An analysis of evolutionary constraints, gene duplication and essentiability in the yeast metabolic network demonstrates that the structure and function of a metabolic network shapes the evolution of its enzymes. Most studies of molecular evolution are focused on individual genes and proteins. However, understanding the design principles and evolutionary properties of molecular networks requires a system-wide perspective. In the present work we connect molecular evolution on the gene level with system properties of a cellular metabolic network. In contrast to protein interaction networks, where several previous studies investigated the molecular evolution of proteins, metabolic networks have a relatively well-defined global function. The ability to consider fluxes in a metabolic network allows us to relate the functional role of each enzyme in a network to its rate of evolution.Our results, based on the yeast metabolic network, demonstrate that important evolutionary processes, such as the fixation of single nucleotide mutations, gene duplications, and gene deletions, are influenced by the structure and function of the network. Specifically, central and highly connected enzymes evolve more slowly than less connected enzymes. Also, enzymes carrying high metabolic fluxes under natural biological conditions experience higher evolutionary constraints. Genes encoding enzymes with high connectivity and high metabolic flux have higher chances to retain duplicates in evolution. In contrast to protein interaction networks, highly connected enzymes are no more likely to be essential compared to less connected enzymes.The presented analysis of evolutionary constraints, gene duplication, and essentiality demonstrates that the structure and function of a metabolic network shapes the evolution of its enzymes. Our results underscore the need for systems-based approaches in studies of molecular evolution. Saccharomyces cerevisiae [Escherichia coli [Molecular networks and the genes encoding their building blocks represent two different levels of biological organization that interact in evolution. On the one hand, genetic changes such as point mutations, gene deletions, and gene duplications influence the structure and evolution of these networks. Conversely, network function may constrain the kinds of mutations that can be tolerated, and thus how genes evolve. Existing work on the structure and evolution of molecular networks has mainly focused on protein interaction networks -6. Such revisiae ,10 and Ehia coli . Their fSaccharomyces cerevisiae [In the present study, we ask how the topology of a metabolic network and the metabolic fluxes  through reactions in the network influence the evolution of metabolic network genes through point mutations and gene duplication. Our results suggest that both network structure and function need to be understood to fully appreciate how metabolic networks constrain the evolution of their parts. The present study has become possible with the recent publication of a comprehensive compendium of metabolic reactions in the yeast revisiae . This co2, NAD, glutamate, NADP, NADH, NADPH, AMP, NH3, and CoA [Using the stoichiometric equations that describe chemical reactions, we calculate the connectivity of an enzyme as the number of other metabolic enzymes that produce or consume the enzyme's products or reactants . In other words, a metabolic enzyme A and a metabolic enzyme B are connected if they share the same metabolite as either a product or reactant. Highly connected enzymes in this representation are enzymes that share metabolites with many other enzymes. Including the most highly connected metabolites and cofactors such as ATP or hydrogen in a network representation would render the network structure dominated by these few nodes, and would obscure functional relationships between enzymes. We thus excluded the top 14 most highly connected metabolites: ATP, H, ADP, pyrophosphate, orthophosphate, CO and CoA . The resa/Ks) that occurred in the gene coding for the enzyme [a/Ks ratio suggests higher evolutionary constraints on the enzyme, that is, a smaller fraction of accepted amino acid substitutions. In our analysis, we used the average ratio Ka/Ks of unambiguous orthologs in four sequenced Saccharomyces species: S. cerevisiae, S. paradoxus, S. bayanus, and S. mikatae [a/Ks values used in the main analysis were taken from the study by Kellis et al. [We will first discuss how network structure - specifically, an enzyme's position in the network - influences enzyme evolution. Generally, enzymes in central parts of metabolism such as the tricarboxylic acid cycle will have more neighbors than enzymes in peripheral metabolic pathways Figure . The core enzyme . A small mikatae . The aves et al. . We alsos et al.  and obtaa/Ks . The inset in Figure a and gene connectivity . The reason is that using only Ka, instead of the preferable Ka/Ks, as a measure of evolutionary constraints does not compensate for gene-specific differences in synonymous substitution rates and thus introduces additional noise in the data. Additional data file 3 shows that synonymous (silent) substitutions Ks and enzyme connectivity are not significantly correlated . This is to be expected, as synonymous substitutions do not cause amino acid changes and are thus selectively neutral for the purpose of our analysis.Figure a/Ks)? One possibility is that this correlation is primarily mediated by the corresponding gene expression level [a/Ks and mRNA expression levels . Information on mRNA expression of metabolic genes was obtained from the study by Holstege et al. [P = 4.6 × 10-2). Nevertheless, a partial correlation analysis - controlling for mRNA expression levels - between gene connectivity and evolutionary constraint Ka/Ks shows that enzymes in highly connected parts of the network evolve slowly independent of expression levels .Why do highly connected enzymes show greater evolutionary constraint , we used the isoenzyme with the smallest ratio Ka/Ks. The growth conditions we used vary in the available carbon sources and in different uptake rates for oxygen . The correlation is clearly non-linear, and has an exponential shape. The results summarized in Table a/Ks in other growth conditions on glucose and fructose - two natural carbon sources for yeast. Interestingly, the correlations between evolutionary constraint Ka/Ks and flux are substantially lower, and statistically insignificant, for acetate, a carbon source that may not dominate the natural yeast environment [To relate metabolic flux and the rate of enzyme evolution, we used flux balance analysis to calculate metabolic fluxes in the yeast metabolic network , maximizen Table . The calironment . As we dP = 1.4 × 10-4). This result suggests that duplicates of enzymes producing or consuming widely used metabolites are more likely to be retained in evolution. Figure et al. [Gene duplications have effects opposite from those of most amino acid changes: they may increase rather than reduce flux through an enzymatic reaction. We established that highly connected enzymes and enzymes with high associated flux are especially sensitive to amino acid changes Figures  and 3. Ae et al. .et al. [Saccharomyces genome database (SGD) [Evolutionary constraints on enzymes are indirect indicators of metabolic robustness to amino acid changes, changes that a metabolic network tolerated for well over millions of years of evolution. Another type of biological robustness is that against complete gene deletions. Robustness against gene deletions can be derived from laboratory studies in which the effects of gene deletions on growth rate and other indicators of fitness are studied ,24. Theset al. , and usese (SGD)  to colleP = 4.0 × 10-4). No statistically significant difference in connectivity of essential and non-essential enzymes is observed if all metabolites are used to establish network connections. Consequently, highly connected metabolic enzymes are no more likely to be essential than low connected enzymes. Similarly, as Mahadevan et al. [P = 8 × 10-49). In addition to gene duplications, flux rerouting may provide another mechanism to make highly connected genes less essential. In highly connected parts of a metabolic network, metabolic fluxes may be rerouted through alternative pathways after a loss-of-function mutation [Our analyses of essential enzyme-coding genes show the clearest deviations from behavior observed in protein interaction networks. In the metabolic network from which the most highly connected metabolites (such as ATP or hydrogen) have been excluded, the mean connectivity of essential enzymes is significantly smaller than the mean connectivity of non-essential enzymes Figure . For then et al.  demonstrn et al. . Indeed,mutation ,28,29. TIn sum, we demonstrate that both highly connected enzymes and enzymes that carry high metabolic fluxes in the yeast metabolic network have tolerated fewer amino acid substitutions in their evolutionary history. Why are enzymes carrying larger fluxes more constrained? The likely answer comes from the observation that most mutations affecting enzymatic activity may reduce rather than increase flux. Enzymes carrying high fluxes tend to have reaction products that enter a large number of metabolic pathways. Consequently, a mutational reduction in the activity of such enzymes should be more detrimental than a reduction in the activity of enzymes with lower flux.We also show that the genes encoding enzymes with high flux have more duplicates. Importantly, we do not argue that duplications arise more frequently for genes whose products carry high flux, but that such duplications are more likely to be preserved in evolution, because of the advantage - higher flux - they provide. While a gene's duplicates can initially be preserved through an advantageous increase in metabolic flux, after divergence they may provide other functional benefits . DivergeE. coli metabolic network more strongly than nitrogen limitation [We found that the association between predicted enzymatic flux and evolutionary rate is most pronounced for carbon sources that dominate the natural environment of yeast. This suggests that one can use the association between flux and evolutionary constraint to search for conditions that dominated the evolution of metabolic networks. Similar analyses, which use genomic data to infer the environment that has shaped an organism's evolution, have been used before to show that carbon limitation may have influenced the evolution of the mitation , and to mitation .et al. [E. coli metabolic network there exists no statistically significant association between enzyme connectivity and evolutionary constraint. We emphasize that any contradiction between this earlier work and our results is only apparent. First, the earlier study was based on a much smaller set of enzymes (n = 108 as opposed to n = 350 here), and thus had less statistical power. Nevertheless, two different statistical measures in the previous study showed, like we do here, a negative association between connectivity and evolutionary constraint, albeit not at P < 0.05. Second, because of the lack of sufficient sequence information for a closely related sister species of E. coli, the previous study used only amino acid divergence Ka and not the preferable Ka/Ks to indicate evolutionary constraint. In fact, the correlation between connectivity and Ka is very similar between the present study and the previous work .A previous study by Hahn et al.  reportedIt should not be surprising that the observed associations are weak in magnitude. The reason for the low magnitude is that many other factors influence the evolution of enzyme-coding genes. Two of these factors are gene expression levels (discussed in the paper) and constraints stemming from the tertiary and quaternary structure of enzymes, which may differ among enzymes (little is known about such constraints). The key point is that besides all these other factors, metabolic network function and structure also has a clear influence on protein evolution.How do our results on the yeast metabolic network relate to earlier work on protein interaction networks? There, a similar relationship between protein connectivity and evolutionary constraint has been suggested ,5; howevIn conclusion, our analysis of evolutionary constraints, gene duplication, and essentiality demonstrates that the structure and function of a metabolic network shapes the evolution of its enzymes. In the long run, system analyses of biological networks will allow us to increasingly place the evolution of genes in the larger context in which they operate, as building blocks of cellular networks.S. cerevisiae metabolic reactions by Foster et al. [We used a comprehensive collection of the yeast r et al.  to calcuet al. [in silico network of yeast metabolism includes central carbon metabolism, transmembrane transport reactions, pathways responsible for the synthesis and degradation of amino acids, nucleic acids, vitamins, cofactors, and lipids. In total, the network consists of 733 metabolites and 1,175 metabolic reactions. In the flux-balance analysis, the constraints limiting nutrient uptake, reaction irreversibility, and steady-state conservation of metabolite concentrations are applied. The fluxes optimal for growth are then obtained by maximization of biomass production using linear optimization. Linear optimization was performed using the GNU Linear Programming Kit [Flux balance analysis (FBA) was used to obtain metabolic flux distribution as described previously ,17,19. Tet al.  was usedS. cerecisae genome using a previously described whole-genome analysis tool [s) and the number of substitutions per non-synonymous site (Ka) using the maximum likelyhood models of Muse and Gaut [We identified duplicates in the sis tool . Brieflysis tool  and aligsis tool . Putativand Gaut  and Goldand Gaut .a/Ks, Ka, and Ks values used in the analysis were obtained from the study by Kellis et al. [The average Ks et al. . In a cos et al.  and obtadpl1 gene (solid edges), as defined by the reactions shown in (a). The dpl1 gene has a total of six metabolic connections: two established through ethanolamine phosphate (red edges); and four through sphinganine 1-phosphate (blue edges). Metabolic connections between other enzymes are show by dashed edges. Additional data file a. Spearman's rank correlation r = -0.13, P = 1.6 × 10-2. Additional data file s. Spearman's rank correlation r = -0.056, P = 0.30. Additional data file a/Ks, Ka, Ks) for yeast metabolic enzymes.The following additional data are available with the online version of this paper. Additional data file dpl1 gene (solid edges), as defined by the reactions shown in (a). The dpl1 gene has a total of six metabolic connections: two established through ethanolamine phosphate (red edges); and four through sphinganine 1-phosphate (blue edges). Metabolic connections between other enzymes are show by dashed edges.(a) An example of the metabolic reaction network from sphingoglycolipid metabolism; metabolites are drawn as small circles  and enzyme-encoding genes are shown in rectangles. (b) Metabolic connectivity of the Click here for filea. Spearman's rank correlation r = -0.13, P = 1.6 × 10-2The relationship between enzyme connectivity and the average amino acid divergence KClick here for files. Spearman's rank correlation r = -0.056, P = 0.30.The relationship between enzyme connectivity and the average silent divergence KClick here for fileMaximal uptake rate for glucose 15.3 mmol/g dry weight/h and for oxygen 0.2 mmol/g dry weight/h. Note the small number of fluxes - representing glycolysis - with disproportionately large magnitudes. Similar flux distributions were also obtained for other growth conditions.Click here for fileThe correlation between non-zero enzymatic flux through a reaction and the number of duplicates of the respective enzyme's coding gene.Click here for filea/Ks, Ka, Ks) for yeast metabolic enzymes.Connectivity and evolutionary parameters (KClick here for file


The progress in the "-omic" sciences has allowed a deeper knowledge on many biological systems with industrial interest. This knowledge is still rarely used for advanced bioprocess monitoring and control at the bioreactor level. In this work, a bioprocess control method is presented, which is designed on the basis of the metabolic network of the organism under consideration. The bioprocess dynamics are formulated using hybrid rigorous/data driven systems and its inherent structure is defined by the metabolism elementary modes.The metabolic network of the system under study is decomposed into elementary modes (EMs), which are the simplest paths able to operate coherently in steady-state. A reduced reaction mechanism in the form of simplified reactions connecting substrates with end-products is obtained. A dynamical hybrid system integrating material balance equations, EMs reactions stoichiometry and kinetics was formulated. EMs kinetics were defined as the product of two terms: a mechanistic/empirical known term and an unknown term that must be identified from data, in a process optimisation perspective. This approach allows the quantification of fluxes carried by individual elementary modes which is of great help to identify dominant pathways as a function of environmental conditions. The methodology was employed to analyse experimental data of recombinant Baby Hamster Kidney (BHK-21A) cultures producing a recombinant fusion glycoprotein. The identified EMs kinetics demonstrated typical glucose and glutamine metabolic responses during cell growth and IgG1-IL2 synthesis. Finally, an online optimisation study was conducted in which the optimal feeding strategies of glucose and glutamine were calculated after re-estimation of model parameters at each sampling time. An improvement in the final product concentration was obtained as a result of this online optimisation.The main contribution of this work is a novel bioreactor optimal control method that uses detailed information concerning the metabolism of the underlying biological system. Moreover, the method allows the identification of structural modifications in metabolism over batch time. Knowledge of intracellular metabolic fluxes is crucial to understand how different pathways interact and their relative importance within the overall metabolic processes. Metabolic flux analysis (MFA) is an established methodology that allows the quantification of such intracellular fluxes. In MFA, intracellular fluxes are calculated by applying steady-state material balances around intracellular metabolites. In general the number of reactions exceeds the number of metabolites resulting in undetermined systems of algebraic equations . Such syThe determination of metabolic flux distribution in undetermined systems may also be obtained by flux-balance analysis (FBA) . In FBA,Metabolic Pathway Analysis (MPA) is another flux-based analysis method. MPA, unlike FBA, do not look only at the properties of solutions selected by the statement of an objective, but study the full range of achievable biochemical network states that are provided by the solution space. Network-based MPA has focused on two approaches, elementary modes (EMs) and extreme pathways (EPs) -10. ThesAlthough flux-based analysis methods have been mainly used for metabolic engineering ,12, theyThe proposed methodology for bioprocess monitoring and control is represented schematically in Figures Once the model is properly validated, it can be used for on-line intracellular flux distribution monitoring and for on-line process performance optimisation Figure . So the m metabolites and q reactions such as the network represented in Figure We consider a generic metabolic network with N = {nij} a m × q stoichiometric matrix and v = {vj} the vector of q metabolic fluxes with vj denoting the net specific rate of reaction j. Some of the q reactions are irreversible, thus the respective fluxes must be nonnegative: vk > 0 with k denoting the irreversible reactions in the metabolic network. The universe of solutions of system (1) forms a convex polyhedral cone in the solution space T.     (10)K, is established from the elementary modes of Table th column) and product formation (6th column) as lumped equations of glucose, glutamine and alanine. It should be noted that the 5th EM was not included in K because preliminary results showed that this EM has negligible flux. This observation is in agreement with some published works T,     (11)ϕ(c) = [Xwhereas the vector of unknown kinetics is given by:1 ρ2 ρ3 ρ4 μ π]T = ρ     (12)ρ = T.Experimental data of seven experiments (three batch and four fed-batch cultures) were used for the identification of the EM kinetics. Data of 5 experiments were used for parameter calibration whereas data of 2 experiments were used for model validation. The concentrations in the state space vector (eq. 10) were analyzed off-line according to methods published elsewhere . The neuThe hybrid modeling results in terms of measured and predicted state variables are presented in Figure Figure 1 and EM4). The fed-batch culture, which had lower glucose and glutamine concentrations during the cell growth phase, started with substantially lower consumption rates of these nutrients, while maintaining the flux to biomass.In general, the metabolic activity of BHK cells during the cell growth phase is higher in the batch culture than in the fed-batch culture. The EM fluxes in the fed-batch experiments seem to be much more controlled than in the batch experiments. The high levels of glucose and glutamine at the beginning of the batch culture are mostly directed toward the overflow metabolism, i.e., waste production of lactate and alanine  is much higher in the batch experiment. These results are in agreement with several published works for other mammalian cells T) in the sense of maximising the total amount of IgG1-IL2 produced at the end of the experiment (13).After the parameters retuning step, the glucose and glutamine feeding rates were re-optimised (max (see methods section). The increase in volume is negligible in this problem, thus it was not considered in optimisation (13). The risk constraint states that the average risk must not exceed a given maximum level defined by the user. This restricts the feasibility domain to low risk regions and is essential for process optimisation supported by hybrid models since the black-box model  predictions degrade in regions of the input space with sparse measurements.The optimisation (13) is subjected to the constraints of the process dynamics (equation 9), upper and lower bounds for the glucose and glutamine feeding rates, and the maximum risk of model unreliability, RISKRISKmax = 0.35) was adopted during the on-line optimisation experiment. At each sampling time, the flow rates of the feeding pumps were updated according to the re-optimised feeding profiles of glucose and glutamine.The micro-genetic algorithm coded by  was useda priori optimised trajectories), 46 h, 75 h, 92 h and 195 h. The comparison of predicted and measured concentrations shows a very satisfactory performance for the on-line optimisation. Furthermore, although the measured product concentration was only available at the end of the experiment, the predicted time course of this variable follows closely the product measurements. The final product titre obtained was 16.4 mg/1 corresponding to a 10% increase in relation to previous experiments carried out with the same medium  and without on-line optimisation.Figure Higher productivities are likely if the initial medium composition is optimised. The optimisation results of initial glucose and glutamine concentrations along with the corresponding feeding strategies are shown in Figure IgG) increases during the cell death phase as a consequence of the increase of the glutamine concentration.The kinetics of the elementary modes for the optimal strategy is presented in Figure The current work presents a hybrid modeling approach for bioprocesses that integrates information concerning intracellular metabolic fluxes, bioreactor transport phenomena and measured data. First, elementary flux analysis is used to reduce the biosystem metabolic network into a set of macroscopic reactions relating extracellular components only. A bioreactor dynamical model is then established from material balance equations of the compounds which intervene in the final reaction set. Nonparametric techniques are used for identification of the elementary modes kinetics from measured data. The method was successfully applied to a recombinant BHK-21A cell line producing a fusion glycoprotein. A significant result is the achievement of the flux distribution over the runtime of a bioprocess. The so obtained information can be used to identify conditions that favour product formation. A fed-batch BHK culture performed with on-line optimisation supported by the proposed methodology allowed a 10% increase in the final productivity. Higher productivities are expected if starting nutrients concentrations are optimised. The developed tool promises to be advantageous for optimising the productivity of fed-batch biochemical processes since the transfer and adaptation to different cell lines is reasonably straightforwardA batch least squares criteria of residuals in concentrations was adoptedP the number of measured patterns, n the number of state variables, t) and c the measured and model state vectors. The concentrations were previously scaled by the measurements standard deviations. A quasi-Newton algorithm with conjugate gradient with line search (MATLAB™ optimisation toolbox) was adopted.with w can be evaluated using the sensitivities equations, which can be obtained by differentiating eq. (3a) in order to w, yielding, after some manipulations, the following linear time-varying system:The gradients ∂e/∂withc/∂w)t = 0 = 0 because the initial state is independent of parameters w. The evaluation of matrices A and B require the sensitivities ∂ϕj/∂c, ∂ρj/∂w and ∂ρj/∂c. The first term is obtained by analytical derivation of known functions ). The other two matrices are obtained by backpropagating the identity matrix through the neural network. The backpropagation of a given identity matrix column 'i' results in the evaluation of vectors ∂ρij/∂w and ∂ρij/∂c.These equations must be integrated along with hybrid model eqs. (3)–(4). The initial value is , of being unreliable. The use of unreliable model predictions for process control should be avoided. For this reason, the value of the risk is used as a constraint to the optimisation (13). Here, the trust region was defined by nc ellipsoidal clusters of the form:The trust region is the subspace of the input domain, where the model was properly validated with experimental data, showing low modeling error. Model predictions, mj the cluster centres and ∑ = diag{σi2} the diagonal covariance matrix. The choice of the clusters number (nc) is done by trial and error according to the criteria of goodness of measurement density representation. The rule nc = P/3, with P the number of measured patterns normally provides acceptable results. The interpolation tolerances (IT) defines the distance to measured pattern such that yC > 0.5. The clusters width is then given by:with σ = [-2 ln(0.5/n)]-1/2 IT     (M6)mj. The final set of clusters forms a continuous density function f:c→v by applying the maximum operator:The k-means algorithm was used to calculate c) is a scalar between 0 and 1 that can be interpreted as the degree of membership of c to the data set used for training the black-box model (4). Low v values (i.e. c vectors out of the interpolation tolerance) are an indication of high risk of the black box model outputs being unreliable. Finally, the risk of black box model unreliability is given by: RISK(c) = 1 - v(c)     (M8)The output v(BHK – Baby Hamster Kidney cellsCHO – Chinese Hamster Ovary cellsEM – Elementary Flux ModeEP – Extreme PathwayFBA – Flux Balance AnalysisIgG1-IL2 – Imunoglobulin type one linked to interleukin type twoLP – Linear ProgrammingMFA – Metabolic Flux AnalysisMPA – Metabolic Pathway AnalysisAll authors read and approved the final manuscript. RO developed the software. APT, CA and RO participated in the model implementation. APT performed experimental work. PA, MJTC and RO designed and coordinated the study. APT and RO drafted the manuscript.


S). Singular value decomposition (SVD) of the ExPa matrix P has previously been used to characterize the metabolic regulatory problem in the human red blood cell (hRBC) from a network perspective. The calculation of ExPas is NP-hard, and for genome-scale networks the computation of ExPas has proven to be infeasible. Therefore an alternative approach is needed to reveal regulatory properties of steady state solution spaces of genome-scale stoichiometric matrices.Extreme pathways (ExPas) have been shown to be valuable for studying the functions and capabilities of metabolic networks through characterization of the null space of the stoichiometric matrix (W) formed of random samples from the steady-state solution space of the hRBC metabolic network gives similar insights into the regulatory properties of the network as was obtained with SVD of P. This new approach has two main advantages. First, it works with a direct representation of the shape of the metabolic solution space without the confounding factor of a non-uniform distribution of the extreme pathways and second, the SVD procedure can be applied to a very large number of samples, such as will be produced from genome-scale networks.We show that the SVD of a matrix (These results show that we are now in a position to study the network aspects of the regulatory problem in genome-scale metabolic networks through the use of random sampling.Contact: palsson@ucsd.edu In recent years, chemically-detailed metabolic networks capable of simulating growth have been reconstructed at the genome-scale for nearly twenty organisms . Any metP), resulting in two key insights. First, it was shown that SVD of P can compute the effective dimensionality of the regulatory problem from a network perspective. Second, it was shown that the 'eigenpathways' that correspond to the modes of the SVD identified key network branch points that can represent key control points (reactions) in the network. These points constitute a minimal set of reactions whose control sets the steady state flux state for the entire network, represented as a specific location in the steady-state solution space. SVD rank orders these key branch points in accordance with the shape of the solution space and thus rank orders the effectiveness by which the branch points can be used to move the network's flux distribution around in the solution space. These features were demonstrated by showing that the dominant features of the first five eigenpathways correspond to actual major points of control in the hRBC and can effectively locate the solution in a 5-dimensional space . We used two related rank correlation measures [-15 and a rho value of 0.97 with P-value 4.0*10-12. Based on these results, we claim that the sampling methodology produces similar insight into identifying the most important reactions for flux control as did the ExPa-based approach.Scores allow us to impart a rank ordering on the reactions. The degree to which the SVD of measures : KendallW were combined to reconstruct the physiological flux distributions and how the difference between the reconstructed and actual distributions diminished as more singular modes were used to reconstruct the desired solution. The reconstruction and difference computations were performed in the same manner as described previously . Additionally, we assessed how the efficiency of these reconstructions compared with those based on the eigenpathways. The results are shown in Figure We determined how the modes computed from SVD of Helicobacter pylori metabolic reconstruction [While the sampling approach detailed herein used the hRBC for demonstration purposes, the intended application for the method is genome-scale metabolic networks. A requirement for application to such networks is the demonstration of sufficient sampling of their much higher-dimensional steady-state solution spaces. In this section we demonstrate that sufficient sampling is feasible on the genome scale, in the same manner as was done for the hRBC in the Results section above. We use the bacterium truction , which iP. We present three main results. First, we found that the sampling procedure did lead to the same estimates of the dimensionality of the regulatory problem for the metabolic network studied. Secondly, we demonstrated that the resulting PRMs reconstruct physiological flux distributions as efficiently as did the eigenpathways. Thirdly, we showed that the flux adjustments represented by the PRMs were similar, and similarly interpretable, as those represented by the eigenpathways. In particular, we found that the ranking of reactions by their regulatory "importance" in the two approaches was highly correlated with high statistical significance.In previous work  it was da priori reason for believing that SVD of P would be superior to SVD of W. Thus, while the focus of this work was to explain the results of SVD of W in the context of what had already been shown for SVD of P, it is not necessary that an effort be made to force SVD of W to reproduce SVD of P. Instead, what was shown herein is that the approaches can be thought of as attacking the same problem and that SVD of W provides a way forward into genome-scale matrices to the line of inquiry that was begun with SVD of P, since sampling of genome-scale metabolic states has been accomplished at the genome-scale for E. coli and H. pylori [P for these networks has proven infeasible. The results of the two methods will not generally be identical. In the case of SVD of P not only the shape of the space, but, more precisely, the distribution of the pathways along the edges of the space is what is being characterized. Because the samples are generated uniformly in the solution space, it is the shape of the solution space alone that is being characterized in this approach. Thus, the interpretation of SVD of W is more straightforward than is the interpretation of SVD of P.It is important to note that there is no . pylori ,30 – wheWe used physiological flux distributions computed from a kinetic model and compared how well PRMs and eigenpathways reconstructed them. As can be seen in Figure We compared the PRM and eigenpathway flux distributions in two ways. Figures The flux maps in Figures W have a slightly different interpretation than the eigenpathways derived from P. Mathematically, the modes of W are the orthogonal directions of maximal variance in the steady-state solution space of the stoichiometric matrix. Biochemically, they correspond to the directions of greatest flexibility in achieving a network-wide flux distribution. The unequal variances in these orthogonal directions imply a hierarchy, wherein control of the more dominant modes gives the cell more control of achieving a desired flux distribution.The PRMs of From these results we conclude that SVD of a sample of uniformly obtained steady state flux distributions for a metabolic network leads to a similar means of characterizing the regulatory problem as was obtained previously with ExPas. This was a necessary demonstration on a small system, before applying the technique to genome-scale networks. With a number of genome-scale metabolic networks now significantly reconstructed for a number of organisms, the results presented in this work indicate that it should now be possible – given that feasible genome-scale sampling has been demonstrated ,30 – to For this work we utilized a previously-constructed kinetic model  and a stin silico kinetic model [+/K+ pump), 2.5 mM/h for the NADPH load, and 1.7 mM/h for the NADH load.Four steady-state flux distributions were calculated using an ic model , as detaic model , and useic model . These mThe null space of the stoichiometric matrix of the hRBC metabolic network is a  unbounded space. To transform this space into one that is physiologically relevant, we applied, as before, constraints by bounding the minimum and maximum fluxes for each reaction. All irreversible reactions were constrained with a minimum flux of 0 mM/h and a maximum flux of 2000 mM/h unless experimentally-determined maximum flux values were available. Such maximum reaction rates were available for 22 reactions, and have been published previously . AdditioW and were used to calculate an approximate center s of the solution space. The third step calculates the sample points. Beginning with an initial, randomly selected point xm from W, a direction vector is constructed by choosing a randomly selected point y from W and moving along the direction vector d = y-s from xm resulting in a new point xm+1. After each such move xm+1 was randomly substituted into W and the approximate center recalculated. For the results presented herein, we saved a sample point every 1000 moves in the space. It should be noted that the red blood cell metabolic network does not have any non-trivial biochemical loops (type III ExPas) [We use a variant of an artificial centering hit-and-run algorithm  to samplI ExPas) , and thuI ExPas) ,37 that I ExPas) . Note thI ExPas) , it was I ExPas) .max values for the reactions in the hRBC have been previously published [max amongst all reactions that it contained. All of the extreme pathways had at least one reaction with an experimentally defined Vmax.The extreme pathways were calculated from the stoichiometric matrix of the hRBC using a previously published algorithm . Vmax vaublished ,32. Eachu and v, the angle was computed as Θ = cos-1 (u·v/||u|| ||v||).Angles between modes were computed using standard linear algebra. For two vectors ij is the contribution of reaction j to eigenvector i, and there are m eigenvectors, then the loading vector for reaction j is lj = . The score for reaction j is then given by the l2-norm of the loading vector: | lj | = (vj12 + vj22 + ... + vmj2)1/2.The result of applying the SVD to either the matrix of ExPas or the matrix of uniformly sampled flux distributions is a set of eigenvectors. Each reaction has a corresponding loading vector, which is the vector of the reaction's contribution to each of the eigenvectors. Specifically, if vSVD, singular value decomposition; ExPa, extreme pathway; PRM, principal regulatory modeCB and NP performed the sampling and interpretation of the results. CB drafted the manuscript and NP and BP provided critical edits.


Staphylococcus aureus is a pathogenic bacterium that has evolved resistance to many antibiotics, representing a significant health care concern. We present the first manually curated elementally and charge balanced genome-scale reconstruction and model of S. aureus' metabolic networks and compute some of its properties.Several strains of bacteria have sequenced and annotated genomes, which have been used in conjunction with biochemical and physiological data to reconstruct genome-scale metabolic networks. Such reconstruction amounts to a two-dimensional annotation of the genome. These networks have been analyzed with a constraint-based formalism and a variety of biologically meaningful results have emerged. S. aureus strain N315. This reconstruction, termed iSB619, consists of 619 genes that catalyze 640 metabolic reactions. For 91% of the reactions, open reading frames are explicitly linked to proteins and to the reaction. All but three of the metabolic reactions are both charge and elementally balanced. The reaction list is the most complete to date for this pathogen. When the capabilities of the reconstructed network were analyzed in the context of maximal growth, we formed hypotheses regarding growth requirements, the efficiency of growth on different carbon sources, and potential drug targets. These hypotheses can be tested experimentally and the data gathered can be used to improve subsequent versions of the reconstruction.We reconstructed a genome-scale metabolic network of iSB619 represents comprehensive biochemically and genetically structured information about the metabolism of S. aureus to date. The reconstructed metabolic network can be used to predict cellular phenotypes and thus advance our understanding of a troublesome pathogen. Staphylococcus aureus is a pathogenic gram-positive bacterium that causes a variety of disease conditions, some life-threatening, both in hospital settings and in the community at large. Moreover, various strains of this organism have evolved resistance to some of the most clinically useful antibiotics, including methicillin and vancomycin[S. aureus, especially under carefully controlled environmental conditions in chemically-defined media. While some research has been performed towards this goal[S. aureus contains much more readily-available specific information regarding the organism's metabolism than does a compilation of literature data[ancomycin. Althougthis goal, its scoture data.The annotated genome of a microorganism, in conjunction with biochemical and physiological data, can be used to reconstruct the metabolic network of that organism,5. Such in silico analytical methods[Genome-scale reconstructions (GENREs) represent a biochemically and genetically structured database that can be queried and interrogated using l methods. With thl methods. This ral methods-11.Escherichia coli[Saccharomyces cerevisiae[Methylobacterium extorquens [Mannheimia succiniciproducens[Helicobacter pylori[Haemophilus influenzae[E. coli and S. cerevisiae have been analyzed by groups around the world . These analyses have led to several publications of general interest that focus on such diverse topics as the causes of enzyme dispensability[GEMs using the constraint-based modeling formalism have been constructed for a number of microorganisms, including chia coli, Saccharerevisiae,14, Methtorquens , Mannheiproducens, Helicober pylori and Haemnfluenzae, and recnfluenzae. GEMs arnfluenzae. Importansability, the recnsability, and thensability.in silico model allows us to identify areas where our understanding of an organism is inadequate and where additional experimentation is needed[GEMs do not only give valuable computational results, but they also provide a wealth of hypotheses that can be experimentally tested,23-26. Tis needed.S. aureus, termed iSB619 following a previously described naming convention[S. aureus, the work reported contains the most comprehensive metabolic reaction list available for this significant pathogen that is consistent with known phenotypic functions.The work reported herein describes the first manually curated genome-scale elementally and charge balanced metabolic reconstruction and model for the important pathogen onvention, represeS. aureus strain N315 consisting of 619 genes, 537 proteins, 640 reactions, and 571 metabolites. The entire reaction list of this GENRE is included in the supplemental material [see . A set of metabolic maps graphically representing the GENRE is also available at the same web address. This GENRE was built without the benefit of an earlier manually curated reconstruction and model and should be considered a first-draft to the two-dimensional annotation of the S. aureus genome. Unlike other initial GENREs[S. aureus GENRE is nearly completely elementally and charge balanced. All but three reactions produce no net change in terms of chemical elements and charge. Of these three reactions, one  is never used because it is a dead-end (see below), one  is a weighted combination of various fatty acids to form an average phosphatidate molecule for this organism, and one  participates in methionine and spermidine metabolism.We have formulated a GENRE for rial see  and is aal GENREs,18,28, tS. aureus. We provide the most comprehensive reaction list to date, including reactions that were added based on systemic analysis [see . Basic properties of the reconstructed network are summarized in Table Most of the reactions 91%) in the GENRE are associated with one or more genes with only 9% (59) of the reactions included without a known gene. The reactions that do not have a gene association are principally transport reactions, allowing metabolites to cross the cell membrane, and reactions that involve the formation of lipids and other cell wall components see . The incysis see  – the GEysis see , which aysis see  and in g% in the S. aureus. Subsequent additions to the model will likely close some of these gaps. These reactions are further detailed in the supplemental material [see Particularly because this is a first-pass reconstruction, the network has a significant number of dead-end metabolites, as other GENREs do,17,18,25rial see .S. aureus could not be located in the literature, data from Bacillus subtilis was substituted where necessary[S. aureus accounts for only a small fraction of the biomass function. Although a comprehensive biomass function has been published for E. coli and used to analyze the networks of other initial reconstructions[S. aureus because of the differences between gram-positive and gram-negative bacteria. The relative quantities of each required metabolite were included in the biomass function when information existed, but many of the trace compounds were included in small ratios that are not quantitatively accurate. It has been shown previously that the calculated biomass production is relatively insensitive to the exact ratios used in the biomass function[A reaction representing biomass formation, consisting of 58 metabolites required for cellular growth, has been defined and is detailed comprehensively in the supplemental material see . Key comnecessary; quantittructions,18, this function. The bioS. aureus. A literature search revealed experimental growth requirements[S. aureus and they are compared with the computational predictions  allows for the computation of feasible steady-state fluxes through a reaction network that maximize a particular objective and satisfy various constraints, including stoichiometry, thermodynamics, and enzyme capacity,31-33. Suirements,35 for Sns Table . These riSB619 does not account for regulatory effects and as such predicts media requirements as if regulatory processes allow any gene to be expressed at the needed levels. The transcriptional regulatory network reduces the functionality of a constraint-based metabolic model by limiting which reactions are active at a given time[The primary difference between the computationally predicted growth requirements and those from experimental data is the amino acid requirement. Kuroda et. al report tiven time. If enzyS. aureus is said to require an organic source of nitrogen provided by amino acids[S. aureus has shown the ability to grow without amino acids previously thought to be required[We experimentally predict growth if any one of the nine amino acids or derivatives listed in the table is provided. Each of these compounds can either provide nitrogen through a deamination reaction or be directly converted into another compound that can provide nitrogen. ino acids, so the ino acids listing  required; thus thin silico the effect of adding each amino acid individually to the predicted minimal media listed . The individual results are shown in Figure S. aureus can grow more efficiently by uptaking certain amino acids rather than synthesizing them, even though its genome encodes that functionality. Although one might intuitively think that this would be the case for all amino acids, the results in Figure To explore the discrepancy between experimental results and computational predictions, we elected to study S. cerevisiae (14.6%-27.8%) [In order to determine the effects of the deletion of a reaction from the network, as would occur in a gene knock-out experiment, FBA is used with the additional constraint that the flux through a particular reaction is zero. This allows for the rapid prediction of the results of gene deletions and also reaction deletions, as occur when a selective enzyme inhibitor is used. We calculated the effects of all single reaction and gene deletions both on minimal medium and on a rich media . We found that, on rich media, 130 reaction deletions and 88 gene deletions are computationally predicted to be lethal. On minimal media, 230 reaction deletions and 168 gene deletions are predicted to be lethal. These predictions are detailed in the supplemental material see . There a%-27.8%) . We noteS. aureus to the best of our knowledge. These computational predictions can be experimentally tested with targeted gene deletions or the inhibitors listed. Due to the diversity of biological systems in which the inhibitors were initially discovered, gene deletions would be expected to agree with our predictions more than the use of chemical inhibitors. There are a variety of reasons why any one of these inhibitors listed may have no utility whatsoever as a drug. For example, there may be no way for the inhibitor to actually enter a cell. Nevertheless, when considered as an initial guess at potential drugs, Table S. aureus before.We took the results of the reaction deletion study on rich media and searched PubMed for chemical inhibitors of each of those reactions Table . Most ofS. aureus[B. subtilis[S. aureus, a comparison between experimental data and the predictions detailed herein can easily serve as validation for this model and identify problem areas. The absence of a comprehensive, publicly-available data set regarding gene essentiality in this organism is in itself powerful motivation to undertake the reconstruction detailed in this paper, as the reconstruction can rapidly predict essential metabolic genes that can later be screened experimentally as potential drug targets.Additionally, we considered comparing the results of our gene deletion study with the results of an existing experimental approach to determine gene essentiality. Unfortunately, although we are aware of two large-scale studies on gene essentiality in S. aureus,39, ther subtilis, but thiWe computed the sensitivity of growth rate to oxygen uptake on a variety of different carbon sources Figure . The carS. aureus and represents the first draft to its two-dimensional genome annotation. The GENRE with the GPR associations represents a chemically and genetically structured database derived from the underlying data. When the properties of this GENRE are analyzed using FBA, the model computationally predicts phenotypic states. Based on the surprising paucity of physiological growth data available for this organism, especially under carefully defined conditions, the predictions made by the GEM are best viewed as hypotheses. These hypotheses can be experimentally tested, with similar results serving as validation for the GEM and dissimilar results describing failure modes of the GEM. These failure modes can be more interesting than correct predictions because they provide direction for improvement of the GEM and point to areas in metabolism and regulation that need further investigation. The analysis of failure modes and subsequent improvement of the GEM constitute a cycle of iterative model building[E. coli and S. cerevisiae have made substantial improvements over initial genome-scale reconstructions. These enhancements include greater coverage of metabolism, explicit GPR associations, more detailed enzyme localization, and better enforcement of elemental and charge balancing. We expect that future enhancements to iSB619 will improve the accuracy of the reaction network and the GPR associations.The work reported is the first genome-scale metabolic reconstruction for the pathogenic bacterium  building,27, withS. aureus, any advancement of knowledge is welcome.The scope of hypotheses that can be devised with a constraint-based metabolic model ranges from the relatively mundane, such as prediction of relative growth on different carbon sources, to the potentially groundbreaking, such as the determination of novel prospective drug targets. With an organism as harmful as S. aureus strain N315 was downloaded from the Comprehensive Microbial Resource (CMR) at The Institute for Genomic Research (TIGR) website[A genome annotation for ) website and used) website was thenB. subtilis and E. coli. Two books proved particularly helpful for this process[S. aureus could not be produced without the addition of reactions in various pathways present in B. subtilis. Membrane-bound transporters were added whenever evidence existed that a metabolite could enter and/or exit the cell. For example, if data indicated that a given carbon source could be used, a transporter was added to the reconstruction. Potential genes for some of these reactions were located by best-hit BLAST analysis against E. coli and B. subtilis. Each reaction without a gene association was checked against the genome annotations for E. coli and B. subtilis to determine if a gene exists in either organism for a given function. A homology search was used with any gene located in E. coli or B. subtilis against the entire S. aureus N315 genome. A reaction was putatively associated with a gene based on the E value provided by BLAST and the annotation information for that gene from both KEGG and TIGR. If a S. aureus gene did not have any functional information included in either annotation, a BLAST E value of 0.05 was considered sufficient to make an association. If functional information was present and differed from the specific reaction under consideration, a better E value would be necessary; the precise E value required was a judgment call and depended on the specificity of the annotation information. An association based on a relatively large E value should not be considered as a claim that a gene product catalyzes a reaction, but a suggestion as to a candidate gene [see S with each column representing a reaction and each row a metabolite, with each element representing the stoichiometric coefficient. This reconstruction process, including accounting for GPR associations, and the subsequent analysis described below were done using the software program SimPheny™ .After assembling the network based on genomic data, missing functions were noted based on physiological data regarding this organism, as well as s process,43. Likegene see . TranspoS. aureus, the relative production of metabolites required for growth was taken to be similar to that published for the related gram-positive organism B. subtilis[S. aureus[Because no thorough biomass composition has been published for  subtilis. The fatS. aureus. Small aS. aureus.v) that are consistent with all constraints, namely steady-state network operation (eq. 1 below) and reaction rate limitations (eq. 2 below), and which maximize the production of biomass components in the defined ratio. This corresponds to the following linear programming problem:With the reconstructed metabolic network and biomass function defined, flux-balance analysis (FBA)  was usedgrowthmax Z = vSubject toS • v = 0     (1)i < vi < βi     (2)αS is the stoichiometric matrix described above, and αi and βi define the minimum and maximum allowable fluxes through each reaction vi. The flux range was set arbitrarily high for all internal reactions so that no internal reaction restricted the network, with the exception of reactions known to be irreversible, which have a minimum flux of zero. The inputs to the system were restricted where necessary . The function vgrowth is a special reaction taking as substrates all biomass metabolites, ATP and water, and producing ADP, protons, and phosphate (as a result of the non-growth associated ATP maintenance requirement). SimPheny™  was used for all FBA calculations.where The value of Z computed with the above procedure can either be zero or greater than zero depending on the inputs and outputs that are allowed, corresponding to the nutrients provided in the media. A zero value is a computational prediction of no growth; this commonly occurs when an essential nutrient like a carbon source is not provided. Any value greater than zero corresponds to cellular growth.For the deletion study, each reaction was individually constrained to have zero flux and maximal biomass production was computed. Reactions were considered essential if no biomass could be produced without their usage. Gene deletions were computed in a similar manner, but all reactions requiring the presence of a given gene were simultaneously restricted to zero flux prior to computing maximal biomass production.The minimal media was determined computationally with the systematic testing of distinct inputs. In short, different combinations of molecules were allowed to enter the reaction network until the minimal group that allowed biomass production, or non-zero Z, was found. Importantly, the minimal media computed here does not discriminate between extremely slow, inefficient growth and rapid growth; it is only concerned that some amount of biomass production is calculated.CMR: Comprehensive Microbial ResourceEC: Enzyme CommissionFBA: flux-balance analysisGEM: genome-scale modelGENRE: genome-scale reconstructionGPR: gene-protein-reactionKEGG: Kyoto Encyclopedia of Genes and GenomesLP: linear programmingTIGR: The Institute for Genomic ResearchSAB carried out all aspects of the work and drafted the manuscript. BOP conceived of the study, participated in its design and coordination, and helped draft the manuscript. All authors read and approved the final manuscript.Cellular biomass demand table This is a detailed, quantitative listing of the macromolecules required for cellular growth.Click here for fileCompound abbreviations This is a listing of the compound abbreviations used in the reconstruction and the corresponding formal compound names.Click here for fileDead-end reactions This is a listing of the reactions that the current version of the model will never use because they involve a dead-end metabolite.Click here for fileLethal reaction deletions on minimal media This is a listing of all of the reactions that are predicted to be essential for growth on minimal media and their corresponding gene associations.Click here for fileNetwork reaction list This is a comprehensive list of all of the reactions that are in the model.Click here for fileReactions without any gene association This is a list of the reactions that are included in the model without any gene association and rationale for their inclusion.Click here for fileLethal reaction deletions on rich media This is a listing of all of the reactions that are predicted to be essential for growth on rich media and their corresponding gene associations.Click here for fileBoolean gene-reaction associations This is a listing of reaction abbreviations along with the genes that are required for those reactions, in a Boolean form.Click here for fileReactions that were added based on systemic evidence This is a list of the reactions that were added to the reconstruction based on systemic analysis (they do not appear in an obvious fashion in the genome annotation), and the gene(s) with which they are associated in cases where a gene could be located based on homology searches.Click here for file


So far dynamic metabolic models generally have been based on mechanistic rate equations which often contain so many parameters that their identifiability from experimental data forms a serious problem. Recently, approximative rate equations, based on the linear logarithmic (linlog) format have been proposed as a suitable alternative with fewer parameters.Dynamic modeling of metabolic reaction networks under Saccharomyces cerevisiae based on mechanistic rate equations. Elasticities are estimated from the generated data, which define the complete linlog kinetic model of the glycolysis. The effect of data noise on the accuracy of the estimated elasticities is presented. Finally, identifiable subset of parameters is determined using information on the standard deviations of the estimated elasticities through Monte Carlo (MC) simulations.In this paper we present a method for estimation of the kinetic model parameters, which are equal to the elasticities defined in Metabolic Control Analysis, from metabolite data obtained from dynamic as well as steady state perturbations, using the linlog kinetic format. Additionally, we address the question of parameter identifiability from dynamic perturbation data in the presence of noise. The method is illustrated using metabolite data generated with a dynamic model of the glycolytic pathway of Saccharomyces cerevisiae, and the determination of the control coefficients. MC simulations revealed that certain elasticities are potentially unidentifiable from dynamic data only. Addition of steady state perturbation of enzyme activities solved this problem.The parameter estimation within the linlog kinetic framework as presented here allows the determination of the elasticities directly from experimental data from typical dynamic and/or steady state experiments. These elasticities allow the reconstruction of the full kinetic model of  Metabolic activities of living cells are accomplished by a well regulated, highly coupled network of numerous enzyme catalyzed reactions and selective membrane transport systems. To engineer such systems, enzymatic, transport and regulatory functions of the cells are manipulated via the use of recombinant DNA technology . Within In this context, we focus on the metabolic reaction network level. Modeling metabolic reaction systems is usually based on stoichiometric sometimes followed by kinetic modeling.One of the initial steps in the modeling of metabolic reaction networks is to determine the structure and steady state characteristics of a given network using stoichiometric information alone. Steady state models describe time invariant fluxes, gathered from steady state experiments; hence they reflect the structural characteristics of the system. Metabolic Flux Analysis (MFA) and Metabolic Network Analysis (MNA) were developed as powerful tools to analyze such flux data. At steady state, the mass balances over the metabolites in the metabolic network yield a set of linear relations between the metabolic fluxes which can be expressed as:S·v = 0     (1)S is the (m × r) stoichiometric matrix and v is the (r × 1) vector of metabolic fluxes, where m is the number of balanced metabolites and r is the number of fluxes. Here, the system is in most of the cases underdetermined so that there are an infinite number of possible solutions. The realized solution depends on the kinetic properties of the involved reactions; this information is seldom known. To bypass the need of information on kinetics of individual reactions, alternative mathematical approaches have been proposed in the past to obtain a unique solution. An example is the constraint based optimization approach which is based on assumed optimality criteria, e.g. maximum growth, given biochemical, thermodynamic and irreversibility constraints and maximal reaction rates 40]. At tIt is worth mentioning that the obtained parameters (elasticities) are true kinetic parameters that reflect the properties of the enzymes with respect to the corresponding metabolites. They can be assumed to remain invariant as long as the enzyme keeps its properties in response to the changes in the environmental conditions. Additionally, since the linlog kinetics provides an approximation to the actual rate of the corresponding reaction over certain interval, the elasticity parameter that performs best may be different than the theoretical value of the corresponding elasticity. This should be kept in mind when comparing some of the less well determined elasticity values to the theoretical ones.in vivo conditions using data obtained from perturbation experiments remains still a challenging problem in the area of systems biology. In this contribution, we presented a method which allows the determination of the elasticities directly from experimental data from typical dynamic and/or steady state perturbation experiments. These elasticities allow the reconstruction of the full kinetic model of the glycolysis of Saccharomyces cerevisiae, and the determination of the control coefficients. We further show by a posteriori parameter identifiability analysis that a subset of elasticities could not be identified using dynamic perturbation data only. Introduction of additional experimental information, i.e. steady state experiments, solved this parameter identification problem.Constructing dynamic models of metabolic reaction networks under Considering the description of the dynamics, results of the cross validation studies and the final values of the elasticities, we conclude that linlog kinetics, although being an approximate kinetic format, performs very satisfactorily for estimating elasticities from data obtained from dynamic simulation of a mechanistic model, that was realistic in terms of biochemical complexity (glycolysis), noise added and sampling frequency.In linlog kinetics, all rate equations have the same mathematical structure: proportionality to the enzyme level and linearity in the parameters  + Ec ln(c'))     (3)v is the (r × 1) rate vector, r being the number of rates, J0 is the square diagonal matrix containing the reference state fluxes , e' is the square diagonal matrix containing relative enzyme levels , i is the (r × 1) vector of ones, Ex and Ec are the (r × mx) and (r × mc) elasticity matrices, mx and mc being the number of intracellular and extracellular metabolites respectively, and x' and c' are the (mx × 1) and (mc × 1) vectors containing relative concentrations of the intracellular and extracellular metabolites respectively . In each term, the superscript '0' indicates the reference state values.where the vector Ji's) using the mass balances and the measured metabolite concentrations (xj), we can directly use the equation (3) for the estimation of the elasticities. When it is rearranged, the equation (3) can be presented in the following standard linear model:Given that for steady state perturbation experiments, we obtain information on fluxes (a = Y·b     (4)a is the (r × 1) measurement vector that contains the measured normalized fluxes and normalized enzyme levels (a = (J0e')-1 v - i), b the (p × 1) vector that contains the non-zero elasticities of the original elasticity matrices Ex and Ec, and Y is the (r × p) design matrix of which each ith row contains as nonzero elements ln(xj/ck/b. The equation (4) can be solved to obtain the elasticities using linear regression according to:where b = (YTY)-1 YT·a     (5)xj/ck/Y) enzyme activities (presented in matrix e') and steady state fluxes (presented in the vector v and the matrix J0).This shows that estimation of elasticities from steady state perturbations requires data on metabolite levels (ln(xi's) in treating dynamic perturbation data. The elasticities are, in this case, estimated via non-linear parameter estimation procedure, in which the objective function to be minimized is the weighed squared error between the experimental response and simulation results. The general form of the objective function is given in equation(6):Since in a dynamic perturbation experiment, the rate information can not be obtained directly, we will follow different procedure here and we will only make use of the time profiles of the measured metabolites , χexp is the experimentally measured, χsim is the simulated (using linlog kinetics) metabolite matrix containing intracellular and extracellular metabolite concentrations. Two additional parameters are introduced here:the limits of the summations, γ to weigh the effect of uncertainty in different metabolites and λ to weigh the effect of two different sources of data, namely steady state and dynamic perturbation data. In the second term of the equation(6), J'sim and J'exp represent the experimental and simulated steady state fluxes. The equation (6) contains the rate information implicitly, since the χsim results from the integration of the set of ode's representing the dynamics of the system. Note also that although we already explained the treatment of the steady state perturbation data in the previous section, we explicitly included the second term in equation (6) again; to state clearly that this form of the objective function allows also the integration of data from different types of experiments.et al. presented a comparison of global optimization methods used for parameter estimation in biochemical pathways  in equation(3)) defines the number of elasticities to be estimated. Specifically, the elasticity matrix has 72 entries (8 reactions × 9 metabolites) and it followed that 16 of these were nonzero for the Galazzo and Bailey model which was extended with the alterations on the uptake reaction, mentioned in the Appendix. For the present network, the structure of the elasticity matrix is presented in Figure The number of non-zero entries of the elasticity matrix ([ext) at t = 0 from 2666 μmol L-1 to 8000 μmol L-1, where after the relaxation of the system was simulated. The levels of the metabolites were recorded until the system returned to the reference state.To obtain transient data, the reference steady state Table  was pertth minute until the 10th minute, 5 samples were taken, and from the 10th minute until the end (400 min) 30 uniformly distributed samples were taken. Metabolites for which concentrations were assumed to be available at these times are intracellular glucose, G6P, FdP, PEP, ATP and extracellular metabolites glucose, ethanol, glycerol and polysaccharides.In choosing the number of data points, i.e. the sampling frequency, special attention had been paid to use a realistic number of data points that an experimenter can obtain from a typical rapid sampling experiment (for a short time period) or from a dynamic pulse experiment (for a longer time period). The data were obtained as follows: in the first 120 seconds, where the initial dynamics after the glucose uptake are important, 20 samples uniformly distributed over time were taken, in the next 120 seconds, 10 samples and in the last minute 5 samples were taken. For long term experiments, from the 5For the steady state perturbations, the enzyme levels were modulated and the system was allowed to reach a new steady state. The new steady state conditions  were measured.PK and VGol reactions that have three non-zero elasticities (E9–E10–E11 and E13–E14–E15 respectively).The minimum number of independent steady state perturbations needed in order to identify all the elasticities for a reaction equals the number of non-zero elasticities for that rate expression. From Figure IN was chosen because it is at the beginning of the glycolysis and therefore affects the entire pathway. As the second reaction to be perturbed, VATPase was chosen because it directly affects the level of ATP which is the effector of 6 out of 8 reactions. Lastly, VGAPD was chosen for two reasons: 1) to resolve the branch point relations around G6P and 2) to resolve the feed forward effect of FdP on VPK. Note that since the production rate of glycerol (VGol) was coupled  to the production of ethanol (VPK), the steady state branch point relations around FdP can be reformulated as follows:As the reaction to be perturbed, VJPFK = JGAPD + JGol and JGAPD = JPK/2. Additionally, as described above, JGol = k·JPK, hence JPFK = (1 + 2·k)·JGAPD where the value of k is given in the Appendix. Therefore, there is no need to introduce additional steady state perturbations to resolve the branch point relations around FdP. Table ATPase by 10% and the third represents the data after the inhibition of VGAPD by 20%.At steady state, e JPFK =  + 2·k·JGIn order to inspect whether the estimated elasticities are robust against experimental noise in metabolite concentrations, we performed Monte Carlo (MC) simulations. To this end, random errors of 10% are repeatedly added to form 50 sets of "noisy data", which were used in the non-linear parameter estimation procedure. This yields a distribution of estimated elasticities which allows calculating the relative standard deviation for each elasticity as the ratio of standard deviation to the mean of that elasticity distribution.μ/ was used for generating the IEN set up the methodology, performed the calculations and drafted the manuscript. WvW supervised the calculations, the organization of the manuscript and revised the manuscript. WvG contributed in organizing and revising the manuscript. JJH supervised the whole project. All of the authors read and approved the final manuscript.intracellular, G6P, FdP, PEP and ATP), 4 extracellular  metabolites and 8 reactions. In their work, a mathematical model for the description of ethanol production by non-growing Saccharomyces cerevisiae has been constructed using non-linear mechanistic rate equations. The glucose uptake is modeled by symmetrical carrier model (see below for the changes in the uptake reaction), the hexokinase reaction is described by a two substrate single displacement mechanism, the phosphofructokinase and pyruvate kinase reactions are described by allosteric kinetics according to Monod-Wyman-Changeux, polysaccharide formation is explained by Hill type kinetics and the glyceraldehyde 3-phosphate dehydrogenase reaction is described by a kinetic expression which takes into account the crossed product inhibition by G3P and NAD+ and competitive inhibition by AMP, ADP and ATP. The ATP consumption is assumed to have first order kinetics.In this paper, the model of Galazzo and Bailey describing the glycolysis of yeast cells is used. This section describes the details of the metabolic model. The model consists of 5 intracellular (glucose+ and NADH levels are assumed to be constant, as given below.One property of the original model is that the glycerol production is assumed to be stoichiometrically coupled to ethanol formation, i.e. there is no separate dynamics for the glycerol formation. Additionally, the NADAs it was originally published, the glucose influx was a linear function of the glucose-6-phosphate concentration only; hence, it was insensitive to the changes in extracellular glucose concentration. To mimic realistic fermentation conditions, where one can perturb the organism by adding a specific compound such as glucose to a chemostat, we have modified the glucose uptake reaction and modeled the glucose uptake by a symmetrical carrier model, suggesting that the transport of glucose across the cell membrane occurs via facilitated diffusion, as proposed by Teusink et al., . The lisThe rate equations of the reactions in Figure withR = 1 + F6P + 16.67 ATP + 16.67F6P·ATPT = 1 + 5.10-4 F6P + 16.67 ATP + 0.00833F6P·ATP0 is pH dependentwithζi,i=1,2,3 = AMP, ADP, ATP     Ki,i=1,2,3 = 1.1, 1.5, 2.5withR = 1 + 159.88PEP + 0.2ADP + 3.1976PEP·ADPT = 1 + 0.02PEP + 0.2ADP + 0.004PEP·ADP0 pH dependentVGol = 0.06742·VPKVATPase = 12.1·ATPThe following metabolites were assumed in equilibrium throughout the simulated time window:The concentrations of the extracellular metabolites are described by the following mass balances:


Translating a known metabolic network into a dynamic model requires rate laws for all chemical reactions. The mathematical expressions depend on the underlying enzymatic mechanism; they can become quite involved and may contain a large number of parameters. Rate laws and enzyme parameters are still unknown for most enzymes.We introduce a simple and general rate law called "convenience kinetics". It can be derived from a simple random-order enzyme mechanism. Thermodynamic laws can impose dependencies on the kinetic parameters. Hence, to facilitate model fitting and parameter optimisation for large networks, we introduce thermodynamically independent system parameters: their values can be varied independently, without violating thermodynamical constraints. We achieve this by expressing the equilibrium constants either by Gibbs free energies of formation or by a set of independent equilibrium constants. The remaining system parameters are mean turnover rates, generalised Michaelis-Menten constants, and constants for inhibition and activation. All parameters correspond to molecular energies, for instance, binding energies between reactants and enzyme.Convenience kinetics can be used to translate a biochemical network – manually or automatically - into a dynamical model with plausible biological properties. It implements enzyme saturation and regulation by activators and inhibitors, covers all possible reaction stoichiometries, and can be specified by a small number of parameters. Its mathematical form makes it especially suitable for parameter estimation and optimisation. Parameter estimates can be easily computed from a least-squares fit to Michaelis-Menten values, turnover rates, equilibrium constants, and other quantities that are routinely measured in enzyme assays and stored in kinetic databases. Dynamic modelling of biochemical networks requires quantitative information about enzymatic reactions. Because many metabolic networks are known and stored in databases ,2, it woA ↔ B. At substrate concentration a and product concentration b (measured in mM), the reaction rate readsA prominent example of a saturable kinetics is the reversible form of the traditional Michaelis-Menten kinetics  for a reE, turnover rates -1), the shortcuts ã = a/b/kM was a dissociation constant. Later, Briggs and Haldane presented a different derivation that assumes a quasi-steady state for the enzyme-substrate complex and defines kM as the sum of rate constants for complex degradation, divided by the rate constant for complex production, kM = (k-1 + k2)/k1. Other kinetic laws have been derived from specific molecular reaction mechanisms )/[EA] describes the balance of bound and unbound A in chemical equilibrium and can be computed from the Gibbs free energy (in kJ/mol)The reaction proceeds from left to right; the free enzyme RT ≈ 2.490 kJ/mol.with EA] = ã [E0], [EX] = E0], [EAX] = ãE0]. By analogy, we obtain expressions for the product complexes on the right hand side: [EB] = E0], [EY] = E0], [EBY] = E0]. The total enzyme concentration E is the sum over the concentrations of all enzyme complexesWe now make a simplifying assumption: the binding energy of A does not depend on whether X is already bound. With analogous assumptions for binding of X and with the abbreviations EAX and EBY with forward and backward rate constants We next assume a reversible conversion between the complexes kM are actually dissociation constants, related to the binding energies between reactants and enzyme. The terms in the denominator represent the enzyme complexes in the reaction scheme shown above. Equation (8) also shows why the term -1 in formulae (2) and (3) is necessary: the two product terms in the denominator represent all complexes shown in the reaction scheme. However, when summing up the terms from both sides, we counted the free enzyme E0 twice, so we have to subtract it once.which is exactly the convenience rate law (2). The derivation has shown that the turnover rates The same kind of argument can be applied to reactions with other stoichiometries; let us consider a reaction with the left-hand side 2 A + X ↔ ...EAAX gives rise to the first term ã2The substrate complex ã + ã2) and .The activation and inhibition terms in the prefactor can also be justified mechanistically: in addition to binding sites for reactants, the enzyme contains binding sites for activators and inhibitors. Only those enzyme molecules to which all activators and none of the inhibitors are bound contribute to the reaction mechanism; all other enzyme molecules are inactive. Again, we assume that the Gibbs free energies for binding do not depend on whether other modifiers are bound, and they determine the j in eqn. (3) vanish and we obtain the irreversible rate lawTo define a convenience kinetics for irreversible reactions, we assume that all product constants kM value in Michaelis-Menten kinetics (1) has a simple mathematical meaning: it denotes the substrate concentration that leads to a half-maximal reaction velocity if the product is absent. A similar rule holds for the substrate and product constants in convenience kinetics. Let us first assume that all stoichiometric coefficients are ±1; if the product concentrations vanish (bj = 0), then rate law (2) can be factorised intoBesides being a dissociation constant, the am are kept fixed, the rate law readsIf in addition, all substrate concentrations except for a certain am → ∞, the fraction approaches 1, while for am = For αm = 2, we obtain the velocityWhat if the stoichiometric coefficient is larger than one? Applying the same argument for am = αi, we can conclude: at am = αi.At N and regulation matrix W , it is practical to arrange the kinetic parameters in vectors and matrices. The enzyme concentration of a reaction l reads El, and the turnover rates are called nil ≠ 0) comes with a value wli = 1) and inhibition (wli = -1) are quantified by values kM, kA and kI values for non-existing interactions (where nil = 0 or wli = 0) remain unspecified or can be assigned a value of 1, i.e., a logarithmic value of 0.To parametrise an entire metabolic network with stoichiometric matrix c, the convenience kinetics can now be written asWith metabolite concentrations arranged in a vector N+ = (N- = (N. The matrices W+ and W- are derived from W in the same way.with the abbreviation c and in the structure matrices N or W. (ii) A reaction is always catalysed by a specific enzyme; we describe isoenzymes by distinct reactions. (iii) If the sign of a regulatory interaction is unknown, we may consider terms for both activation and inhibition. (iv) To describe indirect regulation, e.g. by transcriptional control, the production and degradation of enzymes has to be modelled explicitly by chemical reactions.Let us add some remarks, (i) It is common to describe some of the metabolite concentrations by fixed values rather than by a balance equation. In the present framework, these metabolites are included in the concentration vector l is defined asThe convenience kinetics (14) has a major drawback: its parameters are constrained by the second law of thermodynamics. The equilibrium constant of reaction ceq is a vector of metabolite concentrations in a chemical equilibrium state. By setting eqn. (3) to zero, we obtain the Haldane relationship  and from a log-uniform distribution on the same interval. Figure In contrast to the convenience rate law (8), the denominators contain mixed terms between substrates and products, and in the ping-pong kinetics, the term +1 is missing. The ordered mechanism yields smaller reaction rates than the ping-pong and the convenience kinetics because its denominator is always larger. To compare the three rate laws, we sampled metabolite concentrations and kM values, kI values, kA values, and turnover rates, and merge their logarithms in a large vector x*. The vector can contain multiple values for a parameter, it can contain thermodynamically dependent parameters, and of course, many parameters from the model will be missing. We try to determine a vector θ of logarithmic system parameters that yields a good match between the resulting parameter predictions x (θ) and the data x*. Solving x* ≈ θ for θ by the method of least squares yields an estimate of the system parameters. Using eqn. (23) again, consistent values of all kinetic parameters can be computed from the estimated system parameters. Contradictions in the original data are resolved; in addition, we can employ a prior distribution representing typical parameter ranges to compensate for missing data. A more general estimation procedure, which can also integrate measured metabolic concentrations and fluxes, is described in the companion article [The parameters in convenience kinetics – the independent and the resulting dependent ones – can be measured in experiments. The linear relationship (23) makes it particularly easy to use such experimental values for parameter fitting: given a metabolic network, we mine the literature for thermodynamic and kinetic data, in particular Gibbs free energies of formation, reaction Gibbs free energies, equilibrium constants,  article .kM, kA, and kI represent concentrations that lead to half-maximal -1 -maximal) effects: the kM values also indicate the threshold between low substrate concentrations that lead to linear kinetics and high concentrations at which the enzyme works in saturation.Convenience kinetics can be used for modelling biochemical systems in a simple and standardised way. In contrast to ad-hoc rate laws such as linlog or generalised mass-action kinetics, the convenience kinetics is biochemically justified as a direct generalisation of the Michaelis-Menten kinetics; it is saturable and allows for activation and inhibition of the enzyme. The parameters kM, kA, and kI values represent dissociation constants between the enzyme and the reactant or modifier, while kV represents the velocity of the transformation step. The system parameters also provide a sensible basis for describing variability in cell populations: the Gibbs free energies of formation depend on the composition of the cytosol, for instance its pH and temperature, and can be expected to show small, possibly correlated variations. The remaining parameters reflect interaction energies, which depend on the enzyme's amino acid sequence; we can expect that these energies vary between cells, and probably more independently than, for instance, the forward and backward turnover rates.The convenience kinetics represents a rapid-equilibrium random-order enzyme mechanism. When all substrates are bound, they are converted in a single step into the products, which then dissociate from the enzyme. The The convenience kinetics does not differ strikingly from established kinetic laws: in a comparison with the ordered and ping-pong mechanisms, the convenience kinetics resembled the ping-pong mechanism, and the similarity between them was greater than that between the ordered and ping-pong mechanisms. Mathematically, the three rate laws differ in their denominators: in convenience kinetics, we find all combinations of substrate concentrations and all combinations of product concentrations, but no mixed terms containing both substrate and product concentrations. The single terms reflect the reactant complexes formed by the enzyme.keq by Gibbs free energies of formation or we choose a set of independent equilibrium constants. This trick is of course not limited to the convenience kinetics: independent parameters and equations of the form (23) can also be used with many other kinetic laws, in particular those that share the denominator of the convenience rate law; also other modes of activation and inhibition can be treated in the same manner as long as the modifiers do not affect the chemical equilibrium.The second concern of this paper was the incorporation of thermodynamic constraints: in pathway-based methods -23, propThe choice of rate laws and parameter values is a main bottleneck in kinetic modelling. Standard rate laws such as the convenience kinetics can facilitate the automatic construction and fitting of large kinetic models. For transcriptional regulation, a general saturable law has been proposed . For metIn kinetic modelling, every chemical reaction has to be characterised by a kinetic law and by the corresponding parameters. The convenience kinetics applies to arbitrary reaction stoichiometries and captures biologically relevant behaviour  with a small number of free parameters. It represents a simple molecular reaction mechanism in which substrates bind rapidly and in random order to the enzyme, without energetic interaction between the binding sites. The same holds for the dissociation of products.For reactions with a single substrate and a single product, the convenience kinetics equals the well-known Michaelis-Menten kinetics. By introducing a set of thermodynamically independent system parameters, we obtained a form of the rate law that ensures thermodynamic correctness and is notably suited for parameter fitting and optimisation.N and W. The coefficients nil contained in the stoichiometric matrix N describe how many molecules of type i are produced in reaction l; negative elements describe consumption of molecules. The elements of the regulation matrix W describe enzyme regulation between metabolites i and enzymes l: wli = 1 indicates activation, wli = -1 represents inhibition, and wli = 0 no interaction.The structure of a metabolic network is defined by the lists of metabolites and reactions and by two structural matrices, In the setting of deterministic differential equations, the substance concentrations in a biochemical system follow the balance equationsc, v, and k contain the metabolite concentrations (in mM), reaction velocities (in mM/s), and system parameters, respectively. External or buffered metabolites with fixed concentrations are contained in the parameter vector k.The vectors W) to the reaction kinetics, we first assume a hypothetical kinetic law without regulation; in this law, the reaction velocity depends only on the substrate and product concentrations. In the real rate law, a metabolite is an activator if (i) it increases the rate although it is not a reactant, or (ii) it increases the rate more strongly than it would by just being a reactant. Inhibition is defined analogously.To relate activation and inhibition  are constrained by fundamental thermodynamic laws that relate the metabolite concentrations in steady state to molecular energies [l changes the Gibbs free energy of the system byThe kinetic laws energies . A singlμi denotes the chemical potential of metabolite i (in kJ/mol). In an ideal mixed phase at pressure P and absolute temperature T, the chemical potential of substance i with concentration ci readswhere the sum runs over all metabolites and R ≈ 8.314 J/(mol K) is Boltzmann's gas constant. In (34), the ci are dimensionless numbers denoting concentrations in mM. In real mixed phases, there would be an additional term +RT ln fi. We neglect this term, assuming an ideal mixed phase without mixture effects on volume or energy; we also neglect effects of changing pressure or electric charges.where l is defined asThe equilibrium constant of reaction ceq is the vector of metabolite concentrations in a chemical equilibrium state. According to the second law of thermodynamics, the equilibrium state of a chemical system is characterised by a minimum of the Gibbs free energy. This implies that each chemical reaction in equilibrium satisfies ΔGl = 0. From eqs. (33), (34), and (35) followswhere where P = 1.015 bar and T = 298.15 K. Equations (36) and (37) constitute the relation (18) between equilibrium constants and the Gibbs free energies of formation.in terms of the Gibbs free energies of formation N and join them in a matrix N into a matrix product N = N = L NR that is used in metabolic control analysis to remove dependent metabolites.Dependencies between equilibrium constants can be treated in a similar manner to the linear dependencies that constitute the conservation relations between metabolites . To chooTo be thermodynamically feasible, the equilibrium constants have to satisfy eqn. (18) or, in vector form,keq = -NT ln kG     (38)ln kG. Let us first assume that kG is given; then the equilibrium constants of the independent reactions readfor at least one choice of the vector kind = -T ln kG,     (39)ln and with the definitionwe can writekind are indeed thermodynamically independent; or in other words, that for any vector ln kind, there exists a vector kG such that eqn. (39) holds; and this is indeed the case because T has full row rank.Hence, eqn. (27) is satisfied and the matrix The authors declare that they have no competing interests.Both authors conceived the convenience kinetics and carried out calculations. W.L. wrote the manuscript, E. K. identified the underlying molecular mechanism and revised the manuscript. Both authors read and approved the final manuscript.The supplement file contains a list of the mathematical symbols used, the derivation of equation (22), and a detailed explanation of the sensitivity matrix Click here for file