Logic backbone of a transcription network

A great part of the effort in the study of coarse-grained models of transcription networks concentrates on their dynamical features. In this Letter, we consider their equilibrium properties, showing that the backbone underlying the dynamic descriptions is an optimization problem. It involves N variables, the gene expression levels, and M constraints, the effects of transcriptional regulation. In the case of Boolean variables and constraints, we investigate the structure of the solutions and derive phase diagrams. Notably, the model exhibits a connectivity transition between a regime of simple gene control, where the input genes control O(1) other genes, and a regime of complex control, where some core input genes control O(N) others.