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Genetic regulatory circuits universally cope with different sources of noisethat limit their ability to coordinate input and output signals. In many cases,optimal regulatory performance can be thought to correspond to configurationsof variables and parameters that maximize the mutual information between inputsand outputs. Such optima have been well characterized in several biologicallyrelevant cases over the past decade. Here we use methods of statistical fieldtheory to calculate the statistics of the maximal mutual information (the`capacity') achievable by tuning the input variable only in an ensemble ofregulatory motifs, such that a single controller regulates N targets. Assuming(i) sufficiently large N, (ii) quenched random kinetic parameters, and (iii)small noise affecting the input-output channels, we can accurately reproducenumerical simulations both for the mean capacity and for the wholedistribution. Our results provide insight into the inherent variability ineffectiveness occurring in regulatory systems with heterogeneous kineticparameters.

Statistics of optimal information flow in ensembles of regulatory motifs