Fig. 3: Antithetic proportional-integral (aPI) feedback controllers.
Three different classes of aPI controllers are designed by appending the standalone aI controller with three inhibitions. Three biologically relevant inhibition mechanisms are considered. Additive Inhibition: The inhibitor species produces the inhibited species separately at a decreasing rate. For instance, in the case of aPI Class 1 with additive inhibition, both controller species Z1 and output XL produce the input X1 separately, but Z1 acts as an activator while XL acts as a repressor. This separate inhibition can be modeled as the production of the input X1 as a positive actuation reaction \({{{{{{{{\mathcal{R}}}}}}}}}_{a}^{+}\) with an additive Hill-type propensity given by \({h}^{+}({z}_{1},{x}_{L})=k{z}_{1}+\frac{\alpha }{1+{({x}_{L}/\kappa )}^{n}}\), where n, α and κ denote the Hill coefficient, maximal production rate and repression coefficient, respectively. This aPI is the closest control architecture to33 and35, since the P and I components are additive and separable (see Fig. 1c, d). Multiplicative inhibition: the inhibitor competes with an activator over a production reaction. In the case of aPI Class 1 with multiplicative inhibition, the output XL inhibits the production of the input X1 by the controller species Z1. This can be modeled as the production of X1 with a multiplicative Hill-type propensity given by \({h}^{+}({z}_{1},{x}_{L})=k{z}_{1}\times \frac{1}{1+{({x}_{L}/\kappa )}^{n}}\). Observe that in this scenario, the proportional (P) and integral (I) control actions are inseparable, and the actuation reaction \({{{{{{{{\mathcal{R}}}}}}}}}_{a}^{+}\) encodes both PI actions simultaneously. Degradation inhibition: the inhibitor invokes a negative actuation reaction that degrades the inhibited species. For instance, in the case of aPI Class 1 with degradation inhibition, the controller species Z1 produces the input X1 (positive actuation reaction \({{{{{{{{\mathcal{R}}}}}}}}}_{a}^{+}\)), while the output XL degrades it (negative actuation reaction \({{{{{{{{\mathcal{R}}}}}}}}}_{a}^{-}\)). The dynamics can be captured by using a positive actuation with propensity h+(z1) = kz1 and a negative actuation with propensity \({h}^{-}({x}_{1},{x}_{L})=\delta {x}_{L}\frac{{x}_{1}}{{x}_{1}+{\kappa }_{1}}\). The total actuation propensity is defined as h(z1, x1, xL) ≔ h+(z1) − h−(x1, xL). The three classes with different inhibition mechanisms give rise to eight controllers that are compactly represented by the closed-loop stoichiometry matrix Scl and propensity function λcl by choosing the suitable h and g functions from the tables.
