  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.
