Sustained oscillations in glycolysis: an experimental and theoretical study of chaotic and complex periodic behavior and of quenching of simple oscillations

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.

From biological data to oscillator models using SINDy

Periodic changes in the concentration or activity of different molecules regulate vital cellular processes such as cell division and circadian rhythms. Developing mathematical models is essential to better understand the mechanisms underlying these oscillations. Recent data-driven methods like SINDy have fundamentally changed model identification, yet their application to experimental biological data remains limited. This study investigates SINDy's constraints by directly applying it to biological oscillatory data. We identify insufficient resolution, noise, dimensionality, and limited prior knowledge as primary limitations. Using various generic oscillator models of different complexity and/or dimensionality, we systematically analyze these factors. We then propose a comprehensive guide for inferring models from biological data, addressing these challenges step by step. Our approach is validated using glycolytic oscillation data from yeast.

Chemical Memory with Discrete Turing Patterns Appearing in the Glycolytic Reaction

Memory is an essential element in information processing devices. We investigated a network formed by just three interacting nodes representing continuously stirred tank reactors (CSTRs) in which the glycolytic reaction proceeds as a potential realization of a chemical memory unit. Our study is based on the 2-variable computational model of the reaction. The model parameters were selected such that the system has a stable limit cycle and several distinct, discrete Turing patterns characterized by stationary concentrations at the nodes. In our interpretation, oscillations represent a blank memory unit, and Turing patterns code information. The considered memory can preserve information on one of six different symbols. The time evolution of the nodes was individually controlled by the inflow of ATP. We demonstrate that information can be written with a simple and short perturbation of the inflow. The perturbation applies to only one or two nodes, and it is symbol specific. The memory can be erased with identical inflow perturbation applied to all nodes. The presented idea of pattern-coded memory applies to other reaction networks that allow for discrete Turing patterns. Moreover, it hints at the experimental realization of memory in a simple system with the glycolytic reaction.

Kinetic and data-driven modeling of pancreatic β-cell central carbon metabolism and insulin secretion

Pancreatic β-cells respond to increased extracellular glucose levels by initiating a metabolic shift. That change in metabolism is part of the process of glucose-stimulated insulin secretion and is of particular interest in the context of diabetes. However, we do not fully understand how the coordinated changes in metabolic pathways and metabolite products influence insulin secretion. In this work, we apply systems biology approaches to develop a detailed kinetic model of the intracellular central carbon metabolic pathways in pancreatic β-cells upon stimulation with high levels of glucose. The model is calibrated to published metabolomics datasets for the INS1 823/13 cell line, accurately capturing the measured metabolite fold-changes. We first employed the calibrated mechanistic model to estimate the stimulated cell's fluxome. We then used the predicted network fluxes in a data-driven approach to build a partial least squares regression model. By developing the combined kinetic and data-driven modeling framework, we gain insights into the link between β-cell metabolism and glucose-stimulated insulin secretion. The combined modeling framework was used to predict the effects of common anti-diabetic pharmacological interventions on metabolite levels, flux through the metabolic network, and insulin secretion. Our simulations reveal targets that can be modulated to enhance insulin secretion. The model is a promising tool to contextualize and extend the usefulness of metabolomics data and to predict dynamics and metabolite levels that are difficult to measure in vitro. In addition, the modeling framework can be applied to identify, explain, and assess novel and clinically-relevant interventions that may be particularly valuable in diabetes treatment.

Agent-based models for detecting the driving forces of biomolecular interactions

Agent-based modelling and simulation have been effectively applied to the study of complex biological systems, especially when composed of many interacting entities. Representing biomolecules as autonomous agents allows this approach to bring out the global behaviour of biochemical processes as resulting from local molecular interactions. In this paper, we leverage the capabilities of the agent paradigm to construct an in silico replica of the glycolytic pathway; the aim is to detect the role that long-range electrodynamic forces might have on the rate of glucose oxidation. Experimental evidences have shown that random encounters and short-range potentials might not be sufficient to explain the high efficiency of biochemical reactions in living cells. However, while the latest in vitro studies are limited by present-day technology, agent-based simulations provide an in silico support to the outcomes hitherto obtained and shed light on behaviours not yet well understood. Our results grasp properties hard to uncover through other computational methods, such as the effect of electromagnetic potentials on glycolytic oscillations.

A predictive dynamic yeast model based on component, energy, and electron carrier balances

The present study describes a novel yeast model for the prediction of yeast fermentation. The proposed model considers the possible metabolic pathways of yeast. For each pathway, the time evolution of components, energy (ATP/ADP), and electron carriers (NAD⁺ /NADH) are expressed with limitation factors for all quantities consumed by each respective pathway. In this manner, the model can predict the partition of these pathways based on the growth conditions and their evolution over time. Several biological pathways and their stoichiometric coefficients are well known from literature. It is important to note that most of the kinetic parameters have no effect as the actual kinetics are controlled by the balance of limiting factors. The few remaining parameters were adjusted and compared with the literature when the data set was available. The model fits our experimental data from yeast fermentation on glucose in a nonaerated batch system. The predictive ability of the model and its capacity to represent the intensity of each pathway over time facilitate an improved understanding of the interactions between the pathways. The key role of energy (ATP) and electron carrier (NAD⁺ ) to trigger the different metabolic pathways during yeast growth is highlighted, whereas the involvement of mitochondrial respiration not being associated with the TCA cycle is also shown.

Chaos, mitochondria and type 2 diabetes; does type 2 diabetes arise from a metabolic dysrhythmia?

The increasing incidence of type 2 diabetes transcends all cultures, largely due to populations transitioning from traditional diets and manual occupations, to sedentary, calorific lifestyles. Excess calorie intake leads to intramuscular fat accumulation and insulin resistance. Physical inactivity causes underutilization of mitochondria causing dysfunction and inflammation. Both insulin resistance and mitochondrial dysfunction mechanisms are known to be closely related and to antagonise one another, although the precise nature of the relationship has eluded characterization. It is poorly understood why this mutual dysfunction progresses on to clinical diabetes in only some patients, why progression is often stepwise and why diabetes control only weakly predicts future cardiovascular disease in individuals. Clinical prediction in patients is therefore currently unsatisfactory and current linear assumptions require challenging. Cells contain networks of oscillating ionic fluxes. Cellular activity is characterised by complex patterns of fluctuation with sudden transitions between patterns. The non-linear nature of these oscillations is well characterised in neuronal activity, cardiac impulses and more recently mitochondria, but not previously in relation to diabetes. Cells under metabolic stress demonstrate complex fluctuations of mitochondrial distribution, coupling strength and synchronisation resulting in periodic or chaotic oscillations of function, causing accumulation of intracellular fat and excess reactive oxygen species (ROS), which exacerbates insulin resistance. Glucose, insulin and HbA1c in patients are also known to oscillate in complex patterns but the mechanisms and significance are largely unknown. Drawing on existing evidence and models from other diseases, a nonlinear, dynamical hypothesis of diabetes onset and progression is proposed. Insulin receptor pathways and mitochondria are treated as two populations of coupled, phase oscillators. Health or disease states depend on system stability or instability and reflect the balance of substrate supply and energy demand. The implication of this novel mechanism is that diabetes and the complications are not the consequence of a distinct pathological agent or pathway, but more an evolving dysrhythmia of normal cellular energetics systems, resulting from accumulated adverse lifestyle conditions. This hypothesis is proposed with the intention of stimulating research into non-linear dynamical constructs as an alternative to current linear models, to improve risk prediction and trajectory analysis in type 2 diabetes.

Recent advances in biomedical simulations: a manifesto for model engineering

Biomedical simulations are widely used to understand disease, engineer cells, and model cellular processes. In this article, we explore how to improve the quality of biomedical simulations by developing simulation models using tools and practices employed in software engineering. We refer to this direction as model engineering. Not all techniques used by software engineers are directly applicable to model engineering, and so some adaptations are required. That said, we believe that simulation models can benefit from software engineering practices for requirements, design, and construction as well as from software engineering tools for version control, error checking, and testing. Here we survey current efforts to improve simulation quality and discuss promising research directions for model engineering.

Mitochondrial chaotic dynamics: Redox-energetic behavior at the edge of stability

Mitochondria serve multiple key cellular functions, including energy generation, redox balance, and regulation of apoptotic cell death, thus making a major impact on healthy and diseased states. Increasingly recognized is that biological network stability/instability can play critical roles in determining health and disease. We report for the first-time mitochondrial chaotic dynamics, characterizing the conditions leading from stability to chaos in this organelle. Using an experimentally validated computational model of mitochondrial function, we show that complex oscillatory dynamics in key metabolic variables, arising at the “edge” between fully functional and pathological behavior, sets the stage for chaos. Under these conditions, a mild, regular sinusoidal redox forcing perturbation triggers chaotic dynamics with main signature traits such as sensitivity to initial conditions, positive Lyapunov exponents, and strange attractors. At the “edge” mitochondrial chaos is exquisitely sensitive to the antioxidant capacity of matrix Mn superoxide dismutase as well as to the amplitude and frequency of the redox perturbation. These results have potential implications both for mitochondrial signaling determining health maintenance, and pathological transformation, including abnormal cardiac rhythms.

A Digitally Programmable Cytomorphic Chip for Simulation of Arbitrary Biochemical Reaction Networks

Prior work has shown that compact analog circuits can faithfully represent and model fundamental biomolecular circuits via efficient log-domain cytomorphic transistor equivalents. Such circuits have emphasized basis functions that are dominant in genetic transcription and translation networks and deoxyribonucleic acid (DNA)-protein binding. Here, we report a system featuring digitally programmable 0.35 μm BiCMOS analog cytomorphic chips that enable arbitrary biochemical reaction networks to be exactly represented thus enabling compact and easy composition of protein networks as well. Since all biomolecular networks can be represented as chemical reaction networks, our protein networks also include the former genetic network circuits as a special case. The cytomorphic analog protein circuits use one fundamental association-dissociation-degradation building-block circuit that can be configured digitally to exactly represent any zeroth-, first-, and second-order reaction including loading, dynamics, nonlinearity, and interactions with other building-block circuits. To address a divergence issue caused by random variations in chip fabrication processes, we propose a unique way of performing computation based on total variables and conservation laws, which we instantiate at both the circuit and network levels. Thus, scalable systems that operate with finite error over infinite time can be built. We show how the building-block circuits can be composed to form various network topologies, such as cascade, fan-out, fan-in, loop, dimerization, or arbitrary networks using total variables. We demonstrate results from a system that combines interacting cytomorphic chips to simulate a cancer pathway and a glycolysis pathway. Both simulations are consistent with conventional software simulations. Our highly parallel digitally programmable analog cytomorphic systems can lead to a useful design, analysis, and simulation tool for studying arbitrary large-scale biological networks in systems and synthetic biology.

Parameters Estimation in Phase-Space Landscape Reconstruction of Cell Fate: A Systems Biology Approach

The thermodynamical formalism of irreversible processes offers a theoretical framework appropriate to explain the complexity observed at the macroscopic level of dynamic systems. In this context, together with the theory of complex systems and systems biology, the thermodynamical formalism establishes an appropriate conceptual framework to address the study of biological systems, in particular cancer.The Chapter is organized as follows: In Subheading 1, an integrative view of these disciplines is offered, for the characterization of the emergence and evolution of cancer, seen as a self-organized dynamic system far from the thermodynamic equilibrium. Development of a thermodynamic framework, based on the entropy production rate, is presented in Subheading 2. Subheading 3 is dedicated to all tumor growth, as seen through a "phase transitions" far from equilibrium. Subheading 4 is devoted to complexity of cancer glycolysis. Finally, some concluding remarks are presented in Subheading 5.

Mathematical model of chaotic oscillations and oscillatory entrainment in glycolysis originated from periodic substrate supply

We study the influence of periodic influx on a character of glycolytic oscillations within the forced Selkov system. We demonstrate that such a simple system demonstrates a rich variety of dynamical regimes (domains of entrainment of different order (Arnold tongues), quasiperiodic oscillations, and chaos), which can be qualitatively collated with the known experimental data. We determine detailed dynamical regimes exploring the map of Lyapunov characteristic exponents obtained in numerical simulations of the Selkov system with periodic influx. In addition, a special study of the chaotic regime and the scenario of its origin in this system was evaluated and discussed.

Discerning mechanistically rewired biological pathways by cumulative interaction heterogeneity statistics

Changes in response of a biological pathway could be a consequence of either pathway rewiring, changed input, or a combination of both. Most pathway analysis methods are not designed for mechanistic rewiring such as regulatory element variations. This limits our understanding of biological pathway evolution. Here we present a Q-method to discern whether changed pathway response is caused by mechanistic rewiring of pathways due to evolution. The main innovation is a cumulative pathway interaction heterogeneity statistic accounting for rewiring-specific effects on the rate of change of each molecular variable across conditions. The Q-method remarkably outperformed differential-correlation based approaches on data from diverse biological processes. Strikingly, it also worked well in differentiating rewired chaotic systems, whose dynamics are notoriously difficult to predict. Applying the Q-method on transcriptome data of four yeasts, we show that pathway interaction heterogeneity for known metabolic and signaling pathways is indeed a predictor of interspecies genetic rewiring due to unbalanced TATA box-containing genes among the yeasts. The demonstrated effectiveness of the Q-method paves the way to understanding network evolution at the resolution of functional biological pathways.

Nutrient regulation of insulin secretion and action

Pancreatic β-cell function is of critical importance in the regulation of fuel homoeostasis, and metabolic dysregulation is a hallmark of diabetes mellitus (DM). The β-cell is an intricately designed cell type that couples metabolism of dietary sources of carbohydrates, amino acids and lipids to insulin secretory mechanisms, such that insulin release occurs at appropriate times to ensure efficient nutrient uptake and storage by target tissues. However, chronic exposure to high nutrient concentrations results in altered metabolism that impacts negatively on insulin exocytosis, insulin action and may ultimately lead to development of DM. Reduced action of insulin in target tissues is associated with impairment of insulin signalling and contributes to insulin resistance (IR), a condition often associated with obesity and a major risk factor for DM. The altered metabolism of nutrients by insulin-sensitive target tissues (muscle, adipose tissue and liver) can result in high circulating levels of glucose and various lipids, which further impact on pancreatic β-cell function, IR and progression of the metabolic syndrome. Here, we have considered the role played by the major nutrient groups, carbohydrates, amino acids and lipids, in mediating β-cell insulin secretion, while also exploring the interplay between amino acids and insulin action in muscle. We also focus on the effects of altered lipid metabolism in adipose tissue and liver resulting from activation of inflammatory processes commonly observed in DM pathophysiology. The aim of this review is to describe commonalities and differences in metabolism related to insulin secretion and action, pertinent to the development of DM.

The effect of unhealthy β-cells on insulin secretion in pancreatic islets

Background

Insulin secreted by pancreatic islet β-cells is the principal regulating hormone of glucose metabolism and plays a key role in controlling glucose level in blood. Impairment of the pancreatic islet function may cause glucose to accumulate in blood, and result in diabetes mellitus. Recent studies have shown that mitochondrial dysfunction has a strong negative effect on insulin secretion.

Methods

In order to study the cause of dysfunction of pancreatic islets, a multiple cell model containing healthy and unhealthy cells is proposed based on an existing single cell model. A parameter that represents the function of mitochondria is modified for unhealthy cells. A 3-D hexagonal lattice structure is used to model the spatial differences among β-cells in a pancreatic islet. The β-cells in the model are connected through direct electrical connections between neighboring β-cells.

Results

The simulation results show that the low ratio of total mitochondrial volume over cytoplasm volume per β-cell is a main reason that causes some mitochondria to lose their function. The results also show that the overall insulin secretion will be seriously disrupted when more than 15% of the β-cells in pancreatic islets become unhealthy.

Conclusion

Analysis of the model shows that the insulin secretion can be reinstated by increasing the glucokinase level. This new discovery sheds light on antidiabetic medication.

Metabolic regulation of insulin secretion

Regulation of metabolic fuel homeostasis is a critical function of β-cells, which are located in the islets of Langerhans of the animal pancreas. Impairment of this β-cell function is a hallmark of pancreatic β-cell failure and may lead to development of type 2 diabetes mellitus. β-Cells are essentially "fuel sensors" that monitor and react to elevated nutrient load by releasing insulin. This response involves metabolic activation and generation of metabolic coupling factors (MCFs) that relay the nutrient signal throughout the cell and induce insulin biosynthesis and secretion. Glucose is the most important insulin secretagogue as it is the primary fuel source in food. Glucose metabolism is central to generation of MCFs that lead to insulin release, most notably ATP. In addition, other classes of nutrients are able to augment insulin secretion and these include members of the lipid and amino acid family of nutrients. Therefore, it is important to investigate the interplay between glucose, lipid, and amino acid metabolism, as it is this mixed nutrient sensing that generate the MCFs required for insulin exocytosis. The mechanisms by which these nutrients are metabolized to generate MCFs, and how they impact on β-cell insulin release and function, are discussed in detail in this article.

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.

Mathematical model of metabolism and electrophysiology of amino acid and glucose stimulated insulin secretion: in vitro validation using a β-cell line

We integrated biological experimental data with mathematical modelling to gain insights into the role played by L-alanine in amino acid-stimulated insulin secretion (AASIS) and in D-glucose-stimulated insulin secretion (GSIS), details important to the understanding of complex β-cell metabolic coupling relationships. We present an ordinary differential equations (ODEs) based simplified kinetic model of core metabolic processes leading to ATP production (glycolysis, TCA cycle, L-alanine-specific reactions, respiratory chain, ATPase and proton leak) and Ca(2+) handling (essential channels and pumps in the plasma membrane) in pancreatic β-cells and relate these to insulin secretion. Experimental work was performed using a clonal rat insulin-secreting cell line (BRIN-BD11) to measure the consumption or production of a range of important biochemical parameters (D-glucose, L-alanine, ATP, insulin secretion) and Ca(2+) levels. These measurements were then used to validate the theoretical model and fine-tune the parameters. Mathematical modelling was used to predict L-lactate and L-glutamate concentrations following D-glucose and/or L-alanine challenge and Ca(2+) levels upon stimulation with a non metabolizable L-alanine analogue. Experimental data and mathematical model simulations combined suggest that L-alanine produces a potent insulinotropic effect via both a stimulatory impact on β-cell metabolism and as a direct result of the membrane depolarization due to Ca(2+) influx triggered by L-alanine/Na(+) co-transport. Our simulations indicate that both high intracellular ATP and Ca(2+) concentrations are required in order to develop full insulin secretory responses. The model confirmed that K(+) ATP channel independent mechanisms of stimulation of intracellular Ca(2+) levels, via generation of mitochondrial coupling messengers, are essential for promotion of the full and sustained insulin secretion response in β-cells.

Exploring the genetic control of glycolytic oscillations in Saccharomyces cerevisiae

BACKGROUND

A well known example of oscillatory phenomena is the transient oscillations of glycolytic intermediates in Saccharomyces cerevisiae, their regulation being predominantly investigated by mathematical modeling. To our knowledge there has not been a genetic approach to elucidate the regulatory role of the different enzymes of the glycolytic pathway.

RESULTS

We report that the laboratory strain BY4743 could also be used to investigate this oscillatory phenomenon, which traditionally has been studied using S. cerevisiae X2180. This has enabled us to employ existing isogenic deletion mutants and dissect the roles of isoforms, or subunits of key glycolytic enzymes in glycolytic oscillations. We demonstrate that deletion of TDH3 but not TDH2 and TDH1 (encoding glyceraldehyde-3-phosphate dehydrogenase: GAPDH) abolishes NADH oscillations. While deletion of each of the hexokinase (HK) encoding genes (HXK1 and HXK2) leads to oscillations that are longer lasting with lower amplitude, the effect of HXK2 deletion on the duration of the oscillations is stronger than that of HXK1. Most importantly our results show that the presence of beta (Pfk2) but not that of alpha subunits (Pfk1) of the hetero-octameric enzyme phosphofructokinase (PFK) is necessary to achieve these oscillations. Furthermore, we report that the cAMP-mediated PKA pathway (via some of its components responsible for feedback down-regulation) modulates the activity of glycoytic enzymes thus affecting oscillations. Deletion of both PDE2 (encoding a high affinity cAMP-phosphodiesterase) and IRA2 (encoding a GTPase activating protein- Ras-GAP, responsible for inactivating Ras-GTP) abolished glycolytic oscillations.

CONCLUSIONS

The genetic approach to characterising the glycolytic oscillations in yeast has demonstrated differential roles of the two types of subunits of PFK, and the isoforms of GAPDH and HK. Furthermore, it has shown that PDE2 and IRA2, encoding components of the cAMP pathway responsible for negative feedback regulation of PKA, are required for glycolytic oscillations, suggesting an enticing link between these cAMP pathway components and the glycolysis pathway enzymes shown to have the greatest role in glycolytic oscillation. This study suggests that a systematic genetic approach combined with mathematical modelling can advance the study of oscillatory phenomena.

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.

Systems biology of the metabolic network regulated by the Akt pathway

Cancer has been proposed as an example of systems biology disease or network disease. Accordingly, tumor cells differ from their normal counterparts more in terms of intracellular network dynamics than single markers. Here we shall focus on a recently recognized hallmark of cancer, the deregulation of cellular energetics. The constitutive activation of the phosphatidylinositol 3-kinase (PI3K)/Akt pathway has been confirmed as an essential step toward cell transformation. We will consider how the effects of Akt activation are connected with cell metabolism; more precisely, we will review existing metabolic models and discuss the current knowledge available to construct a kinetic model of the most relevant metabolic processes regulated by the PI3K/Akt pathway. The model will enable a systems biology approach to predict the metabolic targets that may inhibit cell growth under hyper activation of Akt.

Robustness of glycolysis in yeast to internal and external noise

Glycolysis is one of the most essential intracellular networks, found in a wide range of organisms. Due to its importance and due to its wide industrial applications, many experimental studies on all details of this process have been performed. Until now, however, to the best of our knowledge, there has been no comprehensive investigation of the robustness of this important process with respect to internal and external noise. To close this gap, we applied two complementary and mutually supporting approaches to a full-scale model of glycolysis in yeast: (a) a linear stability analysis based on a generalized modeling that deals only with those effective parameters of the system that are relevant for its stability, and (b) a numerical integration of the rate equations in the presence of noise, which accounts for imperfect mixing. The results suggest that the occurrence of metabolite oscillations in part of the parameter space is a side effect of the optimization of the system for maintaining a constant adenosine triphosphate level in the face of a varying energy demand and of fluctuations in the parameters and metabolite concentrations.

Minimum protein oscillator based on multisite phosphorylation∕dephosphorylation

The authors propose a novel minimum oscillator whereby a protein with multiple phosphorylation sites directly embedded in a negative feedback loop can exhibit oscillation. They demonstrate that if the fully phosphorylated substrate inhibits the first phosphorylation step in a cooperative manner, multisite substrates can exhibit oscillatory behaviour at the presence of a kinase and phosphatase. With a fixed number of sites, the non-linearity of the negative feedback and the substrate∕enzyme ratio must be above certain threshold values to generate undamped oscillation. There is an inverse relationship between the number of phosphorylation sites and the minimum non-linearity of the negative feedback required for oscillation; that is, the ultrasensitivity and time delay rooted in multisite phosphorylation compensate for the explicit non-linearity in the negative feedback. The period and amplitude of oscillation are mainly determined by the number of phosphorylation sites and the substrate∕enzyme ratio. The authors' results suggest that a multisite protein can be exploited for the construction of a synthetic protein oscillator featuring simplicity, robustness and tunability.

Quantitative analysis of cellular metabolic dissipative, self-organized structures

One of the most important goals of the postgenomic era is understanding the metabolic dynamic processes and the functional structures generated by them. Extensive studies during the last three decades have shown that the dissipative self-organization of the functional enzymatic associations, the catalytic reactions produced during the metabolite channeling, the microcompartmentalization of these metabolic processes and the emergence of dissipative networks are the fundamental elements of the dynamical organization of cell metabolism. Here we present an overview of how mathematical models can be used to address the properties of dissipative metabolic structures at different organizational levels, both for individual enzymatic associations and for enzymatic networks. Recent analyses performed with dissipative metabolic networks have shown that unicellular organisms display a singular global enzymatic structure common to all living cellular organisms, which seems to be an intrinsic property of the functional metabolism as a whole. Mathematical models firmly based on experiments and their corresponding computational approaches are needed to fully grasp the molecular mechanisms of metabolic dynamical processes. They are necessary to enable the quantitative and qualitative analysis of the cellular catalytic reactions and also to help comprehend the conditions under which the structural dynamical phenomena and biological rhythms arise. Understanding the molecular mechanisms responsible for the metabolic dissipative structures is crucial for unraveling the dynamics of cellular life.

Global self-regulation of the cellular metabolic structure

BACKGROUND

Different studies have shown that cellular enzymatic activities are able to self-organize spontaneously, forming a metabolic core of reactive processes that remain active under different growth conditions while the rest of the molecular catalytic reactions exhibit structural plasticity. This global cellular metabolic structure appears to be an intrinsic characteristic common to all cellular organisms. Recent work performed with dissipative metabolic networks has shown that the fundamental element for the spontaneous emergence of this global self-organized enzymatic structure could be the number of catalytic elements in the metabolic networks.

METHODOLOGY/PRINCIPAL FINDINGS

In order to investigate the factors that may affect the catalytic dynamics under a global metabolic structure characterized by the presence of metabolic cores we have studied different transitions in catalytic patterns belonging to a dissipative metabolic network. The data were analyzed using non-linear dynamics tools: power spectra, reconstructed attractors, long-term correlations, maximum Lyapunov exponent and Approximate Entropy; and we have found the emergence of self-regulation phenomena during the transitions in the metabolic activities.

CONCLUSIONS/SIGNIFICANCE

The analysis has also shown that the chaotic numerical series analyzed correspond to the fractional Brownian motion and they exhibit long-term correlations and low Approximate Entropy indicating a high level of predictability and information during the self-regulation of the metabolic transitions. The results illustrate some aspects of the mechanisms behind the emergence of the metabolic self-regulation processes, which may constitute an important property of the global structure of the cellular metabolism.

A scalable and integrative system for pathway bioinformatics and systems biology

MOTIVATION

Progress in systems biology depends on developing scalable informatics tools to predictively model, visualize, and flexibly store information about complex biological systems. Scalability of these tools, as well as their ability to integrate within larger frameworks of evolving tools, is critical to address the multi-scale and size complexity of biological systems.

RESULTS

Using current software technology, such as self-generation of database and object code from UML schemas, facilitates rapid updating of a scalable expert assistance system for modeling biological pathways. Distribution of key components along with connectivity to external data sources and analysis tools is achieved via a web service interface.

AVAILABILITY

All sigmoid modeling software components and supplementary information are available through: http://www.igb.uci.edu/servers/sb.html.

Nutrient regulation of insulin secretion and beta-cell functional integrity

Pancreatic beta-cells are often referred to as "fuel sensors" as they continually monitor and respond to dietary nutrients, under the modulation of additional neurohormonal signals, in order to secrete insulin to best meet the needs of the organism. beta-cell nutrient sensing requires metabolic activation, resulting in production of stimulus-secretion coupling signals that promote insulin biosynthesis and release. The primary stimulus for insulin secretion is glucose, and islet beta-cells are particularly responsive to this important nutrient secretagogue, It is important to consider individual effects of different classes of nutrient or other physiological or pharmacological agents on metabolism and insulin secretion. However, given that beta-cells are continually exposed to a complex milieu of nutrients and other circulating factors, it is important to also acknowledge and examine the interplay between glucose metabolism and that of the two other primary nutrient classes, the amino acids and fatty acids. It is the mixed nutrient sensing and outputs of glucose, amino and fatty acid metabolism that generate the metabolic coupling factors (MCFs) involved in signaling for insulin exocytosis. Primary MCFs in the beta-cell include ATP, NADPH, glutamate, long chain acyl-CoA and diacylglycerol and are discussed in detail in this article.

The number of catalytic elements is crucial for the emergence of metabolic cores

Background

Different studies show evidence that several unicellular organisms display a cellular metabolic structure characterized by a set of enzymes which are always in an active state (metabolic core), while the rest of the molecular catalytic reactions exhibit on-off changing states. This self-organized enzymatic configuration seems to be an intrinsic characteristic of metabolism, common to all living cellular organisms. In a recent analysis performed with dissipative metabolic networks (DMNs) we have shown that this global functional structure emerges in metabolic networks with a relatively high number of catalytic elements, under particular conditions of enzymatic covalent regulatory activity.

Methodology/Principal Findings

Here, to investigate the mechanism behind the emergence of this supramolecular organization of enzymes, we have performed extensive DMNs simulations (around 15,210,000 networks) taking into account the proportion of the allosterically regulated enzymes and covalent enzymes present in the networks, the variation in the number of substrate fluxes and regulatory signals per catalytic element, as well as the random selection of the catalytic elements that receive substrate fluxes from the exterior. The numerical approximations obtained show that the percentages of DMNs with metabolic cores grow with the number of catalytic elements, converging to 100% for all cases.

Conclusions/Significance

The results show evidence that the fundamental factor for the spontaneous emergence of this global self-organized enzymatic structure is the number of catalytic elements in the metabolic networks. Our analysis corroborates and expands on our previous studies illustrating a crucial property of the global structure of the cellular metabolism. These results also offer important insights into the mechanisms which ensure the robustness and stability of living cells.

A kinetic core model of the glucose-stimulated insulin secretion network of pancreatic beta cells

The construction and characterization of a core kinetic model of the glucose-stimulated insulin secretion system (GSIS) in pancreatic beta cells is described. The model consists of 44 enzymatic reactions, 59 metabolic state variables, and 272 parameters. It integrates five subsystems: glycolysis, the TCA cycle, the respiratory chain, NADH shuttles, and the pyruvate cycle. It also takes into account compartmentalization of the reactions in the cytoplasm and mitochondrial matrix. The model shows expected behavior in its outputs, including the response of ATP production to starting glucose concentration and the induction of oscillations of metabolite concentrations in the glycolytic pathway and in ATP and ADP concentrations. Identification of choke points and parameter sensitivity analysis indicate that the glycolytic pathway, and to a lesser extent the TCA cycle, are critical to the proper behavior of the system, while parameters in other components such as the respiratory chain are less critical. Notably, however, sensitivity analysis identifies the first reactions of nonglycolytic pathways as being important for the behavior of the system. The model is robust to deletion of malic enzyme activity, which is absent in mouse pancreatic beta cells. The model represents a step toward the construction of a model with species-specific parameters that can be used to understand mouse models of diabetes and the relationship of these mouse models to the human disease state.

Synchronization in coupled cells with activator-inhibitor pathways

The functional dynamics exhibited by cell collectives are fascinating examples of robust, synchronized, collective behavior in spatially extended biological systems. To investigate the roles of local cellular dynamics and interaction strength in the spatiotemporal dynamics of cell collectives of different sizes, we study a model system consisting of a ring of coupled cells incorporating a three-step biochemical pathway of regulated activator-inhibitor reactions. The isolated individual cells display very complex dynamics as a result of the nonlinear interactions common in cellular processes. On coupling the cells to nearest neighbors, through diffusion of the pathway end product, the ring of cells yields a host of interesting and unusual dynamical features such as, suppression of chaos, phase synchronization, traveling waves, and intermittency, for varying interaction strengths and system sizes. But robust complete synchronization can be induced in these coupled cells with a small degree of random coupling among them even where regular coupling yielded only intermittent synchronization. Our studies indicate that robustness in synchronized functional dynamics in tissues and cell populations in nature can be ensured by a few transient random connections among the cells. Such connections are being discovered only recently in real cellular systems.

Simple and complex spatiotemporal structures in a glycolytic allosteric enzyme model

Pattern formation in glycolysis is studied with a classical reaction-diffusion allosteric enzyme model. It is found that, similar to recent experimental reports in the yeast extracts, a small magnitude local perturbation can induce transient target waves in a two dimensional oscillatory medium. An above threshold stimulation generates target waves which eventually evolve into spatiotemporal chaos upon collisions with the boundary or other wave activities. Detailed simulation studies show that the studied simple glycolytic reaction-diffusion model can support three types of spatiotemporal behaviors which are independent of the boundary conditions: (1) a spatially uniform stable steady state, (2) periodic global oscillations and (3) spatiotemporal chaos.

Towards building the silicon cell: a modular approach

Systems Biology aims to understand quantitatively how properties of biological systems can be understood as functions of the characteristics of, and interactions between their macromolecular components. Whereas, traditional biochemistry focused on isolation and characterization of cellular components, the challenge for Systems Biology lies in integration of this knowledge and the knowledge about molecular interactions. Computer models play an important role in this integration. We here discuss an approach with which we aim to link kinetic models on small parts of metabolism together, so as to form detailed kinetic models of larger chunks of metabolism, and ultimately of the entire living cell. Specifically, we will discuss techniques that can be used to model a sub-network in isolation of a larger network of which it is a part, while still maintaining the dynamics of the larger complete network. We will start by outlining the JWS online system, the silicon cell project, and the type of models we propose. JWS online is a model repository, which can be used for the storage, simulation and analysis of kinetic models. We advocate to integrate a top-down approach, where measurements on the complete system are used to derive fluxes in a detailed structural model, with a bottom-up approach, consisting of the integration of molecular mechanism-based detailed kinetic models into the structural model.

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.

The Silicon Cell initiative: working towards a detailed kinetic description at the cellular level

The Silicon Cell initiative aims to understand cellular systems on the basis of the characteristics of their components. As a tool to achieve this, detailed kinetic models at the network reaction level are being constructed. Such detailed kinetic models are extremely useful for medical and biotechnological applications and form strong tools for fundamental studies. Several recently constructed detailed kinetic models on metabolism (glycolysis), signal transduction (EGF receptor), and the eukaryotic cell cycle (Saccharomyces cerevisiae) have been used to exemplify the Silicon Cell project. These models are stored and made accessible via the JWS Online Cellular Systems Modeling project, a web-based repository of kinetic models. Using a web-browser the models can be interrogated via a user-friendly graphical interface. The goal of the two projects is to combine models on parts of cellular systems and ultimately to construct detailed kinetic models at the cellular level.

Glycolytic oscillations and waves in an open spatial reactor: Impact of feedback regulation of phosphofructokinase

An open spatial reactor has been designed for the investigation of spatio-temporal dynamics of glycolysis. The reactor consists of a diffusive layer made of gel-fixed yeast extract which is in contact with a continuously stirred reservoir to supply this layer with substrates. The coupling between reaction and diffusion in the gel layer enables the formation of spatio-temporal patterns. Temporal oscillations of glycolysis are simply induced by feeding the yeast extract with sugar. Under properly chosen conditions, these oscillations sustain for more than 12 h. A necessary prerequisite for the generation of oscillations is that the ATP concentration in the feeding solution must be high enough to allow for negative feedback of phosphofructokinase. Otherwise, the interplay between ATP-consuming and ATP-producing reactions leads to an unfavorable low ATP/AMP ratio. The generation of travelling NADH-waves is observed in the diffusive layer, when feeding the yeast extract with substrates. Break-up of circular-shaped waves is repeatedly observed, resulting in the formation of rotating NADH-spirals.

Full-scale model of glycolysis in Saccharomyces cerevisiae

We present a powerful, general method of fitting a model of a biochemical pathway to experimental substrate concentrations and dynamical properties measured at a stationary state, when the mechanism is largely known but kinetic parameters are lacking. Rate constants and maximum velocities are calculated from the experimental data by simple algebra without integration of kinetic equations. Using this direct approach, we fit a comprehensive model of glycolysis and glycolytic oscillations in intact yeast cells to data measured on a suspension of living cells of Saccharomyces cerevisiae near a Hopf bifurcation, and to a large set of stationary concentrations and other data estimated from comparable batch experiments. The resulting model agrees with almost all experimentally known stationary concentrations and metabolic fluxes, with the frequency of oscillation and with the majority of other experimentally known kinetic and dynamical variables. The functional forms of the rate equations have not been optimized.

Finite wavelength instabilities in a slow mode coupled complex ginzburg-landau equation

In this Letter, we discuss the effect of slow real modes in reaction-diffusion systems close to a supercritical Hopf bifurcation. The spatiotemporal effects of the slow mode cannot be captured by traditional descriptions in terms of a single complex Ginzburg-Landau equation (CGLE). We show that the slow mode coupling to the CGLE introduces a novel set of finite wavelength instabilities not present in the CGLE. For spiral waves, these instabilities highly affect the location of regions for convective and absolute instability. These new instability boundaries are consistent with transitions to spatiotemporal chaos found by simulation of the corresponding coupled amplitude equations.