Fig. 2. Summary of the qualitative behavior of the Γ-OU and CIR models.

a Qualitative behavior can be visualized in a two-dimensional parameter space, with κ/(κ + β + γ) on one axis and the gain ratio θ/(θ + a) on the other. The four limits discussed in the text correspond to the four corners of this space. When a ≫ θ, we obtain Poisson-like behavior (green). When a ≪ θ, we obtain overdispersed distributions (orange). b Dynamics of limiting models. The Γ-OU and CIR models were simulated using four parameter sets close to the limiting regimes; transcription rates are visualized using trajectories and cell cartoons, where transcription rate is a logarithmic function of cell color. Ten thousand samples from the joint RNA count distribution are depicted in the rightmost column. Both models reduce to the constitutive model in the fast reversion and low gain limits, where the transcription rate K(t) is effectively constant in time and identical for all cells in the population. Both reduce to the mixture model in the slow reversion limit, so that K(t) is inhomogeneous across the population but constant in time for individual cells. In the high gain limit, the Γ-OU and CIR models yield different heavy-tailed distributions, with the CIR limiting model appearing to be uncharacterized. In both cases, K(t) exhibits sporadic large fluctuations within single cells.