The less a competitor is saturated by the target, the more sensitive it is to changes in the concentration of the target, as well as to other competitors. As a consequence, most of the mRNAs in yeast turn out to respond ultrasensitively to changes in ribosome concentration. Finally, applying the theory to a genome-wide dataset we observe that high and low response mRNAs exhibit distinct Gene Ontology profiles.Often, in living cells different molecular species compete for binding to the same molecular target. Typical examples are the competition of genes for the transcription machinery or the competition of mRNAs for the translation machinery. Here we show that such systems have specific regulatory features and how they can be analysed. We derive a theory for molecular competition in parallel reaction networks. Analytical expressions for the response of network fluxes to changes in the total competitor and common target pools indicate the precise conditions for ultrasensitivity and intuitive rules for competitor strength. The calculations are based on measurable concentrations of the competitor-target complexes. We show that kinetic parameters, which are usually tedious to determine, are not required in the calculations. Given their simplicity, the obtained equations are easily applied to networks of any dimension. The new theory is illustrated for competing sigma factors in bacterial transcription and for a genome-wide network of yeast mRNAs competing for ribosomes. We conclude that molecular competition can drastically influence the network fluxes and lead to negative response coefficients and ultrasensitivity. Competitors that bind a large fraction of the target, like bacterial σ One of the main challenges of current systems biology is to understand the properties of large and complex molecular networks It is expected that natural selection minimizes the cost of producing abundant catalytic machinery, for example the large ribosomal complexes involved in translation or RNA polymerase in transcription E. coli, seven different types of sigma factors exist, each directing transcription of a specific set of genes 70. σ54  confers specificity to RNAP for transcribing genes regulated by the availability of nitrogen and some stress response genes 28 (or σF) is needed for the expression of the flagellum and chemotaxis genes 38 (σS) accumulates during the stationary phase and directs expression of genes related to stress management and maintenance H, σE and σFecI, act in heat shock response, extra- cytoplasmic stress and iron-transport respectively e.g.Another example of molecular competition is the so-called sigma cycle in bacterial transcription . This tye.g. flux) towards a small fractional change in an external parameter . For instance, if a specific translation flux depends proportionally on the specific mRNA concentration, then the corresponding response coefficient To understand how strongly regulation in these and analogous networks is affected by molecular competition, we here develop a general theoretical framework. Based on the theory of Metabolic Control Analysis The response coefficient corresponds to the infinitesimal form of the classical ‘amplification factor’ used by Goldbeter and Koshland to describe an input-output (stimulus-response) relation in any biochemical system ϕ the response and S the stimulus.The finite version  is less suitable than the continuous version to derive generic formula, given the nonlinear dependence on the size of the variation in stimulus.measurable concentrations of the target-competitor complexes. This will be much more convenient than a classical, direct measurement of response coefficients, which requires a tedious titration of specific macromolecules, such as ribosomes, mRNAs or sigma factors. The resulting equations are analyzed and applied to both a small-scale and large-scale network: the sigma cycle and translation, respectively. This yields new insights in the regulation of these networks.In the following we will specifically focus on the sensitivity of the steady state fluxes of the competing pathways towards changes in the levels of the competitors, as well as towards changes in the levels of the target for which they compete. We will first derive the general theory that allows for quantification of response coefficients based on 1c and 2c, and their common target t  at rates 1β and 2β into the products  and the products dissociate from the complex. Simultaneously the competitors are released from the target. The case of two competitors is generalized below to any number of competitors ‘n’, and to simultaneous binding of multiple identical target molecules to one competitor. The latter applies to translation, when a single mRNA (competitor) binds multiple ribosomes (the targets). In n competitors for binding to a single pool of target molecule are listed. The following assumptions are made:To explain the basic structure of the type of networks considered in our analysis we first turn to the most simple reaction scheme with two competitors target t . In the The binding of the competitor to the target is described by reversible mass-action kinetics,The subsequent production step is described by irreversible mass-action kinetics,s to be constant and subsume it into the rate constant of the production reaction to yield an apparent rate constant We assume the substrate concentration e.g. nucleotides) prior to its conversion to product (e.g. mRNA), the rates β would be better described by Michaelis-Menten equations, with 1tc and 2tc as the enzyme concentrations. Also then, the rates β remain proportional to 1tc and 2tc. We should be aware that a hyperbolic substrate function is then hidden in the apparent reaction constants of the production reaction. However, as substrate effects are not the focus of this paper, we will not discuss this any further.The irreversibility of the production step (assumption 2) is warranted by the free-energy requirement of the elongation steps in transcription and translation, for which we have tailored the model. If we would explicitly consider reversible binding of the substrate  external molecular species as:For this type of networks, we assess the influence of molecular competition for a common target molecule on the fluxes of the competitors. Our approach is based on the theoretical framework of Metabolic Control Analysis and applies to steady states. In the example of V in response to a small  change in p.This can be understood as the  change of p will refer to the concentrations of the moieties. Upon a change in such a parameter, the steady state will change and the following internal response coefficients can be defined for the response of a system variable V to the change in the steady state of a concentration of a species kS occurring in a moiety conserved cycle.kS could substitute for any of the molecular species t, 1c, 2c, 1tc or 2tc. Sauro V (e.g. flux) to an initial change in the concentration of a species kS involved in a conserved sum iT relates to the external response coefficients towards all the moieties in which this species participates:ikn the stoichiometry of moiety k occurring in cycle i. The summation is over all moieties.We follow the theory developed in i were derived (cf. Based on the above the following equations for the flux ‘through’ an arbitrary competitor ived cf. .(2)(3)T and specific competitorc iC and on the concentration of specific target-competitor complexes itc. Thus, to evaluate the response coefficients in a real competition network, the underlying rates and rate constants need not be determined.These are the central equations in our analysis and will give insights into how molecular competition shapes key control and regulatory properties of these networks. Note that each of these response coefficients depends only on the concentrations of conserved moieties of target E. coli only σ70, σ54 and σ28 are considered here. They represent by far the three most abundant sigma factors in the exponential phase of E. coli.We will first illustrate the use of equations (2)-(4) for the example of the sigma cycle. Of the seven different types of sigma factors in et al.E. coli W3110 cells:Maeda Target (RNAP: further shortened to ‘E’): 700 molecules/cell (no free molecules);70: 545 out of 700 molecules/cell are bound to RNAP;Competitor σ28: 100 out of 370 molecules/cell are bound to RNAP;Competitor σ54: 55 out of 110 molecules/cell are bound to RNAP;Competitor σ28 towards the total concentrations of RNAP and the various sigma factors:Based on this limited information and equations (2)-(4) one calculates the response coefficients of the flux through σThe responses of the fluxes through the other sigma factors are calculated analogously. The full result is presented in i/Citc. Second, increasing a specific competitor concentration leads to a decrease of any of the other fluxes, i.e. all fluxes have a negative response coefficient towards changes in the competing sigma factors. The latter follows from equation (4) if one takes into account that all response coefficients to the target are positive. This competition is stronger, i.e. the negative response coefficient towards another competitor becomes larger, when the sigma factor of interest is less saturated with the target. At the same time, the negative response coefficient towards a competitor increases when the latter binds an increasing amount of target. Indeed, comparing the response of the σ70 flux to changes in σ54 (55/110  =  50% saturation) and σ28 (100/370 = 27% saturation), we find that the absolute amount of target bound is determining the competitor strength, and not the saturation. Noteworthy, the strongest competitor has the weakest response to a change in its own concentration.From this table we can already see a number of principles emerging. First, the high sensitivity of the flux of the least saturated sigma factor to an increase in RNAP is remarkable. This follows directly from equation (2). Only the numerators are different for the response coefficients of the different sigma factors to the target and they increase with decreasing i-th competitor-binding flux in terms of the amount of the target bound to the different competitors. This is expressed as fractions of the total competitor concentrations iC occupied by target  and the fractions of the total target concentration bound by each competitor (i/Ttc). The value of the numerator equals the fraction of competitor i not occupied by target. The denominator has an intuitive symmetrical structure. Based upon this formula, only positive values are possible for these response coefficients (i is present in the bound form. The second condition is satisfied as long as the bulk of the target is not bound to (nearly) saturated competitors.Equation (2) gives the response of the ficients . Values cf.As a general demonstration of the validity of the formula we have calculated the response coefficients for the network from i.e. its upper limit, while the numerator approaches 0. Also when at low total target concentration the denominator becomes 1 and the response coefficient becomes nearly equal to the value of the numerator, i.e. the fraction of free competitor i. This is also shown in In j w.r.t. competitor i:Interestingly, as illustrated by our examples, response coefficient values higher than 1 are found. This indicates that this process is more sensitive to a change in target concentration than a classical first order reaction or Michaelis-Menten enzyme would be. This phenomenon has been termed ultrasensitivity T is defined by:In accordance with equation (2), ultrasensitivity towards the total target concentration This condition is fulfilled if:t:Or, since i can only occur if the available free target is below a threshold determined by the degree of saturation of the competitors. This upper bound will increase with an increased binding of the target to the other competitors  and/or with an increased fractional occupation of those other competitors . For both small-scale and large-scale cellular networks it definitely suggests that at least for the least saturated competitors ultrasensitivity to target changes might be a normal phenomenon. Furthermore, since t can only be positive, at least one factor ijρ has to be higher than 1 for ultrasensitivity to occur. This means that at least one competitor other than i must have a higher fractional occupation by the target than i itself. In other words, the flux through the strongest binder can never respond ‘ultrasensitively’ to the target concentration. It also follows that ultrasensitivity cannot occur when all competitors are equally strong and therefore have the same degree of saturation. For example, in the case of two competitors, ultrasensitivity of the flux through 1tc occurs when the free target complies with:This inequality indicates that ultrasensitivity of flux 12ρ>1 (competitor 2 is more saturated with target than competitor 1) ultrasensitivity to the target is possible for the flux through competitor 1 . If 12ρ<1 then the opposite is true.If Since the denominator in (2) is the same regardless of which competitor response considered, the ratio between two of these response coefficients equals:54 and σ28 in the example; cf.This means that the relative ratio of the response coefficients of the different competitors to the target is equal to the ratio of the free competitor fractions. The less saturated competitor will always have the highest response to the common resource. This implies that whenever under a particular condition one competitor exhibits ultrasensitivity, then all less saturated competitors will be ultrasensitive too and even more pronounced. See for example Equation (3) describes the response of a competitor flux towards changes in its own total concentration, while equation (4) describes the cross-talk of competitors to competing fluxes.T bound to that competitor. Based on equations (2) and (3) only values in the interval  are possible for T . This equation describes the mutual regulatory influence or crosstalk of the respective competitors. The response of the flux of a competitor i towards a change in another competitor k equals the response coefficient towards a change in target multiplied by the fraction of total target bound by competitor k. According to this equation, only values i to the target, whereas the second factor is specific for competitor k. The latter competitor will therefore only have a strong effect if it has bound enough of the target and at the same time the other competitor is sensitive to the target concentration. Intuitively, this makes sense. In absolute values the response coefficient towards a competitor will therefore always be smaller than the response coefficient to the target  is that for large numbers of (comparably strong) competitors i can only be ultrasensitive to changes in some competitor k if it is ultrasensitive to the target. The concomitant reduction of ultrasensitivity as compared to that towards the target depends on how much of the target is bound to competitor k. Re-writing the condition for ultrasensitivity (cf.Equation (4) indicates that the flux through ty , cf. leads tot will only exist if the saturation of competitor k is higher than that of competitor i, and at the same time the total amount of complex ktc is sufficiently high.This condition is complex and not immediately insightful. Since the first term is negative, a solution with positive From equations (2) and (4) we can easily derive the following simple relations:28 affects the flux of σ70 almost twofold more than σ54 does, ‘because’ there are almost twice as many molecules of target bound to σ28 than to σ54 (100 versus 55).Equation (6) shows that the relative sensitivity of one flux towards different competitors depends solely on the absolute amount of the target bound to these respective competitors. This is remarkable as one might expect it to correspond to their degree of saturation in accordance with the respective binding strengths of the competitors. In other words, a weaker binder that binds more target because of its higher abundance, will be a stronger competitor. Logically, when the same fractional increase occurs for the two competitors, the one which has bound the most target will again sequester the most target. In the example of i and j. Finally, if one competitor flux exhibits ultrasensitive behavior, then all the less saturated competitor fluxes will do so as well and with higher absolute response coefficients (compare the σ54 flux with the σ28 flux in Equation (7) shows that the relative effect of changing one competitor on the fluxes of the other competitors depends solely on the sensitivity of these fluxes to the common molecular target. This reflects the assumption in the model that the cross-talk between the fluxes of the different competitors is mediated solely by the common target. Furthermore, by using equation (4) it can be shown that the ratio of these responses is equal to the ratio of the free fractions of the ‘influenced’ molecules cf.Also in protein synthesis competition occurs, with different mRNAs competing for a common set of ribosomes and translation factors. As compared to the reaction scheme of i is replaced by: The fractional occupation (saturation) by target of competitor iP is the average number of ribosomes bound per molecule of mRNA i. In an experimental context this typically corresponds to the product of the fraction of mRNA i bound to ribosome (‘ribosome occupancy’) and the so-called ‘polysome size’ (the average number of ribosomes in the ribosome-bound fraction of mRNA i). iN is the number of codons of mRNA i, and L the number of codons occupied by one ribosome.i:We perform a similar replacement for the fraction of target bound to competitor iC is the total concentration of mRNA i, T the total concentration of ribosomes.Making the necessary substitutions transforms equations (2-4) into:Since we had to make extra assumptions to derive these equations , we first compared their outcome to a detailed mathematical model of the translation process in which these assumptions have not been made. In this model translation consists of a single-step initiation phase, a multi-step elongation phase (each codon is represented by a separate state variable) and a single-step termination phase . It extei to the ribosome concentration , equation (8) can be used to derive an (approximate) condition for ultrasensitivity of the translation rate of an individual mRNA to the total ribosome concentration. From equation (8) it can be derived that ultrasensitivity of flux cf. equation (4)).Similarly, the conditions for ultrasensitivity towards individual mRNA species can be derived from the conditions for binding of a single target molecule, by making the same replacements as given above. Analogous to the above it follows that, at low free ribosome concentrations, the least saturated mRNAs will respond ultrasensitively to changes in the total concentration of ribosome. These will also respond ultrasensitively to changes in certain competitors if the latter bind a large fraction of the total ribosome pool , we find:This implies that the relative sensitivity of two mRNAs to a third competitor is equal to their relative sensitivity to the target concentration, which is determined by their relative saturation by ribosome (the term ‘free’ mRNA is not instructive here).Noteworthy, one of the most interesting relations is found by adapting equation (6) i (‘competitor strength’) is the same across all mRNA species and conditions. Obviously, this independence of i is also valid in the case of single target binding.This equation shows that the relative competitive strength of two mRNAs influencing the flux of a third competitor, is proportional to the relative amount of ribosome bound. Accordingly, the relative ranking according to influence on each ‘other’ mRNA species et al.Clearly, the power of these equations lies in the possibility to predict the response coefficients for translation of the entire transcriptome, based on the measurable quantities of polysome size , ORF length, individual mRNA and total ribosome levels. There is no need for any kinetic information. We have applied formula (8) to a dataset consisting of yeast mRNA abundance and ribosome footprints (polysome sizes) as reported by Siwiak and Zielenkiewicz cf.Saccharomyces Genome Database’  was used to find out how well certain biological categories are represented in the low and high response groups, relative to the S. cerevisiae whole genome. S. cerevisiae (‘Yeast GO-Slim Process’). Genes related to translation are overrepresented in the low response group, whereas underrepresented in the high response group. That pattern is shared by the vesicle-mediated transport, the cofactor metabolic process, and the cellular protein catabolic process categories. We looked within these biological groups which genes were specifically over- or underrepresented. Within the gene set related to vesicle-mediated transport those necessary for endocytosis are more strongly overrepresented. The other groups have a broader pattern of overrepresentation. For instance, in the translation category we can find more genes related to translation initiation, elongation, regulation, tRNA-amino acylation etc. in the low response group. Cellular amino acid metabolic process , cellular membrane organization , and heterocycle metabolic process are GO categories with a strong overrepresentation in the low response group, yet without the underrepresentation in the high response group. RNA metabolic process, transcription, ribosome biogenesis, and protein complex biogenesis are specifically strongly underrepresented in the high response group. Less categories show the reverse pattern with underrepresentation in the low response group and overrepresentation in the high response group, with the prominent exception of the category of uncharacterized genes . The smaller categories protein modification process, DNA metabolic process, and meiosis share this trend. Interestingly, within the latter two categories genes related to DNA repair, DNA recombination, and telomere maintenance are overrepresented. Comparison with an even higher level set of GO terms (‘Super GO-Slim’) resulted in the same broad patterns , we analyzed the gene composition of the upper and lower 5% bins (231 out of 4621 genes) of the widest distribution (patterns .S. cerevisiae (Yeast Go-Slim Function’). Over- and under-representation for respectively low and high response groups are clear for the protein binding and hydrolase categories. The former category has specifically more genes related to unfolded protein binding and ubiquitin binding. The latter category has specifically more genes with peptidase activity, and with ATPase activity (related to transmembrane movement of substances). Genes with oxidoreductase activity are strongly overrepresented in the low response group. The RNA binding category is strongly underrepresented in the high response group, mainly because of translation related genes. Similar as above, a striking under- and over-representation of unknown molecular function in the low and high response group, respectively, is found. The DNA binding category has the same, albeit less pronounced, pattern.http://www.yeastgenome.org/cgi-bin/GO/goSlimMapper.pl). Finally, we have analyzed the most significant differences based on a set of macromolecular complex terms  38 (σS) HE. coli genes. Firstly, we have found that the most abundant and most saturated sigma factor, σ70, is the least sensitive to RNAP, followed by σ54 (more than twice as sensitive) and σ28 (more than three time as sensitive) in accordance with their lower saturation, respectively. The same order is obeyed for the sensitivity to changes in (different) competitors, however; σ70 is expected to be hardly sensitive to the others. Indeed, it would make sense during exponential growth for the expression of growth-related and housekeeping genes directed by σ70 to be ‘robust’ to changes in accessory sigma factors. Contrary to that, we predict σ70 to induce a pronounced negative response on the σ28 (flagella and chemotaxis) and σ54 (nitrogen and stress responsive) directed genes.Due to the condition that the affecting competitor must bind a relatively large portion of the target pool, the antagonistic regulation of competitors is most easily demonstrated with a relatively small number of competitors, such as in the example of sigma factor competition in 70, which mimics the effect of a σ38 mutation. Both lead to induction of σ70 dependent genes, silencing of σ38-dependent genes, and inhibition of stress resistance E. coli stationary phase. σ38 was not included in our calculations since its presence is negligible in the exponential phase E. coli strain W3110, total levels of the main sigma factors remain the same in stationary versus exponential phase, with the exception of σ38, which increases to up to 30 percent of that of σ70. Interestingly, σ38 mutants are highly motile in comparison with wild-type 70 and σ28 as these are known to direct flagellar gene transcription. According to our theory it is expected that lowering σ38 would affect σ28 more than it affects σ70.In the stationary phase the predicted cross-talk has been confirmed by overproduction of σ38 also results in a growth advantage in stationary phase cultures of E. coli, a phenotype termed GASP 38 and σ54. In fact, the latter competition may be more prevalent than that between σ38 and σ70 in the context of metabolic functions. Using Phenotype Microarray analysis, it has been found that σ38 mutants metabolize 92 different nitrogen sources better than the wild type does, but only 8 carbon sources E. coli σ38 mutants versus wild type cells in the mouse colon 38 and σ54. Based on our framework and assuming that the relative saturation of σ70 and σ54 in the stationary phase is similar to that during exponential growth, we can indeed predict that decreasing σ38 would alter the transcription flux of σ54 directed genes significantly more than that of the σ70 directed genes .Partial loss of sigma σe.g.In transcription and translation networks, competition is taking place between much larger numbers of competitors. Many different genes (or rather promoters) are competing for RNA polymerase and transcription factors, and thousands of different transcripts are competing for a common set of ribosomes and translation factors. In both cases, it has been shown that the target is only available in limiting concentrations e.g. mRNAs encoding ribosomal proteins, developmental transcription programs, etc.). Viruses which make use of the gene expression machinery of a host cell provides another example in which competition can be vital  and this in turn will affect the abundance of large groups of mRNAs simultaneously  coefficients and describe the response of the entire system to small perturbations in parameters, J upon infinitesimal relative change in the value of the parameter jp. The response coefficient is equally well defined for steady-state metabolite concentration changes as for flux changes. Finally, the control coefficient J in response to an infinitesimal change in one of the reaction rates iv of the pathway. To compare the response coefficients calculated with equations (2)-(4) to the values obtained with a detailed kinetic model, the following approximate formula was used to calculate response coefficients numerically from the model: e.g. a one percent change in parameter jp. As a control, for a number of cases the more tedious but analytical ‘Matrix Formalism’ MCA links ‘global’ control properties of a network to ‘local’ properties  and the numerical differential equation solver NDSolve was used for simulation. The steady-state values of the different variables were used to calculate the steady-state fluxes needed for the calculation of the response coefficients (following the procedure described above). Furthermore, the same steady-state concentrations were used as input to equations (2)-(4) and (8)-(10) for comparison.http://www.ploscompbiol.org). These data were downloaded to MS EXCEL™. As in cf.i.e. slightly below the maximum value of 8.33… Although the Siwiak and Zielenkiewicz dataset did not contain ribosome occupancies, we have completed it for more than 98 percent with the ribosome occupancy values from the yeast dataset compiled by Brockmann et al.i.e. 0.75). With some simple algebraic manipulations all Saccharomyces cerevisiae Genome Database’ , which maps annotations of a group of genes to more general terms and/or bins them into broad categories, i.e. GO Slim terms.For the application of expression (8) to a genome-scale data set, we have made use of the data published by Siwiak and Zielenkiewicz Text S1Derivation of equations (2)-(4).(DOC)Click here for additional data file.Text S2Proof for specific relations between control coefficients in competing reactions. Proof for relations between control coefficients in parallel reactions competing for target, needed to derive equation (2)-(4) (cf.-(4) cf..(DOC)Click here for additional data file.Text S3Short proofs for various properties of response coefficients.(DOC)Click here for additional data file.Text S4Ultrasensitivity condition for . Derivation of ultrasensitivity condition for (DOC)Click here for additional data file.Text S5Equations for the general case of multiple target binding. The formulas for multiple competitors and multiple target binding can be derived analogously to equations (2)-(4) in (DOC)Click here for additional data file.Figure S1Flux responses in a network of three competitors: effect of large perturbations. Comparison of the flux response coefficients (towards changes in target concentration) calculated with expression (2) (depicted by the big dots), to the values obtained with the basic model (cf.70), 370 molecules/cell for competitor 2 (σ54) and 110 molecules/cell for competitor 3 (σ28). Other parameters needed for simulation are as in odel cf. for thre(TIF)Click here for additional data file.Figure S2Input-output relations in a network of three competitors. Comparison of the fluxes (in molecules.min−1) obtained through simulation of the basic model (cf.70), 370 molecules/cell for competitor 2 (σ54) and 110 molecules/cell for competitor 3 (σ28). Other parameters needed for simulation were the reaction rate constants: α1,fk  = 24, α2,fk  = 8, α3,fk  = 11 (molecules−1.min−1), α1,rk  = 3, α2,rk = 6, α3,rk = 30 (min−1), and β1’k = β2’k = β3’k = 5 (min−1). The values were selected to fit the sigma factor example at a total target concentration of 700 molecules/cell (cf.odel cf. for threcell cf..(TIF)Click here for additional data file.Figure S3Input-output relations in a network of three competitors. (A). Flux response coefficients calculated for three competitors, simulated at different levels (1-3000 molecules/cell) of competitor 1. By analogy with the sigma factor example in the main text the total competitor levels that were fixed, were taken to be 370 molecules/cell for competitor 2 (σ54) and 110 molecules/cell for competitor 3 (σ28). The total target level was set at 700 molecules/cell. Other parameters needed for simulation were as in −1) relation corresponding to (A).(TIF)Click here for additional data file.Figure S4Input-output relations in a network of three competitors. (A). Flux response coefficients calculated for three competitors, simulated at different levels (1-3000 molecules/cell) of competitor 2. By analogy with the sigma factor example in the main text the total competitor levels that were fixed, were taken to be 700 molecules/cell for competitor 1 (σ70) and 110 molecules/cell for competitor 3 (σ28). The total target level was set at 700 molecules/cell. Other parameters needed for simulation were as in −1) relation corresponding to (A).(TIF)Click here for additional data file.Figure S5Input-output relations in a network of three competitors. (A). Flux response coefficients calculated for three competitors, simulated at different levels (1-3000 molecules/cell) of competitor 3. By analogy with the sigma factor example in the main text the total competitor levels that were fixed, were taken to be 700 molecules/cell for competitor 1 (σ70) and 370 molecules/cell for competitor 2 (σ54). The total target level was set at 700 molecules/cell. Other parameters needed for simulation were as in −1) relation corresponding to (A).(TIF)Click here for additional data file.Figure S6Translation model. The validity of expressions (8-10) was tested by comparison with a protein synthesis model based on Heinrich and Rapoport’s model (TIF)Click here for additional data file.Figure S7Gene Ontology mapping  of high and low response groups. Histogram indicating the relative frequency (as %) of GO classes in high and low response gene sets. The genes corresponding to mRNAs with the 5% highest  or lowest  response coefficients were pooled and mapped with the GO Slim Mapper tool (http://www.yeastgenome.org/cgi-bin/GO/goSlimMapper.pl), based on the ‘Super GO-Slim Process’ GO set. This is a small set of very broad, high level GO Biological Process terms, useful for binning groups of genes in general categories. The corresponding percentages for the whole yeast genome  are represented by green bars (‘Genome’).(TIF)Click here for additional data file.Figure S8Gene Ontology mapping (in terms of molecular complex) of high and low response groups. Histogram indicating the relative frequency (as %) of GO classes in high and low response gene sets. The genes corresponding to the mRNAs with the 5% highest  and lowest  response coefficients were pooled and mapped with the GO Slim Mapper tool (http://www.yeastgenome.org/cgi-bin/GO/goSlimMapper.pl), based on the ‘Cellular Component’ ontology. This is a set of granular protein complex terms, useful for determining whether your protein of interest is a member of a particular complex. The corresponding percentages for the whole yeast genome  are represented by green bars (‘Genome’).(TIF)Click here for additional data file.


Analysis of elementary modes (EMs) is proven to be a powerful constraint-based method in the study of metabolic networks. However, enumeration of EMs is a hard computational task. Additionally, due to their large number, EMs cannot be simply used as an input for subsequent analysis. One possibility is to limit the analysis to a subset of interesting reactions. However, analysing an isolated subnetwork can result in finding incorrect EMs which are not part of any steady-state flux distribution of the original network. The ideal set to describe the reaction activity in a subnetwork would be the set of all EMs projected to the reactions of interest. Recently, the concept of "elementary flux patterns" (EFPs) has been proposed. Each EFP is a subset of the support  of at least one EM.We introduce the concept of ProCEMs (Projected Cone Elementary Modes). The ProCEM set can be computed by projecting the flux cone onto a lower-dimensional subspace and enumerating the extreme rays of the projected cone. In contrast to EFPs, ProCEMs are not merely a set of reactions, but projected EMs. We additionally prove that the set of EFPs is included in the set of ProCEM supports. Finally, ProCEMs and EFPs are compared for studying substructures of biological networks.We introduce the concept of ProCEMs and recommend its use for the analysis of substructures of metabolic networks for which the set of EMs cannot be computed. Metabolic pathway analysis is the study of meaningful minimal pathways or routes of connected reactions in metabolic network models ,2. Two cEM and EXPA analysis are promising approaches for studying metabolic networks ,16. HoweProjected Cone Elementary Modes (ProCEMs) for the analysis of substructures of metabolic networks. The organisation is as follows. Firstly, the mathematical concepts used in the text are formally defined. Secondly, we review the studies which have tried to investigate (some of) the EMs or EXPAs of large-scale networks. In the next step, we present the concept of ProCEMs and propose a method to compute them. Finally, we compare ProCEMs with elementary flux patterns (EFPs) from the mathematical and computational point of view, and analyse some concrete biological networks.The goal of the present paper is to introduce the new concept of N with m internal metabolites and n reactions. Formally, we describe N by its stoichiometric matrix S ∈ ℝm × n and the set of irreversible reactions Irr ⊆ {1, . . ., n}. If steady-state conditions hold, i.e., there is no net production or consumption of internal metabolites, the set of all feasible flux distributions defines a polyhedral coneWe consider a metabolic network flux cone [which is called the (steady-state) lux cone ,2.polyhedral cone in canonical form is any set of the form P = {x ∈ ℝn | Ax ≤ 0}, for some matrix A ∈ ℝk × n. To bring (1) in canonical form, we can replace the equalities Sv = 0 by the two sets of inequalities S · v ≤ 0 and -S · v ≤ 0. Furthermore, the inequalities i v≥ 0, i ∈ Irr are multiplied by -1. Any non-zero element x ∈ P is called a ray of P. Two rays r and r' are equivalent, written r ≅ r', if r = λr', for some λ > 0. A ray r in P is extreme if there do not exist rays r', r"∈ P, r' ≇ r" such that r = r' + r".A v ∈ ℝn, the set supp(v) = {i ∈ {1, . . ., n} | i v≠ 0 } is called the support of v.For every e ∈ C is called an elementary mode (EM) [v ∈ C \ {0} such that supp(v) ⊊ supp(e). Thus, each EM represents a minimal set of reactions that can work together in steady-state.A flux vector ode (EM) ,4 if theE = {e1, e2, . . ., se}, generates the cone C [C can be written as a non-negative linear combination of the vectors in E.The set of all pairwise non-equivalent EMs, e cone C . This meQ ⊆ × n of dimension p resp. q with p + q = n, the projection of Q onto Given a set Q = {v}, we simply write In the special case p + q reactions and a subnetwork p "interesting" reactions. For the flux cone C of C on the subspace projected cone on projected cone elementary mode (ProCEM). The projection e ∈ C to the subspace projected elementary mode (PEM). As we will see in the sequel, the two concepts of PEM and ProCEM are closely related but different.Now consider a metabolic network If the subnetwork As mentioned above, the set of EMs of a genome-scale network may be large, and in general, cannot be computed with the available tools. Even if this is possible, one cannot simply extract interesting information from it. Therefore, a subset of EMs  should be computed to reduce the running time and/or output size of the algorithm. Several approaches to this problem have been proposed in the literature. These strategies can be classified into four main categories:yield analysis [The first strategy is to constrain the complete set of EMs (or EXPAs) to a subset describing a phenotype space or a set of phenotypic data. For example, Covert and Palsson  showed tanalysis . In thisA second strategy to focus on the EMs (or EXPAs) of interest is to select a (possibly disconnected) subsystem, rather than the complete metabolic model, by assuming all other reactions and metabolites to be "external", and computing the EMs (or EXPAs) of this selected subsystem. This idea, i.e., cutting out subsystems or splitting big networks into several subsystems, is broadly used in the literature . In H. pylori network [Although this strategy is useful, it can result in serious errors in the computational analysis of network properties . For exa network  results  network  suggest  network  can resuTo better understand this problem, we consider Figure flux pattern is defined as a set of reactions in a subnetwork that is included in the support of some steady-state flux vector of the entire network [elementary flux pattern if it cannot be generated by combination of two or more different flux patterns. Each EFP is the support of (at least) one PEM. It is suggested that in many applications, the set of EFPs can be used instead of EMs [We observed that some errors may appear in the analysis of isolated subsystems. One possible solution to this problem is to compute a "large" subset of PEMs, or alternatively, as suggested by Kaleta et al. , to comp network . A flux d of EMs .Although EFPs are promising tools for the analysis of metabolic pathways, they also have their own shortcomings. The first important drawback of EFPs is that they cannot be used in place of EMs in certain applications , where ti and j are partially coupled [e and f, such that ie/j e≠ i/fj f[i, j}. From the theoretical point of view, finding all EMs that correspond to a certain EFP is computationally hard .17, while the subnetwork SuN induces a 9-dimensional subspace G is the set of EMs in Figure supp(p8) = E4 ∪ E5, supp(p9) = E3 ∪ E5, and supp(p10) = E1 ∪ E2. Hence, the flux patterns corresponding to these PEMs are not elementary. Therefore, some EMs may exist in the network which have no corresponding EFP on a certain subnetwork. This means that by EFP analysis possibly many EMs of the original network cannot be recovered. Informally speaking, we ask whether the set of EFPs is the largest set of PEM supports which can be computed without enumerating all EMs.Every EFP is related to at least one EM in the original metabolic network. However, one of the limitations of EFP analysis is that EFPs are activity patterns of some EMs, not necessarily all of them. We will show this by an example. In Figure x1, x2, x3 correspond to reactions 1, 2, 3, thus the flux cone is the open triangle shown in light gray). This network has two EMs, which are the generating vectors of the flux cone, g1 and g2. Now, if we are interested in a subnetwork composed of reactions 1 and 2, then we can project the flux cone to the 2D subspace produced by these two reactions. This is comparable to light projection on a 3D object to make 2D shadows. The projected cone is shown in dark gray. When the flux cone is projected onto the lower-dimensional space, new generating vectors may appear. In this example, g1 and g3 (in 2D space) are the generating vectors of the projected cone. Intuitively, one can think about g3 as the projected flux vector through reaction 1 and 3. This projected flux cone is certainly different from the flux cone of a network made by deleting reaction 3 , which are in principle the extreme rays of the cone obtained by projecting the original flux cone onto the subspace of boundary reactions. They suggest that the extreme rays of this "conversion" cone, i.e., the ECMs, can be computed even for large networks [Historically, the idea of flux cone projection has already been used in some papers. Wiback and Palsson  suggestenetworks .Following this idea, we introduce the ProCEM set ("Projected Cone Elementary Mode" set), which is the set of EMs of the projected flux cone. In contrast to , we formS ∈ ℝm×n of the network is Irr ⊆ {1, . . ., n}, and the set of reactions ∑ ⊆ {1, . . ., n} in the subnetwork of interest, while as an output it will return the complete set of ProCEMs. The computation of ProCEMs is achieved in three main consecutive steps.Our algorithm needs three input objects: the stoichiometric matrix Step 1 - Preprocessing: The aim of this step is to remove inconsistencies from the metabolic network and to transform it into a form suitable for the projection in Step 2. First, based on ∑ we sort the columns of S in the form:i-th column belongs to ∑ iff the i-th column is in Ā. Next, the blocked reactions [where the reaction corresponding to the eactions  are remoA represent the "interesting" reactions after splitting reversible reactions and removing the blocked reactions. In the following, we assume that A (resp. B) has p (resp. q) columns. Given S', the steady-state flux cone in canonical form will look as followswhere the columns of G (resp. H) represent the columns to be kept (resp. eliminated):where matrix p Idenotes the p × p identity matrix, and 0p,q the p × q zero matrix.Here Step 2 - Cone Projection: In this step, the flux cone is projected, eliminating the reactions corresponding to columns in H. Several methods have been proposed in the literature for the projection of polyhedra [block elimination method [projection cone. In our case, the projection cone is defined asolyhedra . For ourn method . This meT Hdenotes the transpose of H.where r1, r2, . . ., kr} of W using the double description method [We enumerate the extreme rays {n method . The prowhereW, thus a large number of them might be redundant [This representation of the projected cone contains as many inequalities as there are extreme rays in edundant . These rStep 3 - Finding ProCEMs: In the final step, the extreme rays of the projected cone, i.e., the ProCEMs, are enumerated. Similarly as in Step 2, the double description method is employed to enumerate the extreme rays of t subsets and then iteratively execute Step 2, eliminating every subset of reactions one by one. By proceeding in this fashion, the intermediary projection cones, W1, W2, . . ., t Wget typically smaller, thus enumerating their extreme rays requires less memory. On the other side, the more sets we partition into, the slower the projection algorithm usually gets.With the block elimination algorithm, it is also possible to perform the projection in an iterative manner. This means that rather than eliminating all the "uninteresting" reactions in one step, we can partition these in The ProCEM enumeration algorithm has been implemented in MATLAB v7.5. In our implementation, polco tool v4.7.1 ,13 is usArabidopsis thaliana [The metabolic network model of red blood cell (RBC)  is used thaliana   | u ∈ J}.Proof. Suppose that for some F ∈ E, there exists no v ∈ J such that F = supp(v). Since F is an EFP, there exists p ∈ P such that F = supp(p). It follows p ∉ J, but C is the flux cone. Therefore, there exist r ≥ 2 different ProCEMs, say u1,..., r u∈ J, such that k >c0 for all k. Since k u≥ 0, for all k, we have supp(ku) ≠ supp(p) for all k. Since supp(ku) is a flux pattern for all k, this is a contradiction with F being an EFP. □Here, we present a simple algorithm to show that it is possible to compute the set of EFPs when the set of ProCEMs is known. Table u is a flux pattern Z. In the main procedure, we check whether every such flux pattern is elementary or not. If Z is not elementary, then it is equal to the union of some other flux patterns. Therefore, if all other flux patterns which are subsets of supp(u) are subtracted from Z, this set becomes empty. This algorithm has the complexity q is the number of ProCEMs and n is the number of reactions.We know that the support of every ProCEM In order to compare our approach (computation of ProCEMs) with the enumeration of EFPs, we tested these methods for analysing subnetworks of the RBC model . Again, When we computed the EFPs of this network by EFPTools , only 90r reactions was selected. Then, EFPs and ProCEMs were computed. The task was repeated for different subnetwork sizes. The computational results can be found in Figure In order to compare the computation of EFPs and ProCEMs, the following task was performed on the RBC model . In eachr increases, computation of ProCEMs does not become slower, while computation of EFPs significantly slows down. This is an important observation, because the difference between the number of EFPs and ProCEMs also increases with r.From Figure ProCEM analysis becomes important when PEMs cannot be computed. This may happen frequently in the analysis of large-scale metabolic networks, as memory consumption is a major challenge in computation of EMs . In suchA. thaliana plastid was studied . The set of ProCEMs covers more PEMs than EFPs. Therefore, ProCEMs contain more information than EFPs. The set of ProCEMs is computable without enumerating all EMs. Is there a bigger set of vectors that covers even more PEMs and does not require full enumeration of EMs? This question is yet to be answered. One possible extension to this work is to use a more efficient implementation of polyhedral projection. With such an implementation, analysis of different subnetworks in genome-scale network models using ProCEMs is an interesting possibility for further research. For example, the ProCEMs can be used in the identification of all pathways with optimal yield  and in tThe authors declare that they have no competing interests.The original idea was presented by SAM, LD and AB. The mathematical results are presented by SAM, and improved by all authors. Implementation of the ProCEM method and performing the computational experiments are done by LD. The manuscript was originally drafted by SAM, and improved by all authors. The final version of the manuscript was read and approved by all authors.Plastid metabolic network. In the first tab of this Excel file, general information about the plastid network of A. thaliana is mentioned. In the second tab, stoichiometric matrix and the set of reversible reactions (as a binary vector) is included. In the third tab, the reactions involved in sugar and starch metabolism are listed.Click here for file


Metabolic reconstruction is the computational-based process that aims to elucidate the network of metabolites interconnected through reactions catalyzed by activities assigned to one or more genes. Reconstructed models may contain inconsistencies that appear as gap metabolites and blocked reactions. Although automatic methods for solving this problem have been previously developed, there are many situations where manual curation is still needed.iCG238, the genome-scale metabolic model for the bacterium Blattabacterium cuenoti, obligate endosymbiont of cockroaches.We introduce a general definition of gap metabolite that allows its detection in a straightforward manner. Moreover, a method for the detection of Unconnected Modules, defined as isolated sets of blocked reactions connected through gap metabolites, is proposed. The method has been successfully applied to the curation of B. cuenoti iCG238 is a more accurate model version named as B. cuenoti iMP240.We found the proposed approach to be a valuable tool for the curation of genome-scale metabolic models. The outcome of its application to the genome-scale model  This relational information can be organized in the so-called gene-protein-reaction association tables (GPR) that codes the enzyme with activity EC 1.3.1.76 and the gene BLBBGE_278 [GenBank: CP001487] which codes an enzyme that catalyzes activities EC 2.1.1.107 and EC 4.99.1.4.After an inspection of Since siroheme is an important cofactor involved in sulfur and nitrogen metabolism, the cell would have to be able to maintain certain pool of this cofactor. Hence, during the bacterial growth phase, the organism will need some production of siroheme. In order to take into account this fact, siroheme was included into the biomass equation. After including this modification to the model, FBA was applied to find a metabolic state that maximizes biomass production. As expected, it was found that the four reactions included in UM5 and the four new reactions added to the model showed non-zero flux under optimal state  and L-glutamate 5-semialdehyde dehydratase annotated to be spontaneous. Both reactions were annotated as belonging to the proline biosynthetic pathway. In this scenario suggest that the pathway only lacks the last reaction step in order to be able to produce proline  could not be found in the BRENDA database. However it was found as an entry in BiGG database. For the2+ and Mg2+ and, the associated exchange fluxes. Both ions are included in the biomass equation as described by other authors, and its inclusion in the new model predicts that (S)-Dihydroorotate can be produced by the metabolism of the endosymbiont.The iMP240 also suggest six new compounds that need to be present in the medium in order for the organism to be able to grow. These sets of metabolites include: i) mevalonate, which is needed to synthesize menaquinol and 2-demetyl-menaquinol; ii) glycerol, which is phosphorylated by the activity EC 2.7.1.30, and used in the biosynthesis of phosphatidylglycerol species. In model iGC238 glycerol-3-phosphate is produced from dihydroxyacetonephosphate through the reaction EC 1.1.5.3 operating in reverse sense. However, it was not possible to found any evidence suggesting that the reaction EC 1.1.5.3 could operate in a reversible manner. Hence, if the reaction is considered as irreversible, then the cell cannot produce glycerol-3-phosphate. Then glycerol should be uptaken from the medium and phosphorylated inside the cell. iii) Porphobilinogen that is converted into hydroxymethylbilane, the first precursor in the synthesis of siroheme, which in turn requires Fe2+; iv) the case of K+ and Mg2+ are trivial: these ions become required in the minimal medium after their inclusion into the biomass equation.The predictions using the model in-silico analysis of the model iMP240. These results were then compared with the set of essential genes previously predicted for model iCG238[iCG238 remains as essential in iMP240. In addition, a set of 30 genes was predicted as essential in iMP240, resulting in total of 202 metabolic genes predicted as essential (~84%). As it is depicted in Figure iCG238), while the remaining 5 are new genes.The set of essential metabolic genes were computed by an el iCG238. Figure iCG238 because the cofactor was not included in the biomass equation. However, the cofactor must be essential to the organism, due to its strict aerobiosis, and then it should be included as a biomass component. As a consequence most of the genes coding the biosynthetic pathway of menaquinol are predicted as essential in the new scenario. Moreover, the 5 new genes included in iMP240 that are predicted as essentials, include two genes that code for three steps in the biosynthesis of siroheme, two genes that code for the transport of nicotinamide and Mg+2, and the gene that encode Ftsl, an essential cell division protein.This increase in the predicted network fragility obtained after model curation is explained, in greater extent, due to the addition of new components to the biomass equation. For example, the genes coding for the biosynthetic pathway of menaquinol where predicted as non-essential in model Unconnected Modules (UM), defined as isolated sets of blocked reactions connected through gap metabolites have been proposed. The visual representation of UM can shed useful information that may help the model’s curator to solve inconsistencies. Furthermore, the present approach can be combined with existing tools in order to find a set of model modifications that solves the inconsistencies and thus improves model’s predictions.In this paper we have introduced a general definition of gap metabolite that allows its detection in a straightforward manner, even for the cases of upstream-non-produced and downstream-non-consumed metabolites. Moreover, a method for the detection of B. cuenoti iCG238. In this way every blocked reaction detected in the model was successfully unblocked or alternatively removed, in those cases where there was not enough information supporting the existence of such reactions. As an example, those reactions found as blocked and with no gene association were excluded. Moreover, new reactions were added to the model based on the careful revision of the genome annotation that allows the identification of gene functions previously not included, as well as the incorporation of new compounds into the biomass equation. As a consequence of model curation a new GSM version of B. cuenoti, named as iMP240, is proposed. The impact of these modifications, with respect to some structural properties of the networks, was analyzed by performing different in-silico analysis over each model’s version.This method was applied to the curation of the GSM of the cockroach endosymbiont As a final commentary, the method here presented can be considered as a semi-automatic approach that has the advantage of allowing a quick representation of the gaps of the model but that needs the supervision of an expert in the biology of the studied organism. This issue may be seen as a drawback of the method but, as in the case of automatic genome annotation, there is a trade-off between the degree of automation of the metabolic reconstruction and the quality of the generated model.RNP: Root-non-produced; RNC: Root-non-consumed; DNP: Downstream-non-produced; UNC: Upstream-non-consume; UM: Unconnected module; FBA: Flux balance analysis; CBM: Constraint based modeling; GSM: Genome scale model; RS: Reactions subset; LP: Linear programming; MILP: Mixed integer linear programming; GPR: Gene-protein-reaction; EC: Enzyme commission number; TC: Transport commission number.The authors declare that they have no competing interests.in-silico experiments. All authors contributed in designing research, analyzing the data, and writing the paper. All authors have read and approved the manuscript.JP, FM and MPL presented the original idea of the work and designed the study. MPL conceived the algorithmic procedure, implemented the code, and carried out the Schematic representation of UMs. The file contains schemes and a brief description of each UM.Click here for fileNew GSM model of B. cuenoti iMP240. Spreadsheet with the model description. The file contains three sheets corresponding to Metabolites, Reactions, and Exchange Fluxes, respectively.Click here for fileComparative table iCG238 vs iMP240. Spreadsheet describing the main differences between both model versions. The file contains three sheets: the first one includes the genes added as well the genes that were removed; the second one presents the removed reactions; and the third one shows the set of added reactions.Click here for file


Yarrowia lipolytica, a model oleaginous yeast, which can be employed to accumulate large amounts of lipids that could be further converted to biodiesel. In order to understand the metabolic characteristics of Y. lipolytica at a systems level and to examine the potential for enhanced lipid production, a genome-scale compartmentalized metabolic network was reconstructed based on a combination of genome annotation and the detailed biochemical knowledge from multiple databases such as KEGG, ENZYME and BIGG. The information about protein and reaction associations of all the organisms in KEGG and Expasy-ENZYME database was arranged into an EXCEL file that can then be regarded as a new useful database to generate other reconstructions. The generated model iYL619_PCP accounts for 619 genes, 843 metabolites and 1,142 reactions including 236 transport reactions, 125 exchange reactions and 13 spontaneous reactions. The in silico model successfully predicted the minimal media and the growing abilities on different substrates. With flux balance analysis, single gene knockouts were also simulated to predict the essential genes and partially essential genes. In addition, flux variability analysis was applied to design new mutant strains that will redirect fluxes through the network and may enhance the production of lipid. This genome-scale metabolic model of Y. lipolytica can facilitate system-level metabolic analysis as well as strain development for improving the production of biodiesels and other valuable products by Y. lipolytica and other closely related oleaginous yeasts.With the emergence of energy scarcity, the use of renewable energy sources such as biodiesel is becoming increasingly necessary. Recently, many researchers have focused their minds on  Nowadays with the rapidly developed world's economy, a series of problems especially energy scarcity come out and become more and more serious. As a finite and unsustainable fuel source Yarrowia lipolytica is now long used as a model organism for lipid accumulation partly because it is the only one with developed genetic tools Recently many researchers have been focusing their minds on oleaginous microorganisms that are able to synthesize and accumulate oil at least 25% of their dry weight and therefore act as ideal candidates for fuel production Y. lipolytica, formerly known as Candida, Endomycopsis or Saccharomycopsis lipolytica, is often found in environments rich in hydrophobic substrates, such as alkanes, fatty acids and oils Y. lipolytica yeast have been isolated and can be efficiently cultivated on various substrates such as glucose (but not sucrose), alcohols and acetate. Via de novo synthesis pathway, this yeast is able to accumulate large amounts of lipids, in some cases to the level exceeding 50% of cell dry weight [DW] Y. lipolytica was determined by B. Dujon et al. Y. lipolytica have been established, making rapid gene deletion and other molecular-level research possible. From the above, Y. lipolytica has been chosen by many researchers as the target microorganism for mechanism and regulation studies of lipid accumulation. However, even with existing tools and knowledge about Y. lipolytica, slow progress has been made towards understanding of lipid accumulation with classical metabolic engineering approaches.As the only ascosporic member of the genus Yarrowia, the yeast in silico modeling of an organism's metabolic capabilities E. coli was released in 2000 http://synbio.tju.edu.cn/GSMNDB/gsmndb.htm). Though the standardized metabolic network reconstruction process has been well established The large quantity of information featured in public databases, like details about genomes, pathways and proteins has led to metabolic network reconstructions and Y. lipolytica iYL619_PCP was reconstructed based on a combination of genome annotation and the more detailed biochemical knowledge. Using this metabolic reconstruction, we firstly predicted the substrates of in silico model and then performed gene knock-out and flux variability analysis to predict the essential genes in iYL619_PCP using glucose as the sole carbon source and help design new strains. Overall, iYL619_PCP provides a valuable tool and platform for the system-level metabolic analysis of Y. lipolytica and may also provide important metabolic basis and feasible engineering ways for the improvement of fatty acid synthesis in many other oleaginous yeasts.In this study the metabolic network model of Y. lipolytica, information about genome, reaction and relevant protein was mined from multiple available databases, i.e. KEGG Up to now, there are a large quantity of public databases that can be used to construct a genome-scale metabolic network Y. lipolytica iYL619_PCP was constructed. Because of lots of mistakes in databases such as the missing of proton in one reaction, manual curation was needed. With the help of pathway in KEGG, the draft reconstruction was refined and assembled in a pathway-by-pathway manner. In this step, we compiled programs to find the deadend metabolites , were calculated from various sources including published articles and available experimental data , i.e., related to polymerization of protein, or non-growth associated (NGAM) that is related to maintaining membrane potential. In order to estimate the maintenance requirements, experimental data obtained from continuous cell growth on glycerol was used in silico cell growth rate or production rates of biotechnologically important metabolites such as ATP.FBA As mentioned above, FBA relies on the imposition of a series of constraints including equations that balance reaction inputs and outputs (Eq. 2) and inequalities that define the maximum and minimum allowable fluxes of the reactions, namely the directionality and enzymatic capacity constraints (Eq. 3). These balances and bounds imposed on model define an allowable solution space of a linear optimization  problem of maximize or minimize a cellular objective function such as the rate of biomass synthesis (Eq. 1) FBA can be performed using COBRA Toolbox Y. lipolytica. Consulting some available articles in silico minimal media was determined, i.e. the production of biomass is not zero. With the minimal media, each possible carbon source was allowed to flow into the in silico model one by one by adding exchange reactions (if there was no corresponding one) for the sake of simulating the growth of model under different environmental conditions. The results of substrate utilization prediction were compared with those obtained experimentally elsewhere Using FBA, we systematically predicted a minimal media composition capable of supporting growth of FVA is a frequently used computational tool to calculate the full range of numerical values for each reaction flux under a given simulation condition while maintaining some states of the network, e.g., supporting maximum biomass production rate i-th reaction in the wild-type strain; i-th reaction in the mutant strain with j-th gene deleted; j-th gene deletion, which was used to weigh the flux variation when the j-th gene was deleted.In order to design new strains, FVA was used to measure the variation of flux of a single-gene-deleted mutant strain compared with that of the wild-type strain with the equation introduced below:Y. lipolytica  was used to be compared with those in other databases. The number of genes and proteins obtained from each database was showed  and NG (substrate that can't be assimilated in experiment but can in silico). There are four GN-substrates including D-mannitol, D-glucitol, succinate and hexadecane and only one NG-substrate of trehalose. These instances of GN indicate that the model iYL619_PCP has a weaker ability than that it should have, whereas the instance of NG just in opposite, indicate the model has a stronger ability. There might be at least two reasons for four GN-substrates in the model iYL619_PCP. On one hand, some reactions associated with the assimilation of these four substrates were overlooked, resulting in gaps in biosynthetic pathways. On the other hand, the equation of biomass reaction might be responsible for four wrong predictions. For example, if a component was not a part of biomass but was wrongly included into the biomass equation, the predicted biomass synthesis flux will be zero in case that this component can not be synthesized from a certain carbon source. This implied that further validation and improvement of biomass compositions are required for the constructed model system. Apropos of trehalose, the only NG-substrate, a reaction catalyzed by the product of gene YALI0D15598g was found responsible for the synthesis of glucose from trehalose. Meanwhile, by referring to some articles Utilization of various substrates by tal data . During Y. lipolytica, a single gene deletion study with FBA was conducted with model iYL619_PCP in the glucose-minimal media described above. As shown in Y. lipolytica.In practical experiments, genetic manipulation has become an indispensable tool to design mutant strains and validate phenotypic outcomes. However, the process of gene deletion is very time-consuming. In order to reduce the scope of experiments and quickly investigate essentiality of each gene in i-th reaction in the wild-type strain was zero, j-th gene resulted in a non-zero flux of the i-th reaction in the mutant strain, the j-th gene might be a good candidate for strain engineering to achieve improved yield of the target product. For example, reaction “R0272” shows that serine is synthesized from 3-Phosphoglycerate. FVA results indicated that the minimum flux value of this reaction was zero in the wild-type strain, whereas the knockout of either gene “YALI0F16819g” or gene “YALI0F09185g” resulted in a minimum flux of 19.7162 mmol/g DCW/h and 19.6913 mmol/g DCW/h respectively. Although in this example Flux variability analysis was used to help design strains that can produce more lipids. Because lipids such as triglyceride are intracellular products, the production of lipids is in great relation to biomass production. Here in order to help design mutant strains that produce a large mount of lipids, the biomass production was set at the optimal value Y. lipolytica with the help of available knowledge in public databases and scientific publications. The model iYL619_PCP is the first genome-scale metabolic reconstruction for oleaginous yeasts with information directly derived from multiple databases. Both the SBML representation Click here for additional data file.Additional file S2A database of all protein and reaction associations sorted from KEGG and Expansy-Enzyme.(XLSX)Click here for additional data file.Additional file S3The reconstructed genome-scale metabolic model iYL619_PCP (XLS), including metabolite list, genome annotation, detail of reactions and metabolites, eliminated reactions and added reactions during the curation and some characters of the model.(XLSX)Click here for additional data file.Additional file S4in silico minimal media.Results of flux balance analysis and flux variability analysis and information used to help determine the (XLSX)Click here for additional data file.Additional file S5The model iYL619_PCP in SBML format(XML)Click here for additional data file.Additional file S6Yarrowia lipolytica.Determination of biomass composition and estimation of energy requirements of (DOC)Click here for additional data file.


The choice of carbon labeling pattern is an important consideration when designing these experiments. We present a novel Monte Carlo algorithm for finding the optimal substrate input label for a particular experimental objective (flux or flux ratio). Unlike previous work, this method does not require assumption of the flux distribution beforehand.Carbon-13 (E. coli isotopomer model, different commercially available substrate labeling patterns were tested computationally for their ability to determine reaction fluxes. The choice of optimal labeled substrate was found to be dependent upon the desired experimental objective. Many commercially available labels are predicted to be outperformed by complex labeling patterns. Based on Monte Carlo Sampling, the dimensionality of experimental data was found to be considerably less than anticipated, suggesting that effectiveness of 13C experiments for determining reaction fluxes across a large-scale metabolic network is less than previously believed.Using a large 13C analysis is a useful tool in systems biology, high redundancy in measurements limits the information that can be obtained from each experiment. It is however possible to compute potential limitations before an experiment is run and predict whether, and to what degree, the rate of each reaction can be resolved.While  In vivo metabolic reaction flux data provides insight into the dynamic function of the cell  glucose and [6-13C] glucose are plotted in Figure 13C] glucose scores higher at measuring the pentose phosphate pathway and most of lower glycolysis, whereas [1-13C] glucose glucose scores much higher at measuring the glyoxylate shunt. The results suggest that there is no single label that yields a high score for all experimental objectives. For example, the exchange of formate (EX_for) could be easiest measured with a  glucose; however, this labeling pattern is bested by [1-13C] glucose for the measurement of reaction formyltetrahydrofolate deformylase (FTHFD) , which gives a much higher Z-score than the best commercially available label,  glucose (Z = 18.5). Thus, there may be motivation to synthesize compounds with more complex labeling patterns than commonly used. Additionally, there are certain reactions which are predicted to be difficult to measure with any labeling pattern. For example, the Z-scores for each possible labeled glucose substrate for the reaction pyruvate oxidase (POX) all lie within the level of noise, as determined by the comparison with random hi-lo experiment Z-scores.To illustrate the differences in label-dependent reaction resolving capacity, two sets of Z-scores corresponding to 1-1C glucose) Figure . This no13C experiments show a clear pathway bias regardless of labeling pattern. The maximum Z-score of all labeling patterns was found for each reaction, giving a metric for the maximum potential for reaction flux determination using 13C-labeled glucose , no additional information is given from the data. Similarly, a dimensionality of one indicates that the data can specify one degree of freedom. Singular value decomposition (SVD) is a data reduction technique that allows the estimation of data dimensionality , consisting of all sample points generated from Monte Carlo Sampling, is decomposed into M = U · Σ · T V where U and V are orthonormal bases and Σ is a diagonal matrix containing singular values in descending order. The singular values are effectively weightings that describe the information content of the corresponding vectors in U and V towards reconstructing the full matrix M. A partial reconstruction of M is possible by taking only a subset of the singular values greater than some threshold. These thresholds have a direct interpretation as the uncertainty with which a data point can be measured. For example, a threshold cutoff of 0.01 indicates that the remaining uncertainty of the data falls within 0.01 or 1% error in the measurement of isotope enrichment.The Monte Carlo sampling approach enables the determination of the dimensionality of simulated y Figure . A data 13C fragments derived from uniformly sampled flux distribution sets for several glucose labels. The results are summarized in Figure 13C] glucose with 73 dimensions. The three labels for which experimental data was measured in the subsequent section, [1-13C] glucose, [6-13C] glucose, and 20% [U-13C] glucose, had dimensions 53, 53 and 35, respectively, at this cutoff. These values are all significantly lower than the best label, and, in particular, the uniform labeled experiment only produces half of the dimensionality as the optimal experiment. This result is significant. While 139 dimensions (the number of undetermined dimensions for the model used) are required to specify a unique flux vector, the dimensionality of the 13C data for each label is significantly lower. The best labeling experiment specifies just over half (73/139 = 0.52) the degrees of freedom required, and 20% [U-13C] glucose only specifies about one fourth of the possible degrees of freedom (0.26). It is worth noting that SVD is a linear operation used to approximate properties of a non-linear system and the true degrees of freedom may be even lower than reported. SVD serves as a useful upper bound on the dimensionality of data for non-linear systems, but the difference between SVD dimensionality and true dimensionality may grow to be unacceptable for large systems. For the system studied here, SVD was found to be of practical use.To determine the dimensionality of the isotopomer data, SVD was performed on E. coli. Flux distributions that best explain each set of 13C data were calculated using a non-linear optimization problem:In order to assess the agreement of computationally predicted flux elucidation capacity with experimental data, we took fluxomic measurements for three labeling patterns in The function Error(v) is a score of how well a given flux distribution fits the experimental data. It is defined as:i is the measured fractional enrichment of fragment i, fragment i(v) is the computed fractional enrichment of fragment i as a function of the flux distribution v, and α = 0:014 is the standard deviation of the fragments as calculated from experimental replicates.where measured 13C data and all combinations thereof. Confidence intervals for reaction rates were computed by maximizing and minimizing the value of each reaction in turn subject to a slightly relaxed score.Reaction flux confidence ranges were then computed for all reactions using all three sets of i c=  is a vector of all zeros with a 1 in position i, and max Errorwas set based on the confidence value. Because different data sets provide different levels of consistency, Errormax was chosen to be 30 units greater than the minimum error found.where <8.6 × 10-34). A receiver operating characteristic (ROC) curve suggests that the Z-scores can identify with both sensitivity and specificity the reactions that can be elucidated in a label-specific manner, with better performance predicting ranges that are restricted more by data , while adding 13C data reduces the ranges and shifts the curve left. With almost no exception, including one experiment yields larger confidence intervals than any combination of two carbon sources which in turn is a larger range than including all three sets. Of the single experiment curves, the 20% [U-13C] glucose curve provides notably worse ranges than the other two experiments, consistent with the finding that 20% [U-13C] substrate provides data with the smallest number of dimensions.The number of reactions elucidated at particular confidence intervals was then found Figure . Using d-1·h-1 (a relatively non-stringent cutoff), 85 reactions are specified simply from uptake rate data without any 13C data. Performing the least informative 13C experiment, using 20% [U-13C] substrate, yields 105 reactions that meet the confidence criterion, whereas the combination of all three 13C experiments yields 125 reactions that meet the criterion. In other words, performing all three experiments will increase the number of elucidated reactions by 40 reactions or about 50%. As the model used contains 278 carbon-tracking reactions and reaction groups, the increase in knowledge at 1 mmol · gDW-1·h-1 confidence from 85 to 125 reactions from using 13C data indicates that a large gap in the knowledge remains. It seems apparent that other methods must be developed to obtain flux information at the genome scale from single experiments, as would ultimately be desirable. However, as noted in the above section, the majority of reactions that are elucidated by 13C-labeled glucose experiments lie in central metabolic pathways, which tend to be both of high interest and not-specifiable by constraints alone.At a reaction confidence of 1 mmol · gDW13C experiments using Monte Carlo Sampling. This allows us to predict the success of experiments before performing them. The method used here 1) does not require experimental identification of the 'real' flux state a priori [We introduce a new framework for calculating the uncertainty inherent to a priori  and 2) ra priori . This fr13C] glucose or [6-13C] glucose. Thus, the amount of information likely to be obtained by a 13C experiment can be predicted in a reaction-specific manner before having to carry out an experiment.• The choice of input label is important, as different labels perform better than others. In particular, the commonly-used 20% mixture of uniform label + 80% natural label was shown computationally and experimentally to resolve significantly fewer reaction fluxes than either  glucose and  glucose, are predicted to perform better than commercially available single labels for many reactions.• There is no universally best label. The best label depends on the experimental objective. Certain reactions are more precisely measured with some labels than others, and no label is best at elucidating all reactions. Certain hypothetical 13C data dimensionality is less than anticipated. Whereas each 13C experiment can measure 186 pieces of information at a time, there is a high degree of interdependence. We measured the true data dimensionality to be between 35 and 50 dimensions for commercially available labels and as high as 73 for exotic labels. This high data redundancy can partially explain why 13C experiments yield many reaction rates with high uncertainties.• The 13C analysis using solely amino acids due to the lower than expected dimensionality of the isotopomer data. However, steady-state 13C analysis is clearly still useful elucidating reaction fluxes in E. coli metabolism. Notably, as the study was conducted using only protein-derived amino acids, it would be of immediate interest to determine the additional benefit of measuring other classes of labeled metabolites, as well as the benefit of more recently developed experimental techniques such as multi-substrate -labeled, 100% [6-13C]-labeled, or a mixture of 20% uniformly [U-13C]-labeled with 80% natural glucose (which is randomly 1% 13C). Cells were grown to mid-log phase, corresponding to OD600 of 0.6. 3 ml of each culture was harvested by centrifugation at 4°C. The media was aspirated and analyzed with HPLC to determine the remaining glucose concentration. Cell pellets were placed at -80°C prior to further analysis.Prior to labeling, single colonies of μl each of tetrahydrofuran and N-tert-butyldimethylsilyl-N-methyltrifluoroacetamide (Aldrich), and incubated for 1 hr at 80°C to derivatize amino acids. Samples were filtered through 0.2 μm PVDF filters and injected into a Shimadzu QP2010 Plus GC-MS (0.5 μl with 1:50 split ratio). GC injection temperature was 250°C and the GC oven temperature was initially 130°C for 4 min, rising to 230°C at 4°C/min and to 280°C at 20°C/min with a final hold at this temperature for 2 min. GC flow rate with helium carrier gas was 50 cm/s. The GC column used was a 15 m × 0.25 mm × 0.25 m SHRXI-5ms (Shimadzu). GC-MS interface temperature was 300 degrees with 70 eV ionization voltage. The mass spectrometer was set to scan an m/z range of 50 to 600.Cells were resuspended in 0.1 ml 6 M HCl and transferred to glass vials. Protein was digested into amino acids under a nitrogen atmosphere for 18 hr at 105°C in an Eldex H/D Work Station. Digested samples were dried to remove residual HCl, resuspended with 75 Mass data were retrieved from the GC-MS for fragments of 14 derivatized amino acids: cysteine and tryptophan were degraded during amino acid hydrolysis; asparagine and glutamine were converted respectively to aspartate and glutamate; arginine was not stable to the derivatization procedure. For each fragment, these data comprised mass intensities for the base isotopomer , and isotopomers with increasing unit mass (up to M+6) relative to that of M+0. These mass distributions were normalized by dividing by the sum of M+0 to M+6, and corrected for naturally-occurring heavy isotopes of the elements H, N, O, Si, S, and (in moieties from the derivatizing reagent) C, using matrix-based probabilistic methods as described ,38 imple13C data as described in the Results. In calculating the best fit flux values from experimental data, a small variation was introduced to reduce the number of variables and remove constraints. Let N be a basis for the null space of S. Then all valid fluxes can be written as:Reaction rates were computed from v| = 335 to |α| = 139. Optimization was performed with the Tomlab/SNOPT package. This method is an iterative local optimization and is therefore not guaranteed to find the optimal solution. To address the issue of local minima, the procedure was run with many randomly generated starting points and the lowest minimum was taken.This reduced the number of variables from |The code was written in the MATLAB environment and the COBRA toolbox. Linear Programming was done with the Tomlab/CPLEX package and nonlinear optimization with the TOMLAB/SNOPT interface. The EMU and cumomer method were written in native Matlab but generated in Perl. Computations were performed on a Dell Studio XPS desktops (2.6 Ghz core i7 with 9-12 GB ram) and a custom Rocks cluster .E. coli samples. DS performed MS fragmentation and natural abundance correction. JLR generated the 13C E. coli model with 13C mappings. AO and BOP were responsible for the strategic vision. All authors have read and approved the manuscript.JS devised methods and drafted manuscript. DZ performed additional analysis and completed manuscript. WC and SM implemented methods in Matlab and performed computational analyses. VP grew Z-scores, confidence intervals, and isotopomer model. This file contains the absolute and relative Z-scores for individual reactions across all glucose labeling patterns tested, confidence intervals calculated using experimental 13C tracing data, and details of the model that was used in calculations.Click here for file