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.

Hopf Bifurcation of a Type of Neuron Model with Multiple Time Delays

2019 ◽  
Vol 29 (12) ◽  
pp. 1950163 ◽  
Author(s):  
Suqi Ma
Keyword(s):  
Hopf Bifurcation ◽  
Parameter Space ◽  
Time Delays ◽  
Neuron Model ◽  
Multiple Time ◽  
Multiple Time Delays ◽  
The Stability ◽  
Geometrical Scheme ◽  
Delay Differential ◽  
Two Parameter

By applying a geometrical scheme developed to tackle the eigenvalue problem of delay differential equations with multiple time delays, Hopf bifurcation of Hopfield neuron model is analyzed in two-parameter space. By the introduction of two new angles, the calculation of imaginary roots is carried out analytically and effectively. By increasing the parameter to cross over the Hopf bifurcation lines, the stability switching direction is confirmed. The method is a useful tool to show the partition of stable and unstable regions in two-parameter space and detect double Hopf bifurcation further. The typified dynamical behaviors based on nearby double Hopf points are analyzed by applying the normal form technique and center manifold method.

scholarly journals The Notch pathway regulates both the proliferation and differentiation of follicular cells in the panoistic ovary of Blattella germanica

Open Biology ◽  
2016 ◽  
Vol 6 (1) ◽  
pp. 150197 ◽  
Author(s):  
Paula Irles ◽  
Nashwa Elshaer ◽  
Maria-Dolors Piulachs
Keyword(s):  
Cell Fate ◽  
Ovarian Follicle ◽  
Notch Pathway ◽  
Blattella Germanica ◽  
Active Role ◽  
Follicular Cells ◽  
Cell Fate Decisions ◽  
Proliferation And Differentiation ◽  
Ancestral Function ◽  
Main Components

The Notch pathway is an essential regulator of cell proliferation and differentiation during development. Its involvement in insect oogenesis has been examined in insect species with meroistic ovaries, and it is known to play a fundamental role in cell fate decisions and the induction of the mitosis-to-endocycle switch in follicular cells (FCs). This work reports the functions of the main components of the Notch pathway (Notch and its ligands Delta and Serrate) during oogenesis in Blattella germanica , a phylogenetically basal species with panoistic ovary. As is revealed by RNAi-based analyses, Notch and Delta were found to contribute towards maintaining the FCs in an immature, non-apoptotic state. This ancestral function of Notch appears in opposition to the induction of transition from mitosis to endocycle that Notch exerts in Drosophila melanogaster, a change in the Notch function that might be in agreement with the evolution of the insect ovary types. Notch was also shown to play an active role in inducing ovarian follicle elongation via the regulation of the cytoskeleton. In addition, Delta and Notch interactions were seen to determine the differentiation of the posterior population of FCs. Serrate levels were found to be Notch-dependent and are involved in the control of the FC programme, although they would appear to play no crucial role in panoistic ovary oogenesis.

Impacts of Multiple Time Delays on a Gene Regulatory Network Mediated by Small Noncoding RNA

2020 ◽  
Vol 30 (05) ◽  
pp. 2050069
Author(s):  
Ming Liu ◽  
Fanwei Meng ◽  
Dongpo Hu
Keyword(s):  
Hopf Bifurcation ◽  
Gene Regulatory Network ◽  
Regulatory Network ◽  
Time Delays ◽  
Noncoding Rna ◽  
Multiple Time ◽  
Normal Form Theory ◽  
Small Noncoding Rna ◽  
Multiple Time Delays ◽  
Gene Regulatory

In this paper, the impacts of multiple time delays on a gene regulatory network mediated by small noncoding RNA is studied. By analyzing the associated characteristic equation of the corresponding linearized system, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Furthermore, the explicit formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are given by the center manifold theorem and the normal form theory for functional differential equations. Finally, some numerical simulations are demonstrated for supporting the theoretical results.

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.

Stability and Hopf Bifurcation Analysis in Hindmarsh–Rose Neuron Model with Multiple Time Delays

2016 ◽  
Vol 26 (11) ◽  
pp. 1650187 ◽  
Author(s):  
Dongpo Hu ◽  
Hongjun Cao
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Time Delays ◽  
Single Neuron ◽  
Neuron Model ◽  
Multiple Time ◽  
Bifurcation Diagrams ◽  
Dynamical Behaviors ◽  
Multiple Time Delays ◽  
The Stability

In this paper, the dynamical behaviors of a single Hindmarsh–Rose neuron model with multiple time delays are investigated. By linearizing the system at equilibria and analyzing the associated characteristic equation, the conditions for local stability and the existence of local Hopf bifurcation are obtained. To discuss the properties of Hopf bifurcation, we derive explicit formulas to determine the direction of Hopf bifurcation and the stability of bifurcated periodic solutions occurring through Hopf bifurcation. The qualitative analyses have demonstrated that the values of multiple time delays can affect the stability of equilibrium and play an important role in determining the properties of Hopf bifurcation. Some numerical simulations are given for confirming the qualitative results. Numerical simulations on the effect of delays show that the delays have different scales when the two delay values are not equal. The physiological basis is most likely that Hindmarsh–Rose neuron model has two different time scales. Finally, the bifurcation diagrams of inter-spike intervals of the single Hindmarsh–Rose neuron model are presented. These bifurcation diagrams show the existence of complex bifurcation structures and further indicate that the multiple time delays are very important parameters in determining the dynamical behaviors of the single neuron. Therefore, these results in this paper could be helpful for further understanding the role of multiple time delays in the information transmission and processing of a single neuron.

scholarly journals A Fractional-Order Model for Zika Virus Infection with Multiple Delays

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-20 ◽  
Author(s):  
R. Rakkiyappan ◽  
V. Preethi Latha ◽  
Fathalla A. Rihan
Keyword(s):  
Hopf Bifurcation ◽  
Virus Infection ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Zika Virus ◽  
Epidemic Model ◽  
Time Delays ◽  
Multiple Delays ◽  
Multiple Time ◽  
Zika Virus Infection

Time delays and fractional order play a vital role in biological systems with memory. In this paper, we propose an epidemic model for Zika virus infection using delay differential equations with fractional order. Multiple time delays are incorporated in the model to consider the latency of the infection in a vector and the latency of the infection in the infected host. We investigate the necessary and sufficient conditions for stability of the steady states and Hopf bifurcation with respect to three time delays τ1, τ2, and τ3. The model undergoes a Hopf bifurcation at the threshold parameters τ1∗, τ2∗, and τ3∗. Some numerical simulations are given to show the effectiveness of obtained results. The numerical simulations confirm that combination of fractional order and time delays in the epidemic model effectively enriches the dynamics and strengthens the stability condition of the model.

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