Exponential stability of impulsive stochastic genetic regulatory networks with time varying delays and reaction diffusion

R E S E A R C H

Open Access

Exponential stability of impulsive

stochastic genetic regulatory networks with

time-varying delays and reaction-diffusion

Boqiang Cao

_{, Qimin Zhang}

_{and Ming Ye}

*_{Correspondence:} [email protected] 1_{School of Mathematics and} Statistics, Ningxia University, Yinchuan, 750021, P.R. China Full list of author information is available at the end of the article

Abstract

We present a mean-square exponential stability analysis for impulsive stochastic genetic regulatory networks (GRNs) with time-varying delays and reaction-diffusion driven by fractional Brownian motion (fBm). By constructing a Lyapunov functional and using linear matrix inequality for stochastic analysis we derive sufficient

conditions to guarantee the exponential stability of the stochastic model of impulsive GRNs in the mean-square sense. Meanwhile, the corresponding results are obtained for the GRNs with constant time delays and standard Brownian motion. An example is presented to illustrate our results of the mean-square exponential stability analysis.

Keywords: genetic regulatory networks; exponential stability; impulsive; reaction-diffusion; factional Brownian motion

1 Introduction

A genetic regulatory network (GRN) is a dynamic system to depict the interactions be-tween genes (mRNA) and proteins. Since GRNs play a key role in the area of cell and molecular biology, they have received increasing attention in the community of mathe-matical biology in recent years (see references [–]). An important topic related to math-ematical analysis of GRNs is to investigate the stability of GRNs. Wuet al.[] and Wang [] conducted robust stability analysis of GRNs by using stochastic analysis approach. Wang

et al.[] investigated the mean-square exponential stability of stochastic GRNs with time-varying delays by constructing a Lyapunov-Krasovskii functional. Although this stability analysis leads to conclusions on whether solutions of GRNs converge to an equilibrium point when a GRN system becomes stable, this analysis does not give the convergence rate of the system. For many GRN systems, slow convergence rates are undesirable, and high convergence rates (e.g., exponential rates) are needed. Therefore, it is necessary to study the exponential stability for a GRN system.

The aim of this study is to investigate the mean-square exponential stability of the solu-tion of a GRN model with diffusion process, impulses, degradasolu-tion reacsolu-tions, time-varying delays, and fBm of extrinsic noise. Stability analysis for such a comprehensive GRN model is rare in the literature. For example, Wuet al.[], Wang [], Wanget al.[], and Wang

et al.[] studied the stability and convergence of GRNs with stochastic perturbation and time delays but without diffusion and reaction. Although Maet al.[, ] investigated

the asymptotic stability of GRNs with diffusion, reaction, and delays, the systems of their study [, ] are deterministic. In [], finite-time robust stochastic stability was consid-ered under stochastic perturbation, reaction, diffusion, and delays. In addition, like in [, –], the stochastic perturbation in [] was described using a standard Brownian motion instead of a fractional Brownian motion.

Our study uses fBm to describe extrinsic noise introduced into the GRNs for mean-square exponential stability analysis, which is a novelty of this paper. We denote an fBm as

BH_{(t), whereHis the Hurst parameter. The fBm has a long-memory in comparison with} standard Brownian motion (H=_) []. In [], it was shown that a fractional Brownian motion can be used to describe a subdiffusive dynamics process. Experimental data of chromatin mobilizations show that an fBm is more appropriate to model gene movements than a standard Brownian motion []. Therefore, introducing the long-term correlations described by an fBm in GRNs is an important contribution to the literature.

Another contribution of this study is to introduce impulses into stochastic GRNs to describe sudden changes in the amount of mRNA and proteins. According to [–], an impulse is referred to the phenomenon that a system state is changed abruptly at a given time. The changes may be caused by abrupt change of physical environments, such as intake of drug or nutriment and exertion of the external force. For example, it was pointed out in [] that metaphase progression can be controlled by external mechanical impulses through different mechano-chemical cellular reactions. We have not found references of stability analysis of stochastic GRNs with impulses. Therefore, out study differs from the existing studies (e.g., Zhouet al.[], Wanget al.[], Maet al.[], Hanet al.[]) on the following two aspects: (a) a diffusion-reaction process driven by a fractional Brownian motion is considered, and (b) the impulses are involved.

The rest of this paper is organized as follows: In Section , we introduce the impulsive stochastic GRNs and define the exponential stability. Sufficient conditions of exponential stability in the mean-square sense for trivial solutions of GRNs are established in Section . Section  illustrates our analysis by a numerical example.

2 GRNs and preliminaries

In Section ., we first define a deterministic GRN with time-varying delays and then in-troduce an impulse and stochastic perturbation into the deterministic model to define the model that is investigated in this study. The preliminaries needed for the exponential stability analysis are given in Section ..

2.1 Deterministic and stochastic GRNs

A deterministic GRN is defined as follows []:

⎧ ⎨ ⎩

∂m˜i(t,x)

∂t =

l k=

∂ ∂xk(Dik

∂m˜i(t,x)

∂xk ) –aim˜i(t,x) + n

j=wijgj(p˜j(t–σ(t),x)) +qi, ∂p˜i(t,x)

∂t =

l

k=∂x∂k(D

∗ ik

∂p˜i(t,x)

∂xk ) –cip˜i(t,x) +bim˜i(t–τ(t),x), i= , , . . . ,n,

()

the degradation rates of the mRNA and protein, respectively;wijis defined as follows:

wij=

⎧ ⎪ ⎨ ⎪ ⎩

δij, jis an activator of genei, –δij, jis a repressor of genei, , there is no link from genejtoi,

whereδijis the dimensionless transcriptional rate of transcriptional factorjto genei;gjis the activation function of thegj(s) = s

h

+sh, wherehis the Hill coefficient;qi=

j∈Ijδij, where Ijdenote the set of all repressors of genej; andσ(t) andτ(t) are the time-varying delays satisfying

≤τ(t)≤ ¯τ, τ˙(t)≤μ< ,

≤σ(t)≤ ¯σ, σ˙(t)≤μ< ,

()

whereτ¯,σ¯,μ, andμare nonnegative real numbers, andμ¯ =μ∨μis assumed.

Considering the time delays, we can give the initial conditions associated with () as follows:

˜

mi(t,x) =ϕi(t,x), x∈Q,s∈[–d, ],i= , , . . . ,n,

˜

pi(t,x) =ϕi∗(t,x), x∈Q,s∈[–d, ],i= , , . . . ,n,

whered=σ¯∨ ¯τ, andϕi(t,x),ϕi∗(t,x)∈C([–d, ]×Q,R). Moreover, the following Dirichlet boundary conditions are considered:

˜

mi(t,x) = , x∈∂Q,t∈[–d, ],

˜

pi(t,x) = , x∈∂Q,t∈[–d, ].

Letm∗= (m∗_,m∗_, . . . ,m∗_n) andp∗= (p∗_,p∗_, . . . ,p∗_n) denote the unique solution of () and the equilibrium point (m∗,p∗) of system () to the origin. Using the transformations

mi=m˜i–m∗i andpi=p˜i–p∗i (i= , , . . . ,n), we can transform () into the matrix form as follows:

⎧ ⎨ ⎩

∂m(t,x)

∂t =

l k=

∂ ∂xk(Dk

∂m(t,x)

∂xk ) –Am(t,x) +Wf(p(t–σ(t),x)),

∂p(t,x)

∂t =

l k=

∂ ∂xk(D

∗ k

∂p(t,x)

∂xk ) –Cp(t,x) +Bm(t–τ(t),x),

()

where

A=diag(a,a, . . . ,an), B=diag(b,b, . . . ,bn), C=diag(c,c, . . . ,cn),

Dk=diag(Dk,Dk, . . . ,Dnk), m(t,x) =

m(t,x),m(t,x), . . . ,mn(t,x)

T ,

D∗_k=diagD∗_k,D∗_k, . . . ,D∗_nk, p(t,x) =p(t,x),p(t,x), . . . ,pn(t,x)

T ,

fpt–σ(t),x=fpt–σ(t),x,fpt–σ(t),x, . . . ,fn

pt–σ(t),xT,

fi

pt–σ(t),x=gi

pi

t–σ(t),x+p∗_i–gi

Now we introduce impulses and stochastic perturbation into account, and equations () become

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

dm(t,x) =l_k= ∂ ∂xk(Dk

∂m(t,x)

∂xk )dt–Am(t,x)dt+Wf(p(t–σ(t),x))dt

+S(t,m(t,x),p(t,x))dBH(t), t=tk,t≥,

dp(t,x) =l_k=∂x∂

k(D

∗ k

∂p(t,x)

∂xk )dt–Cp(t,x)dt+Bm(t–τ(t),x)dt, t=tk,t≥,

Δm(tk,x) =m(t_k+,x) –m(tk,x) =Ukm(tk,x), k∈N,

Δp(tk,x) =p(t+k,x) –m(tk,x) =Vkp(tk,x), k∈N,

()

wherem(t+

k,x) =limt→t_k+m(t,x),p(t+k,x) =limt→t_k+p(t,x),tk represent the moments when impulses occur,tk<tk+,limt→∞tk=∞, andBH(t) denotes ann-dimensional fBm with Hurst parameterH∈(,_].

2.2 Preliminaries

In this section, we introduce some necessary definitions, assumptions, and lemmas needed for the subsequent discussion. Let (Ω,F,P) be a complete probability space with filtration{Ft}t≥satisfying the usual conditions (i.e., it is increasing and right continuous,

andFcontains allP-null sets). For convenience, letATdenotes the transpose of a matrix A,λmax(A) andλmin(A) denote the largest and smallest eigenvalues of a square matrixA,

respectively.

The vector norm · is defined as

y(t,x) =

Q

yT(t,x)y(t,x)dx /

, ∀y(t,x)∈C[, +∞]×Q,Rn,

z(t,x) _d=

Q

sup –d≤t≤

zT(t,x)z(t,x)dx /

, ∀z(t,x)∈C[–d, ]×Q,Rn,

and for a real square matrixA= (aij)n×n, its normAp(p= , ,∞) is defined as

A=max

j n

i=

|aij|, A=

λmax

AT_A, A_∞=max i

n

j= |aij|,

and we writeAp=A(p= , ,∞) without causing any confusion.

Definition  According to [], the trivial solution of system () with initial valuesϕ(t,x),

ϕ∗(t,x)∈C_{([–d, ]×Q,R}n_{) is said to be exponentially stable in the mean-square sense if} there exist constantsα,α,M,M>  such that

E m(t,x) ≤Mϕe–αt, E p(t,x) ≤M ϕ∗ e–αt_{, t≥,}

where

ϕ= sup

s∈[–τ¯,],x∈Q

m(s,x) , ϕ∗ = sup

s∈[–σ¯,],x∈Q

p(s,x) .

Lemma  Let R> and R> be a pair of diagonal matrices.Then

positive definite matrix.Then we have

xTAy≤xTAx+yTAy,

i|ani|)are positive definite diagonal matrices, and|a| is the absolute value of a real

number a.

The proof is completed.

To investigate the mean-square exponential stability of trivial solution for system (), we introduce the following conditions:

(A) The functionf(p(t,x))satisfies the Lipschitz condition: there exists a positive

con-stantKsuch that

(A) The noise intensityS(t,m(t,x),p(t,x))in equation () satisfies the condition

traceSTt,m(t,x),p(t,x)St,m(t,x),p(t,x)

≤mT(t,x)Am(t,x) +pT(t,x)Ap(t,x),

whereAandAare known matrices.

(A) There exist positive definite matricesP,P,Q,Q such that the following linear

matrix inequalities are satisfied:

–π



 PDL– PA+ Q+ 

 –μ¯Q+Ht

H–_{Q> ,}

–π



 PD

∗

L– PC+ Q+ K

 –μ¯Q+Ht

H–_{Q> ,}

()

whereQ,Q,Q,Qof the same forms asA,Ain Lemma  are positive definite

diagonal matrices, and

DL=diag

_l

k= Dk

L

k ,

l

k= Dk

L

k , . . . ,

l

k= Dnk

L

k

,

D∗_L=diag _l

k= D∗_k

L

k ,

l

k= D∗_k

L

k , . . . ,

l

k= D∗_nk

L

k

.

(A) ρ≡sup_k∈N(tk–tk–) <∞,

 <λρ< –ln

λ(β+β) + d

 –μ¯(λK+λ)

, ()

whereβ≡supk∈NI+Uk,β≡supk∈NI+Vk,λ=max{λmax(P),λmax(P)},

λ=λmax(Q),λ=λmax(Q).

(A∗) ρ≡sup_k∈N(tk–tk–) <∞, <λρ< –ln[λ(β+β) +d(λK+λ)].

(A) There exists a positive constantηsuch that

ln(/λ) tk–tk–≤

η≤α, k= , , . . . , ()

whereλ=min{λmin(P),λmin(P)}, andαwill be defined in ().

3 Exponential stability

In this section, we establish conditions of the exponential stability for system () by con-structing a suitable Lyapunov function.

Theorem  If assumptions(A)-(A)hold,then the trivial solution of system()is globally exponentially stable in the mean-square sense.

Proof Define a Lyapunov function as follows:

where

Using Itô’s formula, we have

dV= 

According to the Dirichlet boundary conditions, Green’s formula, and Lemma , we have

– 

where n denotes the outer normal vector of∂Q, and

Substituting () and () into () and using Lemma , we obtain

and

where Lemma  is applied. Combining (), (), and () and taking the expectation of both sides, we derive that

and

EVt,m(t,x),p(t,x)≤λ

E m(t,x) +E p(t,x) 

+ λK  –μ¯

tk

tk–σ(tk)

E p(s,x) ds

+ λ  –μ¯

tk

tk–τ(tk)

E m(s,x) ds, k= , , , . . . , ()

whereλ,λ,λ, andλwere defined in assumption (A) and (A), respectively. Thus, we

obtain

E m(t,x) +E p(t,x) ≤ 

λ

EVt,m(t,x),p(t,x). ()

Substituting () into (), we have

EdVt,m(t,x),p(t,x)≤ λ

λ

EVt,m(t,x),p(t,x)dt

≡λEVt,m(t,x),p(t,x)dt, ()

whereλ=λ

λ. Integrating the both sides of inequality () fromtk–tot(t∈(tk–,tk],k=

, , . . .), we obtain

EVt,m(t,x),p(t,x)≤V,m(,x),p(,x)·eλt, t∈[,t], ()

and

EVt,m(t,x),p(t,x)

≤Vt_k+_–,mt_k+_–,x,pt+_k–,x·eλ(t–tk–)_{, t}∈_(t

k–,tk],k= , , . . . . ()

We define the functionras follows:

r(z) = (z+λ)ρ+ln

λ(β+β) + d

 –μ¯(λK+λ)e zd

, z∈[, +∞).

Then, we have

r(z) =ρ+ d

_(λK+λ )ezd

λ( –μ¯)(β+β) +d(λK+λ)ezd

> , z∈[,∞),

andr(z)→+∞(z→+∞). Moreover, from () we haver() < . Thus, there exists a unique positive numberαsuch thatr(α) = , that is,

(α+λ)ρ= –ln

λ(β+β) + d

 –μ¯(λK+λ)e αd

Fort∈[,t], we have by () that

EVt,m(t,x),p(t,x)≤λ

E m(,x) +E p(,x) 

+ λK  –μ¯



–σ()

E p(s,x) ds

+ λ  –μ¯



–τ()

E m(s,x) ds

≤

λ+ d

 –μ¯(λK+λ)

ψeλt_.

Therefore,

E m(t,x) ≤ 

λEV

t,m(t,x),p(t,x)≤ 

λ

λ+ d

 –μ¯(λK+λ)

ψeλt

= 

λ

λ+ d

 –μ¯(λK+λ)

e(λ+α)ρψe–αρ

≤M

λ

ψe–αt, t∈[,t], ()

whereM= (λ+_–d_μ¯[λK+λ])e(λ+α)ρ> , andα is defined in (). Similarly, we can

conclude that

E p(t,x) ≤Mψe–αt, t∈[,t]. ()

Equations () and () then yield

sup

t–τ(t)≤s≤t,x∈Q

E m(t,x) ≤M

λ

ψe–α(t–τ(t))_≤M

λ

ψe–αt_eατ¯ ,

sup

t–σ(t)≤s≤t,x∈Q

E p(t,x) ≤M

λ

ψe–α(t–σ(t))_≤M

λ

ψe–αt_eασ¯ .

()

From the last two equations of system () we conclude that

E mt+_k,x =E (I+Uk)m(tk,x) 

≤ (I+Uk) ·E m(tk,x) ≤βE m(tk,x) ,

E pt_k+,x =E (I+Vk)p(tk,x)  ()

≤ (I+Vk)



·E p(tk,x)



≤βE p(tk,x)



,

k= , , . . . .

Thus, according to ()-(), we have

EVt_+,mt+_,x,pt_+,x

≤λ

+ λK

Now, we will show that

EVt_k+,mt+_k,x,pt_k+,x≤e–(λ+α)ρM

In addition, from () and () it follows directly that

+   –μ¯

ti+

ti+–τ(ti+)

E m(s,t) ds

≤λ

βE m(ti+,x) 

+βE p(ti+,x) 

+ λK  –μ¯

ti+

ti+–σ(ti+)

E p(s,t) ds

+ λ  –μ¯

ti+

ti+–τ(ti+)

E m(s,t) ds

≤

λ(β+β) +

λKσ¯

 –μ¯ e ασ¯

+ λτ¯  –μ¯e

ατ¯

M

λi+ 

ψe–αti+

≤

λ(β+β) + d

 –μ¯(λKλ)e αd

M

λi_+ψ

_e–αti+

=e–(λ+α)ρM

λi+ 

ψe–αti+_,

which shows that () holds fork=i+. Therefore, () is true for everyk= , , . . . . Hence, fort∈(tk,tk+], we can conclude by () that

E m(t,x) ≤ 

λE

Vt,m(t,x),p(t,x)≤ 

λE

Vt_k+,mt+_k,x,pt_k+,xeλ(t–tk)

≤e–(λ+α)ρ M λk_+ψ

_e–αtk_eλρ_=e–αρ M

λk_+ψ

_e–αtk

≤e–α(t–tk)M

λk_+ψ

_e–αtk₌ M

λk_+ψ

_e–αt

≤M

λ

eηtk_ψ_e–αt≤M

λ

ψe–(α–η)t, k= , , . . . . ()

Similarly, we have

E p(t,x) ≤ M

λk_+ψ

_e–αt_≤M λ

ψe–(α–η)t, k= , , . . . , ()

which, together with (), (), and (), show that the trivial solution of system () is

globally exponentially stable in the mean square sense.

Remark  If the time delays are constant functions with respect tot, that is,τ(t) =τ,

σ(t) =σ, then Theorem  reduces to the following result.

Corollary  If assumptions(A)-(A), (A∗),and(A)hold,then the trivial solution of system()withτ(t) =τ,σ(t) =σis globally exponentially stable in the mean-square sense.

Corollary  If assumptions(A), (A), (A),and(A)hold and Eq. ()in(A)is replaced

then the trivial solution of system()is globally exponentially stable in the mean-square sense.

4 Numerical simulations

In this section, we illustrate our results in a numerical example. Without loss of generality, we consider a two-dimensional system (n= ,l= ) and choose the parameters of system () as follows:

By using MATLAB to solve the inequalities given in conditions (A)-(A), we get the fol-lowing feasible solution:

We know that the conditions of Theorem  are satisfied. By Theorem  we can conclude that the trivial solution of system () is exponential stability in the mean-square sense.

Figure 1 Contour plot and sectional view of mRNA concentrationm1(t,x).

Figure 2 Contour plot and sectional view of mRNA concentrationm2(t,x).

Figure 4 Contour plot of and sectional view protein concentrationp2(t,x).

atx= .. The four figures show that the concentrations of both mRNA and protein are exponentially stable, indicating effectiveness of the results derived in Theorem .

5 Conclusions

In this paper, we analyze the mean-square exponential stability for the comprehensive GRNs with (a) diffusion-reaction, (b) time-varying delay, (c) impulsive control, and (d) fBm for extrinsic noise. The stability analysis is a more challenging than the previous analysis reported in the literature that considered only one or two of the four model components. Our derived conditions of the mean-square exponential stability have a simple form and can be used for evaluating the exponential stability of GRNs in a numerically straightfor-ward manner.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors have equal contributions to the writing of this paper. All authors have read and approved the final version of the manuscript.

Author details

1_{School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, P.R. China.}2_{Department of Scientific} Computing, Florida State University, Tallahassee, FL 32306-4120, USA.

Acknowledgements

The research was supported in part by the National Natural Science Foundation of China (Nos. 11461053 and 11661064) and the Innovation Project of Ningxia University (GIP201624). The third author was supported in part by the DOE Early Career Award DE-SC0008272 and NSF-EAR grant 1552329.

Received: 28 September 2016 Accepted: 21 November 2016

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