Four Alternative Views Of A Metabolic Network

In the world of the computer study of the metabolism, we like to represent the entanglement of the biochemical reactions of a metabolic network as… a network. (Well, you probably did not fall off your chair with this one…)

But this term of ‘metabolic network’ can in fact cover different kind of abstractions.

One of these formalisations is the directed unweighted bipartite graph \(G(V, E)\), where \(V\) is the union of the set of reaction nodes \(R\) and the set of metabolite nodes \(M\), and \(E\) is the set of directed edges \((m_1, r_1) \in E\) when \(m_1\) is a reactant of the reaction \(r_1\) and \((r_1, m_2) \in E\) when \(m_2\) is a product of reaction \(r_1\).

When we focus more on the reactions, we can build a reaction graph \(G^{( r)}(V^{( r)}, E^{( r)})\), where we remove the metabolite from the equation. The set of vertices \(V^{( r)}\) becomes the set of reactions, and \((r_1, r_2) \in E^{( r)}\) when the reaction \(r_1\) produces a metabolite \(m\) that is also a reactant of the reaction \(r_2\).

We can also focus on the metabolites, and build a metabolite graph \(G^{(m)}(V^{(m)},E^{(m)})\), where the set of node \(V^{(m)}\) contains the set of metabolites and there is a edge \((m_1, m_2) \in E^{(m)}\) when there exists a reaction \(r\) that has \(m_1\) as reactant and \(m_2\) as product.

One last abstraction we can use is the hypergraph. Recall that a hypergraph is a graph where edges can link more than two nodes. In our case, we could consider an hypergraph where the hyperedges would represent reactions. These hyperedges would be directed¹, and link on one side of the edge the reactant metabolites of the reactions and to the other side, the product metabolite.

A simple example with the “L-valine biosynthesis pathway” (BioCyc Pathway VALSYN-PWY)

https://biocyc.org/pathway?orgid=ECOLI&id=VALSYN-PWY

Unweighted directed bipartite graph

We represent the metabolite nodes with an ellipse and the reaction nodes with a rectangular shape.

Reaction graph

Metabolite graph

Hypergraph of metabolite nodes with reaction hyperedge

What about the BioCyc pathway representation?

(ref)

The BioCyc pathway representation is the representation of a hypergraph, where nodes are metabolites and hyperedges are reactions, some of which are bidirectionals.

In our framework, we would rather avoid bidirectional edges, as it might bring confusion in rectant / products, and would make two alternative edges for the two directions of the reaction, as we did above.

References

• On support for hypergraphs in Graphviz. Graphviz project issue tracker. Available: https://gitlab.com/graphviz/graphviz/-/work_items/1911 [Accessed Oct. 14, 2026].
• The L-valine biosynthesis pathway in Escherichia coli K-12. EcoCyc. Available: https://biocyc.org/pathway?orgid=ECOLI&id=VALSYN-PWY. [Accessed Oct. 14, 2026].