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

Kinetic modelling of glycolytic oscillations

Glycolytic oscillations have been studied for well over 60 years, but aspects of their function, and mechanisms of regulation and synchronisation remain unclear. Glycolysis is amenable to mechanistic mathematical modelling, as its components have been well characterised, and the system can be studied at many organisational levels: in vitro reconstituted enzymes, cell free extracts, individual cells, and cell populations. In recent years, the emergence of individual cell analysis has opened new ways of studying this intriguing system.

Personalised modelling of clinical heterogeneity between medium-chain acyl-CoA dehydrogenase patients

BACKGROUND

Monogenetic inborn errors of metabolism cause a wide phenotypic heterogeneity that may even differ between family members carrying the same genetic variant. Computational modelling of metabolic networks may identify putative sources of this inter-patient heterogeneity. Here, we mainly focus on medium-chain acyl-CoA dehydrogenase deficiency (MCADD), the most common inborn error of the mitochondrial fatty acid oxidation (mFAO). It is an enigma why some MCADD patients-if untreated-are at risk to develop severe metabolic decompensations, whereas others remain asymptomatic throughout life. We hypothesised that an ability to maintain an increased free mitochondrial CoA (CoASH) and pathway flux might distinguish asymptomatic from symptomatic patients.

RESULTS

We built and experimentally validated, for the first time, a kinetic model of the human liver mFAO. Metabolites were partitioned according to their water solubility between the bulk aqueous matrix and the inner membrane. Enzymes are also either membrane-bound or in the matrix. This metabolite partitioning is a novel model attribute and improved predictions. MCADD substantially reduced pathway flux and CoASH, the latter due to the sequestration of CoA as medium-chain acyl-CoA esters. Analysis of urine from MCADD patients obtained during a metabolic decompensation showed an accumulation of medium- and short-chain acylcarnitines, just like the acyl-CoA pool in the MCADD model. The model suggested some rescues that increased flux and CoASH, notably increasing short-chain acyl-CoA dehydrogenase (SCAD) levels. Proteome analysis of MCADD patient-derived fibroblasts indeed revealed elevated levels of SCAD in a patient with a clinically asymptomatic state. This is a rescue for MCADD that has not been explored before. Personalised models based on these proteomics data confirmed an increased pathway flux and CoASH in the model of an asymptomatic patient compared to those of symptomatic MCADD patients.

CONCLUSIONS

We present a detailed, validated kinetic model of mFAO in human liver, with solubility-dependent metabolite partitioning. Personalised modelling of individual patients provides a novel explanation for phenotypic heterogeneity among MCADD patients. Further development of personalised metabolic models is a promising direction to improve individualised risk assessment, management and monitoring for inborn errors of metabolism.

Exploring the evolution of biochemical models at the network level

The evolution of biochemical models is difficult to track. At present, it is not possible to inspect the differences between model versions at the network level. Biochemical models are often constructed in a distributed, non-linear process: collaborators create model versions on different branches from novel information, model extensions, during curation and adaption. To discuss and align the versions, it is helpful to abstract the changes to the network level. The differences between two model versions can be detected by the software tool BiVeS. However, it cannot show the structural changes resulting from the differences. Here, we present a method to visualise the differences between model versions effectively. We developed a JSON schema to communicate the differences at the network level and extended BiVeS accordingly. Additionally, we developed DiVil, a web-based tool to represent the model and the differences as a standardised network using D3. It combines an automatic layout with an interactive user interface to improve the visualisation and to inspect the model. The network can be exported in standardised formats as images or markup language. Our method communicates the structural differences between model versions. It facilitates the discussion of changes and thus supports the collaborative and non-linear nature of model development.

Availability and implementation: DiVil prototype: https://divil.bio.informatik.uni-rostock.de, Code on GitHub: https://github.com/Gebbi8/DiVil, licensed under Apache License 2.0.

Contact:url="tom.gebhardt@uni-rostock.de.

Kinetic Modeling of Saccharomyces cerevisiae Central Carbon Metabolism: Achievements, Limitations, and Opportunities

Central carbon metabolism comprises the metabolic pathways in the cell that process nutrients into energy, building blocks and byproducts. To unravel the regulation of this network upon glucose perturbation, several metabolic models have been developed for the microorganism Saccharomyces cerevisiae. These dynamic representations have focused on glycolysis and answered multiple research questions, but no commonly applicable model has been presented. This review systematically evaluates the literature to describe the current advances, limitations, and opportunities. Different kinetic models have unraveled key kinetic glycolytic mechanisms. Nevertheless, some uncertainties regarding model topology and parameter values still limit the application to specific cases. Progressive improvements in experimental measurement technologies as well as advances in computational tools create new opportunities to further extend the model scale. Notably, models need to be made more complex to consider the multiple layers of glycolytic regulation and external physiological variables regulating the bioprocess, opening new possibilities for extrapolation and validation. Finally, the onset of new data representative of individual cells will cause these models to evolve from depicting an average cell in an industrial fermenter, to characterizing the heterogeneity of the population, opening new and unseen possibilities for industrial fermentation improvement.

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 in studying 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.

Intercellular communication induces glycolytic synchronization waves between individually oscillating cells

Many organs have internal structures with spatially differentiated and sometimes temporally synchronized groups of cells. The mechanisms leading to such differentiation and coordination are not well understood. Here we design a diffusion-limited microfluidic system to mimic a multicellular organ structure with peripheral blood flow and test whether a group of individually oscillating yeast cells could form subpopulations of spatially differentiated and temporally synchronized cells. Upon substrate addition, the dynamic response at single-cell level shows glycolytic oscillations, leading to wave fronts traveling through the monolayered population and to synchronized communities at well-defined positions in the cell chamber. A detailed mechanistic model with the architectural structure of the flow chamber incorporated successfully predicts the spatial-temporal experimental data, and allows for a molecular understanding of the observed phenomena. The intricate interplay of intracellular biochemical reaction networks leading to the oscillations, combined with intercellular communication via metabolic intermediates and fluid dynamics of the reaction chamber, is responsible for the generation of the subpopulations of synchronized cells. This mechanism, as analyzed from the model simulations, is experimentally tested using different concentrations of cyanide stress solutions. The results are reproducible and stable, despite cellular heterogeneity, and the spontaneous community development is reminiscent of a zoned cell differentiation often observed in multicellular organs.

Differential equation-based minimal model describing metabolic oscillations in Bacillus subtilis biofilms

Biofilms offer an excellent example of ecological interaction among bacteria. Temporal and spatial oscillations in biofilms are an emerging topic. In this paper, we describe the metabolic oscillations in Bacillus subtilis biofilms by applying the smallest theoretical chemical reaction system showing Hopf bifurcation proposed by Wilhelm and Heinrich in 1995. The system involves three differential equations and a single bilinear term. We specifically select parameters that are suitable for the biological scenario of biofilm oscillations. We perform computer simulations and a detailed analysis of the system including bifurcation analysis and quasi-steady-state approximation. We also discuss the feedback structure of the system and the correspondence of the simulations to biological observations. Our theoretical work suggests potential scenarios about the oscillatory behaviour of biofilms and also serves as an application of a previously described chemical oscillator to a biological system.

Emergence of collective oscillations in adaptive cells

Collective oscillations of cells in a population appear under diverse biological contexts. Here, we establish a set of common principles by categorising the response of individual cells against a time-varying signal. A positive intracellular signal relay of sufficient gain from participating cells is required to sustain the oscillations, together with phase matching. The two conditions yield quantitative predictions for the onset cell density and frequency in terms of measured single-cell and signal response functions. Through mathematical constructions, we show that cells that adapt to a constant stimulus fulfil the phase requirement by developing a leading phase in an active frequency window that enables cell-to-signal energy flow. Analysis of dynamical quorum sensing in several cellular systems with increasing biological complexity reaffirms the pivotal role of adaptation in powering oscillations in an otherwise dissipative cell-to-cell communication channel. The physical conditions identified also apply to synthetic oscillatory systems.

There are many examples of cell populations exhibiting density-dependent collective oscillatory behaviour. Here, the authors show that sustained collective oscillations emerge when cells anticipate variation in signal and attempt to amplify it, a property that can be linked to adaptation.

Effect of macromolecular crowding on the kinetics of glycolytic enzymes and the behaviour of glycolysis in yeast

Water is involved in all aspects of biological activity, both as a solvent and as a reactant. It is hypothesized that intracellular water is in a highly structured state due to the high concentrations of macromolecules in the cell and that this may change the activity of intracellular enzymes due to altered binding affinities and allosteric regulations. Here we first investigate the kinetics of two glycolytic enzymes in artificially crowded aqueous solutions and show that crowding does indeed change their kinetics. Based on our kinetic measurements we propose a new model of oscillating glycolysis that instead of Michaelis-Menten or Monod-Wyman-Changeux kinetics uses the Yang-Ling adsorption isotherm introduced by G. Ling in the frame of the Association-Induction (AI) hypothesis. Using this model, we can reproduce previous experimental observations of the coupling of glycolytic oscillations and intracellular water dynamics, e.g., (i) during the metabolic oscillations, the latter variable oscillates in phase with ATP activity, and (ii) the emergence of glycolytic oscillations largely depends on the extent of intracellular water dipolar relaxation in cells in the resting state. Our results support the view that the extent of intracellular water dipolar relaxation is regulated by the ability of cytoplasmic proteins to polarize intracellular water with the assistance of ATP, as suggested in the AI hypothesis. This hypothesis may be relevant to the interpretation of many other biological oscillators, including cell signalling processes.

Phosphofructokinase controls the acetaldehyde-induced phase shift in isolated yeast glycolytic oscillators

The response of oscillatory systems to external perturbations is crucial for emergent properties such as synchronisation and phase locking and can be quantified in a phase response curve (PRC). In individual, oscillating yeast cells, we characterised experimentally the phase response of glycolytic oscillations for external acetaldehyde pulses and followed the transduction of the perturbation through the system. Subsequently, we analysed the control of the relevant system components in a detailed mechanistic model. The observed responses are interpreted in terms of the functional coupling and regulation in the reaction network. We find that our model quantitatively predicts the phase-dependent phase shift observed in the experimental data. The phase shift is in agreement with an adaptation leading to synchronisation with an external signal. Our model analysis establishes that phosphofructokinase plays a key role in the phase shift dynamics as shown in the PRC and adaptation time to external perturbations. Specific mechanism-based interventions, made possible through such analyses of detailed models, can improve upon standard trial and error methods, e.g. melatonin supplementation to overcome jet-lag, which are error-prone, specifically, since the effects are phase dependent and dose dependent. The models by Gustavsson and Goldbeter discussed in the text can be obtained from the JWS Online simulation database: (https://jjj.bio.vu.nl/models/gustavsson5 and https://jjj.bio.vu.nl/models/goldbeter1).

Dissipative structures and biological rhythms

Sustained oscillations abound in biological systems. They occur at all levels of biological organization over a wide range of periods, from a fraction of a second to years, and with a variety of underlying mechanisms. They control major physiological functions, and their dysfunction is associated with a variety of physiological disorders. The goal of this review is (i) to give an overview of the main rhythms observed at the cellular and supracellular levels, (ii) to briefly describe how the study of biological rhythms unfolded in the course of time, in parallel with studies on chemical oscillations, (iii) to present the major roles of biological rhythms in the control of physiological functions, and (iv) the pathologies associated with the alteration, disappearance, or spurious occurrence of biological rhythms. Two tables present the main examples of cellular and supracellular rhythms ordered according to their period, and their role in physiology and pathophysiology. Among the rhythms discussed are neural and cardiac rhythms, metabolic oscillations such as those occurring in glycolysis in yeast, intracellular Ca⁺⁺ oscillations, cyclic AMP oscillations in Dictyostelium amoebae, the segmentation clock that controls somitogenesis, pulsatile hormone secretion, circadian rhythms which occur in all eukaryotes and some bacteria with a period close to 24 h, the oscillatory dynamics of the enzymatic network driving the cell cycle, and oscillations in transcription factors such as NF-ΚB and tumor suppressors such as p53. Ilya Prigogine's concept of dissipative structures applies to temporal oscillations and allows us to unify within a common framework the various rhythms observed at different levels of biological organization, regardless of their period and underlying mechanism.

Targeting glycolysis in the malaria parasite Plasmodium falciparum

Glycolysis is the main pathway for ATP production in the malaria parasite Plasmodium falciparum and essential for its survival. Following a sensitivity analysis of a detailed kinetic model for glycolysis in the parasite, the glucose transport reaction was identified as the step whose activity needed to be inhibited to the least extent to result in a 50% reduction in glycolytic flux. In a subsequent inhibitor titration with cytochalasin B, we confirmed the model analysis experimentally and measured a flux control coefficient of 0.3 for the glucose transporter. In addition to the glucose transporter, the glucokinase and phosphofructokinase had high flux control coefficients, while for the ATPase a small negative flux control coefficient was predicted. In a broader comparative analysis of glycolytic models, we identified a weakness in the P. falciparum pathway design with respect to stability towards perturbations in the ATP demand.

DATABASE

The mathematical model described here has been submitted to the JWS Online Cellular Systems Modelling Database and can be accessed at http://jjj.bio.vu.nl/database/vanniekerk1. The SEEK-study including the experimental data set is available at DOI 10.15490/seek.1.

INVESTIGATION

56 (http://dx.doi.org/10.15490/seek.1.

INVESTIGATION

56).

Rites of passage: requirements and standards for building kinetic models of metabolic phenotypes

The overarching ambition of kinetic metabolic modeling is to capture the dynamic behavior of metabolism to such an extent that systems and synthetic biology strategies can reliably be tested in silico. The lack of kinetic data hampers the development of kinetic models, and most of the current models use ad hoc reduced stoichiometry or oversimplified kinetic rate expressions, which may limit their predictive strength. There is a need to introduce the community-level standards that will organize and accelerate the future developments in this area. We introduce here a set of requirements that will ensure the model quality, we examine the current kinetic models with respect to these requirements, and we propose a general workflow for constructing models that satisfy these requirements.

Validation and selection of ODE based systems biology models: how to arrive at more reliable decisions

Background

Most ordinary differential equation (ODE) based modeling studies in systems biology involve a hold-out validation step for model validation. In this framework a pre-determined part of the data is used as validation data and, therefore it is not used for estimating the parameters of the model. The model is assumed to be validated if the model predictions on the validation dataset show good agreement with the data. Model selection between alternative model structures can also be performed in the same setting, based on the predictive power of the model structures on the validation dataset. However, drawbacks associated with this approach are usually under-estimated.

Results

We have carried out simulations by using a recently published High Osmolarity Glycerol (HOG) pathway from S.cerevisiae to demonstrate these drawbacks. We have shown that it is very important how the data is partitioned and which part of the data is used for validation purposes. The hold-out validation strategy leads to biased conclusions, since it can lead to different validation and selection decisions when different partitioning schemes are used. Furthermore, finding sensible partitioning schemes that would lead to reliable decisions are heavily dependent on the biology and unknown model parameters which turns the problem into a paradox. This brings the need for alternative validation approaches that offer flexible partitioning of the data. For this purpose, we have introduced a stratified random cross-validation (SRCV) approach that successfully overcomes these limitations.

Conclusions

SRCV leads to more stable decisions for both validation and selection which are not biased by underlying biological phenomena. Furthermore, it is less dependent on the specific noise realization in the data. Therefore, it proves to be a promising alternative to the standard hold-out validation strategy.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0180-0) contains supplementary material, which is available to authorized users.

Multiplicity of steady states in glycolysis and shift of metabolic state in cultured mammalian cells

Cultured mammalian cells exhibit elevated glycolysis flux and high lactate production. In the industrial bioprocesses for biotherapeutic protein production, glucose is supplemented to the culture medium to sustain continued cell growth resulting in the accumulation of lactate to high levels. In such fed-batch cultures, sometimes a metabolic shift from a state of high glycolysis flux and high lactate production to a state of low glycolysis flux and low lactate production or even lactate consumption is observed. While in other cases with very similar culture conditions, the same cell line and medium, cells continue to produce lactate. A metabolic shift to lactate consumption has been correlated to the productivity of the process. Cultures that exhibited the metabolic shift to lactate consumption had higher titers than those which didn't. However, the cues that trigger the metabolic shift to lactate consumption state (or low lactate production state) are yet to be identified. Metabolic control of cells is tightly linked to growth control through signaling pathways such as the AKT pathway. We have previously shown that the glycolysis of proliferating cells can exhibit bistability with well-segregated high flux and low flux states. Low lactate production (or lactate consumption) is possible only at a low glycolysis flux state. In this study, we use mathematical modeling to demonstrate that lactate inhibition together with AKT regulation on glycolysis enzymes can profoundly influence the bistable behavior, resulting in a complex steady-state topology. The transition from the high flux state to the low flux state can only occur in certain regions of the steady state topology, and therefore the metabolic fate of the cells depends on their metabolic trajectory encountering the region that allows such a metabolic state switch. Insights from such switch behavior present us with new means to control the metabolism of mammalian cells in fed-batch cultures.

Entrainment of heterogeneous glycolytic oscillations in single cells

Cell signaling, gene expression, and metabolism are affected by cell-cell heterogeneity and random changes in the environment. The effects of such fluctuations on cell signaling and gene expression have recently been studied intensively using single-cell experiments. In metabolism heterogeneity may be particularly important because it may affect synchronisation of metabolic oscillations, an important example of cell-cell communication. This synchronisation is notoriously difficult to describe theoretically as the example of glycolytic oscillations shows: neither is the mechanism of glycolytic synchronisation understood nor the role of cell-cell heterogeneity. To pin down the mechanism and to assess its robustness and universality we have experimentally investigated the entrainment of glycolytic oscillations in individual yeast cells by periodic external perturbations. We find that oscillatory cells synchronise through phase shifts and that the mechanism is insensitive to cell heterogeneity (robustness) and similar for different types of external perturbations (universality).

Understanding bistability in yeast glycolysis using general properties of metabolic pathways

Glycolysis is the central pathway in energy metabolism in the majority of organisms. In a recent paper, van Heerden et al. showed experimentally and computationally that glycolysis can exist in two states, a global steady state and a so-called imbalanced state. In the imbalanced state, intermediary metabolites accumulate at low levels of ATP and inorganic phosphate. It was shown that Baker's yeast uses a peculiar regulatory mechanism--via trehalose metabolism--to ensure that most yeast cells reach the steady state and not the imbalanced state.

RESULTS

Here we explore the apparent bistable behaviour in a core model of glycolysis that is based on a well-established detailed model, and study in great detail the bifurcation behaviour of solutions, without using any numerical information on parameter values.

CONCLUSION

We uncover a rich suite of solutions, including so-called imbalanced states, bistability, and oscillatory behaviour. The techniques employed are generic, directly suitable for a wide class of biochemical pathways, and could lead to better analytical treatments of more detailed models.

How informative is your kinetic model?: using resampling methods for model invalidation

Background

Kinetic models can present mechanistic descriptions of molecular processes within a cell. They can be used to predict the dynamics of metabolite production, signal transduction or transcription of genes. Although there has been tremendous effort in constructing kinetic models for different biological systems, not much effort has been put into their validation. In this study, we introduce the concept of resampling methods for the analysis of kinetic models and present a statistical model invalidation approach.

Results

We based our invalidation approach on the evaluation of a kinetic model’s predictive power through cross validation and forecast analysis. As a reference point for this evaluation, we used the predictive power of an unsupervised data analysis method which does not make use of any biochemical knowledge, namely Smooth Principal Components Analysis (SPCA) on the same test sets. Through a simulations study, we showed that too simple mechanistic descriptions can be invalidated by using our SPCA-based comparative approach until high amount of noise exists in the experimental data. We also applied our approach on an eicosanoid production model developed for human and concluded that the model could not be invalidated using the available data despite its simplicity in the formulation of the reaction kinetics. Furthermore, we analysed the high osmolarity glycerol (HOG) pathway in yeast to question the validity of an existing model as another realistic demonstration of our method.

Conclusions

With this study, we have successfully presented the potential of two resampling methods, cross validation and forecast analysis in the analysis of kinetic models’ validity. Our approach is easy to grasp and to implement, applicable to any ordinary differential equation (ODE) type biological model and does not suffer from any computational difficulties which seems to be a common problem for approaches that have been proposed for similar purposes. Matlab files needed for invalidation using SPCA cross validation and our toy model in SBML format are provided at http://www.bdagroup.nl/content/Downloads/software/software.php.

Allosteric regulation of phosphofructokinase controls the emergence of glycolytic oscillations in isolated yeast cells

Oscillations are widely distributed in nature and synchronization of oscillators has been described at the cellular level (e.g. heart cells) and at the population level (e.g. fireflies). Yeast glycolysis is the best known oscillatory system, although it has been studied almost exclusively at the population level (i.e. limited to observations of average behaviour in synchronized cultures). We studied individual yeast cells that were positioned with optical tweezers in a microfluidic chamber to determine the precise conditions for autonomous glycolytic oscillations. Hopf bifurcation points were determined experimentally in individual cells as a function of glucose and cyanide concentrations. The experiments were analyzed in a detailed mathematical model and could be interpreted in terms of an oscillatory manifold in a three-dimensional state-space; crossing the boundaries of the manifold coincides with the onset of oscillations and positioning along the longitudinal axis of the volume sets the period. The oscillatory manifold could be approximated by allosteric control values of phosphofructokinase for ATP and AMP.

DATABASE

The mathematical models described here have been submitted to the JWS Online Cellular Systems Modelling Database and can be accessed at http://jjj.mib.ac.uk/webMathematica/UItester.jsp?modelName=gustavsson5. [Database section added 14 May 2014 after original online publication].

Heterogeneity of glycolytic oscillatory behaviour in individual yeast cells

There are many examples of oscillations in biological systems and one of the most investigated is glycolytic oscillations in yeast. These oscillations have been studied since the 1950s in dense, synchronized populations and in cell-free extracts, but it has for long been unknown whether a high cell density is a requirement for oscillations to be induced, or if individual cells can oscillate also in isolation without synchronization. Here we present an experimental method and a detailed kinetic model for studying glycolytic oscillations in individual, isolated yeast cells and compare them to previously reported studies of single-cell oscillations. The importance of single-cell studies of this phenomenon and relevant future research questions are also discussed.

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.

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

MOTIVATION

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.

RESULTS

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.

AVAILABILITY

The complete python code is available as Supplementary Information.

CONTACT

bjornhald@gmail.com or pgs@kiku.dk

SUPPLEMENTARY INFORMATION

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.

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.

DATABASE

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.

DATABASE

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.