Stability and Bifurcation Analysis in a Model of Hes1 Selfregulation with Time Delay

The dynamics of a delayed mathematical model for Hes1 oscillatory expression are investigated. The linear stability of positive equilibrium and existence of local Hopf bifurcation are studied. Moreover, the global existence of large periodic solutions has been established due to the global bifurcation theorem.

Authors:

• Kejun Zhuang
• Hailong Zhu

Keywords:

• Hes1
• Hopf bifurcation
• time delay
• transcriptional repression loop

References:

[1] B.C. Goodwin. Oscillatory behavior in enzymatic control processes. Adv.
Enzyme Regul. 3 (1965), 425-439.
[2] Samuel Bernard, Branka Cajavec, Laurent Pujo-Menjouet, Michael C
Mackey, Hanspeter Herzel. Modelling transcriptional feedback loops: the
role of Gro/TLE1 in Hes1 oscillations. Phil. Trans. R. Soc. A 364 (2006),
1155-1170.
[3] Hiroshi Momiji, Nicholas A.M. Monk. Dissecting the dynamics of the
Hes1 genetic oscillator. J. Theor. Biol. 254 (2008), 784-798.
[4] Nicholas A.M. Monk. Oscillatory expression of Hes1, p53, and NF-╬║B
deriven by transcriptional time delays. Curr. Biol. 13 (2003), 1409-1413.
[5] Anael Verdugo, Richard Rand. Hopf bifurcation in a DDE model of gene
expression. Commun. Nonlinear Sci. Numer. Simul. 13 (2008), 235-242.
[6] Min Xiao, Jinde Cao. Genetic oscillation deduced from Hopf bifurcation
in a genetic regulatory network with delays. Math. Biosci. 215 (2008),
55-63.
[7] Junjie Wei, Chunbo Yu. Hopf bifurcation analysis in a model of oscillatory
gene expression with delay. P. Roy. Soc. Edinb. A 139 (2009),
879-895.
[8] H. Hirata, S. Yoshiura, T. Ohtsuka etal. Oscillatory expression of the
bHLH factor Hes1 regulated by a negative feedback loop. Science 298
(2002), 840-843.
[9] Shigui Ruan, Junjie Wei. On the zeros of transcendental function with
applications to stability of delayed differential equations with two delays.
Dynam. Cont. Discrete Impuls. Syst. A 10 (2003), 863-874.
[10] B.D. Hassard, N.D. Kazarinoff, Y.H. Wan. Theory and Applications of
Hopf bifurcation. Cambridge: Cambridge University Press, 1981.
[11] Jianhong Wu. Symmetric functional differential equations and neural
networks with memory. Transactions of the AMS 350 (1998), 4799-4838.