Robust stochastic stability analysis of genetic regulatory networks with disturbance attenuation

Neurocomputing 79 (2012) 39–49

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Robust stochastic stability analysis of genetic regulatory networks with disturbance attenuation Yonghui Sun a,b,n, Gang Feng b, Jinde Cao c a

College of Energy and Electrical Engineering, Hohai University, Nanjing 210098, China Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong, China c Department of Mathematics, Southeast University, Nanjing 210096, China b

a r t i c l e i n f o

abstract

Article history: Received 25 October 2009 Received in revised form 10 September 2011 Accepted 19 September 2011 Communicated by S. Hu Available online 11 November 2011

This paper studies the robust stochastic stability of uncertain stochastic genetic regulatory networks with disturbance attenuation. A novel delay-dependent robust stability condition with disturbance attenuation, in the form of linear matrix inequalities (LMIs) is derived for the uncertain stochastic genetic networks with time-varying delays and intrinsic and extrinsic noises. These stability conditions can be tested efficiently by the available commercial software packages such as Matlab LMI Control Toolbox. Two numerical examples with simulations are given to illustrate the effectiveness and validity of the derived theoretical results. & 2011 Elsevier B.V. All rights reserved.

Keywords: Robust stochastic stability Genetic regulatory networks Disturbance attenuation Extrinsic noises

1. Introduction In the post-genomic era, one of the main challenges is to understand and study the gene functions, for example, how genes and proteins interact to form a complex network that performs complicated biological functions in living organisms. Genetic regulatory networks (GRNs) have thus attracted considerable attention from researchers in various fields recently. Scientists have tried to employ variety of methods to find, understand and predict the gene regulations. One of the popular methods is to study modeling and simulation of the genetic regulatory networks, see [1–5]. Some of genetic regulatory network models have been successfully constructed recently, and it is shown that the dynamical model can be an effective method to predict the regulations of genes and proteins. Basically, there are two types of genetic network models, the Boolean model [1,2], where the activity of each gene has two states, and the differential equation model, where the variables describe the concentrations of gene products, such as mRNAs and proteins [6–9]. In this paper, we will focus on the differential equation model of genetic regulatory networks with special regulation logics. From many experiment results, it is understood that time delays are inevitable due to the transcription, translation, diffusion, and

n

Corresponding author. Tel./fax: þ 86 25 58099077. E-mail address: [email protected] (Y. Sun).

0925-2312/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2011.09.023

translocation processes of genes, which will affect the entire dynamics of the biochemical systems [10,11]. Hence, time delays should be taken into account when modeling the biochemical systems. In [10], the authors employed a reduced model to show the role of feedback loops and time delays in the Drosophila circadian oscillator. Chen and Aihara [12] presented a functional differential equation model for genetic regulatory networks with time delays, then analyzed its local stability and bifurcation. Genetic regulatory networks model with the SUM logic was presented in [13], some stability conditions in the form of LMIs were derived for the genetic regulatory networks with time-varying delays. In [14], Ren and Cao studied the robust stability of genetic networks with time-varying delays. It is worth noting that molecular events in cells are subject to significant thermal fluctuations and noisy process with transcriptional control, alternative splicing, translation, diffusion and chemical modification reaction, thus gene expression is best viewed as a stochastic process [15–19]. Stochastic dynamic models are the ideal tools for the investigations of gene networks [20–22]. Chen et al. [23] revealed cooperative behaviors in a general coupled noisy system with time delays. Robust stability of stochastic interval genetic networks was studied by Wang et al. [24]. In [25], the authors studied the stochastic stability with disturbance attenuation for stochastic genetic regulatory networks. Chen and Wang [26] investigated the attenuation of molecular noises in genetic regulatory networks. By using the fuzzy interpolation approach, the stabilization problem for stochastic gene networks was considered in [27], where the time delay and specific regulation function have been ignored.

40

Y. Sun et al. / Neurocomputing 79 (2012) 39–49

Furthermore, it is generally believed that robustness is a fundamental property of biological systems [31,32], cellular functioning is robust to a broad range of perturbations [33]. As pointed in [33], systemic diseases such as cancer and diabetes can be classified as failures in the robustness mechanisms of biological systems, which has also been named as genetic uncertainties consisting of mutations [34–36]. Thus, when modeling and analyzing genetic regulatory networks, it is desirable to take into account these uncertainties and extrinsic disturbances from system viewpoint [37–39], which is an important and promising way to quantify the robustness of the genetic regulatory networks. To the best of our knowledge, when the parameter uncertainties as well as extrinsic noises appear simultaneously in stochastic genetic regulatory networks, the problem of the robust stochastic stability of the uncertain genetic regulatory networks with disturbance attenuation have not been fully investigated. Motivated by the above discussions, we will investigate robust stochastic stability of uncertain genetic regulatory networks with intrinsic and extrinsic noises. A novel delay-dependent robust stability condition will be derived, and how to find the optimal attenuation level of extrinsic noises for genetic regulatory networks is also discussed. These results and observations will help understand the process of gene regulations. The rest of this paper is organized as follows. In Section 2, the uncertain stochastic genetic regulatory networks model with extrinsic noises and parameter perturbations is introduced, some necessary assumptions and a lemma which will be used in the proof of the main results are also given. In Section 3, delaydependent robust stochastic stability conditions are derived, and some comparisons with the existing results are presented as well. In Section 4, two numerical examples with simulations are given to illustrate the effectiveness of the obtained results. At last, this paper is completed with a conclusion and some discussions. In this study the following notations stand. For any matrix A, A 40 means that A is symmetric positive definite. Efg stands for the mathematical expectation operator. The superscript ‘T’ represents the transpose of the matrix. JxJ is used to denote a vector P 2 norm defined by JxJ ¼ ð ni¼ 1 9xi 9 Þ1=2 . I is the identity matrix and oðtÞ is an one-dimensional Brownian motion defined on the probability space.

2. Genetic network model and preliminaries In a genetic network, lots of genes interact and regulate the expression of other genes by proteins and the gene derivatives. The change in expression of a gene is controlled by the stimulation and inhibition of proteins in the process of transcription, translation and post-translation. In this paper, we start from the following genetic networks model [12,43]: ( _ i ðtÞ ¼ ai mi ðtÞ þ b i ðp1 ðtÞ,p2 ðtÞ, . . . ,pn ðtÞÞ, m ð1Þ p_ i ðtÞ ¼ ci pi ðtÞ þdi mi ðtÞ, i ¼ 1; 2, . . . ,n, where mi ðtÞ,pi ðtÞ A R are the concentrations of mRNA and protein of the ith node, respectively. The parameters ai and ci are the degradation rates of mRNA and protein, respectively; di is the translation rate, and b i represent the feedback regulation of the protein on the transcription, which is generally a nonlinear function but has a form of monotonicity with its variables [2,6]. Generally, the form of b i may be very complicated, depending on all biochemical reactions involved in this regulation. Here for convenience, we only consider the SUM logic [4] representing the case that each transcription factor acts additively to regulate the

ith gene, which is considered in [13,14,24]. That is b i ðp1 ðtÞ,p2 ðtÞ, . . . ,pn ðtÞÞ ¼

n X

b ij ðpj ðtÞÞ:

ð2Þ

j¼1

Without loss of generality, following Hill form 8  hj pj ðtÞ > > > aj > > > b  hj if > ij > p ðtÞ > > 1 þ aj j > > > < b ij ðpj ðtÞÞ ¼ > > 1 > > bij >  hj if > > p ðtÞ > > 1 þ aj j > > > > :

the function is expressed by the

transcription factor j is an activator of gene i, transcription factor j is an repressor of gene i,

where hj is the Hill coefficient and aj is a positive constant, and bij is a bounded constant, which is the dimensionless transcriptional rate of transcription factor j to i. Note that 

 pj ðtÞ hj aj

1þ



¼ 1

 pj ðtÞ hj

1þ

aj

1  h j , pj ðtÞ

aj

hence we can rewrite (1) as follows: 8 n X > > > : p_ ðtÞ ¼ c p ðtÞ þd m ðtÞ, i i i i i

ð3Þ

i ¼ 1; 2, . . . ,n,

where  hj x

f j ðxÞ ¼

aj

 hj 1 þ axj

P is a monotonically increasing function. li ¼ j A Ii bij and Ii is the set of all j nodes which are repressors of gene i. B ¼ ðbij Þ A Rnn is defined as follows: 8 bij if transcription factor j > > > > > is an activator of gene i, > < 0 if there is no link from node j to i, bij ¼ > > > b if transcription factor j  > ij > > : is an repressor of gene i: Taking into account the transcriptional time delay, we have the following genetic regulatory networks model with SUM regulatory logic [13,14,25]: 8 n X > > > : p_ ðtÞ ¼ c p ðtÞ þd m ðttðtÞÞ, i ¼ 1; 2, . . . ,n, i i i i i where tðtÞ and sðtÞ are time-varying delays satisfying 0 r tðtÞ r t and 0 r sðtÞ r s, respectively. Rewrite model (4) into the compact form ( _ mðtÞ ¼ AmðtÞ þ Bf ðpðtsðtÞÞÞþ L, ð5Þ _ ¼ CpðtÞ þ DmðttðtÞÞ, pðtÞ where mðtÞ ¼ ½m1 ðtÞ,m2 ðtÞ, . . . ,mn ðtÞT , pðtÞ ¼ ½p1 ðtÞ,p2 ðtÞ, . . . ,pn ðtÞT , A ¼ diagða1 ,a2 , . . . ,an Þ, C ¼ diagðc1 ,c2 , . . . ,cn Þ, D ¼ diagðd1 ,d2 , . . . ,dn Þ and L ¼ ½l1 ,l2 , . . . ,ln T .

Y. Sun et al. / Neurocomputing 79 (2012) 39–49

In fact, gene regulation is intrinsically a noisy process, which is subject to intracellular and extracellular noises perturbations and environment fluctuations [15–18]. Such noises will undoubtedly affect the dynamics of the networks [25,26]. In this study, we consider a stochastic dynamic model of genetic regulatory networks with intrinsic and extrinsic noises as follows: 8 > < dmðtÞ ¼ ½AmðtÞ þ Bf ðpðtsðtÞÞÞ þEvðtÞ þ L dt þHðmðtÞ,pðtÞ,mðttðtÞÞ,pðtsðtÞÞÞ doðtÞ, ð6Þ > : dpðtÞ ¼ ½CpðtÞ þ DmðttðtÞÞ dt, where oðtÞ is an one-dimensional Brownian motion and v(t) is the extrinsic noise belonging to L2 ½0,1Þ [27]. Taking into account all possible variations of the biochemical parameters in the cell, e. g., due to gene silencing [33], and the perturbation from the environment during the gene regulation [34], we have the following model: 8 ^ > < dmðtÞ ¼ ½ðA þ DAÞmðtÞ þ ðB þ DBÞf ðpðtsðtÞÞÞ þ EvðtÞ þ L dt > :

þHðmðtÞ,pðtÞ,mðttðtÞÞ,pðtsðtÞÞÞ doðtÞ,

dpðtÞ

¼

½ðC þ DCÞpðtÞ þðD þ DDÞmðttðtÞÞ dt, ð7Þ

where DA, DB, DC and DD can be seen as the unknown constant biochemical parameter uncertainties, which are assumed to be of the following form: ½DA, DB, DC, DD ¼ WF 0 ½Ea ,Eb ,Ec ,Ed :

ð8Þ

Here, W,Ea ,Eb ,Ec and Ed are the known real constant matrices with appropriate dimensions and F0 satisfy F T0 F 0 r I:

ð9Þ

It is assumed that vectors mn and pn are the equilibrium point vectors of (7) with vðtÞ  0. For convenience, we shift an intended equilibrium point ðmn ,pn Þ of (7) to the origin by letting xðtÞ ¼ mðtÞmn ,yðtÞ ¼ pðtÞpn . Thus, we have 8 > < dxðtÞ ¼ ½ðA þ DAÞxðtÞ þ ðB þ DBÞgðyðtsðtÞÞÞ þEvðtÞ dt þ HðxðtÞ,yðtÞ,xðttðtÞÞ,yðtsðtÞÞÞ doðtÞ, ð10Þ > : dyðtÞ ¼ ½ðC þ DCÞyðtÞ þðD þ DDÞxðttðtÞÞ dt, with Hðmn ,pn ,mn ,pn Þ ¼ 0, where gðyðtÞÞ is defined as gðyðtÞÞ ¼ f ðyðtÞ þ pn Þf ðpn Þ. Since fi is a monotonically increasing function with saturation, it satisfies 0r

f i ðaÞf i ðbÞ r k, ab

ð11Þ

for all a,b A R with a ab, thus gi satisfies the sector condition g i ðaÞðg i ðaÞkaÞ r0:

ð12Þ

And the noise intensity HðxðtÞ,yðtÞ,xðttðtÞÞ,yðtsðtÞÞÞ is assumed to satisfy the following assumption: 2 6 6 6 6 6 6 6 6 6 6 6 6 6 S¼6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

X1 % %

0

FT

0

X2 P 2 D M T % X3 0 % % X4

Assumption 1. Noise intensity HðxðtÞ,yðtÞ,xðttðtÞÞ,yðtsðtÞÞÞ in (10) satisfies the following conditions: trace½HT ðx1 ,x2 ,x3 ,x4 ÞHðx1 ,x2 ,x3 ,x4 Þ r

4 X

xTi Ni xi ,

ð13Þ

i¼1

where N i , i ¼ 1; 2,3; 4 are the known matrices. The main aim of this paper is to investigate the robust stochastic stability of the gene networks (10) with a given disturbance attention level. Next we will give the corresponding definition. Definition 1 (Xu and Chen [37]). For a given scalar g 4 0, uncertain stochastic regulatory networks (10) is said to be robustly stochastically stable with disturbance attenuation g if it is robustly stochastically stable in mean square with vðtÞ  0 and moreover under zero initial conditions, the norm of z(t) satisfies the following condition: JzðtÞJE2 o gJvðtÞJ2 ,

ð14Þ

for all nonzero v(t) belonging to L2 ½0,1Þ, where zðtÞ ¼ ½xT ðtÞ,yT ðtÞT R1 and JzðtÞJE2 ¼ ðEð 0 JzðtÞJ2 dtÞÞ1=2 . The following lemma will be used in the proof of our results. Lemma 1 (Xie [40]). Given matrices Q ¼ Q T ,H,E and 0 o R ¼ RT of appropriate dimensions, Q þ HFE þ ET F T HT o0,

ð15Þ

for all F satisfying F T F r R, if and only if there exists some l 4 0 such that Q þ lHHT þ l

1 T

E RE o0:

ð16Þ

3. Robust stochastic stability analysis of GRNs In this section, we will present the main results on the robust stochastic stability of the uncertain stochastic genetic regulatory networks with intrinsic and extrinsic noises (10). Theorem 1. Given a scalar g 4 0, the uncertain stochastic genetic regulatory networks (10) is robustly stochastically stable with disturbance attenuation g, if there exist scalars a 40, r1 40, r2 40 and matrices 0 o P1 ,0 oP 2 ,0 o R,0 oQ ,0 o S,0 o L,F and M such that the following LMIs hold:

P1 B

P1 E

tAT R

0

0

0

0

pffiffiffi 2P 1 W

0 0

0 0

0 0

0 F

0 tF

sC T Q sDT Q

0 0

0 0

kL

0

0

0

0

0

sMQ

0

%

%

%

%

X5

0

0

0

0

0

0

%

%

%

%

%

g2 I

0

0

0

0

%

%

%

%

%

%

tBT R tET R tR

0

0

0

0

0 pffiffiffi 2tRW

%

%

%

%

%

%

%

S

0

0

0

0

%

%

%

%

%

%

%

%

tR

0

0

0

%

%

%

%

%

%

%

%

%

sR

0

%

%

%

%

%

%

%

%

%

%

sQ

0 0

%

%

%

%

%

%

%

%

%

%

%

aI

%

%

%

%

%

%

%

%

%

%

%

%

%

41

0

3

pffiffiffi 2P 2 W 0

7 7 7 7 7 7 7 0 7 7 0 7 7 7 0 7 7 7 o 0, 0 7 7 0 7 7 7 0 7 pffiffiffi 2sQW 7 7 7 7 0 7 7 0 5 aI

ð17Þ

42

Y. Sun et al. / Neurocomputing 79 (2012) 39–49

P 1 r r1 I,

ð18Þ

S r r2 I,

ð19Þ

where

2xT ðttðtÞÞFxðttðtÞÞ Z t xT ðttðtÞÞFf 1 ðsÞ ds2xT ðttðtÞÞ 2 ttðtÞ t

Z F

T

ttðtÞ

ETa Ea ,

X1 ¼ P 1 AA P1 þ r1 N1 þ tr2 N1 þ I þ a

Z 

X2 ¼ P 2 CC T P2 þ r1 N2 þ tr2 N2 þ I þ aETc Ec , T

t

ttðtÞ

Z

t

þ

ETd Ed ,

X3 ¼ F F þ r1 N 3 þ tr2 N 3 þ a

H1 ðsÞ doðsÞ

ttðtÞ

H1 ðsÞ doðsÞ

ttðtÞ t

Z

Vðt,xðtÞ,yðtÞÞ ¼ V 1 ðt,xðtÞ,yðtÞÞ þ V 2 ðt,xðtÞ,yðtÞÞ þ V 3 ðt,xðtÞ,yðtÞÞ þ V 4 ðt,xðtÞ,yðtÞÞ,

 xT ðttðtÞÞF þ ð20Þ

where

ttðtÞ

þRf 1 ðsÞ ds þ½xT ðttðtÞÞFS1 ½F T xðttðtÞÞ " Z T #

Z h  F T xðttðtÞÞ þ S

V 3 ðt,xðtÞ,yðtÞÞ ¼

V 4 ðt,xðtÞ,yðtÞÞ ¼

t

Z

Z

Z

t

t

tþy

T f 2 ðsÞQf 2 ðsÞ

ds dy,

HT1 ðsÞSH1 ðsÞ

ttðtÞ

þxT ðttðtÞÞFS1 F T xðttðtÞÞ Z t T Z þ H1 ðsÞ doðsÞ S

ds dy,

ttðtÞ

T

y ðtÞP2 ðC þ DCÞyðtÞ þ y ðtÞP 2 ðD þ DDÞxðttðtÞÞ þ trace½HT ðxðtÞ,yðtÞ,xðttðtÞÞ,yðtsðtÞÞÞ P1 HðxðtÞ,yðtÞ,xðttðtÞÞ,yðtsðtÞÞÞ

t ttðtÞ

 H1 ðsÞ doðsÞ :

þ r1 ½xT ðtÞN 1 xðtÞ þyT ðtÞN 2 yðtÞ þ x ðttðtÞÞN 3 xðttðtÞÞ þy ðtsðtÞÞN4 yðtsðtÞÞ: ð21Þ By introducing a free weighting matrix F of appropriate dimensions, one also has Z t T T f 1 ðsÞRf 1 ðsÞ ds LV 2 ðt,xðtÞ,yðtÞÞ ¼ tf 1 ðtÞRf 1 ðtÞ tt Z t T T f 1 ðsÞRf 1 ðsÞ ds r tf 1 ðtÞRf 1 ðtÞ ttðtÞ

 Z þ 2x ðttðtÞÞF xðtÞxðttðtÞÞ t

H1 ðsÞ doðsÞ



 Z þ 2y ðtsðtÞÞM yðtÞyðtsðtÞÞ

t ttðtÞ

f 1 ðsÞ ds

 f 2 ðsÞ ds

2yT ðtsðtÞÞMyðtsðtÞÞ Z t yT ðtsðtÞÞMQ 1 M T yðtsðtÞsÞ ds þ Z

tsðtÞ t

tsðtÞ T

T

½yT ðtsðtÞÞM þ f 2 ðsÞQ Q 1 ½yT ðtsðtÞÞM T

þ f 2 ðsÞQ T ds r sf 2 ðtÞQf 2 ðtÞ þ2yT ðtsðtÞÞMyðtÞ 2yT ðtsðtÞÞMyðtsðtÞÞ þ syT ðtsðtÞÞMQ 1 M T yðtsðtÞÞ:

ð23Þ

Moreover, from Assumption 1 and the condition (19), it is easy to derive Z t HT1 ðsÞSH1 ðsÞ ds LV 4 ðt,xðtÞ,yðtÞÞ ¼ tHT1 ðtÞSH1 ðtÞ tt Z t r tHT1 ðtÞSH1 ðtÞ HT1 ðsÞSH1 ðsÞ ds ttðtÞ

r tr2 ½xT ðtÞN1 xðtÞ þ yT ðtÞN2 yðtÞ þ xT ðttðtÞÞN3 xðttðtÞÞ

T

¼ tf 1 ðtÞRf 1 ðtÞ Z t T f 1 ðsÞRf 1 ðsÞ dsþ 2xT ðttðtÞÞFxðtÞ 

t

tsðtÞ



T

T

tsðtÞ

T

T

yT ðtÞP2 ðC þ DCÞyðtÞ þ yT ðtÞP 2 ðD þ DDÞxðttðtÞÞ T

Similarly, after introducing another matrix M, one further has Z t T T f 2 ðsÞQf 2 ðsÞ ds LV 3 ðt,xðtÞ,yðtÞÞ ¼ sf 2 ðtÞQf 2 ðtÞ ts Z t T T r sf 2 ðtÞQf 2 ðtÞ f 2 ðsÞQf 2 ðsÞ ds

r sf 2 ðtÞQf 2 ðtÞ þ 2yT ðtsðtÞÞMyðtÞ

r 2½xT ðtÞP 1 ðA þ DAÞxðtÞ þ xT ðtÞP 1 ðB þ DBÞgðyðtsðtÞÞÞ

ttðtÞ

 H1 ðsÞ doðsÞ

ð22Þ

T

ttðtÞ

t

T

LV 1 ðt,xðtÞ,yðtÞÞ ¼ 2½xT ðtÞP1 ðA þ DAÞxðtÞ þ xT ðtÞP 1 ðB þ DBÞgðyðtsðtÞÞÞ



T Z S



r tf 1 ðtÞRf 1 ðtÞ þ2xT ðttðtÞÞFxðtÞ

with f 1 ðtÞ,f 2 ðtÞ and H1 ðtÞ defined by f 1 ðtÞ ¼ ðA þ DAÞxðtÞ þðB þ DBÞ gðyðtsðtÞÞÞ, f 2 ðtÞ ¼ ðC þ DCÞyðtÞ þ ðD þ DDÞxðttðtÞÞ and H1 ðtÞ ¼ HðxðtÞ,yðtÞ,xðttðtÞÞ,yðtsðtÞÞÞ, respectively. Then by the Itˆo-differential rule [41] and the condition (18), one has

Z

H1 ðsÞ doðsÞ

H1 ðsÞ doðsÞ

2xT ðttðtÞÞFxðttðtÞÞ þ txT ðttðtÞÞFR1 F T xðttðtÞÞ

t

tþy

0

ttðtÞ

T

tþy

s

ttðtÞ

S S1

H1 ðsÞ doðsÞ

t

t

þ

Z

0

Z

f 1 ðsÞRf 1 ðsÞ ds dy,

t

ttðtÞ

V 1 ðt,xðtÞ,yðtÞÞ ¼ xT ðtÞP 1 xðtÞ þ yT ðtÞP2 yðtÞ, t

ttðtÞ

 H1 ðsÞ doðsÞ

T



Z

t



½xT ðttðtÞÞF þf 1 ðsÞRR1 ½F T xðttðtÞÞ

Proof. Firstly, we show the robust stochastic stability of the gene networks (10) with vðtÞ  0. Consider the following Lyapunov– Krasovskii functional:

0

ttðtÞ

T Z H1 ðsÞ doðsÞ S

H1 ðsÞ doðsÞ

T

X5 ¼ LLT þ aETb Eb :

Z

t

r tf 1 ðtÞRf 1 ðtÞ þ 2xT ðttðtÞÞFxðtÞ2xT ðttðtÞÞFxðttðtÞÞ Z t þ ½xT ðttðtÞÞFR1 ½F T xðttðtÞÞ ds

X4 ¼ MT M þ r1 N4 þ tr2 N4 ,

V 2 ðt,xðtÞ,yðtÞÞ ¼

T Z S

þ yT ðtsðtÞÞN 4 yðtsðtÞÞ

Z

t ttðtÞ

HT1 ðsÞSH1 ðsÞ ds: ð24Þ

Y. Sun et al. / Neurocomputing 79 (2012) 39–49

By combining Eqs. (21)–(24), and using the condition (12), one can get LVðt,xðtÞ,yðtÞÞ r 2½xT ðtÞP 1 ðAþ DAÞxðtÞ þxT ðtÞP1 ðB þ DBÞgðyðtsðtÞÞÞ T

T

y ðtÞP 2 ðC þ DCÞyðtÞ þy ðtÞP 2 ðD þ DDÞxðttðtÞÞ T

T

þ r1 ½x ðtÞN1 xðtÞ þ y ðtÞN2 yðtÞ þ xT ðttðtÞÞN3 xðttðtÞÞ þ yT ðtsðtÞÞN4 yðtsðtÞÞ T f 1 ðtÞRf 1 ðtÞ þ2xT ðt

þt

T

tðtÞÞFxðtÞ2x ðttðtÞÞFxðttðtÞÞ

þ txT ðttðtÞÞFR1 F T xðttðtÞÞ þ xT ðttðtÞÞFS1 F T xðttðtÞÞ

Z

t

þ ttðtÞ

T Z H1 ðsÞ doðsÞ S

t

ttðtÞ

þ tr2 ½xT ðtÞN 1 xðtÞ þyT ðtÞN 2 yðtÞ þxT ðttðtÞÞN 3 xðttðtÞÞ

n X

2

ttðtÞ

HT1 ðsÞSH1 ðsÞ ds

^ 1 þt ¼ x^ ðtÞ½G

T G^ 4 Q G^ 4 þ

þs

Z

t

þ ttðtÞ

Z

t

 ttðtÞ

%

X3

%

%

%

X4

%

%

%

%

X5

%

%

%

%

%

%

%

%

%

%

%

0 0 F 0 0 0 S

%

%

%

%

%

%

%

0 0 tF 0 0 0 0 tR

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

t

H1 doðsÞ

%

0 0 0 sMQ 0 0 0 0 0

%

%

sQ

%

%

%

0 aI

%

%

%

%

3 0 pffiffiffi 7 2P 2 W 7 7 7 0 7 7 7 0 7 7 0 7 7 7 0 7 o0, 7 0 7 7 7 0 7 pffiffiffi 7 2sQW 7 7 7 0 7 7 0 5 aI

By using the known Schur complement [42] and Lemma 1 introduced in Section 2, LMI condition (26) is equivalent to the following inequality for a scalar a 4 0:

where x^ ¼ ½xT ðtÞ,yT ðtÞ,xT ðttðtÞÞ,yT ðtsðtÞÞ,g T ðyðtsðtÞÞÞT , 2 3 ^1 P 0 FT 0 P 1 ðB þ DBÞ 6 7 ^ 2 P 2 ðD þ DDÞ M T 7 6 % P 0 6 7 6 7 ^ G^ 1 ¼ 6 % 7, % P3 0 0 6 7 7 6 % % % X4 kL 4 5 T % % % % LL

T

T

ð27Þ

^ ¼ diagðEa ,Ec ,Ed ,0,Eb ,0; 0,0; 0,0Þ, F T

T

T

^ ,O ^ ,0; 0,0, O ^ ,0; 0, O ^ ,0ÞT , ^ ¼ ðO C 1 2 3 4



HT1 ðsÞSH1 ðsÞ ds,

^ 2 ¼ P 2 ðC þ DCÞðC þ DCÞT P2 þ r N 2 þ tr N 2 , P 1 2

pffiffiffi 2P 1 W 0 0 0 0 pffiffiffi 2tRW 0 0 0

0 sC T Q sDT Q 0 0 0 0 0 sR

X^ 2 ¼ P2 CC T P 2 þ r1 N 2 þ tr2 N 2 þ aETc Ec :

T

t

^ 1 ¼ P 1 ðAþ DAÞðA þ DAÞT P1 þ r N 1 þ tr N 1 , P 1 2

P1 B 0 0 kL

F^ ¼ diagðF 0 ,F 0 , . . . ,F 0 Þ,

T G^ 3 R1 G^ 3

s

ttðtÞ

%

tAT R 0 0 0 t BT R tR

0 MT 0

where

T G^ 5 Q 1 G^ 5 x^ ðtÞ

T Z H1 doðsÞ S

FT P2 D

T

i¼1 T T G^ 2 RG^ 2 þ G^ 3 S1 G^ 3 þ

0 X^ 2

X^ 1

^ þC ^ F^ F ^ o 0, ^ F^ C ^ þF O

li gðyi ðtsðtÞÞÞ½gðyi ðtsðtÞÞÞkyi ðtsðtÞÞ

T

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

X^ 1 ¼ P1 AAT P 1 þ r1 N 1 þ tr2 N 1 þ aETa Ea ,

2yT ðtsðtÞÞMyðtsðtÞÞ þ syT ðtsðtÞÞMQ 1 M T yðtsðtÞÞ

t

2

where

T

Z

On the other hand, it follows from the condition (17) that the following LMI holds:

ð26Þ

 H1 ðsÞ doðsÞ

þ sf 2 ðtÞQf 2 ðtÞ þ 2yT ðtsðtÞÞMyðtÞ

þ yT ðtsðtÞÞN 4 yðtsðtÞÞ

43

^ 1 ¼ ½P 1 W,0; 0,0,P 1 W,0; 0,0; 0,0, O ð25Þ

^ 2 ¼ ½0,P2 W,P 2 W,0; 0,0; 0,0; 0,0, O ^ 3 ¼ ½tRW,0; 0,0, tRW,0; 0,0; 0,0, O ^ 4 ¼ ½0, sQW, sQW,0; 0,0; 0,0; 0,0, O 2 6 6 6 6 6 6 6 6 6 ^ O¼6 6 6 6 6 6 6 6 6 6 4

~1 P %

0 ~2 P

FT P2 D ~3 P

0 MT

P1 B 0

tAT R 0

0 0

0 0

0 sC T Q

0

F 0 0 0

tF

sD T Q

0 0 0

0 0 0

%

%

%

%

%

X4

%

%

%

%

0 kL LLT

%

%

%

%

%

0 0 tBT R tR

%

%

%

%

%

%

S

%

%

%

%

%

%

%

0 tR

%

%

%

%

%

%

%

%

0 0 sR

%

%

%

%

%

%

%

%

%

0 0

3

7 7 7 0 7 7 7 sMQ 7 7 0 7 7 7, 0 7 7 0 7 7 7 0 7 7 0 7 5 sQ

ð28Þ

^ 3 ¼ F T F þ r N 3 þ tr N 3 , P 1 2 with

G^ 2 ¼ ½ðA þ DAÞ,0; 0,0,B þ DB, G^ 3 ¼ ½0; 0,F T ,0; 0,

~ 1 ¼ P1 AAT P 1 þ r N 1 þ tr N 1 , P 1 2 ~ 2 ¼ P2 CC T P 2 þ r N 2 þ tr N 2 , P 1 2 ~ 3 ¼ F T F þ r N3 þ tr N3 : P 1 2

G^ 4 ¼ ½0,ðC þ DCÞ,D þ DD,0; 0,

By using the Schur complement [42], it follows from inequality (27) that T

G^ 5 ¼ ½0; 0,0,MT ,0:

T

T

T

T

G^ ¼ G^ 1 þ tG^ 2 RG^ 2 þ G^ 3 S1 G^ 3 þ tG^ 3 R1 G^ 3 þ sG^ 4 Q G^ 4 þ sG^ 5 Q 1 G^ 5 o 0: ð29Þ

44

Y. Sun et al. / Neurocomputing 79 (2012) 39–49

G~ 5 ¼ ½0; 0,0,MT ,0; 0:

Furthermore, it is noted that (Z T Z t ) t E H1 doðsÞ S H1 doðsÞ ttðtÞ

¼E

Z

Noting the condition (17), by using the Schur complement and Lemma 1 introduced in Section 2, we know the following matrix inequality holds for a scalar a 4 0:

ttðtÞ

HT1 ðsÞSH1 ðsÞ ds :

t ttðtÞ

ð30Þ

Then, by taking the mathematical expectation of both sides of (25), one can deduce that T ^ x^ ðtÞg o 0: EfLVðt,xðtÞ,yðtÞÞgr Efx^ ðtÞG

ð31Þ

Based on the Lyapunov stability theory of stochastic differential equation, it can be concluded from the condition (31) that the gene networks (10) with parameter uncertainties is stochastically asymptotically stable in the mean square with vðtÞ  0. Next, we will show the disturbance attenuation. Consider the same Lyapunov functional (20) and just modify f 1 ðtÞ in V 2 ðt,xðtÞ,yðtÞÞ as f 1 ðtÞ ¼ ðAþ DAÞxðtÞ þðB þ DBÞgðyðtsðtÞÞÞþ EvðtÞ. Under zero initial conditions, we have Z t

LVðs,xðsÞ,yðsÞsÞ ds : ð32Þ EfVðt,xðtÞ,yðtÞÞg ¼ E 0

ð33Þ

0

It follows from Eqs. (32) to (33) that Z t

½xT ðsÞxðsÞ þ yT ðsÞyðsÞg2 vT ðsÞvðsÞ þLVðs,xðsÞ,yðsÞÞ ds JðtÞ ¼ E 0

EfVðt,xðtÞ,yðtÞÞg Z t

 T x ðsÞxðsÞ þ yT ðsÞyðsÞg2 vT ðsÞvðsÞ þ LVðs,xðsÞ,yðsÞÞ ds : rE 0

ð34Þ

St :¼ xT ðtÞxðtÞ þ yT ðtÞyðtÞg2 vT ðtÞvðtÞ þLVðt,xðtÞ,yðtÞÞ Z t T HT1 ðsÞSH1 ðsÞ ds r x ðtÞGxðtÞ þ ttðtÞ

t

ttðtÞ

 H1 doðsÞ ,

ð35Þ

where xðtÞ ¼ ½xT ðtÞ,yT ðtÞ,xT ðttðtÞÞ,yT ðtsðtÞÞ,g T ðyðtsðtÞÞÞ,vT ðtÞT , T

T

T

T

G ¼ G~ 1 þ tG~ 2 RG~ 2 þ G~ 3 S1 G~ 3 þ tG~ 3 R1 G~ 3 þ sG~ 4T Q G~ 4 þ sG~ 5 Q 1 G~ 5 , 2 6 6 6 6 ~ G1 ¼ 6 6 6 6 6 4

P1

0

FT

0

P 1 ðB þ DBÞ

%

P2

MT

0

%

%

P 2 ðD þ DDÞ ^3 P

0

0

%

%

%

X4

kL

%

%

%

%

LLT

%

%

%

%

%

P1 E

P 2 ¼ P2 ðC þ DCÞðC þ DCÞT P 2 þ r1 N 2 þ tr2 N 2 þI,

~ 3 ¼ ½0; 0,F T ,0; 0,0, ðG

G~ 4 ¼ ½0,ðC þ DCÞ,D þ DD,0; 0,0,

3

7 0 7 7 0 7 7 7, 0 7 7 0 7 5 g2 I

P 1 ¼ P1 ðAþ DAÞðA þ DAÞT P 1 þ r1 N 1 þ tr2 N 1 þI,

G~ 2 ¼ ½ðA þ DAÞ,0; 0,0,B þ DB,E,

2 6 6 6 6 6 6 6 6 6 6 6 O¼6 6 6 6 6 6 6 6 6 6 6 4

P1

0

FT

0

P1 B

P1 E

tAT R

0

0

0

%

P2

P2 D ^3 P

MT

0

0

0

0

0

sC T Q T

%

%

0

0

0

0

F

tF

sD Q

%

%

%

X4

%

%

%

%

kL LLT

0 0

0 tBT R

0 0

0 0

0 0

%

%

%

%

%

g2 I

%

%

%

%

%

%

tET R tR

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

0

0

0

%

0 S

0 0

0 0

%

%

tR

0

%

%

%

sR

%

%

%

%

0

3

7 7 7 0 7 7 7 sMQ 7 7 7 0 7 7 0 7 7, 7 0 7 7 0 7 7 7 0 7 7 0 7 5 sQ 0

P1 ¼ P1 AAT P1 þ r1 N 1 þ tr2 N 1 þI, P2 ¼ P2 CC T P2 þ r1 N 2 þ tr2 N 2 þI,

F ¼ diagðEa ,Ec ,Ed ,0,Eb ,0; 0,0; 0,0; 0Þ, C ¼ ðOT1 , OT2 ,0; 0,0; 0, OT3 ,0; 0, OT4 ,0ÞT , O1 ¼ ½P1 W,0; 0,0,P 1 W,0; 0,0; 0,0; 0, O2 ¼ ½0,P2 W,P 2 W,0; 0,0; 0,0; 0,0; 0, O3 ¼ ½tRW,0; 0,0, tRW,0; 0,0; 0,0; 0, O4 ¼ ½0, sQW, sQW,0; 0,0; 0,0; 0,0; 0:

G o 0:

ttðtÞ

T Z H1 doðsÞ S

where

Noting the condition (36) and applying the Schur complement [42], one obtains

Hence

t

ð36Þ

F ¼ diagðF 0 ,F 0 , . . . ,F 0 Þ,

For a given g 40, we define Z t

JðtÞ ¼ E ½xT ðsÞxðsÞ þ yT ðsÞyðsÞg2 vT ðsÞvðsÞ ds :

Z

O þ CF F þ FT F T CT o 0,

ð37Þ

Then it is straightforward to get Z t

xT ðsÞGxðsÞ ds o 0, JðtÞ r E

ð38Þ

0

which implies that the condition (14) in Definition 1 is satisfied. In other words, the genetic networks model (10) with parameter uncertainties is stochastically stable with disturbance attenuation g. This completes the proof. & In fact, the optimal attenuation level gn with parameter uncertainties can be obtained by solving the following constrained optimization problem:

d0 ¼

min

a, r1 , r2 ,P 1 ,P 2 ,R,Q ,S, L,F,M

subject to

d with d ¼ g2

r1 40, r2 40,P1 4 0,P2 40,R 4 0,

Q 4 0,S 4 0, L 4 0,F,M and ð17Þ2ð19Þ:

ð39Þ 1=2

Then the optimal performance level g ¼ ðd0 Þ n

.

Remark 1. By solving the above constrained optimization problem (39), the optimal noise attenuation level can be obtained. The obtained results are more general than those results in [14], where intrinsic and extrinsic noises were not considered, those in [26] where neither parameter uncertainties nor time delays were considered, and those in [24,28,30] where no attenuation levels were obtained.

Y. Sun et al. / Neurocomputing 79 (2012) 39–49

In the case when there are no parameter uncertainties in the system (10), Theorem 1 is specialized as follows. Corollary 1. Given a scalar g 4 0, the stochastic genetic regulatory networks (10) without parameter uncertainties is stochastically stable with disturbance attenuation g, if there exist scalars r1 4 0, r2 4 0 and matrices P1 40,P2 4 0,R 40,Q 4 0,S 4 0, L 4 0,F and M such that the following LMIs hold:

O o 0,

45

function, it is easy to get k¼0.65. Time delays are set as tðtÞ ¼ 0:4 þ 0:39cosðtÞ9, sðtÞ ¼ 0:49sinðtÞ9, which are both not differentiable at some points, it is easy to check that t ¼ 0:7 and s ¼ 0:4. Without loss of generality, the noise intensity is set as HðxðtÞ,yðtÞ,xðttðtÞÞ,yðtsðtÞÞÞ ¼ O1 xðtÞ þ O2 xðttðtÞÞ þ O3 yðtÞ þ O4 yðtsðtÞÞ,

ð45Þ

ð40Þ for simplicity, we just let

P 1 r r1 I,

ð41Þ

S r r2 I,

ð42Þ

2

0:2

O1 ¼ 0, O2 ¼ 6 4 0

0:1

0

0:1

0

0

3

0 7 5, 0:2

2

0:1

O3 ¼ 6 4 0

0:1

0 0:2 0

0

3

0:1 7 5 0:1

where O is defined in (36). Remark 2. In [25], the authors investigated stochastic stability of stochastic gene networks with disturbance attenuation, where time delays tðtÞ and sðtÞ are required to be differentiable and satisfy t_ ðtÞ o 1 and s_ ðtÞ o 1. However, the delay dependent results in our paper have removed these restrictions and it is thus expected that our results are less conservative than those results in [25].

and O4 ¼ 0. It is easy to verify that N 1 ¼ 0, N 2 ¼ 2OT2 O2 , N3 ¼ 2OT3 O3 and N 4 ¼ 0. Let extrinsic noises be vðtÞ ¼ e0:0005t sinð0:02tÞ and E ¼ ½0:04,0:04,0:04T . After solving the constrained optimization problem (39) without parameter uncertainties via LMI control toolbox, we can get the optimal disturbance attenuation level gn ¼ 0:1672, and the

Remark 3. There have been some results considering stochastic stability of genetic networks with time-varying delays, such as [28,29], which made some generalizations compared to some of existing results. It should be pointed out that, all the stability conditions obtained in the above mentioned results need the conditions that the time-varying delays are differentiable and the derivatives of them are less than a constant. However, the results here do not need these restrictions and are thus less conservative than those results provided in [28,29].

1.2 1 0.8 0.6

In this section, two numerical examples are employed to illustrate the effectiveness of the obtained results.

0.4

Example 1. Consider a gene network model reported in [43], which comprises three repressor genes (i ¼ lacl,tetR and cl) and their promoters (j ¼ cl,lacl and tetR): 8 dxi a > > > < dt ¼ xi þ 1 þ yn þ a0 , j ð43Þ > dyi > > ¼ bðxi yi Þ: : dt

where A ¼ diagð1; 1,1Þ, C ¼ diagð0:5,0:5,0:5Þ, D ¼ diagð0:5,0:5,0:5Þ and the coupling matrix 2 3 0 0 1 6 7 0 5, B ¼ 0:2  4 1 0 0 1 0 with 0.2 being the coupling intensity, and correspondingly, L ¼ ½0:2,0:2,0:2T . The gene regulation function in this example is chosen as f ðxÞ ¼ x2 =ð1 þ x2 Þ with the Hill coefficient 2. For this regulation

0.2 0

0

5

10

15

20

Times (sec) Protein concentrations

0.6

y1 y2 y3

0.55 0.5 0.45 0.4 y(t)

Taking into account the time delay and the noise disturbances, we get the following model in compact form: 8 > < dxðtÞ ¼ ½AxðtÞ þ Bf ðyðtsðtÞÞÞþ EvðtÞ þ L dt þ HðxðtÞ,yðtÞ,xðttðtÞÞ,yðtsðtÞÞÞ doðtÞ, ð44Þ > : dyðtÞ ¼ ½CyðtÞ þ DxðttðtÞÞ dt,

x1 x2 x3

x(t)

4. Illustrative examples

mRNA concentrations

1.4

0.35 0.3 0.25 0.2 0.15 0.1

0

5

10 Times (sec)

15

20

Fig. 1. (a) mRNA concentrations xi , i ¼ 1; 2,3. (b) Protein concentrations yi , i ¼ 1; 2,3.

46

Y. Sun et al. / Neurocomputing 79 (2012) 39–49

corresponding feasible solutions as follows: 2 3 12:0692 1:2527 1:2752 6 1:2527 10:6045 2:4589 7 P1 ¼ 4 5, 1:2752 2:4589 10:2563

mRNA concentrations

1.4

x1 x2

1.2

x3

1

14:2980 6 R ¼ 4 3:1044

0:1650 7 5,

6 Q ¼ 4 0:0007 0:0012 2

15:7896 6 S ¼ 4 0:0505 0:0082

0:0022

0:0028 7 5, 0:0038

0:0505

4:6309

10 Times (sec)

15

20

Protein concentrations y1 y2 y3

0.5 0.45 0.4

3

0:8794 7 5,

0:9158

5

0.55

15:0105

0:6021

0

0.6

0:0082 7 0:7604 5,

0:7604

6 F ¼ 4 0:5517 0:5796

0

3

15:3188

0:4381

0.2

3

0:0012

4:9085

0.4

13:1965

0:0007 0:0028

0.6

3 1:3461 5:0525 7 5,

5:0525

0:0009

0.8

7:0363

3:1044 13:5740

1:3461

2

0:1326

0:1650

2

2

1:0139 5:4172

x(t)

7:9523 6 P 2 ¼ 4 1:0139 0:1326

3

y(t)

2

0.35 0.3

4:5066

0.25

2

1:0283 6 M ¼ 4 0:3840 0:1342

0:5252 0:7436 0:0499

3

0.2

0:1055 7 5, 0:6478

0.15

0:1327

0.1

L ¼ diagð1:4164,1:4164,1:4164Þ, r1 ¼ 12:9102, r2 ¼ 15:9573. Thus, it can be concluded that the system is stochastically stable with the disturbance attenuation level gn ¼ 0:1672. In the simulations, the Euler–Maruyama numerical scheme is used to simulate stochastic genetic regulatory networks with extrinsic noises (44). With the given initial conditions xð0Þ ¼ ½0:8,1:2,0:4T and yð0Þ ¼ ½0:3,0:1,0:6T of the networks (44), Fig. 1(a) and (b) is the time responses of the mRNA and protein concentrations, respectively. Fig. 2(a) and (b) is the time responses of the mRNA and protein concentrations without intrinsic noises, respectively. It is noted that time delays tðtÞ and sðtÞ in this example are not differentiable, which do not satisfy the conditions required in [25], so the results in [25] fail to be used to study this example. Furthermore, in order to make fair comparison with results obtained in [25], we also consider the same time delays as in [25], that is, both of them are differentiable. In this case, after applying the constrained optimization algorithm proposed in this paper, one can get the optimal attenuation level gn ¼ 0:1522, which is much smaller than the level g ¼ 4:5 obtained in [25]. Example 2. Consider the following synthetic uncertain stochastic genetic regulatory networks with extrinsic noises 8 > < dxðtÞ ¼ ½ðA þ DAÞxðtÞ þ ðB þ DBÞgðyðtsðtÞÞÞ þEvðtÞ þ L dt þ ½O1 yðtÞ þ O2 xðttðtÞÞ doðtÞ, > : dyðtÞ ¼ ½ðC þ DCÞyðtÞ þðD þ DDÞxðttðtÞÞ dt, ð46Þ

0

5

10 Times (sec)

15

20

Fig. 2. (a) mRNA concentrations xi ði ¼ 1; 2,3Þ without intrinsic noises. (b) Protein concentrations yi ði ¼ 1; 2,3Þ without intrinsic noises.

Fig. 3. Synthetic gene network (activation: -; repression:,).

where A ¼ diagð1:2,1:6,1:6,2,1:6,1:8,1:8,1:5,1:6,1:4Þ, C ¼ diagð0:8, 0:8,0:8,0:8,0:8,0:8,0:8,0:8,0:8,0:8Þ, D ¼ diagð0:3,0:3,0:3,0:3,0:3,0:3, 0:3,0:3,0:3,0:3Þ, the topology of the model can be seen in Fig. 3,

Y. Sun et al. / Neurocomputing 79 (2012) 39–49

Ed ¼ diagð0:2,0:3,0:1,0:1,0:2,0:2,0:1,0:3,0:6,0:2Þ:

and the coupling matrix 2

0 6 0 6 6 6 0 6 6 0 6 6 6 0 B ¼ 0:3  6 6 0 6 6 6 1 6 6 0 6 6 4 0 0

47

1

1

0

1

1

0

0

1

0

0

1

0

1

0

1

0

0

0

0

1

0

1

0

0

0

0

0

1

0

1

0

0

1

0

0

0

1

0

1

0

0 0

1 0

0 0

0 0

0 0

0 0

1 0

0 1

0

1

0

0

0

1

0

1

0

0

1

0

0

0

0

0

1

0

0

0

1

0

0

0

0

3

W ¼ 0:1nI,

1 7 7 7 0 7 7 1 7 7 7 0 7 7: 1 7 7 7 0 7 7 0 7 7 7 1 5

F 0 ¼ I and correspondingly,

L ¼ ½0:28,0:57,0,0:31,0; 0,0:29,0:59,0,0:32T : For simplicity, the regulation function and time delays tðtÞ and sðtÞ are chosen to be the same as those in the last example.

0

The bounds of the uncertain parameter uncertainties are given as Ea ¼ diagð0:2,0:2,0:5,0:3,0:4,1,0:1,0:2,0:3,0:2Þ,

Furthermore, the noise intensity here are set as O1 ¼ diagð0:2,0:1, 0,0:3,0:1,0; 0,0:4,0:2,0:1Þ and O2 ¼ diagð0:3,0:2,0:2,0:1,0:2, 0,0:3,0:1,0:2,0:3Þ. The noise disturbance v(t) is assumed to be 1=ð0:5þ 1:6tÞ,t Z 0. Applying the constrained optimization algorithm (39), we can get the optimal disturbance attenuation level gn ¼ 0:1225. It thus can be concluded that model (46) is robustly stochastically stable with disturbance attenuation gn ¼ 0:1225. Employing the same numerical method, the simulation results are shown as follows. Fig. 4(a) and (b) is the time responses of the mRNA and protein concentrations, respectively. Fig. 5(a) and (b) is for the noise free cases.

Ec ¼ diagð0:1,0:1,0:2,0:1,0:2,0:1,0:2,0:2,0:1,0:2Þ, mRNA concentrations

10

x1 x2

9

8

x3

8

7

x5

x4

6

x6

5

x8

6

x7 x9

4

5 4

x10

3

3

2

2

1

1

0

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10

7

x(t)

x(t)

9

mRNA concentrations

10

0

5

10

15

0

20

0

5

Times (sec) Protein concentrations

10

y3

7

y5

7

6

y6

6

5

y8

8

y4

y7

y(t)

y(t)

8

y9

4 3

5 4

y10

3

2

2

1

1

5

10 Times (sec)

20

15

y1 y2 y3 y4 y5 y6 y7 y8 y9 y10

9

y2

0

15

Protein concentrations

10

y1

9

0

10 Times (sec)

20

Fig. 4. (a) mRNA concentrations xi , i ¼ 1; 2, . . . ,10. (b) Protein concentrations yi , i ¼ 1; 2, . . . ,10.

0

0

5

10

15

20

Times (sec) Fig. 5. (a) mRNA concentrations xi ði ¼ 1; 2, . . . ,10Þ without intrinsic noises. (b) Protein concentrations yi ði ¼ 1; 2, . . . ,10Þ without intrinsic noises.

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Y. Sun et al. / Neurocomputing 79 (2012) 39–49

It should be pointed out that the time-varying delays in this example are not differentiable or even unknown, therefore, these stochastic robust stability conditions developed in [28,29], where they need the delay to be differentiable and the derivatives of them to be less than a constant, can not be applied onto this example.

5. Conclusions In this paper, we have studied the robust stochastic stability of uncertain stochastic genetic regulatory networks with extrinsic noises. Based on the Lyapunov stability theory, by using the free weighting matrix method, a novel delay-dependent robust stochastic stability condition is derived for those genetic regulatory networks, which could be easily solved by using the Matlab LMI control toolbox. The derived results do not require the differentiability of the time-varying delays and the derivatives of them to be less than a constant, which could be less conservative than some of the existing results. Two numerical examples are finally presented to illustrate the application and effectiveness of the obtained results. It is noted that other interesting topics, such as multistability, toggle switch and relaxation oscillators found in gene networks, gene networks with discrete and distributed delays [44,45], the network model with polytopic uncertainties, need much more effort and will be studied in future.

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Yonghui Sun received the MS degree in applied mathematics from Southeast University, Nanjing, China, in 2007, and the PhD degree in control theory and application from City University of Hong Kong, in 2010. His research interests include stochastic control, complex networks, systems biology, and fuzzy modeling and control. Dr. Sun is an active reviewer for many international journals.

Y. Sun et al. / Neurocomputing 79 (2012) 39–49 Gang Feng received the B.Eng and M.Eng degrees in Automatic Control from Nanjing Aeronautical Institute, China in 1982 and in 1984 respectively, and the PhD degree in Electrical Engineering from the University of Melbourne, Australia in 1992. He has been with City University of Hong Kong since 2000 where he is now Chair Professor of Mechatronic Engineering. He is also ChangJiang Chair Professor at Nanjing University of Science and Technology, awarded by Ministry of Education, China. He was lecturer/senior lecturer at School of Electrical Engineering, University of New South Wales, Australia, 1992–1999. He was awarded an Alexander von Humboldt Fellowship in 1997–1998, and the IEEE Transactions on Fuzzy Systems Outstanding Paper Award in 2007. His current research interests include hybrid systems and control, modeling and control of energy systems, and intelligent systems & control. Prof. Feng is an IEEE Fellow, an associate editor of IEEE Transactions on Fuzzy Systems, and was an associate editor of IEEE Transactions on Automatic Control, IEEE Transactions on Systems, Man & Cybernetics, Part C, Mechatronics and Journal of Control Theory and Applications.

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Jinde Cao (M’07-SM’07) received the BS degree from Anhui Normal University, Wuhu, China, the MS degree from Yunnan University, Kunming, China, and the PhD degree from Sichuan University, Chengdu, China, all in mathematics/applied mathematics, in 1986, 1989, and 1998, respectively. He was with Yunnan University from 1989 to 2000. Since 2000, he has been with the Department of Mathematics, Southeast University, Nanjing, China. From 2001 to 2002, he was a Post-Doctoral Research Fellow with the Department of Automation and Computer-Aided Engineering, Chinese University of Hong Kong, Shatin, Hong Kong. He was a Visiting Research Fellow and a Visiting Professor with the School of Information Systems, Computing and Mathematics, Brunel University, Middlesex, UK, from 2006 to 2008. He is the author or co-author of more than 160 research papers and five edited books. His current research interests include nonlinear systems, neural networks, complex systems, complex networks, stability theory, and applied mathematics. Dr. Cao was an Associate Editor of IEEE Transactions on Neural Networks from 2006 to 2009. He is an Associate Editor of Journal of the Franklin Institute, Mathematics and Computers in Simulation, Neurocomputing, International Journal of Differential Equations, Discrete Dynamics in Nature and Society, and Differential Equations and Dynamical Systems. He is a reviewer of Mathematical Reviews and Zentralblatt-Math.