  We present an analytical treatment of a genetic switch model consisting of
two mutually inhibiting genes operating without cooperative binding of the
corresponding transcription factors. Previous studies have numerically shown
that these systems can exhibit bimodal dynamics without possessing two stable
fixed points at the deterministic level. We analytically show that bimodality
is induced by the noise and find the critical repression strength that controls
a transition between the bimodal and non-bimodal regimes. We also identify
characteristic polynomial scaling laws of the mean switching time between
bimodal states. These results, independent of the model under study, reveal
essential differences between these systems and systems with cooperative
binding, where there is no critical threshold for bimodality and the mean
switching time scales exponentially with the system size.
