This comparison results in two types of model-prediction inconsistencies; either the model predicts growth when no growth is observed in the experiment (GNG inconsistencies) or the model predicts no growth when the experiment reveals growth (NGG inconsistencies). Here we propose an optimization-based framework, GrowMatch, to automatically reconcile GNG predictions  and NGG predictions . We use GrowMatch to resolve inconsistencies between the predictions of the latest in silico Escherichia coli (iAF1260) model and the in vivo data available in the Keio collection and improved the consistency of in silico with in vivo predictions from 90.6% to 96.7%. Specifically, we were able to suggest consistency-restoring hypotheses for 56/72 GNG mutants and 13/38 NGG mutants. GrowMatch resolved 18 GNG inconsistencies by suggesting suppressions in the mutant metabolic networks. Fifteen inconsistencies were resolved by suppressing isozymes in the metabolic network, and the remaining 23 GNG mutants corresponding to blocked genes were resolved by suitably modifying the biomass equation of iAF1260. GrowMatch suggested consistency-restoring hypotheses for five NGG mutants by adding functionalities to the model whereas the remaining eight inconsistencies were resolved by pinpointing possible alternate genes that carry out the function of the deleted gene. For many cases, GrowMatch identified fairly nonintuitive model modification hypotheses that would have been difficult to pinpoint through inspection alone. In addition, GrowMatch can be used during the construction phase of new, as opposed to existing, genome-scale metabolic models, leading to more expedient and accurate reconstructions.Genome-scale metabolic reconstructions are typically validated by comparing  Escherichia coli model with data available at the Keio database. Despite the highly curated nature of the Escherichia coli model, GrowMatch identified and resolved a large number of model prediction inconsistencies by taking advantage of available compilations of experimental data.Over the past decade, mathematical models of cellular metabolism have been constructed for describing existing metabolic processes. The gold standard for testing the accuracy and completeness of these models is to compare their cellular growth predictions  across different scenarios with available experimental data. Although these comparisons have been used to suggest model modifications, the key step of identifying these modifications has often been performed manually. Here, we describe an automated procedure GrowMatch that addresses this challenge. When the model overpredicts the metabolic capabilities of the organism by predicting growth in contrast with experimental data, we use GrowMatch to restore consistency by suppressing growth enabling biotransformations in the model. Alternatively, when the model underpredicts the metabolic capabilities of the organism by predicting no growth  in contrast with available data, we use GrowMatch to restore consistency by adding growth-enabling biotransformations to the model. We demonstrate the use of GrowMatch by reconciling growth prediction inconsistencies of the latest  Escherichia coliSaccharomyces cerevisiaeHomo Sapiens metabolic map There are currently 700 completely sequenced genomes along with extensive compilations of data in vivo, down-regulation or inhibition of genes/enzymes under the experimental conditions, or absence of biomass constituents from the in silico biomass description. Conversely in NGG cases, the model under-predicts the metabolic capabilities of the organism due to the absence of relevant functionalities/reactions in the model. In this study, we introduce optimization-based techniques to systematically suggest modifications  in genome-scale metabolic reconstructions in order to reconcile experimental and computational growth predictions across different mutants.As shown in E. coli K-12 BW25113. Several studies are already available that use gene essentiality data available at the Keio database and other sources to suggest targeted improvements in existing metabolic reconstructions in silico models of increasing complexity were successively contrasted against in vivo datasets of differing size to correct the predictive capabilities of the models. Recently, Joyce et al. in vivo in a glycerol supplemented minimal medium and then compared them with the corresponding in silico predictions to suggest improvements in the model in vivoin silico models, the key step of resolving these gaps was performed manually.The proposed method makes use of gene essentiality data sets currently available for many microorganisms E. coliin silico growth consistent with in vivo data across various carbon/nitrogen substrates. Alternatively, methods to identify and fill gaps in metabolic models based on connectivity information have also been described and applied to the genome scale models of E. coli and S. cerevisiaeThe need to develop automated procedures to improve the accuracy of existing metabolic reconstructions has been recognized and has led to the development of a number of computational procedures. To this end, Reed et al. global suppression meaning that it can be imposed universally for all experimental perturbations . Alternatively, if an identified restriction clashes with one or more GG predictions then it is referred to as a conditional suppression meaning that it is imposed only in the mutant strain associated with the GNG prediction for which it is correcting.In this paper, we supplement previous efforts global additions; otherwise, we refer to them as conditional additions. In the next section we discuss the results obtained by applying GrowMatch to the most recent genome-scale model of E. coli, iAF1260 E. coli reconstruction was chosen as the focus of this study to benchmark the ability of GrowMatch to identify model corrections even for a very well curated model. Using GrowMatch, we improved the growth prediction consistency of the iAF1260 model with the data available at the Keio database from 90.6% to 94.6% when considering only globally valid corrections and to 96.7% when additionally considering conditional corrections.Similarly, NGG inconsistencies are corrected one-by-one to GG by identifying the minimal set of model modifications  that enable biomass formation (above a pre-specified cutoff). If none of these modifications affect any of the consistent NGNG cases, we refer to them as in silico model of E. coliin silico growth on minimal glucose medium with the in vivo OD measured after 48 hours on minimal glucose. To account for the genetic differences between MG1655 (the strain used to construct the in silico model) and BW25113 (the strain used in the in vivo study), we eliminated five reactions from the in silico model  that are associated with genes (araBAD and rhaBAD) not present in the BW25113 strain. Characterizing a single gene-deletion mutant as a ‘Grow’ (G) or a ‘No-Grow’ (NG) mutant requires a cutoff for the computed (for the in silico model) and observed (for the in vivo experiment) values of growth. In this study, we adopted as the growth cutoff (i.e. in silico side and in vivo side) the one proposed in the recent study by Joyce and co-workers average growth exhibited by all the single gene deletions under consideration. We use the same growth cutoff definition for both in vivo and in silico mutant classifications. For the in vivo growth classifications, we determined the growth cutoff using the data in the Keio database. For mutants with no OD measurements available, we checked the essentiality scores .Here, we demonstrate the use of GrowMatch to resolve growth prediction inconsistencies between the latest in silico/in vivo growth predictions. Almost 70% of these inconsistencies are GNG implying that the iAF1260 model, when in error, tends to over rather than under-predict the metabolic capabilities of E. coli. Note that all the abbreviations used in this section are identical to the ones used in the in silico model of E. coliin silico that are unavailable in vivo. in vivo these deletions are fatal. In these cases, we hypothesize that the in silico growth can be negated by simply deactivating the reaction that is catalyzed by the corresponding isozymes. In fifteen out of the twenty-two cases, the suppression of the isozymes  negates growth thus converting the GNG mutants into NGNG mutants. It appears that in vivo, under the specific experimental conditions (aerobic glucose), the alternative isozyme does not exhibit sufficient activity to restore the activity of the deleted isozyme. Note that all these reaction suppressions are conditional suppressions as the reactions are essential for growth in all GG mutants. We define complementary (non-complementary) isozymes as pairs of isozymes that satisfy the following two conditions: (a) at least one of the isozymes is encoded by a gene associated with a GG (GNG) mutant and (b) the isozymes catalyze an essential reaction (under aerobic glucose conditions). We checked the sequence similarity of complementary and non-complementary isozymes using the BlastP algorithm. The results are available in thrA, metL] and ) as all these genes are associated with GNG mutants. The 95% confidence intervals  for this expression data are tabulated in To see if the genes that code for non-complementary isozymes are inactive under aerobic minimal glucose, we checked their expression levels. Specifically, we examined the relative expression levels for these pairs of genes (deleted gene and gene associated with non-complementing isozyme) available at Covert et al., ΔmetL and ΔthrA, can be eliminated by suppressing any of the two associated essential reactions, aspartate kinase (ASPK) or homoserine dehydrogenase (HSDy)  see . TherefoΔubiG, ΔuxaB) correspond to reactions that are conditionally blocked for a glucose minimal medium  encode for enzymes that catalyze blocked reactions in the metabolic network. Blocked reactions are defined as reactions that cannot carry any flux under given substrate conditions ΔbioB, ΔbioD, ΔbioF, ΔcaiT, ΔalsB, Δint) can be restored by adding components produced by the corresponding reactions to the biomass equation  are resolved by making global modifications. Also, seventeen of these 26 GNG mutants correspond to reactions involved in tRNA charging reactions. GrowMatch converted these seventeen GNG mutants into NGNG mutants by modifying the biomass equation by explicitly including the charged and the uncharged tRNA molecules in place of the amino acids. For example, in the GNG mutant ΔleuS, the deleted reaction LEUTRS  is blocked. This reaction is unblocked by including leutrna (charged tRNA) and trnaleu (uncharged tRNA) as a reactant and product in the biomass equation, respectively. This restores flux through the reaction LEUTRS and converts ΔleuS into an NGNG mutant. We note that the consistency of these seventeen GNG mutants is restored by making global modifications, as adding these components to biomass does not affect any correct model predictions. For the remaining three GNG mutants, we first attempted to restore flow connectivity using (GapFill) before using GrowMatch. However, GapFill was unable to restore flow through any of these reactions by filling functionalities using reactions from the multi-organism databases of MetaCyc GrowMatch resolved 23 of these 26 inconsistencies by suitably adding biomass components to the biomass equation. Specifically, consistency to six GNG mutants  are resolved by suppressing two additional reactions in the network. Also, for ten of these GNG mutants, GrowMatch identified alternative suppression candidates  biosynthesis, both of which are essential metabolites for biomass formation. GrowMatch restores consistency in ΔglyA either by suppressing serine production  or alternatively by disabling mlthf production (by suppressing the Glycine Cleavage System). In ΔserA and ΔserB, GrowMatch suggests blocking serine production by disallowing the reversibility of glycine hydroxymethyltransferase (glyA)  . AlternaΔcarA and ΔcarB. These genes encode for a multi-domain protein that catalyzes the reaction carbamoyl phosphate synthase (CBPS) (glutamine-hydrolysing), which is involved in the production of carbamoyl-phosphate. As shown in ΔcydC by suppressing GLYAT and GLYCL (Glycine Cleavage System) to prohibit biomass formation  and S-Adenosyl-L-methionine (amet), are not produced in this mutant network. Closer investigation reveals that the reaction uroporphyrinogen methyltransferase, which is a reaction that consumes amet and is involved in the siroheme biosynthesis pathway, cannot carry any flux when these suppressions are carried out in ΔcydC. This results in no production of these biomass precursors resulting in zero biomass formation in silico. All the examples highlighted above lead to model modification that would have been difficult to come up with by inspection without the aid the alternatives provided by GrowMatch.ormation . Note thE. coli K12 genome. These results are summarized in −13) with other open reading frames in E. coli. For example, the gene argD whose deletion results in a NGG mutant, shares high sequence similarity with astC . Also, the gene aspC whose deletion results in a NGG mutant, shares a high sequence similarity  with tyrB, which transcribes to form a subunit of tyrosine aminotransferase. Hence, it is possible that it encodes for the activities of these genes in the respective NGG mutants in vivo thereby preserving growth.Restoring growth for the NGG predictions requires that production routes be established in the metabolic model for all 63 precursor metabolites to biomass. ΔluxS, had alternative means of consistency restoration, one by adding reactions and the other by allowing the secretion of a metabolite. Three (including ΔluxS) were resolved by adding reactions from KEGG and MetaCyc ΔluxS) by allowing the secretion of metabolites from the cell into the extracellular space. None of the inconsistencies could be resolved by modifying the directionality of existing reactions in the model.We next use GrowMatch to resolve the NGG inconsistencies by adding pathways using one or more of the three mechanisms discussed previously. GrowMatch identified consistency-restoring hypotheses for 5/38 mutants. Interestingly, one NGG mutant single reactions from the multi-organism databases of KEGG and MetaCyc. Specifically, ΔluxS is corrected by adding the reaction putative adenosylhomocysteinase (from the organism Rhizobium leguminosarum) and Δasd is corrected by adding the reaction catalyzed by Protein APA1 (from the organism Saccaromyces cerevisiae). We note, however, that proteins catalyzing these reactions have low sequence similarity with the E. coli K12 genome (BLAST score = 28.1 bits with gene product of ybcK and 29.6 bits with gene product of yshA respectively) and that the validity of these hypotheses, like all those generated by GrowMatch, must be explored experimentally. Consistency in one NGG mutant (ΔcysN) is achieved by adding the reaction catalyzed by sulfate adenylyltransferase, the activity of which is documented in EcoCyc but was not included in the iAF1260 reconstruction global additions.The first three NGG resolutions were corrected by adding ΔfolD is resolved by allowing the secretion of 3,4-dihydroxy-2-butanone 4-phosphate that serves as the biosynthetic precursor for the xylene ring of riboflavin. Glycolaldehyde and S-ribosyl-L-homocysteine are reactants in the reactions catalyzed by aldA and luxS respectively. To resolve the NGG mutants ΔaldA and ΔluxS, GrowMatch hypothesizes the presence of secretion mechanisms (currently absent from the model) for glycolaldehyde and S-ribosyl-L-homocysteine, respectively (E. coli−4 mmol/gDW hr), it is possible that its toxic accumulation is prevented either by the (possibly non-specific) activity of a transporter that is already present or by its diffusion out of the cell.The other three resolutions see  are all ectively . Interesin silico/in vivo growth prediction inconsistencies in single gene-deletion mutants. In GNG mutants, GrowMatch restores consistency by suppressing reactions to prohibit growth. In NGG mutants, GrowMatch restores consistency by adding growth-enabling pathways. We demonstrated this procedure by reconciling the growth prediction inconsistencies between the most recent in silico model of E. coli, iAF1260 in vivo growth data available at the Keio mutant collection E. coli, we were able to improve the consistency from 90.6% to 94.6%. In addition, by carrying out conditional modifications in the specific mutants, we further improve the overall consistency in growth predictions to 96.7%. Moreover, specificity has been recently proposed to be an important measure to determine the effectiveness of in silico simulations as a screen in computational gene essentiality predictions Here we have developed an automated procedure, GrowMatch, to resolve ΔluxS) had alternative means of consistency restoration.GrowMatch resolved eighteen GNG inconsistencies by suggesting suppressions in the mutant metabolic networks whereas fifteen inconsistencies were resolved by suppressing isozymes in the metabolic network. The remaining 23 inconsistencies corresponding to blocked genes were repaired by simply adding component(s) of the associated blocked reactions to the biomass equation . GrowMatE. coli but absent in the in silico model iAF1260 aldA and epd that were not associated with them in the iAF1260 model . GrowMatch also resolved two NGG mutants by indirectly preventing the toxic accumulation of metabolites. Surprisingly, in the case of NGG mutants, none of the resolutions were achieved by allowing the reversibility of irreversible reactions in the model. This result is in contrast to previous results in which a large proportion of connectivity problems in the previous version of the E. coli genome-scale model were resolved by expanding reversibility of reactions in the model E. coli model In this study, we were able to pinpoint missing functionalities that may have been overlooked during model reconstruction. In one such example, were able to resolve a NGG mutant by adding a reaction  with documented evidence of its being present in in silico models E. coli with data available in the MetaCyc and EcoCyc databases In line with recent explanations for GNG inconsistencies in in silico growth prediction consistency with the available data.It is important to note that GrowMatch makes use of parsimony criteria to prioritize alternative model correcting hypotheses. Therefore, biologically relevant hypotheses that involve more than the selected maximum allowed limit of model modifications will be missed. Also, using alternate cellular objectives such as MOMA E. coli model presented here, our group has recently used it  during the construction phase of the genome-scale metabolic model of Mycoplasma genitalium iPS189.In summary, we believe that GrowMatch, in conjunction with GapFill, are useful model-refinement tools during the reconstruction of new metabolic models or testing/curation of existing ones. In addition to the use of GrowMatch to restore growth inconsistencies for the latest i|i = 1, 2… M},{j|j = 1, 2… N} and {k|k = 1, 2… K} that span the M metabolites, N reactions and K genes, respectively present in the metabolic network. Furthermore, we define the index set {l|l = 1, 2… L} to represent the L in vivo experiments under consideration. Set lKO is defined to include genes that are knocked out in experiment l. We define a set Model to include all reactions in the existing genome-scale metabolic reconstruction. We maximize the formation of biomass subject to the available substrate feed and mass balance constraints implied by the stoichiometric model. . The in silico predictions are then compared with in vivo data. ijS is the stoichiometric coefficient of metabolite i in reaction j and parameters j to genes k as follows:First, we define the sets, parameters and variables that are common to the mathematical procedures formulated to resolve NGG and GNG inconsistencies. To this end, we define the index sets, {k1 and k2 for reaction j then k1 and k2 then These definitions imply that if there exists two isozymes jUB and LBj, were chosen not to exclude any physiologically relevant metabolic flux values. The upper bound for all reactions was set to 1,000. Unless specified otherwise, the lower bound was set equal to zero for irreversible reactions and to −1,000 for reversible reactions. The flux in reaction j is denoted by variable jv and is restricted to vary between lower and upper bounds jLB and jUB, respectively. Using these definitions,we will now discuss the mathematical procedures developed to resolve GNG and NGG inconsistencies.Upper and lower bounds, in vivo. This could be due to the erroneous presence in the model of pathways that produce biomass precursor metabolites. The aim here is to identify the minimum number of suppressions that need to be imposed for a given experiment l* corresponding to a GNG mutant to ensure that the maximum biomass formation is zero. These suppressions are carried out by either (a) restricting flux in transport/ intracellular reactions or (b) restricting the reversibility of reactions defined as reversible in the model. The description of these suppressions requires the definition of the binary variable jy to pinpoint them in the network.A GNG single gene deletion mutant occurs when the model predicts growth whereas no growth is observed in vivo experiment l* are identified by solving the following bilevel optimization problem GrowMatch:The suppressions required to ensure that the maximum biomass formation is below the imposed cut-off min-max formulation. Specifically, the inner optimization problem identifies the maximum possible amount of biomass formation by redirecting metabolic fluxes subject to stoichiometry, uptake and ATP maintenance. The outer optimization problem minimizes biomass formation by choosing a pre-specified number n* of reactions in the network to suppress. A zero objective function value implies that the n* selected reaction suppressions  successfully prevent the network from forming biomass. This converts the GNG occurrence for in vivo experiment l* into NGNG restoring consistency of prediction. Alternative ways of restoring prediction consistency can be obtained by imposing successive integer cuts global suppressions. On the other hand, if any of these suppressions restrict biomass production in any of the GG mutants, they are referred to as conditional suppressions. The identified set of suppressions  is finally tested by contrasting them against literature evidence regarding the presence or absence of activity of the suppressed reaction under the experimental conditions.The aim of GrowMatch is to identify the minimal number of reaction suppressions needed to zero the maximum biomass formation. We do this by ensuring that there is no biomass formation even when fluxes in the network are systematically re-apportioned so as biomass formation is maximized. This leads to a in silico growth thereby restoring consistency to the mutant. For GNG mutants associated with blocked genes, we check if adding a component from the corresponding reaction to the biomass equation converts it into an NGNG mutant.For GNG mutants associated with genes encoding isozymes, we check if simply deleting the associated reaction prohibits in silico despite growth in vivo. This means that at least one precursor metabolite in the biomass equation cannot be produced. The aim is to modify the existing genome-scale model by adding pathways so as to restore biomass production that may achieve this. To this end, we first construct a database of reactions consisting of (a) reactions from an external database of reactions, (b) irreversible reactions from the original genome-scale model with their directionalities reversed, and (c) transport reactions that enable secretion pathways for metabolites. We define the set Database to represent the reactions that populate this database. For the external databases of reactions, we use the multi-organism databases, MetaCyc NGG mutants are characterized by the lack of growth in vivo experiment l* using the following optimization formulation GrowMatch:Based on these definitions, we next identify the minimal number of modifications required to correct a single NGG mutant corresponding to the l*. The next constraint imposes stoichiometric balance on all metabolites in the model. The requirement of meeting a minimum amount of biomass, quantified by parameter jy = 1 for a reaction j from the database, then there is a non-zero flux through it. The optimal solution to GrowMatch identifies the reactions that need to be added from the database and/or the metabolites that need to be secreted from the metabolic network to ensure a minimum necessary biomass production in the NGG mutant. As in the case of GNG mutants, GrowMatch can be used to identify exhaustively all sets of reactions that need to be added to resolve a particular NGG mutant using integer cuts.In GrowMatch, the objective function minimizes the number of modifications (addition of reactions or activation of secretion of metabolites) in the metabolic model. The first constraint enforces zero flux through reactions that are rendered absent through the elimination of the genes that are knocked out in experiment E. coli). For irreversible reactions selected to be made reversible, we query for such evidence in the literature and also estimate the ΔG values We test the hypotheses generated to resolve the NGG mutant using the following two criteria. For reactions added from the database, we check the two-way protein-protein BLAST expectation value between the enzyme that catalyzes that reaction and the genome of interest  in restricted domains of reactions that reduce biomass production Click here for additional data file.Table S2Blattner numbers of genes associated with NGG mutants(0.44 MB XLS)Click here for additional data file.Table S3BLAST scores and expression data for complementary and non-complementary isozymes(0.05 MB XLS)Click here for additional data file.Table S4E. coli genomeSequence similarity between genes associated with NGG mutants and alternative genes in the (0.04 MB XLS)Click here for additional data file.Table S5in silico biomass equation to resolve GNG mutants associated with blocked genesComponents added to (0.02 MB XLS)Click here for additional data file.


We describe a function-driven approach to the analysis of metabolism which takes into account the phylogenetic origin of biochemical reactions to reveal subtle lineage-specific metabolic innovations, undetectable by more traditional methods based on sequence comparison. The origins of reactions and thus entire pathways are inferred using a simple taxonomic classification scheme that describes the evolutionary course of events towards the lineage of interest. We investigate the evolutionary history of the human metabolic network extracted from a metabolic database, construct a network of interconnected pathways and classify this network according to the taxonomic categories representing eukaryotes, metazoa and vertebrates.It is demonstrated that lineage-specific innovations correspond to reactions and pathways associated with key phenotypic changes during evolution, such as the emergence of cellular organelles in eukaryotes, cell adhesion cascades in metazoa and the biosynthesis of complex cell-specific biomolecules in vertebrates.This phylogenetic view of metabolic networks puts gene innovations within an evolutionary context, demonstrating how the emergence of a phenotype in a lineage provides a platform for the development of specialized traits. Comparative genomics studies provide a global view of the innovations in the gene repertoire within a lineage, and they are most useful in identifying novel gene families, or families that have been massively expanded. The comparative analysis of gene families and their lineage-specific expansions thus provide fundamental insights into the innovations of a gene repertoire for a taxonomic group under investigation. In many cases, the evolutionary expansion of a family, in terms of numbers of genes within a species, can be directly linked to a characteristic trait of the lineage. For example, genes that have been characterized as eukaryotic-specific are mostly involved in the organization of DNA within the nucleus, the compartmentalization of the cytoskeleton, membrane transport, the control of the cell-cycle, various regulatory processes, and RNA splicing . Gene faA limitation of this sequence-driven approach is that subtle changes, which do not necessarily involve massive expansions, cannot, in many cases, be deciphered in such comparative studies. Nonetheless, this type of change might reflect an important adaptation of a species to its environment that cannot be readily detected by sequence information alone. Examples include the adaptation of the colobine monkey RNASE1B to the pH environment in its small intestine, probably as a response to its unique diet . AnotherFor enzymatic proteins, the well-studied relationship between sequence and function allows us to translate genomic information into relatively concise functional predictions -8. ThereHere, we focus on the examination of lineage-specific innovations during the evolution of the human metabolic network. Primarily, we aim to gain a better understanding of the relationship between the evolution of the human metabolic set and the development of new, tissue-specific pathways. Several factors indicate that tissue differentiation is, at least in part, accompanied by tissue specialization of metabolism. First, most of the enzymatic genes have specialized expression patterns and they are not globally expressed across mammalian tissues . Second,ftp://ftp.genome.jp/pub/kegg/ligand/). In total, KEGG provides gene assignment (in at least a single species) for 2,272 reactions, out of which 971 reactions are predicted to occur in human (4 digit EC numbers).The distribution of enzymatic reactions (EC numbers) in species was retrieved from the KEGG database ,16 , also participate in other pathways, such as the oxidative phosphorylation pathway. ATP synthase is a universal enzyme which is widespread among species and domains of life, but it takes part in different pathways across various lineages. This example points to two limiting factors, arising from the structure of the KEGG pathway diagrams, which make the inference of a species' pathway repertoire based on the species' reaction repertoire a highly non-trivial exercise. First, the reactions are not necessarily unique to a single pathway, and about half of the reactions occur in multiple pathways. Second, the pathways in KEGG are reference pathways – i.e., they are a composite collection of all relevant reactions found in all species [To address this problem and focus on those pathways which are pertinent to our analysis, we have considered only pathways where at least a single human reaction is either pathway-specific, or vertebrate-specific: the rationale being that each of those reactions is thus either biochemically specific or taxonomically relevant, respectively . We have also excluded pathways where less than three participating reactions were found in the human species. This results in a set of 78 pathways detected in human to which 652 human enzymes are assigned.ftp://ftp.genome.jp/pub/kegg/pathway/map. We used this information to construct the network representation of the metabolic pathways in human. Each pathway is a node in the network and is connected to other pathways according to the links in the KEGG diagram.The KEGG scheme provides information about the links between different pathway maps . Pathway A and pathway B are linked if a product of a reaction from pathway A is the substrate of a successive reaction from pathway B. The information was retrieved from the .map files downloaded from All human reactions considered here were further classified into four groups, describing the putative lineage where those reactions have emerged. This classification was performed according to predictions in KEGG for the distribution of reactions in the collection of species (as described earlier in the section 'Retrieving a set of the human metabolic reactions'). Vertebrate-specific reactions are those human reactions which are predicted to be limited to vertebrates; metazoan-specific reactions are those human reactions which are predicted to occur only in metazoan species (including metazoan species other than vertebrates); eukaryotic-specific reactions are those human reactions which are predicted to be limited to eukaryotes (including eukaryotic species other than metazoa); universal reactions are human reactions which are widely distributed in prokaryotic species .For each pathway, we have estimated its lineage of origin according to the phylogenetic age of its reaction-members, where pathways were assigned according to the classification of their most ancient unique reaction.Methods), forming the previously reported extensive conserved core of metabolic enzymes common to archaea, bacteria and eukaryotes [The distribution of the human reactions from the 78 selected pathways into the phylogenetic groups is shown in Table karyotes ,18 , have a universal origin Table . These uPathways which appear to have a eukaryotic origin include pathways which are involved in the biosynthesis of characteristic components of the eukaryotic membrane, such as sphingolipids or glycan structures, or pathways involved in inter- and intra-cellular signaling . Many other pathways of a putative eukaryotic origin are involved in processes which are characteristic of higher taxa including animals, such as androgen and estrogen metabolism, C21-steroid hormone metabolism, and bile acid biosynthesis. In these pathways, the majority of reactions belong to more recent lineage groups, and only a few reactions are specific to eukaryotes , absorption of ingested lipids in the gastrointestinal track (taurine metabolism), cartilage differentiation (chondroitin sulfate and kertan sulphate biosynthesis), and blood cell recognition (glycosphingolipid biosynthesis). The evolution of these eukaryotic-, metazoan-, and vertebrate-specific pathways and their functional significance is discussed below.The human pathways were linked to form a network (see Methods section) where each pathway is a node. The constructed network is shown in Figure With the exception of two pathways (linoleic acid metabolism and selenoamino acid metabolism), all universal pathways form a connected component of the network Figure . The mosWhen constructing a network only from the lineage-specific pathways, one can observe that, unlike the case of universal pathways, lineage-specific pathways do not form a single-component network Figure . The linOur analysis demonstrates that the conserved core of metabolism, reported to be common to all domains of life, forms a structural core component of the metabolic network , which is transferred to nascent polypeptide chains  stems from homologous processes in the plasma membrane of archaea or bacteria . The earIn most eukaryotic species, the early steps in N-glycan processing in the ER are conserved. The addition of further saccharides to the ancestral oligosaccharide during eukaryotic evolution is probably driven by the internalization of glycoprotein biosynthesis from the plasma membrane to the ER and the concomitant need to export newly synthesized proteins to the cell surface . Later mThe importance of complex N-glycans in the evolution of multicellular species might be related to the role of glycosaminoglycans in the formation of the extracellular matrix. Though, as can be predicted from their great abundance and structural variety, the functions of proteoglycans in vertebrates are far more diverse. The main groups of glycosaminoglycans in vertebrates play different roles in different tissues: Keratan sulfate and chondroitin sulfate are important structural components of connective tissues (cartilage and bone); Heparan sulfate regulates hematopoietic processes; Dermatan sulfate is found mostly in skin .In the network of pathways Figure , the eukGlycolipids are sugar-containing lipids found in the membrane of cells, where in eukaryotic cells, glycolipids are derived from sphingosine . The classification of reactions into pathways is somewhat subjective, since it can be implemented in a varied manner. Second, the characterization of the set of reactions in species, as well as the classification of reactions into phylogenetic groups is dependent both on the sensitivity of the homology assignment procedure and the range of species analyzed. Third, this analysis concerns only a limited set of reactions whose biochemical function and exact position in the metabolic network are well defined (fully-assigned EC reactions). Such information is not available for a substantial part of the lineage specific reactions which were therefore not considered in this work. These performance issues affect our ability both to reconstruct the complement of the reaction set and to delineate the contribution of each lineage towards the development of the metabolic repertoire during evolution. Yet, despite these limitations, the findings of this analysis are entirely consistent with current knowledge (as shown in  and hereOur analysis indicates that a universal metabolic core vertically inherited from a pre-eukaryotic ancestor remained highly conserved all the way up to mammals. Compatible with previous studies which have characterized a universal metabolic 'skeleton' ,20,21, tThe phylogenetic-structure of the metabolic network, which is described here for the first time, emphasizes the gradual evolution of processes which are specific to a tissue, whereas the development of a phenotype in an ancient lineage provides a platform for the evolution of more recent traits. As the phylogenetic representation of the metabolic network puts gene innovations within an evolutionary context, it demonstrates in a quasi-quantitative manner the importance of events such as the development of the eukaryotic membrane, cellular organelles and the system of transport between them, towards the emergence of multicellular life. While the basic mechanisms of inter-cellular communication are common between invertebrates and vertebrates, many of the vertebrate-specific reactions are involved in tissue-specific functions. The evolution of these pathways must have coincided with the emergence of cell types and organs.Finally, this network approach highlights evolutionary junctions, at which ancestral species gained the ability to catalyze a new reaction type or use a new substrate. Examples of such junctions from the analysis performed here are cholesterol and sphingiosine, molecules whose biosynthesis and function are lineage-specific. The study of such junctions can contribute towards our understanding of the co-evolution between enzymes and substrates, and the mechanisms of biochemical innovation during evolution.SF conceived and performed the analysis, and drafted the versions of the manuscript. LG assisted in the reconstruction and analysis of the network, and contributed to preparation of the final manuscript. CAO assisted in the organization and writing of the manuscript and contributed with critical review. JMT contributed with discussion of the draft versions and critical review. All authors read and approved the manuscript.


Saccharomyces cerevisiae is able to adjust to external oxygen availability by utilizing both respirative and fermentative metabolic modes. Adjusting the metabolic mode involves alteration of the intracellular metabolic fluxes that are determined by the cell's multilevel regulatory network. Oxygen is a major determinant of the physiology of S. cerevisiae but understanding of the oxygen dependence of intracellular flux distributions is still scarce.The yeast S. cerevisiae CEN.PK113-1A growing in glucose-limited chemostat cultures at a dilution rate of 0.1 h-1 with 20.9%, 2.8%, 1.0%, 0.5% or 0.0% O2 in the inlet gas were quantified by 13C-MFA. Metabolic flux ratios from fractional [U-13C]glucose labelling experiments were used to solve the underdetermined MFA system of central carbon metabolism of S. cerevisiae.Metabolic flux distributions of 2. Respirative ATP generation provided 59% of the ATP demand in fully aerobic conditions and still a substantial 25% in 0.5% oxygenation. An extensive redistribution of fluxes was observed in anaerobic conditions compared to all the aerobic conditions. Positive correlation between the transcriptional levels of metabolic enzymes and the corresponding fluxes in the different oxygenation conditions was found only in the respirative pathway.While ethanol production was observed already in 2.8% oxygen, only minor differences in the flux distribution were observed, compared to fully aerobic conditions. However, in 1.0% and 0.5% oxygen the respiratory rate was severely restricted, resulting in progressively reduced fluxes through the TCA cycle and the direction of major fluxes to the fermentative pathway. A redistribution of fluxes was observed in all branching points of central carbon metabolism. Yet only when oxygen provision was reduced to 0.5%, was the biomass yield exceeded by the yields of ethanol and CO13C-constrained MFA enabled quantitative determination of intracellular fluxes in conditions of different redox challenges without including redox cofactors in metabolite mass balances. A redistribution of fluxes was observed not only for respirative, respiro-fermentative and fermentative metabolisms, but also for cells grown with 2.8%, 1.0% and 0.5% oxygen. Although the cellular metabolism was respiro-fermentative in each of these low oxygen conditions, the actual amount of oxygen available resulted in different contributions through respirative and fermentative pathways. Saccharomyces cerevisiae is a facultative anaerobic organism. It is able to respond to external oxygen availability by utilizing both respirative and fermentative metabolic modes and it grows at a fast rate even when aerobic respiration is limited or completely prevented  a. (2001) . On the . (2001)  observed. (2001) . Howeverrylation .mit. Frick and Wittmann (2005) observed considerably increased anaplerotic fluxes in S. cerevisiae at high growth rates  in aerobic chemostats compared to low growth rate (D = 0.15 h-1) and the increases in the anaplerotic fluxes were accompanied by high malic enzyme fluxes -1 min-1 (vvm). Chemostat cultivations were performed with five different oxygenation conditions: 20.9%, 2.8%, 1.0%, 0.5% and 0.0% oxygen of the chemostat inlet gas. For cultures which received less than 20.9% O2 in the gas stream, O2 was replaced with the equivalent volume of N2, so that total gas flow was maintained constant for all experiments. The N2 gas used was 99.999% pure. Gas concentration  was analysed continuously in an Omnistar quadrupole mass spectrometer , calibrated with 3% CO2 in Ar. 13C labelled CO2 was taken into account in the determination of CERs during feeding with13C glucose.Cells were grown in 0.8 to 1 l medium in Biostat CT (2.5 l working volume) bioreactors. Cultures were inoculated to an initial OD2 to prevent additional O2 being added by diffusion into the medium in the anaerobic and low oxygen cultures. The kLa  for the bioreactor in the cultivation conditions was 0.035–0.039s-1 (in pure water). The DOT was 83% in cultures receiving 20.9% O2 and 0% in all the other cultures glucose . During steady state growth the active pathways in the cells will determine how the yeast biomass becomes 13C-labelled. After approximately 1.5 residence times biomass samples, 50 ml of culture broth, corresponding to 0.27 to 0.05 g CDW, were harvested by centrifugation. The cell pellets were suspended into 10 ml of 6 M HCl and the biomass was hydrolysed in sealed glass tubes at 110°C for 22 h. The suspensions were dried and dissolved in H2O for filtration through 0.2 μm filters. The filtrates were vacuum-dried and dissolved in D2O for NMR experiments. The pH of the samples was below 1 due to residual HCl.13C-labelling of amino acids is based on the assumption that both a metabolic and an isotopomeric steady state exist. To establish a cost-effective protocol for a larger number of 13C-labelling experiments, 13C-labelled substrate was fed to a chemostat operating in a metabolic steady state for the duration of 1.5 volume changes glucose labelling during metabolic steady state, 13C-HSQC NMR measurements and software FCAL for the integration of 13C-scalar fine structures of the amino acid carbon signals in the 13C-HSQC NMR spectra and the calculation of relative abundances of fragmentomers using probabilistic equations relating the 13C-scalar fine structures and the intact carbon chain fragments [f-values for the replicate experiments and their standard deviations are given in columns. For nomenclature of f-values, see Methods.ragments . The f-vClick here for fileConfidence intervals (95%) for the metabolic net fluxes. 95% confidence intervals for the net fluxes in the central carbon of S. cerevisiae CEN.PK113-1A in glucose-limited chemostats, D = 0.1 h-1, in different oxygenation conditions. The lower and upper bounds for the confidence intervals of the replicate experiments are given in μmol/(g CDW h) for each net flux. ND stands for not determined because the reactions were excluded from the stoichiometric models according to the metabolic flux ratio (METAFoR) analysis data.Click here for filePearson correlations between the transcriptional levels of genes encoding metabolic enzymes and the corresponding fluxes. The Pearson correlation between the transcriptional levels of the S. cerevisiae genes encoding metabolic enzymes, whose levels were measured in Wiebe et al. [13C-MFA model, and flux values in five different oxygenation conditions . The pairs of genes and fluxes that showed positively correlated behaviour (Pearson correlation >0.60) are marked with green.e et al. , mapped Click here for fileS. cerevisiaeStoichiometric model of the central carbon metabolism of . Reactions in the stoichiometric model of the central carbon metabolism of Saccharomyces cerevisiae, including also anabolic fluxes from metabolic intermediates to biosynthesis, transport reactions across the mitochondrial membrane and uptake and excretion reactions, applied in the 13C-MFA determination of the metabolic net fluxes in different oxygenation conditions.Click here for file


Constraint-based models enable structured cellular representations in which intracellular kinetics are circumvented. These models, combined with experimental data, are useful analytical tools to estimate the state exhibited (the phenotype) by the cells at given pseudo-steady conditions.Pichia pastoris, a workhorse for heterologous protein expression, is validated against several experimental available datasets. Firstly, maximum theoretical growth yields are calculated and compared to the experimental ones. Secondly, possibility theory is applied to quantify the consistency between model and measurements. Finally, the biomass growth rate is excluded from the datasets and its prediction used to exemplify the capability of the model to calculate non-measured fluxes.In this contribution, a simplified constraint-based stoichiometric model of the metabolism of the yeast This contribution shows how a small-sized network can be assessed following a rational, quantitative procedure even when measurements are scarce and imprecise. This approach is particularly useful in lacking data scenarios. Escherichia coli, Saccharomyces cerevisiae, Pseudomonas putida [The collection of biochemical reactions involved in the metabolism of a cell can be assembled in networks in order to carry out studies under a system-level approach . Such ans putida -4, and as putida -7.m internal metabolites:Given a metabolic network, a matrix equation can be used in order to describe the mass balances around the nodes, the c is a vector of metabolite concentrations and v is the vector of reaction rates, or fluxes, representing the mass flow through each of the n reactions in the network [in which  network .In order to avoid reaction kinetics, still rarely known, the internal metabolites are often assumed not to accumulate and thus (1) turns into a system of linear equations. Then, other constraints can be imposed; for instance, it is common to consider particular enzyme kinetics , thermodBy combination of this model and a set of measurable fluxes, the remaining ones can be estimated performing a metabolic flux analysis (MFA) . It is eBesides, these medium-sized networks are derived from the known biochemical reactions involved in the metabolism of a cell, and rely necessarily on reductionist hypothesis, being their validation often insufficient. They are seldom validated against datasets different from the one of interest, which is thus inconveniently used both to validate the model and to perform the MFA analysis. Herein we discuss a procedure seeking for further validation of these networks.Pichia pastoris is world-wide recognized as a reference platform for the expression of recombinant proteins in eukaryotes, due to the possibility to grow cultures to very high cell densities, its ability to produce post-translational modifications, and the good protein yield/cost ratio. Heterologous genes are cloned under P. pastoris strong and tightly regulated alcohol oxidase promoter, and thus expressed when the cells grow on methanol as sole or combined carbon source.The methylotrophic yeast P. pastoris has been usually addressed heuristically. Only a few publications [Pichia growth and protein production. Among these, semi-structured, metabolism-based models representing intracellular behaviour are particularly rare [The optimization of recombinant protein expression in ications -19 descrrly rare ,21.P. pastoris will be described and validated against the available experimental data. Then, its ability to predict non-measured fluxes will be illustrated by estimating the biomass growth rate. The potential use of the model for the estimation of intracellular fluxes will also be discussed. In summary, this work applies a systematic, yet simple, procedure to provide further validation for a small-sized model of P. pastoris, using only data from extracellular measurements.In the following sections, a constraint-based model of A constraint-based model, assuming that internal metabolites are at steady-state and considering the irreversibility of some reactions, can be described with a set of model constraints .Where v that fulfill (2) are considered valid cellular states.The constraints in (2) define a space of feasible steady-state flux distributions, or flux states, which ideally comprises every theoretically possible phenotype: only flux vectors exactly satisfy the constraints. Such difficulty is overcome by taking into account uncertainty, as follows:The simplest consistency analysis could be performed checking that the flux states shown by cells fulfill the constraints imposed by the model. However, this simple approach would be impractical because measurements are imprecise and do not em represents the error or deviation between the actual fluxes vm and the measured values wm.where v fulfilling (2) and (3) for "reasonably small" deviations em. Otherwise, we will conclude that model and measurements are inconsistent. An easy way to evaluate consistency is to find the flux vector v fulfilling (2) and (3) that minimises the (variance-weighted) sum of errors:Model and measurements can be consistent if there is a vector em are distributed normally with zero mean value and have a variance-covariance matrix F. If only linear equality constraints are considered in χ2-distribution, and therefore a χ2-test can be used to detect and evaluate the inconsistency. The χ2-test is based upon statistical hypothesis testing to determine if the deviation is within expected experimental error [χ2-test cannot be used because its assumptions are not fulfilled (ϕ does not follows a χ2-distribution anymore). Yet, the residual ϕ provides at least a rough indication of consistency.Where it is assumed that al error . HoweverThe consistency analysis can also be formulated as a possibilistic constraint satisfaction problem, as it has been recently proposed in . The basem by two pairs of non-negative decision variables :ε1, μ1, ε2 and μ2 relax the basic assertion wm = vm, conforming a set of possibility distributions in  associated to some cost index J. Among different possible choices, a simple -yet sensible- one is the linear cost index:These decision variables α≥0 and β≥0, which are row vectors of measurement reliability coefficients.with v, is given by the value of the cost index:The possibility π of each solution δ of (2) and (5), which corresponds to a particular flux vector wm = vm is fully possible; the more wm differs from vm, the less possible such situation is". See the article for further technical details [The interpretation of (5-7) may be: " details .2max and μ2max and the weights α and β. This way, each measurement is represented by a distribution of possibility . For instance, the user can choose a band of 10% around the measured value. The values α and β define the decreasing possibility to assign to values out of this interval (details below).Defining two pairs of decision variables, there is more flexibility to represent the measurements in possibilistic terms: the user can assign the bounds εmples in ). The bovmp corresponding to a given set of measurements is obtained solving a linear programming (LP) problem:At this point, the maximum possibility (minimum-cost) flux vector The possibility of the most possible solution being, This degree of possibility provides an indication of the consistency between model , which contains the most possible value for each flux. However, a point-wise estimate is limited when multiple combinations might be reasonably possible. In this situation, a possibilistic interval estimate is a better choice.The interval of values with conditional possibility higher than for a given variable, The upper bound ramework .P. pastoris growth on glucose, which has been extended with reactions representing methanol and glycerol metabolism. This is a simplified representation whose objective is not to accurately describe the full biochemistry of the yeast but to generate a model in which to apply methodologies of interest aimed to process analysis, monitoring and control.The metabolic network presented in Figure P. pastoris  are represented for growth on the substrates mainly used for its culture: glucose, glycerol and methanol. In this case, a mean biomass equation derived from the macromolecular composition of the yeast is used to summarize the anabolic pathways according to [Saccharomyces cerevisiae models coming from the literature [P. pastoris biomass compositions, if available.The main catabolic pathways of the yeast rding to . Key metterature ,23,24 weThe model contains 45 compounds and 44 metabolic reactions. The balanced growth condition can be applied to 36 internal metabolites, resulting in a 36 × 44 stoichiometric matrix with 8 degrees of freedom . For instance, the theoretical yield for growth on glucose is 4.93, whereas the experimental one is 3.98 (Cmmol DW/mmol). The maximum yield on glycerol and methanol is 2.25, and the experimental ones at different ratios of glycerol and methanol range between 1.31 and 0.63. It also seems that the experimental yields decrease for combinations of substrates with lower theoretical yields.Thus, no experimental yield violates the maximum theoretical ones . However, the experimental yields tend to be lower than theoretical ones. There are several reasons for this deviation: (a) the model does not consider restrictions on energy cofactors, such as ATP, nor the resources devoted to recombinant protein production, (b) the EM analysis does not take into account the ratio between the different substrates in mixed cases, and (c) even if optimal pathways exist, the actual behaviour of cells does not necessarily makes use of them in terms of growth .The datasets in Table F in (4) is defined accordingly.In all weighted least squares problems, a standard deviation of 10% is assigned to each measurement of the set trying to capture their uncertainty. The variance-covariance matrix In the Possibilistic MFA problems, the uncertainty of the measurements was represented as follows:(a) Full possibility (π = 1) is assigned to values near the measured ones, less than ± 5% deviation, to account for random errors.(b) A decreasing possibility is assigned to larger deviations so that values with a deviation equal to ± 20% have a possibility of π = 0.1 .ε2max, μ2max) and weights  for each measurement wm. Due to (a), the bounds are defined as ε2max = μ2max = 0.05·wm. Then we operate with equations (5-7) to achieve (b). From (5) we have that, 0.2·wm = ε120% + ε2max, and from (6) and (7), log(0.1) = -α·ε120%. As a result we get that, α = -log(0.1)/(0.2-0.05)/wm. Since uncertainty is symmetric, β = α.This representation is achieved choosing the necessary bounds (mpv) are given. The last column provides another indicator of consistency: the degree of measurements uncertainty needed to find a flux vector in full agreement with the model constraints (π = 1). All the computations were performed with MATLAB , and YALMIP toolbox [The results for each dataset are shown in Table  toolbox  was usedThe consistency between model and experimental measurements is very high, but for a small set. In these cases, the inconsistency pinpoints especial characteristics of these sets of data, as explained below.Pichia growing on glucose, shows very good agreement. The measured data has full possibility (π = 1), meaning that there is a flux vector compatible with model and measurements. In fact, as shown in the last column, a band of 1% around the measured values is sufficient to enclose this flux vector. Notice also that the residual is very low.The dataset D1, which corresponds to Datasets A1 and A2, which correspond to cultures growing totally or mainly on glycerol and producing a small amount of protein, also show a good agreement. The discrepancy between measurements and model is larger for A3 (π = 0.25), but still a band of 10% of deviation around measurements encloses a flux vector compatible with the model. Dataset A3 corresponds to a culture growing mainly on methanol, but supplemented on glycerol, and producing larger amounts of protein. The discrepancy is larger for A4, which corresponds to a scenario with high protein productivity.Similar results are obtained with cultures at a higher growth rate (datasets B1-B3), B1 and B2 are highly consistent, while protein producing B3 shows similar behaviour to A3-A4. This suggests the existence of non-modelled phenomena, probably related with protein production. The agreement is quite good for the three datasets C1-C3, but the increase of the discrepancy along with higher protein expression is also noticeable.P. pastoris cultures. These datasets were defined taking random combinations of values for each flux within predefined bounds . Results are depicted in Figure The estimated growth rate is found to be in very good agreement with the measured one for the vast majority of the analysed scenarios , which correspond to cultures at different growth rates, using different substrates, and coming from three independent literature references. For two other scenarios (A2 and C3), the most possible estimate is still accurate.The fact that, although limited, the model has predictive capacity provides further validation for this constraint-based representation. This conclusion is strengthened if we consider that the growth rate is highly interconnected along the whole network, since the biomass equation takes into account several metabolic precursors, and thus accurate correspondence between substrate uptake, respiratory fluxes and growth cannot be inferred in a straight-forward way from the network.Once the model has been validated, possibilistic MFA could be used to estimate all the non-measured fluxes, either intracellular or extracellular, as done with the growth rate in the previous section. For illustration purpose, the flux distributions for each scenario are given in the additional file Notice that these estimations cannot be done by means of traditional MFA because the measurements would be insufficient to get a determined system.The network has 8 degrees of freedom (44 fluxes and 36 linear equations) and there are 9 measured fluxes. However, these measurements introduce only 7 independent additional linear constraints, so the system remains under-determined with 1 degree of freedom . Possibiv2, v3 and v4 belonging to the glycolysis pathway, are positive as expected in cultures grown in glucose, and appear inverted in glycerol and/or methanol fed cultures.- Fluxes v21, v22 and v23 represent the isomerization of R5P into Ru5P and Xu5P. Note how v23 inverts its direction at growing methanol fluxes, as increased methanol consumption demands higher amounts of Xu5P thus requiring more R5P precursor.- Fluxes v32, v33 and v34 represent the branchpoint related to methanol usage, that is, how this flux is split between direct oxidation and catabolic pathways. High methanol fluxes are necessarily conducted via CO2 generation and thus flux v34 becomes distinct from zero in A4, B4, C2 and C3 scenarios.- Fluxes In this way, these results further validate the predictive capability of the model.Pichia pastoris has been validated in several experimental scenarios resulting in good agreement between estimations and measurements. In addition, the predictive capacity of the model for cell growth rate, an attractive target for industrial fermentation monitoring and control, has been verified. Interestingly, the accuracy of predictions worsens for higher protein producing scenarios, showing how the model, derived for a wild-type strain, is increasingly less precise as wider resources are devoted to recombinant protein generation.The consistency of a constraint-based model of It must be highlighted that the model has been strictly constructed upon first-principles and sensible hypothesis. At this point, the model can be curated, extended, and its parameters tuned in order to improve the consistency with the investigated scenarios. Particularly, energy requirements, strongly related to protein expression, are not yet considered within the model and future work will address this issue.This contribution shows how a small-sized network can in general be assessed following a rational, quantitative procedure even when measurements are scarce. Possibilistic MFA becomes a useful tool to systematize this procedure. This approach enables validation considering the stoichiometric balances and also reactions reversibilities, and accounting for measurements imprecision. The use of Possibilistic MFA also makes it possible to predict non-measured fluxes without removing the network under-determinancy. There is, however, a challenge when validating networks with higher number of degrees of freedom because there may be many flux vectors compatible with the (few) available measurements. It is expected that the datasets will be highly consistent, so the approach in this case would be to check if the model rejects the artificially generated invalid datasets.When a validated model is available, ideally incorporating measurements for some intracellular fluxes, the kind of comparative analysis proposed herein will provide a insight on how the internal state of the cells determines its external behavior, and potentially lead intervention within cells, suggesting target metabolites or biochemical branch-points and also allowing optimization through manipulation of extracellular variables, such as feeding strategies and substrate selection.MTS, FLL and JPM designed the research and conceptualized the manuscript. MTS elaborated the metabolic network; FLL designed the consistency analysis method. MTS and FLL analyzed the results and drafted the manuscript. JPM supervised and coordinated the project. All authors read and approved the final manuscript.nd and 3rd authors are grateful to grants DPI2008-06880-C03-01 and A/016560/08). FLL is recipient of a fellowship from the Spanish Ministry of Science and Innovation (FPU AP2005-1442). The authors are grateful to the Company Biopolis for his support to this research.This research has been partially supported by the Spanish Government (2P. pastorisMetabolic network for . This includes the list of reactions, metabolites and stoichiometric matrix.Click here for fileElementary mode analysis. This file includes the whole set of elementary modes, the corresponding macroreactions and the calculation of the theoretical yields.Click here for fileComplete flux distribution per scenario. This file includes the figures representing the estimation of each intracellular flux for all datasets.Click here for file


Flux variability analysis is often used to determine robustness of metabolic models in various simulation conditions. However, its use has been somehow limited by the long computation time compared to other constraint-based modeling methods.We present an open source implementation of flux variability analysis called fastFVA. This efficient implementation makes large-scale flux variability analysis feasible and tractable allowing more complex biological questions regarding network flexibility and robustness to be addressed.Networks involving thousands of biochemical reactions can be analyzed within seconds, greatly expanding the utility of flux variability analysis in systems biology. Flux balance analysis (FBA) ,2 is conS is an m × n stoichiometry matrix with m metabolites and n reactions and c is the vector representing the linear objective function. The decision variables v represent fluxes, with V ⊆ ℝn and vectors vl and vu specify lower and upper bounds, respectively. The constraints Sv = 0 together with the upper and lower bounds specify the feasible region of the problem.where the matrix Flux variability analysis (FVA)  is used Applications of FVA for molecular systems biology include, but are not limited to, the exploration of alternative optima of (1) , studyinw represent some biological objective such as biomass or ATP production. After solving (1) with c = w, FVA solves two optimization problems for each flux i vof interestLet Z0 = wTv0 is an optimal solution to (1), γ is a parameter, which controls whether the analysis is done w.r.t. suboptimal network states (0 ≤ γ <1) or to the optimal state (γ = 1). Assuming that all n reactions are of interest, FVA requires the solution of 2n LPs. While FVA is clearly an embarrassingly parallel problem and is therefore ideally suited for computer clusters, this note focuses on how FVA can be run efficiently on a single CPU. A multi-CPU implementation of fastFVA can be done in the same way as for FVA, i.e., by distributing subsets of the n reactions to individual CPUs. It is expected to give almost linear speedup for sufficiently large problems.where n reactions and solves the two optimization problems in (2) from scratch each time by calling a specialized LP solver, such as GLPK [i, i = 1, 2, ..., n all elements of c are zero except i c= 1. Since the only difference between each iteration is a change in the objective function, i.e., the feasible region does not change, solving the LPs from scratch is wasteful. Each time a LP is solved, the solver has to spend some effort in finding a feasible solution. Once a feasible solution is found, the solver then proceeds to locate the optimum. The small changes in the objective function suggest that, on average, the optimum for iteration i does not lie far away from the optimum for iteration i + 1. With Simplex-type LP algorithms, this property can be exploited by solving problem (1) from scratch and then solving the subsequent 2n problems of (2) by starting from the last optimum solution each time (warm-starts). It should be noted that the default behavior of some Simplex-type solvers is to use warm-starts when a sequence of LPs is solved within the same application call. However, current implementations of FVA do not make use of this option . FurthePSetup problem (1), denote it by P from scratch to get v0 and Z0Solve wTv ≥ γZ0 to PAdd the constraint for i = 1 to ni c= 1 and j c= 0, ∀j ≠ i   Let P, starting from vi-1   Maximize vi and iZ                     to get i maxFlux= iZ   v0 = vn.Once all maximization problems have been solved, the minimization problems are solved in the same way, starting from n reactions.An important difference between the various LP solvers available is their ability to exploit multiple core CPUs or multi-processor CPUs to increase performance. The GLPK solver, for example, is a single threaded application. When running on a quad-core machine with hyperthreading enabled, the CPU load is only at 12-13%. On multi-core machines, a significant speedup can often be achieved by simply running multiple instances of fastFVA, each working on a different subset of the The fastFVA package runs within the Matlab environment, which will facilitate the use of fastFVA by users less experienced in programming. In addition, many biochemical network models can be imported into Matlab using the Systems Biology Markup Language (SBML) and the COBRA toolbox .The fastFVA code is written in C++ and is compiled as a Matlab EXecutable (MEX) file  machinery of de Table . The bioThe fastFVA code was tested on a DELL T1500 desktop computer with a 2.8 GHz quad core Intel i7 860 processor with hyperthreading enabled and Windows 7.The running times are given in Table γ = 0 in (2). The α-spectrum [γ in (2) is set to zero and the S matrix is replaced by a matrix P containing the extreme pathways as its columns.The fastFVA code can be used to compute the flux-spectrum , a variaspectrum , which hWith this efficient FVA tool in hand, new questions can be addressed to study the flexibility of biochemical reaction networks in different environmental and genetic conditions. It is now possible to design computational experiments requiring hundreds or even thousands of FVAs.http://notendur.hi.is/ithiele/software/fastfva.html together with pre-compiled binaries for Linux and Microsoft Windows. The fastFVA code runs under Matlab and relies on third-party solvers to solve linear optimization problems. Two such solvers are supported, the open source GLPK [The fastFVA package is freely available at rce GLPK  and the IT conceived and designed the study. SG implemented fastFVA and carried out the experiments. Both authors wrote the manuscript and approved its final version.This file contains the C++ source code, the pre-compiled binaries, an example on how to use fastFVA and scripts for carrying out the experiments described above.Click here for fileThis file contains the six metabolic networks used in the experiments.Click here for file


Optimization aims to make a system or design as effective or functional as possible. Mathematical optimization methods are widely used in engineering, economics and science. This commentary is focused on applications of mathematical optimization in computational systems biology. Examples are given where optimization methods are used for topics ranging from model building and optimal experimental design to metabolic engineering and synthetic biology. Finally, several perspectives for future research are outlined. To optimize means to find the best solution, the best compromise among several conflicting demands subject to predefined requirements . Mathematical optimization has been extremely successful as an aid to better decision making in science, engineering and economics.Optimization and optimality are certainly not new concepts in biology. The structures, movements and behaviors of animals, and their life histories, have been shaped by the optimizing processes of evolution or of learning by trial and error ,2. Moreodecision variables (those which can be varied during the search of the best solution), the objective function , and the constraints . Decision variables can be continuous , resulting in continuous optimization problems, or discrete (represented by integer numbers), resulting in integer optimization  problems. In many instances, there is a mix of continuous and integer decision variables.First, I will introduce several basic concepts that can help readers unfamiliar with mathematical optimization. The key elements of mathematical optimization problems are the objective function to minimize is the cost of the food, the decision variables are the amounts of each type of food to be purchased (assumed as continuous variables), and the constraints are the nutritional needs be satisfied, like total calories, or amounts of vitamins, minerals, etc., in the diet.As an illustrative example, consider the "diet problem", one of the first modern optimization problems , studiedThe "diet problem" has certain interesting properties: it is a continuous problem where both the objective function  and the constraints are linear with respect to the decision variables, so this problem belongs to the important class of linear programming, or LP . These linear constraints define a feasible space (space of decision variables where constraints are satisfied) which is a convex polyhedron, so it is a convex problem. Convex optimization problems  are partNon linear programming (NLP) deals with continuous problems where some of the constraints or the objective function are nonlinear. In contrast to LP, NLP problems are much more difficult to solve. Further, the presence of nonlinearities in the objective and constraints might imply nonconvexity, which results in the potential existence of multiple local solutions . Thus, in nonconvex problems one should seek the globally optimal solution among the set of possible local solutions. For the simple case of only two decision variables, one can visualize the objective function of a multimodal problem as a terrain with multiple peaks. Simple examples of unimodal and multimodal surfaces are presented in Figure The solution of multimodal problems is studied by the subfield of global optimization -10. ManyModel-based optimization is a key methodology in engineering, helping in the design, analysis, construction and operation of all kind of devices. Since engineering approaches are playing a significant role in the rapid evolution of systems biology -14, it iIn fact, optimization is already playing a key rôle. Examples of applications of optimization in systems biology, classified by the type of optimization problem, are given in Table in silico predictions of Escherichia coli metabolic capabilities [Saccharomyces cerevisiae metabolic network [Optimization methods have been applied in both metabolic control analysis ,16 and bbilities , or the  network .in vitro metabolic pathways using synthetic biology, as the key component in rational redesign and directed evolution [Metabolic engineering exploits an integrated, systems-level approach for optimizing a desired cellular property or phenotype . New optvolution -26.Coupling constraint-based analysis with optimization has been used to generate a consistent framework for the generation of hypotheses and the testing of functions of microbial cells using genome-scale models . Extensii.e. "which are the criteria (objective functions) being optimized in these networks?", a question which has been addressed in detail recently [A particularly interesting question in this context concerns the principles behind the optimal metabolic network operation, recently ,34. Consrecently . Moreoverecently -38, and recently , an apprReverse engineering in systems biology aims to reconstruct the biochemical interactions from data sets of a particular biological system. Optimization has been used for inferring important biomolecular networks, such as e.g. transcriptional regulatory networks , gene reSystem identification ,51 is a The problem of parameter estimation in biochemical pathways, formulated as a nonlinear programming problem subject to the pathway model acting as constraints, has also received great attention -59. SincSince biological experiments are both expensive and time consuming, it would be ideal if one could plan them in an optimal way, i.e. minimizing their cost while maximizing the amount of information to be extracted from such experiments. This is the purpose of optimal experimental design and optimal identification procedures -66, a toAlthough, as already mentioned, it would be desirable to formulate all the optimization problems as convex ones, in many occasions this is not possible, so we face the solution of global optimization problems, most of which belong to the class of NP-hard problems , where oAnother important issue is the stochasticity that is inherent in biomolecular systems ,72. Thisin silico by optimizing design parameters targeting a selected objective function. The optimization of such synthetic biological systems is in fact receiving increasing attention: optimization algorithms could search for the components  and find the best configurations optimizing the dynamic behaviour according to predefined design objectives [As stated in , it wouljectives . A promijectives , which cjectives .Moreover, optimization could also be used after the design and construction phases, inside a model predictive control framework , to optiFinally, it should be recognized that standard optimization can be sometimes insufficient for gaining deeper insights regarding certain aspects of systems biology, such as in the evolution of biological systems. While evolving towards optimal properties, the environment may change or organisms may even change their own environment, which in turn alters the optimum. In an evolutionary system, continuing development is needed so as to maintain its fitness relative to the systems it is co-evolving with. In other words, everyone has to keep improving in order to survive, which is known as the "Red Queen" effect . Thus, gSutherland  claims t