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In prokaryotic genomes the number of transcriptional regulators is known toquadratically scale with the total number of protein-coding genes. Toolboxmodel was recently proposed to explain this scaling for metabolic enzymes andtheir regulators. According to its rules the metabolic network of an organismevolves by horizontal transfer of pathways from other species. These pathwaysare part of a larger "universal" network formed by the union of allspecies-specific networks. It remained to be understood, however, how thetopological properties of this universal network influence the scaling law offunctional content of genomes. In this study we answer this question by firstanalyzing the scaling properties of the toolbox model on arbitrary tree-likeuniversal networks. We mathematically prove that the critical branchingtopology, in which the average number of upstream neighbors of a node is equalto one, is both necessary and sufficient for the quadratic scaling. Conversely,the toolbox model on trees with exponentially expanding, supercritical topologyis characterized by the linear scaling with logarithmic corrections. We furthergeneralize our model to include reactions with multiple substrates/products aswell as branched or cyclic metabolic pathways. Unlike the original model thenew version employs evolutionary optimized pathways with the smallest number ofreactions necessary to achieve their metabolic tasks. Numerical simulations ofthis most realistic model on the universal network from the KEGG database againproduced approximately quadratic scaling. Our results demonstrate why, in spiteof their "small-world" topology, real-life metabolic networks are characterizedby a broad distribution of pathway lengths and sizes of metabolic regulons inregulatory networks.

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