Elsevier

Neurocomputing

Volume 73, Issues 16–18, October 2010, Pages 2882-2892
Neurocomputing

Stability and bifurcation of genetic regulatory networks with delays

https://doi.org/10.1016/j.neucom.2010.08.009Get rights and content

Abstract

A four-dimension genetic regulatory network model with double genes and four delays is considered. The existence of Hopf bifurcation is first considered and then explicit formulas are derived by using the normal form method and center manifold theory to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Finally, numerical simulations are carried out to support the theoretical analysis of the research.

Introduction

The genetic regulatory networks (GRNs) are attracting more and more attention from biology, engineering and other research fields over past few years. Genetic regulatory networks are structured by networks of regulatory interactions between DNA, RNA, proteins inhibiting the expression of other genes for gaining insight into the underlying processes of living systems at the molecular level. Recent studies on genetic regulatory networks are fruitful, and many important results have been reported in the literature [1], [2], [5], [6], [13], [14], [24].

Generally speaking, so far, two types of genetic network models have been proposed and utilized [3], [4], [10], [11], [18]: Boolean model (discrete model) and differential equation model (continuous model). In the discrete model, only two states (that is, ON and OFF) are used to express the activity of each gene, and the state of a gene is determined by a Boolean function of the states of other related genes. In the continuous model, usually a differential equation is utilized to describe the whole network, with the state variables describing the concentrations of gene products, such as mRNAs and proteins.

When modelling the genetic networks, time delay is an important factor, due to slow biochemical reactions such as gene transcription and translation, which should be considered. Recently, much work has been done on the stability problem of genetic regulatory networks with time delays [9], [12], [15], [19], [21], [22]. In [9], the authors discussed robust stability of genetic regulatory networks with distributed delay. Different from other papers, distributed delay is induced. Based on Lyapunov stability theory and linear matrix inequality (LMI), sufficient conditions for genetic regulatory networks to be global asymptotic stability and robust stability are derived in terms of LMI. In [22], the authors investigated the robust asymptotic stability problem of genetic regulatory networks with time-varying delays and polytopic parameter uncertainties. Both cases of differentiable and nondifferentiable time-delays are considered, and the convex polytopic description is utilized to characterize the genetic network model uncertainties. By using a Lyapunov functional approach and linear matrix inequality (LMI) techniques, the stability criteria for the uncertain delayed genetic networks are established in the form of LMIs. In [15], the authors presented a robust analysis approach to asymptotic stability of the delayed genetic regulatory networks (GRNs) with SUM regulatory logic in which each transcription factor acts additively to regulate a gene, i.e., the regulatory function sums over all the inputs. Based on the Lyapunov stability theory and linear matrix inequality (LMI), sufficient conditions are given to ensure the stability of the GRNs. In [21], the robust exponential stability problem is considered for a class of stochastic genetic networks with uncertain parameters. Sufficient conditions are derived to guarantee the robust exponential stability in the mean square of stochastic genetic networks for all admissible parameter uncertainties.

We consider the GRNs described by the following differential equations [3], [15], [18]: M˙i(t)=aiMi(t)+gi(P1(tτ),P2(tτ),,Pn(tτ))P˙i(t)=ciPi(t)+diMi(tτ),i=1,2,,n,where Mi(t) and Pi(t) denote the concentration of mRNA and protein of the ith node at time t, respectively. ai and ci are positive real numbers that present the degradation rates of mRNA and protein, respectively. di is the translation rate. τ is the node delay, and the functions gi denotes the feedback regulation of the protein on the transcription, which is generally nonlinear function but has a form of monotonicity of each variable. In most existed paper, function gi is of SUM logic form [3], [9], [15], which is gi=j=1nGij(Pj(t)). The functions Gij(Pj(t)) are usually expressed by the Hill formGij(Pj(t))=αij(Pj(t)/βj)Hj1+(Pj(t)/βj)Hjif transcription factorjis an activator of genei,αij11+(Pj(t)/βj)Hjif transcription factorjis a repressor of genei,where Hj is the Hill coefficients, bj is positive constants, αij are bounded constants, while are the dimensionless transcription rate of transcription factor j to i. From this, one can rewrite (1.1) as follows: M˙i(t)=aiMi(t)+j=1nbijfj(Pj(tτ))+BiP˙i(t)=ciPi(t)+diMi(tτ),i=1,2,,n,where fi(xj)=(xj/βj)Hj/(1+(xj/βj)Hj), Bi=Iiαij and Ii is the set of all the j which is a repressor of gene i, bij are defined as follows:bij=αijif transcription factorjis an activator of genei,0if there is no link fromjtoi,αijif transcription factorjis a repressor of genei.

As is well known, the studies on GRNs not only involve a discussion of stability properties, but also involve other dynamic behavior, such as periodic oscillatory behavior, chaos and bifurcation. In many applications, the periodic oscillatory behavior is of great interest. In differential equations with delays, periodic oscillatory behavior can arise through the Hopf bifurcation. The study of bifurcations on GRNs is quite important. On the one hand, the bifurcations, which involve emergence of oscillatory behaviors, may provide an explanation for the parameter sensitivity observed in practice in many realistic genetic regulatory networks; and on the other hand, if we understand more about the bifurcation behaviors of various genetic regulatory networks, we can apply the existing effective bifurcation control methods to achieve some desirable system behaviors that benefit the networks [17], [20], [23]. Motivated by the above discussions, in this paper, a four-dimension genetic regulatory network model with double genes and four delay has been considered, and it is a more general model. The existence of Hopf bifurcation is first considered and then explicit formulas are derived by using the normal form method and center manifold theory to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Finally, numerical simulations are carried out to support the theoretical analysis of the research.

This paper is organized as follows. In Section 2, the local stability is discussed, and some sufficient conditions for the existence of Hopf bifurcation are derived. In Section 3, a formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions will be given by using the normal form method and center manifold theorem introduced by [8]. Numerical simulations are given in Section 4.

Section snippets

Existence of Hopf bifurcation

We consider the following double genes regulatory genetic networks with four delays M˙1(t)=a1M1(t)+b11f1(P1(tσ3))+b12f2(P2(tσ3))+B1M˙2(t)=a2M2(t)+b21f1(P1(tσ4))+b22f2(P2(tσ4))+B2P˙1(t)=c1P1(t)+d1M1(tσ1)P˙2(t)=c2P2(t)+d2M2(tσ2).We assume that σ1+σ3=σ2+σ4 and denote τ=σ1+σ3. Let u1(t)=M1(tσ1), u2(t)=M2(tσ2), u3(t)=P1(t) and u4(t)=P2(t), then system (2.1) is equivalent to the following system: u˙1(t)=a1u1(t)+b11f1(u3(tτ))+b12f2(u4(tτ))+B1u˙2(t)=a2u2(t)+b21f1(u3(tτ))+b22f2(u4(tτ))+

Direction and stability of the bifurcation

In this section, we will study the direction of Hopf bifurcation and stability of bifurcating periodic solutions by employing the normal form method and center manifold theorem introduced by Hassard [8].

For convenience, let t=sτ, xi(t)=ui(tτ), (i=1,2,3,4) and τ=τ0+μ, where τ0 is defined in (2.13), μR. Then we can write (2.2) in C([−1,0],R4) as follows:x˙(t)=Lμ(xt)+F(μ,xt),where C([1,0],R4)={φ:[1,0]R4 with φ is continuous on [−1,0]}, xt(θ)=x(t+θ)C, Lμ:CR and F:R×CR are given,

Numerical example

In this section, we carry out numerical simulations on system (2.1) to demonstrate the Hopf bifurcation results in Sections 2 and 3 by using the Matlab software. In particular, we consider a particular case of system (2.1) in the following form: M˙1(t)=M1(t)+2.4P12(tσ3)1+P12(tσ3)3.35P22(tσ3)1+P22(tσ3)+2M˙2(t)=1.3M2(t)+2.6P12(tσ4)1+P12(tσ4)+1.5P22(tσ4)1+P22(tσ4)+1P˙1(t)=2.2P1(t)+3M1(tσ1)P˙2(t)=3.1P2(t)+2M2(tσ2),we assume that σ1+σ3=σ2+σ4 and denote τ=σ1+σ3. Let u1(t)=M1(tσ1), u2(t

Conclusion

In this paper, a four-dimension genetic regulatory network model with double genes and four delays is considered. Hopf bifurcation analysis on genetic regulatory networks is a useful approaches that can provide much information about periodic solutions near a destabilized steady state, in terms of the system's parameters. By choosing the delay as a bifurcation parameter and analyzing the associated characteristic equation, we obtained the condition which can assure the local stability and

Kai Wang was born in Xinjiang, China, in 1982. He received the B.S. degree and M.S. degree in Mathematics from Xinjiang University, Xinjiang, China, in 2005 and 2008, respectively. Now, he is working toward the Ph.D. degree in Mathematics from Xinjiang University, Urumqi, China. His current research interests include stability theory, neural networks, complex networks, mathematical biology and epidemiology.

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Kai Wang was born in Xinjiang, China, in 1982. He received the B.S. degree and M.S. degree in Mathematics from Xinjiang University, Xinjiang, China, in 2005 and 2008, respectively. Now, he is working toward the Ph.D. degree in Mathematics from Xinjiang University, Urumqi, China. His current research interests include stability theory, neural networks, complex networks, mathematical biology and epidemiology.

Lei Wang was born in Xinjiang, China, in 1981. She received the B.S. degree and M.S. degree from the Department of Mathematics, Xinjiang Normal University, Urumqi, China, in 2003 and 2006, respectively. She is currently working toward the Ph.D. degree in College of Mathematics and System Sciences at Xinjiang University. Her research interests are neural networks, mathematical biology and epidemiology.

Zhidong Teng was born in Xinjiang, China, in 1960. He received the B.S. degree in Mathematics from Xinjiang University, Xinjiang, China, in 1982, and the Ph.D. degree in Applied Mathematics and Mechanics from Kazakhstan State University, Kazakhstan, in 1995.

He is a professor and Doctoral Advisor of Mathematics and System Sciences of Xinjiang University, Xinjiang, China. His current research interests include nonlinear dynamics, delay differential equations, dynamics of neural networks, mathematical biology, and epidemiology.

Haijun Jiang was born in Hunan, China, in 1968. He received the B.S. degree from the Mathematics Department, Yili Teacher College, Yili, Xinjiang, China; the M.S. degree from the Mathematics Department, East China Normal University, Shanghai, China, and the Ph.D. degree from the College of Mathematics and System Sciences, Xinjiang University, China, in 1990, 1994, and 2004, respectively. He was a Post doctoral Research Fellow in the Department of Southeast University, Nanjing, China, from 2004.9 to 2006.9.

He is a professor and Doctoral Advisor of Mathematics and System Sciences of Xinjiang University, Xinjiang, China. His present research interests include nonlinear systems, mathematical biology, epidemiology, and dynamics of neural networks.

Supported by The National Natural Science Foundation of P.R. China [10961022,60764003,10901130], The Natural Science Foundation of Xinjiang [2010211A07], The Scientific Research Programmes of Colleges in Xinjiang [XJEDU2007G01, XJEDU2008S10, XJEDU2009S21], Xinjiang Medical University Scientific Research and Innovation Foundation [2007-18].

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