Ultrasensitivity and sharp threshold theorems for multisite systems

We study the ultrasensitivity of multisite binding processes where ligand molecules can bind to several binding sites, considering more particularly recent models involving complex chemical reactions in phosphorylation systems such as allosteric phosphorylation processes, or substrate-catalyst chain reactions and nucleosome mediated cooperativity. New statistics based formulas for the Hill coefficient and the effective Hill coefficient are provided and necessary conditions for a system to be ultrasensitive are exhibited. We then assume that the binding process is described by a density dependent birth and death process. We provide precise large deviation results for the steady state distribution of the process, and show that switch-like ultrasensitive responses are strongly related to the multi-stability of the associated dynamical system. Ultrasensitivity occurs if and only if the entropy of the dynamical system has more than one global minimum for some critical ligand concentration. In this case, the Hill coefficient is proportional to the number of binding sites, and the systems is highly ultrasensitive. We also discuss the interpretation of an extension $I_q$ of the effective Hill coefficient $I_{0.9}$ for which we recommend the computation of a broad range of values of $q$ instead of just the standard one corresponding to the 10% to 90% variation in the dose-response. It is shown that this single choice can sometimes mislead the conclusion by not detecting ultrasensitivity. This new approach allows a better understanding of multisite ultrasensitive systems and provides new tools for the design of such systems.