Variance reduction in stochastic gene expression under integral feedback control

Given a stochastic biomolecular chemical reaction network, a reference input and a designated molecular species to be regulated, the antithetic integral feedback molecular motif introduced in [1] was shown to guarantee network ergodicity, robust disturbance rejection, and a zero steady-state tracking error (robust perfect adaptation). While these favorable properties are guaranteed for the mean dynamics, the controlled network may in fact have an increased output variance due to the inevitable molecular noise introduced by the controller. In this paper we show that, in the context of gene expression, the stationary protein variance can be decreased and the system performance improved by combining the antithetic integral feedback motif with a negative feedback control strategy. It is demonstrated the smaller the variance of the proteins, the larger the settling-time of its mean trajectory, which underscores a basic trade-off between these two objectives.