Sparse essential interactions in model networks of gene regulation

Gene regulatory networks typically have low in-degrees, whereby any given gene is regulated by few of the genes in the network. What mechanisms might be responsible for these low in-degrees? Starting with an accepted framework of the binding of transcription factors to DNA, we consider a simple model of gene regulatory dynamics. In this model, we show that the constraint of having a given function leads to the emergence of minimum connectivities compatible with function. We exhibit mathematically this behavior within a limit of our model and show that it also arises in the full model. As a consequence, functionality in these gene networks is parsimonious, i.e., is concentrated on a sparse number of interactions as measured for instance by their essentiality. Our model thus provides a simple mechanism for the emergence of sparse regulatory networks, and leads to very heterogeneous effects of mutations.