Approximate statistical dynamics of a genetic feedback circuit

Auto-regulation, a process wherein a protein negatively regulates its own production, is a common motif in gene expression networks. Negative feedback in gene expression plays a critical role in buffering intracellular fluctuations in protein concentrations around optimal value. Due to the nonlinearities present in these feedbacks, moment dynamics are typically not closed, in the sense that the time derivative of the lower-order statistical moments of the protein copy number depends on high-order moments. Moment equations are closed by expressing higher-order moments as nonlinear functions of lower-order moments, a technique commonly referred to as moment closure. Here, we compare the performance of different moment closure techniques. Our results show that the commonly used closure method, which assumes a priori that the protein population counts are normally distributed, performs poorly. In contrast, conditional derivative-matching, a novel closure scheme proposed here provides a good approximation to the exact moments across different parameter regimes. In summary our study provides a new moment closure method for studying stochastic dynamics of genetic negative feedback circuits, and can be extended to probe noise in more complex gene networks.