5.5.2 Graph analysis of the core E. coli metabolic model
Figure 5.3. The E. coli core metabolic model.
(a) Map of the E. coli core metabolic model (Orth, Fleming, and Palsson 2010) created with the online tool Escher (King et al. 2015b).
(b) Reactions of the model coloured by pathways.
Figure 5.3. The E. coli core metabolic model.
(c) Undirected reaction graph A from equation (5.4). Since the graph is constructed ignoring directionality, each node represents forwards and backwards reactions.
(d) Probabilistic Reaction Flux Graph Dp of the metabolic model constructed from equation (5.9).
Reversible reactions are represented by two overlapping nodes (one blue node for the forward reaction and another read for the backward when possible).
The layout of the reactions in the map and the graphs is the same.
The graphs described in the previous section are constructed and analysed here for the E. coli core metabolic model (Orth, Fleming, and Palsson 2010).
This model is relatively small, with 72 metabolites (20 extracellular and 52 intracellular) and 95 reactions (20 exchange reactions, 25 transport reactions, 49 metabolic reactions and one biomass reaction).
The canonical way to describe metabolic reactions is in terms of subsystems or pathways that consist of reactions that serve a specific function (Folch-Fortuny et al. 2015; Schilling, Letscher, and Palsson 2000; Schuster, Fell, and Dandekar 2000).
For example, the reactions that form glycolysis convert D-glucose into pyruvate and produce adenosine triphosphate (ATP) and nicotinamide adenine dinucleotide (NADH).
In this model, the reactions are grouped into 11 metabolic pathways that represent the main biochemical routes in the central carbon metabolism (Figure 5.3a).
The notion of directed flow is at the core of the construction of the graphs proposed in this chapter.
Therefore the interest resides in studying the directed pattern of flow between reactions in these networks and observing how they change in different contexts.
Such patterns can be understood as a generalisation of the notion of pathway, which are tailored to specific contexts.
This analysis will help us understand which reactions form groups of interaction, under what circumstances, and what are their defining features.
The Flux-Balance Graphs Mv proposed can help to transition from analyses that do not incorporate networks nor context, to a framework of flexible, context-dependent, graph-based analyses of the cell’s metabolism.
Markov Stability community detection framework (Delvenne, Yaliraki, and Barahona 2010; Delvenne et al. 2013) is used to extract groups of reactions in the different graphs.
This framework employs diffusion processes (flows) of different duration on graphs, which is ideally suited to study the proposed reaction graphs.
Markov Stability defines a community in precisely the way that interests us: a group of nodes in which flows are retained at specific scales.
The duration of the diffusion process acts as a resolution parameter on the size of the communities (Schaub et al. 2012).
A consequence of defining communities in terms of flowretention is that Markov Stability can naturally incorporate the directionality of the connections (Beguerisse-Díaz et al. 2014; Lambiotte, Delvenne, and Barahona 2014) (see last subsection in Section 2.3.1), which is crucial to analyse metabolism in a realistic way.
Therefore, communities are studied in reaction graphs: groups of reactions that are tightly linked by the flow of metabolites they produce and consume.
Each community is formed by reactions that retain metabolites as much as possible.
The community structure of the graphs obtained from the core E. coli metabolic model (A, Dp, and the metabolic graphs Mv for a selection of v) is analysed in the next subsections.
This analysis will enable to answer questions such as: How does context affect the community structure of the network?, is it useful to describe the networks in terms of the same pathways in very different scenarios?, or what is the multiscale organisation of metabolism and how does it relate to the standard pathways?
Probabilistic flux reaction graph of the E.coli core metabolic model As previously mentioned, the graph A, obtained from equation (5.4), contains connections between reactions that share metabolites in any capacity and does not distinguish between reversible and irreversible reactions (Fig. 5.3c).
This graph has 95 reactions and 1158 connections.
In contrast, the graph Dp (Fig. 5.3d) contains 154 reactions (i.e., all forward reactions and all legitimate reverse reactions) and 1,604 connections.
Due to its construction in equation (5.9), the connections created by pool metabolites are weighted correctly, so that more weight is placed on connections that describe the flow of less-abundant, yet more informative, metabolites.
Graphs A and Dp reveal the underlying complexity of the connectivity of the reactions which is typically absent from pathway representations.
Annex 7.5 highlights additional differences between A and Dp in pathway composition (Figure 7.27a-b), pagerank (Figure 7.27c-d) and community structure (Figure 7.27e-f).
The community structures of the graphs A and Dp are extracted.
The undirected graph A has a robust partition into seven communities (see Figure 7.27e in Annex 7.5 for a detailed description of each community).
These communities are largely determined by the connections created by pool metabolites.
For example, community C1A is mainly formed by reactions that consume/produce ATP and water.
The biomass reaction (the largest consumer of ATP) is not a member of this community.
This reaction uses the majority of the ATP produced in the cell for cellular growth; however, the construction of the graphs considers any connection that involves ATP equally.
Other communities in this graph are also determined by pool metabolites such as NAD+ and NADP+ (C3A).
The community structure in the network A highlights its limitations due to the absence of biological context and the overwhelming amount of uninformative connections.
Figure 7.27f in Annex 7.5 shows the PRG Dp.
This graph has a robust partition into five communities.
The communities in this graph emphasise flows of metabolites that are important for specific functions, and can be described in terms of pathways.
For example community C1Dp contains the pentose phosphate pathway and the first steps of glycolysis.
Communities contain reactions that tend to belong to the same pathways.
Although an improvement on A, the graph Dp is still context-independent and thus not suitable to study metabolism in specific scenarios.
Flux-Balance Graphs of the E. coli core metabolic model
Figure 5.4. Metabolic graphs and their corresponding flux maps of E. coli in different contexts. Each of these graphs is obtained from equation (5.15) and the solution of an FBA problem in four different contexts:
(a) aerobic growth with D-glucose as a source of carbon;
(b) aerobic growth with ethanol; the inactive reactions in each context (i.e., with zero flux) are shown in grey.
The thickness of the connections is proportional to the edge weights within each graph.
Figure 5.4. Metabolic graphs and their corresponding flux maps of E. coli in different contexts.
Each of these graphs is obtained from equation (5.15) and the solution of an FBA problem in four different contexts:
(c) anaerobic with D-glucose;
(d) aerobic growth with D-glucose but limited ammonium and phosphate; the inactive reactions in each context (i.e., with zero flux) are shown in grey.
The thickness of the connections is proportional to the edge weights within each graph.
Four different Flux-Balance Graphs from the E. coli core metabolic model are now examined.
These graphs illustrate how different circumstances force important changes in the pattern of metabolite flows.
The scenarios analysed are:
• Mglc: Aerobic growth with D-glucose as a carbon source.
• Metoh: Aerobic growth with ethanol.
• Manaero: Anaerobic growth on D-glucose.
• Mlim: Aerobic growth with D-glucose, and limited phosphate and ammonium.
In each case an FBA problem in which the constraints encode a different scenario is solved.
The optimal reaction fluxes represent the state of the cell’s metabolism in each context.
The graphs are constructed using each optimal flux v_ and equation (5.15).
In all cases, the connected components of the graphs have fewer nodes and connections than Dp because the solution of the FBA contains numerous reactions with zero net flux.
Figure 5.4 shows that the four networks are remarkably different from each other: the nodes in the giant component, the weights of the connections are different, as well as the pagerank of the reactions.
The flux map of each FBA solution is located next to each network to facilitate interpretation and analysis.
The community structure of each graph Mv is shown in Figure 5.5.
Additionally, a Sankey diagram (Rosvall and Bergstrom 2010) between the traditional pathways and the communities found in each partition is provided as well.
The main features of the communities of each graph are explained below.
Annex 7.6 describes in detailed all communities and contains the output figures for the Markov Stability process (Figure 7.28 in Annex 7.6).
The graphMglc has a robust partition into three communities (see Figure 5.5a and Annex 7.6.3) with a concrete interpretation: community C1glc contains reactions in charge of processing carbon from D-glucose to pyruvate.
Community C2glc harbours the bulk of the cell’s ATP production.
Community C3glc contains the reactions in charge producing NADH and NADPH (cell’s reductive power).
The graph from growth in ethanol (Metoh Figure 5.5b) also has a partition into three communities resembling those in Mglc with subtle, yet important differences.
For example, in C1etoh the change of source of carbon from D-glucose to ethanol has transformed glycolysis into gluconeogenesis by a reversal of the flux in the reactions in C1glc.
Moreover, this community contains reactions in charge of the production of growth precursors (e.g., ME2, PPCK, GLUDy, and ICDHyr).
The biomass reaction is now contained in C1etoh due the increased flow of precursors relative to ATP production.
The absence of oxygen has, predictably, a profound impact in the cell.
The graph Manaero (Figure 5.5c) reflects this new metabolic regime.
The connectivity and the communities in this graph are different from the other aerobic scenarios.
For example, the reactions CYTBD and NADH16 which are the first two steps of the electron transport chain are absent.
This has a profound effect (due to their high connectivity in normal circumstances) in the flow of metabolites and, consequently, on the community structure of the graph.
Figure 5.5. Community structure and pathway composition of metabolic graphs of E. coli in different contexts.
(a) Aerobic growth with D-glucose.
(b) aerobic growth with ethanol. The reactions are coloured according to their community (see Annex 7.6, subsections 7.6.3 and 7.6.4 for a detailed analysis of each partition).
Next to each graph a Sankey diagram (Rosvall and Bergstrom 2010) shows the pathway composition of each community found.
Figure 5.5. Community structure and pathway composition of metabolic graphs of E. coli in different contexts.
(c) anaerobic with D-glucose.
(d) aerobic growth with D-glucose and limited ammonium and phosphate.
The reactions are coloured according to their community (see Annex 7.6, subsections 7.6.5 and 7.6.6 for a
detailed analysis of each partition).
Next to each graph a Sankey diagram (Rosvall and Bergstrom 2010) shows the pathway composition of each community found.
The graph Mlim (Figure 5.5d) also depicts the metabolic network under severe conditions.
In this scenario, the community structure of the FBG reflects a phenomenon known as overflow metabolism (Basan et al. 2015; Vemuri et al. 2007), which occurs when the cell takes in more carbon than it can process.
In this case overflow metabolism is due to limited phosphate and NH4: the over-abundance of carbon is secreted from the cell, there is a strong decrease in the growth rate, and a partial shut-down of the TCA cycle.
Community C3lim is similar to C3glc and C3etoh, with the addition of the secretion routes of acetate and formate.