Identification of Network Differences via CONGA

CONGA identifies functional differences between two networks by comparing network reconstructions aligned at the gene level. The constraint-based method identifies gene deletion strategies leading to different optimal flux distributions in the two networks. CONGA calculates the flux difference between two reactions in different models (e.g., Flux 1 in Species A minus Flux 2 in Species B) and identifies deletions such that the specified flux difference is maximized while both models are simultaneously maximizing biomass (Figure 1).

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Figure 1. Conceptual structure of the CONGA formulation.

CONGA employs a bilevel optimization problem to identify genetic perturbations with nonidentical effects in each of two networks. The outer problem is an MILP which finds gene deletions maximizing the difference in flux value between two reactions in two different models. The inner problems (in italics) are flux-balance analysis (FBA) problems which ensure the flux difference is maximized while both models are maximizing biomass. An optional tilt can be added to the inner problem which forces the flux in the outer problem to the lowest value that still support maximum biomass production. FBA imposes constraints based on reaction stoichiometry, reaction directionality, and enzyme capacities. GPR constraints associate genes to reactions and are used to enforce the reaction deletions associated with the gene deletions in the outer problem. CONGA can select any genes for deletion, with the restriction that orthologous genes present in both models be deleted simultaneously from both models. Finally, a limit may be imposed on the total number of gene deletions.

https://doi.org/10.1371/journal.pone.0034670.g001

We refer to a solution identified by CONGA as a gene deletion set. CONGA can select any genes for deletion, with the restriction that orthologous genes present in both models be deleted simultaneously from both models. We note that while CONGA can calculate the flux difference between any two reactions, we believe that selecting equivalent reactions (e.g., biomass) provides the most useful objective for comparing models. Via manual investigation of the results, we are able to classify gene deletion sets identified by CONGA as arising due to one of four types of functional network differences:

1. genetic differences, in which gene-protein-reaction (GPR) relationships differ between models;
2. orthology differences, in which genes encoding enzymes with identical functions cannot be assigned as orthologs (e.g., due to sequence dissimilarity);
3. metabolic differences, where one organism has additional reactions which enable it to carry out unique biochemical transformations; and
4. mixed differences, which arise due to some combination of types 1–3.

Using two example networks, we demonstrate the types of functional differences CONGA can identify (Figure 2). Each reaction network catalyzes the conversion of substrate (S) to biomass (BM) and some by-product (P) (Figure 2A). We refer to the two species as A and B, and the biomass- and by-product-producing reactions as and , respectively. Each pathway producing biomass gives different yields for BM and P (Figure 2B), though the optimal flux distributions maximizing biomass without any gene deletions are identical in the two organisms (Figure 2C). By applying CONGA with different objective functions, we can identify gene deletion conditions under which network differences become apparent (Figure 2D).

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Figure 2. Application of CONGA to an example pair of metabolic networks.

(A) In these two example networks, substrate (S) is utilized to produce biomass (BM) and some by-product (P). We refer to the two species as A and B, and the biomass- and product-producing reactions as and , respectively. (B) List of genes and reactions present or absent in each network. All shared reactions have orthologs present in both networks, except for the reaction associated with genes G23a and G23b, which are not orthologs. (C) A schematic view of the wildtype network behaviors in which flux through is maximized. (D) Gene deletion sets identified by CONGA for the stated CONGA objectives The first three objectives maximize in Species B over Species A. The last objective maximizes in Species A over Species B. The type of model difference (genetic, orthology, or metabolic) associated with each deletion set is also given. (E through H) Schematic views of the flux distributions associated with each gene deletion set in D. The optimal flux distributions in the example networks change as a result of the gene deletion sets in D. Differences in the optimal flux distributions are due to differences in the two networks.

https://doi.org/10.1371/journal.pone.0034670.g002

We first used CONGA to compute gene deletion sets maximizing in Species B over Species A (). This objective will be greatest when a gene deletion set is predicted to be lethal in Species A and not in Species B. One such deletion set contains the ortholog G12, which is present in both models (Figure 2E). Under this deletion, growth becomes impossible in Species A, whereas Species B has additional reactions which allow it to convert I1 to B via metabolite I4. Thus, this gene deletion set points to a metabolic difference between the two models. CONGA can also be used to identify genetic differences (Figure 2F). For instance, the deletion of GS1 is lethal only in Species A, because Species B has an additional isozyme (GS1a) which carries out the same transformation. Thus, this deletion set points to a genetic difference. Other deletion sets point to orthology differences (Figure 2G). For example, genes G23a and G23b are not orthologs even though they carry out the same reaction. Thus, the deletion of G2B and G23a is lethal in Species A, but Species B can still carry flux through the reaction associated with G23b.

CONGA can also identify how metabolic differences affect cellular phenotypes other than growth rate (Figure 2H). In this example, the objective is to maximize the difference in flux through in Species A over Species B (). (The resulting phenotypes for each model are analogous to production phenotypes predicted by OptORF [35].) Deleting G2B forces Species A to utilize the lower reaction pathway, producing 0.06 BM and 0.2 P per S. However, the optimal flux distribution for Species B uses the upper reaction pathway, as this route produces more biomass (0.08 BM per S vs 0.06 BM per S via the lower pathway). As a consequence, Species A produces more by-product: 0.2 P per S in Species A vs. 0.1 P per S in Species B.

Because production values may not be unique at the maximum growth rate, CONGA can artificially inflate flux differences between models. This can only occur when the fluxes whose difference is being maximzed (e.g., chemical production rates) differ from the fluxes maximized by each model (e..g, biomass). In this case, we impose a a tilt on the objective of the inner problem. This tilt forces CONGA to identify deletions such that the specified flux difference is maximized when the individual fluxes through each reaction are at their lowest values that still support maximum biomass production. See Methods for additional details.

Comparison of E. coli Metabolic Models

We first used CONGA to compare two genome-scale metabolic models of E. coli, the iJR904 model [15] and the iAF1260 model [16]. The iAF1260 model extends the iJR904 model by compartmentalizing the network (separating the cytoplasm and periplasm), improving the biomass composition, and adding new metabolic reactions. The iJR904 model has been used frequently for metabolic engineering studies [36], but to our knowledge no studies have examined the extent to which the iAF1260 model's additional metabolic content affects computationally derived strain designs.

To explore the effect of the iAF1260 model's larger network, we used CONGA to identify gene deletion strategies for three commonly studied fermentation products–ethanol, lactate, and succinate–seeking identical knockout conditions where the iAF1260 model predicted higher production rates than the iJR904 model, and vice versa. We refer to such strategies as model-dominant strategies. For example, an iAF1260-dominant strategy is one in which the same gene deletion set predicts higher chemical production in the iAF1260 model than in the iJR904 model. Because some of these knockout strategies result in nonunique chemical production rates, model-dominant strategies were identified with respect to the lowest possible production rate consistent with the maximum growth rate.

Our initial CONGA results revealed a need to reconcile the fermentation pathways between the two models, due to changes in representation made in the iAF1260 model. We thus modified the iJR904 model to reflect these changes and repeated the simulations using the reconciled models. (See Dataset S2 for details.) For ethanol, succinate, and lactate, we identified the top three model-dominant strategies for each model for up to three, four, and five knockouts, respectively. We observed that multiple deletions are necessary to detect differences in production of these latter metabolites, and the difference in yield does not improve significantly beyond four or five knockouts, depending on the model and product. We also employed OptORF [35], without transcriptional regulation, to identify the top three deletion strategies for each model and product, for each number of gene deletions. We refer to these strategies as OptORF strategies. These strategies were then compared to the model-dominant strategies identified by CONGA, to determine if optimal OptORF strategies are likely to give similar or different predictions between the two models.

The CONGA results for the model-dominant strategies for ethanol production are presented in Figure 3A. We observed that only 4 of the 16 (25%) model-dominant strategies were also OptORF strategies (red bars), and none of the triple-deletion model-dominant strategies were OptORF strategies. This suggests that, when examining optimal OptORF strategies for higher numbers of gene knockouts, either model's predictions are likely to be similar at the maximum growth rate. However, the models may predict different ethanol production rates using the same gene deletion set for strategies which do not result in the maximum level of chemical production.

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Figure 3. Model-dominant production strategies for ethanol.

(A) Deletion strategies for ethanol production. Each bar represents the absolute difference in predicted ethanol yields between the iJR904 and iAF1260 models as a fraction of the maximum theoretical yield (2 ethanol/glucose). Left side: Strategies for which the iAF1260 model predicts higher production. Right side: Strategies for which the iJR904 model predicts higher production. Corresponding gene deletion strategies involving 1, 2, or 3 genes are given below the figure. Numbers above each bar indicate the fraction of the theoretical maximum yield obtained by each model, with the dominant model listed first. Some strategies have a nonunique ethanol production phenotype, in which multiple ethanol production values can occur at the maximum growth rate. For these scenarios, the production difference calculated by CONGA is from the lowest expected level of ethanol production in each model, and such strategies are indicated in green. Strategies for which the yield of the dominant model meets or exceeds the yield for the third-best OptORF strategy for that model are known as OptORF strategies, and such strategies are indicated in red. (B) The same gene deletion strategies after reconciliation of the iJR904 and iAF1260 networks with respect to metabolic differences.

https://doi.org/10.1371/journal.pone.0034670.g003

The CONGA results for model-dominant strategies for the production of lactate and succinate were quite different (Figure S1A and S2A). Here, 15 of the 30 model-dominant strategies are also OptORF strategies. Of these 15 strategies, 13 are iJR904-dominant strategies, with 11 involving the deletion of mphF and adhC (thereby removing acetaldehyde dehydrogenase). When these two genes are deleted, ethanol synthesis is no longer possible in the iJR904 model, while the iAF1260 model can synthesize ethanol via a second pathway (Figure 4A). The double deletion of mphF and adhC enables iJR904-dominant strategies for lactate and succinate production, with additional deletions determining whether lactate or succinate is the dominant product. We also observed that the iAF1260-dominant strategies for succinate production are all of low-yield (less than 10% the theoretical maximum). In fact, the iAF1260 model requires five gene deletions to obtain yields greater than 10% of the theoretical maximum, while the iJR904 model requires only two gene deletions. These results demonstrate that CONGA can also be used to identify differences in the ease of coupling growth to chemical production in different models or organisms.

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Figure 4. Flux maps illustrating differences in metabolic pathways in E. coli GENREs.

The text above each map indicates the pathway responsible for the phenotypic difference, the phenotype with which the strategy is associated, and the gene deletion for which the phenotype occurs. (A–B) Schematic views of the flux distributions associated with the indicated gene deletion set. Metabolites are represented in plain text. Metabolic transformations are indicated via arrows, with thicker arrows indicating higher flux. In some instances, multiple transformations are combined into a single dashed arrow or lumped into a subsystem. Subsystems are indicated by plain text enclosed in a grey rectangle. Fluxes active in the iAF1260 network are in red, fluxes active in the iJR904 network are in blue, and inactive fluxes are in grey. If gene (reaction) deletions occur in the fermentation pathway, they are indicated by black ‘X’s. Fluxes crossing the dashed boundary indicate transport to the extracellular environment. Metabolite abbreviations: 12 ppd, 1,2-propanediol; 2aobut, L-2-Amino-3-oxobutanoate; actp, acetyl phosphate; athr, allo-threonine. All other abbreviations match those used in the iSyp611 metabolic model (see Dataset S2).

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We then set out to investigate which network differences between iJR904 and iAF1260 account for the production differences associated with each gene deletion set found by CONGA. Of the 46 total model-dominant strategies, 34 (74%) could be attributed to at least one of six metabolic differences between the two models (Table 1). The remaining 12 model-dominant strategies predicted production differences of less than 10% the theoretical maximum yield, and in many cases much less. Two of the network differences (1,2-propanediol synthesis and hexokinase) were associated only with iJR904-dominant ethanol production strategies, while others were responsible for more than one set of model-dominant strategies. For example, differences in the succinate transport mechanism were implicated in strategies associated with iAF1260-dominant production of ethanol and lactate, and with iJR904-dominant production of succinate.

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Table 1. Explanation of metabolic differences between the iJR904 and iAF1260 models of E. coli.

https://doi.org/10.1371/journal.pone.0034670.t001

Many of these network differences affect the balance of possible fermentation products (Figure 4 and Figure S3). For example, the iAF1260 network contains an additional pathway to convert acetyl-CoA to ethanol via L-2-amino-3-oxobutanoate and allo-threonine (Figure 4A). As noted above, this extra pathway for ethanol synthesis in the iAF1260 model carries flux in many of the iJR904-dominant lactate and succinate production strategies, demonstrating that a single network difference can be found under multiple simulation conditions. In other instances, network differences affect flux balances outside the central fermentation pathways (Figure 4B). For example, when the genes edd (or eda), tpiA, and fsaB are deleted, disrupting glycolysis and the Entner-Doudoroff pathway, the iJR904 and iAF1260 models produce different products. The iJR904 model converts glucose into ribose-5-phosphate (r5p) via the oxidative and non-oxidative branches of the pentose phosphate pathway. The r5p is then converted to deoxyribose-5-phosphate and broken down into glyceraldehyde-3-phosphate (g3p), which enters glycolysis, and acetaldehyde (acald), which gets converted to ethanol. In contrast, the iAF1260 model converts glucose to g3p and dihydroxyacetone phosphate (dhap). As in the iJR904 model, g3p enters glycolysis, while dhap enters the methylglyoxal (mthgxl) pathway. Some of the mthgxl is converted to 1,2-propanediol (12 ppd) via a unique 12 ppd synthesis pathway, while the remaining mthgxl continues through the pathway to make pyruvate.

After identifying the metabolic differences that lead to model-dominant strategies, we modified the iJR904 and iAF1260 networks to contain identical representations of each pathway (Dataset S2) and re-evaluated the phenotype predictions of each knockout strategy. After the network reconciliation, we found that all but one of the knockout mutants are now predicted to have similar production rates (Figure 3B, Figures S1B and S2B).

While other studies have identified functional differences between the iJR904 and iAF1260 models with respect to growth phenotypes (e.g., gene essentiality predictions [16]) using an enumerative approach, here we compared the two reconstructions with respect to their metabolic engineering predictions using an algorithmic approach that identifies just those conditions resulting in different model predictions. We hypothesized that coupling of metabolites to biomass would be more difficult in the larger iAF1260 model, and that the model might have higher production levels (or larger production ranges if multiple products are possible), due to the larger network containing more ways to balance internal fluxes. These hypotheses were not borne out (with the notable exception of coupling succinate production to biomass), as we were able to predict similar production levels using both models. In fact, the production differences we did observe were due to only 21 reactions that represent just 3.5% of the 594 unique metabolic reactions in the iAF1260 model (described previously in [16]).

Cyanobacterial Metabolic Differences

Having analyzed two models of the same organism, we then sought to analyze two models of closely related but distinct organisms, and to examine organisms less well-studied than E. coli, to see if CONGA can be used to generate new physiological insights. For this application, we selected two cyanobacteria, Synechococcus sp. PCC 7002 and Cyanothece sp. ATCC 51142. Very few genome-scale metabolic reconstructions of cyanobacteria have been published to date [37]–[39], and our group has recently developed two more, the iSyp611 model of Synechococcus (this paper) and the iCce806 model of Cyanothece [23]. In order to gain insight into the metabolic similarities and differences between these two cyanobacterial strains, we used CONGA to identify gene deletion sets that were predicted to be lethal in only one cyanobacterial metabolic model, as well as to improve our draft Synechococcus reconstruction.

We first applied CONGA to the draft iSyp611 model. Some of the gene deletion sets identified by CONGA arose due to missing genes in the draft iSyp611 model. For example, CONGA identified gene deletion sets containing protein synthesis enzymes present only in the iCce806 network. Synechococcus also has these proteins, but they had not been included in the model. Other network differences arose due to incomplete GPR associations in the draft iSyp611 model. For example, the iCce806 model associated HisB with both histidinol-phosphatase and imidazoleglycerol-phosphate dehydratase, while the draft iSyp611 network only associated the protein with histidinol-phosphatase. The original annotation indicated the gene was bifunctional, and the draft iSyp611 model was updated accordingly. This approach increased the size of the iSyp611 model from 542 to 611 genes, an increase in gene content of 13%. This increase in gene content is comparable to that seen in metabolic network reconciliation [9], which was used to expand the gene content of genome-scale models of Pseudomonas aeruginosa and Pseudomonas putida by 3% and 18%, respectively.

After refining the draft model based on these results, the resulting model (iSyp611) was compared again to iCce806 using CONGA. We identified 30 gene deletion sets that are lethal only in the iSyp611 model and 36 gene deletion sets that are lethal only in the iCce806 model (Table 2). We found that in many instances different gene deletion sets mapped to the same set of reaction deletions (or reaction deletion set). For example, we identified six gene deletion sets lethal in the iSyp611 model that all mapped to photosystem II. As a result of these and other redundancies, the 30 gene deletion sets for the iSyp611 model reduced to 20 unique reaction deletion sets, and the 36 gene deletion sets for the iCce806 model reduced to 18 unique reaction deletion sets.

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Table 2. Number of lethal gene deletion sets for the cyanobacterial models iSyp611 and iCce806. Numbers in parentheses correspond to unique reaction deletion sets.

https://doi.org/10.1371/journal.pone.0034670.t002

Of the four types of functional network differences, we were most interested in metabolic differences, although the other types are also important. For example, genetic differences may occur because the genes encoding an essential protein have not yet been identified in one organism. In total, the metabolic differences accounted for 4 of 30 gene deletion sets (or 2 of 20 reaction deletion sets) for the iSyp611 model and 10 of 36 gene deletion sets (or 5 of 18 reaction deletion sets) for the iCce806 model (Table 3).

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Table 3. Explanation of metabolic differences between the cyanobacterial models iSyp611 (Synechococcus) and iCce806 (Cyanothece).

https://doi.org/10.1371/journal.pone.0034670.t003

Two of the reaction deletion sets which are lethal only in the iCce806 model require deletion of two reactions from both models (Figure 5). In the first deletion set (Figure 5A), deletion of glutamate dehydrogenase and glutamate synthase prevents the iCce806 model from synthesizing glutamate. The iSyp611 model has a unique reaction, valine amino-transferase (VPAMT), which allows it to recover from this double deletion (blue arrows). In the second deletion set (Figure 5B), deletion of pyruvate kinase and malate dehydrogenase prevents the iCce806 model from making pyruvate. The iSyp611 model has another unique reaction, aspartase (ASPT), which enables it to produce pyruvate and recover from the double deletion. A search of the Cyanothece genome failed to reveal candidate genes for ASPT and VPAMT, lending support to the hypothesis that they may be true metabolic differences between the two cyanobacteria.

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Figure 5. Identified metabolic differences in cyanobacteria.

(A) Top: Pathways for synthesis of glutamate (glu) from alpha-ketoglutarate (akg) used in iCce806. Bottom: Pathway predicted by the iSyp611 model when glutamate dehydrogenase (GLUD) and glutamate synthase (GLUS) are deleted. Valine aminotransferase (VPAMT) enables the synthesis of glutamate from pyruvate (pyr). (B) Top: Pathway for conversion of phosphoenolpyruvate (pep) to pyruvate when pyruvate kinase (PYK) is deleted from iCce806. Bottom: Pathway predicted by the iSyp611 model when malate dehydrogenase (MDH) is also deleted. Aspartase (ASPT) allows malate (mal) to be synthesized entirely from fumarate (fum), rather than from fumarate and oxaloacetate (oaa). (A and B) Red arrows indicate flux in the iCce806 model. Blue arrows represent flux in the iSyp611 model under the indicated knockout condition. Black arrows indicate inactive reactions and reaction deletions are indicated by black ‘X’s. Gray arrows (top panels) indicate reactions not present in the iCce806 model. Arrow thickness corresponds to relative flux levels. Reaction and metabolite abbreviations are identical in the iSyp611 and iCce806 models and are given in Dataset S2.

https://doi.org/10.1371/journal.pone.0034670.g005

CONGA reveals differences that can be used to reconcile and improve genome-scale metabolic models of closely-related species. We intend to use the remaining genetic and orthology differences found by CONGA as a starting point in further updating our reconstruction, as they may indicate missing or incorrectly annotated genes. CONGA can also identify differences in metabolic capabilities between models: our analysis here indicates Synechococcus and Cyanothece share a significant number of pathways, with important differences in central and amino acid metabolism.

Drug Targeting in Human Pathogens

While were able to idenitfy metabolic differences between the two cyanobacteria, many of the differences identified by CONGA were not due to reaction-level differences. We thus sought to use CONGA to explore differences in metabolic capabilities between two dissimilar oganisms, and to exploit those differences to identify organism-specific drug targets. For this application, we applied CONGA to existing models of two phylogenetically distant human pathogens, the iNJ661 model of M. tuberculosis [28] and the iSB619 model of S. aureus [29], in order to explore differences in pathogenicity and drug resistance based on differences in reaction and gene content. As with our analysis of the cyanobacterial models, we sought genetic perturbation strategies that were predicted to be lethal in only one organism.

Our preliminary analysis identified a total of 168 unique gene deletion sets, of which 139 (83%) could be traced in whole or in part to genetic or orthology differences. As these differences made up the majority of identified differences, we manually evaluated the quality of the orthology assignments and the original GPR associations. This analysis resulted in the modification of the GPR associations for 19 reactions in the iSB619 model and 36 reactions in the iNJ661 model (Dataset S2). As a result of these changes, 7 genes were eliminated from and 3 added to the iSB619 model, with 10 genes eliminated from and 4 added to the iNJ661 model.

A number of these initial genetic- and orthology-related gene deletion sets arose due to different representations of the glycine cleavage complex (GCC) and pyruvate dehydrogenase system (PDH) in the two models (Figure 6A). Both GCC and PDH are composed of three separate enzymes (a, b, and c), each of which carries out a distinct catalytic activity. Deletion of GCC is predicted to be lethal in both organisms, and because one subunit is shared by GCC and PDH, deletions to one complex may affect the other. In its original form, the iSB619 reconstruction modeled PDH as an overall reaction, and GCC via its three individual reactions (Figure 6B). In contrast, the iNJ661 model represented both PDH and GCC as individual and overall reactions (Figure 6B). Due to these differences, a number of ortholog deletions are lethal in only one model. For example, deletion of the ortholog pair (SA0945, Rv2495) deletes PDH from the iSB619 network, but only deletes PDHb from the iNJ661 network. The deletion is lethal only in the iSB619 model. We thus revised the GPR associations for these complexes to give a consistent representation between the two models (Figure 6C). These changes also required changes to the stoichiometric matrices in each model. (See Dataset S2 for details.)

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Figure 6. Example adjustment of pathogen models following preliminary analysis.

(A) Original model annotations for the glycine cleavage (GCC) and pyruvate dehydrogenase (PDH) complexes. Green boxes represent reactions and gray boxes represent genes. S. aureus loci are in blue text and M. tuberculosis loci are in red text. Dashed lines indicate orthologs and solid lines connect genes to reactions. SA1365 and SA1366 are orthologous to the N-terminus and C-terminus of Rv1832, respectively, and together are orthologous to the entire Rv1832 sequence. A ‘+’ sign between genes indicates a complex; a ‘/’ sign indicates isozymes. (B) The two models were inconsistent in their representation of these two enzyme complexes. This table indicates the presence or absence of individual (a, b, c) and lumped (GCC, PDH) reactions before and after model adjustments. The shaded gray boxes indicate the presence of a particular function, and the small black text indicates that model's specific reaction. (C) Revised model annotations for the GCC and PDH complexes. The color scheme is the same as in A.

https://doi.org/10.1371/journal.pone.0034670.g006

We applied CONGA again after this initial reconciliation, and identified 71 gene deletion sets lethal only in the iSB619 model and 84 gene deletion sets lethal only in the iNJ661 model (Table 4). Of these, a total of 99 gene deletion sets (64%) were still due to genetic or orthology differences. Nevertheless, CONGA identified 18 gene deletion sets arising from metabolic differences which were lethal only in the iSB619 model, and 38 such gene deletion sets lethal only in the iNJ661 model. As with the cyanobacteria, in some instances multiple gene deletion sets mapped to the same reaction deletion set (Table 4). Of these, we examined only those gene deletion sets arising from metabolic differences, and identified 17 unique reaction deletion sets lethal only in the iSB619 model and 28 unique reaction deletion sets lethal only in the iNJ661 model.

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Table 4. Number of lethal gene deletion sets for the human pathogen models iSB619 and iNJ661. Numbers in parentheses correspond to unique reaction deletion sets.

https://doi.org/10.1371/journal.pone.0034670.t004

These 45 unique reaction deletion sets served as the starting set of potential drug targets. We employed a multi-step process to reduce these reaction deletion sets to a set of candidate antibiotic targets. First, because genes may be associated with more than one reaction, we eliminated from each unique reaction deletion set any reactions that were nonessential to the set. For example, CONGA identified the deletion of SA1487 as lethal in S. aureus, leading to the reaction deletion set DHFS and THFGLUS. However, the deletion of THFGLUS is not lethal, so THFGLUS was removed from the reaction deletion set, giving the reduced reaction deletion set DHFS. We then examined the reduced reaction deletion sets and eliminated those sets where more than one reaction deletion was required to give a lethal prediction. Such reaction deletion sets are likely to be poor candidates for potential drug targets, because they may require development of a multiple-drug treatment strategy. For example, CONGA identified the reaction deletion set RNDR1, RNDR4 as being lethal in M. tuberculosis, with both reaction deletions necessary to give a lethal prediction. This set was subsequently eliminated from the set of candidate antibiotic targets. Finally, we eliminated those reactions included in the Recon 1 genome-scale metabolic model of human metabolism [40], as drugs targeting these reactions may cause adverse side-effects in humans. This procedure yielded 10 reactions as candidate antibiotic targets in S. aureus and 37 reactions as candidate antibiotic targets in M. tuberculosis (Table 5).

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Table 5. Potential drug targets in the human pathogens S. aureus and M. tuberculosis.

https://doi.org/10.1371/journal.pone.0034670.t005

Many of the candidate antibiotic targets are already targeted by existing antibiotics (Table 5), demonstrating that our approach can correctly identify candidate metabolic functions for drug targeting. Most of the reactions for which antimicrobials exist are involved in cell wall and cell membrane synthesis. While both organisms require these biosynthetic capabilities, their cell walls and membranes are structurally different, and so different proteins and reactions are required. These differences are reflected in the standard antimicrobial treatments for these two pathogens. For example, vancomycin binds to the D-alanine terminus of peptidoglycan and prevents the incorporation of teichoic acids into the matrix [41]. Mycobacteria, such as M. tuberculosis, have structurally distinct cell walls, for which isoniazid, ethambutol, and pyrazinamide are required treatments [42]–[44]. We were also able to find reports of small molecule inhibitors of fatty acid synthesis in both S. aureus [45], [46] and M. tuberculosis [47].

We also identified a variety of other metabolic functions which antibiotics do not yet target. For example, the iSB619 model requires tetrahydrofuran (THF) and NAD to produce biomass. Unfortunately, many staphylococci are already resistant to inhibitors of THF synthesis [48], while inhibitors of the nicotinamidases S. aureus uses for NAD synthesis have only recently been identified [49]. However, M. tuberculosis can grow in media lacking THF and NAD [28], suggesting the lack of THF and NAD in the iNJ661 biomass equation may reflect a model development choice, rather than a biological difference. We identified M. tuberculosis' unique use of siderophores for iron transport, for which biosynthesis inhibitors have been identified [50]. We also identified mycobacteria's use of unique glycolipids, but we were unable to identify inhibitors that have been reported in the literature, making glycolipid synthesis a potential new target for new M. tuberculosis-specific antibiotics. Of the remaining organism-specific metabolic functions, two candidate antibiotic targets (nicotinamidase in S. aureus and siderophore synthesis in M. tuberculosis) had not been identified by previous computational studies of these models [28], [29].

By comparing pathogens against each other, we are able to identify essential functions unique to a particular pathogen. This enables the identification of narrow-spectrum antibiotics tailored to individual pathogens. It is believed that the use of such antibiotics can overcome multi-drug resistance through novel mechanisms of action [32], [33] and slow the rate of resistance transfer across species [31], [34]. We believe our framework provides a rapid means of identifying unique metabolic functions as possible targets for new antimicrobials, and will provide a useful tool for combating the rapid rise of multi-drug resistant bacteria.

Assessment of Ortholog Calling Methods

Before CONGA can be applied to a pair of metabolic models, a gene-level alignment must be performed. We perform this alignment by identifying the orthologous genes between the two models, and we force CONGA to select ortholog pairs as a single unit. Prior to applying CONGA to the pathogen models, we examined three methods for identifying orthologous genes (Figure 7). The first method utilized a BLAST search [51] to identify those pairs of M. tuberculosis and S. aureus genes which were mutual best-BLAST hits of each other, called bidirectional best-BLAST hits. An E-value of was employed as a cutoff. The second method used OrthoMCL [52] to identify pairs of genes belonging to the same ortholog group (a cross-taxa group of genes in which all genes are bidirectional best-BLAST hits of one another). The last method utilized the SEED [13] to identify genes belonging to the same FIGfam (sets of proteins homologous along their entire length).

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Figure 7. Comparison of ortholog identification methods for S. aureus and M. tuberculosis.

(A) Number of model genes identified as orthologs by each of the three methods. Only the orthologs present in both models are included in the diagram. Overlapping areas indicate orthologs identified by one or more methods. (B) Number of false positive orthology assignments made by each of the three methods. A false positive orthology assignment indicates the two genes are associated with different reactions in their respective models. Overlapping areas indicate false positives identified by one or more methods. (C) Overlap of gene content between the two pathogens, based on SEED FIGfams. Smaller circles represent genetic content of the two models, with the larger circles representing the entire genome. Numbers within overlapping areas indicate numbers of orthologs.

https://doi.org/10.1371/journal.pone.0034670.g007

We first identified ortholog pairs where both the M. tuberculosis and S. aureus genes were included in the iNJ661 and iSB619 models, respectively. We found that the number and content of ortholog calls depended on the method used (Figure 7A). The bidirectional best-BLAST search identified a total of 287 of a possible 619 genes. SEED identified the fewest, with only 229 orthologs. Of these, 175 orthologs were common to all methods, with smaller numbers of orthologs being shared by pairs of methods.

We also analyzed the three methods for false positive ortholog calls (Figure 7B). A false positive ortholog call is one in which two orthologs are associated with different reactions in their respective models. We found that all three methods identified 7 ortholog pairs for which model annotations were distinctly different (Table 6). SEED identified the fewest additional false positives, giving 14 total. Full details of orthologs assigned by each method can be found in Dataset S3. We then analyzed the effect of each ortholog calling method on the gene deletion sets identified by CONGA. We found that using orthologs identified by bidirectional best-BLAST and OrthoMCL yielded numerous gene deletion sets containing false positive ortholog pairs. In contrast, the number of gene deletion sets containing true ortholog pairs was relatively insensitive to the method used to call orthologs. We thus chose to perform all simulations using SEED orthologs.

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Table 6. False positive ortholog calls in the iSB619 (S. aureus) and iNJ661 (M. tuberculosis) human pathogen models.

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Using the orthologs identified by SEED, we then assessed the metabolic overlap between the two models (Figure 7C). In addition to the 224 orthologs present in both models, the iSB619 model contains 33 genes with orthologs that are not included in the iNJ661 model, and the iNJ661 model contains 13 genes with orthologs that are not in the iSB619 model. These 46 genes can likely be used to expand the scope of each model. Additionally, we identified 253 orthologs included in neither model. Using SEED, we were able to classify these 253 orthologs into subsystems and found that 45% were involved in protein, DNA, or RNA metabolism, while 15% were involved in non-metabolic functions such as cell division, regulation, and the stress response. An additional 22% were of unknown or uncertain function. The remaining 18% were spread across a variety of metabolic subsystems, with 35 of the 253 (8%) orthologs being involved in vitamin and cofactor synthesis. Many of these 35 genes are involved in the assembly of metal clusters and would not generally be included in a metabolic model. Finally, we observed that metabolic genes are enriched for members of an ortholog pair: 37% (229 of 619) of genes in the iSB619 model had orthologs in the iNJ661 model, while only 21% (528 of 2515) of genes in the S. aureus genome had orthologs in the M. tuberculosis genome ().

Formulation of Bilevel Optimization Problem for Identification of Gene Deletion Sets

The CONGA framework employs a bilevel optimization problem to identify genetic perturbations which disproportionately change flux through a selected reaction (e.g., growth or by-product secretion) in one organism over another (Figure 1). The outer problem is a mixed-integer linear program (MILP) which finds gene deletions maximizing the flux difference between two reactions in different models. The two inner problems (one for each model) are flux-balance analysis (FBA) problems [64], linear programs (LPs) which maximize growth subject to reaction stoichiometry, thermodynamics, and enzyme capacities. We alter the FBA problems using deletions given by the outer problem. Gene-protein-reaction (GPR) constraints associate genes with reactions and are used to enforce the gene deletions given by the outer problem. These constraints are formulated using the logical relationships developed previously [35]. CONGA can select any genes for deletion, with the restriction that orthologous genes present in both models be deleted simultaneously from both models.

The FBA formulation for each model's inner problem is shown below:(1)(2)(3)(4)

Each reaction j in the set of reactions j has a flux given by . The FBA objective is a linear combination of fluxes , where c is a vector of weights. We choose to maximize for biomass alone, in which case is a standard basis vector along biomass, and the objective is written as . Each reaction j consumes and produces some metabolites i in the set of metabolites I, with stoichiometry given by . By conservation of mass, net production of each metabolite across the entire network must be zero at steady-state (equation 2). Each reaction is constrained to have flux within an appropriate range as given by enzyme capacities and thermodynamics (equation 3). For reactions deleted by the outer problem, a binary variable () takes a zero value (), and the corresponding flux is constrained to zero (equation 4).

On-off reaction states are given by the binary variable y and determined by GPR constraints embedded in the outer problem:(5)

Each gene g in the set of genes G, protein p in the set of proteins P, and reaction j in the set of reactions J has a corresponding binary variable z, w, and y, respectively, which determines the gene, protein, or reaction's on-off state. (See [35] for details.) Each reaction j with a known GPR association can be carried out by a subset of enzymes , and each enzyme is specified by the subset of gene products . The outer problem selects one or more genes for deletion (), and the GPR constraints implement the necessary logical relationships to determine the set of deleted reactions ().

To identify lethal gene deletion sets, the outer problem identifies deletions such that the growth rate of one species (A) is maximized with respect to the other (B). So long as growth is unconstrained, an objective of the form(6)will first identify gene deletions lethal only in species B. Finally, additional constraints are added which impose a limit K on the total number of gene deletions,(7)and which ensure that all pairs of orthologous genes are deleted in common:(8)The set of orthologs O contains all pairs of genes and found to be orthologous between Species A and Species B.

The final formulation results from using equation (6) as the outer objective, and accumulating equations (1)–(5), (7), and (8) as constraints. Equations (1)–(5) and (7) must be imposed for each species:

Reformulation to Single-Level Optimization Problem

To facilitate the solution process, we reformulated the bilevel program as single-level MILP by replacing the inner maximization problems with their optimality conditions, in accordance with strong duality [65]. The strong duality theorem for a linear program states that, at optimality, the values of the primal and dual objectives are equal, and the primal and dual variables satisfy the primal and dual constraints, respectively [65]. Thus, each inner problem (equations (1) to (4)) can be replaced by formulating its dual, equating the primal and dual objectives, and accumulating the primal and dual constraints. This reformulation was first proposed for the bilevel strain design problem OptKnock [66] and has since been described for other bilevel problems [35], [60], [67].

This reformulation requires a new variable for each constraint of the inner problem. Each metabolite i must satisfy the mass balance, for which we introduce the unconstrained dual variable . Active reactions are further constrained to be within the range , for which we introduce the positive dual variables and , respectively. In many cases, and are assigned large, arbitrary values. To reduce the size of the reformulation, we eliminated the upper bound constraint () and imposed the lower bound constraint () only on uptake fluxes and irreversible reactions, collectively the set . Finally, reactions removed by gene knockouts are constrained to zero flux, for which we introduce a free dual variable . This allows the dual of each inner problem to be formulated as:(9)(10)(11)(12)(13)(14)

Constraints (11) to (13) can be implemented using big-M constraints [68] or using the GAMS/CPLEX indicator constraint facility (the latter was used in this work).

The single-level formulation can then be constructed by using equation (6) as the outer objective, equating the primal and dual objectives (1) and (9) for each network, including constraints (2) to (5), (7), and (10) to (14) for each network, and adding equation (8). Equating the primal and dual objectives of the inner problem gives(15)so that the final, single-level formulation can be expressed as:

We also implemented integer cut constraints [69] to allow the generation of multiple solutions.

Modifications for Identification of Model-Dominant Strategies

To identify model-dominant chemical production strategies in the E. coli models, we sought gene deletions maximizing chemical production in one model with respect to the other. For these simulations, a few modifications from the previous formulation are required. First, the outer objective, equation (6), was altered to reflect chemical production flux. The vector c was changed to a standard basis vector along the production flux of interest. We denote this objective as .(16)

Some knockout conditions result in a nonunique phenotype for a particular chemical, in which multiple chemical production values can occur at the maximum growth rate. Under such conditions, CONGA can artificially inflate flux differences between models, by choosing a large production rate in one model and a small production rate in the second. We thus imposed a tilted objective function on each inner problem, which maximizes biomass while imposing a small penalty () on chemical production; this causes the inner problem to return the value of representing the lowest expected flux through the reaction [70].(17)

Because flux values in general are not necessarily unique, this tilted objective is necessary whenever the fluxes whose difference is being maximzed (e.g., chemical production rates) differ from the fluxes maximized by each model (e..g, biomass). We found this formulation is sensitive to the value chosen for . If is too small, the tilt does not correctly return the value of representing the lower bound through the reaction, and if is too large, the tilt returns solutions with a slightly suboptimal growth rate. For our comparison of the E. coli models, we found that setting avoided both of these problems. However, the tradeoff between growth rate and chemical production flux varies from model to model and product to product, suggesting our value of may not be generally applicable. Modifying the inner objective in this way requires modifying the weight vector c in equation 15 to include the value where . We also imposed this tilted objective function when using OptORF to identify the top deletion strategies for each model and product.

We also constrained the dual variables associated with the reaction deletion constraint to be between [−1, 1] to improve solver performance [35], [71].(18)

Finally, we constrained both models to have nonzero biomass. The final, single-level formulation can be expressed as:

A sample implementation of CONGA for the network in Figure 2 can be found in Text S1.

Finally, we note that CONGA can be rewritten to consider a reaction alignment and reaction deletions, by redefining the set O (equation (8)) and using reaction instead of gene deletions (equation (7)).

Reducing the Number of Variables

In order to reduce simulation times, we eliminated essential and blocked genes from consideration as possible deletions by CONGA. For each of the six models, we performed single-gene deletions with all possible nutrients provided to the model in order to identify essential genes (those required for cellular growth). Genes whose orthologs were essential in both models as well as essential genes without orthologs were then excluded from consideration by CONGA. Flux-variability analysis (FVA) [72] was also used, again with all possible nutrients provided, to identify blocked reactions (those incapable of carrying flux), and genes encoding only blocked reactions were also excluded from consideration (i.e., if a gene encoded both a blocked and a nonblocked reaction, CONGA was allowed to select that gene for deletion).

For our comparison of the E. coli models, we performed additional steps in order to improve the run-time performance of CONGA. First, we identified essential and blocked genes (as described above) on glucose media under anaerobic conditions (i.e., the simulation conditions). We also removed from consideration all genes associated with membrane transporters. These two steps forced CONGA to consider only metabolic genes that can be active under the simulation conditions. Finally, we developed a procedure to reduce the number of genes needed to determine the on-off state of each reaction. For a set of isozymes catalying the same set of reactions, all but one isozyme were fixed to the off state, such that deleting the remaining isozyme forced the set of reactions to the off state. Likewise, for a set of subunits which are all components of the same protein, all but one subunit were fixed to the on state, such that deleting the remaining subunit forced the protein to the off state. These sets of isozymes and subunits were identified using a mixed-integer programming (MIP) approach (Text S2). Finally, as in CONGA, orthologs present in both models were forced to have the same state. This ensures an internally consistent selection of isozymes and subunits (e.g., if one member of an ortholog pair is fixed to the on (or off) state, the other must be also). The complete set of isozyme and subunit sets for the two E. coli models can be found in Dataset S3. The same genes that were fixed to be on or off in CONGA were also fixed to be on or off in the OptORF simulations used to identify the top deletion strategies for each model and product.

Identification of Orthologs

A gene-based alignment of two networks requires a method for identifying orthologous genes between two genomes. Since the E. coli simulations studied two models of the same organism, we were able to immediately match gene loci. For the cyanobacterial simulations, we used the set of bidirectional best-BLAST hits identified during the first step of the iSyp611 reconstruction process. Genes added during the manual reconstruction process were checked against the final iCce806 model, and additional orthologs were identified. For the pathogen studies, we used SEED to identify orthologs, as this method identified the smallest number of false positive ortholog pairs (Table 6 and Discussion). A full summary of ortholog pairs used in each simulation can be found in Dataset S3.

Construction of the iSyp611 Metabolic Network

We have formulated a genome-scale network reconstruction of the photosynthetic cyanobacterium Synechococcus sp. PCC 7002 consisting of 611 genes, 533 proteins, 552 reactions, and 542 metabolites (Table 7). A total of 517 reactions (94%) are associated with genes and proteins, represented by gene-protein-reaction (GPR) associations.

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Table 7. Comparison of iSyp611 (Synechococcus) and iCce806 (Cyanothece) cyanobacterial models.

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The model was constructed from a draft version of the iCce806 reconstruction of Cyanothece sp. ATCC 51142 via a gene-level comparison. The Synechococcus sp. PCC 7002 genome sequence was downloaded from the GenBank database at the National Center for Biotechnology Information (NCBI) website [73]. A bidirectional best-BLAST search was used to identify potential orthologs between the two genomes. The validity of the associations was manually assessed using annotation information available from NCBI, KEGG, [12], and SEED [13]. For those genes deemed highly probable orthologs, protein and reaction associations were copied from the iCce806 model to create a draft reconstruction using SimPheny (Genomatica Inc., San Diego, CA).

After assembling the draft network, missing functions were added to ensure production of all biomass components. Candidate reactions were selected based on pathways in other cyanobacterial strains. Potential genes for these reactions were located by best-hit BLAST analysis against other cyanobacterial genomes as well as annotation information obtained from NCBI, KEGG, and SEED. In cases where genomic information was unavailable, reactions were selected based on their frequency of occurrence in related strains. This draft model contained 542 genes, of which 497 were orthologous to genes in Cyanothece (Table 7).

We also applied CONGA to our draft reconstruction, and use the results to add new subunits and isozymes to existing reactions. In all, nearly 70 genes were added to the reconstruction. A complete list of reactions and GPR associations in the iSyp611 model is included in the Supporting Information, in Microsoft Excel and SBML formats (Datasets S1 and S2).

Wherever possible, the reactions used to represent RNA, DNA, protein, fatty acid, and lipid synthesis were updated to reflect the particulars of Synechococcus sp. PCC 7002. DNA and RNA composition was based on genomic GC content, and protein composition was obtained from amino acid counts of the proteome. Fatty acid composition was taken from Synechococcus sp. PCC 7002 [74], [75], and lipid composition was taken from Synechococcus sp. PCC 7942 [76]. The biomass equation was formulated using weight fractions of macromolecules (DNA, RNA, protein, lipid, fatty acids, glycogen) measured from Synechosystis sp. PCC 6803 in batch culture [77], and the composition of the soluble pool was copied from the iJR904 model [15].

The final metabolic reconstruction was used to formulate a constraint-based model of Synechococcus metabolism. Experimentally, Synechococcus is able to grow phototrophically using light, carbon dioxide, and ammonium. Our constraint-based model was capable of predicting growth under photoautotrophic and glycerol heterotrophic conditions via FBA.

Models and Simulation Conditions

The CONGA analysis of the two published models of E. coli, iJR904 [15] and iAF1260 [16] were performed under anaerobic, glucose-limited conditions (uptake rate 18.5 mmol/gDW/hr). All reported chemical production levels were normalized to the theoretical maximum (2 mol ethanol/mol glucose, 2 mol lactate/mol glucose, and 1.71 mol succinate/mol glucose).

For the iSyp611 and iCce806 comparisons, several reactions in the iSyp611 model were replaced with their iCce806 equivalents, including biomass, ATP synthase, DNA, RNA, lipid, and protein synthesis, and cytochrome oxidases unique to Cyanothece (Dataset S2). Simulations were performed under carbon-limited photoautotrophic conditions, with maximum uptake fluxes for photons for both photosystems, carbon dioxide, and ammonium constrained to 100 mmol photons/gDW/hr, 20 mmol/gDW/hr, and 10 mmol/gDW/hr, respectively. Unconstrained uptake of inorganic phosphate, oxygen, magnesium(II), protons, sulfate, and water was also allowed. Growth-associated and non-growth associated ATP maintenance requirements were set to zero.

For the iNJ661 and iSB619 models, simulations were performed on nine distinct minimal media with different carbon and nitrogen sources (Dataset S2). The set of gene deletion sets common to all conditions was then analyzed for potential drug targets.

All simulations were performed using CPLEX 12 (IBM, Armonk, NY) accessed via the General Algebraic Modeling System (GAMS, GAMS Development Corporation, Washington, DC). Simulations were performed on a Red Hat Enterprise Linux server with 2.66 GHz Intel Xeon processors and 8 GB of RAM. CONGA can identify a lethal gene deletion set containing just one gene in less than a second. Lethal gene deletion sets containing two or three genes took on average 3 minutes to identify. Identifying model-dominant chemical production strategies is more time-consuming, with a model-dominant strategy containing two genes requiring on average 2 minutes. Model-dominant strategies containing three, four, or five genes took an average of 15 minutes, 75 minutes, and 5 hours to identify, respectively. A full summary of differences identified by each simulation can be found in Dataset S3.