Hybrid modeling of noise reduction by a negatively autoregulated system

We analyze the reduction of intrinsic noise caused by transition of a promoter between its active and inactive state in a negatively regulated genetic network, i.e., transcription of the gene is inhibited by its own gene product. To measure the noise attenuation, we compare its behavior to an inducible gene for which activation and deactivation of the gene take place at constant rates. As a model, we choose a hybrid approach in which some of the reaction channels are modeled as discrete events, and other reactions are modeled as continuous processes. Such a model is appropriate for investigations of noise caused by low reactant numbers. By focusing on intrinsic noise originating from the switching behavior of the regulatory system of a particular gene, we model only the transition between two different promoter states as a discrete event. We show that the stationary distributions of the unregulated and the autoregulated system are given as a solution of two coupled ordinary differential equations. Also, beside the distribution densities, the first two central moments are derived in closed analytical forms. We give conditions on the parameters when one or the other system shows lower fluctuations.