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The antithetic integral feedback motif recently introduced in Briat, Gupta &Khammash (Cell Systems, 2017) is known to ensure robust perfect adaptation forthe mean dynamics of a given molecular species involved in a complex stochasticbiomolecular reaction network. However, it was observed that it also leads to ahigher variance in the controlled network than that obtained when using aconstitutive (i.e. open-loop) control strategy. This was interpreted as thecost of the adaptation property and may be viewed as a performancedeterioration for the overall controlled network. To decrease this variance andimprove the performance, we propose to combine the antithetic integral feedbackmotif with a negative feedback strategy. Both theoretical and numerical resultsare obtained. The theoretical ones are based on a tailored moment closuremethod allowing one to obtain approximate expressions for the stationaryvariance for the controlled network and predict that the variance can indeed bedecreased by increasing the strength of the negative feedback. Numericalresults verify the accuracy of this approximation and show that the controlledspecies variance can indeed be decreased, sometimes below its constitutivelevel. Three molecular networks are considered in order to verify the wideapplicability of two types of negative feedback strategies. The main conclusionis that there is a trade-off between the speed of the settling-time of the meantrajectories and the stationary variance of the controlled species; i.e.smaller variance is associated with larger settling-time.

Variance reduction for antithetic integral control of stochastic reaction networks