Effect of magnitude and variability of energy of activation in multisite ultrasensitive biochemical processes.

Protein activity is often regulated by ligand binding or by post-translational modifications such as phosphorylation. Moreover, proteins that are regulated in this way often contain multiple ligand binding sites or modification sites, which can operate to create an ultrasensitive dose response. Here, we consider the contribution of the individual modification/binding sites to the activation process, and how their individual values affect the ultrasensitive behavior of the overall system. We use a generalized Monod-Wyman-Changeux (MWC) model that allows for variable conformational free energy contributions from distinct sites, and associate a so-called activation parameter to each site. Our analysis shows that the ultrasensitivity generally increases as the conformational free energy contribution from one or more sites is strengthen. Furthermore, ultrasensitivity depends on the mean of the activation parameters and not on their variability. In some cases, we find that the best way to maximize ultrasensitivity is to make the contribution from all sites as strong as possible. These results provide insights into the performance objectives of multiple modification/binding sites and thus help gain a greater understanding of signaling and its role in diseases. Author summary Multisite protein modification is ubiquitous in gene regulation and signal transduction, often in the form of multisite phosphorylation. Many models of multisite ultrasensitivity are available in the literature, but they usually assume that all sites contribute equally to the activation of the multisite target. In this work, we relax this assumption and carry out computational and mathematical analysis of a multisite system in which the conformational free energy contribution varies across sites. We find that the ultrasensitivity of the system tends to increase (with some exceptions) when the conformational free energy contributed by any given site is strengthened. Our analysis predicts that all active sites should have approximately the same conformational free energy contribution, a property observed in proteins with unstructured modification domains and bulk electrostatics. We were also able to predict from first principles an energy range of -2 to -4 kcal/mol per site that effectively maximizes ultrasensitive behavior. This prediction is consistent with experimental measurements in phosphorylation sites. Another strategy predicted by some of our models is to select a subset of the sites and activate them uniformly, while silencing other modification sites in the protein. This strategy is also observed experimentally in many multisite phosphorylation proteins.