  A new type of cooperativity termed temporal cooperativity [Biophys. Chem. 105
585-593 (2003), Annu. Rev. Phys. Chem. 58 113-142 (2007)], emerges in the
signal transduction module of phosphorylation-dephosphorylation cycle (PdPC).
It utilizes multiple kinetic cycles in time, in contrast to allosteric
cooperativity that utilizes multiple subunits in a protein. In the present
paper, we thoroughly investigate both the deterministic (microscopic) and
stochastic (mesoscopic) models, and focus on the identification of the source
of temporal cooperativity via comparing with allosteric cooperativity.
  A thermodynamic analysis confirms again the claim that the chemical
equilibrium state exists if and only if the phosphorylation potential
$\triangle G=0$, in which case the amplification of sensitivity is completely
abolished. Then we provide comprehensive theoretical and numerical analysis
with the first-order and zero-order assumptions in
phosphorylation-dephosphorylation cycle respectively. Furthermore, it is
interestingly found that the underlying mathematics of temporal cooperativity
and allosteric cooperativity are equivalent, and both of them can be expressed
by "dissociation constants", which also characterizes the essential differences
between the simple and ultrasensitive PdPC switches. Nevertheless, the degree
of allosteric cooperativity is restricted by the total number of sites in a
single enzyme molecule which can not be freely regulated, while temporal
cooperativity is only restricted by the total number of molecules of the target
protein which can be regulated in a wide range and gives rise to the
ultrasensitivity phenomenon.
