  Gene regulatory networks typically have low in-degrees, whereby any given
gene is regulated by few of the genes in the network. They also tend to have
broad distributions for the out-degree. What mechanisms might be responsible
for these degree distributions? Starting with an accepted framework of the
binding of transcription factors to DNA, we consider a simple model of gene
regulatory dynamics. There, we show that selection for a target expression
pattern leads to the emergence of minimum connectivities compatible with the
selective constraint. As a consequence, these gene networks have low in-degree,
and "functionality" is parsimonious, i.e. is concentrated on a sparse number of
interactions as measured for instance by their essentiality. Furthermore, we
find that mutations of the transcription factors drive the networks to have
broad out-degrees. Finally, these classes of models are evolvable, i.e.
significantly different genotypes can emerge gradually under mutation-selection
balance.
