  The antithetic integral feedback motif recently introduced in Briat, Gupta &
Khammash (Cell Systems, 2017) is known to ensure robust perfect adaptation for
the mean dynamics of a given molecular species involved in a complex stochastic
biomolecular reaction network. However, it was observed that it also leads to a
higher variance in the controlled network than that obtained when using a
constitutive (i.e. open-loop) control strategy. This was interpreted as the
cost of the adaptation property and may be viewed as a performance
deterioration for the overall controlled network. To decrease this variance and
improve the performance, we propose to combine the antithetic integral feedback
motif with a negative feedback strategy. Both theoretical and numerical results
are obtained. The theoretical ones are based on a tailored moment closure
method allowing one to obtain approximate expressions for the stationary
variance for the controlled network and predict that the variance can indeed be
decreased by increasing the strength of the negative feedback. Numerical
results verify the accuracy of this approximation and show that the controlled
species variance can indeed be decreased, sometimes below its constitutive
level. Three molecular networks are considered in order to verify the wide
applicability of two types of negative feedback strategies. The main conclusion
is that there is a trade-off between the speed of the settling-time of the mean
trajectories and the stationary variance of the controlled species; i.e.
smaller variance is associated with larger settling-time.
