It has been reported that the number of transcription factors encoded in a prokaryotic genome scales approximately quadratically with the total number of its genes. As a result in small genomes (<500 genes) the fraction of regulators among all proteins is less than 0.5 %, while in the largest genome (~10,000 genes) this fraction reaches as high as 10 %.
We proposed a simple mode explaining this empirical scaling law. In our model the repertoire of proteins encoded in the genome of a prokaryotic organism is viewed as its collection of tools. Adapting to a new environmental condition (e.g. learning to use a new nutrient source) involves acquiring or evolving new genes/enzymes as well as reusing some of the tools that are already encoded in the genome. As the toolbox of an organism grows larger, it can reuse its existing enzymes more often, and thus needs to acquire fewer new enzymes to master each new functional task. From this analogy follows that, in general, the number of functional tasks an organism sould accomplish increases faster than linearly with its number of protein-coding genes. Under the assumption that the number of regulators is proportional to the number of functional tasks our model explains the quadratic scaling between the number of transcription factors and the number of all genes in prokaryotic genomes. Our model also includes transcriptional regulation of metabolic enzymes, which is assumed to be tightly coordinated with their associated pathways. The distribution of length of co-regulated metabolic pathways in our model is in agreement with the empirically observed broad distribution of regulon sizes in E.coli.