Controlling event timing precision for gene expression with external disturbances

Gene expression is inherently a stochastic process, leading to randomness in the timing of events. A problem of interest is to understand how cells achieve temporal precision. Previous work has shown that if a gene product (protein) is stable then a no feedback strategy provides minimum noise in timing. Here we investigate the form of optimal feedback strategy when the protein production rate is affected by an external dynamic disturbance (noise). We specifically consider a telegraph process (two state) for the external disturbance and show that the best strategy is still no feedback. Using this strategy, we provide an exact formula for the minimum achievable noise. Finally, we examine the effect of model parameters on the minimum noise.