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We study the ultrasensitivity of multisite binding processes where ligandmolecules can bind to several binding sites, considering more particularlyrecent models involving complex chemical reactions in phosphorylation systemssuch as allosteric phosphorylation processes, or substrate-catalyst chainreactions and nucleosome mediated cooperativity. New statistics based formulasfor the Hill coefficient and the effective Hill coefficient are provided andnecessary conditions for a system to be ultrasensitive are exhibited. We thenassume that the binding process is described by a density dependent birth anddeath process. We provide precise large deviation results for the steady statedistribution of the process, and show that switch-like ultrasensitive responsesare strongly related to the multi-stability of the associated dynamical system.Ultrasensitivity occurs if and only if the entropy of the dynamical system hasmore than one global minimum for some critical ligand concentration. In thiscase, the Hill coefficient is proportional to the number of binding sites, andthe systems is highly ultrasensitive. We also discuss the interpretation of anextension $I_q$ of the effective Hill coefficient $I_{0.9}$ for which werecommend the computation of a broad range of values of $q$ instead of just thestandard one corresponding to the 10% to 90% variation in the dose-response. Itis shown that this single choice can sometimes mislead the conclusion by notdetecting ultrasensitivity. This new approach allows a better understanding ofmultisite ultrasensitive systems and provides new tools for the design of suchsystems.

Ultrasensitivity and sharp threshold theorems for multisite systems