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 24 and mapped onto corresponding reactions in the iAF1260 metabolic model 25. Gene expression data for the parental E. coli strain grown aerobically in glucose minimal medium from an earlier study 26 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) 24.

RELATCH accurately predicted the metabolic flux distributions in four knockout mutants before undergoing adaptive evolution, while existing methods (FBA, MOMA, and ROOM) often over-predicted the flux values (Figure 2a). RELATCH effectively predicted the initial responses to the genetic perturbations, including the limited use of latent pathways and rerouting of fluxes around the metabolic defect, with an overall effect of decreasing network throughput. Only RELATCH predicted a significant reduction of growth and glucose uptake rates in three unevolved mutants (Δpgi, Δppc, and Δtpi), as well as slight increase in glucose uptake rate for the Δpta mutant (Additional file 3). In the Δpgi mutant, activation of the glyoxylate shunt, decreased flux through lower glycolysis (approximately 23% of the parental strain), and unaltered flux through pentose phosphate and Entner-Doudoroff pathways were successfully predicted by RELATCH. Activation of the glyoxylate shunt was also accurately predicted for the Δppc mutant, enabling an alternative mechanism for synthesizing citric acid cycle intermediates. The most notable initial response in the Δpta mutant was secretion of pyruvate instead of acetate, which was only predicted by RELATCH (MOMA and ROOM predicted acetate production similar to the parent strain using alternative acetate production routes). RELATCH's pyruvate secretion prediction in the Δpta mutant is due to the minimization of relative flux changes, since lowering the penalty for latent pathway activation and/or removing enzyme contribution constraints still led to pyruvate secretion predictions. In agreement with the MFA measurements and in vitro enzyme assays 24, RELATCH predicted activation of methylglyoxal pathway with limited capacity in the Δtpi mutant. Interestingly, RELATCH predicted no growth for the Δppc and Δtpi mutants when a very high penalty for latent pathway activation was used (Additional file 2), suggesting that the activation of latent pathways is needed to enable growth in these mutants.

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 27 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 24. 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 27. 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 24. 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 24.

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 28, iAF1260 25, and iJO1366 29). 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.

To further evaluate the accuracy of the different methods for predicting growth rates, we subsequently grew 22 single knockout mutants of E. coli K-12 BW25113 in glucose minimal medium (Additional file 1), and compared the measured growth rates to the predictions by RELATCH and other existing methods. To better estimate the flux distribution in the parental strain, we used MFA measurements and physiological parameters from a different study 30 where glucose uptake, acetate secretion and growth rates were more consistent with our parental strain growth curves. MOMA and RELATCH predictions were closest to experimentally observed growth rates (each with an average squared error across all mutants of 0.01 h^-2), while FBA and ROOM generally predicted higher growth rates as noted in a previous study 22 (Figure 3; Additional file 7). The growth rate predictions for the Δppc mutant differed significantly among the four methods where FBA, MOMA, and ROOM predicted higher growth rates and RELATCH predicted no growth (note the RELATCH growth predictions differ from earlier Δppc predictions because the reference states differ). The growth of a Δppc mutant requires suppressor mutations in the glyoxylate shunt 31 and RELATCH predicted no growth due to inactivation of the glyoxylate shunt. If the glyoxylate shunt was activated in this mutant, then RELATCH would predict positive growth (0.41 h^-1).

We also performed growth phenotype predictions for 1,260 E. coli single gene knockout strains. We compared the growth phenotype predictions by FBA, MOMA, and RELATCH to the list of experimentally essential genes from the iAF1260 study 25. For RELATCH and MOMA, we used the reference flux distribution determined for Figure 2. The results show that RELATCH found more true negative cases than FBA without any additional false negative cases (Table 1). Although MOMA found six more true negative cases, it also had nine more false negative cases (Additional file 8 for details).

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 233233, and gene expression datasets for parental strains grown aerobically in glucose minimal medium were obtained from the same study if available 23 or related studies 3435.

We first tested the capability of RELATCH to predict fluxes in E. coli mutants grown in chemostat cultures 23. The results indicated that RELATCH can effectively predict the metabolic responses to the deletion of single enzymes, including isozymes, by cells grown in chemostat cultures (Figure 4a). For 21 out of 23 mutants, the SSE was less than 1 and the average SSE across all mutants was 0.32 (see Additional file 9 for details). Interestingly, the ΔtktA and ΔrpiB mutants had decreased biomass yields (approximately 65% and 70% of the parental strain's, respectively), which was not predicted by RELATCH. In E. coli, transketolase and ribose-5-phosphate isomerase are essential for growth on glucose 3637, and E. coli has two isozymes for each (TktA/TktB and RpiA/RpiB, respectively). Since TktA is known to be responsible for the major transketolase activity 38, it seems that the activity of TktB alone in the ΔtktA mutant is not sufficient for efficient growth (the ΔtktB mutant had a similar behavior to the wild type's). However, the behavior of the ΔrpiB mutant was unexpected since RpiB is thought to be the minor isozyme under this growth condition 37, and RELATCH predicts a flux distribution similar to the parent strain for this mutant.

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 32 using the iMM904 metabolic model 39. 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 40. 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 41, 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 33. 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 42 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 24. 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).

We further tested the ability of RELATCH, FBA, and MOMA to predict shifts in metabolic fluxes due to environmental perturbations, where cells are grown in different media or reactor conditions. First, we analyzed the metabolic flux distributions when cells move from glucose to galactose minimal medium in aerobic batch cultures 30. In contrast to glucose, E. coli is not well adapted to and grows slowly on galactose, but can improve growth after undergoing adaptive laboratory evolution 43. We used the MFA and expression data for the E. coli strain grown aerobically in glucose minimal medium 30 to predict flux distributions on galactose. The RELATCH prediction was much more accurate than the predictions from other methods, and the results suggest that galactose metabolism is limited by the capacity of galactose utilization pathways (Figure 5a). Interestingly, when the relaxed parameters were used instead of the tight parameters, the predicted flux distribution was close to experimental values for a NagC transcription factor mutant that had derepressed galactose uptake and thus increased growth rate 30 (Additional file 11).

Next, we predicted flux distributions when cells are grown anaerobically on glucose in batch culture 44. 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 44 and expression 26 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 45. The metabolic flux distribution 46 and expression data 23 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 46, 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 47.

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 23 were used to estimate the reference state, and the MFA data at D = 0.1, 0.4, 0.5, and 0.7 h^-1 23 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.

We introduced new variables for enzyme contributions ( W j , n e n z ) for all reaction (j)-enzyme (n) pairs using gene-to-protein-to-reaction (GPR) associations. To estimate the flux distribution (w) and enzyme contributions (W^enz) in a reference state, we utilized MFA measurements and gene expression data. The sum of squared differences between flux variables and MFA estimates (w^exp) weighted by the reciprocal of confidence intervals (w^conf), and the sum of squared enzyme contributions weighted by the reciprocal of enzyme expression values (E_n) were minimized to calculate the flux distribution and corresponding enzyme contribution (Equation 1). Using Equation 1, isozymes with higher expression will have higher enzyme contributions unless the associated flux is zero. If a reaction j has known GPR ( j ∈ J G P R ) and is associated with multiple isozymes, N(j), the flux value through the reaction is constrained by the sum of enzyme contributions for all the associated isozymes (Equation 2). If multiple reactions are associated with a multi-functional enzyme, different W^enz variables are assigned to each enzyme-reaction pair. External flux values (for example, glucose uptake rates) were constrained using the physiological measurements from the reference state (see Additional file 1 for details):

min w , W e n z ∑ j ∈ J M F A w j e x p - w j w j c o n f 2 + ∑ j ∈ J G P R ∑ n ∈ N ( j ) ( W j , n e n z ) 2 E n

- ∑ n ∈ N ( j ) W j , n e n z ≤ w j ≤ ∑ n ∈ N ( j ) W j , n e n z ∀ j ∈ J G P R

Using the estimated metabolic flux distribution (w) and enzyme contribution (W^enz) in a reference state, the metabolic flux distribution (v) and enzyme contribution (V^enz) in a perturbed microbial system was predicted by minimizing the relative flux changes from a reference state for reactions active in the reference state (J_act) and the enzyme contribution increases for enzymes inactive in the reference state (N\N_act) with a penalty α (Equation 3). In addition to enzyme contribution constraints on fluxes (Equation 4), we imposed a limit (γ) on contribution increases for active enzymes (N_act) (Equation 5). We used two different sets of values for these parameters (tight parameter values of α = 10 and γ = 1.1, or relaxed parameter values of α = 1 and γ = ∞) when predicting metabolic behaviors to different perturbations (see Additional file 1 for details). To simulate cases where γ = ∞, Equation 5 is omitted since the V^enz values are not constrained by the W^enz values. A gene knockout was simulated by setting the contribution of associated enzymes to zero, and an environmental perturbation was simulated by changing what metabolites are allowed to be taken up.

min v , V e n z ∑ j ∈ J a c t w j - v j w j 2 + α ∑ j ∈ J G P R ∑ n ∉ N a c t ( j ) V j , n e n z

- ∑ n ∈ N ( j ) V j , n e n z ≤ v j ≤ ∑ n ∈ N ( j ) V j , n e n z ∀ j ∈ J G P R

V j , n e n z ≤ γ W j , n e n z ∀ j ∈ J G P R , ∀ n ∈ N a c t ( j )

The estimated flux distribution (w) used by RELATCH was also used for MOMA and ROOM calculations. For RELATCH, MOMA, and ROOM calculations, none of the exchange fluxes (except for growth rates in the chemostat cases since these are known a priori) were constrained to the measured values in the perturbed states, but they were instead predicted by the methods. To evaluate the differences between model predictions (v) and experimental MFA results (v^exp) we calculated the sum of squared errors per flux (SSE) using the following equation:

S S E = ∑ j ∈ J M F A ( v j e x p - v j ) 2 J M F A

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 51 can be found in Additional file 12.