An important goal of medical research is to develop methods to recover the loss of cellular function due to mutations and other defects. Many approaches based on gene therapy aim to repair the defective gene or to insert genes with compensatory function. Here, we propose an alternative, network-based strategy that aims to restore biological function by forcing the cell to either bypass the functions affected by the defective gene, or to compensate for the lost function. Focusing on the metabolism of single-cell organisms, we computationally study mutants that lack an essential enzyme, and thus are unable to grow or have a significantly reduced growth rate. We show that several of these mutants can be turned into viable organisms through additional gene deletions that restore their growth rate. In a rather counterintuitive fashion, this is achieved via additional damage to the metabolic network. Using flux balance-based approaches, we identify a number of synthetically viable gene pairs, in which the removal of one enzyme-encoding gene results in a non-viable phenotype, while the deletion of a second enzyme-encoding gene rescues the organism. The systematic network-based identification of compensatory rescue effects may open new avenues for genetic interventions.
Advances in studies of genome-level cellular networks have highlighted striking properties in their large-scale structure, such as a heavy-tailed connectivity distribution and hierarchical or modular organization (Barabási and Oltvai, 2004). However, to capture functional aspects of biological systems, it is necessary to take into account dynamical processes, such as enzymatic activity in the case of metabolism.
Here, we use metabolic reaction fluxes as a representation of cellular phenotypes to develop network-based strategies to recover metabolic function that may have been lost due to defective genes. A recent study of cascading failures in generic complex networks (Motter, 2004) suggested that these cascades can be mitigated through the intentional removal of selected links. In the context of biological networks, this result raises the rather counterintuitive possibility that damage to cellular phenotypes, such as growth, can be alleviated by the targeted removal or downregulation of selected genes. In this study, we have developed a systematic approach to identify such rescue gene knockouts in genome-wide metabolic networks.
Focusing on the metabolism of single-cell organisms, we demonstrate our approach by computationally analyzing reconstructed metabolic networks of Escherichia coli (Edwards and Palsson, 2000) and Saccharomyces cerevisiae (Duarte et al, 2004). We identify genes whose removal can improve the growth of knockout mutants with reduced growth performance relative to the wild type. This is achieved by forcing the cell to either bypass the functions affected by the initial gene loss or to compensate for the lost function. In the extreme case of mutants with zero growth, our analysis predicts that it is sometimes possible to identify single genes whose removal will make it possible for the organism to regain the ability to grow. Consequently, our results suggest the possibility of synthetic-rescue genes, and thus, promise to represent a paradigm shift in the study of gene essentiality.
Figure 1 is a schematic description of our framework to identify possible cases of metabolic rescue. It is based on a combination of the constraint-based approach of flux balance analysis (FBA) (Edwards and Palsson, 2000) and the related method of minimization of metabolic adjustment (MOMA) (Segrè et al, 2002). Simply stated, FBA is a computational technique that identifies possible steady-state reaction fluxes in genome-scale metabolic networks, and MOMA is a variant that predicts postmutational flux states.
We use MOMA to predict the growth of mutants, since the decrease in an organism's growth rate that often follows a gene deletion frequently could be just a transient effect: experiments in fixed media show that after many generations both wild-type and mutant strains typically increase their growth rate (Edwards and Palsson, 2000; Fong and Palsson, 2004) through the accumulation of appropriate regulatory changes and mutations that bring the metabolic system to an optimal steady state. Consequently, FBA is appropriate for predicting the growth phenotype of adapted wild-type strains, as well as the maximum potential for growth recovery in mutant strains.
Our analysis has identified multiple examples of the rescue effect, for both growing and non-growing mutants (Figure 4). We refer to the former case as a suboptimal recovery and the latter as the Lazarus effect. In particular, for E. coli cells in a minimal medium with glucose as the single carbon source, we predict that mutants with the lethal deletion of gene fbaA, pfk, or tpiA can be rescued through the concurrent deletion of other genes (Figure 4A). Sometimes, such as in the case of fbaA mutants in arabinose medium, the Lazarus effect comes along with the deletion of a single additional gene (Figure 4B). Counterintuitively, however, we find that the strength of the recovery generally increases with the number of genes that can be deleted.
The mechanism underlying the rescue effect is general and does not depend on the specific details of MOMA or FBA. In particular, any computational or experimental methodology that can help estimate metabolic fluxes can be used to identify candidates for rescue deletions. This study thus suggests a promising approach to restore metabolic function and identify genetic compensatory interactions, with potentially important implications for disease treatment research.
Recent advances in systems and network biology indicate that specific cellular functions are rarely carried out by single genes, but rather by groups of cellular components, including genes, proteins, and metabolites (Elena and Lenski, 1997; Hartwell et al, 1999; Vogelstein et al, 2000; Barabási and Oltvai, 2004; Bonhoeffer et al, 2004; Albert, 2005; Segrè et al, 2005). Such a network-based view changes the way we think about the impact of mutations and other genetic defects: the damage caused by a malfunctioning protein or gene is often not localized, but spreads through the cellular network, leading to a loss of cellular function by incapacitating one or several functional modules (Barabási, 2007; Goh et al, 2007). The increasingly sophisticated experimental tools that help us systematically map various cellular interactions offer hope that in the future we will be able to focus not only on the individual components but also monitor and explore the global changes in the cellular network induced by the defective gene or protein. Such network-based approaches indicate that the loss of proteins involved in a large number of protein–protein interactions often results in the death of the organism, a finding that may be useful for the design of antibiotics or cancer drugs. Yet for most genetic diseases, particularly those caused by germ-line mutations, the goal is not to kill the cell, but to recover the lost cellular function or limit the existing damage. This raises an important question: can we develop network-based strategies to predict how to recover function that may have been lost due to defective genes?
In single-cell organisms, the frequently observed reduction in an organism's growth rate following a gene deletion often represents only a transient effect, reflecting the fact that the metabolic network of the mutant operates in a suboptimal regime until appropriate regulatory changes and mutations accumulate to bring the metabolic system to a new optimal steady state (Fong and Palsson, 2004; Herring et al, 2006). Experiments in fixed nutrient environments show that after many generations mutants typically increase their growth rate, converging through adaptation to a new optimal value predicted by flux balance analysis (FBA) (Edwards and Palsson, 2000). If the growth rate in this optimal state is zero, then the organism cannot grow, indicating that the deleted gene is essential. We will refer to these genes as optimally essential (Box 1).
Box 1 Summary of new terminology and effects associated with the identification of genetic rescue interactions
Synthetically viable gene sets are defined analogously for interactions involving more genes.
These genes are essential for growth regardless of the state of the other genes.
The deletion of these genes is lethal but the genes themselves are not essential.
Gene deletion leads to zero growth rate but growth is possible in optimal states
MOMA and FBA
Gene deletion restores the growth of otherwise non-viable mutants
MOMA and FBA
Gene deletion increases the growth of already growing strains
MOMA and FBA
Often experiments observe no growth for mutants missing a metabolic enzyme that are predicted to be viable by FBA, prompting us to classify the deleted gene as essential. One potential explanation for the observed discrepancy is that the gene may have an unknown function, regulatory or other, whose absence inhibits growth. Yet for some enzymes an equally compelling explanation is the following: an important challenge of each mutant is to reproduce until the evolutionary tuning of its regulatory system approaches the new optimal growth state. Thus, even if FBA predicts a non-zero optimal growth rate, some mutants may not survive due to their inability to grow in the suboptimal state right after the gene deletion (Figure 1A, red line). The growth rate of the organisms shortly after a gene deletion can be effectively calculated using the minimization of metabolic flux adjustment (MOMA) method (Segrè et al, 2002), a variant of FBA. In the following, we will call a gene suboptimally essential if the optimal (FBA predicted) growth rate is non-zero in its absence, while the MOMA-predicted growth rate is zero. Therefore, experiments will probably classify organisms missing a suboptimally essential gene as unable to grow. However, in contrast with the optimally essential genes, a mutant missing a suboptimally essential gene would be determined as viable if its metabolism and regulatory system had the chance to re-adjust to its environment.
Here, we show that the growth rate of an organism lacking a suboptimally essential gene may be restored via the removal of other enzyme-encoding genes. We will refer to this as the Lazarus effect, as it restores the growth of mutants initially classified as non-viable by experiments since they displayed zero growth rates. We also discuss suboptimal recovery, a weaker manifestation of the proposed mechanism, which forces viable mutants to increase their growth rate following additional gene deletions. Our approach is inspired by a method proposed in Motter (2004) to control cascading failures in complex networks and by microbial optimization methods for the targeted production of metabolites (Burgard et al, 2003; Pharkya and Maranas, 2006).
The principle underlying the proposed rescue effect is illustrated schematically in Figure 1B–E. Consider the situation where, in the wild-type organism, the optimal growth state corresponds to the utilization of the M1 → M2 → M4 pathway, i.e. the flux of reactions involving the M3 metabolite is either zero or close to zero. In the early state after the deletion of the enzyme catalyzing the M2 → M4 reaction, metabolism operates suboptimally (Figure 1C) by minimizing the necessary flux rearrangement compared to the optimal wild-type flux state (Figure 1B). The optimal postdeletion state, however, requires more drastic flux reorganization, sending most of the flux through the M1 → M3 → M4 pathway (Figure 1D). It would take additional regulatory and metabolic adjustments to reach this new optimal state. This process can be facilitated by deleting the enzyme catalyzing the M1 → M2 reaction, forcing the cell to use the optimal M1 → M3 → M4 pathway (Figure 1E). Therefore, by suppressing stoichiometrically inefficient pathways, we can force the cell to enhance the activity of a more efficient set of reactions, resulting in an increased growth rate. Our goal is to show that such additional deletions, whose role is to enhance the activity of the most efficient pathways, can be predicted by systematically comparing the suboptimal and the optimal fluxes under the same conditions.
To implement the approach described in Figure 1, we developed an algorithm to identify rescue deletions for all mutants missing an enzyme-encoding gene. For this, we use MOMA to determine the suboptimal fluxes ν1MOMA characterizing the mutant shortly after a gene deletion (Figure 1C) and FBA to predict a flux state ν1FBA compatible with optimal growth for the mutant (Figure 1D). If the mutant's metabolism operates suboptimally after the gene deletion, the FBA-predicted growth rate for the mutant is larger than the MOMA-predicted growth rate, and thus, we have a chance to intervene and increase the suboptimal growth rate. In this case, based on the difference in flux pattern between ν1FBA and ν1MOMA (see Materials and methods), we test a set of secondary rescue gene deletions that aim to reduce the difference between the suboptimal and the optimal growth rates (Figure 1E) by using MOMA to determine the new metabolic flux state ν2MOMA. If appropriate rescue gene deletions are identified, the obtained growth rate G2MOMA is higher than the growth rate of the original mutant, concluding our procedure. If the MOMA-predicted growth rate for the original mutant is zero, the rescue deletions can bring along the Lazarus effect, inducing a non-zero growth rate; if it is non-zero, the rescue deletions may induce a suboptimal recovery, increasing the mutant's growth rate toward its optimal FBA-predicted value. Note that the identified rescue deletions do not change the optimal growth rate, but affect only the suboptimal growth rate (see Materials and methods). The new terminology related to this recovery mechanism is summarized in Box 1.
We illustrate the proposed procedure in Figure 2 for the TCA cycle of Escherichia coli MG1655 fed arabinose as the sole carbon source (see Materials and methods and Supplementary Information). MOMA predicts that the deletion of the fbaA gene rearranges the fluxes throughout the whole cycle and inhibits the production of phenylalanine, tyrosine, and L-lysine (dotted reactions in Figure 2B), which represent necessary building blocks of the biomass (cf. Figure 2A). Thus, the suboptimal growth rate of this mutant is zero, a prediction supported by experiments in arabinose media (Fraenkel, 1987). In contrast, FBA indicates that a non-zero growth rate can be achieved by a global rearrangement of the flux states (Figure 2C), resulting in changes in flux magnitudes and directions (e.g. the sucCD reaction). Consequently, the organism could grow if it could get past its suboptimal state when, soon after the gene deletion, its growth rate is zero. We can force the organism to approach the new optimal state by deleting, for example, the genes aceA and sucAB, which catalyze reactions that are active in the suboptimal state (Figure 2B) but are not active in the optimal state (Figure 2C). These two rescue deletions will activate the production of all biomass components after rerouting the fluxes through the pentose phosphate pathway (Figure 2D), and result in a non-zero growth rate, rescuing the otherwise non-viable mutant.
The growth rate of the fbaA mutant can be further enhanced by deleting additional genes that catalyze reactions that are inactive in the optimal state (see Materials and methods). We illustrate this in Figure 3A, which shows the predicted suboptimal growth rate of the fbaA-deficient E. coli mutant after the concurrent removal of several genes in addition to aceA and sucAB. While the rescue deletion of aceA is sufficient to recover growth, the additional deletion of sucAB, tnaB, xapB, and prr further enhances the growth rate, with a large enhancement predicted after the removal of tnaB. The biomass production reaches a plateau of about 67% of the wild-type biomass production rate after the deletion of 40 genes. The situation is similar for suboptimal recovery: as we show in Figure 3B for the case of the nuoA mutant with glucose as the carbon source, additional gene deletions can increase the growth rate of the mutant, eventually approaching 59% of its wild-type optimal value.
Systematically applying our method to the E. coli metabolism in glucose minimal medium, we identified 6 suboptimally essential genes, which represent candidates for the Lazarus effect, and 17 candidates for suboptimal recovery (see Figure 4A). Most of the mutants miss genes involved in the central metabolism, while a few miss genes that participate in amino-acid metabolism and transport processes. Of particular interest are mutants with the genes pfk, fbaA, or tpiA deleted, whose essentiality has been tested and is supported by experiments (Fraenkel, 1987). As we show in Supplementary Table SI and Figure 4A, the growth rate of these mutants is restored by additional targeted gene deletions that increase the suboptimal growth rate from zero to more than 45% of the wild-type growth rate.
In Figure 4B we show that, for various media, the increase in the biomass production rate obtained after the deletion of a single rescue gene can be more than 10% of the wild-type rate. In other cases, however, we need to simultaneously delete several genes to rescue growth. This is illustrated in Figure 3B, where we show that the growth performance of non-viable tpiA-deficient mutants in a glucose medium can be restored only through the concurrent deletion of six genes, aceA, gadA, gadB, lpdA, tynA, and gpt, representing a six-viable set, which is the converse of the k-robust set necessary to suppress cellular growth (Deutscher et al, 2006). The suboptimal tpiA mutant uses the glyoxylate pathway, which is shut down by these rescue deletions. Our prediction, that the glyoxylate pathway is not needed in the optimal state, is supported by a recent experimental observation (Fong et al, 2006). This observation indicates that the flux of the glyoxylate pathway in viable but not fully evolved tpiA mutants is initially non-zero. However, over the course of a few weeks of adaptive evolution in glucose media, the glyoxylate flux converges to zero (Fong et al, 2006). Once the six genes are absent, the concurrent deletion of additional genes can further increase the organism's growth rate (Figure 3B).
Note that, while the proposed rescue procedure works in all media, the list of mutants that can be rescued by additional deletions as well as the necessary rescue deletions depends on the tested medium. Indeed, we find that the number of E. coli mutants whose growth rate increases by more than 10% of the wild-type growth rate after rescue deletions is 8, 21, and 25 in minimal acetate, minimal glucose, and rich media, respectively. Therefore, the rescue effect is more frequent in richer media, where the increased availability of substrates in the environment increases the number of non-essential metabolic genes that can be deleted to improve performance. Furthermore, the proposed rescue mechanism is expected to work for all organisms, allowing us to predict rescue deletions each time an accurate metabolic reconstruction is available. To show this, we determined all single-gene rescues that can recover the growth rate by more than 1% of the wild-type rate in glucose media for deletion mutants of three reconstructed organisms with very different genomes: Helicobacter pylori (341 enzyme-encoding genes), E. coli (660), and Saccharomyces cerevisiae (750). Interestingly, the obtained number of mutant-rescue combinations for these organisms, 58, 94 and 58, respectively, is consistently large and to some extent comparable despite the significant differences in their metabolism.
In our analysis of the most significant cases for the eukaryote S. cerevisiae, we predict the Lazarus effect for 3 mutants and suboptimal recovery for 11 other gene deletions in a glucose minimal medium (Figure 4C). It is interesting to note that several of these genes are human orthologs (Steinmetz et al, 2002; BiGG, 2007), including genes pfk, tpi1, lpd1, and mir1. Of the three mutants predicted to exhibit the Lazarus effect, two of them have been experimentally verified to be non-viable, while positive growth has been observed in experiments for the third one (SGD, 2007). The observed small disagreement with our predictions is probably due to the incompleteness of the reconstructed model or the fact that the organisms in the experiments were not fully adapted to the medium modeled in our computations. As in the case of E. coli, the intensity of the recovery generally increases with the number of genes in the rescue set and best recovery may involve up to 50 genes in the examples shown in Figure 4C. However, we predict that a comparable recovery can be obtained with significantly fewer deletions (Supplementary Information). In particular, we also find numerous examples in several media of single-gene rescue deletions resulting in a significant increase of biomass production in S. cerevisiae mutants (Figure 4D).
The focus so far in this work has been on developing an approach to computationally predict synthetic rescue in metabolic networks. Importantly, the founding hypothesis of this approach, that the suboptimal growth rate of an organism can be improved by the removal of properly selected genes, is consistent with experiments. To demonstrate this, in Figure 5 we reanalyze experimental results (Fong and Palsson, 2004; Fischer and Sauer, 2005) for the growth rate of several mutants in their suboptimal state, before and after a gene deletion. The compiled data in Figure 5A indicate that the suboptimal growth rates of E. coli MG1655 can indeed improve considerably after the deletion of selected enzyme-encoding genes, an effect observed in multiple environments. In Figure 5B, we show similar results for B. subtilis 168, following the removal of genes involved in various cellular functions. Note that the metabolism of the wild-type strains in these experiments is not fully adapted to the media and operates in a suboptimal regime. This experimental evidence, together with the power of FBA (Edwards et al, 2001; Ibarra et al, 2002) and MOMA (Segrè et al, 2002; Shlomi et al, 2005) to predict the optimal and suboptimal growth rates of an organism in agreement with experimental data, supports our hypothesis that properly selected gene deletions can improve the growth rate of an organism that has not yet adapted to its environment. To further substantiate this claim, we calculated the reaction fluxes determined by experimental uptake and growth rates (Fong and Palsson, 2004) as well as the corresponding optimal reaction fluxes. We used these flux distributions to test our assumption that gene deletions that increase (not increasing) growth tend to be associated with reactions whose fluxes are much larger (smaller) than the optimal fluxes. As shown in Supplementary Table SIV, a total of 20 out of 22 E. coli mutants analyzed are correctly predicted with this assumption, in support of the proposed rescue mechanism.
The mechanism behind the rescue effect introduced above does not depend on the specific details of MOMA or FBA; in fact, any computational or experimental methodology that can help us estimate the metabolic fluxes can be used to identify candidates for rescue deletions. For example, one could use 13C-tracer techniques (Sauer, 2004) to experimentally determine the reaction fluxes of the suboptimal gene-deficient strain and an optimal, or close to optimal, version of the same strain. Candidates for rescue deletions typically correspond to genes catalyzing reactions that are active in the suboptimal state but inactive in the optimal state. By identifying these reactions experimentally, one could minimize biases due to inaccurate modeling in the identification of the candidate rescue deletions. However, we find that our in silico predictions are robust to parameter choices and do not rely on the fine tuning of metabolic fluxes or environmental conditions (see Supplementary Information). Furthermore, we predict that the rescue set in an impaired cell is not unique, and the number of rescue combinations that lead to the same effect generally increases with the number of genes in the set (Supplementary Information). These observations corroborate the feasibility of systematic experimental implementation of synthetic rescues. Indeed, the main difficulties expected in verifying our predictions, namely the inaccuracies in matching real genetic and environmental conditions as well as potential side effects of rescue deletions due to, e.g. unknown function, are substantially alleviated by the robustness and flexibility of the rescue interactions. This generality, which transcends particular computational methods, could serve as a bridge to implementations of our approach in multi-cellular organisms, as it facilitates the control of undesirable effects in the recovery of specific cellular functions.
The possibility of rescuing a mutant using additional gene deletions is a general mechanism not limited to metabolism. For example, the removal of comA and sigD genes enhances the growth rate of Bacillus subtilis (see Figure 5), despite the fact that they have no known enzymatic functions. Additionally, it has been observed in E. coli that edd-deficient mutants grow at a reduced rate and eda-deficient mutants do not grow at all in a gluconate medium, while the double edd/eda mutant is viable. In this case, the mechanism for the rescue effect is different from the one discussed above: upon the deletion of eda, the cells accumulate toxic compounds; this accumulation stops when edd is also deleted (Fraenkel, 1987). Gene deletion-induced rescue processes have been observed previously in mammalian cells as well. For example, Irs2 knockout mice develop diabetes in 6–8 weeks (Kushner et al, 2004; Hahnfeldt and Hlatky, 2005). Yet the additional knockout of Ptp1b partially compensates for the lack of Irs2, doubling the survival time. Similar effects were documented for mutations in HK1.ros, HK1.fas, and HK1.TGK-α in tandem with the loss of p53 gene (Wang et al, 2000; Hahnfeldt and Hlatky, 2005). The mechanisms behind these examples involve mostly local gene–gene interactions, as opposed to the global effect we have systematically unveiled here. They indicate, however, that organisms could be characterized in general by potentially extensive sets of synthetically viable double knockouts, representing gene pairs for which the double mutant is viable while one of the single mutants is not. High-throughput techniques, increasingly used to identify synthetically lethal pairs (Tong et al, 2001; Ooi et al, 2003), could be used to uncover such synthetically viable gene pairs as well. Other techniques may be developed to identify similar interactions between gene sets, as proposed above. The results could be used to detect new genetic compensatory mechanisms and would offer a better understanding of cellular functions, just as synthetically lethal gene pairs have deepened our understanding of genetic interactions (Wong et al, 2004; Boone et al, 2007). Furthermore, our results force us to adjust the current paradigm of gene essentiality: even if the deletion of a gene is lethal, the gene is not necessarily essential to support life (Kobayashi et al, 2003; Glass et al, 2006; Hashimoto et al, 2005; Pál et al, 2006) because the organism's ability to metabolize biomass may be restored by additional gene deletions.
Finally, our findings may also offer a new alternative to restore the loss of cellular function caused by specific mutations. Indeed, current approaches based on gene therapy (Ho and Commins, 2001; Kaiser, 2005; Kimmelman, 2005) may trigger abnormal activity associated with the vector and insertion site, such as oncogenesis, or reinforce the activity of pathways encoding malfunctioning products of the faulty gene, such as misfolding proteins. From a drug design and therapy perspective, it may be more advantageous to block the activity of selected pathways rather than trying to restore the activity of a faulty gene or protein. Specific previous experimental studies that can be related to the recovery mechanism reported here corroborate the feasibility of such an approach. It has been observed, for example, that Myc deletions rescue the Apc deficiency in murine small intestine. This presumably takes place because Myc is required for gene activation involved in cancer development often following Apc inactivation (Sansom et al, 2007). In a different study, the combination of antibiotics exhibiting hyper-antagonistic interactions, where the combined effect of two antibiotics is weaker than at least one alone, has been shown to select against resistant strains (Chait et al, 2007). In the context of our work this means that, in the two-drug sublethal medium, the ‘deficient’ bacterial cells (non-resistant strain) prevail. Another example is found in studies of E. coli mutants unable to grow anaerobically on glucose and other hexoses when gene adh (ethanol production) or gene pta (acetic acid production) is inactivated, but the mutant with both genes deactivated will grow through the production of lactic acid as the major fermentation product (Gupta and Clark, 1989). On the other hand, a non-fermenting mutant of E. coli, NZN111, is rescued to ferment glucose through the inactivation of the ptsG gene, resulting in the production of succinate, acetate, and ethanol by rerouting fluxes that would go through the partially blocked pathways of pyruvate in NZN111 (Chatterjee et al, 2001). In addition, it has been shown that the concurrent deletion of genes zwf, sfcA, maeB, ndh, ldhA, and frdA maximizes the biomass yield in wild-type E. coli MG1655 by eliminating the elementary metabolic modes associated with low biomass yield (Trinh et al, 2006). The latter study also demonstrates the feasibility of creating mutants with several targeted rescue deletions (Causey et al, 2003), in support of our suggestions. These examples are not limited to a single organism and can be interpreted as different manifestations of a common rescue mechanism. Therefore, a combination of experimental and computational studies aimed at systematically uncovering synthetically viable gene pairs and gene sets, as well as the underlying rescue effects, may open new avenues for the next generation of therapeutic strategies.
Materials and methods
For a network with m metabolites and n reactions, the stoichiometric constraints are represented by where S=(Sij) is the m × n matrix of stoichiometric coefficients and ν=(νj) is the vector of fluxes. The individual fluxes are limited by thermodynamic constraints, substrate availability, and the maximum reaction rates supported by the catalyzing enzymes and transporting proteins, as
where αj=βj=0 for uptake reactions of substrates not available in the medium. The biomass production is incorporated as an additional reaction biomass unit, where the stoichiometric coefficient ci corresponds to the experimentally measured biomass composition of metabolite xi (Edwards and Palsson, 2000). FBA consists of finding a metabolic state that satisfies these constraints while maximizing the biomass flux G. The deletion of genes responsible for the production of the enzymes involved in reaction j corresponds to imposing the bounds αj=βj=0 in Equation (1). MOMA aims to find a solution νMOMA, compatible with the constraints imposed to the mutant, while being closest to the original metabolic state νFBA in terms of Euclidean distance in the space of fluxes. Our implementations of FBA and MOMA are based on the optimization softwares GNU Linear Programming Kit (Makhorin, 2001) and Object-Oriented Quadratic Programming Package (Gertz and Wright, 2001), respectively, and have been tested using independent implementations of the CPLEX solver (ILOG CPLEX, Version 10.2.0, www.ilog.com).
Identifying rescue gene deletions
Consider a strain generated by the deletion of a metabolic gene that constrains at least one of the non-zero metabolic fluxes of the wild-type organism and such that the biomass flux after this deletion is G1MOMA<G1FBA. To increase G1MOMA, we compute the vector of all metabolic fluxes ν1FBA=(νj1FBA) predicted by FBA and use them to define a second gene deletion. This deletion is defined by identifying the minimum number of metabolic genes that deactivate most or all reactions j with νj1FBA=0. These gene deletions force the metabolic system to operate closer to the optimal regime predicted by FBA, while they do not change the FBA fluxes and the predicted steady-state biomass production (i.e. G2FBA=G1FBA). The corresponding changes in the MOMA-predicted fluxes are expected to increase the biomass flux from G1MOMA to G2MOMA>G1MOMA. We recursively discard the deletions that have no impact on the MOMA-predicted biomass production. The number of gene deletions is further reduced through recursively activating genes from the rescue set that contribute the least to the increase in G2MOMA. Note that this approach increases the biomass production itself, which is not necessarily related to the biomass yield considered in metabolic engineering studies (Causey et al, 2003; Trinh et al, 2006).
Algorithmically, we start with a mutant strain defined by the deletion of one or more metabolic genes, and identify the sets of rescue deletions by adhering to the following procedure:
Calculate the FBA optimal flux vector νwtFBA for the wild-type strain.
Calculate the FBA optimal flux vector ν1FBA for the mutant strain.
Calculate the MOMA flux vector ν1MOMA for the mutant strain, using νwtFBA as a reference flux vector.
Continue if biomass flux G1MOMA<G1FBA, as a set of rescue deletions may exist.
Identify reaction set K consisting of all reactions j such that νj1MOMA≠0 and νj1FBA=0. Potential recovery is implemented by setting (αj, βj)=(0, 0) for every j ∈ K.
Identify K* as self-consistent subset of K by obeying the gene–enzyme relationships.
Incrementally reduce K* by identifying and activating recursively the gene from the rescue set whose deletion contributes the least to the increase of the biomass flux. Gene activation is implemented by restoring the original (αj, βj) of the corresponding reactions.
This study was partially supported by NSF-MRSEC program (DMR-0520513) at the Materials Research Center of Northwestern University (AEM), DOE under contract DE-AC52-06NA25396 (NG), LLNL-LDRD office (06-ERD-061) and DOE under contract W-7405-Eng-48 (EA), and NIH U01 A1070499-01 and 1P20 CA11300-01 (ALB).