OSCILLATORY DYNAMICS INDUCED BY MULTI-DELAYS IN GENE EXPRESSION

2011 ◽  
Vol 14 (03) ◽  
pp. 451-469 ◽  
Author(s):  
QI WANG ◽  
BO LIU ◽  
SHIWEI YAN
Keyword(s):  
Gene Expression ◽  
Mathematical Method ◽  
Hopf Bifurcation ◽  
Positive Feedback ◽  
Time Delays ◽  
Chaotic Dynamics ◽  
Biochemical Reaction ◽  
Reaction Systems ◽  
Oscillatory Dynamics ◽  
Model Equations

We discuss the existence of Hopf bifurcation, the dynamical stability breaking of the molecular concentrations of two generic biochemical reaction systems and their sensitivity to the changes of the parameters incorporated in the model equations. It is found that the oscillatory dynamics can be expected for the systems due to the inclusion of time delays. The chaotic dynamics and the periodic windows in the chaotic domains can exist in the case of the system with two time delays. The proposed mathematical method may have the significance in the problems where the negative and/or positive feedback dynamics, as well as the time delays have the characteristic physical and biological background.

Oscillatory Dynamics of P53 Network With Time Delays

2018 ◽  
Vol 21 (6) ◽  
pp. 411-419 ◽  
Author(s):  
Conghua Wang ◽  
Fang Yan ◽  
Yuan Zhang ◽  
Haihong Liu ◽  
Linghai Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Cell Fate ◽  
Time Delays ◽  
Sustained Oscillation ◽  
Multiple Time ◽  
Cell Fate Decisions ◽  
Oscillatory Dynamics ◽  
Main Components ◽  
Parameter Values ◽  
P53 Network

Aims and Objective: A large number of experimental evidences report that the oscillatory dynamics of p53 would regulate the cell fate decisions. Moreover, multiple time delays are ubiquitous in gene expression which have been demonstrated to lead to important consequences on dynamics of genetic networks. Although delay-driven sustained oscillation in p53-based networks is commonplace, the precise roles of such delays during the processes are not completely known. Method: Herein, an integrated model with five basic components and two time delays for the network is developed. Using such time delays as the bifurcation parameter, the existence of Hopf bifurcation is given by analyzing the relevant characteristic equations. Moreover, the effects of such time delays are studied and the expression levels of the main components of the system are compared when taking different parameters and time delays. Result and Conclusion: The above theoretical results indicated that the transcriptional and translational delays can induce oscillation by undergoing a super-critical Hopf bifurcation. More interestingly, the length of these delays can control the amplitude and period of the oscillation. Furthermore, a certain range of model parameter values is essential for oscillation. Finally, we illustrated the main results in detail through numerical simulations.

scholarly journals Stability and Hopf Bifurcation Analysis of a Gene Expression Model with Diffusion and Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yahong Peng ◽  
Tonghua Zhang
Keyword(s):  
Gene Expression ◽  
Hopf Bifurcation ◽  
Local Stability ◽  
Time Delays ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Expression Model ◽  
The Stability ◽  
And Diffusion ◽  
Gene Expression Model

We consider a model for gene expression with one or two time delays and diffusion. The local stability and delay-induced Hopf bifurcation are investigated. We also derive the formulas determining the direction and the stability of Hopf bifurcations by calculating the normal form on the center manifold.

scholarly journals Turing–Hopf bifurcation and spatiotemporal patterns in a Gierer–Meinhardt system with gene expression delay

2021 ◽  
Vol 26 (3) ◽  
pp. 461-481
Author(s):  
Shuangrui Zhao ◽  
Hongbin Wang ◽  
Weihua Jiang
Keyword(s):  
Gene Expression ◽  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Time Delays ◽  
Spatiotemporal Patterns ◽  
Third Order ◽  
Spatially Inhomogeneous ◽  
Math Biol ◽  
Formation Systems ◽  
Expression Time

In this paper, we consider the dynamics of delayed Gierer–Meinhardt system, which is used as a classic example to explain the mechanism of pattern formation. The conditions for the occurrence of Turing, Hopf and Turing–Hopf bifurcation are established by analyzing the characteristic equation. For Turing–Hopf bifurcation, we derive the truncated third-order normal form based on the work of Jiang et al. [11], which is topologically equivalent to the original equation, and theoretically reveal system exhibits abundant spatial, temporal and spatiotemporal patterns, such as semistable spatially inhomogeneous periodic solutions, as well as tristable patterns of a pair of spatially inhomogeneous steady states and a spatially homogeneous periodic solution coexisting. Especially, we theoretically explain the phenomenon that time delay inhibits the formation of heterogeneous steady patterns, found by S. Lee, E. Gaffney and N. Monk [The influence of gene expression time delays on Gierer–Meinhardt pattern formation systems, Bull. Math. Biol., 72(8):2139–2160, 2010.]

Feedback-Induced Variations of Distribution in a Representative Gene Model

2015 ◽  
Vol 25 (07) ◽  
pp. 1540008
Author(s):  
Peijiang Liu ◽  
Zhanjiang Yuan ◽  
Lifang Huang ◽  
Tianshou Zhou
Keyword(s):  
Gene Expression ◽  
Power Law ◽  
Positive Feedback ◽  
Bimodal Distribution ◽  
Stochastic Bifurcation ◽  
Protein Distribution ◽  
Identical Cell ◽  
Induced Changes ◽  
Gene Switching ◽  
Cell To Cell Variability

Gene expression is inherently noisy, implying that the number of mRNAs or proteins is not invariant rather than follows a distribution. This distribution can not only provide the exact information on the dynamics of gene expression but also describe cell-to-cell variability in a genetically identical cell population. Here, we systematically investigate a two-state model of gene expression, a model paradigm used to study expression dynamics, focusing on the effect of feedback on the type of mRNA or protein distribution. If there is no feedback, then the distribution may be bimodal, power-law tailed, or Poisson-like, depending on gene switching rates. However, we find that feedback can tune or change the type of the distribution in each case and tends to unimodalize the distribution as its strength increases. Specifically, positive feedback can change not only a power-law tailed distribution into a bimodal or Poisson-like distribution but also a bimodal distribution into a Poisson-like distribution (implying that stochastic bifurcation can take place). In addition, it can make a Poisson-like distribution become more peaked but does not change the type of this distribution. In contrast to positive feedback, negative feedback has less influence on the shape of the distributions except for the bimodal case. In all cases, the noise-feedback curve used extensively in previous studies cannot well reflect the feedback-induced changes in the shape of distributions. Feedback-induced variations in distribution would be important for cell survival in fluctuating environments.

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