  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.
