5.6 Conclusions
Metabolic networks have been described by multiple and fundamentally different graphs, unlike other cellular networks that have a natural correspondence between species-interactions and nodes-edges of a graph (Alon 2007).
This has been long acknowledged, and in essence there are three types of metabolic graphs (Palsson 2006): a graph with reactions as nodes, a graph with metabolites as nodes, and a graph with both reaction and metabolites as nodes.
Moreover, each one of these graphs can be directed or undirected, and the edge weights may be computed according to different principles.
Another source of confusion is that metabolic pathways are typically drawn as maps of metabolites with directed arrows to represent enzymatic reactions among them (King et al. 2015b).
Although these diagrams resemble a graph, they are not amenable to network-theoretic analyses unless we ascribe a specific meaning to the arrows between nodes.
These subtleties in the construction of metabolic graphs are important because they can affect the interpretation of results (Arita 2004; Ouzounis and Karp 2000).
Yet there is still a lack of consensus in the community and, as a result, comparisons across studies are difficult and cast doubt on the generality of the observed topological properties.
The objective in this chapter was to address some of the missing elements in the construction of graphs for metabolic networks.
The construction of two types of graphs were presented where nodes represent reactions and directed edges represent metabolites produced by one reaction and consumed by another.
In the Probabilistic Flux Reaction Graph (PRG), the edge-weights describe the probability that a any two reactions produce-consume a molecule of any metabolite.
This probabilistic formulation can tame the overwhelming number of connections generated by pool metabolites without the need to remove them from the network description, as commonly done in the literature (Kreimer et al. 2008; Ma and Zeng 2003b; Samal and Martin 2011; Silva et al. 2008).
To incorporate the effect of the environment, the Flux Balance Graph (FBG) is proposed, in which edge weights are the total flux of metabolites between reactions predicted by Flux Balance Analysis (FBA).
By computing FBA solutions for different exchange fluxes between the cell and its environment, one can systematically build metabolic graphs for different compositions of the growth media.
When applied to the core E. coli metabolic model the topology of the FBG effectively captures known metabolic adaptations such as the glycolytic-gluconeogenic switch, overflow metabolism, and the effects of anoxia.
The proposed FBG draws a novel connection between modern network theory (which studies graphs) and constraint-based methods widely employed in metabolic modelling (Orth, Fleming, and Palsson 2010; Rabinowitz and Vastag 2012.
Previous attempts to incorporate constraint-based models into metabolic graphs include, for example, the use of FBA solutions to search for node clusters (Samal et al. 2006), and the study of network robustness upon removal of FBA-constrained reaction nodes (Smart, Amaral, and Ottino 2008).
Using a graph built from a genome-scale metabolic model S. cerevisiae, it has been shown that the connectivity of reaction nodes does not correlate with FBA fluxes that maximise growth (Vitkup, Kharchenko, and Wagner 2006).
This brings into question the amount of physiologically-relevant information contained in graphs that are built from the whole metabolic blueprint of a cell but that are blind to the environmental and biological context.
In contrast, the FBG exploits the physiological predictions from FBA to construct metabolic graphs that are more informative of cell physiology and are directly grounded on specific environmental conditions.
The resulting graphs are smaller and less connected that those built from the complete metabolic blueprint, but they shed further light on the organisation of metabolic activity in realistic physiological conditions.
A number of promising applications of these results open up.
First, the most immediate application of the FBG is to study how environmental inputs shape the community structure and evolution of metabolic networks (Takemoto 2013; Zhou and Nakhleh 2012).
Such analysis has potential in biomedicine, for example, by finding metabolic conditions that maximise the efficacy of drugs treatments Chang et al. 2010; Csermely, Ágoston, and Pongor 2005).
Second, the FBG can be readily used to quantify metabolic robustness via graph statistics upon node (e. g. , reaction) removal (Smart, Amaral, and Ottino 2008).
Third, the proposed approach can be extended to include dynamic adaptations of metabolic activity, for example, by using dynamic extensions of FBA (Mahadevan, Edwards, and Doyle 2002; Rügen, Bockmayr, and Steuer 2015; Waldherr, Oyarzún, and Bockmayr 2015), or by incorporating static (Colijn et al. 2009) and time-varying (Oyarzún 2011) enzyme concentrations.
Fourth, the FBG could provide a novel route for robustness analysis of FBA solutions (Gudmundsson and Thiele 2010).
A common challenge is that FBA solutions are not unique, but instead they belong to a solution space containing an infinite number of flux distributions that are equally optimal (Orth, Thiele, and Palsson 2010).
The connectivity of the FBG depends on the particular FBA solution used to construct it, and thus one can exploit non-uniqueness to quantify the robustness of solutions in a space of graphs.