Fig. 3: Antithetic Integral Controller in Cyberloop.

a, b Cyberloop implementation of the Antithetic Integral Controller motif, and the associated controller reactions²⁰. The cellular readout, X_L, is used to update the reaction propensities of the controller network involving species Z₁ and Z₂. The applied light input (control input) intensity is proportional to the copy-number of controller species Z₁, which is obtained from stochastic simulation in the computer. c Demonstration of set-point tracking. Left: three Cyberloop runs with different reference values (dashed lines) were performed. Thin lines indicate cumulative time averages of nascent RNA counts in individual cells, while thick lines represent the population average. Right: distribution of the average nascent RNA count per cell over the course of the experiment (Experimental parameters: \(k=0.1\,{\min }^{-1},\,\eta =5\,{\min }^{-1},\,\theta =0.02\,{\min }^{-1}\) and \(\mu =(7,14,21)\times \theta \,{\min }^{-1}\); Number of cells: red - 94, blue - 76, green - 110). d Single-cell output traces of three randomly chosen cells in the Cyberloop experiment shown in (c) with ref. = 21. e Effect of annihilation reaction rate on the closed-loop system dynamics. Five independent experiments with differing annihilation rates η were performed. The plot shows the time-course evolution of average nascent RNA counts in those experiments. Lower values of η (lighter color) led to longer transients, but did not affect steady-state properties of the system (Experimental parameters: \(k=0.1\,{\min }^{-1},\,\theta =0.02\,{\min }^{-1}\) and \(\mu =14\times \theta \,{\min }^{-1}\); Number of cells: in increasing order of η values - 79, 80, 90, 104, 100). f Effect of controller species production rates. Three independent set-point tracking experiments with a common reference (dashed line), but differing values of θ and μ were performed. Left: time-course evolution of average nascent RNA counts in those experiments. Higher θ and μ values led to larger overshoot and longer settling times. Right: steady-state distribution of Z₁ and Z₂ copy-numbers over all the cells. Lower θ and μ values resulted in lower copy-numbers of Z₁ and Z₂ at steady-state. The Antithetic Integral Controller operates well, even in very low (<10) copy-number regime of controller species (Experimental parameters: \(k=0.01\,{\min }^{-1},\,\eta =5\,{\min }^{-1},\,\theta =2\,{\min }^{-1}\) and \(\mu =28\,{\min }^{-1}\); Number of cells: red - 91, blue - 87, green - 104). g Effect of physiological dilution on the operation of Antithetic Control motif - addition of degradation reactions (inset) for the controller species of the Antithetic Controller. The degradation rate δ was set as \(\frac{\,{{\mbox{log}}}(2)}{{{\mbox{Doubling Time}}}\,}\), that is, the dilution rate of the intended host cell type (S. cerevisiae). In these Cyberloop experiments, this dilution rate was found to have negligible impact on the controller performance. Left: shows the averaged population response over time of two experiments with (red) or without (blue) degradation reactions (Experimental parameters: \(k=0.1\,{\min }^{-1},\,\eta =5\,{\min }^{-1},\,\theta =0.02\,{\min }^{-1}\) and μ = 14 × θ; Number of cells: red - 81, blue - 76). Center: contains the histogram of average nascent RNA counts per cell at steady-state for the same experiments shown in (Left). Right: contains the distribution of steady-state errors for three parameter combinations. Inset percentage values are the percentage mean errors from the reference, and the error bars denote mean + / − standard error. (Experimental parameters: \(k=0.1\,{\min }^{-1},\,\eta =5\,{\min }^{-1},\,\theta =1\,{\min }^{-1}\) and \(\mu =14\,{\min }^{-1}\); Number of cells: from left to right in the violin plot - 76, 81, 67, 93, 97, 114). Source data are provided as a Source Data file.