The effect of site-to-site variability in ultrasensitive dose responses

In this paper we study the ultrasensitive behavior of multisite phosphorylation or ligand binding systems, under site-to-site variations in the modification rates. Using computational methods and mathematical analysis, we prove that the Hill coefficient reaches its maximum value when all sites are identical to each other. This is shown for a non-cooperative multisite system with arbitrary activation function as well as for the well known MWC model. We also show that the Hill coefficient of the dose response is locally robust to variations in individual modification rates. The results suggest that maximal ultrasensitivity is reached when sites are similar to each other but not necessarily identical, a conformation found in unstructured modification domains present in many experimental systems.