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.
1. 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 [1–4]. The miR-17-92 cluster as a polycistronic gene located in human chromosome 13 ORF 25 (c13orf25) is composed of 7 mature miRNAs [1]. MiRNAs play critical roles in biological processes, as posttranscriptional regulators of gene expression [2]. MicroRNA-based regulation has been simulated by specially designed mathematical models [5–7]. 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 [8–11]). 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 [12–14]. 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 parameter in the network is incorporated into ordinary differential equations to give rise to periodic solutions at certain critical values.
The organization of the paper is as follows: we describe the Myc/E2F/miR-17-92 network with time delay in Section 2. In Section 3.1, we discuss local stability of the solution and Hopf bifurcation of the network with respect to negative feedback strength . Under different values of , local stability and Hopf bifurcation of time delay are studied in Section 3.2. In Section 3.3, a formula is given to determine the direction and stability of the Hopf bifurcation of the time delay. In Section 4, four examples on influence of time delay on different states of negative feedback strength are given to support our results. Finally, the conclusions are drawn in Section 5.
2. 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.
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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 and 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. is the coefficient of protein expression and 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 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 defineWith these substitutions, system (2) can be rewritten asThe initial values for system (4) take the form ofwhere , and all the parameters are positive.
3. 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).
3.1. Local Stability and Hopf Bifurcation of the Negative Feedback Strength
Firstly, at the time delay , we consider Hopf bifurcation of the negative feedback strength 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 thenEliminating 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 transformsystem (4) becomeswhereFrom (9), it is easy to get the following characteristic equation. That is,thus, the two degree polynomial equation is obtainedIfthen all the roots of (12) have negative real parts. So the equilibrium of system (4) is locally stable.
On the other hand, we assumeand are the roots of (12).
Viewing the negative feedback strength as a bifurcation parameter, the sufficient conditions for the occurrence of the Hopf bifurcation of system (4) are obtained in the following results.
Theorem 1. If and hold, there exist , such thatthen system (4) can exhibit Hopf bifurcation.
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.
3.2. 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 , , then system (9) becomeswhereThen we obtain the characteristic equation of system (14).
That is,When , it becomes
is the root of (16) if and only if satisfiesSeparating the real and imaginary parts, we getwhich is equivalent toLet , then it is transformed to bewhere
Assume (H4) equation (21) has at least one positive real root.
From (21), we can obtainThus, ifthen none of 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 obtainWithout loss of generality, (21) is assumed to have two positive roots, defined by , respectively. Thus, (20) has two positive roots .
So we haveIt follows thatwhere .
DefineTaking the derivative of with respect to , we can getThen,whereIf the conditionholds, thenThat is, the transversality condition is satisfied.
Based on the analysis above, the following results are obtained.
Theorem 2. Let be defined by (26); one has
(i) if , , and hold, then the equilibrium of system (4) is asymptotically stable for all ;
(ii) if , , and hold, then the equilibrium is locally asymptotically stable for and unstable for ; moreover, if , then system (4) undergoes a Hopf bifurcation at the equilibrium when .
3.3. Direction and Stability of the Hopf Bifurcation at Time Delay
In this section, we further consider direction and stability of Hopf bifurcation assuming at by using the methods in Hassard et al. [17]. For convenience, let and denote , then system (14) becomesClearly, is a Hopf bifurcation value of system (32).
Its linear part is given byand nonlinear part isFor convenience of studying the problems of Hopf bifurcation, we rewrite system (32) aswhere and for ; are given, respectively, byHere, h.o.t. denotes higher order terms.
For convenience analysis, letwhere ; , .
Then,
According to the Riesz representation theorem, there exists a 2 × 2 matrix function , such thatHere, we choosewhere is defined by
For , defineandFor , the adjoint operator of is defined byAnd define a bilinear inner product as follows:where .
From the discussion in Section 3.2, we obtain that are eigenvalues of . At the same time, they are also eigenvalues of since and are adjoint operator. In the following, we calculate the eigenvectors of and corresponding to and , respectively. Suppose is eigenvector of . Through the discussion above, we haveThat is,We can obtainThus, or .
On the other hand, we assume is the eigenvector of and then obtainand
Then, normalize and by the following conditions:In order to get , we need to determine the value of .Thus, we choose assuch that .
Next, we employ the algorithms in [17] and a computation process similar to [18–20] to calculate some important coefficients as follows:where
The coefficients , and can be obtained if the parameters in system (4) are given, while the coefficient needs to compute and by a calculation process similar to [18–20] as follows:andwhere are determined by the following equations, respectively:whereIt follows thatwhereSimilarly, we can getwhere
From (56), (57), and (58), we can get . Furthermore, we can obtain the following values:
Through the discussion above, we obtain the following result.
Theorem 3. For system (4), the coefficient determines the direction of Hopf bifurcation; it is supercritical (subcritical) if ; the periodic solutions are stable (unstable) on the manifold if ; and the period increases (decreases) if .
4. 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 is also investigated.
Example 1. By increasing the negative feedback strength , system (4) undergoes a transition from monostability to oscillation and then back to monostability. That is, Hopf bifurcation occurs when the parameter 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 and when and hold, respectively, as well as limit cycles for between the two Hopf bifurcations when . Also, the equilibrium is asymptotically stable for or . The equilibrium of system (66) at is stable (see Figure 2(b)) and oscillations of limit cycles at are presented in Figure 2(c), and then the oscillation behavior vanishes and the equilibrium tends to be stable again at (see Figure 2(d)).
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Example 2. We consider dynamic changes induced by time delay for the negative feedback strength when system (4) has a positive equilibrium at a high steady state in Example 1 in Figure 2. Other parameters are same as Example 1.
In Theorem 2, we get , , , , and . Then conditions , , and hold. Therefore, we obtain that the steady state of system (4) is stable for all (see Figures 3(a), 3(b), 3(c), and 3(d)).
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Example 3. A limit cycle and an equilibrium can coexist in system (4) for at the negative feedback strength 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 , 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 and to determine the stability of periodic solutions bifurcating from equilibrium and direction of the Hopf bifurcation at the critical point . When , we can compute and by means of software Maple. Further, we can get . 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), 4(b), and 4(c)).
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Example 4. Fixing the negative feedback strength , system (4) has a positive equilibrium 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 (see Figure 5).
For the arbitrary negative feedback strength and the parameters given in Example 1, system (4) is always stable. For example, at , we have , , , , , and . It means that conditions , , and hold. By the Theorem 2, we know that system (4) is stable for all . 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.
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5. 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 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 . 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 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 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 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 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 , , and hold, then the equilibrium of system (4) is asymptotically stable for all ; if , , and hold, the equilibrium is locally asymptotically stable for ; moreover, if , system (4) undergoes a Hopf bifurcation at the equilibrium when . 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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (nos. 11872084, 11472009, and 11801224).