Genetic Toggle Switch in the Absence of Cooperative Binding: Exact Results

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 nonbimodal 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.