RELATCH: relative optimality in metabolic networks explains robust metabolic and regulatory responses to perturbations
© Kim and Reed; licensee BioMed Central Ltd. 2012
Received: 5 July 2012
Accepted: 26 September 2012
Published: 26 September 2012
Predicting cellular responses to perturbations is an important task in systems biology. We report a new approach, RELATCH, which uses flux and gene expression data from a reference state to predict metabolic responses in a genetically or environmentally perturbed state. Using the concept of relative optimality, which considers relative flux changes from a reference state, we hypothesize a relative metabolic flux pattern is maintained from one state to another, and that cells adapt to perturbations using metabolic and regulatory reprogramming to preserve this relative flux pattern. This constraint-based approach will have broad utility where predictions of metabolic responses are needed.
Computational modeling of metabolic networks has been useful in studying microbial metabolism and developing tools for many applications. Among different computational approaches, constraint-based models utilize genome-scale metabolic networks to predict metabolic flux distributions in microbial cells, and they have been used to guide metabolic engineering , drug discovery , and adaptive evolution  studies. For example, flux balance analysis (FBA) predicts metabolic flux distributions in optimally growing microbes, by maximizing biomass yields [4, 5]. FBA can also predict the effects of gene deletions on metabolic behaviors by removing the associated reactions from the network, and its predictions are shown to be consistent with experimental observations for parental and gene knockout strains of Escherichia coli that have undergone adaptive evolution [6, 7]. Recently, FBA has been used to discover drug targets by identifying essential metabolic functions in different growth conditions representing the host environment [8, 9]. Incorporation of additional molecular crowding constraints, which restrict total enzyme levels and thus flux capacities, into FBA improves growth rate predictions of parental and mutant E. coli strains in different environmental conditions . Genomics-driven constraints, such as grouping reaction constraints  have also been incorporated to improve flux and growth rate predictions in genetically or environmentally perturbed strains.
Several constraint-based approaches for integrating omics data (for example, transcriptomics, proteomics, or metabolomics) with metabolic models have been developed to predict metabolic flux distributions in different environmental or genetic conditions [12–16]. For example, E-Flux uses relative gene expression levels to place upper and lower bounds on individual fluxes in the models , while another approach, gene inactivity moderated by metabolism and expression (GIMME), instead uses expression data in the objective function to penalize the use of fluxes based on the magnitude of the flux and how far a gene's expression falls below a chosen threshold . Another approach, integrative omics-metabolic analysis (IOMA), was developed to predict flux distributions by integrating proteomic and metabolomic data into kinetic constraints that are included in the models . Unlike FBA, all of these omics-based approaches require data from the genetically or environmentally perturbed states to predict fluxes in that state, which is often not available ahead of time.
Since FBA assumes that cells grow optimally, other approaches are used to predict the behavior of perturbed strains that exhibit suboptimal growth (for example, unevolved mutants) due to regulatory restrictions or other metabolic limitations, without requiring any data from the perturbed state. One such approach is minimization of metabolic adjustment (MOMA), which predicts the behaviors of unevolved mutants by minimizing the sum of squared differences in flux distributions between mutant and parent strains . MOMA has been used to improve production of valuable biochemicals [18, 19], study epigenetic interactions associated with genetic diseases , and describe cooperative interactions between microbes . Another approach, regulatory on/off minimization (ROOM) minimizes the number of significant flux changes in mutant strains relative to the parental strain . Both MOMA and ROOM use a reference (for example, wild type) flux distribution to predict a perturbed flux distribution by minimizing the Euclidean and Hamming distances, respectively, and hence the reference flux distribution significantly affects their predictions. The reference flux distribution is normally determined in both approaches by FBA; however, a more accurate description of the reference state can be obtained from available experimental data as proposed here. Neither MOMA nor ROOM consider flux fold changes in their minimization procedures, and so large fold changes in flux may be predicted if the Euclidean or Hamming distances are minimized. While flux predictions with MOMA and ROOM show good correlation to experimental measurements [17, 22] they can still be quantitatively inaccurate, which may guide valuable experimental efforts in the wrong direction. Therefore, there is still a need for a more accurate approach that accounts for metabolic and regulatory adjustments at a genome-scale to predict flux distributions in genetically and environmentally perturbed microbial systems, without requiring experimental data from perturbed states.
To predict the flux distributions in perturbed systems, we employed two parameters - a penalty (α) for latent pathway activation and a limit on enzyme contribution increases (γ) for active enzymes (see Materials and methods). The underlying rationale for the RELATCH parameters is that strains would initially adjust to new perturbations with limited metabolic and regulatory adjustments, whereas over time they could adapt to these conditions by increasing the capacity of previously active and latent pathways. Here, we considered the recent loss of metabolic enzymes (for example, unevolved knockout mutants) or a change in growth conditions that cells are less accustomed to (for example, galactose) as conditions the cells are not adapted to. For these non-adapted conditions, we used tight parameter values, including a high penalty for latent pathway activation (α = 10) and restricted enzyme contribution increases in active enzymes from the parental state (γ = 1.1). We considered cells to be adapted to a condition if knockout strains were adaptively evolved, if parental strains were grown in conditions cells were accustomed to (for example, anaerobic growth), or if strains were grown in a chemostat (since strains grown in chemostat can rapidly accumulate beneficial mutations during the pre-culture and stabilization period ). For these adapted conditions, we used relaxed parameter values, including a low penalty for latent pathway activation (α = 1) and no restriction on enzyme contribution increases in active enzymes (effectively, γ = ∞). These two sets of parameter values were determined by analyzing four knockout E. coli strains before and after adaptive laboratory evolution described in the next section (Additional files 1 and 2) and were applied systematically to the other datasets.
Flux predictions in E. coli mutant strains before and after adaptive laboratory evolution
We first used RELATCH to predict flux distributions in knockout mutants of E. coli K-12 MG1655 (Δpgi, Δppc, Δpta, and Δtpi) before and after adaptive laboratory evolution. MFA flux values for the parental strain were taken from an existing study  and mapped onto corresponding reactions in the iAF1260 metabolic model . Gene expression data for the parental E. coli strain grown aerobically in glucose minimal medium from an earlier study  and MFA estimates were then combined to estimate the genome-scale flux distribution and enzyme contribution in the parental strain. This reference flux distribution was then used to predict flux distributions in unevolved and evolved knockout mutants and compared to the reported MFA flux estimates for these mutants by calculating the sum of squared errors per flux (SSE; Equation 6) and Pearson's correlation coefficient (r) .
For adaptively evolved knockout mutants, RELATCH also predicted mutant flux distributions (Figure 2b,c) and growth and glucose uptake rate recovery (Additional file 4) with the greatest quantitative accuracy. We should note that the predicted unevolved and evolved mutant flux distributions are different for RELATCH (due to use of tight and relaxed parameter values), while they are the same for the other three methods. The relaxed parameter values allowed further increases in enzyme contributions for active enzymes, as well as increased use of latent pathways, in order to compensate for the gene deletion. For the evolved Δpgi mutant, RELATCH predicted further increase in fluxes through the NADPH-producing pentose phosphate pathway and subsequent conversion of excess NADPH to NADH via NAD(P)H transhydrogenase (catalyzed by SthA), as compared to the unevolved mutant. This suggests that redox balancing is a bottleneck in the unevolved Δpgi mutant, which is consistent with a previous study  where whole-genome resequencing of evolved Δpgi strains found frequent mutations in the SthA (converting NADPH to NADH) and PntAB (converting NADH to NADPH) transhydrogenases. RELATCH also predicted a decrease in pyruvate kinase flux, which is consistent with gene expression data . Interestingly, one of the two evolved Δpgi strains (E1) had slower growth and considerably higher acetate secretion compared to the other evolved Δpgi strain (E2) (Additional file 4), indicating that the Δpgi E1 strain is less optimal with respect to growth than Δpgi E2. It seems that the Δpgi E1 strain had evolved to a local optimum since another study showed that ten independently evolved Δpgi strains had similar growth and acetate secretion rates as the Δpgi E2 strain . The flux values predicted by RELATCH were closer to the flux values in the Δpgi E2 strain (SSE = 0.32) than the flux values in the Δpgi E1 strain (SSE = 1.59). For the evolved Δppc and Δpta mutants, all methods predicted the flux patterns well (Figure 2b), but the extracellular fluxes and growth rates were most accurately predicted by RELATCH (Additional file 4). Increased glyoxylate shunt flux and decreased citric acid cycle flux from isocitrate to succinate in the evolved Δppc mutant were also correctly predicted by RELATCH, which is consistent with expression changes in the genes involved in these pathways . For the Δtpi mutant, FBA incorrectly predicted high fluxes through the Entner-Doudoroff pathway to maximize biomass production, while RELATCH instead predicted increased use of the methylglyoxal pathway to produce pyruvate, and activation of the non-phosphotransferase system glucose transporters due to the decreased production of phosphoenolpyruvate. In this case, the RELATCH prediction agreed with MFA, enzyme activity, and gene expression measurements .
Robustness of the flux predictions
We performed multiple sensitivity analyses using the same four E. coli mutant strains to demonstrate the robustness of RELATCH predictions. First, we investigated the effect of the reference flux distribution on predictions using four scenarios where less experimental data from the reference state is used. The results indicate that including more experimental data to estimate the reference flux distribution improves the accuracy of predictions by RELATCH, MOMA, and ROOM (Additional file 5). In all four scenarios, RELATCH outperformed existing approaches regardless of the availability of experimental data. Second, we investigated how the flux predictions change when different metabolic models of E. coli are used (iJR904 , iAF1260 , and iJO1366 ). The predictions were not sensitive to the metabolic models, except for ROOM predictions for the Δpgi mutant, and RELATCH predictions were still more accurate with all three models (Additional file 6). Third, we performed a sensitivity analysis of the two parameters (α and γ) in RELATCH, where we varied the values of the parameters and investigated their effects on the accuracy of flux predictions (Additional file 1 for details). The results indicate that the predictions were not significantly affected by the parameter values as long as α and γ were in the same order of magnitude (Additional file 2).
We also investigated the effects of alternative optimal solutions on the flux predictions. We first found alternative reference flux distributions by solving the reference flux estimation problem (Additional file 1, Equations S1 to S6) and subsequently minimizing and maximizing each flux variable with the fixed optimal objective function value. Among the 2,383 reactions in the iAF1260 model, only 80 total reactions had flux variability in the reference state, with 35 reactions (mostly in the nucleotide salvage pathway) having variability between 0.01 and 0.22 mmol/gDW/h and 45 reactions (mostly transporters) having variability less than 0.01 mmol/gDW/h. The variability is due to the redundancy and multi-functionality of enzymes in the nucleotide salvage pathway and multiple transport reactions, and is much lower than is often found across alternative FBA solutions (90 reactions with flux variability more than 1 mmol/gDW/h and 134 total reactions with variability). We subsequently used ten alternative reference state solutions from the five reactions with the most variability to predict the flux distributions in the unevolved and evolved four E. coli mutant strains, but we did not find any instances where the alternative reference states affected prediction errors (SSE values were the same). We also investigated the existence of alternative optimal solutions in perturbed states using the four E. coli mutant strains. We solved the flux prediction problem (Additional file 1, Equations S7 to S14) given a fixed reference state, and subsequently minimized and maximized each flux variable after fixing the optimal objective function value. The reactions included in the MFA dataset (whose experimental flux values are available for comparison) did not have any flux variability, which was also confirmed by the fact that slightly increasing or decreasing each flux value led to an increase in the objective function value. Together, these results indicate that there is little variation across alternative solutions for the reference state and that alternative solutions for the reference and perturbed states would likely result in similar SSE values.
Growth rate and phenotype predictions for single knockout E. coli mutants
A summary of growth phenotype predictions for 1,260 single gene knockout E. coli mutants
Predicting metabolic responses to the complete and partial loss of metabolic functions
Next, we used RELATCH to predict the effects of gene knockouts, resulting in the total or partial loss of reaction activities, in three different organisms. Existing methods like FBA, MOMA, and ROOM cannot predict the effects of isozyme deletions since they are based on reaction deletions; however, RELATCH can estimate the flux each isozyme contributes towards the total flux through a reaction and predict the consequences of removing individual isozymes. Large-scale MFA datasets for parental and knockout strains of E. coli, S. cerevisiae, and B. subtilis were taken from previous studies [23, 32, 33], and gene expression datasets for parental strains grown aerobically in glucose minimal medium were obtained from the same study if available  or related studies [34, 35].
To examine whether these results were unique to E. coli, we subsequently used RELATCH to predict metabolic flux distributions in 35 knockout S. cerevisiae mutants of central metabolic genes grown in batch cultures  using the iMM904 metabolic model . The predictions were in good agreement with experimental measurements for most mutants, including the mutants (for example, zwf1Δ, ald6Δ, pda1Δ, and oac1Δ) whose fluxes deviated significantly from the parental strain (Figure 4b; Additional file 9). Further investigation into a few mutants with less accurate predictions (rpe1Δ, fum1Δ, and mdh1Δ) was done to identify potential model improvements. The rpe1Δ mutant had the lowest measured glucose uptake rate among all 35 mutants (approximately 26% of parental value), but RELATCH predicted a larger glucose uptake rate (approximately 65% of parental value) and reversal of the second transketolase reaction (TKT2, so that erythrose 4-phosphate is produced). However, reversal of TKT2 was not experimentally observed at a statistically significant flux level in any of the 35 mutants. When the TKT2 reaction was constrained to be irreversible in the forward direction consuming erythrose 4-phosphate, the RELATCH prediction improved significantly for the rpe1Δ mutant (SSE decreased from 22.3 to 0.9), suggesting that the TKT2 reaction may proceed only in the forward direction in this condition, which is supported by a recent thermodynamic analysis study . Two mutants in the TCA cycle, fum1Δ and mdh1Δ, experimentally had very similar flux distributions and impaired growth, but RELATCH predicted higher growth rates for these two mutants by using alternative routes involving the glyoxylate shunt and malic enzyme. Mdh1 is a component of malate-oxaloacetate and malate-aspartate shuttles, which regulate the NADH/NAD ratio in mitochondria and cytosol , and this regulation is not accounted for in the current metabolic model. Therefore, the transport of oxaloacetate could be affected in the mdh1Δ mutant, as well as the fum1Δ mutant, which would result in limited availability of mitochondrial malate. Interestingly, the oac1Δ mutant, which lacks the mitochondrial oxaloacetate transporter, also had a very similar flux distribution to the mdh1Δ and fum1Δ mutants experimentally, which supports the hypotheses that deletion of mdh1Δ and fum1Δ alters oxaloacetate transport activity. Overall, RELATCH predicted the metabolic responses to a number of genetic perturbations in yeast, as well as led to potential model improvements based on discrepancies (see Additional file 10 for a comparison of prediction methods for mutants that do not involve isozymes).
We also analyzed a large-scale MFA dataset for parental and mutant strains of B. subtilis grown in batch cultures . In comparison to the MFA datasets for other organisms, the B. subtilis MFA results contained fewer flux estimates (four internal fluxes and two external fluxes), which could affect our metabolic flux distribution and enzyme contribution estimates for the reference state. Among the 137 viable mutants in the dataset, 63 of the deleted genes were in the iYO844 metabolic model  and the effects of these 63 deletions were predicted by RELATCH (see Additional file 10 for a comparison of prediction methods for mutants that do not involve isozymes). The predictions were consistent with the MFA measurements for most B. subtilis mutants involving different metabolic pathways, except for the Δpgi, Δzwf, ΔtreA, and ΔacoA mutants (Figure 4c; Additional file 9). The B. subtilis Δpgi mutant had significant flux through the pentose phosphate pathway, allowing the mutant to grow slightly slower than the parent strain (approximately 82% of growth rate), which is very similar to the behavior of the evolved E. coli Δpgi mutant . The Δzwf mutant also exhibited a flexible response by increasing glycolytic flux and acetate production. The flux distributions for both of these two mutants were more accurately predicted if the relaxed RELATCH parameter values were used instead of the tight parameter values to account for the mutant's robust responses (SSE decreased from 5.57 to 1.28 for the Δpgi mutant and from 6.54 to 1.81 for the Δzwf mutant). The ΔtreA and ΔacoA mutants lack enzymes involved in trehalose and acetoin catabolism, respectively, which were both absent from the medium. RELATCH predicts these genes would be dispensable under the condition tested and it is unclear why these mutants exhibit a significant growth defect (reduced glucose uptake rate and growth rate).
Predicting flux redistribution in response to environmental perturbations
Next, we predicted flux distributions when cells are grown anaerobically on glucose in batch culture . E. coli is well adapted to anaerobic glucose catabolism since it is naturally found in the intestinal tract of mammals, but the metabolic flux distributions are quite distinct from those in aerobic conditions due to transcriptional regulation, and redox and energy metabolism differences. For this case, we used the MFA  and expression  data for the E. coli strain grown aerobically in glucose minimal medium for the reference state. In the MFA flux measurements, aerobic and anaerobic conditions were simulated using two different metabolic networks. Instead of modifying the metabolic network, here we compared the sum of the predicted pyruvate dehydrogenase (producing CO2) and pyruvate formate lyase (producing formate) fluxes to the measured flux from pyruvate to acetyl-CoA and production of C1 compounds. Surprisingly, RELATCH was able to predict an almost two-fold increase in glycolytic fluxes, as well as significant ethanol production in anaerobic conditions, in agreement with experimental data (Figure 5b). However, the favored use of pyruvate formate lyase over pyruvate dehydrogenase was not predicted by MOMA or RELATCH, indicating that additional transcriptional regulatory knowledge is needed to predict such behaviors.
We also tested RELATCH's capability to predict responses to carbon source changes (glucose to acetate) in chemostat cultures. Aerobically, E. coli grows efficiently on acetate as a sole carbon source (low growth rate but high biomass yield) and rapidly adapts from growth on glucose to acetate . The metabolic flux distribution  and expression data  for E. coli BW25113 grown on glucose in chemostat with a dilution rate of 0.2 h-1 were used to first estimate a glucose reference state. Since there are two acetate utilization pathways in E. coli (via Ack-Pta and Acs) and MFA cannot distinguish between them, we used the sum of the predicted fluxes through both pathways in our comparison (Figure 5c). The predictions by different methods were all similar to experimental observations; however, RELATCH accurately predicted use of the oxidative pentose phosphate pathway to make pentose phosphates, which is supported by a zwf gene knockout study , while FBA and MOMA predicted use of the non-oxidative pentose phosphate pathway. In addition to the glyoxylate shunt, RELATCH predicted use of the glycerate pathway (Gcl-GlxR-GlxK) feeding glyoxylate into glycolysis/gluconeogenesis, which is experimentally up-regulated during growth on acetate .
Finally, we analyzed the metabolic flux distributions when cells are grown in chemostat at different dilution rates (D). The MFA and expression measurements for E. coli BW25113 at D = 0.2 h-1  were used to estimate the reference state, and the MFA data at D = 0.1, 0.4, 0.5, and 0.7 h-1  were compared to predicted flux values. When the change was moderate (D = 0.1, 0.4, and 0.5 h-1), the model predictions were in good agreement with the data (Figure 5d). However, the experimental flux distribution at D = 0.7 h-1 was significantly different from the others, including increased citric acid cycle flux and acetate production, which all three methods failed to predict. Also, the biomass yield in this condition was much lower, indicating cells were growing suboptimally due to acetate overflow.
Genome-scale metabolic models are being rapidly developed for many organisms, and their applications in biological discovery, metabolic engineering, evolution, and drug discovery continue to expand . Constraint-based models and methods are useful tools to investigate the metabolic potential of an organism and predict its cellular behavior. These models describe the possible metabolic behaviors within given physicochemical constraints, but do not necessarily provide a single metabolic state of the system of interest. Based on an optimal growth assumption, FBA predictions are shown to be well correlated to experimental data for evolved cells , but are less accurate for the behaviors of unevolved cells, which can grow suboptimally. Alternatively, MOMA and ROOM were developed to predict such behaviors of unevolved genetically perturbed systems without requiring any data from a perturbed state. While these latter two methods show good correlation between predicted and observed fluxes, they are still limited in their ability to predict flux distributions with high quantitative fidelity. Also, intracellular flux distributions predicted by existing methods have not yet been rigorously evaluated against genetic perturbations in organisms besides E. coli or against environmental perturbations.
In this work, we presented a new approach, RELATCH, to predict the quantitative metabolic behaviors of genetically or environmentally perturbed microbial systems. RELATCH utilizes available information to estimate the metabolic state before perturbations (MFA, physiological, and transcriptomic measurements) and predict the effects of perturbations using two parameters whose values are chosen according to the nature of the perturbations. We demonstrated RELATCH's prediction capability using large-scale datasets from different perturbation experiments for three model organisms, including unevolved/evolved mutants, batch/chemostat cultures, and genetic/environmental perturbations. Our results show that RELATCH dramatically outperforms existing methods with regard to predicting intracellular flux distributions in gene knockout strains. In addition, RELATCH predictions for environmentally perturbed E. coli strains were significantly more accurate compared to other methods, especially for strains grown in batch cultures (approximately 5- to 50-fold lower SSE) where substrate uptake and growth rates are difficult to predict. It was previously suggested that B. subtilis maintains suboptimal metabolism for the sake of robustness, which led to flexible responses by maintaining the relative metabolic flux distributions . This is consistent with our assumption that perturbed strains would have minimal relative changes in metabolic fluxes with limited regulatory adjustment. While not done here, it is possible that RELATCH predictions could be further improved if organism-specific parameter values were found by training the algorithm on a small dataset. Parameter differences, if identified, would characterize how organisms achieve metabolic robustness.
While the constraint-based metabolic models may not provide a detailed description of the dynamic metabolic and regulatory mechanisms in response to perturbations, they can still provide accurate snapshots of metabolic states during adaptation at a genome-scale level without the need for detailed kinetic parameters. The underlying metabolic and regulatory responses can be inferred from further analyses of the changes in metabolic flux distributions. We showed here that RELATCH can accurately describe such changes using two simple parameters - a penalty for latent pathway activation and a limit on enzyme contribution increases in active pathways. In addition to consistency with MFA datasets, RELATCH predictions were also consistent with previous experimental data, including transcriptomics (for example, increased expression of glyoxylate shunt in evolved Δppc mutants), enzyme assay (for example, increased activity of methylglyoxal pathway in Δtpi mutants), and whole-genome resequencing (for example, mutations in SthA and PntAB in evolved Δpgi mutants). Using RELATCH, the metabolic changes arising from adaptation to perturbations were postulated by probing the flux space using these two parameters. This allows for the generation of hypotheses regarding the importance of metabolic changes, which is useful for metabolic engineering or drug targeting.
Given the ability of RELATCH to predict metabolic responses arising from perturbations with significantly greater quantitative accuracy, the approach can be used to improve the production of biofuels, therapeutics, and commodity chemicals, as well as to identify drug targets for human pathogens. Further integration of RELATCH into computational strain design approaches as a quadratic cellular objective is also possible . In addition, RELATCH could potentially be used to predict metabolic flux distributions in higher eukaryotes (for example, plant or mammalian cells) where biological objective functions are not always obvious. With RELATCH's predictive accuracy, general applicability, and low data requirements, this computational approach will benefit a wide variety of fields.
Materials and methods
Estimating the flux distribution and enzyme contribution in a reference state
Predicting the flux distribution in a genetically or environmentally perturbed state
In all instances, only the w and W enz from the reference state are used to estimate v, and none of the v exp fluxes were used to predict v. An implementation of RELATCH using the COBRA Toolbox for MATLAB  can be found in Additional file 12.
flux balance analysis
metabolic flux analysis
minimization of metabolic adjustment
regulatory on/off minimization
sum of squared errors per flux.
We thank Nattapol Arunrattanamook for conducting the E. coli mutant growth phenotyping experiments that were used in the comparison of growth rate predictions. We also thank Wai Kit Ong for his help editing the manuscript. This work was funded by the US Department of Energy Great Lakes Bioenergy Research Center (DOE BER Office of Science DE-FC02-07ER64494).
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