A simple kind of model trying to account for switch-like behavior is where a set of class B proteins activate their own transcription. If class A proteins are considered to be expressed in a constitutive way, and not subjected to regulated degradation, they are present at a constant level. Only the time-evolution of each of the class B species is thus of interest. Calling , the concentrations of (class B species), the equations are
with , , where is the concentration of complex at which transcription is half-maximal, is the total quantity of class A proteins, and are respectively the maximal synthesis rate and the degradation rate of , and where each is normalized with respect to the dissociation constant for the complex (this normalization leads to each maximal synthesis rate being divided by the dissociation constant of the complex, see Appendix A). The equations assume that for all the quantity of complexes is negligible compared to the total quantity of (see Cinquin, 2006, for a relaxation of that assumption).
This set of equations is the same as derived by Cinquin (2005), without the restriction . We perform a steady state analysis of the system, assuming that it equilibrates over a time scale much shorter than that of cellular differentiation; this assumption is supported by the fact that transcription factors commonly have very short half-lives, which can be as low as a few minutes, while cellular differentiation often takes place over the course of hours or days.