TY - JOUR
T1 - Global Hopf bifurcation in the ZIP regulatory system
AU - Claus, Juliane
AU - Ptashnyk, Mariya
AU - Bohmann, Ansgar
AU - Chavarría-Krauser, Andrés
N1 - Funding Information:
This work was supported by: German Research Foundation Grant number CH 958/1-1, Excellence Initiative II Heidelberg University “Mobilitätsmaßnahmen im Rahmen internationaler Forschungskooperationen 2013–2014” project number D.80100/13.009, and EPSRC Grant number EP/K036521/1.
Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg.
PY - 2015/10/14
Y1 - 2015/10/14
N2 - Regulation of zinc uptake in roots of Arabidopsis thaliana has recently been modeled by a system of ordinary differential equations based on the uptake of zinc, expression of a transporter protein and the interaction between an activator and inhibitor. For certain parameter choices the steady state of this model becomes unstable upon variation in the external zinc concentration. Numerical results show periodic orbits emerging between two critical values of the external zinc concentration. Here we show the existence of a global Hopf bifurcation with a continuous family of stable periodic orbits between two Hopf bifurcation points. The stability of the orbits in a neighborhood of the bifurcation points is analyzed by deriving the normal form, while the stability of the orbits in the global continuation is shown by calculation of the Floquet multipliers. From a biological point of view, stable periodic orbits lead to potentially toxic zinc peaks in plant cells. Buffering is believed to be an efficient way to deal with strong transient variations in zinc supply. We extend the model by a buffer reaction and analyze the stability of the steady state in dependence of the properties of this reaction. We find that a large enough equilibrium constant of the buffering reaction stabilizes the steady state and prevents the development of oscillations. Hence, our results suggest that buffering has a key role in the dynamics of zinc homeostasis in plant cells.
AB - Regulation of zinc uptake in roots of Arabidopsis thaliana has recently been modeled by a system of ordinary differential equations based on the uptake of zinc, expression of a transporter protein and the interaction between an activator and inhibitor. For certain parameter choices the steady state of this model becomes unstable upon variation in the external zinc concentration. Numerical results show periodic orbits emerging between two critical values of the external zinc concentration. Here we show the existence of a global Hopf bifurcation with a continuous family of stable periodic orbits between two Hopf bifurcation points. The stability of the orbits in a neighborhood of the bifurcation points is analyzed by deriving the normal form, while the stability of the orbits in the global continuation is shown by calculation of the Floquet multipliers. From a biological point of view, stable periodic orbits lead to potentially toxic zinc peaks in plant cells. Buffering is believed to be an efficient way to deal with strong transient variations in zinc supply. We extend the model by a buffer reaction and analyze the stability of the steady state in dependence of the properties of this reaction. We find that a large enough equilibrium constant of the buffering reaction stabilizes the steady state and prevents the development of oscillations. Hence, our results suggest that buffering has a key role in the dynamics of zinc homeostasis in plant cells.
KW - Hopf bifurcation
KW - Periodic orbits
KW - Stability
KW - Transport processes in plants
KW - Zinc uptake
UR - http://www.scopus.com/inward/record.url?scp=84941366851&partnerID=8YFLogxK
U2 - 10.1007/s00285-014-0836-1
DO - 10.1007/s00285-014-0836-1
M3 - Article
C2 - 25312412
AN - SCOPUS:84941366851
SN - 0303-6812
VL - 71
SP - 795
EP - 816
JO - Journal of Mathematical Biology
JF - Journal of Mathematical Biology
IS - 4
ER -