The Logic Backbone of a Transcription Network

A great part of the effort in the study of coarse grained models of transcription networks is directed to the analysis of their dynamical features. In this letter, we consider the \emph{equilibrium} properties of such systems, showing that the logic backbone underlying all dynamic descriptions has the structure of a computational optimization problem. It involves variables, which correspond to gene expression levels, and constraints, which describe the effect of \emph{cis-}regulatory signal integration functions. In the simple paradigmatic case of Boolean variables and signal integration functions, we derive and discuss phase diagrams. Notably, the model exhibits a connectivity transition between a regime of simple, but uncertain, gene control, to a regime of complex combinatorial control.