Stability and Hopf Bifurcation Analysis in a Delayed Myc / E 2 F / miR-1792 Network Involving Interlinked Positive and Negative Feedback Loops

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 andMyc/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 viaHopf 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.


Introduction
Due to the development of large-scale experimental and computational techniques, a posttranscriptional regulation by small noncoding microRNAs (miRNAs) has been discovered in many cellular processes, including cell growth, development, differentiation, and apoptosis [1][2][3][4].The miR-17-92 cluster as a polycistronic gene located in human chromosome 13 ORF 25 (c orf ) is composed of 7 mature miRNAs [1].MiRNAs play critical roles in biological processes, as posttranscriptional regulators of gene expression [2].MicroRNAbased regulation has been simulated by specially designed mathematical models [5][6][7].The transcription factors E2F and Myc act as tumor suppressors or oncogenes and participate in the control of cell proliferation and apoptosis [1].Aguda et al. proposed a simple model involving miR-17-92, E2F, and Myc, which is composed of a positive feedback (E2F/Myc) and a negative feedback (Myc/E2F/miR-17-92) [1].It presents a bistable switch behavior and a one-way switch in the network, which corresponds to the bistability and monostability, respectively [1,2].Subsequently, Li et al. illustrated an abstract model of the network presented by Aguda et al. and focused on the physiological significance of miRNAs [2].It was found that the existence of miRNAs improves the ability of the bistable switches in the network [2].Zhang et al. further analyzed this abstract model and suggested that the interlinked positive and negative feedback loops buffer noise effects rather than only amplifying or suppressing the noise [3].
Dynamical analysis of the system with time delay is an essential topic in many fields, especially for the models of gene expression (see [8][9][10][11]).Time delay is inevitable in Myc/E2F/miR-17-92 network since feedback loops involve many intermediate processes such as transcription, translation, and posttranslational modifications [12][13][14].Time delay influences the time-dependent dynamics even for the simplest circuits with one and two gene elements, which can give rise to rich dynamical behaviors such as periodic and chaotic dynamics [15].
In this work, we consider the effect of time delay in the delayed Myc/E2F /miR-17-92 network.As negative feedback is often used in biochemistry to generate oscillations to achieve homeostasis, the steady state may lose stability and be replaced by oscillations [16].So an inhibition efficiency

Model Description
An illustration of network involving miR-17-92, E2F, and Myc and abstract structure of this network are depicted in Figure 1, where P is the protein module and M denotes the miRNA cluster module.The positive feedback in module P (E2Fs and Myc) as an autocatalytic process can lead to the transcription of M but be inhibited by M.
Aguda B D et al. presented a model of interaction between miR-17-92, E2F and Myc, which is described by the following differential equations [1]: where [P] and [M] represent the concentrations of P and M, respectively.  denotes the P-independent constitutive transcription of M, and   describes the constitutive protein expression.Parameters   and   are the degradation rates of P and M, respectively.Γ 1 is the coefficient of protein expression and Γ 2 is the inhibition efficiency parameter.  stands for the rate constant, and   represents the constant of protein expression.
For this model, we consider the inhibition efficiency parameter Γ 2 to explore stability of equilibria and oscillatory behaviors via Hopf bifurcation through its critical value as negative feedback usually leads to oscillation.Moreover, we investigate the effect of the time delay in the network when the concentrations of P and M are described by the following delayed differential equations (DDEs): For the convenience of analysis, we define With these substitutions, system (2) can be rewritten as The initial values for system (4) take the form of where ( 1 ,  2 ) ∈ ([−, 0], [0, +∞) 2 ), and all the parameters are positive.

Main Results
System (4) exhibits dynamics behaviors of the steady state and periodic phenomenon in the different parameter regimes.
Here, we focus especially on theoretical analysis on the stability and existence of Hopf bifurcation of system (4).

. . Local Stability and Hopf
Bifurcation of the Negative Feedback Strength Γ 2 .Firstly, at the time delay  = 0, we consider Hopf bifurcation of the negative feedback strength Γ 2 as the measure of the miRNA inhibition through the critical values and local stability of equilibrium before the bifurcation in system (4).
Let  * ( * ,  * ) be the equilibrium of system (4), and then Eliminating  * from above equations, we get the following equation on  * : Obviously, the equation has at most three positive real roots.
Setting  * ( * ,  * ) is one of the three roots.Secondly, we translate the equilibrium  * to the origin.By the linear transform where (1) =   +    2 ( − ) then all the roots of ( 12) have negative real parts.So the equilibrium  * of system (4) is locally stable.
Viewing the negative feedback strength Γ 2 as a bifurcation parameter, the sufficient conditions for the occurrence of the Hopf bifurcation of system (4) are obtained in the following results.
Time delay is inevitable and plays an important role in negative feedback loop of the Myc/E2F/miR-17-92 network due to the transcription and translation.So time delay on this network is considered in the next sections.

. . Local Stability and Hopf Bifurcation of Time Delay.
In this section, we take the time delay  as a bifurcating parameter to investigate the stability and existence of Hopf bifurcation of time delay in system (4).For convenience, let  1 =   ,  2 =   ,  3 =   ,  4 =   , then system (9) becomes where Then we obtain the characteristic equation of system (14).That is, When  = 0, it becomes ( > 0) is the root of ( 16) if and only if  satisfies Separating the real and imaginary parts, we get which is equivalent to Let  =  2 , then it is transformed to be where Assume (H4) equation ( 21) has at least one positive real root.
From (21), we can obtain Thus, if then none of  1 ,  2 is positive.So (21) has no positive roots.It means that there are no imaginary roots in characteristic equation (16).
From those discussed above, we obtain Without loss of generality, ( 21) is assumed to have two positive roots, defined by  1 ,  2 , respectively.Thus, (20) has two positive roots It follows that where  = 1, 2,  = 0, 1, 2, . ... Define Taking the derivative of  with respect to , we can get where If the condition That is, the transversality condition is satisfied.
Based on the analysis above, the following results are obtained.

Numerical Simulations
In this section, we present four examples to verify our theoretical results.At same time, the effect of time delay at different situations of negative feedback strength Γ 2 is also investigated.
Example .By increasing the negative feedback strength Γ 2 , system (4) undergoes a transition from monostability to oscillation and then back to monostability.That is, Hopf bifurcation occurs when the parameter Γ 2 increases to the critical values.Here, we give an example when some important parameters are taken as follows [1,2]: Then, system (4) becomes the following ordinary differential equations: Through bifurcation analysis of system (66) in Figure 2(a), we obtain two Hopf bifurcations at Γ 2 = 0.04011 and 0.06554 when (H2) and (H3) hold, respectively, as well as limit cycles for 0.04011 < Γ 2 < 0.06554 between the two Hopf bifurcations when   +   −  < 0. Also, the equilibrium  * is asymptotically stable for Γ 2 < 0.04011 or Γ 2 > 0.06554.The equilibrium of system (66) at Γ 2 = 0.01 is stable (see Figure 2(b)) and oscillations of limit cycles at Γ 2 = 0.05 are presented in Figure 2(c), and then the oscillation behavior vanishes and the equilibrium tends to be stable again at Γ 2 = 0.09 (see Figure 2(d)).
Example .We consider dynamic changes induced by time delay  ≥ 0 for the negative feedback strength Γ 2 = 0.01 when system (4) has a positive equilibrium  1 (1.37151, 27.93015) at a high steady state in Example 1 in Figure 2. Other parameters are same as Example 1.
Example .A limit cycle and an equilibrium can coexist in system (4) for  = 0 at the negative strength Γ 2 = 0.04 in Figure 2.So system (4) appears to be periodic solution or steady state depending on different initial conditions.
As time delay  passes through the critical value  0 ≈ 2.23, the periodic solution transits to steady-state solution via Hopf bifurcation when the initial point is chosen near the limit cycle (see Figure 4).Next, we calculate the values of  2 and  2 to determine the stability of periodic solutions bifurcating from equilibrium  2 (0.72821, 15.06414) and direction of the Hopf bifurcation at the critical point  0 .When  =  0 ≈ 2.23, we can compute Re( 1 (0)) ≈ −0.20118,  2 ≈ −0.40236 and  +  ≈ 0.01847 > 0 by means of software Maple.Further, we can get Re(  ( 0 )) ≈ 17.96347 > 0,  2 ≈ 0.0112 > 0. By Theorem 3, we know that the direction of Hopf bifurcation is supercritical, and the periodic solutions are stable on the manifold (see Figures 4(a Example .Fixing the negative feedback strength Γ 2 = 0.07, system (4) has a positive equilibrium  3 (0.08174, 2.13478) at a low steady state in Figure 2. The periodic solutions bifurcate from the equilibrium via Hopf bifurcation when time delay  increases to the critical value  0 ≈ 35(see Figure 5).
For the arbitrary negative feedback strength Γ 2 ∈ (0, 0.04) and the parameters given in Example 1, system (4) is always stable.For example, at Γ 2 = 0.02, we have It means that conditions (H1), (H2), and (H5) hold.By the Theorem 2, we know that system (4) is stable for all  ≥ 0. It can explain that delayed negative feedback could not lead to oscillations in the high state in Figure 2. In a word, the time delay leads to oscillation behaviors when system (4) is at the low steady state but not at the high one in Figure 2.

Conclusions and Discussions
In this paper, the Myc/E2F/miR-17-92 network with time delay is considered.Occurrence of Hopf bifurcations associated with oscillation of the inhibition efficiency parameter Γ 2 and further the time delay , respectively, are investigated by combined local stability and bifurcation theory with numerical simulations.Furthermore, the direction and stability of Hopf bifurcation are also studied by the center manifold theorem and normal form method as well as a numerical example supporting the results.
We find that the time delay has a destabilizing role by choosing appropriate parameters in this network.Besides that, initial condition is important as periodic solution and steady state coexist under some circumstances (see Figure 2).If the initial values are chosen near the positive equilibrium, the steady state of system (4) remains unchanged when the parameter  less than the critical value  0 .However, if the initial values are chosen near the limit cycle, system (4) appears to be periodic solution and then transits to steady-state solution via Hopf bifurcation of time delay  (see Figure 4).With the negative feedback strength Γ 2 increasing, system (4) undergoes a transition from the high steady state to oscillations and then to low steady state (see Figure 2(a)).We consider the effect of time delay on the dynamics of the network at every different state.When the negative feedback strength Γ 2 is chosen at the high steady states, there will be no periodic oscillations with increasing time delay  (see Figure 3).However, when the negative feedback strength Γ 2 is chosen between the two Hopf bifurcation points in Figure 2(a), the periodic solutions transit to steady-state solutions via Hopf bifurcation with increasing time delay (see Figure 4).The periodic solutions bifurcate from the equilibrium of system (4) with increasing time delay  when the negative feedback strength Γ 2 is chosen at the low steady states (see Figure 5).The values of the parameters are chosen in the special way above or they need to satisfy the conditions in Theorem 2. If (H1), (H2), and (H5) hold, then the equilibrium of system (4) is asymptotically stable for all  ≥ 0; if (H1), (H2), and (H4) hold, the equilibrium is locally asymptotically stable for 0 ≤  <  0 ; moreover, if + ̸ = 0, system (4) undergoes a Hopf bifurcation at the equilibrium when  =  0 .At the same time, the direction and stability of Hopf bifurcation can be determined by Theorem 3.This work can further provide a theoretic instruction for exploring the dynamics of the network.
Noise is inevitable in gene regulatory networks and plays important roles in circuits' dynamics.It may induce bistability, oscillations, and bifurcations which are not present in the deterministic model [7,21,22].How does stochastic noise affect dynamics of the network with time delay?This is a very valuable and worth exploring problem.We leave it as the future work.

Figure 1 :
Figure 1: An illustration of the network involving miR-17-92, E2F and Myc (a) and its abstract model (b), where P and M represent the protein module (E2Fs and Myc) and miR-17-92 cluster, respectively.

Figure 2 :
Figure 2: (a) Bifurcation diagram of [P] with Γ 2 as a bifurcation parameter.The red solid lines depict stable steady states and the black dashed line depicts unstable steady states.The maxima and minima are depicted by the blue dots and green dots for the stable and unstable limit cycle, respectively.The bifurcation points of the subcritical Hopf bifurcation are marked as sub and LPC as fold limit cycle bifurcation points.(b), (c), and (d) are time courses diagram of  and  for the parameter Γ 2 =0.01(b), 0.05(c), and 0.09 (d), respectively.The initial conditions: (0) = 0.1, (0) = 0.1.

Figure 4 :
Figure 4: (a) Bifurcation diagram of [P] with time delay  as the bifurcating parameter when fixing the negative feedback strength Γ 2 = 0.04.The stable steady state is depicted by the red solid line, and the maxima and minima are depicted by the blue dots for the stable limit cycle.(b) and (c) are time courses diagram of [] for the parameter  = 1 (b) and 4.5 (c), respectively.Initial conditions: (0) = 0.1, (0) = 0.1.

Figure 5 :
Figure 5: (a) Bifurcation diagram of [P] with time delay  as the bifurcating parameter when fixing the negative feedback strength Γ 2 = 0.07.The red solid line depicts stable steady states and the blue dots depict the maxima and minima for the stable limit cycle.(b) and (c) are time courses diagram of [] for the parameter  = 1 (b) and 40 (c), respectively.Initial conditions: (0) = 0.1, (0) = 0.1.