  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.
