Journal of Mathematical Biology

, Volume 57, Issue 1, pp 29–52

On the number of steady states in a multiple futile cycle


  • Liming Wang
    • Department of MathematicsRutgers University
    • Department of MathematicsRutgers University

DOI: 10.1007/s00285-007-0145-z

Cite this article as:
Wang, L. & Sontag, E.D. J. Math. Biol. (2008) 57: 29. doi:10.1007/s00285-007-0145-z


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.


Futile cyclesBistabilitySignaling pathwaysBiomolecular networksSteady states

Mathematics Subject Classification (2000)


Copyright information

© Springer-Verlag 2007