Abstract
MicroRNAs are currently considered as key regulators of a wide variety of biological pathways and regulate many processes of life and obtained more and more attention in recent years. In this paper, we investigate the dynamics of gene network regulated by miR-34a (microRNA) involved in triple negative feedback loop. As we know that the p53 network involve in cancer, How the cancer arise is unclear. We investigate this negative feedback network by using mathematical model and drive the theoretical results of globally asymptotic stability and provide the sufficient conditions for the periodic oscillation. These results are propitious to understand how p53 network involved in miR-34a induces the cancer.
1. Introduction
MicroRNAs (miRNAs) are a family of small regulatory RNAs whose function is to regulate the activity and stability of specific mRNA targets through posttranscriptional regulatory mechanism and play a role in repressing translation of mRNA or degrading mRNAs [1–9]. Recent studies show that microRNAs play a central role in many biological (cellular) processes, including developmental timing, cell proliferation, apoptosis, metabolism, cell differentiation, somitogenesis and tumour genesis [1]. In this paper, we focus on miR-34a which may behave as inhibitor depending on biological context. Resecting normal tumor tissues of 25 human hepatocellular carcinoma patients demonstrated an inverse correlation between miR-34a and c-Met-protein [2]. MiR-34a is proposed as a marker for the activity of the p53 pathway in chronic lymphocytic leukemia [3] (Table 1). The induction of miR-34a was most pronounced among all differential regulations. Also expression of the primary miR-34a transcript was induced after p53 activation [4]. MiR-34a may play a cytoprotective role in cell survival [5]. In U87 cell, miR-34a plays a negative role in the regulation of Dll1 target gene through downregulating Dll1 protein but not mRNA [6]. Expression of miR-34a is induced which may mediate the target gene and result in the cells arresting in G1 and G2 phase to repair DNA [7].
In order to understand further the miR-34a involved in the network with p53 and Sirt1, we have planned to model this network with mathematical model. It is well known that the time delay is quite ubiquitous in nature, so we also investigate the relationship between the time delay and the network with miR-34a, p53, and Sirt1. We know that the delay is often caused by a finite signal transmission speed and memory effects, so the time delay can sometimes destabilize the stable unique equilibrium. If the time delay reaches a threshold value, the system will generate the phenomenon with self-oscillation. In nature, the oscillation often occurs in physiological regulatory systems with time delay which can induce the complex behaviors.
In recent years, many scientists deemed that mathematical modeling could be used to investigate the differences at the dynamical level between healthy and pathologic configurations of biological pathways [10]. By using the mathematical model, the researchers can detect the key points regulating main properties of biological system and find the methods to solve the different diseases.
In this paper, we will investigate the dynamics of the gene network composed of miR-34a, p53, and Sirt1 and reveal how the dynamics of microRNA regulation is affected by time delay associated with translation degradation of mRNA. In Section 2, we give the gene network represented by mathematical model and some theoretical results. In Section 3, we show the numerical analysis of the network. Finally, we summarize the results.
2. Gene Regulatory Network Mediated by MicroRNAs
2.1. A Simple Mathematical Model of Gene Regulation with Time Delay
In [11], a mathematical description of miRNA regulation process is presented. Figure 1 shows the relationship among the related factors comprised in this model [11]. Using the network in Figure 1, Lai et al. [11] gave the ODE model and investigated different possible designs of the silencing mechanism exerted by miR-34a on Sirt1. The network is composed of four main parts: the activation of p53 in response to DNA damage-mediated signals; the positive feedback loop integrated by p53, miR-34a, and Sirt1; the DNA damage induced enhancement in DBC1 activity and its inhibitory effect on Sirt1; and the negative feedback loop between Mdm2 and active p53. And the variables are as follows: expression level of the p53-inhibitor MDM2, fraction of nonactive (deacetylated) p53, fraction of active (acetylated) p53 (p53*), expression level of the miR-34a, expression level of the p53 acetylation inhibitor Sirt1, and expression level of the Sirt1-activity inhibitor DBC1. The dynamics of the network is described by the following system:
(a)
(b)
Assume that the activation rates are , and for Mdm2, p53, miR-34a, and DBC1, the degradation rates are , and for Mdm2, p53, miR-34a, Sirt1, and DBC1, the synthesis rates are , and for Mdm2, p53, miR-34a, Sirt1, and DBC1, and the translational repression rates are for p53*, p53. In addition, the inhibition rate is .
2.2. Oscillation Induced by Time Delay
In this subsection, we consider the system (1) with () as state variables. Based on Hopf bifurcation theory, we investigate the stability of the time delay. Then the linearized system of (1) at equilibrium () is as follows: where
Then we can obtain the characteristic equation of (1) at the equilibrium () as follows: where is the identity matrix, and the characteristic equation (4) has the following form: where , , , , , , , , , = + , + − − , + + + , , , , , , , , , , , , /, , , , , , and In order to simplify the equation, we can suppose , and we only consider If we assume that , we will have Considering the transcendental equation (7), clearly is the root of (7) if and only if Now separating the real and imaginary parts, we have Adding up the squares of both equations above, we obtain where . Let , and then (10) becomes where Let
Lemma 1. If , (13) has at least one positive root.
Proof. Clearly, , and . Hence, there exists a , so that . This completes the proof.
Lemma 2. If , all the roots of (13) have negative real parts if and only if and , .
The sufficient and necessary condition is obvious by the Routh-Hurwitz criterion.
Lemma 3. If , then the sufficient condition for (13) to have positive real roots is , or , .
Proof. From (13), we have , and suppose that have four real roots and , and is the local minimum value, and if , or , , there exists or , so that ; this completes the proof.
Suppose that (13) has five positive roots, denoted by , and , respectively. Hence, (10) has five positive roots, say , , , , and . From (9), we can get where = + , = + + , = − , = , . = .
Define , and let be the root of (7) satisfying , , and if , then Hopf bifurcation occurs. From (7) when , we can obtain where , , , , and , and by the Routh-Hurwitz criterion, all roots of (15) have negative real parts if and only if , , , .
Lemma 4. Consider the exponential polynomial where and is polynomial about . As vary, the sum of the orders of the zeros of on the open right half plane can change only if a zero appears on or across the imaginary axis.
Then, we have the following theoretical results.
Theorem 5. Suppose that , , , is true.(i)If , and , , then all roots of (7) have negative real parts for all ; thus, the steady state () of (7) is absolutely stable.(ii)If or , , then all the roots of (7) have negative real parts when ; thus, the steady state of (7) is asymptotically stable.(iii)If the condition of (ii) is satisfied, = , and , = , then is a pair of simple purely imaginary roots of (13) and all other roots have negative real parts. Moreover, > , Thus, (7) exhibits the Hopf bifurcation at ().
The another case we can discuss as above, and ignore it here.
3. Numerical Analysis
In this section, we present some numerical results of system (1) to verify the analytical predictions obtained in Section 2. Without loss of generality, if we take , , , , , , , , , , , , , , , , then the system (1) becomes which has a positive equilibrium which satisfies the conditions indicated in Section 2. When , is asymptotically stable (see Figure 2).
When , the system (17) is illustrated by the computer simulations (see Figure 2). When passes through the critical value , the positive equilibrium loses its stability and the Hopf bifurcation occurs (see Figures 3 and 4).
Figure 3 shows the behaviors and phase portraits of system (17) with , and the positive equilibrium is asymptotically stable.
Figure 4 shows the behaviors and phase portraits of system (17) with and Hopf bifurcation occurs from the positive equilibrium as showed in Figure 5, we can know that the expression of protein and the oscillation occurs when the time delay overpasses the critical points. Above the critical value of time delay, the expression of small RNA also oscillates periodically and provides the capabilities inside and outside cell to communicate.
In system (1), the time delay approximately represents transportation or diffusion process from nucleus to cytoplasm of mRNA and from cytoplasm to nucleus of protein, respectively. These delays play different roles in the dynamical behaviors of the system (1).
4. Discussion
From the above discussion, we know that and affect the dynamical behaviors in the form of due to the cyclic structure of system (1), and we used the sum of and as a bifurcation parameter and then found that the stability depends on the sum of delay. Here we just gave a very simplified model to describe the dynamical behaviors of p53, Sirt1, and miR-34a. In this paper, we have given a detailed analysis of mathematical properties of the system. Our results indicate that the steady state is asymptotically stable for all delay values under certain conditions. However, if these conditions are not satisfied, there exists a critical value of the time delay . When , the steady state is asymptotically stable; when , there are periodical oscillations and the Hopf bifurcation appears.
From the above discussion, we understand how the microRNA regulate the negative feedback loop in cancer signalling network (Figure 1), so we can use these results to explain how the prostate cancer stem cells are regulated by miR-34 [12] and give a clear understanding on the possible reason of prostate cancer.
Finally, it is worth noting that microRNA-mediated regulation has gained recent attention, and computational studies have revealed various regulatory properties unique to microRNAs. These findings will be helpful for our understating of the operating mechanisms and biological implications of microRNA-mediated regulation. They also have great potential for biotechnological and therapeutic applications and synthetic biology.
5. Conclusions
We analyzed a simple model of the interactions between miR-34a and target protein p53 and Sirt1 and others. Our goal is to explore the oscillatory dynamics and how the microRNAs repress its target protein. Finally, we derive explicit conditions on how the dynamics of a time delay model of the inter-action between the microRNA (miR-34a) cluster and p53 and Sirt1 depends on system parameters. Our analysis reveals the complex behavior of the network and there is a limit cycle after a Hopf bifurcation for the time delay parameter and it shows that the analytical results agree with numerical simulations.
Conflict of Interests
The authors declare that there is no conflict of interests.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (11272277), Program for New Century Excellent Talents in University (NCET-10-0238), the Key Project of Chinese Ministry of Education (211105), Innovation Scientists and Technicians Troop Construction Projects of Henan Province (134100510013), and Innovative Research Team in University of Henan Province (13IRTSTHN019).