Analytical results for a generalized model of bursty gene expression with molecular memory

The activation of a gene is a complex biochemical process and could involve small steps, creating a memory between individual events. However, the effect of this molecular memory was often neglected in previous work. How the molecular memory affects gene expression remains not fully explored. We analyze a stochastic model of bursty gene expression, where the waiting time from inactivation to activation is assumed to follow a nonexponential (in fact, Erlang) distribution. We derive the analytical expression for the gene-product distribution, which explicitly traces the effect of molecule memory. Interestingly, we find that the effect of molecular memory is equivalent to the introduction of feedback. In addition, we analytically show that the stationary distribution is always super-Poissonian, independent of the detail of the waiting-time distribution, and there is the optimal step size that minimizes the Fano factor for any given mean burst size and is a decreasing function of the mean burst size. These analytical results indicate that molecular memory is an unneglectable factor affecting gene expression.