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We consider the dynamics of a population of organisms containing two mutuallyinhibitory gene regulatory networks, that can result in a bistable switch-likebehaviour. We completely characterize their local and global dynamics in theabsence of any noise, and then go on to consider the effects of either noisecoming from bursting (transcription or translation), or Gaussian noise inmolecular degradation rates when there is a dominant slow variable in thesystem. We show analytically how the steady state distribution in thepopulation can range from a single unimodal distribution through a bimodaldistribution and give the explicit analytic form for the invariant stationarydensity which is globally asymptotically stable. Rather remarkably, thebehaviour of the stationary density with respect to the parameterscharacterizing the molecular behaviour of the bistable switch is qualitativelyidentical in the presence of noise coming from bursting as well as in thepresence of Gaussian noise in the degradation rate. This implies that onecannot distinguish between either the dominant source or nature of noise basedon the stationary molecular distribution in a population of cells. We finallyshow that the switch model with bursting but two dominant slow genes has anasymptotically stable stationary density.

The Limiting Dynamics of a Bistable Molecular Switch With and Without Noise