Fig 5. The Hill coefficient H_robust that gives the best positional resolution for the widest range of steady state transcription periods for the equilibrium model with N = 6 binding sites.

Since the regulatory parameters are unlikely to change between nc 11-13, we look for the model quantified by its Hill coefficient that results in the lowest values of the positional resolution in the largest range of T. Inset: H_robust is calculated as the minimum of the upper bound of the confidence interval of the optimal Hill coefficient CI(H*), where the optimal Hill coefficient H* is defined in Fig 4A, and the confidence intervals are the solid blue lines in Fig 4B. H_robust is shown as a function of the OS number (N) and number of readout genes (M). Also plotted for reference are H = N (dashed line) and H = N/2 (solid line). H_robust for M = 1 is much less than the maximum steepness allowed by the model H = N and with a lot of spatial averaging (M = 10) H_robust approaches N.