Stability and Hopf Bifurcation Analysis in a Delayed Myc/E2F/miR-17-92 Network Involving Interlinked Positive and Negative Feedback Loops

Abstract

MiR-17-92 plays an important role in regulating the levels of the Myc/E2F protein. In this paper, we consider a coupling network between Myc/E2F/miR-17-92 delayed negative feedback loop and Myc/E2F positive feedback loop described by a two-dimensional delay differential equation. Based on linear stability analysis and bifurcation theory, sufficient conditions for stability of equilibria and oscillatory behaviors via Hopf bifurcation are derived when choosing time delay as well as negative feedback strength associated with oscillations as bifurcation parameters, respectively. Furthermore, direction and stability of Hopf bifurcation of time delay are studied by using the normal form method and center manifold theorem. Finally, several numerical simulations are performed to verify the results we obtained.

Novel extended mixed controller design for bifurcation control of fractional‐order Myc/E2F/miR‐17‐92 network model concerning delay

“…Thus, delay will be introduced into model (). Based on this viewpoint, Wang and Yang [14] built a delayed Myc/E2F/miR‐17‐92 network model which takes the following form: $$$$ \left\{\begin{array}{l}\frac{d\left[\mathcal{P}\right]}{dt}={a}_P+\frac{\kappa_P{\left[\mathcal{P}\left(t-\rho \right)\right]}^2}{{\mathrm{Y}}_1+{\left[\mathcal{P}\left(t-\rho \right)\right]}^2+{\mathrm{Y}}_2\left[\mathcal{M}\left(t-\rho \right)\right]}-{b}_P\left[\mathcal{P}\right],\\ {}\frac{d\left[\mathcal{M}\right]}{dt}={a}_M+{\kappa}_M\left[\mathcal{P}\right]-{b}_M\left[\mathcal{M}\right],\end{array}\right. $$$$ where $$$ \mathcal{P},\mathcal{M} $$$ stand for the protein module (E2Fs and Myc) and miR‐17‐92 cluster, respectively; $$$ \left[\mathcal{P}\right],\left[\mathcal{M}\right] $$$ stand for the concentrations of $$$ \mathcal{P},\mathcal{M} $$$, respectively.…”

“…$$$ {\kappa}_M $$$ is the rate constant, $$$ {\kappa}_P $$$ is the constant of protein expression, and $$$ \rho $$$ is a delay. By selecting the delay $$$ \rho $$$ as bifurcation parameter, Wang and Yang [14] explored the stability and Hopf bifurcation of model ().…”

“…Thus, delay will be introduced into model (1.1). Based on this viewpoint, Wang and Yang [14] built a delayed Myc/E2F/miR-17-92 network model which takes the following form:…”

“…It is worth pointing out that all the involved publications above on chemical models (see previous studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14]) are merely concerned with integer-order dynamical equations. In recent years, a lot of studies demonstrate that fractional-order differential equation is a better powerful tool to describe the objective reality in our world than the conventional integer-order differential equation.…”

mRNA and miRNA Expression Analyses of the MYC/E2F/miR-17-92 Network in the Most Common Pediatric Brain Tumors

“…Experimental studies have shown that there is a negative feedback loop among members of the miR-17-92 cluster and E2F and MYC transcription factors [ 5 , 6 ]. Many mathematical models of the MYC / E2F /miR-17-92 network were created, estimating how overexpression of the miR-17-92 cluster affects different types of cancers [ 5 , 7 , 8 ]. The consecutive studies provide evidences that the overexpression of miR-17-92 members is involved in the development of many solid tumors, including lung [ 9 ], breast [ 10 ], colon [ 11 ], hepatocellular [ 12 ], and stomach cancer [ 13 ].…”