Toolbox model of evolution of metabolic pathways on networks of arbitrary topology

TL;DR

This paper investigates how the topology of universal metabolic networks influences the scaling laws of genome content, proving that tree-like structures with critical branching produce quadratic scaling, aligning with empirical observations.

Contribution

It analytically demonstrates the influence of network topology on metabolic pathway scaling, extending the toolbox model to arbitrary topologies and more realistic metabolic pathways.

Findings

Quadratic scaling occurs only in critical branching topologies.

Supercritical topologies lead to linear scaling with logarithmic corrections.

Realistic metabolic network simulations also show quadratic scaling.

Abstract

In prokaryotic genomes the number of transcriptional regulators is known to quadratically scale with the total number of protein-coding genes. Toolbox model was recently proposed to explain this scaling for metabolic enzymes and their regulators. According to its rules the metabolic network of an organism evolves by horizontal transfer of pathways from other species. These pathways are part of a larger "universal" network formed by the union of all species-specific networks. It remained to be understood, however, how the topological properties of this universal network influence the scaling law of functional content of genomes. In this study we answer this question by first analyzing the scaling properties of the toolbox model on arbitrary tree-like universal networks. We mathematically prove that the critical branching topology, in which the average number of upstream neighbors of a node is equal to one, is both necessary and sufficient for the quadratic scaling. Conversely, the toolbox model on trees with exponentially expanding, supercritical topology is characterized by the linear scaling with logarithmic corrections. We further generalize our model to include reactions with multiple substrates/products as well as branched or cyclic metabolic pathways. Unlike the original model the new version employs evolutionary optimized pathways with the smallest number of reactions necessary to achieve their metabolic tasks. Numerical simulations of this most realistic model on the universal network from the KEGG database again produced approximately quadratic scaling. Our results demonstrate why, in spite of their "small-world" topology, real-life metabolic networks are characterized by a broad distribution of pathway lengths and sizes of metabolic regulons in regulatory networks.