Mixed Poisson distributions in exact solutions of stochastic autoregulation models

In this paper we study the interplay between stochastic gene expression and system design using simple stochastic models of autoactivation and autoinhibition. Using the Poisson representation, a technique whose particular usefulness in the context of nonlinear gene regulation models we elucidate, we find exact results for these feedback models in the steady state. Further, we exploit this representation to analyze the parameter spaces of each model, determine which dimensionless combinations of rates are the shape determinants for each distribution, and thus demarcate where in the parameter space qualitatively different behaviors arise. These behaviors include power-law-tailed distributions, bimodal distributions, and sub-Poisson distributions. We also show how these distribution shapes change when the strength of the feedback is tuned. Using our results, we reexamine how well the autoinhibition and autoactivation models serve their conventionally assumed roles as paradigms for noise suppression and noise exploitation, respectively.

Figure 1

The top left shows that the response of the protein distribution to increasing activation strength in the $\varphi <1,\beta <1$ quadrant in the autoactivation model resembles the classic binary response associated with autoactivation systems. The bottom left shows that the effect of increasing activation strength $a$ on the protein distribution in the fourth quadrant ($\varphi <1,\beta >1$) is graded. The top right shows the effect of increasing autorepression strength $r$ on the Fano factor of the protein distribution in the autorepression model. For the values chosen, both $\beta$ and the Fano factor go through maximum values at (different) intermediate values of $r$, before the distribution becomes sub-Poissonian after the threshold value $r=r_0$; this happens exactly when $\beta =0$. The bottom right shows the effect of increasing autorepression strength $r$ on the protein distribution in the autorepression model. Six different points from the above figure are chosen from the range where $\beta$ remains positive. The other rates are the same as the ones used in the top right figure. Here $r$ increases from lighter to darker values.