  A stochastic model of autoregulated bursty gene expression by Kumar et al.
[Phys. Rev. Lett. 113, 268105 (2014)] has been exactly solved in steady-state
conditions under the implicit assumption that protein numbers are sufficiently
large such that fluctuations in protein numbers due to reversible
protein-promoter binding can be ignored. Here we derive an alternative model
that takes into account these fluctuations and hence can be used to study low
protein number effects. The exact steady-state protein number distributions is
derived as a sum of Gaussian hypergeometric functions. We use the theory to
study how promoter switching rates and the type of feedback influence the size
of protein noise and noise-induced bistability. Furthermore we show that our
model predictions for the protein number distribution are significantly
different from those of Kumar et al. when the protein mean is small, gene
switching is fast, and protein binding is faster than unbinding.
