Bulletin of Mathematical Biology

, Volume 75, Issue 11, pp 2028–2058 | Cite as

Multistationarity in Sequential Distributed Multisite Phosphorylation Networks

  • Katharina Holstein
  • Dietrich Flockerzi
  • Carsten ConradiEmail author
Original Article


Multisite phosphorylation networks are encountered in many intracellular processes like signal transduction, cell-cycle control, or nuclear signal integration. In this contribution, networks describing the phosphorylation and dephosphorylation of a protein at n sites in a sequential distributive mechanism are considered. Multistationarity (i.e., the existence of at least two positive steady state solutions of the associated polynomial dynamical system) has been analyzed and established in several contributions. It is, for example, known that there exist values for the rate constants where multistationarity occurs. However, nothing else is known about these rate constants.

Here, we present a sign condition that is necessary and sufficient for multistationarity in n-site sequential, distributive phosphorylation. We express this sign condition in terms of linear systems, and show that solutions of these systems define rate constants where multistationarity is possible. We then present, for n≥2, a collection of feasible linear systems, and hence give a new and independent proof that multistationarity is possible for n≥2. Moreover, our results allow to explicitly obtain values for the rate constants where multistationarity is possible. Hence, we believe that, for the first time, a systematic exploration of the region in parameter space where multistationarity occurs has become possible. One consequence of our work is that, for any pair of steady states, the ratio of the steady state concentrations of kinase-substrate complexes equals that of phosphatase-substrate complexes.


Sequential distributed phosphorylation Mass-action kinetics Multistationarity Sign condition Rate constants 



KH and CC acknowledge financial support from the International Max Planck Research School in Magdeburg and the Research Center “Dynamic Systems” of the Ministry of Education of Saxony-Anhalt, respectively. Finally, we would like to thank the diligent reviewers for their valuable suggestions.


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Copyright information

© Society for Mathematical Biology 2013

Authors and Affiliations

  • Katharina Holstein
    • 1
  • Dietrich Flockerzi
    • 1
  • Carsten Conradi
    • 1
    Email author
  1. 1.Max-Planck-Institut Dynamik komplexer technischer SystemeMagdeburgGermany

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