  Living things benefit from exquisite molecular sensitivity in many of their
key processes, including DNA replication, transcription and translation,
chemical sensing, and morphogenesis. At thermodynamic equilibrium, the basic
biophysical mechanism for sensitivity is cooperative binding, for which it can
be shown that the Hill coefficient, a sensitivity measure, cannot exceed the
number of binding sites. Generalizing this fact, we find that for any kinetic
scheme, at or away from thermodynamic equilibrium, a very simple structural
quantity, the size of the support of a perturbation, always limits the
effective Hill coefficient. This support bound sheds light on and unifies
diverse sensitivity mechanisms, ranging from kinetic proofreading to a
nonequilibrium Monod-Wyman-Changeux (MWC) model proposed for the E. coli
flagellar motor switch, and represents a simple, precise bridge between
experimental observations and the models we write down. In pursuit of
mechanisms that saturate the support bound, we find a nonequilibrium binding
mechanism, nested hysteresis, with sensitivity exponential in the number of
binding sites, with implications for our understanding of models of gene
regulation.
