Bulletin of Mathematical Biology

, Volume 81, Issue 6, pp 1829–1852 | Cite as

Emergence of Oscillations in a Mixed-Mechanism Phosphorylation System

  • Carsten Conradi
  • Maya MinchevaEmail author
  • Anne Shiu


This work investigates the emergence of oscillations in one of the simplest cellular signaling networks exhibiting oscillations, namely the dual-site phosphorylation and dephosphorylation network (futile cycle), in which the mechanism for phosphorylation is processive, while the one for dephosphorylation is distributive (or vice versa). The fact that this network yields oscillations was shown recently by Suwanmajo and Krishnan. Our results, which significantly extend their analyses, are as follows. First, in the three-dimensional space of total amounts, the border between systems with a stable versus unstable steady state is a surface defined by the vanishing of a single Hurwitz determinant. Second, this surface consists generically of simple Hopf bifurcations. Next, simulations suggest that when the steady state is unstable, oscillations are the norm. Finally, the emergence of oscillations via a Hopf bifurcation is enabled by the catalytic and association constants of the distributive part of the mechanism; if these rate constants satisfy two inequalities, then the system generically admits a Hopf bifurcation. Our proofs are enabled by the Routh–Hurwitz criterion, a Hopf bifurcation criterion due to Yang, and a monomial parametrization of steady states.


Multisite phosphorylation Monomial parametrization Oscillation Hopf bifurcation Routh–Hurwitz criterion 



AS was partially supported by the NSF (DMS-1312473/1513364 and DMS-1752672) and the Simons Foundation (#521874). AS thanks Jonathan Tyler for helpful discussions. CC was partially supported by the Deutsche Forschungsgemeinschaft DFG (DFG-284057449).

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Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.HTW BerlinBerlinGermany
  2. 2.Northern Illinois UniversityDeKalbUSA
  3. 3.Texas A&M UniversityCollege StationUSA

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