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Metabolism has frequently been analyzed from a network perspective. A major question is how network properties correlate with biological features like growth rates, flux patterns and enzyme essentiality. Using methods from graph theory as well as established topological categories of metabolic systems, we analyze the essentiality of metabolic reactions depending on the growth medium and identify the topological footprint of these reactions. We find that the typical topological context of a medium-dependent essential reaction is systematically different from that of a globally essential reaction. In particular, we observe systematic differences in the distribution of medium-dependent essential reactions across three-node subgraphs (the

How topology shapes dynamics is a long-standing question in the field of network theory [

Furthermore, robustness of metabolism against gene or reaction deletions has been explored using flux-balance analysis (FBA) [

Along similar lines of research, metabolic reactions have been classified in several ways based on topological information [

UPUC metabolites have been introduced by Samal

Synthetic accessibility (SA), defined by Wunderlich and Mirny [

Network context of topological reaction categories. (

In the example in

A third category of reactions comes from a sampling of random environmental conditions and predicting steady-state fluxes that optimize biomass production using FBA. The set of reactions predicted to be active in all conditions has been termed metabolic core (MC) [

Although experimental data from systematic knockout studies is available for

An alternative approach of exploring the relationship between network architecture and function is based on the enumeration of few-node subgraphs. It has been shown that the subgraph composition of functionally related networks tends to be similar [

Here we explore the question if a topological understanding of reaction essentiality can be established by integrating the

We start by introducing the

In

See

Summarizing statistics of reaction categories and essentialities.

Reactions | UPUC | SA | MC | No category | |
---|---|---|---|---|---|

Overall/Cytosol | 1284/707 | 296/193 | 238/230 | 231/197 | 820/375 |

Non-essential (Overall/Cytosol) | 789/402 | 137/57 | 27/25 | 25/16 | 608/312 |

Conditional lethal (Overall/Cytosol) | 326/162 | 90/74 | 73/73 | 85/79 | 198/55 |

Essential (Overall/Cytosol) | 169/143 | 69/62 | 138/132 | 121/102 | 14/8 |

In order to subdivide the metabolic reactions into essentiality classes, namely ^{4} 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

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.

Almaas

Outcome of combinatorial minimal media simulations. (

(^{1}-pyrroline-5-carboxylate dehydrogenase (

Using the relative essentiality profile (see

Reaction categories and essentiality classes. The proportions of the three different essentiality classes determined for UPUC, SA and MC component (for absolute numbers see

Next, we explore the composition of essentiality classes in combinatorial sets of reaction categories (see ^{c}^{c}^{c}

Enrichment of combinatorial reaction category sets for essentiality classes. Combinatorial sets sorted on the basis of (

Unfortunately, no clear separation of

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.

Enrichment on three-node subgraphs. The statistical over- and under-representation of (

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

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

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

The clustering is defined by the conditional probability of a reaction

The genome-scale metabolic reconstruction

For a given metabolic model, flux-balance analysis (FBA) [

with an objective coefficient vector _{min}_{max}_{min}

Combinatorial minimal media were constructed using the following procedure. (i) All experimentally verified nutrients in the

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 _{max}

Reactions are assigned to the metabolic core if they were active in all wild-type simulation, following the definition of Almaas

The synthetic accessibility of all reactions in the system was computed according to [

The UPUC reactions were determined in analogy to the algorithm published in [

Three-node motifs as well as static networks were enumerated using the software

Using a variety of topological descriptors, we have been able to characterize the network properties of reactions displaying medium-dependent essentiality in a large-scale combinatorial minimal media screen employing flux-balance analysis.

The two classification schemes for metabolic reactions are (1) occurrence in directed three-node subgraphs and (2) functional categories of metabolic reactions motivated by network topology and FBA. We observe that the distribution of the three classes of metabolic reactions derived from relative essentiality is non-random across the three-node subgraphs. At the same time the distribution of essentiality classes across the three functional categories (UPUC, SA and MC) is highly diverse for the conditional lethal reactions, but far more homogeneous for the other two classes. Putting all these observations together leads to an accurate topological characterization of medium-dependent essential reactions.

These two topological characterizations are quite independent. In particular, when distributing the reaction categories across the three-node subgraphs, we see almost no differences between the three reaction categories in their subgraph preference profile.

Among the diverse combinatorial sets defined from the established topological categories, several very different ones contain a large number of

With the wide range of FBA models available, a natural next step of our investigation is to analyze these categories of essential reactions and their topological implementation also in other organisms. Also, other categories of metabolic reactions derived from large FBA screens can be topologically assessed, for example reactions that are active only in a very small number of environmental conditions (rarely active reactions). We expect that the topological implementation of such rarely active reactions can shed light on the robustness of metabolic systems against environmental variations.

Lastly, further validating the results with gene expression data can be an interesting line of investigation, starting from our previous work on effective networks derived from gene expression patterns [

The authors thank ZebaWunderlich for providing additional information on the synthetic accessibility approach. Furthermore, the authors would like to thank Areejit Samal for discussions about topological implications of reaction essentiality and Moritz Beber for providing his python wrapper for mfinder. MH acknowledges support from Volkswagen Foundation (grant I/82717) and Deutsche Forschungsgemeinschaft (grant HU 937/6). NS acknowledges support by Jacobs University Bremen in form of a PhD scholarship. CM acknowledges support from the Helmholtz Alliance on Systems Biology (project “CoReNe”).

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