Figure 1. Illustration of a hierarchical metabolic network and its coarse-graining.

(A) White rectangle represents the whole metabolic network $𝒩$. Example subnetworks μ and ν are represented by the dark and light gray rectangles. Only metabolites and reactions that belong to these subnetworks are shown; other metabolites and reactions in $𝒩$ are not shown. Metabolites 1 and 5 may be adjacent to other metabolites in $𝒩$; this fact is represented by short black lines that do not terminate in metabolites. Subnetworks μ and ν are both modules because there exists a simple path connecting their I/O metabolites that lies within μ and ν and contains all their internal metabolites (dashed blue line). (B) Network $𝒩$ can be coarse-grained by replacing module μ at steady state with an effective reaction between its I/O metabolites 1 and 2, with the rate constant is $y_{\mu }$. (C) Network $𝒩$ can be coarse-grained by replacing module ν at steady state with an effective reaction between its I/O metabolites 1 and 5, with the rate constant is $y_{\nu }$.