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 miR34a (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 miR34a induces the cancer.
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 [
Notation for species concentrations.
Name  Notation  Description 

Mdm2 

Murine double minute 
p53 

Deacetylated (inactive) p53 
p53* 

Acetylated (active) p53 
miR34a 

MicroRNA34a 
Sirt1 

Silent information regulator 
DBC1 

Deleted gene in breast cancer 
In order to understand further the miR34a 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 miR34a, 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 selfoscillation. 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 [
In this paper, we will investigate the dynamics of the gene network composed of miR34a, p53, and Sirt1 and reveal how the dynamics of microRNA regulation is affected by time delay associated with translation degradation of mRNA. In Section
In [
Schematic diagram of the complex network which illustrates the mechanisms. (a) Original model and (b) abstract model.
Assume that the activation rates are
In this subsection, we consider the system (
Then we can obtain the characteristic equation of (
If
Clearly,
If
The sufficient and necessary condition is obvious by the RouthHurwitz criterion.
If
From (
Suppose that (
Define
Consider the exponential polynomial
Then, we have the following theoretical results.
Suppose that
If
If
If the condition of (ii) is satisfied,
The another case we can discuss as above, and ignore it here.
In this section, we present some numerical results of system (
An asymptotically stable equilibrium for
When
An asymptotically stable equilibrium for
A periodic solution bifurcated from equilibrium for
Figure
Figure
Bifurcation diagram with total time delay as a parameter when
In system (
From the above discussion, we know that
From the above discussion, we understand how the microRNA regulate the negative feedback loop in cancer signalling network (Figure
Finally, it is worth noting that microRNAmediated 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 microRNAmediated regulation. They also have great potential for biotechnological and therapeutic applications and synthetic biology.
We analyzed a simple model of the interactions between miR34a 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 interaction between the microRNA (miR34a) 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.
The authors declare that there is no conflict of interests.
This work is supported by the National Natural Science Foundation of China (11272277), Program for New Century Excellent Talents in University (NCET100238), 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).