In order to subdivide the metabolic reactions into essentiality classes, namely non-essential, conditional lethal, and essential, we quantify the relative essentiality of a reaction by computing optimal, i.e., maximizing biomass production, steady-state flux distributions for over more than 7 × 10⁴ combinatorial minimal media conditions. Furthermore, all subsequent single reaction knockouts of active (non-zero flux carrying) reactions are performed to identify for each medium condition the set of essential reactions (see Methods for details). An illustrative example of this concept, involving E. coli central carbon metabolism [31], is provided in the supplementary materials.

The relative essentiality of a particular reaction is then defined as the number of lethal outcomes due to its removal divided by the number of environmental conditions under which it has been active. An alternative definition of relative essentiality would be to normalize the number of lethal outcomes to the total number of media sampled. In this case, however, essential reactions that are rarely active would give an unrealistically low essentiality value.

Figure 2a shows the sorted relative essentiality profile of all reactions in the E. coli model, which have been active at least once during the FBA simulations; blocked reactions [32] have thus been eliminated; see also Methods). In Figure 2a the three essentiality classes are clearly visible: The removal of most reactions has no or only small consequences for the production of biomass (non-essential). Some reactions are globally essential (essential) and a third set is only medium-dependent essential (conditional lethal). Excluding non-essential reactions, Figure 2b depicts the relative essentiality profile in a semilog plot. The stepwise appearance of the conditional lethal curve indicates three major groups of reactions that exhibit very similar relative essentiality scores, connected by some intermediate reactions.

Almaas et al. [21] reported a high, global essentiality for MC reactions. This finding suggests a higher essentiality for more active reactions. However, the activity and relative essentiality profiles exhibit no such correlation (see Figure 2c,d). Moreover, in contrast to [21], MC reactions are not exclusively essential. This deviation from previously reported results seems to be due to model differences (Almaas et al. utilized the older E. coli model iJR904 [33]; see Supplementary Figure S2) and not due to the changes in simulation procedures (combinatorial minimal media in our study vs. random media sampling in [21]; see Supplementary Figure S3).

Figure 2e displays the computed relative essentiality profile as well as the co-occurrence frequency of all cytosolic reactions in E. coli metabolism. Albeit visually appealing, no clear pattern emerges from this type of visualization, substantiating the need for a more rigorous topological analysis.

Figure 3 illustrates the rationale behind our investigation by summarizing the diverse topological representations of groups of metabolic reactions. These different representations (bipartite network representation → reaction-centric representation → established topological categories and three node subgraphs) are compared, either individually or in combination, with the essentiality classes introduced above, in order to understand typical topological “implementations” of conditional lethal reactions.

Using the relative essentiality profile (see Figure 2), we determine the amount of reactions belonging to the three essentiality classes for each of the three reaction categories. The results in Figure 4 show that the three categories incorporate different amounts of reactions belonging to each of the three essentiality classes. The SA reaction set seems to be composed of a mixture of conditional lethal and essential reactions whereas the UPUC reactions exhibit a high amount of non-essential reactions. As already mentioned, in contrast to previous findings ([21], see previous Section 2.1), the MC is not exclusively composed of essential reactions, but rather shows a similar essentiality class composition as the SA category.

Next, we explore the composition of essentiality classes in combinatorial sets of reaction categories (see Figure 5). For all 127 non-empty intersections and unions of UPUC, SA and MC, we perform an enrichment analysis for the three reaction essentiality classes. Besides a few exceptional cases, non-essential reactions seem to be strongly underrepresented in the majority of combinatorial sets (Figure 5a), coinciding with the capability of these reaction categories to predict essentiality. The exceptions to this observation have a tendency to include the exclusively UPUC set UPUC∩SA^c∩MC^c where X^c denotes the absolute complement. On the other hand, the majority of combinatorial sets is significantly enriched for essential reactions (Figure 5c). More importantly, more than half of the combinatorial sets exhibit a clear separation of essential from conditional lethal and non-essential classes. Comparing Figure 5a and Figure 5c reveals that the sequence of combinatorial sets in the sorted non-essential enrichment resembles the essential sequence in reverse order (e.g., the exclusive UPUC set being visually absent for high essential reaction enrichment). This observation provides evidence for a strong negative association between these two essentiality classes in the context of the UPUC, SA and MC categories.

Unfortunately, no clear separation of conditional lethal from non-essential and essential reactions is achieved by this combinatorial approach (Figure 5b). These results indicate that UPUC, SA and MC, albeit good essentiality predictors, do not provide the means for a topological characterization of medium-dependent essentiality.

In the following we will now quantify whether the established topological categories or the three-node subgraphs contain more information about medium-dependent essential reactions.

Figure 6 shows the statistical over- and under-representation of the three established topological categories (Figure 6a) and the three essentiality classes (Figure 6b) across all possible three-node subgraphs of the reaction-centric metabolic network (Figure 3). The striking result is that the three established topological categories display very similar subgraph associations, while the three essentiality classes show strong differences in their subgraph associations. Counter-intuitively, subgraphs thus perform better in distinguishing essentiality classes than in distinguishing the established topological categories discussed above.

Conditional lethal reactions have a fundamentally different “footprint” when mapped onto subgraphs. The most important building block is the bidirectional V-in (i.e., the V-in with one of the links being bi-directional). Non-essential reactions, on the other hand, are suppressed in chains, but elevated in V-in and V-out subgraphs. Surprisingly, all the topology-motivated reaction categories (UPUC, SA, MC) display very similar occurrence patterns across the motifs.

In contrast to the majority of works on network motifs, we do not take the motif composition of the total (“static”) network into account, but rather compute the subgraph associations medium by medium from each effective network spanned by all reactions with non-zero flux optimizing biomass production. Supplementary Figure S4 shows the same analysis as Figure 6, but for the subgraphs extracted from the total, static network. It is seen that the signal (e.g., the discrimination between essentiality classes) is much weaker there.

This is conceptually more plausible since the reactions comprising a subgraph in the static network may in fact be never active together and, consequently, such a subgraph may functionally never be available (see Supplementary Figure S5 for a distribution of Hamming distances between subgraph occurrence profiles from the static and effective networks).

The topological “footprint” of the different essentiality classes cannot be affected by the number of occurrences of three-node subgraphs in the metabolic network, as the null model of randomly drawn sets of reactions compensates for this. It could be, however, that the clustering of reactions in one of the reaction categories or a bias in the degree distribution may induce a systematic skew in the distribution of these reactions over the three-node subgraphs. We checked for these distortions of our result by computing the amount of clustering in each of the essentiality classes (see Supplementary Figure S6).

The clustering is defined by the conditional probability of a reaction r being in this class C (e.g., conditional lethal) given that a neighboring reaction r' is in this class: c(C) = P(r ∈ C|r' ∈ C) = P(r, r' ∈ C)/P(r' ∈ C), r' ∈ N(r). Essential reactions exhibit the highest amount of clustering, but non-essential and conditional lethal reactions show very similar distributions (see Supplementary Figure S6). On this basis we expect that the results shown in Figure 6 are not severely distorted by clustering.

The genome-scale metabolic reconstruction iAF1260 [37] of E. coli was used in all our experiments. Each reversible reaction was replaced by two irreversible reactions acting in opposite directions. For our topological analyses, first a bipartite graph representation was generated from the stoichiometry of the model and then projected onto a reaction centric network (see [38] for a review on network representations of metabolism).

For a given metabolic model, flux-balance analysis (FBA) [11] enables the computation of a steady-state flux distribution that maximizes a specific biological objective Z (e.g., maximal biomass production). Generally, the linear optimization problem defined in FBA can be stated as follows:

with an objective coefficient vector c, the stoichiometric matrix S, the flux vector v and the constraint vectors v_min and v_max. As we are considering reversible reactions as two independent unidirectional reactions, we set v_min to zero. Problems like Equation 1 can be efficiently solved using linear programming. In order to avoid thermodynamically infeasible loops, we utilized pFBA [39], effectively using the solution of Equation 1, to fix the objective to its maximum value and minimize the L1-norm of all other fluxes in a second optimization.

Combinatorial minimal media were constructed using the following procedure. (i) All experimentally verified nutrients in the iAF1260 model were classified as sources for elemental carbon, nitrogen, sulfur and phosphate (see also Supplementary Table S1). Some compounds fall hereby into multiple categories, e.g., glucose-6-phosphate is both a carbon and a phosphate source. (ii) Combinations of nutrients were then chosen such that only one of each elemental source was included in the medium, e.g., no additional phosphate source was provided in a medium containing glucose-6-phosphate. Steady-state fluxes that optimize biomass production have been calculated for all possible substrate combinations leading to a total of 72468 analyzed minimal-media conditions.

For each simulation, the essentiality of all active reactions was determined by fixing the respective fluxes to zero and recomputing the maximal biomass flux for the mutants. A reaction was classified as essential if the biomass flux dropped to zero.

We removed all globally blocked reactions from the model to give the topological methods described in this article (UPUC, SA) the opportunity to work on the same information content as their dynamical counterpart (MC). A high (not as high as the default flux boundaries v_max) maximal uptake and secretion rate was assigned to all available transporters in the system and then blocked reactions were confirmed by flux variability analysis [32]. These globally blocked reactions cannot carry a flux under any environmental conditions and consequently are not available to methods that use FBA.

Reactions are assigned to the metabolic core if they were active in all wild-type simulation, following the definition of Almaas et al. [21]. In contrast to [21], however, we use a finite number of combinatorial minimal media instead of randomly sampled conditions. Consequently, the size of the metabolic core in this study is larger than in the original work.

The synthetic accessibility of all reactions in the system was computed according to [20]. The needed outputs were defined to be the substrates of the biomass function and the ingredients of a glucose minimal medium were defined to be the inputs of the system. As a variation to [20] we decided to include no further additional compounds that ensure that all outputs are reached in the wild type. Instead we used a set of bootstrapping metabolites [40] that permit a proper functioning of the algorithm but are not the starting points of the breadth first search.

The UPUC reactions were determined in analogy to the algorithm published in [19]. We determined all metabolites with an in-degree and out-degree of one (UPUC metabolites) in the bipartite graph representation of the metabolism of iAF1260. Then we computed the set of reactions (UPUC reactions) that are associated with the set of UPUC metabolites for further analysis.

Three-node motifs as well as static networks were enumerated using the software mfinder [28]. There are two sorting schemes for subgraph types in the literature. We employed the one from Milo et al., where subgraphs are grouped according to criteria (cyclic versus acyclic; then connectivity or number of bidirectional links), rather than the one, where three-node subgraphs are sorted according to their “identifier” (the adjacency matrix of the subgraph, read as a binary number). In all subgraph-related figures, this subgraph identifier is also indicated in the corresponding subgraph pictogram.