acetaldehyde concentration can be modeled as a small perturbation on the "pacemaker" whose effect on the period of the oscillations of cells in the same suspension is indeed such that a synchronization develops.     

Control analysis as a tool to understand the formation of the las operon in Lactococcus lactis

In Lactococcus lactis the enzymes phosphofructokinase (PFK), pyruvate kinase (PK) and lactate dehydrogenase (LDH) are uniquely encoded in the las operon. We used metabolic control analysis to study the role of this organization. Earlier studies have shown that, at wild-type levels, LDH has no control over glycolysis and growth rate, but high negative control over formate production (C(Jformate)LDH=-1.3). We found that PFK and PK exert no control over glycolysis and growth rate at wild-type enzyme levels but both enzymes exert strong positive control on the glycolytic flux at reduced activities. PK exerts high positive control over formate (C(Jformate)PK=0.9-1.1) and acetate production (C(Jacetate)PK=0.8-1.0), whereas PFK exerts no control over these fluxes at increased expression. Decreased expression of the entire las operon resulted in a strong decrease in the growth rate and glycolytic flux; at 53% expression of the las operon glycolytic flux was reduced to 44% and the flux control coefficient increased towards 3. Increased las expression resulted in a slight decrease in the glycolytic flux. At wild-type levels, control was close to zero on both glycolysis and the pyruvate branches. The sum of control coefficients for the three enzymes individually was comparable with the control coefficient found for the entire operon; the strong positive control exerted by PK almost cancels out the negative control exerted by LDH on formate production. Our analysis suggests that coregulation of PFK and PK provides a very efficient way to regulate glycolysis, and coregulating PK and LDH allows cells to maintain homolactic fermentation during glycolysis regulation.     

'Slave' metabolites and enzymes. A rapid way of delineating metabolic control.

Although control of fluxes and concentrations tends to be distributed rather than confined to a single rate-limiting enzyme, the extent of control can differ widely between enzymes in a metabolic network. In some cases, there are enzymes that lack control completely. This paper identifies one surprising origin of such lack of control: If, in a metabolic system, there is a metabolite that affects the catalytic rate of only one enzyme, the corresponding enzyme cannot control any metabolic variable other than the concentration of that metabolite. We call such enzymes 'slave enzymes', and the corresponding metabolites 'slave metabolites'. Implications of the existence of slave enzymes for the control properties of enzymes further down the metabolic pathway are discussed and examined for the glycolytic pathway of yeast. Inadvertent assumptions in metabolic models may cause the latter incorrectly to calculate absence of metabolic control. The phenomenon of slave enzymes may well be important in enhancing metabolic signal transduction.