  From a coarse-grained perspective the motif of a self-activating species,
activating a second species which acts as its own repressor, is widely found in
biological systems, in particular in genetic systems with inherent oscillatory
behavior. Here we consider a specific realization of this motif as a genetic
circuit, in which genes are described as directly producing proteins, leaving
out the intermediate step of mRNA production. We focus on the effect that
inherent time scales on the underlying fine-grained scale can have on the
bifurcation patterns on a coarser scale in time. Time scales are set by the
binding and unbinding rates of the transcription factors to the promoter
regions of the genes. Depending on the ratio of these rates to the decay times
of the proteins, the appropriate averaging procedure for obtaining a
coarse-grained description changes and leads to sets of deterministic
equations, which differ in their bifurcation structure. In particular the
desired intermediate range of regular limit cycles fades away when the binding
rates of genes are of the same order or less than the decay time of at least
one of the proteins. Our analysis illustrates that the common topology of the
widely found motif alone does not necessarily imply universal features in the
dynamics.
