Optimal bang–bang feedback for bursty gene expression

Stochasticity in gene expression poses a critical challenge to the precise control of cellular function. In this paper we examine how precisely can a stochastically expressed protein attain a given target expression level. We consider a protein which is produced in bursts and which is able to control its expression via a negative feedback loop; we specifically focus on feedback of a bang–bang type which turns off the production of the protein whenever its concentration exceeds a given threshold. Using a piecewise deterministic mathematical formalism, we derive explicit expressions for the probabilistic distribution of the protein concentration, and for the mean square deviation from the target level. Employing a combination of analytic and numerical optimization, we identify the optimal value of the bang–bang threshold, in terms of minimising the deviation, and examine the dependence of the optimal value on the target level and the sub-threshold burst frequency. The systematic analysis allows us to formulate a number of quantitative and qualitative conclusions about the controllability of burst like gene expression. Finally, we outline directions for future research into the topic.