Resource constrained flux balance analysis predicts selective pressure on the global structure of metabolic networks

Background A universal feature of metabolic networks is their hourglass or bow-tie structure on cellular level. This architecture reflects the conversion of multiple input nutrients into multiple biomass components via a small set of precursor metabolites. However, it is yet unclear to what extent this structural feature is the result of natural selection. Results We extend flux balance analysis to account for limited cellular resources. Using this model, optimal structure of metabolic networks can be calculated for different environmental conditions. We observe a significant structural reshaping of metabolic networks for a toy-network and E. coli core metabolism if we increase the share of invested resources for switching between different nutrient conditions. Here, hub nodes emerge and the optimal network structure becomes bow-tie-like as a consequence of limited cellular resource constraint. We confirm this theoretical finding by comparing the reconstructed metabolic networks of bacterial species with respect to their lifestyle. Conclusions We show that bow-tie structure can give a system-level fitness advantage to organisms that live in highly competitive and fluctuating environments. Here, limitation of cellular resources can lead to an efficiency-flexibility tradeoff where it pays off for the organism to shorten catabolic pathways if they are frequently activated and deactivated. As a consequence, generalists that shuttle between diverse environmental conditions should have a more predominant bow-tie structure than specialists that visit just a few isomorphic habitats during their life cycle. Electronic supplementary material The online version of this article (doi:10.1186/s12918-015-0232-5) contains supplementary material, which is available to authorized users.


Analytic calculations of transition points for the toy-network
To calculate transition points for the toy-network, we look for an optimal network that maximizes the flux of biomass reaction for a given switching parameter r. We first consider all possible structures and strategies of regulation that fulfill the constrains in Eq. 1-5 of the main text. Afterwards, by comparing the maximum biomass flux of those different structures and strategies with the same maximum resource investments of ϕ 0 , we are able to find the optimal structure and strategy of regulation for the optimal network that maximizes the growth rate.
For each environmental condition there are two independent pathways that satisfy the constraint of Eq. 1 of the main text. For example, Fig. S1 shows the two independent pathways, P 1 and P 2, for the first environmental condition containing nutrient S1. Using these two pathways, two possible structures for the network can be considered. Note that choosing P 1 for one environmental condition and P 2 for the other environmental condition has no benefit since this would increase the invested metabolic resources of each environment and the investment of the switching conditions at the same time.
In the first transition, considering ν max as the maximum flux of the pathways, the total investment of the enzymatic resources of P 1 is 5 × ν max since the pathway contains 5 reactions and the invested resources for the switching condition is r × 2 × 4 × ν max related to 2 × 4 unshared reactions of the pathways of two different environmental conditions. Similarly, the total investment of enzymatic resources of P 2 is 6 × ν max and the invested resources for the switching condition is r × 2 × 2 × ν max . So the resource constraint of Eq. 4 of the main text related to each pathway is where ν 1 max and ν 2 max are the optimal fluxes using pathway P 1 and P 2 respectively. So the maximum fluxes in both cases are proportional to ϕ 0 and are functions of r: Now if ν 1 max has a grater value, P 1 would be the optimal pathway and if ν 2 max is larger, P 2 would be optimal pathway. The transition, which depend on r, occurs when the right hand sides of Eqs. 2 become equal that happens at r = 0.25. For r < 0.25, P 1 has a larger flux and for r > 0.25, P 2 has a larger flux. In this way, by changing r we change the cost of switching investments and change the structure of the optimized network between two networks of Fig. 3a and Fig. 3b of the main text.
For the second transition, the structure of the pathways of the network are the same (like P 2) but with different regulations. In the first kind of regulation, which is up-and downregulating the needed enzymes, the total investment of the metabolic resources is 6 × ν max and the invested resources for the switching condition is r × 2 × 2 × ν max . But in the second kind of regulation, which is constantly upregulating all enzymes, the total investment of the metabolic resources is 8 × ν max and the cost of the investment is 0 since in this strategy all of the enzymes are expressed simultaneously and there is no cost for regulation in switching condition. So the resource constraint of Eq. 4 of the main text is like where ν 1 max and ν 2 max are the optimal fluxes using first and second regulation strategy respectively. So the maximum fluxes in both case are proportional to ϕ 0 and are a function of r: Now if ν 1 max has a greater value, first regulation strategy is the optimal one and if ν 2 max is larger the second one is optimal strategy. The transition, which depends on r, occurs when the right hand sides of both equations in Eq. 4 become equal that is at r = 0.5. For r < 0.5 the first strategy would have larger flux and for r > 0.5 the second one. In this way, by changing r we change the cost of investment in resource constraint and change the strategy of regulation for the optimized network. These results are in agreement with computational results of the main text.

Generalized random network model
To generalize the simple toy-model, we consider a hierarchical random network as the universe of reactions as shown in Fig. S2. The first layer of the network contains input metabolites and the last layer contains biomass contents. Metabolites in the intermediate levels are intermediate metabolites of the pathways that convert input metabolites to biomass. The structural properties of the random networks is shown in Fig. S3 for different switching parameters, r. Here, the considered universe of reactions has 3 intermediate levels with 15 metabolites for each level, 7 input metabolites and 7 metabolites as the biomass content. Each level is connected to the next level by means of 20 random reactions. To increase the complexity of the network, 20 additional reactions are also considered randomly between any two metabolites of the network.
As it is shown in Fig. S3, any changes in the switching parameter change the structural parameters of the network. The behavior of structural parameters, like growth rate; ASP, number of regulons and number of selected input metabolites, are similar to the results of our simple toy-model and E.coli core metabolic network. However, here we see almost continuous transitions due to an averaging over 400 realizations. Sharp changes can be observed in the results of a single realization such as Fig. S4. P1 biomass S1 S2 P2 biomass S1 S2 Figure S1: Alternative pathways. Bold arrows represent two possible pathways, which preserved flux balance constraint, in presence of substrate S1 as the environmental condition.  Figure S3: Structural parameters for the generalized random model. The considered network has 59 metabolites and 108 reactions as the universe of metabolic networks (3 intermediate levels with 15 metabolites, 7 input metabolites and 7 metabolites in biomass). We average the quantities over 400 realizations. a) average growth rate vs. r. b) average shortest path from an input metabolite to each of the biomass contents for the optimized network vs. r. c) number of selected input metabolites after optimization process vs. r. d) number of nodes of the optimized network vs. r e) number of reactions of the optimized network vs. r. f) number of regulons vs. r.  Figure S5: Structural parameters for the E. coli core metabolic network. Here, α and β values are the same. 5 different environmental conditions that represent different carbon sources with 2 to 6 carbon atoms is used. a) average growth rate vs. r. b) average shortest path from an input metabolite to each of the biomass contents for the optimized network vs. r. c) number of selected input metabolites after optimization process vs. r. d) number of nodes of the optimized network vs. r e) number of reactions of the optimized network vs. r. f) number of regulons vs. r.  Figure S6: Structural parameters for the E. coli core metabolic network. A single realization of α and β with 10 percent fluctuations is used. 5 different environmental conditions that represent different carbon sources with 2 to 6 carbon atoms is used. a) average growth rate vs. r. b) average shortest path from an input metabolite to each of the biomass contents for the optimized network vs. r. c) number of selected input metabolites after optimization process vs. r. d) number of nodes of the optimized network vs. r e) number of reactions of the optimized network vs. r. f) number of regulons vs. r.