We argue that a shift from steady state to dynamic isotopomer analysis is required to deal with these cellular complexities and provide a review of dynamic studies of compartmentalized energy fluxes in eukaryotic cells including cardiac muscle, plants, and astrocytes. Knowledge of complex metabolic behaviour on a molecular level is prerequisite for the intelligent design of genetically modified organisms able to realize their potential of revolutionizing food, energy, and pharmaceutical production. We describe techniques to explore the bidirectionality and compartmentalization of metabolic fluxes using information contained in the isotopic transient, and discuss the integration of kinetic models with MFA. The flux parameters of an example metabolic network were optimized to examine the compartmentalization of metabolites and and the bidirectionality of fluxes in the TCA cycle of Saccharomyces uvarum for steady-state respiratory growth.Isotope labeling is one of the few methods of revealing the  Both experimental and analytical methods enabling dynamic studies that require direct measurement of the mass and/or positional isotopomers and the pool sizes of intermediate metabolites are developing quickly , 2, 3. Tde novo design and assembly of biosynthetic pathways for both natural and unnatural compounds.” [MFA is an important tool for strain improvement in biotechnology  with a vpounds.” . SynthetSaccharomyces uvarum for steady-state respiratory growth fed with 13C1,2 acetate and unlabeled glucose.This paper is a short review of the motivations for moving from MFA using data collected at isotopic steady state to making full use of the information contained in the isotopic transient. Examples are taken from recent studies that make good use of this information followed by a short section on performing this analysis under conditions of unstationary metabolism. An attempt is made to point towards the future of dynamic modeling of cellular systems using predictive kinetic models–The holy grail of modern biology. Simulations of isotopic transients are used to explore the information contained in the isotopic transient and examine techniques to exploit this information. Following this is a short example where the flux parameters are optimized for the TCA cycle in 1.1.et al. [et al. [The majority of MFA studies have been conducted at metabolic steady state, and the majority of these involve measuring isotopomers at isotopic steady state. Recent studies conducted at the metabolic and isotopic steady state include Blank et al.  and Vo e [et al. . These a [et al. , 9 and m [et al.  since a E.coli [Saccharomyces cerevisiae [et al. [13C NMR to track metabolite dynamics. Although little is known about protein turnover rates in vivo prokaryotes are expected to display less protein turnover than eukaryotes [A recent study demonstrates the use of MFA at metabolic steady state using isotopic transient data in the pentose phosphate pathway and citric acid cycle (TCA) of E.coli . Their mrevisiae . den Hol [et al.  measuredkaryotes . Isotopi2.et al. [Solanum tuberosum). Shastri and Morgan [MFA using isotopic transient data is more often applied in eukaryotic systems as it is not so easy to avoid compartmentalization and bi-directional exchange with large metabolic pools. However, since the nature of many of these dynamic processes has yet to be elucidated, MFA using isotopic transient data has been performed mostly on small linear branches of the metabolic networks without accounting for global dynamic behavior . There aet al.  who usedd Morgan  assess td Morgan , 19.Often, the organism of interest cannot be sustained in a steady metabolic state over long periods of time. To overcome this limitation one could resort to simulating the isotopic transient with a non-steady metabolism, or shorten the labeling experiment to less than one minute since the concentrations of enzymes in cells remain constant over short time spans (10 s to 1min) .2.1.et al. [et al. [There has been some progress recently in MFA studies with a non-steady metabolism and a lack of kinetic structure. A few researchers have started the move towards non-stationary MFA, with Wahl et al.  and Baxt [et al.  recently2.2.It is widely accepted that metabolic systems ubiquitously display oscillations in metabolic fluxes through temporal compartmentalization, proposed to be driven by oscillating metabolic cycles . By turnet al. [et al. [In continuous culture yeast can be enticed to grow with a stable oscillating metabolism with a period between 40 minutes to 5 hours. While get al.  measured [et al.  concludeKeeping in mind that enzyme concentrations remain constant over short time spans it is conceivable that one could use a device like the BioScope  to perfo3.in vivo kinetics of many enzyme systems within the metabolic network are well characterized. For many systems this information is not available, so development of kinetic models of metabolic systems is much less common than the use of phenomenological MFA to characterize metabolic activity. However, predictive kinetic models allow us to use the information content of experimental data points measured at one physiological condition to predict the dynamic behavior of the system at another physiological condition.Predictive kinetic models can be created in systems where the The modeling process involves (1) developing a theory of how the biological system operates, (2) representing the system as a set of ordinary and/or partial differential equations with direct physical meaning, (3) fitting the parameters of this system using one dataset, (4) testing the predictive qualities of the system using another related dataset, and (5) adjusting the theory and repeating the process as required. Metabolic models that have passed this kind of scrutiny allow us to predict bi-directional metabolic fluxes and system behavior under conditions where measured data is sparse. Great improvements can be achieved with the use of data gathered decades ago, which is often of high quality and fundamental in nature.The complexity and scope of the model ought to be limited by the quality and amount of measured data used to tune it, so introduction of kinetic parameters into dynamic models must be carefully considered. It is wise to restrict the addition of kinetic parameters to enzyme systems that have been systematically studied such that the kinetic scheme is biologically relevant and the kinetic parameters are known with some level of confidence. This ensures that there is additional data available for the tuning process, and the parameters are physiologically relevant.With this approach it is possible to maintain the structural identifiability of the model while adding more parameters. If many parameter sets can fit the available data, biological insight is severely limited if not impossible, so it is wise to always check the robustness of the solution during parameter optimization. With this in mind, it is not recommended to replace phenomenological MFA with phenomenological kinetic schemes that include more parameters since this only works to reduce the structural identifiability of the model while adding no biological insight.in vivo under a wide range of metabolic conditions. With the availability of additional kinetic insight and data metabolic flux analysis in the heart has progressed along a different path from the microbial and plant systems mentioned above. Predictive kinetic models in the heart are widespread since drug development is only possible with fundamental knowledge of enzyme operation, and this work is best performed in the public domain. With the future shift towards the use of cellulosic biorefineries it is predicted that there will be an increasing economic stimulus to study the fundamentals of exotic metabolisms and thus a resurgence in fundamental kinetic studies in plant and microbial systems.Ultimately, the construction of a predictive kinetic model involves the laborious task of studying each enzyme system et al. [With the complexity of biological systems, predictive models are useful to exclude hypotheses regarding their function. Vendelin et al.  quantifiet al. .et al. [Predictive kinetic models are better suited to exclude hypotheses regarding dynamic metabolism than phenomenological MFA. Selivanov et al.  and Liebet al.  have pub3.1.in vivo is the NMR saturation and inversion transfer technique developed in theory by McConnell [One important tool for probing the mechanisms of complicated kinetic systems cConnell  and in pcConnell . Nuclei 31P NMR to study the energy metabolism in hearts is a good example of how compartmentalization and bi-directionality of reaction steps complicates the analysis of a small network of reactions.A number of reviews discuss techniques for using saturation and inversion transfer for studying the kinetics of complex reaction schemes , 35, 36.T1 value [Early studies observed a discrepancy between the measured forward and reverse rate in the creatine kinase reaction when the myocardium was operating at steady state. To resolve this discrepancy it was concluded that analysis of the NMR data should include either compartmentalization of substrates or enzymes, or include an exchange of ATP with other phosphorus species such as inorganic phosphate , 38. In T1 value .et al. [Since the amount of information available from one single magnitization transfer protocol is insufficient to fit all parameters, Joubert et al.  used fouet al. , which iet al. .in vivo kinetics was developed to study energy metabolism in skeletal muscle using mass spectrometry to follow the enrichment of oxygen isotopes into energy metabolites. Replacing the external cellular environment with 18O atoms as a function of the enrichment of 18O in the water [18O incorporation into the high-energy phosphoryls one can determine the rate of hydrolysis of the energy metabolites [A complimentary method for exploring he water . The sizabolites .18O transient is induced, followed by freeze clamping in liquid nitrogen and a long preparatory procedure prior to analysis in the mass spectrometer.There are a number of technical difficulties when implementing this approach. The analytical work is very laborious and many animals are required for a statistically significant study. Each dynamic data point requires sacrificing one animal where an 18O in the energy metabolites of toad skeletal muscle, Dawis et al. [18O could be incorporated per molecular turnover. They judiciously discussed the issues bi-directional reaction steps within enzymatic complexes and wrote that “In practice, it will be difficult to verify a multiple-reversal model for the intact cell. Consequently, it will not be easy to choose between a multiple reversal model and a compartmentalization model.” Dawis et al. [The analysis of the data is also tricky since phosphotransfer dynamics contain compartmentalized metabolites and bi-directional reaction steps. To simplify the analysis of their transient experimental data on the uptake of s et al.  assumed s et al.  also str18O transfer studies is probably not enough to distinguish between possible reaction networks with various combinations of compartmentalization and bi-directional fluxes. Because of these limitations the above assumption of uni-directional fluxes was applied in a series of papers that explored the kinetics and compartmentalization of energy metabolism in rat skeletal muscle [A proper study of the bi-directionality of phosphotransfer networks has yet to be completed, and the amount of data collected in l muscle , 47, 48.31P NMR can be enhanced by the use of either a 17O or 18O induced isotope shift in the 31P NMR spectra. Pucar et al. [18O assisted 31P NMR method to study energetics in mouse heart. The method was employed in a series of papers exploring compartmentalized energetics [18O transfer kinetics, all with the same assumption of uni-directional fluxes. The development of improved methods utilizing NMR saturation and inversion will extend the range of applicability of this powerful technique [Saturation and inversion using r et al.  introducergetics , 51, 52 echnique , 55 whil4.The isotopic transient contains information about the underlying behavior of the metabolic system. The task is to build a model of the metabolic system that can best reproduce both the isotopic transient and the steady state isotopomer distribution of all metabolites. This involves finding the sizes of metabolic pools, the bi-directional rates of exchange between compartments in the cell, and the effect of bi-directional enzyme reactions on the isotopomer distribution. Of these, only the sizes of metabolic pools do not affect the steady state labeling state of the metabolites and the biomass created from them.4.1.et al. [To aid in the discussion of extracting information from isotopic transient data, we have composed a simple example of the TCA cycle with carbon enrichment found in et al. . PyruvatAnalogous schemes can be drawn for any biological isotope including oxygen, phosphorus, and nitrogen isotopes, although the atom transitions in these networks are less well defined and functional groups containing these elements tend to be more reactive resulting in a network with a significant number of side reactions and sinks that complicate analysis as in the above phosphotransfer network studies.4.2.13C and end at isotopic equilibrium at an enriched 13C state with steady isotopomer population distribution. Thus, the steady state isotopomer distribution for each metabolite is found from the last points of the simulation when the system has reached isotopic steady state.Isotopomer balance equations can be generated from the metabolic network, and using these, an isotopic transient can be simulated. The transient is induced by a step change in any or all members of the isotopomer population distribution of all metabolites that act as inputs or outputs to the system. For isotopomers that act as outputs to the system, the bi-directionality of the exit reaction step will induce isotope labeling in reverse direction to the net flux. The isotopomer distributions of all metabolites in the system begin at the natural labeling state of 1.1% We used the most direct approach to solve for the isotopic transient by numerically solving the full set of isotopomer balances. Various strategies have been devised to transform this system into an equivalent system that is computationally more efficient to solve, including the bondomer approach , decompo12C and 1’s representing 13C.To illustrate the information one can obtain from the isotopic transient, we present two sets of simulations. Our nomenclature for isotopomers in the figures and discussion below can be summarized as follows: The carbons are numbered according to chemical nomenclature and start at the right with 0’s representing The first set was obtained by continuously feeding pyruvate and acetate while performing a step change in the acetate isotopomer population from natural enrichment to 100% fully labeled 13C1,2 acetate. Two simulations were made with two different sets of metabolic pool sizes (A and B). The pool sizes of all metabolites in both sets were selected at random over three orders of magnitude. All net flux and exchange flux parameters were the same in both simulations. Since only metabolic pool sizes were changed between simulations, the steady state isotopomer distribution are identical for both simulations, as expected. The isotopic transients of the most highly enriched isotopomers of mitochondrial citrate from both simulations are given in 13C when acetate is used as the tracer illustrating the usefulness of this inexpensive tracer for studying the TCA cycle.The second set of simulations was obtained by continuously feeding pyruvate and acetate. The three simulations were made by performing (1) a step change to fully labeled acetate as above, (2) a step change from natural enrichment to 100% fully labeled 13C1,2,3 pyruvate, and (3) a step change in both fully labeled acetate and fully labeled pyruvate together. All other parameters, including metabolic pool sizes, net fluxes, and exchange fluxes were the same in all three simulations. The citrate isotopomers from these three simulations are given in Different isotopomers from each of the three simulations display similar dynamics as the metabolic system is operating in the exact same way. Comparing the dynamics of the citrate isotopomers between the acetate and pyruvate switch, three pairs of isotopomers reach the same proportion of the steady state isotopomer population: (1) the unlabeled citrate isotopomers, (2) the 000011 and 111100 complimentary pair, and (3) the 100011 and 011100 complimentary pair. Different isotopic tracers reveal the same underlying metabolic behavior at steady state for the TCA cycle intermediates, with the dynamics revealing complimentary information.When fully labeled acetate is fed to the metabolic system, the 000011 citrate isotopomer reveals similar dynamics as the same isotopomer when both acetate and pyruvate are fed to the metabolic system. When fully labeled pyruvate is fed to the metabolic system, the 011100 citrate isotopomer reveals similar dynamics as the 011111 citrate isotopomer when both acetate and pyruvate are fed to the system.When both labeled acetate and labeled pyruvate enter the metabolic system, we see both types of isotopomer dynamics appear, however in this case when the isotopomer populations of pyruvate and acetate consist of 100% fully labeled compounds all information about the steady state is lost as the system becomes fully labeled. Thus the use of multiple labeling experiments on the same metabolic system under the same growth conditions is useful to study the dynamic behavior of the metabolic system, and is thus useful to gain insight into the metabolic pool sizes, compartmentalization, and the bi-directionality of metabolic fluxes.To make these two example simulations quantitative one must find the appropriate metabolite pool sizes, net fluxes, and exchange fluxes that adequately reproduce a sufficient amount of transient isotopomeric data, possibly supplemented with additional steady state isotopomeric data, measurements of metabolic pool sizes, substrate utilization rates, and biomass production rates.5.Any difference between measured data and model predictions can be used in an optimization routine to find sets of net fluxes, exchange fluxes, and pool sizes that can reproduce the measured data within experimental errors. If the optimization routine cannot obtain a realistic fit with a sufficient amount of data, the metabolic scheme must be adjusted, possibly with the inclusion of compartmentalization and the process repeated. After finding a set of model parameters that can sufficiently reproduce measured data, one can gain insight into the operation of the metabolic network.13C enrichment probability.All types of isotopomeric measurement can be compared with the output from the dynamic solver, including data collected at isotopic steady state: Mass isotopomers from mass spectrometers, NMR positional enrichments, double enrichments, triple enrichments, and beyond all contain information about the operation of the metabolic scheme. Each measurement type requires one to sum up the appropriate pool of simulated isotopomers that correspond to the measured It should be noted that the process of optimization is not restricted to experiments performed with one enriched substrate. Data from multiple experiments at the same metabolic state using different labeled substrates can be combined to optimize one set of parameters. In this case the optimizer must simulate the isotopomer balance equations once for every experiment with a different step change in labeled substrate using the same set of parameters, and comparing each with their respective set of experimental data. The three simulations in 5.1.Since it is difficult to accurately measure many metabolic pools, making the transient simulation quantitative typically requires additional transient isotopic data. Using an optimization routine it is possible to find a realistic set of metabolic pool sizes that best match isotopic transient data and pool size measurements. To accomplish this, the optimizer would be allowed to manipulate all metabolic pool sizes, thus changing the isotopic transient, while attempting to minimize the difference between measured isotopomeric data and measured pool sizes. In practice one would not usually optimize only the metabolic pool sizes as one usually needs to optimize the net flux and exchange flux parameters at the same time.13C1,2 isotopomer of citrate. With this in mind, transient data that is able to capture the shape and timing of major transient curves like this one are useful for constraining not only the net fluxes and bi-directionality of the metabolic network, but also metabolic pool sizes. If the pool size found by optimization does not match that measured during the experiment, it could be a clue that this metabolic pool is compartmentalized. Other clues in the shape of these transients also aid in identifying compartmentalization.5.2.Information about the bi-directionality of fluxes and the compartmentalization of metabolic pools is contained in the isotopic dynamics. Compartmentalization is revealed in a number of ways. Consider a linear pathway:If the labeling in C becomes enriched faster than B, B is compartmentalized. This means that one should optimize the flux parameters for at least two separate pools of B:i with Bj and their pool sizes. ATP exhibits compartmentalization in cardiomyocytes and astrocytes, as evidenced by a 31P NMR saturation and inversion analysis of the creatine kinase reaction[The shape of the isotopic transient depends on the exchange of B reaction:(3)PCr2The kinetic data suggest that ATP exchanges with inorganic phosphorus and participates in other reactions via separate compartments:et al. [et al. [Fitting the data to this kinetic scheme suggests the need to consider both the function of the bound enzymes and restrictions of diffusion in the system, which both may lead to localized compartmentalization. Evidence for diffusional restrictions and compartmentalization of ATP was explored by Sonnewald et al.  who obse [et al.  performeet al. [Localized compartmentalization of energy metabolites in cells with high energy requirements is well known , 64. Kaaet al.  studied et al. , 67.et al. [Vendelin and Birkedal  found diet al.  developeet al. .Compartmentalized metabolic pools may play a role in controlling shifts in metabolism. Separate cytosolic pools of pyruvate in astrocytes have been observed to switch between acting as the precursor for energy production depending on the substrate being consumed . In gene5.3.Saccharomyces uvarum. This example introduces the basic process of extracting information from isotopomeric data and does not include many details in the modeling process such as sensitivity analysis and a through discussion of the flux parameters found. Judicious analysis of this system will require a separate publication.To illustrate the process of extracting information from isotopomeric data using isotopic simulation coupled with optimization, we have included a simple example of the TCA cycle in et al. [et al. [13C NMR absolute and conditional enrichments from the carbon skeleton of proteinogenic amino acids harvested and hydrolyzed at isotopic steady state. This excludes the optimization of pool sizes so they have all been set to be equal to simplify simulation, and all comparisons to measured data were made at the last time point simulated after all isotopic dynamics reached steady state.The metabolic system is given in et al. . We opti [et al.  measuredWe have included measurements of the rate of biomass production from all TCA metabolites in The optimization was carried out with the following reactions set to be bi-directional: malate dehydrogenase (EC 1.1.1.37), fumerase (EC 4.2.1.2), citrate synthase (EC 2.3.3.1), and the three transport reactions for oxaloacetate, pyruvate, and acetyl-coenzyme A. All reactions involving carbon dioxide, except for the bi-directional production of bicarbonate via carbonic anhydrase (EC 4.2.1.1), were set to be uni-directional.By starting at a large number of plausible starting points selected at random over the range of the free flux parameters, the optimizer always settled on one single optimal solution and occasionally stopped at a few other local optima that did not reproduce the data very well. Changing the weighting of measured data points within the optimizer and excluding one or two at random did not significantly change the optimal solution found as this solution matched all available data quite well. The optimal fit to the isotopomeric data is given in With respect to bi-directional reactions, malate dehydrogenase was found to be very bi-directional with The pyruvate fit was the least perfect and the fit required the pyruvate transporter (R200) to be bidirectional. This may be telling us that the assumption that mitochondrial pyruvate is the sole precursor for Ala production is not entirely true, although at least some production of Ala from mitochondrial pyruvate is required to fit the data. Ala is produced from cytosolic pyruvate during fermentative growth so it is possible that both mitochondrial and cytosolic pyruvate act as precursors for Ala production, but this must be confirmed with additional data and future simulations possibly with the inclusion of an additional compartmentalized pool.Pyruvate is a metabolite that participates in a large number of intersecting central metabolic pathways, typically has a low intra-cellular concentration, and has been observed to exhibit multiple cytoplasmic compartments along with mitochondrial compartmentalization , 74. Thiet al. [The steady state isotopomer profiles of the cytosolic and mitochondrial pools of oxaloacetate are given in et al. . These set al. .To make the transient of this optimization quantitative we would have to include slow bi-directional exchange with storage compounds, since this has been found to dramatically influence the time scale of isotopic dynamics. The isotopic dynamics of TCA cycle metabolites such as 2-oxoglutarate, succinate, fumerate, glutamate, and aspartate, are all influenced by reversible aminotransferase reactions that transfer amino groups from α-amino acids to α-keto acids . This ma6.We have shown that dynamic isotopic transients reveal important insights into the operation of metabolic networks, including the bi-directionality of enzyme and transport reactions, and the compartmentalization of metabolites, including localized compartmentalization not separated by a membrane barrier and that caused by diffusional restrictions. Our optimization of the TCA cycle illustrates that using dynamic isotopic models does not complicate the analysis of steady state isotopomeric data if the transient part of the simulation is excluded, and the possibility for additional insight with the inclusion of only a small amount of transient data should not be overlooked. Models that make use of isotopic transient data are expected to become increasingly important as steady state isotopomeric models currently struggle with the realities of compartmentalization.in vivo. Numerical tools are also developing rapidly, however the current state of dynamic modeling continues to grapple with the difficulties of compartmentalization. Teasing out the details of compartmentalization using dynamic models involves the addition of more parameters. When introducing such parameters, the structural identifiability of the model must be preserved so that biological insight can be extracted from the measured data. This is a challenge for large metabolic systems and can only be accomplished by including as much information as possible to constrain the trajectories of the model solution. Examples include thermodynamic constraints, constraints on the pool sizes, integration of known kinetic information, and the fitting of isotopomeric data from as many experiments as possible.The predicted rise in the use of dynamic models is supported by the rapid development of analytical techniques to measure both isotopomeric transients and the kinetics of individual reactions Although a vast amount of kinetic detail is required to build predictive kinetic models, their use within isotopic transient models is expected to improve and expand phenomenological MFA. It is hoped that fundamental kinetic studies will once again become a funding priority and through their continuation support the use of kinetic schemes within realistically sized metabolic models, since the marriage of kinetics and MFA is predicted to become an ever increasingly important tool in systems biology.


Networks coming from protein-protein interactions, transcriptional regulation, signaling, or metabolism may appear to have “unusual” properties. To quantify this, it is appropriate to randomize the network and test the hypothesis that the network is not statistically different from expected in a motivated ensemble. However, when dealing with metabolic networks, the randomization of the network using edge exchange generates fictitious reactions that are biochemically meaningless. Here we provide several natural ensembles of randomized metabolic networks. A first constraint is to use valid biochemical reactions. Further constraints correspond to imposing appropriate functional constraints. We explain how to perform these randomizations with the help of Markov Chain Monte Carlo (MCMC) and show that they allow one to approach the properties of biological metabolic networks. The implication of the present work is that the observed global structural properties of real metabolic networks are likely to be the consequence of simple biochemical and functional constraints. The most commonly used such approach for biological networks is based on performing edge exchanges Social networks exhibit “small world” characteristics in vitro. This corresponds to the first level of “constraints” that should be imposed when randomizing metabolic networks. Other levels can be introduced based on functionality. For instance, to understand the differences between the metabolic network of a given organism and “what might have been expected” in other realizations, one may appeal to the fact that organisms are alive, eat, reproduce etc. The purpose of the present work is to show how, within metabolic networks, one may introduce randomized ensembles; these ensembles can be used as benchmarks, allowing one to meaningfully ask whether a given organism's metabolic network is particularly atypical.Our focus in the present work is on metabolic networks. Previous studies have revealed that metabolic networks of living organisms are highly structured. For example, the degree distribution of the metabolites in these networks has a power law tail The outline of the paper is as follows. We first examine the fat tail in the metabolite degree distribution when using realistic biochemical reactions and investigate the source of this tail. Then we address the randomization problem in metabolic networks and introduce network ensembles subject to increasing levels of constraints. We study the structural properties of metabolic networks in these ensembles, including the clustering coefficient and sizes of the strong components. These results are discussed in the following section, while detailed methods are provided in the last section.etc. This behavior is in fact typical of all metabolites of high degree: they transfer small groups and are therefore generally referred to as “currency” metabolites Given a metabolic network such as illustrated in random biochemical reactions? One could introduce artificial reactions in which randomly chosen metabolites would be transformed into others; however this would not preserve atomic species, and even if one could enforce conservation, it would nearly always lead to reactions which have no existence. A more suitable approach is to restrict ourselves to biochemically realizable reactions. We used a database of such reactions compiled by Rodrigues and Wagner E, coli iJR904 metabolic model E. coli genome-scale metabolic network To address this point in a quantitative framework, we ask what would be expected in a “random” organism, that is, in one using E. coli has far fewer reactions than the 5870 in the KEGG_Hybrid database, and so the maximum degree in E. coli must be smaller; this is visible in E. coli to “random” organisms so we should force the number of reactions to be the same as in E. coli, allowing any of the biochemical reactions in KEGG_Hybrid. This defines a simple ensemble of possible metabolic networks where the biochemical constraint of using real reactions is enforced. We have thus generated 1,000 random genomes  and computed the degree distribution in this ensemble. The result is displayed in E. coli.The metabolic network of E. coli. Quantitatively, we have checked this: among the metabolites of degree at least 50 in KEGG_Hybrid, 94% are also present in E. coli. Furthermore, this same pattern is expected in the Random ensemble simply because choosing reactions at random gives a higher probability of incorporating metabolites that participate in many reactions. Again this can be tested explicitly: for any of the metabolites that have degree at least 50 in KEGG_Hybrid, the Random metabolic networks include them with probability above 0.99. Interestingly, the biochemical nature of these high degree metabolites is quite specific: they are categorized as “carriers” or as “precursors” in the biochemical literature The similarity of the three distributions in reactions, where the degree of a reaction is given by the number of its substrates (metabolites it involves). In contrast to metabolites, reactions do not have high degrees: a typical reaction will involve just a few metabolites, the most frequent number being 4, and very rarely will there be more than 6. This situation is illustrated in not of the currency type.All the above concerns the degree of the metabolites. The same kind of analysis can be performed for the degree of the The “Random” ensemble is a first way to define randomized metabolic networks: it takes into account both the need to use meaningful biochemical reactions and the number of reactions in the genome of interest. We shall consider that the metabolic network (of an organism or of a randomized organism) is specified by its set of enzyme coding genes, and we shall refer to this set as its “genotype”.etc. Just as we fixed the number of reactions n in the genotypes forming the Random ensemble, it is appropriate to include further “macroscopic” constraints to refine the randomization ensembles. For each added constraint, one can expect the statistical characteristics in the ensemble to become closer to those of living organisms, but the hope is that just a few relevant constraints will be sufficient to have a quite satisfactory randomized ensemble. Beyond the constraints already mentioned, there is the simple fact that metabolism of living organisms allows them to grow and reproduce. Although these are incredibly complex tasks, genome-scale metabolic network models R because it is simply based on using a fixed number of random reactions in KEGG_Hybrid. Adding the constraint of the number of metabolites gives the ensemble we label RM and so forth. We now describe these successive ensembles and the computational tools used to sample them. We shall then examine the statistical properties of the networks in each of these ensembles and see the effects of successively adding these constraints.The approximate power law tail found in the degree distribution of metabolites in living organisms can be traced to the contribution of currency metabolites cf. . HoweverR constrains the number of reactions; it consists of genotypes having exactly n reactions in the KEGG_Hybrid list where n is the number of reactions of the “reference” organism which one wants to benchmark in a randomization test; for specificity, the reader can think of this reference organism as being E. coli. The second ensemble further constrains the number of metabolites; the number of metabolites m in a genome is obtained by counting all the distinct metabolites associated with the reactions in that genotype. In practice, the sampling is simpler if one constrains m to be in a range; we shall use m≤Em where Em is the number of metabolites in the reference organism E. coli. We denote this second ensemble which imposes two constraints by RM.  The third ensemble, denoted by uRM, restricts the nature of the reactions in KEGG_Hybrid that we permit ourselves to consider. The motivation for this restriction comes from the fact that KEGG_Hybrid includes many reactions involving “exotic” metabolites which are involved with just one reaction. In such cases, those reactions will necessarily be isolated and thus “afunctional” biochemically; unless the associated metabolites are part of the biomass, such reactions would have no reason to be kept in a biological organism.  A similar situation arises when a reaction requires a metabolite that can only be produced in particular chemical environment that we do not consider; such a reaction will then be blocked. Our working definition of a “blocked” reaction is based on the possible fluxes it can sustain in the steady state; if a reaction is guaranteed to be never used in such conditions, then it is considered as blocked and removed from the KEGG_Hybrid list. In practice, for all the reactions in KEGG_Hybrid we determine whether they are blocked unblocked reactions is then used as input for constructing genotypes. The nomenclature uRM of the ensemble indicates that we use unblocked reactions with constrained numbers of reactions and metabolites.The allowed reactions will always be taken from the aforementioned KEGG_Hybrid list. This guarantees that every reaction satisfies atomic conservation laws. Also, these reactions are either reversible or irreversible, and this is taken into account in the modeling. The first ensemble etc). These are then used as bricks for building proteins, DNA, lipids, etc. Genome-scale metabolic models provide a tested framework to relate genotypes (lists of enzyme-coding genes) to metabolic capabilities and phenotypes i.e., a set of nutrients defining allowed input fluxes into the cell. If a genotype's metabolic network satisfies this constraint, we say that the genotype is “viable” on that chemical environment because the in silico FBA predicts that the cell can grow given those nutrients. This is illustrated in The essence of living organisms is growth and reproduction. Metabolism plays a central role therein, transforming various nutrients brought in from outside the cell into primary metabolites . The associated ensemble thus takes into account viability constraints. These constraints can be considered as being imposed successively: one can force growth first on one chemical environment, then on two, then on three, and so on. We refer to these ensembles as uRM-V1, uRM-V2, uRM-V3, etc. Interestingly, the first step, namely going from uRM to uRM-V1, turns out to be the most stringent as the ones thereafter give rise to only rather small changes.There are many possible choices of nutrients; clearly one needs sources of all major elements . It is common practice to focus on the carbon utilization because it is often limiting; we thus work with “minimal” environments having a single carbon source. In the lab, it is easy to test whether a microorganism grows on a whole panel of different environments, and the corresponding list of growth/no-growth results is referred to as the R can be sampled by drawing genotypes with the correct number of reactions, but the constraints inherent to the other ensembles do not allow such a simple procedure. We thus resort to Markov Chain Monte Carlo (MCMC) as a way to sample each ensemble. This requires obtaining an element of the ensemble as a starting point, and then performing random walks within the ensemble. Each trial step involves doing a reaction swap (exchanging a reaction in the genotype with one that is not) to respect the constraint of having a fixed number of reactions. Then the different constraints of the ensemble of interest are checked; if they are satisfied, the trial step (to a new genotype) is accepted, otherwise it is rejected. The schematic representation of this is given in uRM-V1. Note that by construction, in each ensemble, all of the elements are equiprobable; our ensembles are then just nested sets of genotypes satisfying increasing numbers of constraints. The reduction in size of these sets as each constraint is added can be extremely severe. For instance, when going from all reactions to unblocked reactions, over half of the reactions are discarded, leading to a reduction of the genotype space by approximately a factor 2n. Similarly, it was shown in previous work 22. These numbers drive home the necessity of using MCMC for sampling: direct random sampling is hopelessly inefficient.The ensemble uRM-V10, we find a total of 106 specific reactions that are necessary for any genotype to have nonzero biomass flux. These 106 reactions are then present in every genotype of the ensemble uRM-V10. Nevertheless, two genotypes taken at random in this ensemble tend to be rather different. Specifically, if one takes two genotypes G1 and G2 at random in uRM-V10, we find that on average G2 will have more than 50% of its reactions that are not in G1. This also holds true when we compare a random genotype to that of E. coli. In the ensembles with fewer constraints (eg. uRM-V1), the level of dissimilarity between random genotypes is even higher. Thus, in spite of some shared reactions in all genotypes, the genotypes in our ensembles are not very similar to each other or to E. coli: the ensembles have a high level of genetic diversity.In any of our ensembles sampled by MCMC, two genotypes taken at random will share some reactions and differ in others. For instance, in the ensemble To each genotype is associated a list of reactions and their corresponding metabolites; the whole can be represented by a bipartite graph cf. . From thstructural properties of that (directed) graph. These are as follows. (1) The clustering coefficient C which roughly is a measure of the frequency of triangles in the network. (2) The average path length L between randomly chosen nodes. (3) The probability PC that two randomly selected nodes A and B are connected by a directed path from A to B; this gives an indication of whether a metabolite is involved in another's production. (4) The size of the largest strong component (LSC) which measures the connectivity of the network. In a directed graph, a strong component is defined as a maximal sub-graph such that there exists a directed path between any two of its nodes; in the case of an undirected graph, it is then just a maximal connected component. We focus on the largest of these strong components in this work. (5) The “IN” (respectively the “OUT”) sub-graph for a given strong component is the set of nodes for which there is a directed path to (respectively from) the strong component For each genotype generated and saved in the different ensembles, we have constructed its metabolite-metabolite graph. Then we measure several of the standard E. coli genotype. In C, the average size of the LSC, and the average size of LSC+IN+OUT. To the right of the bar associated with uRM-V10 we show the value for E. coli. Clearly the first constraints strongly affect the structure of the metabolic networks, while increasing the number of environments on which one forces viability gives rise only to modest changes. When considering the other structural properties of networks in the ensembles, we see that for the clustering coefficient, the first constraint (going from R to RM) is the most important  to incorporate real biochemical reactions. We showed that this could be done in practice by using a database of real biochemical reactions compiled from KEGG and iJR904. In this framework, we found that the degree distribution of metabolites had a fat tail very similar to what is seen in real organisms.Over the past decade, genome-scale metabolic networks have been constructed for several organisms. In all cases, the metabolite degree distribution exhibits a fat tail compatible with a power decay One may ask whether the observed fat tail is an artifact of the KEGG database itself which summarizes the reactions in today's organisms. It is quite possible that other reactions and cofactors can act as substitutes to the ones occurring in KEGG, and thereby affect the degree distribution. However, it is likely that selection pressures act against such substitutes, for instance because of efficiency of catalysis or availability of molecular species. In effect, the use of KEGG reflects evolutionary constraints in addition to the purely biochemical ones. Thus, the present work is relevant for natural organisms but much less so for synthetic ones.E. coli biomass reaction formula makes our growth phenotypes insensitive to missing reactions as long as they arise in rare biosynthetic pathways.Another caveat associated with this study is the bias arising due to incompleteness of the KEGG database. Firstly, KEGG is missing transporters of less studied organisms; as a consequence a number of reactions appear to be blocked. Secondly, many biosynthesis pathways are incomplete; this is especially true for rare (or poorly understood) pathways. However, both of these biases can be expected to have only mild consequences within our study. Indeed for our choices of chemical environments, the curation of genome-scale models of different organisms has filled the gaps for transporters. Furthermore, our use of the R showed significant differences with the biological case (where the ensemble R corresponds to choosing randomly a given number of reactions in the database KEGG_Hybrid). Since understanding the topological properties of networks can give insights into their structure-function relationship, it is appropriate to refine the benchmark ensemble. Thus, we successively added further global constraints, in particular by enforcing metabolic capabilities, in this context biomass production. Adding such functional constraints takes into account the growth properties of living organisms and thus the “macroscopic” forces which shape biological metabolic networks. We find that by adding biochemical and functional constraints, the structural properties of the random networks in our ensembles become very close to what is seen biologically as illustrated in Looking at structural properties beyond the degree distribution, we found that the ensemble uRM-V10, the metabolic networks show large differences in reaction usage. Quantitatively, two randomly chosen networks in the ensemble uRM-V10 will typically differ in half of their reaction content. As a cautionary note, it is important to stress that the observed trends here concern global structural measures commonly used in general network analysis. One cannot exclude the possibility that consideration of metabolism-specific observables based for instance on fluxes may lead to a different picture.Perhaps most strikingly, these trends occur within ensembles that maintain a high level of genetic diversity. Indeed even in our most constrained ensemble, In conclusion, the present work indicates that the observed global structural properties of metabolic networks in living organisms are likely to be consequences of the simplest biochemical and functional constraints. Such a possibility has been previously suggested E. coli genome-scale metabolic model iJR904 E. coli genome-scale metabolic model iJR904. In this work, we will refer to the set of 5870 reactions contained in the hybrid database We have used a hybrid database compiled by Rodrigues and Wagner E. coli iJR904 metabolic model; these can be used to transport such metabolites across the cell boundary. The 143 external metabolites were taken to be the set of possible uptake and secreted metabolites in the network. Further, an objective function Z in the form of a biomass reaction, that requires synthesis of all biomass components of E. coli, as defined in the iJR904 model The hybrid database also contains transport reactions for 143 external metabolites in the E. coli iJR904 model, we found 2968 of the 5870 reactions in the hybrid database to be blocked under all environmental conditions Genome-scale metabolic networks typically contain “blocked” reactions that can have only zero flux in every investigated chemical environment under any steady state condition E. coli metabolic model iJR904 E. coli iJR904 model. In this work, we refer to this set of 831 reactions as the E. coli metabolic network.The The metabolic network can be represented as a directed bipartite graph built up of two types of nodes, metabolites and reactions, connected by two types of links. We can distinguish reactant metabolites from product metabolites of a reaction as follows: A link from a metabolite node to a reaction node specifies a reactant while a link from a reaction node to a metabolite node specifies a product. Note that in a bipartite graph, links between similar types of nodes are forbidden. It is important to differentiate between reversible and irreversible reactions in the network. In Starting from a directed bipartite graph of metabolites and reactions, we can construct an associated directed unipartite graph of metabolites, referred to as a metabolite-metabolite graph. It summarizes the metabolic network structure by assigning links from reactant metabolites to product metabolites in each reaction. In the simplest definition, two metabolites will be “neighbors” (connected by a link) if and only if they appear in at least one common reaction etc) of a large compound. Currency metabolites are the co-factors responsible for such transfers, and they are quite ubiquitous. Examples of currency metabolites include ATP, ADP, NADH, NAD+, H2O, H+, Pi that are normally used as carriers for transferring electrons or certain functional groups such as phosphate group, amino group, methyl group, one carbon unit, etc. In our construction of the unipartite graph, we omit links arising due to presence of currency metabolites in each reaction. In The different treatment of currency vs. non currency metabolites is based on the fact that biochemical reactions most often consist of adding or removing a small group  is the number of reactions in which the metabolite i participates either as a reactant or a product in the network. The metabolite degree distribution P(k) is defined as the probability that a randomly selected metabolite node participates in exactly k reactions in the network. We use the bipartite graph representation of the metabolic network to compute the metabolite degree and degree distribution.The degree of a metabolite P(k) ∼k-γIn P(k) gives the probability that a randomly selected reaction has exactly k substrates in it. We use the bipartite graph representation of the metabolic network to compute the reaction degree and degree distribution. The degree of a reaction is the number of substrates that participate either as a reactant or a product in it. The reaction degree distribution NΔ is the number of triangles and N3 is the number of connected triples in the graph. In this work, we have computed the clustering coefficient for each network in our ensemble using the unipartite metabolite graph representation. Note that when computing the clustering coefficient, the graph is considered undirected.The clustering coefficient quantifies the extent to which the neighbors of a node in a graph are connected to each other L> is a measure of the overall navigability in a network. It is defined as the average length of the shortest paths between all pairs of nodes in the directed unipartite metabolite graph. When computing the average path length for a disconnected graph, one considers only the node pairs for which a directed path exists. We have also computed the probability CP that a directed path exists between any two nodes in the unipartite metabolite graph. The clustering coefficient C, average path length <L> and probability CP that a path exists between two nodes in a graph were computed using the igraph library The average path length <i and j in the set there is a directed path from i to j and from j to ilargest strong component (LSC). The associated IN component consists of nodes which have access to LSC nodes via some directed path, but lack access from LSC nodes back to them via any directed path. The OUT component consists of nodes which can be accessed from the LSC nodes via some directed path, but lack access to LSC nodes from them via any directed path. Note that the so-called “bow-tie” architecture of networks is based on these LSC, IN and OUT components; that architecture has been observed both in the World Wide Web (WWW) Given a directed graph, a strongly connected component is a maximal set of nodes such that for any pair of nodes N reactions. A simple representation of a metabolic network genotype G uses a binary string of length N, e.g., i being either present (ib = 1) or absent (ib = 0)  method to sample genotypes in the KEGG_Hybrid reaction set with same number of metabolites and reactions as in E. coli.The th element of the sequence is generated from the kth one using a probabilistic transition rule. We start with the E. coli genotype and then propose a small modification in the genotype; if this modified genotype has its number of metabolites ≤ Em (the number in E. coli), one accepts it as the next genotype of the sequence, otherwise the next genotype is identical to the current genotype. In this work, the modification introduced at each transition step is a reaction swap. That is, each modification adds one reaction from KEGG_Hybrid reaction set and removes another reaction from the current genotype, so as to keep the number of reactions En constant in the genotype. The MCMC thereby produces a walk in the subspace of genotypes of En reactions and at most Em metabolites. Starting from the initial E. coli genotype, we first carried out 105 attempted swaps or Markov chain steps to erase the memory of the starting genotype. After this initial phase, we continued the MCMC procedure to sample genotypes with exactly En reactions and at most Em metabolites. During this phase, it is not useful to keep all of the genotypes produced because they are strongly correlated. We thus saved only every 1000th genotype generated, and we did 106 steps. We refer to the set of 1000 genotypes with En reactions and ≤ Em metabolites within KEGG_Hybrid reaction set as the RM ensemble. We find that the procedure is relatively efficient, with an acceptance rate 0.22 that is not small. We also find that a substantial fraction of the networks have in fact m = Em.The MCMC method produces a sequence of genotypes forming a chain, the term “chain" coming from the property that the (k+1)Ensemble uRM of genotypes with fixed number of reactions and metabolites in the Unblocked KEGG_Hybrid set. “Blocked” reactions can have only zero flux in every investigated chemical environment under steady state conditions, and thus are “afunctional” in all genotypes. As a next step, we enforce the constraint that the genotypes in the ensemble are sampled within the Unblocked KEGG_Hybrid reaction set rather than the KEGG_Hybrid reaction set. We refer to this ensemble of genotypes containing the same number of metabolites Em and reactions En as in the E. coli network within the unblocked reaction set as uRM. We generate the genotypes in the uRM ensemble through a slightly modified MCMC method from that mentioned above to generate the RM ensemble. In this case, at each transition step, we impose a reaction swap to the current genotype that is restricted to the Unblocked KEGG_Hybrid set, i.e., we remove one reaction from the current genotype and add a reaction from the Unblocked KEGG_Hybrid set. The rest of the procedure is exactly the same as above for sampling RM. We have sampled 1000 genotypes in the uRM ensemble with En reactions and at most Em metabolites.E. coli metabolic network has the ability to produce biomass components starting from nutrient metabolites in its environment for growth and maintenance. Thus as a next step, we enforce the additional functional constraint of growth in a chemical environment. We define the ensemble uRM-V1 as that part of uRM in which the genotypes satisfy the functional constraint of non-zero biomass flux in the glucose minimal environment  the number of metabolites in the modified genotype is at most Em, the number in E. coli, and (b) the modified genotype is viable under glucose minimal environment. The rest of the procedure is exactly same as when sampling genotypes in the ensemble uRM; a flowchart of the MCMC algorithm for sampling the ensemble uRM-V1 is shown in uRM-V1 ensemble.The E. coli is able to survive and grow under diverse environmental conditions , we have further generated two ensembles of genotypes satisfying increased functional constraints of (a) viability under 5 specified minimal environments (referred to as the ensemble uRM-V5) and (b) viability under 10 specified minimal environments (referred to as ensemble uRM-V10), respectively. These ensembles can be sampled by a MCMC procedure that is just a slight modification from the one shown in Since, Figure S1Edge exchange randomization is biochemically meaningless. The commonly used edge exchange or link permutation procedure for randomizing biological networks is inappropriate for metabolic networks as the method generates fictitious reactions violating balance of mass, charge and atomic elements. Here, starting with two reactions , we perform an edge exchange associated to metabolites fum and ac that generates two new hypothetical reactions  that violate balance of mass and atomic elements. Note that ac has 2 carbon atoms and fum has 4 carbon atoms.(TIF)Click here for additional data file.Figure S2Schematic summary of the relationship between genotypes and phenotypes. The genotype specifies the list of reactions in a metabolic network. The phenotype is determined by whether the metabolic network can produce biomass components (growth) in a choice of chemical environments; this condition is computed using FBA.(TIF)Click here for additional data file.Figure S3Flowchart of the MCMC algorithm to sample genotypes in the ensemble uRM-V1. The Markov chain starts with the E. coli genotype. We then perform 105 Markov Chain steps to erase the memory of the initial genotype. After this initial phase, we continue the MCMC procedure to sample the genotype network and save every 1000th genotype generated. We terminate the Markov chain after saving 1000 genotypes. Note that the length of the run (and the choice of saving frequency) should be long enough to obtain a meaningful and uncorrelated sample of genotypes using this algorithm.(TIF)Click here for additional data file.Figure S4Clustering coefficient C of the metabolic networks in the different ensembles. Different bars from left to right correspond to network ensembles incorporating an increasing number of constraints and the last bar corresponds to the E. coli metabolic network. The standard deviation is also displayed for each ensemble.(TIF)Click here for additional data file.Figure S5Average path length <L> of the metabolic networks in the different ensembles. Different bars from left to right correspond to network ensembles incorporating an increasing number of constraints and the last bar corresponds to the E. coli metabolic network. The standard deviation is also displayed for each ensemble.(TIF)Click here for additional data file.Figure S6Statistical properties of randomized networks in different ensembles and the E. coli metabolic network using a modified currency list. The three axes are associated with graph characteristics of the networks and are same as in (TIF)Click here for additional data file.Table S1List of currency metabolites used to construct the unipartite graph. This list was built mostly from information in the paper by Ma and Zeng ). The metabolites shaded in grey were absent in the modified currency list used to generate (XLS)Click here for additional data file.


A decade ago, a team of biochemists including two of us, modeled yeast glycolysis and showed that one of the most studied biochemical pathways could not be quite understood in terms of the kinetic properties of the constituent enzymes as measured in cell extract. Moreover, when the same model was later applied to different experimental steady-state conditions, it often exhibited unrestrained metabolite accumulation.ab initio modeling are improved substantially by (i) including appropriate allosteric regulation and (ii) measuring the enzyme kinetic parameters under conditions that resemble the intracellular environment. The following modifications proved crucial: (i) implementation of allosteric regulation of hexokinase and pyruvate kinase, (ii) implementation of maxV values measured under conditions that resembled the yeast cytosol, and (iii) redetermination of the kinetic parameters of glyceraldehyde-3-phosphate dehydrogenase under physiological conditions.Here we resolve this issue by showing that the results of such in vivo metabolism in terms of in vitro biochemistry.Model predictions and experiments were compared under five different conditions of yeast growth and starvation. When either the original model was used , or the enzyme parameters were measured under conditions that were, as usual, optimal for high enzyme activity, fructose 1,6-bisphosphate and some other glycolytic intermediates tended to accumulate to unrealistically high concentrations. Combining all adjustments yielded an accurate correspondence between model and experiments for all five steady-state and dynamic conditions. This enhances our understanding of  ab initio, based on independently measured characteristics of the catalysts (the enzymes), seldom gives reliable results. A key problem in this field is that enzyme characteristics are often studied under non-physiological conditions that do not resemble the environment inside the cell. In this study we measured the enzyme characteristics under physiological conditions and assembled the results into a computational model of yeast energy metabolism. We show that this simple trick greatly improves the predictive value of the computational model. This allowed us to predict correctly how yeast cells adapt to nitrogen starvation, an industrially relevant situation, in which remodeling of the proteome strongly affects cellular energy metabolism.Baker's yeast is widely applied in modern biotechnology, for instance for production of heterologous protein or biofuel. For such applications a thorough understanding of the central energy metabolism of the bug is crucial. Nevertheless, even for this well-known organism, attempts to build models  Indeed, biochemistry and molecular biology have furnished impressive examples where the precise mechanism of action of individual macromolecules, such as DNA and the proton-translocating ATPase was elucidated. At this stage, however, no functional et al.maxV values to the dynamic response of yeast after addition of excess glucose. Hynne et al.maxV) but also their affinity constants (mK) in vivo values of kinetic parameters.The best-researched biological network is the breakdown of glucose. The enzymes of yeast glycolysis have been examined qualitatively and quantitatively for almost a century. Since the 1960s many kinetic computer models of yeast glycolysis have been constructed. The early models focused on the mechanisms underlying oscillations in yeast cultures and cell extracts, and were not much concerned with quantitative precision in vitro and in vivo enzyme kinetics precludes reliable modeling based on detailed enzyme kinetic equations Heijnen and colleagues have argued that the discrepancy between et al. was the first to actually evaluate to what extent biochemical knowledge obtained from in vitro studies could be used to predict the glycolytic flux and the concentrations of glycolytic intermediates in yeast in vitro kinetics did not quite describe the in vivo activity for all of the glycolytic enzymes satisfactorily. In order to obtain a satisfactory description, they had to displace the kinetic parameter values away from their in vitro magnitudes, albeit only in a few cases far beyond the standard error of the means.Teusink et al.maxV values from cultures grown under other conditions into the Teusink et al. model, has not been very successful. Often the model fails to reach a steady state and accumulate intermediate metabolites while the real yeast cultures do reach such a steady state . Reijenga et al.et al.The model of Teusink in vivo dynamics of metabolic pathways. So far enzyme kinetics have hardly been studied under conditions that resemble the intracellular environment. Recently, we therefore developed an assay medium that should resemble the yeast cytosol in vivo-like medium differed substantially from the activities in the ‘optimal’ non-physiological media The observed discrepancies between experiments and models could in principle be caused by various regulatory mechanisms, such as enzyme-enzyme interactions, channeling and rapid posttranslational modifications. Another obvious reason might be that, until recently, enzyme kinetic parameters were most often determined under non-physiological conditions that had been optimized for high enzyme activity but see . This wain vitro enzyme kinetics can be used to predict in vivo yeast glycolysis. To this end the computer model of yeast glycolysis by Teusink et al.maxV values and values of parameters that were measured in the new in vivo-like assay medium. In addition, hitherto missing but known allosteric regulators were implemented in the model. We show that if both innovations are combined, accurate model predictions are obtained. In contrast to earlier studies, a single model could be used to describe five different experimental situations.Here we therefore revisit the question whether Saccharomyces cerevisiae before and after nitrogen starvation. We studied: (i) non-starved cells from a glucose-limited chemostat culture grown under respiratory conditions (dilution rate D = 0.1 h−1), (ii) cells grown under these respiratory conditions and subsequently starved for nitrogen during 4 h, (iii) non-starved cells grown under respirofermentative conditions  and (iv) 4 h nitrogen-starved cells derived from this respirofermentative culture. From each of the four cultures yeast cells were harvested to measure (i) the maximal glycolytic flux and the intracellular metabolite concentrations in an off-line assay under anaerobic glucose-excess conditions (fermentative capacity (ii) the maxV of the glycolytic and fermentative enzymes, and (iii) the kinetic parameters of glucose transport across the plasma membrane. The fluxes and metabolite concentrations have already been reported in maxV values presented in in vivo conditions  fluxes to the side-branches that were fixed in the model ((ii) external metabolite concentrations that were fixed (max(iii) V values of the enzymes ; and (iv) other kinetic parameters of the enzymes and the glucose transporter  exre fixed ; (iii) V(i) the independently measured fluxes through hexose transport-hexokinase, phosphoglucose isomerase-aldolase and glyceraldehyde-3-phosphate dehydrogenase-alcohol dehydrogenase (see (ii) the independently measured glycolytic metabolite concentrations.Model predictions were compared to nase see ; and , which had a substantial effect on both the flux and the intermediate metabolite concentrations. In iK that fitted best was chosen for each condition . In mK for NAD+ of 2.84 mM and for glyceraldehyde 3-phosphate of 0.39 mM under all conditions. The new model reached a steady state close to the experimentally determined steady state for all conditions studied see . To achimaxV – is implemented.Thus we obtained a single model that can describe four different steady-state conditions when the result of altered gene expression – measured at the level of in vivo-like maxV values alone. The dynamic behavior of the new model was evaluated (i) with the in vivo-like maxV values implemented in the original Teusink et al.(ii) with the maxV values measured under conditions that were optimized for each enzyme, but in the new model . The metabolite concentrations measured under these conditions were taken as the initial concentrations. In case of a good fit, the calculated concentrations should therefore stabilize at their initial values. In all cases, the concentrations of glucose 6-phosphate and fructose 6-phosphate dropped rapidly and then stabilized. Fructose 1,6-bisphosphate, however, increased continuously up to very high levels when either the allosteric regulation was lacking  applied in the present study and therefore we used our dataset to model the response to addition of glucose. There are two differences between the experiment and the model simulation. First, in the experiment a single dose of glucose was added at time zero, while the modeled increase of glucose was sustained. However, within the time frame of the experiment the extracellular glucose concentration did not decrease significantly  and trehalose 6-phosphate (−1) were also implemented in the model. In agreement with experiments −1. The model was evaluated at a iK value of 0.04 mM for the inhibition of hexokinase by trehalose 6-phosphate, like the steady-state calculations for the corresponding culture at D = 0.1 h−1  was predominantly regulated by interaction of the enzymes with metabolites. In contrast, the downregulation of the flux upon nitrogen starvation of the respirofermentative culture (D = 0.35 h−1) was mostly regulated by gene expression. If we use the in vivo-like maxV assay on which the present paper is built, we obtain qualitatively the same results as in the previous study  and metabolite concentrations.In previous studies we analyzed the distribution of regulation between gene expression and metabolism upon nitrogen starvation of the same cultures as studied here us study  [23]. In−1 the glycolytic flux  decreased by 16% upon nitrogen starvation. This was reproduced by the model (14% decreased flux). Under these conditions the highest flux control was exerted by the glucose transporter  metabolic regulation mostly counteracts gene-expression or maxV regulation, with the highest negative metabolic regulation coefficient for phosphofructokinase (mρ = −3.4). When we calculate the metabolic regulation based on the changes in the measured metabolites and the rate equation used in the model, we do not find this negative metabolic regulation coefficient back. However, when we used the modeled values of the metabolite concentrations, the regulation coefficient was −3.8. This can be explained from the discrepancy between the measured and modeled concentration of fructose 6-phosphate: in the experiment it declined slightly, but in the model it increased 11-fold from 0.4 to 4.4 mM thus counteracting the decrease of the flux through phosphofructokinase. Also for other enzymes with strong metabolic regulation, i.e. hexokinase, phosphoglucose isomerase, pyruvate kinase and pyruvate decarboxylase, the metabolic regulation was only reproduced with the modeled concentrations. Again this was caused by (sometimes subtle) discrepancies between measured and modeled concentrations. Nevertheless, from the calculations concerning hexokinase, it became clear that trehalose 6-phosphate plays an important role in the metabolic part of the regulation .Under this condition (D = 0.35 h−1) it is less straightforward to link the observed metabolic regulation to specific metabolite concentrations. The reason is that the model underestimates the observed increase in flux upon nitrogen starvation. Again the glucose transporter exerts most of the flux control  was reproduced correctly by the model  i.e. glyceraldehyde-3-phosphate dehydrogenase and alcohol dehydrogenase. When the eqK of alcohol dehydrogenase was changed 10-fold to correct for the pH drop of one unit (from 6.8 to 5.8) the model simulation showed a only a slight difference. In contrast, a 10-fold decrease in the eqK of glyceraldehyde-3-phosphate dehydrogenase led to accumulation of the intermediates in upper glycolysis. However, besides a change in eqK, the maxV and the mK values of the enzymes could also change by a change in pH. The pKa values of some of the intermediate metabolites, e.g. glyceraldehyde 3-phosphate and fructose 6-phosphate, are around 6.5, which implies that a drop in intracellular pH should affect the apparent affinity of the enzymes towards these metabolites. In addition, the isoelectric point of the glycolytic enzymes is in the range of 5–8. Thus a change in pH may affect the protonation of the enzymes, which in principle could alter their maxV as well as their mK values. We did not observe changes in maxV of glycolytic enzymes in a pH range from 6.5 to 7 (data not shown).The in vivo-like assay conditions improves the match between experiment and model substantially, and for a number of different conditions. For an even more stringent test of the paradigm that in vivo metabolism is understandable in terms of in vitro biochemistry, the obvious next step is to reconsider all kinetic parameters, including mK and iK values. Apparent mK and iK values depend strongly on binding equilibria of metabolites to protons and other cations in the cell. Wu et al. took such ion-binding equilibria into account and showed that it improved model predictions of mitochondrial bioenergetics substantially in vivo-like conditions will be unavoidable. The next stage will then be the characterization of pure enzymes and insertion of the properties of isoenzymes into integral models. This requires quantitative proteomics of isoenzymes as well as a strict control of enzyme quality during purification, and has been beyond the scope of this study.We have to stress that even the improved model of yeast glycolysis does not match the experimental fluxes and metabolite concentrations perfectly. The key progress of this study is that being precise about allosteric regulation and in vivo-like enzyme assay conditions will be essential. This outcome may not be surprising, yet we could not have anticipated that it would have been sufficient to get such a good description of glycolytic dynamics. We must be prepared that other pathways may depend on more complex regulation than yeast glycolysis under the conditions of study. In cases where, for instance, enzyme-enzyme complexes, channeling of metabolites or rapid posttranslational modifications determine the pathway dynamics, we will need new means to analyze the in vivo biochemistry quantitatively. This will be the next challenge for molecular systems biology.The Silicon Cell philosophy is to construct models of well-defined pathways, which should subsequently be coupled to each other to expand the network Saccharomyces cerevisiae strain CEN.PK113-7D  was cultivated in an aerobic, glucose-limited chemostat  as described in detail by Van Hoek et al.−1, respectively.The haploid, prototrophic For the nitrogen-starvation experiments the same defined mineral medium was used as for the chemostat culture, except that it lacked ammonium sulfate and contained an excess of glucose. The addition of glucose served to prevent additional starvation for the carbon source. Yeast cells harvested from steady-state chemostats were washed with ice-cold starvation medium and resuspended in starvation medium. Cells were brought back in a new fermentor in batch mode (start volume was around 1 litre) at otherwise the same conditions as during chemostat cultivation H-linked assays, at 30°C in a Novostar spectrophotometer (BMG Labtech) as described in detail in Cell-free extracts were prepared freshly by the FastPrep® method described in Van Eunen in vivo-like assay medium The standardized et al.et al. model The glycolytic model of Teusink maxV values of all glycolytic and fermentative enzymes and the maxV and affinity constant of glucose transport were measured in extracts of cells grown under the growth and starvation conditions of this study. In Van Eunen et al.in vivo-like assays (see above) for the same biological samples  and therefore not included. The trehalose, glycerol and succinate fluxes were fixed at the values measured in our study  mK value of pyruvate decarboxylase for pyruvate of 6.36 mM at 50 mM phosphate and the new value was inserted in the model.Finally, the et al.−1 for rates and mM for concentrations) by assuming a yeast cytosolic volume of 3.75 µl.mg cell protein−1The known activation of pyruvate kinase by fructose 1,6-bisphosphate had not been included in the original model Table S1Fixed parameters under all four conditions.(PDF)Click here for additional data file.Table S2Concentrations of metabolites fixed under all four conditions.(PDF)Click here for additional data file.Table S3Initial metabolite concentrations in mM used in the glycolytic model.(PDF)Click here for additional data file.Table S4Kinetic parameters measured under in vivo-like conditions and implemented in the glycolytic model under the four conditions studied (new measurements).(PDF)Click here for additional data file.Table S5maxV values measured under ‘maxV optimized assay’ conditions.(PDF)Click here for additional data file.Table S6Parameter used to simulate the upshift of the extracellular glucose concentration.(PDF)Click here for additional data file.Table S7−1), compared to experimental data fromPredictions from the model of yeast glycolysis for the non-starved cells from the respiratory culture (D = 0.1 h.(PDF)Click here for additional data file.Table S8−1), compared to experimental data fromPredictions from the model of yeast glycolysis for the 4 h N-starved cells from the respiratory culture (D = 0.1 h.(PDF)Click here for additional data file.Table S9−1), compared to experimental data fromPredictions from the model of yeast glycolysis for the non-starved cells from the respirofermentative culture (D = 0.35 h.(PDF)Click here for additional data file.Table S10−1), compared to experimental data fromPredictions from the model of yeast glycolysis for the 4 h N-starved cells from the respirofermentative culture (D = 0.35 h.(PDF)Click here for additional data file.Table S11Response coefficients of the concentration of F16BP towards the model parameters p.(PDF)Click here for additional data file.Table S12−1, Non-starved.Flux Control Coefficients, D = 0.1 h(PDF)Click here for additional data file.Table S13−1, 4 h nitrogen-starved.Flux Control Coefficients, D = 0.1 h(PDF)Click here for additional data file.Table S14−1, non-starved.Flux Control Coefficients, D = 0.35 h(PDF)Click here for additional data file.Table S15−1, 4 h nitrogen-starved.Flux Control Coefficients, D = 0.35 h(PDF)Click here for additional data file.Table S16Comparison of the regulation coefficients presented in   and those re-calculated with the maxV values measured under in vivo-like conditions, but using the same changes in flux.(PDF)Click here for additional data file.Text S1Concentrations of substrates and coupling enzymes in the in vivo-like assays for each individual enzyme.(PDF)Click here for additional data file.Text S2Ordinary differential equations.(PDF)Click here for additional data file.Text S3Rate equations.(PDF)Click here for additional data file.


Constraint-based analyses of metabolic networks are widely used to simulate the properties of genome-scale metabolic networks. Publicly available implementations tend to be slow, impeding large scale analyses such as the genome-wide computation of pairwise gene knock-outs, or the automated search for model improvements. Furthermore, available implementations cannot easily be extended or adapted by users.in silico on a complete E. coli metabolic model.Here, we present sybil, an open source software library for constraint-based analyses in R; R is a free, platform-independent environment for statistical computing and graphics that is widely used in bioinformatics. Among other functions, sybil currently provides efficient methods for flux-balance analysis (FBA), MOMA, and ROOM that are about ten times faster than previous implementations when calculating the effect of whole-genome single gene deletions Due to the object-oriented architecture of sybil, users can easily build analysis pipelines in R or even implement their own constraint-based algorithms. Based on its highly efficient communication with different mathematical optimisation programs, sybil facilitates the exploration of high-dimensional optimisation problems on small time scales. Sybil and all its dependencies are open source. Sybil and its documentation are available for download from the comprehensive R archive network (CRAN). Constraint-based analyses have become a widely used tool for the study of genome-scale biochemical reaction networks. The mosApart from such constraint-based optimisation methods, several other tools that use different philosophies for metabolic modelling are available. One example is the computation of elementary flux modes to represent the feasible solution space of a metabolic network. AnotherSeveral tools for constraint-based optimisation analyses are currently available (reviewed in-9). The The R computer language has become the standard programming environment in many scientific fields that depend on numerical data analysis, in particular in the analysis of biological high-throughput data. However, R currently offers only very limited options for constraint-based analyses. The R package BiGGR providesWith sybil, which shares some functionality with the COBRA Toolbox and the R packages described above, we aim to establish R as a major platform for constraint-based analyses of biological networks. Besides offering powerful analysis tools in a versatile and freely available environment, sybil aims to supersede previous implementations in terms of calculation speed, flexibility and extensibility.Sybil is implemented in the R programming language as an obSybil is programmed for both speed and memory efficiency; in our experience, about 1 GB of RAM should be sufficient for all types of analyses, even when performed on the largest complete single cell type metabolic models currently available.fluxVar for flux variability calculations[geneDeletion for prediction of gene deletion effects). Another way to use sybil is to directly use "low-level" functions . Methods implemented in class sysBiolAlg provide a particularly comfortable way to execute constraint-based analyses involving optimisation steps : here, the class takes care of the optimisation software without user interference. Sybil’s architecture provides the user with a highly flexible and adjustable framework. Sybil is equally suited for off-the-shelf constraint-based analyses, for building complex analysis pipelines, and for the development of new constraint-based analysis methods.Sybil provides a set of "high-level" functions to access frequently used complex algorithms with a single function call  to generate and solve various types of optimisation problems. This process is handled by class optObj, which provides a large set of methods to generate, modify, and solve mathematical programming problems and to access the results; the user does not need any deeper knowledge about the differences of the various solvers that can be used by sybil. However, if necessary, all parameters available within the solver software can be accessed directly in sybil.In the future, we plan to further extend sybil, e. g. by adding methods that incorporate gene expression data into an FBA approach-29. Two The calculation speed of the optimisations depends on the mathematical optimisation software used. Typically, for large mathematical problems, IBM ILOG CPLEX is slightly faster than the two freely available solvers GLPK and COIN-OR Clp (see below). However, major differences in the running times of different constraint-based analysis tools stem mostly from the overheads produced by the communication between the main program and the solver. This overhead is minimised by sybil through purpose-built fast interfaces to the C-API of each package.in silico gene deletion experiment, or alteration of the objective function during flux variability analysis. To speed up iterations over many such small changes, the optimisation problem is formulated only once; all changes are then applied directly to the pre-formed optimisation problem of the mathematical optimisation software.Most of the implemented algorithms require the generation of an optimisation problem based on the model, the constraints, and the desired algorithm (such as FBA or linear MOMA). During batch calculations, only small changes to the optimisation problem are required, e. g., changes of variable bounds in an E. coli metabolism, containing 2382 reactions and 1261 independent genes[Figuresnt genes.FigureAs can be seen in Figurein silico gene deletion experiments with the same complete model of E. coli metabolism used for the flux variability analyses. Regardless of the details of the experiment , sybil clearly outperforms other implementations in terms of computation speed; this is achieved through the efficient handling of optimisation problems that repeatedly need to be re-optimised, but do not change very much from one optimisation to the other. Sybil was successfully used as the constraint-based core of a machine learning method to reconcile model predictions with genome-scale experimental double-gene knockout data[Figureout data. In thisout data).E. coli model in less than five minutes on our test system.Another fast tool is F2C2, a MATLAE. coli[Sybil can read text-based representations of metabolic networks written in the 'Systems Biology Markup Language’ (SBML), which iE. coli is incluE. coli, is requE. coli are also included in this format  as described in[• ribed in,3,"mtf": for flux-balance analysis and additionally selecting the flux distribution resulting in the smallest absolute total flux,• "moma": for minimisation of metabolic adjustment as described in[• ribed in,"lmoma": for a linear version of the MOMA algorithm similar to the version described in[• ribed in, or"room": for regulatory on/off minimisation as described in[• ribed in.The parameter 'solver’ selects the mathematical optimisation software used by the algorithms. It can be set to "glpkAPI": for GLPK[• for GLPK, via R pfor GLPK,"cplexAPI": for IBM ILOG CPLEX[• OG CPLEX, via R pOG CPLEX,"clpAPI": for COIN-OR Clp[• N-OR Clp, via R pN-OR Clp,"lpSolveAPI": for lp_solve[• lp_solve, via R plp_solve, or"sybilGUROBI": for Gurobi[• r Gurobi, via R pAll R packages are available on CRAN, with thin silico experiments that result in high-dimensional optimisation problems. New algorithms can be easily implemented using the sybil framework and can be distributed as add-on packages to the systems biology community.Sybil is designed to address large scale questions in reasonable time frames, making it possible to generate and run Project name: sybilProject home page:http://CRAN.R-project.org/package=sybilOperating system(s): Platform independentProgramming language: ROther requirements: A mathematical optimisation software License: GNU GPLFBA: Flux-balance analysis; FVA: Flux variability analysis; MOMA: Minimisation of metabolic adjustment; ROOM: Regulatory on/off minimisation; SBML: Systems biology markup language.The authors declare that they have no competing interests.GGD developed the R packages sybil, sybilSBML, glpkAPI, clpAPI and cplexAPI, implemented the algorithms, conceived the handling of the mathematical programming software, and wrote the manuscript. AAD developed sybil add-on packages sybilDynFBA and sybilEFBA and tested and applied sybil and made suggestions for improvements and additions. CJF developed sybil add-on package RSeed and tested and applied sybil and made suggestions for improvements and additions. MJL initated the project, suggested improvements and additions to sybil, and contributed to the writing of the manuscript. All authors read and approved the final manuscript.R package sybil. Archive of the sybil version current at submission.Click here for fileFeatures contained in sybil and other commonly used software packages.Click here for fileR package sybilSBML. Archive of the sybilSBML version current at submission.Click here for fileExample dataset. Archive containing the input files required by sybil for the reconstruction of the central energy metabolism of E. coli in column based format and in SBML format. The files packaged into the archive can be read with the sybil command readTSVmod.Click here for filesybil package vignette. A user guide for sybil in PDF format. It can be accessed from within a running R session with the command vignette("sybil").Click here for file