Stability analysis of delayed genetic regulatory networks with stochastic disturbances

Physics Letters A 373 (2009) 3715–3723

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Physics Letters A www.elsevier.com/locate/pla

Stability analysis of delayed genetic regulatory networks with stochastic disturbances ✩ Qi Zhou a,∗ , Shengyuan Xu a , Bing Chen b , Hongyi Li c , Yuming Chu d a

School of Automation, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, PR China Institute of Complexity Science, Qingdao University, Qingdao 266071, Shandong, PR China c Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, PR China d Department of Mathematics, Huzhou Teacher’s College, Huzhou 313000, Zhejiang, PR China b

a r t i c l e

i n f o

Article history: Received 24 February 2009 Received in revised form 15 July 2009 Accepted 6 August 2009 Available online 19 August 2009 Communicated by A.R. Bishop PACS: 05.45.Xt 02.50.-r 02.30.Ks

a b s t r a c t This Letter considers the problem of stability analysis of a class of delayed genetic regulatory networks with stochastic disturbances. The delays are assumed to be time-varying and bounded. By utilizing Itô’s differential formula and Lyapunov–Krasovskii functionals, delay-range-dependent and rate-dependent (rate-independent) stability criteria are proposed in terms of linear matrices inequalities. An important feature of the proposed results is that all the stability conditions are dependent on the upper and lower bounds of the delays. Another important feature is that the obtained stability conditions are less conservative than certain existing ones in the literature due to introducing some appropriate freeweighting matrices. A simulation example is employed to illustrate the applicability and effectiveness of the proposed methods. © 2009 Elsevier B.V. All rights reserved.

Keywords: Genetic regulatory network Stochastic noise Time-varying delays Linear matrix inequality

1. Introduction During the past decades, significant progress in genome sequencing and gene recognition has accumulated a large volume of experimental data. How to use these data to understand gene functions is one of the main challenges. As a dynamic system, genetic networks can be used to describe cellular system models which contain a lot of different substances such as messenger ribonucleic acid (mRNA), proteins, and so on. So far, there have been two main genetic network models which are utilized in the research on the gene regulation processes in living organisms. The first one is the Boolean model (or discrete model), while the second one is differential equation model (or continuous model), see [1–4] and references therein. In the Boolean model, the activity of each gene is expressed in one of two states (that is, ON and OFF), and a Boolean function of the states of other related genes determines the state of a gene. In the differential equation model, the concentrations of gene products are described by the variables, such as mRNAs, proteins and continuous values of the gene regulation systems. Gene regulatory networks (GRNs) become a new area of research in the biological and biomedical sciences. Many significant results on GRNs are reported in the literatures [5–11]. These experiments, by extracting functional information from observation data, contribute a lot to the discovery of higher order structure of an organism and better understanding of both static and dynamic behaviors of genetic networks. On the other hand, due to the slow processes of transcription, translation, and translocation or the finite switching speed of amplifiers, time delays are inevitably caused. From [6], we know that the observed oscillatory expression and activity of three proteins may be driven by transcriptional delays. Moreover, the delays could generate some impact, not only on the dynamical behavior of models, but also on the numerical parameter prediction. Therefore, it is of both theoretical and practical importance to study the stability analysis of ✩ This work is supported by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20060288021, the Natural Science Foundation of Jiangsu Province under Grant BK2008047, and the National Natural Science Foundation of PR China under Grant 60850005. Corresponding author. E-mail address: [email protected] (Q. Zhou).

*

0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.08.036

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Q. Zhou et al. / Physics Letters A 373 (2009) 3715–3723

genetic networks with the simultaneous consideration of the effects of transcriptional delays. In [9], robust asymptotic stability issues were addressed for the GRNs with time-varying delays and norm bounded uncertainties, where sufficient conditions of robust stability were presented in terms of linear matrix inequalities (LMIs). When polytopic parameter uncertainties appear in a delayed genetic regulatory network, robust asymptotic stability can be found in [11]. Moreover, the gene regulation is an intrinsically noisy process; this is always subject to intracellular and extracellular noise perturbations, which are caused by the random births and deaths of individual molecules, along with extrinsic noise due to environment fluctuations [12,13]. Due to the fact that such cellular noises undoubtedly affect the dynamics of networks both quantitatively and qualitatively, it is important to investigate the stochastic genetic regulatory networks [8,14–16]. A model for GRNs with SUM regulatory functions was presented in [8], in which the problem of stability analysis was studied for a class of genetic networks with time-varying delays and stochastic perturbations, while in [14], the filtering problem of nonlinear genetic regulatory networks was investigated. This Letter deals with the problem of stability analysis of a class of delayed genetic regulatory networks with stochastic disturbances. The delays are assumed to be time-varying and bounded. By utilizing Itô’s differential formula and free-weighting matrix method [17], delay-range-dependent and rate-dependent (rate-independent) stability criteria are obtained, which are expressed by means of LMIs. A simulation example is given to demonstrate the applicability and effectiveness of the proposed methods. 1.1. Notation For convenience, we adopt the following notations. The notation X  Y (respectively X > Y ), where X and Y are symmetric matrices, means that the X − Y is positive-semidefinite (respectively, positive-definite). The superscript T represents the transpose; I is the identity matrix with appropriate dimension; ∗ is used as an ellipsis for terms that are induced by symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions. 2. Model and preliminaries In this Letter, based on the structure of the GRN presented in [8] and [9], we consider a functional differential equation model described by



˙ i (t ) = −ai mi (t ) + G i ( p 1 (t − τ (t )), p 2 (t − τ (t )), . . . , pn (t − τ (t ))), m p˙ i (t ) = −c i p i (t ) + di mi (t − σ (t )),

i = 1, 2, . . . , n ,

where mi (t ) and p i (t ) ∈ R denote the concentrations of mRNA and protein of the ith node, respectively. The bounded functions σ (t ) represent the translation delay and feedback regulation delay, respectively, and satisfy the following conditions:

(1)

τ (t ) and

τm  τ (t )  τM < ∞, τ˙ (t )  τμ < ∞,

(2)

σm  σ (t )  σM < ∞, σ˙ (t )  σμ < ∞.

(3)

The parameters ai and c i are the decay rates of mRNA and protein, respectively; di is the translation rate, and G i is the regulatory function of the ith gene, which is generally a nonlinear function of the variables (p 1 (t − τ (t )), p 2 (t − τ (t )), . . . , pn (t − τ (t ))) but has a form of monotonicity with each variable [2,4]. From (1), for any single node i or gene in the network, there is one output p i (t − τ (t )) to other nodes or genes and multiple inputs p j (t − τ (t )), j = 1, 2, . . . , n from other nodes. Being a monotonic increasing or decreasing regulatory function, n G i is usually of the Michaelis-Menten or Hill form. In this Letter, the function G i is taken as G i ( p 1 (t ), p 2 (t ), . . . , pn (t )) = j =1 G i j ( p j (t )), which is called SUM logic [6,18]. That is, each transcription factor acts additively to regulate the ith gene or node. G i j is a monotonic function of the Hill form. If transcription factor j is an activator of gene i, then





G i j p j (t ) = w i j

( p j (t )/β j ) H j ; 1 + ( p j (t )/β j ) H j

(4)

if transcription factor j is a repressor of gene i, then

Gij



 p j (t ) = w i j



1 1 + ( p j (t )/β j ) H j

= wij

 ( p j (t )/β j ) H j , 1− 1 + ( p j (t )/β j ) H j

(5)

where H j is the Hill coefficients, β j is a positive constant, and w i j is a bounded constant, which is the dimensionless transcriptional rate of transcription factor j to i. Based on (4) and (5), (1) can be rewritten in the following form:



˙ i (t ) = −ai mi (t ) + m

n

j =1 b i j g j ( p j (t

p˙ i (t ) = −c i p i (t ) + di mi (t − σ (t )), where g j =

( p j /β j )

Hj

1+( p j /β j )

Hj

, Wi =



j∈ I i

− τ (t ))) + W i , i = 1, 2, . . . , n ,

(6)

w i j and I i is the set of all the j which is a repressor of gene i, B = (b i j ) ∈ Rn×n is defined as follows:

⎧ if transcription factor j is an activator of gene i , ⎪ ⎨ wij, if there is no link from node j to i , b i j = 0, ⎪ ⎩ − w i j , if transcription factor j is an repressor of gene i .

(7)

Expressing system (6) in the matrix form, one has



˙ (t ) = − Am(t ) + B g ( p (t − τ (t ))) + W , m p˙ (t ) = −Cp (t ) + Dm(t − σ (t )),

(8)

Q. Zhou et al. / Physics Letters A 373 (2009) 3715–3723

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where

⎡

⎡

⎤

m1 (t ) ⎢ m2 (t ) ⎥ ⎢ ⎥

m(t ) = ⎢

⎣

p (t ) = ⎢

.. ⎥ , . ⎦ mn (t ) ⎡





⎣

⎤

m1 (t − σ (t )) ⎢ m2 (t − σ (t )) ⎥ ⎢ ⎥

A = diag(a1 , a2 , . . . , an ),



W = ⎢ . ⎥, .

⎣ . ⎦ Wn

⎡

 



g p t − τ (t )

⎥, .. ⎣ ⎦ . mn (t − σ (t ))

⎤

W1 ⎢ W2 ⎥ ⎢ ⎥

.. ⎥ , . ⎦ pn (t )

m t − σ (t ) = ⎢

Let m∗

⎡

⎤

p 1 (t ) ⎢ p 2 (t ) ⎥ ⎢ ⎥

=⎢ ⎣

C = diag(c 1 , c 2 , . . . , cn ),

= [m∗1 , m∗2 , . . . , mn∗ ] T and p ∗

⎤

g 1 ( p (t − τ (t ))) ⎢ g 2 ( p (t − τ (t ))) ⎥ ⎢ ⎥

⎥, .. ⎦ . gn ( p (t − τ (t ))) D = diag(d1 , d2 , . . . , dn ).

= [ p ∗1 , p ∗2 , . . . , pn∗ ] T be an equilibrium of Eq. (8), that is (m∗ , p ∗ ) is the solution of equation

− Am∗ + B g ( p ∗ ) + W = 0,

(9)

−Cp ∗ + Dm∗ = 0. By using the following transformation

x(t ) = m(t ) − m∗ ,

y (t ) = p (t ) − p ∗ ,

an intended equilibrium point (m∗ , p ∗ ) of system (8) can be shifted to the origin:



x˙ (t ) = − Ax(t ) + B f ( y (t − τ (t ))),

(10)

y˙ (t ) = −C y (t ) + Dx(t − σ (t )), where

⎡

⎡

⎤

x1 (t ) ⎢ x2 (t ) ⎥ ⎢ ⎥

x(t ) = ⎢

. ⎥, ⎣ .. ⎦ xn (t )

⎡

⎤

y 1 (t ) ⎢ y 2 (t ) ⎥ ⎢ ⎥



y (t ) = ⎢



x t − σ (t ) = ⎢

. ⎥, ⎣ .. ⎦ yn (t )

⎡

⎤

x1 (t − σ (t )) ⎢ x2 (t − σ (t )) ⎥ ⎢ ⎥

⎥, .. ⎣ ⎦ . xn (t ) − σ (t )

 



f y t − τ (t )

⎤

f 1 ( y (t − τ (t ))) ⎢ f 2 ( y (t − τ (t ))) ⎥ ⎢ ⎥

=⎢ ⎣

⎥, .. ⎦ . f n ( y (t − τ (t )))

with f i ( y (t )) = g i ( y i (t ) − p ∗i ) − g ( p ∗i ). Since g i is a monotonically increasing function with saturation, it satisfies, for all a, b ∈ R with a = b

0

g i (a) − g i (b) a−b

 ki .

From the relation between f i (·) and g i (·), we know that f i satisfies the sector condition

0

f i ( xi ) xi

or equivalently

 ki ,

∀ x i ∈ R,

xi = 0,

f i (0) = 0,

i = 1, 2, . . . , n





f T (x) f (x) − K x  0,

(11)

(12)

where K = diag{k1 , k2 , . . . , kn } > 0. Hence, the genetic networks (8) can be seen as a kind of Lur’e system, which can be investigated by using the fruitful Lur’e system method in control theory [19]. In this Letter, we consider the following time-delay genetic regulatory network with state-dependent stochastic disturbances:



dx(t ) = [− Ax(t ) + B f ( y (t − τ (t )))] dt + δ( y (t ), y (t − τ (t ))) dω(t ), d y (t ) = [−C y (t ) + Dx(t − σ (t ))] dt ,

(13)

where ω(t ) is a scalar Brownian motion with zero mean value and unit variance; δ( y (t ), y (t − τ (t ))) is called the noise intensity matrix. Here we only consider the noise perturbation from regulations (or inter-nodes perturbations). Since the nodes of genetic networks communicate via the variables y (t ) and y (t − τ (t )), we assume that the noise intensity matrix is a function of y (t ) and y (t − τ (t )), which act on the dynamics of x. As in many stochastic system studies, for example, [8] and [20], we assume







 





trace δ T y (t ), y t − τ (t ) δ y (t ), y t − τ (t )

     y T (t ) H 1 y (t ) + y T t − τ (t ) H 2 y t − τ (t ) ,

(14)

where H 1 and H 2 are known constant matrices with appropriate dimensions. 3. Main results In this section, we develop stability conditions for the model (13). By constructing an appropriate Lyapunov functional and introducing some appropriate free-weighting matrices, stability criteria are obtained in the following theorems.

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Q. Zhou et al. / Physics Letters A 373 (2009) 3715–3723

Theorem 1. Given scalars σm  0, σ M > 0, σμ , τm  0, τ M > 0 and τμ , the system (13) is asymptotically stable in mean square for any delays σ (t ) and τ (t ) satisfying (2)–(3) if there exist matrices P s , Q p > 0, R p > 0, Z 1 > 0, Z 2 > 0, M, N, S, Λ, T , Y and L (s = 1, 2, 3, p = 1, 2, 3, 4) with appropriate dimensions as well as positive scalars ρ > 0, κ1 > 0 and κ2 > 0 satisfying the following LMIs:



Ξ11 + Θ + Θ T ∗ P 1 − ρ I  0,



< 0,

(15) (16)

Z 1 − κ1 I  0 ,

(17)

Z 2 − κ2 I  0 ,

(18)



P1

∗

where

Ξ12 Ξ22

P2 P3



> 0,

(19)

⎤ 0 0 P1 0 0 0 P2 0 φ11 0 0 0 0 0 0 ⎥ ⎢ ∗ −Q 1 0 ⎥ ⎢ ∗ φ33 0 0 0 0 0 0 ⎥ ⎢ ∗ ⎥ ⎢ ∗ ∗ φ44 P 2 0 0 0 0 ⎥ ⎢ ∗ ⎥ ⎢ ∗ ∗ ∗ φ55 0 0 P3 0 ⎥, Ξ11 = ⎢ ∗ ⎥ ⎢ ∗ ∗ ∗ ∗ −Q 3 0 0 0 ⎥ ⎢ ∗ ⎥ ⎢ ∗ ∗ ∗ ∗ ∗ ∗ φ77 0 KΛ ⎥ ⎢ ⎣ ∗ 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ φ88 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −2Λ  √ √ √ √ σM M σM − σm N τM S τM − τm T M N , Ξ12 = ⎡

Ξ22 = diag{ − R 1 − R 2 − R 3 − R 4 − Z 1 − Z 2 }, Θ = [ M −M − N

N

0

S

−S − T

T

0 0]

+ L [ − A 0 0 − I 0 0 0 0 B ] + Y [ 0 0 D 0 −C Λ = diag[ λ1 λ2 · · · λn ],

φ11 = Q 1 + Q 2 ,

0 0

− I 0 ],

φ33 = −(1 − σμ ) Q 2 ,

 φ55 = Q 3 + Q 4 + ρ + σ M κ1 + (σ M − σm )κ2 H 1 , φ44 = σ M R 1 + (σ M − σm ) R 2 ,   φ88 = τ M R 3 + (τ M − τm ) R 4 . φ77 = −(1 − τμ ) Q 4 + ρ + σ M κ1 + (σ M − σm )κ2 H 2 , 

Proof. Define new state variables,

 



g 1 (t ) = − Ax(t ) + B f y t − τ (t ) ,

(20)





 g 2 (t ) = δ y (t ), y t − τ (t ) ,   g 3 (t ) = −C y (t ) + Dx t − σ (t ) ,

(21) (22)

then, the system equations (13) can be represented as

dx(t ) = g 1 (t ) dt + g 2 (t )dω(t ),

(23)

d y (t ) = g 3 (t ) dt .

(24)

Therefore, the following equations can be derived,



t

η1 (t ) = 2e (t ) M x(t ) − x(t − σM ) − 

g 1 (s) ds −

t −σ M





t− σ (t )

g 1 (s) ds −

 η4 (t ) = 2e (t ) T y t − τ (t ) − y (t − τM ) −





= 0,

(26)



g 3 (s) ds t −τ M

(27)



t− τ (t )



η5 (t ) = 2e T (t ) L − Ax(t ) + B f y t − τ (t ) − g1 (t ) = 0,   η6 (t ) = 2e (t )Y −C y (t ) + Dx t − σ (t ) − g3 (t ) = 0, T



g 2 ( s ) dω ( s )

= 0,

g 3 (s) ds



 

(25)

t −σ M

t −τ M



= 0,



t

η3 (t ) = 2e (t ) S y (t ) − y (t − τM ) − 

t− σ (t )

t −σ M

T

T

g 2 ( s ) dω ( s )

t −σ M

 η2 (t ) = 2e (t ) N x t − σ (t ) − x(t − σM ) − T



t

T

= 0,

(28) (29) (30)

Q. Zhou et al. / Physics Letters A 373 (2009) 3715–3723

3719

where



M = M 1T

M 2T



N = N 1T



L=

N 2T

M 3T

M 4T

N 3T

N 4T

N 5T

N 6T

M 8T

N 7T

N 8T

L 8T

L 9T

L 3T

S = S 1T

S 2T

S 3T

S 4T

T=

T 1T

T 2T

T 3T

T 4T

T 5T

T 6T

T 7T

T 8T

Y = Y 1T

Y 2T

Y 3T

Y 4T

Y 5T

Y 6T

Y 7T

Y 8T



L 6T

M 7T

L 2T



L 5T

M 6T

L 1T



L 4T

M 5T

S 5T

L 7T

S 6T

S 8T

S 9T

N 9T

T

T

,

,

, T

T 9T

, T

,  T , Y 9T

      e T (t ) = x T (t ), x T (t − σ M ), x t − σ (t ) , g 1T (t ), y T (t ), y T (t − τ M ), y T t − τ (t ) , g 3T (t ), f T y t − τ (t ) , 

 T

S 7T

T

M 9T

with M, N, S, T , L and Y are constant matrices with appropriate dimensions. For the sector condition (12), the following inequality holds

−2

n 

        λi f i y i t − τ (t ) f i y i t − τ (t ) − ki y i t − τ (t )  0,

i =1

that is

 

 



 









η7 (t ) = −2 f T y t − τ (t ) Λ f y t − τ (t ) + 2 f T y t − τ (t ) Λ K y t − τ (t )  0,

(31)

where Λ = diag[λ1 , λ2 , . . . , λn ]. Choose Lyapunov–Krasovskii functional candidate as follows:

V (t ) = V 1 (t ) + V 2 (t ) + V 3 (t ) + V 4 (t ), where

  P1 V 1 (t ) = x (t ) y (t ) ∗ 

T

P2 P3

T

t





x(t ) , y (t )

t

t

T

V 2 (t ) =

(32)

x (s) Q 1 x(s) ds +

t −σ M

g 1T (s) R 2 g 1 (s) ds dθ +

−σ M t +θ



trace g 2T (s) Z 1 g 2 (s) ds dθ +

−σ M t +θ

g 3T (s) R 3 g 3 (s) ds dθ +

−τ M t +θ − σm  t



−  τm  t

0  t

g 1T (s) R 1 g 1 (s) ds dθ +

V 4 (t ) =

t −τ (t )

− σm  t

−σ M t +θ

y T (s) Q 4 y (s) ds,

y (s) Q 3 y (s) ds +

t −τ M

0  t

0  t

T

x (s) Q 2 x(s) ds +

t −σ (t )

V 3 (t ) =

t

T



g 3T (s) R 4 g 3 (s) ds dθ,

−τ M t +θ



trace g 2T (s) Z 2 g 2 (s) ds dθ.

(33)

−σ M t +θ

Let L be the infinitesimal generator of the systems (23)–(24), then, by Itô’s formula, we have





dV (t ) = L V (t ) dt + 2 x T (t ) P 1 + y T (t ) P 2T g 2 (t ) dω(t ), where

L V (t ) = 2xT (t ) P 1 g 1 (t ) + 2 y T (t ) P 2T g 1 (t ) + 2xT (t ) P 2 g 3 (t ) + 2 y T (t ) P 3 g 3 (t )   + trace g2T (t ) P 1 g 2 (t ) + xT (t )( Q 1 + Q 2 )x(t ) − xT (t − σ M ) Q 1 x(t − σ M )       − 1 − σ˙ (t ) xT t − σ (t ) Q 2 x t − σ (t ) + y T (t )( Q 3 + Q 4 ) y (t )       − y T (t − τ M ) Q 3 y (t − τ M ) − 1 − τ˙ (t ) y T t − τ (t ) Q 4 y t − τ (t )     + g1T (t ) σ M R 1 + (σ M − σm ) R 2 g 1 (t ) + g3T (t ) τ M R 3 + (τ M − τm ) R 4 g 3 (t )       + trace g2T (t )(σ M Z 1 ) g 2 (t ) + trace g2T (t ) (σ M − σm ) Z 2 g 2 (t ) t −σm

t g 1T (s) R 1 g 1 (s) ds

− t −σ M

t −σ M

t − t −σ M

−

trace





g 2T (s) Z 1 g 2 (s)

t −τm

t g 1T (s) R 2 g 1 (s) ds

−

g 3T (s) R 3 g 3 (s) ds

t −τ M t −σm

ds −



−

g 3T (s) R 4 g 3 (s) ds

t −τ M



trace g 2T (s) Z 2 g 2 (s) ds

t −σ M

+ η1 (t ) + η2 (t ) + η3 (t ) + η4 (t ) + η5 (t ) + η6 (t ) + η7 (t ).

(34)

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According to Lemma 1 in [21], for any appropriately dimensioned matrices Z 1 > 0 and Z 2 > 0, we have

t

g 2 (s) dω(s)  e T (t ) M Z 1−1 M T e (t ) + 1T Z 1 1 ,

T

−2e (t ) M

(35)

t −σ M t− σ (t )

g 2 (s) dω(s)  e T (t ) N Z 2−1 N T e (t ) + 2T Z 2 2 ,

T

−2e (t ) N

(36)

t −σ M

where t− σ (t )

t 1 =

2 =

g 2 (s) dω(s),

t −σ M

g 2 (s) dω(s). t −σ M

Then, by (16)–(18) and (34)–(36), it can be seen that

  L V (t )  e T (t ) Ξ11 + Θ + Θ T + Ω e (t ) t



+ 1T Z 1 1 −

t− σ (t )



trace g 2T (s) Z 1 g 2 (s) ds + 2T Z 2 2 −

t −σ M

t −



T

e (t ) M +

−



t −σ M

g 1T (s) R 1



−1 

R1

T

M e (t ) +

R 1T

 g 1 (s) ds −

t −σ M

t



trace g 2T (s) Z 2 g 2 (s) ds t− σ (t )









1 e T (t ) N + g 1T (s) R 2 R − N T e (t ) + R 2T g 1 (s) ds 2

t −σ M









1 e T (t ) S + g 3T (s) R 3 R − S T e (t ) + R 3T g 3 (s) ds − 3

t −τ M

t− τ (t )









1 e T (t ) T + g 3T (s) R 4 R − T T e (t ) + R 4T g 3 (s) ds, 4

(37)

t −τ M

where 1 T −1 T −1 T −1 T −1 T −1 T Ω = σM M R − 1 M + (σ M − σm ) N R 2 N + τ M S R 3 S + (τ M − τm ) T R 4 T + M Z 1 M + N Z 2 N .

Applying the arguments in [20], we have

E



1T

t



Z 1 1 =

trace





g 2T (s) Z 1 g 2 (s)

t −σ M

ds,

E



2T



t− σ (t )





trace g 2T (s) Z 2 g 2 (s) ds.

Z 2 2 = t −σ M

Since R p > 0 (p = 1, 2, 3, 4), the last four parts in (37) are all nonpositive. Thus, by applying Schur Complement,

Ξ11 + Θ + Θ T + Ω < 0 is equivalent to (15). After taking mathematical expectation, one has EL V (t ) < 0, which indicates the system (13) is asymptotically stable in mean square. This completes the proof. 2 Remark 1. In [8,15], delay-independent stability conditions were obtained for the delays τ (t ) and σ (t ) satisfying τ˙ (t )  τμ < 1 and σ˙ (t )  σμ < 1, however, in Theorem 1, the delays σ (t ) and τ (t ) satisfy τ˙ (t )  τμ < ∞ and σ˙ (t )  σμ < ∞, respectively; they are more general than the delays considered in [8,15]. When the differential of

σ (t ) and τ (t ) are unknown or not continuously differentiable and the delays σ (t ) and τ (t ) satisfy

τm  τ (t )  τM < ∞,

(38)

σm  σ (t )  σM < ∞.

(39)

By setting Q 2 = 0 and Q 4 = 0 in (32), from the proof of Theorem 1, we can obtain the delay-range-dependent and rate-independent stability criterion for the system (13) in the following theorem. Theorem 2. Given scalars σm  0, σ M > 0, τm  0 and τ M > 0, the system (13) is asymptotically stable in mean square for any delays σ (t ) and τ (t ) satisfying (38)–(39) if there exist matrices P s , Q 1 > 0, Q 3 > 0, R p > 0, Z 1 > 0, Z 2 > 0, M, N, S, Λ, T , Y and L (s = 1, 2, 3, p = 1, 2, 3, 4) with appropriate dimensions as well as positive scalars ρ > 0, κ1 > 0 and κ2 > 0 satisfying the following LMIs:



Ξˆ 11 + Θ + Θ T ∗ P 1 − ρ I  0, Z 1 − κ1 I  0 ,

Ξ12 Ξ22



< 0,

(40) (41) (42)

Q. Zhou et al. / Physics Letters A 373 (2009) 3715–3723

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Fig. 1. A genetic network model (↑: activation; : repression).

Z 2 − κ2 I  0 ,



P1

∗

P2 P3

(43)



> 0,

(44)

where

⎡

0 φˆ 11 ⎢ ∗ −Q 1 ⎢ ⎢ ∗ ∗ ⎢ ⎢ ∗ ∗ ⎢ Ξˆ 11 = ⎢ ∗ ∗ ⎢ ⎢ ∗ ∗ ⎢ ⎢ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗

0 0 0

P1 0 0

0 0 0 P2

0 0 0 0 0 −Q 3

0 0 0 0 0 0

P2 0 0 0 P3 0 0

⎤

0 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥, 0 ⎥ ⎥ KΛ ⎥ ⎥ 0 ⎦ −2Λ

∗ φ44 ∗ ∗ φˆ 55 ∗ ∗ ∗ ∗ ∗ ∗ ∗ φˆ 77 ∗ ∗ ∗ ∗ ∗ φ88 ∗ ∗ ∗ ∗ ∗ ∗   ˆ φ55 = Q 3 + ρ + σ M κ1 + (σ M − σm )κ2 H 1 ,

φˆ 11 = Q 1 ,

  φˆ 77 = ρ + σ M κ1 + (σ M − σm )κ2 H 2 ,

where Ξ11 , Ξ22 , Θ , φ44 , φ88 and Λ have been defined in Theorem 1. Remark 2. By constructing proper Lyapunov–Krasovskii functionals and using LMI method, sufficient conditions for delayed GRN are given in Theorems 1 and 2 to guarantee the asymptotic stability in mean square. It can be seen that these stability criteria are formulated by no model transformation. By introducing appropriate free-weighting matrices, we develop less conservative stability results for the delayed GRN. 4. Numerical example To illustrate the effectiveness of our results, we consider the following delayed genetic network with stochastic noise, in which there are five nodes representing regulation factors and some lines represent regulation links (see Fig. 1),



dx(t ) = [− Ax(t ) + B f ( y (t − τ (t )))] dt + δ( y (t ), y (t − τ (t ))) dω(t ), d y (t ) = [−C y (t ) + Dx(t − σ (t ))] dt ,

(45)

where

⎡ A = diag[1, 1, 1, 1, 1],

−α −α

0 ⎢ 0

α α α α

⎢

B = ⎢ −α ⎣ 0

0 0

0

−α

⎤

0 0⎥

α⎥ ⎥, α⎦ α

C = diag[1, 1, 1, 1, 1], D = diag[1, 1, 1, 1, 1], −α −α −α  T          trace δ y (t ), y t − τ (t ) δ y (t ), y t − τ (t )  y T (t ) H 1 y (t ) + y T t − τ (t ) H 2 y t − τ (t ) , 0 0 0

H 1 = H 2 = 2I , where f (x) = x2 /(x2 + 1), α = 0.1. It is easy to know that the maximal value of the derivative of f (x) is k = 0.65. The time delays and σ (t ) are assumed to be







τ (t ) = 1 + 2sin(t ) /10,





τ (t )



σ (t ) = 1 + sin(t ) /5.

Therefore, we can get the parameters as follows:

τm = 0.1,

τM = 0.3,

σm = 0.2,

σM = 0.4.

It should be pointed out that, the delay-independent conditions in [8] and [15] are not feasible due to the fact that σ (t ) and τ (t ) are not continuously differentiable. However, it can be checked that the LMIs in Theorem 2 are feasible. Therefore, system (45) is asymptotically stable in mean square. Figs. 2 and 3 show the trajectories of variable xi (t ) and y i (t ) (i = 1, 2, . . . , 5) with the same initial condition [0.8, 0.7, 0.6, 0.5, 0.4] T , while xi (t ) denotes mRNA concentrations and y i (t ) (i = 1, 2, . . . , 5) stands for protein concentrations.

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Fig. 2. Transient response of x1 (t ), x2 (t ), x3 (t ), x4 (t ) and x5 (t ).

Fig. 3. Transient response of y 1 (t ), y 2 (t ), y 3 (t ), y 4 (t ) and y 5 (t ).

5. Conclusions In this Letter, the problem of asymptotic stability in mean square has been considered for delayed GRNs with stochastic perturbation. Based on the free-weighting matrix method and the LMI method, stability conditions have been developed in terms of LMIs. A simulation example has been provided to illustrate the effectiveness of the proposed methods. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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