For our study, we use genome-scale metabolic network modeling. The set of chemical reactions that can take place in an organism and their associated metabolites define the organism's metabolic network. Each reaction is typically catalyzed by an enzyme that allows the transformation of substrate molecules into product molecules. With the advent of genome-scale metabolic network modeling 31 32 33 34 , it has become possible to compute which target products can be synthesized by a given set of enzymes, assuming that the network is in a metabolic steady state, and that specific nutrients are available to the network from the environment. The relevant computational method is based on balancing metabolic flux - the rate at which a reaction converts substrates into products - for all reactions, and is thus called flux balance analysis (FBA) 32 35 . Its predictions are usually in good agreement with experimental data 36 37 38 , except where enzymes are mis-regulated, such that a network cannot attain optimal metabolic fluxes through all its reactions. (Such mis-regulation can be sometimes eliminated during laboratory evolution experiments 37 39 , if growth rate maximization is the sole objective of the experiment.)

An organism's set of enzyme coding genes, identified here with a list of reactions, can be viewed as a discretized binary metabolic genotype; for brevity we refer to it from here on as the organism's genotype or metabolic genotype. Specifically, given a total universe of N possible reactions, any genotype can be represented by a string of N bits, G = (b₁ , b₂ , ..., b_N ) as illustrated in Additional File 1 Figure S1a. If enzyme i is encoded in the organism's genome, then b_i = 1, while b_i = 0 otherwise. In this framework, the space of all metabolic genotypes contains 2 ^N elements. Following previous work 30 40 , we here take the universe of reactions to encompass all known enzyme-catalyzed chemical reactions, as represented in publicly available databases 41 42 . This set of reactions is most likely incomplete, but nevertheless sufficiently comprehensive to produce a vast space of metabolic genotypes, where each genotype contains a subset of these reactions.

If an organism can grow in a specific chemical environment (defined through the nutrients it contains), its metabolic network is able to produce all of its biomass precursors (see Methods); we then call the organism (and by extension its metabolic network) viable. This leads us to define an organism's phenotype via its ability to synthesize biomass in a number of given chemical environments. Note that the mapping from genotype to phenotype in our approach is completely determined by the FBA framework. Previous research has shown that this map is highly degenerate, meaning that a huge number of genotypes will produce the same phenotype; indeed, many reactions in a metabolic network are typically non-essential and can be replaced by other reactions. Furthermore, genotypes of identical phenotype are such that small genotypic changes (a reaction deletion, addition, or exchange) connect these genotypes into a vast graph; we refer to this graph as a genotype network. A consequence of this connectivity property is that gradual evolution of genotypes is possible, while leaving the phenotype unchanged 30 40 . For this reason, genotype networks can facilitate evolutionary changes and adaptation of genotypes 43 . Such properties seem to be generic properties of well-studied genotype to phenotype maps, and have been found in many systems. These include RNA and proteins, where the genotype is the sequence and the phenotype is the secondary or tertiary structure 44 45 46 , as well as gene regulatory networks whose genotype specifies a pattern of genetic interactions and whose phenotype corresponds to a gene expression pattern 47 .

To characterize metabolic networks of a given phenotype, we cannot examine all genotypes because of their astronomical number. Instead, we use a Markov Chain Monte Carlo (MCMC) 48 approach to sample a space of genotypes or subsets thereof. This approach is based on performing random walks within that subspace, as illustrated in Additional File 1 Figure S1c. At each step of such a random walk, a small change is applied to the current genotype and the phenotype of the changed genotype is computed; if the phenotype fulfills a pre-specified criterion, the current genotype is updated; if not, the change is rejected, and the current genotype is kept. With appropriate precautions 30 this procedure will create uniform samples of the accessible space of genotypes with a desired phenotype.

The analysis of modularity in large graphs or networks is a mature field. Not surprisingly, multiple different measures of modularity have been developed 8 49 50 51 52 53 54 . Identifying all modules of a large network can be computationally intractable, that is, NP difficult 55 56 . Fortunately, metabolic networks are special, because their analysis can go beyond graph-based representations. The reason is that metabolic networks synthesize biomass, and this function of metabolic networks can be quantified by studying the flow or flux of matter through each reaction in a network. Doing so permits an analysis of modularity that is based on network function, not just topology. Here we take advantage of the notion of coupling between reaction fluxes to identify sets of reactions that form a metabolic module. Such sets have been referred to as reaction/enzyme subsets or correlated reaction sets or Fully Coupled Sets 25 26 27 28 29 . Hereafter we will use the term Fully Coupled Set (FCS) only. These sets define metabolic network modules that are both biochemically sensible 28 29 57 58 and computationally tractable 28 . By definition, two reactions are in the same FCS if the ratio of their fluxes is fixed when considering all possible steady-state flux distributions through the metabolic network. Determining all FCSs of a large metabolic network can be done efficiently using linear optimization (see Methods). We note that the different FCSs in a metabolic network are disjoint, and that not all of a network's reactions need to belong to an FCS (see Methods).

The simplest possible FCS involves reactions in a linear biochemical pathway, arguably the most intuitive form of a functional module in biochemistry. However pathways with branches and cycles can also form FCSs 28 . For illustration, Figure 1 represents the largest FCS containing a cycle that arises in the E. coli metabolic network; it includes reactions that are involved in cell envelope biosynthesis.

We first asked how modules, as defined by FCSs, relate to conventional biochemical pathways, the classical functional modules of metabolism. To this end, we mapped reactions in many different FCSs onto biochemical pathways, as defined by standardized annotations 41 59 . We relied on annotations in the Kyoto Encyclopedia of Genes and Genomes database (KEGG) 41 , a comprehensive metabolic database that annotates biochemical reactions as belonging to a list of pathways. For the metabolic network of E. coli, we find that reactions in the same FCS typically belong to a common pathway. To quantify whether this property was statistically significant, we implemented the following test.

For each FCS, we identified the pathway annotation for all of its reactions. Because each reaction can be annotated as belonging to multiple pathways, we identified for each FCS the pathway annotation that is shared by most of its reactions. We defined the quantity Q as the fraction of reactions that are annotated as belonging to that pathway, and computed Q for each FCS in the metabolic network of E. coli. We observed that in most FCSs (50 percent, corresponding to 50 of 100 FCSs in E. coli) all reactions belonged to the same pathway, and nearly 75 percent of FCSs had more than 75 percent of their reactions belonging in the same pathway. This strong association of reactions in an FCS with one pathway is not expected by chance alone, as a randomization test shows (P < 0.001). Thus, most of the FCSs in E. coli can be viewed as biochemical pathways or parts thereof.

The same analysis can be applied to random samples of metabolic networks with specific properties, as generated by our MCMC sampling procedure (see Methods). Specifically, we first identified FCSs from 1000 in silico metabolic genotypes viable in all of the 89 carbon source environments we consider (see Methods). In this analysis, we observed that in most FCSs (74 percent, corresponding to 45,893 of 62,148 FCSs examined) all reactions belonged to the same pathway, and nearly 80 percent of FCSs had more than 80 percent of their reactions in the same pathway (see Additional File 1 Figure S2). Just like E. coli, the strong association is not expected by chance alone, as a randomization test shows (P < 0.001; see Methods). Thus, both for E. coli and for our random samples, most FCSs can be viewed as biochemical pathways or as parts thereof. To illustrate such FCSs, Additional File 1 Figure S3 shows the most frequent FCS comprising five or more reactions that we found in our sampling. This FCS occurred in 898 of the 1000 metabolic genotypes. All its reactions belong to histidine metabolism (Q = 1).

We next asked quantitatively how network modularity is affected by environmental versatility. To answer this question, we defined two indices of network modularity, which we call M and s. The first index is the number M of reactions in a network that belong to FCSs. Then we calculate the average <M> for a sample of networks generated by our MCMC procedure, where each network in the sample needs to be viable in a given set V_env of chemical environments (see Methods). We consider V_env as an index of environmental versatility for these metabolic networks. In our analysis, we study up to 89 minimal chemical environments that differ only in the sole carbon source they contain (see Methods and Additional File 2 Table S1). In other words, V_env indicates the number of sole carbon sources from which these networks can synthesize all essential biomass precursors. To see how our observations depended on the sets of carbon sources used, we investigated different choices for these sets, where sets of fewer carbon sources were nested within sets of more carbon sources (see Methods for further details).

In Figure 2a we show how <M> depends on versatility V_env , both on average (yellow dots), and for multiple different nested sets of carbon sources (symbols with different colors and shapes). The analysis is based on metabolic networks with the same number of reactions as E. coli 42 . The data show that greater versatility leads to higher values of the modularity index; this trend is clear when considering the average over all choices of carbon sources, and also when considering the different nested sets.

As a network's versatility rises, does an increase in M - the number of reactions in FCSs - occur through an increase in the size of the FCSs, or through an increase in the number of FCSs, while their size remains constant? To address this question, we next studied the number of FCSs, which we denote by s, our second index of modularity. We applied the procedures we described earlier to the same genotypes as before, averaging now the number of modules (FCSs) rather than the number of reactions in these modules.

Figure 2b shows the average number of modules, which we denote as <s>, for the same 1000 metabolic genotypes, the same choices of V_env , and the same nested sets of environments as above. The figure shows that greater versatility leads to higher values of this modularity index. This holds for the averages over different nested sets (yellow dots), and also without averaging, i.e., for different nested sets of carbon sources (symbols of different shapes and colors). Both <M> and <s> show a monotonic increase with V_env but with possible deviations from linearity.

The results of Figure 2 were obtained from networks whose number n of reactions equaled that of the E. coli metabolic network, i.e., n = 831 42 . Additional File 1 Figures S4 and S5 show that the patterns we see are not sensitive to the number of reactions in a network. Specifically, Additional File 1 Figure S4 shows that the average number of reactions in FCSs, <M>, increases with versatility V_env also for networks with n = 500 (Figure S4a) and n = 700 (Figure S4b) reactions. The sole difference to the data of Figure 2a is that the increase of <M> is beginning to level off as V_env reaches the largest values investigated here, in particular for n = 500. Additional File 1 Figure S5 shows that the average number of modules, <s>, also increases with versatility at n = 500 (Figure S5a) and n = 700 (Figure S5b) reactions. However, in contrast to the trend for <M> in Additional File 1 Figure S4, the increase in <s> does not slow down for the largest values of V_env we have examined.

So far we have shown averages of our modularity measures M and s based on samples of random networks of a given versatility. In such a sample, modularity has a distribution, where some networks are more modular, and others less so. We can use this distribution to ask whether the modularity observed in the metabolic network of an organism such as E. coli is atypically high or low. In other words, the distribution of modularity arising in our samples of in silico metabolic networks can provide a null hypothesis to evaluate whether a biological network shows unusual modularity.

Figure 3a shows the distribution of the total number M of reactions in modules, and Figure 3b shows the distribution of the number s of modules (FCSs) in a sample of 1000 random networks with n = 831 reactions (the same as E.coli 42 ), where each network is able to sustain growth on the 89 different sole carbon sources as given in Ref. 29 . This phenotype constitutes the in silico E. coli metabolic phenotype we use. The figure also shows the values of M and s for the metabolic network of E. coli. The data from the network sample allows us to test the null hypothesis that M or s for E. coli could have been drawn from the sample. We find that M is atypically large, being in the top 3 percentile of our MCMC sample. This allows us to reject the null hypothesis at a P-value of P = 0.028. Based on this analysis, we conclude that the metabolic network of E. coli is more modular than expected.

The architecture of the E. coli metabolic network has higher modularity than anticipated, but the large value of M may come from either a greater number of FCSs or from an increased size of the FCSs. Figure 3b shows that the number of FCSs in E. coli is just slightly above the position of the distribution's peak in our ensemble, well within one standard deviation. From this observation one can conclude that the atypically high modularity of the E. coli network stems from the fact that E. coli has larger modules (FCSs) but not much more modules than typical networks allowing growth on 89 carbon sources.

Thus far, we saw that metabolic networks sustaining growth on more nutrients have higher modularity, that is, more reactions contained in modules and more modules (FCSs) (see Figure 2). We surmised that these additional reactions would be closely linked to the additional nutrients that metabolic networks must utilize as their versatility increases. In other words, these reactions and the modules they reside in presumably are needed to metabolize these nutrients, and may thus occur just downstream of them. To inquire whether this is the case, we compared the FCSs of genotypes with maximal versatility (V_env = 89) to FCSs of genotypes with V_env = 1. Specifically, we first extracted the reactions that belonged to FCSs and that occurred in more than 50 percent of the genotypes in each of the two samples. Call these sets of reactions R₈₉ and R₁, for the ensembles with V_env = 89 and V_env = 1, respectively. At a qualitative level, we find that about 90% of reactions in R₁ also belong to R₈₉. We then examined the reactions that belong to R₈₉ but that are not part of R₁, and called this set of reactions R₈₉\R₁. Are the reactions in R₈₉\R₁ immediately downstream of the nutrients? The notion of downstream can be made quantitative through the Scope algorithm 60 61 . A reaction of scope distance one can use the nutrients as its only substrates, a reaction of scope distance two can use products of reactions at scope distance at most one, and so on. (See Methods for a more detailed explanation of this scope distance.) We applied this algorithm to compare the scope distances of reactions in R₈₉\R₁ to the scope distances of all reactions in our universe of reactions. Figure 4 shows a distribution of these distances for both groups of reactions. It indicates that reactions associated in R₈₉\R₁ generally have smaller scope distance than other reactions. A statistical test (see Methods) shows that this difference is significant with a p-value of 10^-5. In sum, reactions of modules involved in increased versatility tend to be more closely downstream of nutrients, suggesting that they typically belong to pathways metabolizing such nutrients. To illustrate this property with concrete examples, we determined which FCSs in R₈₉\R₁ involved any of the 24 reactions occurring at scope distance 1 in Figure 4b. These FCSs have various sizes that range from 2 to 4 reactions. In Additional File 1 Figure S6 we show the three largest of these FCSs, all of them with 4 reactions, together with the pathways they belong to. These FCSs are linear pathways containing reactions of scope distance 1, 2, 3 and 4; they metabolize the nutrients fucose, rhamnose and 3-hydroxycinnamic acid.

Flux balance analysis (FBA) 32 35 is a computational modeling approach widely used to analyze genome-scale metabolic networks. FBA uses structural information contained in the stoichiometric coefficients of each reaction in a metabolic network to predict the possible steady state fluxes of all reactions and the maximum biomass yield of an organism. FBA does not require knowledge of metabolite concentrations or detailed information of enzyme kinetics. The stoichiometric coefficients of all reactions in a network are encapsulated in a matrix S of dimensions m x n, where m is the number of metabolites and n is the number of reactions. Note that some of these reactions correspond to transport processes, i.e., they import or export metabolites. In a metabolic steady state, such as might be attained in a growing cell population with adequate nutrient supply, the metabolites achieve a dynamic mass balance wherein the vector v of metabolic fluxes through the reactions satisfies the equation

S v = 0

so as to satisfy mass conservation. Eq. 1 represents stoichiometric and mass balance constraints on the metabolic network. For genome-scale metabolic networks, Eq. 1 leads to an under-determined system of linear equations in the metabolic fluxes, leading to a large solution space of allowable fluxes. The space of allowable solutions can be reduced by incorporating thermodynamic constraints associated with irreversible reactions, as well as flux capacity constraints which limit the maximum flux through certain reactions. To obtain a particular solution, linear programming (LP) is used to find a set of flux values - a point in the solution space - that maximizes a biologically relevant linear objective function Z. The LP formulation of the FBA problem can be written as:

max Z = max { c T v | S v = 0 , a ≤ v ≤ b }

where the vector c corresponds to the coefficients of the objective function Z, and vectors a and b contain the lower and upper limits of different metabolic fluxes in v. The objective function Z is often chosen to be the so-called growth flux. This is the flux through an artifactual reaction that reflects the synthesis of biomass, which requires biosynthesis of biomass precursor molecules, such as amino acids and nucleotides, in specific proportions. The stoichiometry of this reaction is based on the experimentally measured biomass composition of an organism. The predictions from the FBA framework and related approaches are often in good agreement with experimental results 36 37 38 68 .

In this work, we have used a hybrid database compiled by Rodrigues and Wagner 40 containing 4816 metabolites and 5870 reactions. This hybrid database was obtained by merging the reactions in the Kyoto Encyclopedia of Genes and Genomes (KEGG) LIGAND 41 with those in the E. coli metabolic model iJR904 42 , followed by appropriate pruning to exclude generalized polymerization reactions. Of the 5870 reactions, 2501 are reversible and 3369 are irreversible. Note that more than 5500 reactions in the hybrid database are contained in KEGG database and less than 300 reactions are specific to the E. coli metabolic model iJR904.

In addition to the 5870 metabolic reactions, the hybrid database has transport reactions for 143 external metabolites contained in the E. coli iJR904 model. These 143 external metabolites were assumed to be the set of possible imported and secreted metabolites. Further, the hybrid database includes an objective function Z in the form of a biomass reaction that reflects synthesis of the E. coli biomass components, as defined in the iJR904 model.

Genome-scale metabolic networks typically contain blocked reactions 28 69 which are dead-ends and necessarily have zero flux for every examined chemical environment under any steady-state condition. Such blocked reactions cannot contribute to any steady-state flux distribution and can be excluded from the hybrid database. For the set of 143 external metabolites, we found 2968 of the 5870 reactions in the hybrid database to be blocked under all environmental conditions we examined. We have excluded this set of 2968 blocked reactions from the hybrid database of 5870 reactions to arrive at a reduced reaction set of 1597 metabolites and 2902 reactions. We thus take this reduced set of N = 2902 reactions as the global reaction set.

The E. coli metabolic model iJR904 has 931 reactions (which of course are contained in the hybrid database of 5870 reactions). Our global reaction set (having 2902 reactions) was derived from the hybrid database by excluding blocked reactions; after this exclusion, the E. coli metabolic model iJR904 is left with 831 reactions. Here, we consider this set of 831 reactions to be the E. coli metabolic genotype.

Any subset of n reactions taken from the global reaction set is considered to specify a discretized binary metabolic genotype. For simplicity, we shall refer to this as a metabolic genotype or as a genotype. Specifically, a metabolic genotype G can be represented by a bit string of length N, i.e., G = (b₁ , b₂ , ..., b_N ), where N is the number of reactions in the global reaction set (see Additional File 1 Figure S1a). Each position in the bit string G corresponds to one reaction in the global reaction set, with the reaction being either present (b_i = 1) or absent (b_i = 0) in the genotype. We denote the set or space of metabolic genotypes with a given number n of reactions as Ω(n).

For any genotype, we can use FBA to determine whether the corresponding metabolic network has the ability to synthesize all biomass components in a given chemical environment (medium). We consider a genotype to be viable in a given environment if and only if the maximum biomass flux predicted by FBA for the genotype is nonzero; otherwise we consider the genotype to be non-viable (see Additional File 1 Figure S1b). In general, in silico metabolic studies take a metabolic network's fitness to be proportional to the maximum biomass growth flux the network can attain in a given environment. The metabolic property considered here is simpler: we ask only whether a network can synthesize all biomass components in a given environment, regardless of the synthesis rate. For all the work we report, we use the E. coli biomass composition to determine the viability of a genotype in a given chemical environment.

For our purpose, the metabolic phenotype of a metabolic network (genotype) is determined by the network's viability in a list of well-defined chemical environments (media). We shall denote the subset of genotypes within Ω(n) that have a specific phenotype - growth on a specific list of environments - as V(n). In this work, we use only aerobic minimal environments containing one carbon source. Each environment also contains unlimited amounts of the following inorganic metabolites: ammonia, iron, potassium, protons, pyrophosphate, sodium, sulfate, water and oxygen. Based on FBA applied to the metabolic model iJR904, it was found earlier that E. coli can support nonzero biomass growth on 89 different aerobic minimal environments 29 57 . The environments we focus on here differ in these 89 carbon sources, which are listed in Additional File 2 Table S1.

The Markov Chain Monte Carlo (MCMC) sampling algorithm (see also below) can be used to explore the set of genotypes having a given phenotype. In our case, this phenotype is viability on a given set of minimal environments; if this set consists of V_env environments, we say that the genotype's environmental versatility index is V_env . Thus, the phenotype V_env = 1 refers to genotypes viable in one specific environment, the phenotype V_env = 2 refers to genotypes viable in two given environments, and so on. We have considered 89 minimal environments whose sole carbon sources, their only distinguishing feature, are listed in Additional File 2 Table S1.

We have used MCMC to sample ensembles of increasingly versatile metabolic networks, i.e., ensembles whose networks have V_env = 1, 2, 5, 10, 20, 30, 40, 50, 70 and 89. The genotypes with V_env = 89 are the most versatile among them as they are viable in all 89 minimal environments. There are many ways of choosing 1, 2, or more specific environments out of 89 environments to sample genotypes having a phenotype with V_env = 1, 2, through 89. The properties of sampled genotypes in an ensemble with a given V_env will depend on the choice of those V_env environments. The computations we carry out are computationally very expensive, and they become more expensive with every additional environment in which viability is determined. To limit this expense, we pursued two strategies. First, we used nested sets of environments to sample genotypes in ensembles with different V_env , e.g., the set of 70 environments chosen for V_env = 70 is a subset of that for V_env = 89, and the set of 50 environments chosen for V_env = 50 is a subset of that used for V_env = 70, and so forth. Second, we used only one subset of 70 environments within V_env = 89 to sample an ensemble with V_env = 70, and only one subset of 50 environments within the choice for V_env = 70 for sampling an ensemble with V_env = 50. For V_env below that, we did tackle the variability coming from different environmental choices; specifically, for V_env = 40, we used 10 different subsets of 40 environments within the choice for V_env = 50; thus we generated 10 different genotype ensembles, where each genotype in each ensemble had V_env = 40. Each of these 10 different choices of 40 environments was then used to create a single nested sequence for V_env = 30, 20, 10, 5, 2, and 1. This allowed us to have 10 different ensembles to sample at each of these V_env and to follow for each sequence of nested sets the consequences of modifying V_env . (See Additional File 1 Figure S7 for a diagram representing two such nested sets.) We computed the average properties of the sampled genotypes as well as their dispersion based on the 10 different samples for each value of V_env .

It was shown in previous work 30 for a single environment, corresponding to V_env = 1, that the size of the subspace V(n) relative to Ω(n) is of the order of 10^-22 for genotypes with n = 2000 reactions. This size decreases even further if one requires viability in multiple environments. Such tiny probabilities of finding a desired phenotype in Ω(n) make it infeasible to sample genotypes in V(n) by simply drawing random genotypes in Ω(n) with the correct number n of reactions, followed by determining the phenotype of each genotype. Thus, we relied on the Markov Chain Monte Carlo (MCMC) method described in Ref. 30 to uniformly sample genotypes in V(n).

This MCMC method starts with a genotype in V(n) and produces a sequence of genotypes, wherein the (k+1)^th genotype in the sequence is generated from the k^th genotype using a probabilistic transition rule. At each transition step, one proposes a small modification to the current genotype in the sequence; if the modified genotype has the correct phenotype, one accepts the modified genotype as the next genotype of the sequence; otherwise the next genotype becomes identical to the current genotype. The modification introduced at each transition step is a reaction swap. It consists of removal of one reaction from the current genotype, followed by addition of new reaction from the global reaction set to generate a modified genotype. Note that the reaction swap preserves the number n of reactions in the genotype (see Additional File 1 Figure S1a). Thus, the MCMC approach produces a walk in the subspace V(n), as illustrated in Additional File 1 Figure S1c. Note that in the limit of long walks, this approach samples uniformly the space of genotypes that are accessible from the first genotype of the MCMC procedure and that have a given phenotype.

In our simulations, starting from an initial genotype in V(n), we have first carried out 10⁵ attempted swaps to erase the memory of the starting genotype. After this initial phase, we continued the MCMC procedure to sample genotypes in V(n). During this later phase, it is not useful to keep all of the genotypes produced, as many of them may be highly similar to one another. We thus saved only every 1000^th genotype generated in a sequence of 10⁶ steps. This procedure produces a random ensemble of 1000 genotypes in V(n) 30 . We sampled genotypes in V(n) for three different values of the number of reactions n = 500, 700 and 831.

To start the MCMC sampling, a first genotype having the correct phenotype is required. To this end, we first determined those reactions in the E. coli metabolic network that have nonzero flux in an optimal flux distribution with maximum biomass flux for each of the 89 minimal environments considered here. (Recall that E. coli is viable on all of our 89 environments). The number of nonzero fluxes is ~300 in a typical optimal flux distribution for each of the 89 environments. We generated a genotype with n reactions and phenotype V_env = 1 (i.e., growth on one specified environment) by starting with the set of nonzero flux reactions for E. coli in that environment, and then adding randomly other reactions until we reached a metabolic genotype with exactly n reactions. We generated a genotype with n reactions and V_env = 2 (i.e., growth in two specific environments) by starting with the union of the two sets of reactions that had nonzero flux when the E. coli metabolic network synthesized biomass in the two different environments; then we added randomly other reactions until we reached a metabolic genotype with exactly n reactions. We generated starting genotypes with V_env = 5, 10, 20, 30, 40, 50, 70 and 89 analogously.

A reaction pair v₁ and v₂ are said to be fully coupled to each other if a nonzero flux for v₁ implies a proportionate (nonzero) flux for v₂ in any steady state and vice versa 28 . A fully coupled set (FCS) in a metabolic network is a maximal set of reactions that are mutually fully coupled to each other (thus, there are no FCSs of size 1). A simple argument shows that FCSs of a network are non-overlapping entities. Indeed, if a reaction were to belong to two FCSs, then all reactions in those two sets would be fully coupled pairwise, resulting in one larger FCS.

We denote the number of FCSs in a metabolic network genotype by s. This is one index of modularity. We also define the modularity index M for a genotype as the number of reactions contained in the FCSs of that genotype (M can vary from zero to the total number of reactions in the network). Burgard et al 28 have proposed a linear programming (LP) based method to determine whether two fluxes in a metabolic network are fully coupled. The LP formulation of the coupling problem can be written as:

Solve  R max = max v 1 | v 2 = 1, S . v = 0, a ≤ v ≤ b

Solve  R min = min v 1 | v 2 = 1, S . v = 0, a ≤ v ≤ b

If R _max = R _min then v ₁ and v ₂ are fully coupled. In the above equations, S is the stoichiometric matrix, and vectors a and b contain the lower and upper limits of different metabolic fluxes in v.

We have used the algorithm of Burgard et al. to determine all FCSs in our metabolic network genotypes. We have computed the coupled reaction pairs under conditions where all external metabolites were allowed to be imported or secreted. Further, coupled reaction pairs were computed without assuming a constant biomass composition to avoid coupling a large set of fluxes to the biomass reaction. Hence, all biomass components were allowed to be synthesized independently of one another, without constraining their stoichiometry in the biomass.

Ebenhöh and colleagues 60 61 have introduced the concept of scope based on a network expansion algorithm for the structural analysis of genome-scale metabolic networks. Their approach calculates for a given metabolic network/reaction database and predefined external metabolites (referred to as seed metabolites) the set of metabolites - the scope - which the reaction network is in principle able to produce. In other words, the scope describes the synthesizing capacity of a given set of seed metabolites given a list of metabolic reactions.

The Scope algorithm iteratively updates a set A of metabolites that a reaction system can synthesize. In the algorithm this set A is initialized to the set of nutrient metabolites. At each iteration i of the algorithm, one takes the current set A(i) of producible metabolites and expands it to set A(i+1) as follows. First one initializes, A(i+1) to contain all metabolites in A(i). Then one considers successively each reaction in the database and adds that reaction's products to A(i+1) if and only if all of its substrates are in A(i). This procedure ends when A(i) = A(i+1), that is when in a given iteration no new molecules can be synthesized. We have used the Scope algorithm to define a distance of a reaction in the global reaction set from nutrient metabolites. Specifically, the distance of a reaction from the nutrient metabolites in the seed set is defined as the iteration number i of the Scope algorithm when that particular reaction contributes its products to A(i+1).

A limitation of the Scope algorithm in comparison to constraint-based frameworks like FBA is its inability to deal properly with the self-generating (autocatalytic) nature of certain cofactor metabolites (e.g., ATP, NADH) in the network 70 71 . The scope of the nutrient metabolites in the seed set is sensitive to the presence or absence of such co-factors in the seed set. Following Kun et al 71 , we included in the seed set the autocatalytic metabolites listed in Additional File 2 Table S2 (in addition to the nutrient metabolites in our minimal environment) when computing the distance of reactions with the Scope algorithm.

We have determined the distance from nutrient metabolites for each reaction in the global reaction set for 89 different seed sets corresponding to the 89 aerobic minimal environments. For each reaction, we have designated the minimum of the 89 distances obtained for the 89 different environments as the scope distance of that reaction from the nutrient metabolites.

Since the modularity index M and the number s of FCSs is larger for sampled genotypes with V_env = 89 than with V_env = 1 (cf. Figure 2a), it is appropriate to identify reactions contributing to additional FCSs in genotypes having V_env = 89. To this end, we combined the lists of FCSs for each of the 1000 sampled genotypes at V_env = 89 (and n = 831 reactions) to create a merged list of FCSs that occur in at least one such sampled genotype. Since the FCSs are non-overlapping entities, multiple copies of a FCS in the merged list signify the FCS's presence in multiple sampled genotypes. We then determined the set of reactions that occurred in at least 500 FCSs in the merged list of FCSs for V_env = 89. We refer to it as the consensus set R₈₉ of FCS reactions for V_env = 89 and n = 831. In a similar way, we obtained the consensus set R₁ of FCS reactions for our sampled genotypes with V_env = 1.

The consensus set R₈₉ for V_env = 89 is larger than the set R₁ for V_env = 1, and the complement set R₈₉\R₁ consisting of the reactions belonging to R₈₉ but not R₁ gives the set of reactions that mostly account for the additional FCSs in V_env = 89. We then considered the scope distances of reactions from nutrient metabolites for two choices of reaction sets. The first set is this complement set R₈₉\R₁, the second is the set of all reactions in the global reaction set. (In Figure 4 we show the corresponding distributions.) The scope distances for the reactions in R₈₉\R₁ are clearly concentrated at much smaller values than when considering all possible reactions. A Kolmogorov-Smirnov (K-S) test allowed us to reject the null hypothesis that the two distributions are the same (P < 10^-5). Further, a two sample Welch t-test allowed us to reject the hypothesis that the means of the two distributions are the same (P < 8.10^-8).

We have classified reactions in our global reaction set into different biochemical pathways using the pathway information 72 for reactions in the KEGG database 41 , along with subsystem information 73 for the remaining reactions in the E. coli metabolic model iJR904 42 . We have used this pathway classification as follows to test whether the majority of reactions in a given FCS belong to a common biochemical pathway.

For a given FCS, we define the quantity Q which is the fraction of reactions sharing the dominant annotation for that FCS. We computed Q for each FCS in the merged list of FCSs from our 1000 sampled genotypes with phenotype V_env = 89 and n = 831 (see the previous section for merged lists of FCSs). We then considered the cumulative distribution of Q for FCSs in the merged list, namely, the probability that Q is at least as large as a given value X. The cumulative distribution of Q for FCSs in the merged list with V_env = 89 and n = 831 is shown in Additional File 1 Figure S2. We also computed the fraction h of FCSs in the merged list with Q ≥ h, a quantity that is analogous to the h-index commonly used to measure scientific productivity 74 . This h-index has a value of h = 0.79, as can be seen from the point of intersection of the cumulative distribution of Q with the bisecting line in Additional File 1 Figure S2.

To test the significance of the h-index obtained from the merged list of FCSs for sampled genotypes with V_env = 89 and n = 831, we performed the following randomization test. Starting from the merged list of FCSs and the pathway annotations of their reactions, we generated 1000 equivalent random lists by swapping the annotations among reactions in different FCSs, while preserving the frequency of each annotation in the merged list. We swapped annotations as follows. We first recorded the multiplicity of each distinct FCS within the merged list. We then randomly picked two FCSs in the merged list with the same multiplicity, and one random reaction in each of the two FCSs, and then swapped the annotations of these two reactions in the FCSs. We performed at least 10⁷ swaps starting from the merged list before saving a random list, that is, a list of FCSs whose reaction annotations had been randomized in this way. None of the 1000 random lists we generated had an h-index greater than 0.79 obtained for the merged list with V_env = 89. On this basis, we can reject the hypothesis that reaction annotations are not similar within FCSs at a p-value of less than 0.001.