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A new type of cooperativity termed temporal cooperativity [Biophys. Chem. 105585-593 (2003), Annu. Rev. Phys. Chem. 58 113-142 (2007)], emerges in thesignal transduction module of phosphorylation-dephosphorylation cycle (PdPC).It utilizes multiple kinetic cycles in time, in contrast to allostericcooperativity that utilizes multiple subunits in a protein. In the presentpaper, we thoroughly investigate both the deterministic (microscopic) andstochastic (mesoscopic) models, and focus on the identification of the sourceof temporal cooperativity via comparing with allosteric cooperativity.  A thermodynamic analysis confirms again the claim that the chemicalequilibrium state exists if and only if the phosphorylation potential$\triangle G=0$, in which case the amplification of sensitivity is completelyabolished. Then we provide comprehensive theoretical and numerical analysiswith the first-order and zero-order assumptions inphosphorylation-dephosphorylation cycle respectively. Furthermore, it isinterestingly found that the underlying mathematics of temporal cooperativityand allosteric cooperativity are equivalent, and both of them can be expressedby "dissociation constants", which also characterizes the essential differencesbetween the simple and ultrasensitive PdPC switches. Nevertheless, the degreeof allosteric cooperativity is restricted by the total number of sites in asingle enzyme molecule which can not be freely regulated, while temporalcooperativity is only restricted by the total number of molecules of the targetprotein which can be regulated in a wide range and gives rise to theultrasensitivity phenomenon.

Sensitivity Amplification in the Phosphorylation-Dephosphorylation  Cycle: Nonequilibrium steady states, chemical master equation and temporal  cooperativity