Subnetwork hierarchies of biochemical pathways

TL;DR

This paper tackles the problem of uncovering the hierarchical organization of biochemical networks by decomposing them into subnetworks using a global betweenness-based criterion. It models networks as directed bipartite graphs and iteratively removes reactions with high effective betweenness $c_B(r)=C_B(r)/k_{in}(r)$ to build hierarchy trees that reveal core subnetworks and outer shells. The main contributions are (i) a scalable, biology-agnostic algorithm extending Girvan-Newman to biochemical reaction nodes, (ii) quantitative measures such as $h_{1/2}/h_{max}$ and $S_2^{max}/S_1^{max}$ that capture universal large-scale organization and variability in subnetworks, and (iii) illustrative subnetworks in multiple organisms including metabolic and non-metabolic pathways. The findings show a universal core-shell architecture with a few core clusters around highly connected substances and a general pattern of shell-dominated organization, offering insight into network robustness and evolutionary constraints.

Abstract

We present a method to decompose biochemical networks into subnetworks based on the global geometry of the network. This method enables us to analyse the full hierarchical organisation of biochemical networks and is applied to 43 organisms from the WIT database. Two types of biochemical networks are considered: metabolic networks and whole-cellular networks (also including e.g. information processes). Conceptual and quantitative ways of describing the hierarchical ordering are discussed. The general picture of the metabolic networks arising from our study is that of a few core-clusters centred around the most highly connected substances enclosed by other substances in outer shells, and a few other well-defined subnetworks.

Figures (7)

• Figure 1: A simple hierarchical clustering tree. A horizontal cut gives the tighter connected subgraphs below, and looser connections above.$S_{i}\left(h_{0}\right)$ is the size of the $i^{\prime}$ th largest connected subgraph at height $h_{0}$. Note that the root is at the top and $h$ grows downwards.
• Figure 2: The hierarchical clustering trees of T. pallidum. (a) shows the tree for the metabolic network, (b) shows the wholecellular network. The squares represent the subnetwork configuration at$h=0.1 h_{\text{max }}$ (the height indicated by the arrow). Sizes of the squares are proportional to the size of the clusters they represent.
• Figure 3: Schematic picture of the two different orderings in hierarchy trees. (a) Community-type ordering-same level core-clusters connected by outer parts of the network. (b) Shell-type-a sequence of core-clusters contained in each other. The squares symbolises the reaction nodes that is deleted at the height marked by the arrow. In (a) three subnetworks of similar sizes gets disconnected when the reaction odes are removed. In (b) many individual metabolite nodes (circles) gets isolated.
• Figure 4: The relative size of the network$N / N_{\text{max }}$; the ratio between the largest values of the second largest and largest connected subgraphs $S_{2}^{\text{max }} / S_{1}^{\text{max }}$; and the relative half-height $h_{1 / 2} / h_{\text{max }}$ for the 43 studied organisms.
• Figure 5: The size of the two largest subgraphs rescaled by their largest values,$\tilde{S_{1}}=S_{1} / S_{1}^{\text{max }}$ and $\tilde{S_{2}}=S_{2} / S_{2}^{\text{max }}$ and the ratio $S_{2} / S_{1}$; all as functions of the height of the hierarchy tree, or event time, $h$. The data is for both metabolic and whole cellular networks of the bacteria T. pallidum and M. pneumoniae, and the nematode C. elegans.