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The multisite phosphorylation-dephosphorylation cycle is a motif repeatedlyused in cell signaling. This motif itself can generate a variety of dynamicbehaviors like bistability and ultrasensitivity without direct positivefeedbacks. In this paper, we study the number of positive steady states of ageneral multisite phosphorylation-dephosphorylation cycle, and how the numberof positive steady states varies by changing the biological parameters. We showanalytically that (1) for some parameter ranges, there are at least n+1 (if nis even) or n (if n is odd) steady states; (2) there never are more than 2n-1steady states (in particular, this implies that for n=2, including singlelevels of MAPK cascades, there are at most three steady states); (3) forparameters near the standard Michaelis-Menten quasi-steady state conditions,there are at most n+1 steady states; and (4) for parameters far from thestandard Michaelis-Menten quasi-steady state conditions, there is at most onesteady state.

A remark on the number of steady states in a multiple futile cycle