5.3 Background
Metabolism consists of the set of reactions responsible for converting nutrients into energy and macromolecules that fuel cellular processes (Berg, Tymoczko, and Stryer 2002).
These reactions are organised in metabolic pathways that are composed of metabolites such as carbon sources and intermediate precursors, and the enzymatic reactions that convert them into one another.
Metabolism is naturally amenable to analysis with network science, which has also been successfully used to describe cellular systems such as protein-protein interactions (Thomas et al. 2003), transcriptional regulation (Alon 2007), and protein structure (Amor et al. 2014).
In the case of metabolism, however, the insights gained from network science are somewhat more dispersed because of the lack of a widely-accepted method to construct metabolic networks.
Most studies have focussed on the topological properties of metabolic pathways, such as their degree distribution (Arita 2004; Jeong et al. 2000; Wagner and Fell 2001) or their community structure (Guimera and Nunes Amaral 2005; Ravasz et al. 2002; Takemoto 2013; Zhou and Nakhleh 2012), but the conclusions are highly dependent on the way the networks are constructed (Winterbach et al. 2013).
The central object of study in network science is a graph that describes the entities in the network (nodes or vertices) and the connections among them (edges or arcs), which can indicate relationship or interaction (Newman 2010).
Note that although in the networks literature the terms graph and network are used interchangeably; this is not the case in contexts such as the study of metabolism.
To maintain consistency with the metabolic analysis literature and to avoid confusion the term network is reserved for the set of metabolic reactions and substrates, and the term graph for the mathematical object formed by a set of nodes and their connections.
In the analysis of metabolism there is no unique way to construct graphs in which the nodes are chemical species or reactions and the connections.
Catalytic enzymes convert multiple reactants into products with the help of other species in the network.
Moreover, some enzymes catalyse several reactions and some reactions are catalysed by multiple enzymes.
One can construct fundamentally different graphs for the same metabolic network.
For example, one could create a graph in which the nodes are metabolites and the edges are the reactions that transform one metabolite into another (Jeong et al. 2000; Ma and Zeng 2003a; Ouzounis and Karp 2000; Wagner and Fell 2001).
Alternatively, we could create a graph where nodes are reactions and edges are the metabolites shared among them (Ma et al. 2004; Samal et al. 2006; Vitkup, Kharchenko, and Wagner 2006), or a bipartite graph with two types of nodes, one type for reactions and another one for metabolites, in which the connections denote metabolites that participate in reactions (Smart, Amaral, and Ottino 2008).
These graph constructions have been described extensively (Palsson 2006).
Metabolic reactions operate in a preferred direction depending on the physiological state of the cell and environmental conditions (Berg, Tymoczko, and Stryer 2002).
For example, in glucose-rich environments some glycolytic enzymes operate preferably in their forward mode, while during gluconeogenesis their catalytic activity reverses direction.
Other reactions can only operate in one direction (i.e., they are irreversible).
Although this notion of directionality is a key feature of metabolism, many of the existing graph constructions do not incorporate the direction of the edges (Palsson 2006; Wagner and Fell 2001).
Undirected graphs neglect important information about the connectivity of the metabolic network, such as the distinction between reactions that compete for the same metabolite, the ones who produce the same metabolites, and those that have a supplier-consumer relationship.
Furthermore, current graph constructions are based on metabolic networks that simultaneously include all interactions from all known pathways in an organism.
Although the resulting networks are the blueprint for the whole metabolic activity of a cell, in reality cells switch specific pathways on/off to sustain their energetic budget in different environments.
Such blueprint graphs therefore contain many node connections that are relevant only in specific growth conditions, which distorts the network topology and the biological insights drawn from it.
In this chapter two new graph constructions are proposed to study metabolic networks and their adaptation to the environment in which the connections among the reactions and their intensity have a clear interpretation.
The first of these graphs is the Probabilistic Flux Reaction Graph (PRG or Dp), a weighted, directed graph in which the nodes are reactions, and connections occur between reactions that have a supplier-consumer relationship.
The weight of the connections corresponds to the probability that a metabolite chosen uniformly at random is produced by the source reaction and consumed by the target in the absence of more information about the environmental context.
The second graph is the Flux-Balance Graph (FBG or Mv), a directed, weighted graph where the nodes are again the reactions, but the weight of the connections is the total flow of metabolites per unit time from the source reaction to the target in a specific environmental context.
To obtain the edge-weights flux distributions from Flux Balance Analysis (Orth, Thiele, and Palsson 2010; Rabinowitz and Vastag 2012) are used.
In both graphs, an edge between nodes indicates that metabolites are produced by the source reaction and consumed by the target reaction.
This definition accounts for metabolic directionality and thus captures the natural flow of chemical mass from carbon sources to metabolic products.
One advantage of the Probabilistic Flux Reaction Graph is that the weight of the connections created by metabolites that appear in many reactions (e.g., pool such as ATP, NADH and other co-factors) is very small, but not zero; this is a result of the probabilistic formulation of the graph connectivity.
Pool metabolites typically lead to graphs dominated by the connections created by them, obscuring other more informative features of the network.
A common workaround in the literature is to prune pool metabolites from the network, but this is done with ad-hoc heuristics (Kreimer et al. 2008; Ma and Zeng 2003b; Samal and Martin 2011; Silva et al.
2008), and arbitrarily destroys information that may affect the interpretation of the results.
The approach presented here, motivated by results in Croes et al. 2006, circumvents this issue by appraising the information contained in the interactions generated by all metabolites in a consistent manner without the need to remove any species from the analysis.
The Flux Balance Graph incorporates the environmental context into the graph connectivity by using Flux Balance Analysis (FBA), a widespread method to predict metabolic fluxes in genome-scale metabolic networks (Orth, Thiele, and Palsson 2010; Rabinowitz and Vastag 2012).
Given upper and lower bounds for the flux of each reaction, FBA predicts flux distributions that maximise an objective function, such as the growth rate of the organism, although others are possible (Schuetz, Kuepfer, and Sauer 2007).
The flux of the reaction is subject constraints that describe the state of the environment.
Optimal environment-dependent fluxes from FBA are incorporated into the FBG by defining the weight of the connections to be the total flux of metabolites produced by the source reaction and consumed by the target reaction.
As a result, the edge weights in the FBG can be directly interpreted as fluxes in units of mass per time ( mmol gDW•h ).
The FBG thus allows for a systematic study of genome-scale metabolic adaptations in response to changing carbon sources and other environmental perturbations.
We showcase the utility of the PRG and FBG in the core model for the metabolism of Escherichia coli metabolism (Orth, Fleming, and Palsson 2010).
Our results show that the structure of the FBG can vary dramatically depending on the environmental context under consideration.
The FBG for different environmental conditions was constructed and Markov stability method (Delvenne, Yaliraki, and Barahona 2010) was used to detect node communities in different timescales.
The results suggest that the structure of metabolic networks varies drastically across different environmental conditions, casting doubt on the utility of a single metabolic blueprint to describe cellular adaptations.
The proposed graphs can be readily applied to study the topology of genome-scale metabolic networks (Ravasz et al. 2002), find network clusters and their link to environmental changes (Ma et al. 2004; Samal et al.
2006; Takemoto 2013), and to assess the robustness of metabolic connectivity (Smart, Amaral, and Ottino 2008).
The objective is that these new constructions for metabolic graphs stimulate new applications of network science to cellular metabolism.