Glycolytic Oscillations In Yeast Research Articles

A General Hybrid Modeling Framework for Systems Biology Applications: Combining Mechanistic Knowledge with Deep Neural Networks under the SBML Standard

In this paper, a computational framework is proposed that merges mechanistic modeling with deep neural networks obeying the Systems Biology Markup Language (SBML) standard. Over the last 20 years, the systems biology community has developed a large number of mechanistic models that are currently stored in public databases in SBML. With the proposed framework, existing SBML models may be redesigned into hybrid systems through the incorporation of deep neural networks into the model core, using a freely available python tool. The so-formed hybrid mechanistic/neural network models are trained with a deep learning algorithm based on the adaptive moment estimation method (ADAM), stochastic regularization and semidirect sensitivity equations. The trained hybrid models are encoded in SBML and uploaded in model databases, where they may be further analyzed as regular SBML models. This approach is illustrated with three well-known case studies: the Escherichia coli threonine synthesis model, the P58IPK signal transduction model, and the Yeast glycolytic oscillations model. The proposed framework is expected to greatly facilitate the widespread use of hybrid modeling techniques for systems biology applications.

Synchronisation of glycolytic activity in yeast cells.

Glycolysis is the central metabolic pathway of almost every cell and organism. Under appropriate conditions, glycolytic oscillations may occur in individual cells as well as in entire cell populations or tissues. In many biological systems, glycolytic oscillations drive coherent oscillations of other metabolites, for instance in cardiomyocytes near anorexia, or in pancreas where they lead to a pulsatile release of insulin. Oscillations at the population or tissue level require the cells to synchronize their metabolism. We review the progress achieved instudying a model organism for glycolytic oscillations, namely yeast. Oscillations may occur on the level of individual cells as well as on the level of the cell population. In yeast, the cell-to-cell interaction is realized by diffusion-mediated intercellular communication via a messenger molecule. The present mini-review focuses on the synchronisation of glycolytic oscillations in yeast. Synchronisation is a quorum-sensing phenomenon because the collective oscillatory behaviour of a yeast cell population ceases when the cell density falls below a threshold. We review the question, under which conditions individual cells in a sparse population continue or cease to oscillate. Furthermore, we provide an overview of the pathway leading to the onset of synchronized oscillations. We also address the effects of spatial inhomogeneities (e.g., the formation of spatial clusters) on the collective dynamics, and also review the emergence of travelling waves of glycolytic activity. Finally, we briefly review the approaches used in numerical modelling of synchronized cell populations.

Promotion and inhibition of synchronous glycolytic oscillations in yeast by chitosan.

Synchronous rhythmic activities play crucial roles in diverse biological systems. Glycolytic oscillations in yeast cells have been studied for 50 years with the aim of elucidating the mechanisms underlying the intracellular oscillations and their synchronization. We investigated the effects of chemical disturbances on the individual and collective glycolytic oscillations in yeast cells encapsulated in alginate microparticles, and demonstrated that the addition of chitosan, an antimicrobial agent, decreased the duration of these oscillations. In contrast, the periods and the synchronicity states showed two different responses to the chitosan treatments. The periods were shown to be prolonged following the treatment with 5-50 mg·L-1 and shortened at 75 mg·L-1 of chitosan. Collective oscillations became more synchronized at 5 mg·L-1 of chitosan, and desynchronized at 25-75 mg·L-1 of this compound. These findings can be explained by the balance between two chitosan features, increasing cell membrane permeability and acetaldehyde scavenging. At low concentrations, chitosan presumably acts as a synchronization promoter that does not mediate the synchronization itself but induces an increase in intercellular coupling. We believe that our findings may provide new insights into the synchronous rhythmic activities in biological systems.

Cyanohydrin reactions enhance glycolytic oscillations in yeast

Synchronous metabolic oscillations can be induced in yeast by addition of glucose and removal of extracellular acetaldehyde (ACAx). Compared to other means of ACAx removal, cyanide robustly induces oscillations, indicating additional cyanide reactions besides ACA to lactonitrile conversion. Here, 13C NMR is used to confirm our previous hypothesis, that cyanide directly affects glycolytic fluxes through reaction with carbonyl-containing compounds. Intracellularly, at least 3 cyanohydrins were identified. Extracellularly, all signals could be identified and lactonitrile was found to account for ~66% of total cyanide removal. Simulations of our updated computational model show that intracellular cyanide reactions increase the amplitude of oscillations and that cyanide addition lowers [ACA] instantaneously. We conclude that cyanide provides the following means of inducing global oscillations: a) by reducing [ACAx] relative to oscillation amplitude, b) by targeting multiple intracellular carbonyl compounds during fermentation, and c) by acting as a phase resetting stimulus.

An experimental study of the regulation of glycolytic oscillations in yeast

We have studied oscillating glycolysis in the strain BY4743 and isogenic strains with deletions of genes encoding enzymes in glycolysis, mitochondrial electron transport and ATP synthesis. We found that deletion of the gene encoding the hexokinase 1 isoform does not affect the oscillations while deletion of the gene encoding the hexokinase 2 isoform results in oscillations with smaller amplitude. The latter is associated with an almost 50% decrease in hexokinase activity. Deletions in the genes encoding the α- and β-subunits of phosphofructokinase abolish the oscillations entirely. This loss in oscillatory activity is associated with a fourfold decrease in phosphofructokinase activity. Deletions of genes encoding subunits of the F1F0 ATPase also inhibit the oscillations in accordance with earlier studies using for example inhibitors. Finally, we identified an apparently new control point involving the mitochondrial cytochrome c oxidase. The latter is difficult to explain as oscillatory activity entails 100% inhibition of this enzyme. The mitochondria of this strain seem to have normal F1F0 ATPase activity. Overall these results support earlier experimental and model studies suggesting that in addition to processes within glycolysis also processes outside this pathway contribute to the control of the oscillatory behaviour.

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.

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.

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.

Dynamical quorum sensing: Population density encoded in cellular dynamics

Mutual synchronization by exchange of chemicals is a mechanism for the emergence of collective dynamics in cellular populations. General theories exist on the transition to coherence, but no quantitative, experimental demonstration has been given. Here, we present a modeling and experimental analysis of cell-density-dependent glycolytic oscillations in yeast. We study the disappearance of oscillations at low cell density and show that this phenomenon occurs synchronously in all cells and not by desynchronization, as previously expected. This study identifies a general scenario for the emergence of collective cellular oscillations and suggests a quorum-sensing mechanism by which the cell density information is encoded in the intracellular dynamical state.

Reduction of a biochemical model with preservation of its basic dynamic properties

The complexity of full-scale metabolic models is a major obstacle for their effective use in computational systems biology. The aim of model reduction is to circumvent this problem by eliminating parts of a model that are unimportant for the properties of interest. The choice of reduction method is influenced both by the type of model complexity and by the objective of the reduction; therefore, no single method is superior in all cases. In this study we present a comparative study of two different methods applied to a 20D model of yeast glycolytic oscillations. Our objective is to obtain biochemically meaningful reduced models, which reproduce the dynamic properties of the 20D model. The first method uses lumping and subsequent constrained parameter optimization. The second method is a novel approach that eliminates variables not essential for the dynamics. The applications of the two methods result in models of eight (lumping), six (elimination) and three (lumping followed by elimination) dimensions. All models have similar dynamic properties and pin-point the same interactions as being crucial for generation of the oscillations. The advantage of the novel method is that it is algorithmic, and does not require input in the form of biochemical knowledge. The lumping approach, however, is better at preserving biochemical properties, as we show through extensive analyses of the models.

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.

Metabolic pathways reconstruction by frequency and amplitude response to forced glycolytic oscillations in yeast

The hypothesis that frequency and amplitude response can be used in a complicated metabolic pathway kinetics model for optimal parameter estimation, as speculated by its successful prior usage for a mechanical oscillator and a heterogeneous chemical system, is tested here. Given the complexity of the glycolysis model of yeast chosen, this question is limited to three kinetics parameters of the 87 in the in vitro model developed in the literature. The direct application of the approach, used with the uninformed selection of operating conditions for the oscillation of external glucose concentration, led to miring the data assimilation process in local minima. Application of linear systems theory, however, identified two natural resonant frequencies that, when excited by external forced oscillations of the same frequency, result in the expression of many harmonics in the Fourier spectra, that is, information-rich experiments. A single such information-rich experiment at one of the resonant frequencies was sufficient to break away from the local minima to find the optimum kinetics parameter estimates. The resonant frequencies themselves represent oscillation modes in glycolysis akin to those previously observed. Furthermore, operation of the bioreactor with large amplitude oscillations of glucose feed (25%) leads to enhanced ethanol average yield by 1.6% at the resonant frequency.

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.

Combined Metabolic and Cell Population Modeling for Yeast Bioreactor Control

Abstract Several investigators recently have explored the use of cell population balance equation (PBE) models for the design of biochemical reactor control strategies. A major disadvantage of the PBE modeling approach is that the incorporation of intracellular reactions needed to accurately describe cellular processes leads to substantial computational difficulties. We investigate an alternative modeling technique in which the cell population is constructed from an ensemble of individual cell models. The average value or the number distribution of any intracellular property captured by the cell model can be computed from ensemble simulation data. To illustrate the basic procedure, a single cell model of yeast glycolytic oscillations is used to construct large cell ensembles for the investigation of cell population synchronization. The potential use of cell ensemble models for bioreactor controller design are discussed.

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.

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.

Transduction of Intracellular and Intercellular Dynamics in Yeast Glycolytic Oscillations

Under certain well-defined conditions, a population of yeast cells exhibits glycolytic oscillations that synchronize through intercellular acetaldehyde. This implies that the dynamic phenomenon of the oscillation propagates within and between cells. We here develop a method to establish by which route dynamics propagate through a biological reaction network. Application of the method to yeast demonstrates how the oscillations and the synchronization signal can be transduced. That transduction is not so much through the backbone of glycolysis, as via the Gibbs energy and redox coenzyme couples (ATP/ADP, and NADH/NAD), and via both intra- and intercellular acetaldehyde.