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. 2008 Jul;57(1):29-52.
doi: 10.1007/s00285-007-0145-z. Epub 2007 Nov 16.

On the number of steady states in a multiple futile cycle

Liming Wang  1 , Eduardo D Sontag
Affiliations

On the number of steady states in a multiple futile cycle

Liming Wang et al. J Math Biol. 2008 Jul.

Abstract

The multisite phosphorylation-dephosphorylation cycle is a motif repeatedly used in cell signaling. This motif itself can generate a variety of dynamic behaviors like bistability and ultrasensitivity without direct positive feedbacks. In this paper, we study the number of positive steady states of a general multisite phosphorylation-dephosphorylation cycle, and how the number of positive steady states varies by changing the biological parameters. We show analytically that (1) for some parameter ranges, there are at least n + 1 (if n is even) or n (if n is odd) steady states; (2) there never are more than 2n - 1 steady states (in particular, this implies that for n = 2, including single levels of MAPK cascades, there are at most three steady states); (3) for parameters near the standard Michaelis-Menten quasi-steady state conditions, there are at most n + 1 steady states; and (4) for parameters far from the standard Michaelis-Menten quasi-steady state conditions, there is at most one steady state.

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