Glycolytic Oscillations Research Articles

Mathematical Model for Temperature Regulation of Self-Sustained Glycolytic Oscillations in a Closed Reactor

We base on the Selkov system [1] to construct the model for temperaturecontrol of glycolytic reaction in a closed spatial reactor. To establish acorrespondence with the experiment [2] we add the slow catalytic term xwhich describes the small value of additional substrate influx and productoutflow and introduce a temperature-dependent coefficient satisfying theArrhenius law... The considered model explains the key experimentally observed phenomena[2]: 1) decaying of the average concentrations of reagents duringthe reaction, 2) Arrhenius-type temperature dependence for frequency of oscillations,3) change of the form of oscillations with the temperature growth,4) modulations of oscillations induced by a periodic temperature variation.The addition of the diffusion terms to the system (1) allows to reproducethe emerging of glycolytic travelling waves observed in a closed reactor inthe presence of a temperature gradient [2]. Comparison of the dynamics of travelling waves in the numerical solution with the experimental data [2]permits to propose a new method to estimate the diffusion coefficients of reagents in the case of a chemical reaction occurring in a dense media

Programming strategy for efficient modeling of dynamics in a population of heterogeneous cells

Heterogeneity is a ubiquitous property of biological systems. Even in a genetically identical population of a single cell type, cell-to-cell differences are observed. Although the functional behavior of a given population is generally robust, the consequences of heterogeneity are fairly unpredictable. In heterogeneous populations, synchronization of events becomes a cardinal problem-particularly for phase coherence in oscillating systems. The present article presents a novel strategy for construction of large-scale simulation programs of heterogeneous biological entities. The strategy is designed to be tractable, to handle heterogeneity and to handle computational cost issues simultaneously, primarily by writing a generator of the 'model to be simulated'. We apply the strategy to model glycolytic oscillations among thousands of yeast cells coupled through the extracellular medium. The usefulness is illustrated through (i) benchmarking, showing an almost linear relationship between model size and run time, and (ii) analysis of the resulting simulations, showing that contrary to the experimental situation, synchronous oscillations are surprisingly hard to achieve, underpinning the need for tools to study heterogeneity. Thus, we present an efficient strategy to model the biological heterogeneity, neglected by ordinary mean-field models. This tool is well posed to facilitate the elucidation of the physiologically vital problem of synchronization. The complete python code is available as Supplementary Information. bjornhald@gmail.com or pgs@kiku.dk Supplementary data are available at Bioinformatics online.

Glycolytic oscillations in a model of a lactic acid bacterium metabolism

Glycolytic oscillations in yeast have been extensively studied. It is still unclear, if these oscillations are caused by the allosteric enzyme phosphofructokinase or the stoichiometry of glycolysis which contains an autocatalysis with respect to ATP. Bacterial glycolysis shows a different stoichiometry, however, also containing a stoichiometric autocatalysis. For Escherichia coli, the regulation of the enzyme phosphofructokinase is also assumed to be a major reason for oscillations to occur. We investigated glycolytic oscillations in a quantitative kinetic model for Streptococcus pyogenes set-up on the basis of experimental data. We found oscillations within physiologically feasible parameter ranges. We investigated the origin of these oscillations and conclude that, again, both the stoichiometry of the system, as well as its allosterically regulated enzymes can give rise to these oscillations. For the analysis we employed established and new optimization methods for finding oscillatory regimes and present these in the context of this study.

Influence of cyanide on diauxic oscillations in yeast

Coherent glycolytic oscillations in Saccharomyces cerevisiae are a multicellular property induced by addition of glucose to a starved cell population of sufficient density. However, initiation of oscillations requires an additional perturbation, usually addition of cyanide. The fate of cyanide during glycolytic oscillations has not previously been studied, and is the subject of the present paper. Using a cyanide electrode, a substantial decrease in cyanide concentration was observed. In the pH range 6-7, we found experimentally that the electrode behaves reasonably well, provided changes in pH are taken into account. To our knowledge, use of a cyanide electrode to study cyanide dynamics in living biological systems is new. Cyanide was found to enter starving yeast cells in only negligible amounts, and did not react significantly with glucose. Thus, cyanide consumption must be explained by reactions with glycolytic intermediates and evaporation. Evaporation and reaction with the signalling substance, extracellular acetaldehyde (ACA(x)) only explains the observed cyanide removal if [ACA(x)] is improbably high. Furthermore, differences in NADH traces upon cyanide addition before or after glucose addition strongly suggest that cyanide also reacts with intracellular carbonyl-containing metabolites. We show that cyanide reacts with pyruvate (Pyr) and dihydroxyacetone phosphate in addition to ACA, and estimate their rate constants. Our results strongly suggest that the major routes of cyanide removal during glycolysis are reactions with pyruvate and ACA. Cyanide removal by all carbonyl-containing intermediates led to a lower mean [ACA(x) ], thereby increasing the amplitude of [ACA(x) ] oscillations.

CMOS weak‐inversion log‐domain glycolytic oscillator: a cytomimetic circuit example

SUMMARYThis paper presents a 3 V, 1.21μW subthreshold log‐domain circuit which mimics the oscillations observed during the biochemical process of glycolysis due to the phosphofructokinase enzyme. The proposed electronic circuit is able to simulate the dynamics of the glycolytic oscillator and represent the time‐dependent concentration changes of the reactants and the products of the chemical process based on nonlinear differential equations which describe the biological system. By modifying specific circuit parameters, which correspond to certain chemical parameters, good agreement between the biochemical and electrical model results has been reached. The paper details the similarities between the equations that describe the biochemical process and the equations derived from the circuit analysis of a transistor and a source‐connected linear capacitor, a topology also known as the Bernoulli Cell. With the use of the Bernoulli Cell formalism, the chemical equations which describe the biochemical system have been transformed into their electrical equivalents. The analog circuit, which implements the whole process, has been synthesised, and simulation results including Monte Carlo analysis are provided, in order to verify the robustness of the proposed circuit and to compare its dynamics with prototype biological behaviour. Copyright © 2012 John Wiley & Sons, Ltd.

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.

Exploring the genetic control of glycolytic oscillations in Saccharomyces Cerevisiae

BackgroundA 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.ResultsWe 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.ConclusionsThe 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.

From steady‐state to synchronized yeast glycolytic oscillations I: model construction

An existing detailed kinetic model for the steady-state behavior of yeast glycolysis was tested for its ability to simulate dynamic behavior. Using a small subset of experimental data, the original model was adapted by adjusting its parameter values in three optimization steps. Only small adaptations to the original model were required for realistic simulation of experimental data for limit-cycle oscillations. The greatest changes were required for parameter values for the phosphofructokinase reaction. The importance of ATP for the oscillatory mechanism and NAD(H) for inter-and intra-cellular communications and synchronization was evident in the optimization steps and simulation experiments. In an accompanying paper [du Preez F et al. (2012) FEBS J279, 2823-2836], we validate the model for a wide variety of experiments on oscillatory yeast cells. The results are important for re-use of detailed kinetic models in modular modeling approaches and for approaches such as that used in the Silicon Cell initiative. The mathematical models described here have been submitted to the JWS Online Cellular Systems Modelling Database and can be accessed at http://jjj.biochem.sun.ac.za/database/dupreez/index.html.

From steady‐state to synchronized yeast glycolytic oscillations II: model validation

In an accompanying paper [du Preez et al., (2012) FEBS J279, 2810-2822], we adapt an existing kinetic model for steady-state yeast glycolysis to simulate limit-cycle oscillations. Here we validate the model by testing its capacity to simulate a wide range of experiments on dynamics of yeast glycolysis. In addition to its description of the oscillations of glycolytic intermediates in intact cells and the rapid synchronization observed when mixing out-of-phase oscillatory cell populations (see accompanying paper), the model was able to predict the Hopf bifurcation diagram with glucose as the bifurcation parameter (and one of the bifurcation points with cyanide as the bifurcation parameter), the glucose- and acetaldehyde-driven forced oscillations, glucose and acetaldehyde quenching, and cell-free extract oscillations (including complex oscillations and mixed-mode oscillations). Thus, the model was compliant, at least qualitatively, with the majority of available experimental data for glycolytic oscillations in yeast. To our knowledge, this is the first time that a model for yeast glycolysis has been tested against such a wide variety of independent data sets. The mathematical models described here have been submitted to the JWS Online Cellular Systems Modelling Database and can be accessed at http://jjj.biochem.sun.ac.za/database/dupreez/index.html.

Sustained glycolytic oscillations in individual isolated yeast cells

Yeast glycolytic oscillations have been studied since the 1950s in cell-free extracts and intact cells. For intact cells, sustained oscillations have so far only been observed at the population level, i.e. for synchronized cultures at high biomass concentrations. Using optical tweezers to position yeast cells in a microfluidic chamber, we were able to observe sustained oscillations in individual isolated cells. Using a detailed kinetic model for the cellular reactions, we simulated the heterogeneity in the response of the individual cells, assuming small differences in a single internal parameter. This is the first time that sustained limit-cycle oscillations have been demonstrated in isolated yeast cells. The mathematical model described here has been submitted to the JWS Online Cellular Systems Modelling Database and can be accessed at http://jjj.biochem.sun.ac.za/database/gustavsson/index.html free of charge.

Phosphofructo-2-kinase/Fructose-2,6-bisphosphatase Modulates Oscillations of Pancreatic Islet Metabolism

Pulses of insulin from pancreatic beta-cells help maintain blood glucose in a narrow range, although the source of these pulses is unclear. It has been proposed that a positive feedback circuit exists within the glycolytic pathway, the autocatalytic activation of phosphofructokinase-1 (PFK1), which endows pancreatic beta-cells with the ability to generate oscillations in metabolism. Flux through PFK1 is controlled by the bifunctional enzyme PFK2/FBPase2 (6-phosphofructo-2-kinase/fructose-2,6-bisphosphatase) in two ways: via (1) production/degradation of fructose-2,6-bisphosphate (Fru2,6-BP), a potent allosteric activator of PFK1, as well as (2) direct activation of glucokinase due to a protein-protein interaction. In this study, we used a combination of live-cell imaging and mathematical modeling to examine the effects of inducibly-expressed PFK2/FBPase2 mutants on glucose-induced Ca2+ pulsatility in mouse islets. Irrespective of the ability to bind glucokinase, mutants of PFK2/FBPase2 that increased the kinase:phosphatase ratio reduced the period and amplitude of Ca2+ oscillations. Mutants which reduced the kinase:phosphatase ratio had the opposite effect. These results indicate that the main effect of the bifunctional enzyme on islet pulsatility is due to Fru2,6-BP alteration of the threshold for autocatalytic activation of PFK1 by Fru1,6-BP. Using computational models based on PFK1-generated islet oscillations, we then illustrated how moderate elevation of Fru-2,6-BP can increase the frequency of glycolytic oscillations while reducing their amplitude, with sufficiently high activation resulting in termination of slow oscillations. The concordance we observed between PFK2/FBPase2-induced modulation of islet oscillations and the models of PFK1-driven oscillations furthermore suggests that metabolic oscillations, like those found in yeast and skeletal muscle, are shaped early in glycolysis.

Oscillations in glycolysis in Saccharomyces cerevisiae: The role of autocatalysis and intracellular ATPase activity

We have investigated the glycolytic oscillations, measured as NADH autofluorescence, in the yeast Saccharomyces cerevisiae in a batch reactor. Specifically, we have tested the effect of cell density and a number of inhibitors or activators of ATPase activity on the amplitude of the oscillations. The amplitude dependence on cell density shows the same behavior as that observed in cells in a CSTR. Furthermore, the amplitude decreases with increasing inhibition of the three ATPases (i) F0F1 ATPase, (ii) plasma membrane ATPase (Pma1p) and (iii) vacuolar ATPase (V-ATPase). The amplitude of the oscillations also decreases by stimulating the ATPase activity, e.g. by FCCP or Amphotericin B. Thus, ATPase activity strongly affects the glycolytic oscillations. We discuss these data in relation to a simple autocatalytic model of glycolysis which can reproduce the experimental data and explain the role of membrane-bound ATPases . In addition we also studied a recent detailed model of glycolysis and found that, although this model faithfully reproduces the oscillations of glycolytic intermediates observed experimentally, it is not able to explain the role of ATPase activity on the oscillations.

Measurements of intracellularATP provide new insight into the regulation of glycolysis in the yeast Saccharomyces cerevisiae

Glycolysis in the yeast Saccharomyces cerevisiae exhibits temporal oscillation under anaerobic or semianaerobic conditions. Previous evidence indicated that at least two membrane-bound ATPases, the mitochondrial F(0)F(1) ATPase and the plasma membrane P-type ATPase (Pma1p), were important in regulating the glycolytic oscillation. Measurements of intracellular ATP provide a unique tool to understand the role of these membrane ATPases and how their activities are regulated. We have constructed a new nanobiosensor that can perform time-resolved measurements of intracellular ATP in intact cells. Measurements of the temporal behaviour of intracellular ATP in a yeast strain with oscillating glycolysis showed that, in addition to oscillation in intracellular ATP, there is an overall slow decrease in intracellular ATP because the ATP consumption rate exceeds the ATP production in glycolysis. Measurements of the temporal behaviour of intracellular ATP in yeast strains lacking either of the two membrane bound ATPases have confirmed that F(0)F(1) ATPase and Pma1p contribute significantly to the ATP consumption in the cell and to the regulation of glycolytic oscillation. Furthermore, our measurements also demonstrate that ATPase activity is under strict control. In the absence of glucose ATPase activity is switched off, and the intracellular ATP concentration is high. When glucose is added to the cells the ATP concentration starts to decrease, because ATP consumption exceeds ATP production by glycolysis. Finally, when glucose is used up, the ATP consumption stops immediately. Thus, glucose or some compound derived from glucose must be involved in controlling the activity of these two ATPases.

Dynamical responses of oscillating yeast cell suspensions to periodic forcing.

Fermenting S. cerevisiae cells suspensions undergo synchronized metabolic oscillations whose amplitude and frequency depends on the population density. The asymptotic population-level dynamics is qualitatively and quantitatively accounted for by mathematical models describing the intracellular glycolytic oscillations and the coupling through the external medium. Such dynamical systems often display low-dimensional dynamics for the parameters that quantitatively fit the experimental observations. However, the high dimensionality of the chemical space and the possible existence of intermingled time scales can potentially support more complex dynamics, in particular when the system is perturbed by external forcings. Here, we explore experimentally and by means of numerical models the response of cellular populations to different kinds of periodic forcing: resonant forcing of diluted, nonoscillating populations and strongly nonresonant forcing of populations in oscillating regimes with different dynamical features. We show that, in all cases, low-dimensional semi-quantitative models reproduce the observed dynamics. Both a nonlinear analysis of the experimental time series and the models indicate that complex-looking time series correspond to quasiperiodic and not to chaotic regimes. The fact that low-dimensional dynamical systems are able to reproduce the response of biological populations in different regimes of external periodic forcing supports the use of theoretical models for inquiring the dynamical behaviour of collectively oscillating cells.

Automated refinement and inference of analytical models for metabolic networks

The reverse engineering of metabolic networks from experimental data is traditionally a labor-intensive task requiring a priori systems knowledge. Using a proven model as a test system, we demonstrate an automated method to simplify this process by modifying an existing or related model-–suggesting nonlinear terms and structural modifications–-or even constructing a new model that agrees with the system's time series observations. In certain cases, this method can identify the full dynamical model from scratch without prior knowledge or structural assumptions. The algorithm selects between multiple candidate models by designing experiments to make their predictions disagree. We performed computational experiments to analyze a nonlinear seven-dimensional model of yeast glycolytic oscillations. This approach corrected mistakes reliably in both approximated and overspecified models. The method performed well to high levels of noise for most states, could identify the correct model de novo, and make better predictions than ordinary parametric regression and neural network models. We identified an invariant quantity in the model, which accurately derived kinetics and the numerical sensitivity coefficients of the system. Finally, we compared the system to dynamic flux estimation and discussed the scaling and application of this methodology to automated experiment design and control in biological systems in real time.

Glycolytic Oscillations and Limits on Robust Efficiency

Both engineering and evolution are constrained by trade-offs between efficiency and robustness, but theory that formalizes this fact is limited. For a simple two-state model of glycolysis, we explicitly derive analytic equations for hard trade-offs between robustness and efficiency with oscillations as an inevitable side effect. The model describes how the trade-offs arise from individual parameters, including the interplay of feedback control with autocatalysis of network products necessary to power and catalyze intermediate reactions. We then use control theory to prove that the essential features of these hard trade-off "laws" are universal and fundamental, in that they depend minimally on the details of this system and generalize to the robust efficiency of any autocatalytic network. The theory also suggests worst-case conditions that are consistent with initial experiments.

Simple model for temperature control of glycolytic oscillations

We introduce the temperature-dependent autocatalytic coefficient into the Merkin-Needham-Scott version of the Selkov system and consider the resulting equations as a model for temperature-controlled, self-sustained glycolytic oscillations in a closed reactor. It has been shown that this simple model reproduces key features observed in the experiments with temperature growth: (i) exponentially decreasing period of oscillations; (ii) reversal of relative duration leading and tail fronts. The applied model also reproduces the modulations of oscillations induced by the periodic temperature change.

6-Phosphofructo-2-Kinase/Fructose-2,6-Bisphosphatase (PFKFB) Modulates Slow Oscillations in Pancreatic Islets

Glucose-dependent insulin secretion from beta-cells of pancreatic islets is pulsatile, but the source of the underlying oscillations remains unclear. We have developed a computational model of the beta-cell, termed the ‘Dual Oscillator Model’ (DOM), based on the hypothesis that slow oscillations in insulin secretion reflect slow oscillations in glycolysis, which then interact with fast oscillations arising from membrane electrical activity and Ca2+. One version of the DOM predicts that glycolytic oscillations are generated by phosphofructokinase-1 (PFK1) via allosteric activation by its product fructose-1,6-bisphosphate (FBP), and terminated by depletion of the substrate fructose-6-phosphate (F6P). To directly determine whether PFK1 is a control point for slow metabolic oscillations, we experimentally manipulated the flux through PFK1 by altering the activity of 6-phosphofructo-2-kinase/fructose-2,6-bisphosphatase (PFKFB). This bifunctional enzyme, which contains an N-terminal kinase domain (PFK2) and C-terminal phosphatase domain (FBPase2), strongly activates PFK1 activity via conversion of F6P to fructose-2,6-bisphosphate, a potent allosteric activator of PFK1. PFKFB has also been proposed to bind and directly activate glucokinase, further accelerating glycolytic flux. Using optical measurements of NAD(P)H and Ca2+, we found that increasing the level of PFK2, but not FBPase2, in islets strongly decreased both the period and amplitude of slow oscillations by ∼40%. In many of the PFK2-expressing islets slow oscillations (period >120s) were either converted to fast oscillations (period <120s) or were abolished. These data are consistent with the predictions of the DOM after modification of the model to include regulation of PFK1 by PFKFB, and furthermore support the hypothesis that slow oscillations are driven by oscillations in glycolysis. Supported by F32DK085960 (M.J.M.), NSF-DMS0917664 (R.B.), NIH/NIDDK Intramural Research Program (A.S.), and R01DK46409 (L.S.).

Periodic orbits in glycolytic oscillators: From elliptic orbits to relaxation oscillations

We consider the Sel'kov model of glycolytic oscillator for a quantitative study of the limit cycle oscillations in the system. We identify a region of parameter space where perturbation theory holds and use both Linstedt Poincaré technique and harmonic balance to obtain the shape and frequency of the limit cycle. The agreement with the numerically obtained result is excellent. We also find a different extreme, where the limit cycle is of the relaxation oscillator variety, has a large time period and it is seen that, as a particular parameter in the model is varied, the time period increases indefinitely. We characterize this divergence numerically. A calculational method is devised to capture the divergence approximately.

Scaling in Singular Perturbation Problems: Blowing Up a Relaxation Oscillator

A detailed geometric analysis of the Goldbeter–Lefever model of glycolytic oscillations is given. In suitably scaled variables the governing equations are a planar system of ordinary differential equations depending singularly on two small parameters $\varepsilon$ and $\delta$. In [L. Segel and A. Goldbeter, J. Math. Biol., 32 (1994), pp. 147–160] it was argued that a limit cycle of relaxation type exists for $\varepsilon\ll\delta\ll1$. The existence of this limit cycle is proved by analyzing the problem in the spirit of geometric singular perturbation theory. The degeneracies of the limiting problem corresponding to $(\varepsilon,\delta)=(0,0)$ are resolved by a novel variant of the blow-up method. It is shown that repeated blow-ups lead to a clear geometric picture of this fairly complicated two-parameter multiscale problem.