Fig. 7. Simulation of the integral controller’s response to disturbance.

a Resource availability in the cell is disturbed by inducing the expression of an additional disturbing synthetic gene dist. The resulting change in burden is counteracted by the controller. b, c Evolution of simulated mRNA concentrations over time upon the induction of a disturbing gene with the parameters given in Supplementary Table 9. d–f Simulated trajectories for the sensor protein’s concentration p_sens (the controlled variable), the cell’s growth rate λ, and the resource competition denominator D upon the induction of a disturbing gene. For comparison, the response of an open-loop system (i.e., in absence of the actuator, the annihilator, and the amplifier genes) is plotted on the same axes. All variables are plotted relative to their steady-state values. g Dependence of the steady-state value of p_sens, relative to that for an undisturbed cell, on the magnitude of disturbance—that is, the concentration of the disturbing gene’s DNA in the cell. Open-loop responses are plotted on the same axes for comparison. Outside of the calculated operation range, the controller no longer mitigates resource couplings, so the closed-loop value of p_sens starts falling with increasing c_sens at the same rate as the open-loop value. h The steady-state value of p_sens maintained by the controller for various values of the amplifier gain χ and the disturbing gene’s DNA concentration c_dist. The adaptation error increases as the disturbance rises but can be mitigated by increasing the amplifier gain. Unless specified otherwise, simulation parameters for all panels are given in Supplementary Table 9, and the ODEs used to simulate the controller are Supplementary Eqs. (121)–(129) in Supplementary Note S4.4.1. Source data are provided as a Source Data file.