  A theory for qualitative models of gene regulatory networks has been
developed over several decades, generally considering transcription factors to
regulate directly the expression of other transcription factors, without any
intermediate variables. Here we explore a class of models that explicitly
includes both transcription and translation, keeping track of both mRNA and
protein concentrations. We mainly deal with transcription regulation functions
that are steep sigmoids or step functions, as is often done in protein-only
models, though translation is governed by a linear term. We extend many aspects
of the protein-only theory to this new context, including properties of fixed
points, description of trajectories by mappings between switching points,
qualitative analysis via a state-transition diagram, and a result on periodic
orbits for negative feedback loops. We find that while singular behaviour in
switching domains is largely avoided, non-uniqueness of solutions can still
occur in the step-function limit.
