Zou et al. Advances in Difference Equations (2017) 2017:250 DOI 10.1186/s13662-017-1262-7

RESEARCH

Open Access

Robust stability of interactional genetic regulatory networks with reaction-diffusion terms Chengye Zou1 , Xiaopeng Wei1* , Qiang Zhang1* and Changjun Zhou2 *

Correspondence: [email protected]; [email protected] 1 Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian, 116024, China Full list of author information is available at the end of the article

Abstract We have analyzed the stability of interactional genetic regulatory networks with reaction-diffusion terms under Dirichlet boundary conditions in this article. Corresponding to interaction between unstable genetic regulatory networks and stable genetic regulatory networks, the model is given, and a stability criterion is proposed through construction of appropriate Lyapunov-Krasovskii functions and linear matrix inequalities (LMI). By means of a numerical simulation, we have proved the effectiveness and correctness of the theorem, and we analyzed the factors that influence the stability for interactional genetic regulatory networks. Keywords: interactional genetic regulatory networks; Dirichlet boundary; reaction-diffusion; linear matrix inequalities

1 Introduction Since , the Alon research group has proposed the network module [–], network modules with several nodes become a hot research topic. Modeling and dynamic analysis of these genetic regulatory networks (GRNs), which are considered as important submodule of complex biology network, because a GRN can clarify the mechanism of biological network. At present, the models of genetic regulatory network include directed graphs [–], the Boolean model [–], the Bayesian model [–] and the differential equations model [–]. In these models, the differential equations model has obvious advantages, for example, the differential equations model is more accurate than the Boolean model in describing GRNs, and it has less computational complexity than the Bayesian model. The differential equations model is an open model, because a great deal about dynamic systems can be directly applied to this model, which has attracted a large number of other field experts to join the relevant research, and reaped rich fruits. A real biological network contains tens of thousands of nodes, while the simulations of these studies are based on a few nodes as the research object, the theoretical basis of its simplification is that the number of molecules involved in the chemical reaction is usually very low at a given moment. Although there are many outstanding achievements in the study of differential equations as a model of the GRN, there are still some problems that needed further research. © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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On the one hand, the spatial diffusion phenomenon exists widely in the fields of physics, chemistry, biology, and so on, but most of the current studies on GRNs in terms of the spatial homogeneity of concentrations are for cell components. This proposition can lead to missing a lot of space information, but there are only five articles [, , , , ] about the study of GRNs with reaction-diffusion terms, and the five articles have only discussed the influence of reaction-diffusion terms on the time-delay conservatism,but they did not explore the essence of reaction-diffusion terms. On the other hand, corresponding to the different biological signals, the change of the expression of a protein will affect the expression of gene, which makes the network achieve stability ultimately. Lest life activity is only the result of a local GRN, it also has extensive connections with the surrounding GRN, and the dynamical properties of these GRNs also directly affect the survival of the living body. For example, a virus cannot survive independently, and its replication and transmission must be completed by the synthesis system, the replication system and the protein transport system of the host cell. A large number of studies showed that, after the host is infected by the virus, when the virus is active, enzymes and mRNA concentrations for viral replication will be unstable; when the virus is repressed or dormant, the virus replication related enzymes and mRNA concentrations will tend to stability [–]. In recent years, it was found that there is a wide and complex relation between the virus and the host at the molecular level based on the proteins atlas of interaction between viral and host [–]. In this article we have proposed a model of interactional genetic regulatory networks (GRNs), and we analyzed the stability of interactional GRNs, the theoretical support for the above method is given from the aspect of dynamics. By numerical simulation, three significant conclusions are obtained.

2 Problem formulation Two different nonlinear delayed GRNs are described by equations () and (), respectively: ⎧ ⎨ dui (t) = –a u (t) + n ω f (v (t – σ (t))), i i j= ij j j dt ⎩ dvi (t) = –ci vi (t) + bi ui (t – τ (t)), i = , , . . . , n ,

()

dt

⎧ ⎨ dup (t) = –a u (t) + n  g (v (t – σ  (t))), p p q= pq q q dt ⎩ dvp (t) = –cp vp (t) + bp up (t – τ  (t)), p = , , . . . , n ,

()

dt

where GRN () is stable, and GRN () is unstable, ui (t), vi (t) ∈ Rn and up (t), vp (t) ∈ Rn are the concentrations of mRNA and protein of the ith and the pth nodes at the time t respectively; the parameters ai and ap are the degradation rates of the mRNA, ci and cp are the degradation rates of the protein, bi and bp are the translation rate; fi (x) and gp (y) are the Hill form regulatory functions, which represent the feedback regulation of the protein on the transcription, their forms are described by equation () ⎧ ⎪ ⎪ ⎪ ⎨fj (x) = ⎪ ⎪ ⎪ ⎩gq (y) =

H ( mx ) j j

H +( mx ) j

,

j y ( n q )Hq y +( nq )Hq

() ,

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where Hj and Hq are the Hill coefficients, mj and nq are positive constants, τ (t), τ  (t), σ (t) and σ  (t) are time-varying delays satisfying ⎧ ⎪  ≤ τ ≤ τ (t) ≤ τ ,  ≤ σ ≤ σ (t) ≤ σ , ⎪ ⎪ ⎪ ⎪ ⎨λ ≤ τ˙ ≤ λ , λ ≤ σ˙ ≤ λ ,     ⎪ ⎪  ≤ τ ≤ τ (t) ≤ τ ,  ≤ σ ≤ σ  (t) ≤ σ , ⎪ ⎪ ⎪ ⎩ η ≤ τ˙  ≤ η , η ≤ σ˙  ≤ η ,

()

where W = (ωij ) ∈ Rn ×n and W = (pq ) ∈ Rn ×n are described as equations () and (), αij and βpq are the dimensionless transcriptional rates of transcriptional factor j to gene i and transcriptional factor q to gene p respectively. ⎧ ⎪ ⎪ ⎨αij

if transcription factor j is an activator of gene i,

ωij =  if there is no link from node j to i, ⎪ ⎪ ⎩ –αij if transcription factor j is a repressor of gene i, ⎧ ⎪ if transcription factor q is an activator of gene p, ⎪βpq ⎨ pq =  if there is no link from node q to p, ⎪ ⎪ ⎩ –βpq if transcription factor q is a repressor of gene p.

()

()

Considering the diffusion term, equations () and () can be rewrite as ⎧ ⎨ ∂ui (t,l) = L k= ∂t ⎩ ∂vi (t,l) = L ∂t

 ∂ i (t,l) (Dik ∂u∂l ) – ai ui (t, l) + nj= ωij fj (vj (t ∂lk k ∂ ∗ ∂vi (t,l) k= ∂lk (Dik ∂lk ) – ci pi (t) + bi ui (t – τ (t)),

– σ (t), l)),

()

i = , , . . . , n

⎧ ∂u (t,l)  n ∂up (t,l) L p ∂  ⎪ ⎪ ⎨ ∂t =  k= ∂lk (dpk ∂lk ) – ap up (t, l) + q= pq gq (vq (t – σ (t), l)), ∂vp (t,l) ∗ ∂vp (t,l) = Lk= ∂l∂ (dpk ) – cp vp (t, l) + bp up (t – τ  (t), l), ∂t ∂lk k ⎪ ⎪ ⎩ p = , , . . . , n

()

where l = (l , l , . . . , lL )T ∈  ⊂ Rc ,  = {l||lk | ≤ Lk }, Lk is constant, k = , , . . . L, Dik = Dik (t, l) > , D∗ik = D∗ik (t, l) > , denote the transmission diffusion operator along the ith

∗ ∗ = dpk (t, l) > , denote the gene of mRNA and protein, respectively, dpk = dpk (t, l) > , dpk

transmission diffusion operator along the pth gene of mRNA and protein, respectively. The initial conditions are given by ⎧ ⎨u (s, l) = ψ (s, l), s ∈ (–∞, ], i = , , . . . , n i i  ⎩vi (s, l) = ψ ∗ (s, l), s ∈ (–∞, ], i = , , . . . , n i ⎧ ⎨u (s, l) = ψ (s, l), s ∈ (–∞, ], p = , , . . . , n p p  ⎩vp (s, l) = ψ ∗ (s, l), s ∈ (–∞, ], p = , , . . . , n

()

()

p

∗ Here ψi (s, l), ψi∗ (s, l), ψp (s, l) and ψp (s, l) are bounded and continuous on (–∞, ] × .

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Dirichlet boundary condition is considered: ⎧ ⎨u (t, l) = , l ∈ ∂, t ∈ [–κ, +∞), i ⎩vi (t, l) = , l ∈ ∂, t ∈ [–κ, +∞), ⎧ ⎨u (t, l) = , l ∈ ∂, t ∈ [–κ, +∞), p ⎩vp (t, l) = , l ∈ ∂, t ∈ [–κ, +∞).

()

()

The system () and system () can be rewritten in vector-matrix form: ⎧ ⎨ ∂u (t,l) = l k= ∂t ⎩ ∂v (t,l) = l

()

∂t

∂ (Dk ∂u∂x (t,l) ) – A u (t, l) + W F(v (t – σ (t), l)), ∂lk k ∂ ∗ ∂v (t,l) i = , , . . . , n k= ∂lk (Dk ∂lk ) – C v (t, l) + B u (t – τ (t), l),

()

∂t

∂  (t,l) (dk ∂u∂x ) – A u (t, l) + W G(v (t – σ  (t), l)), ∂lk k ∂ ∗ ∂v (t,l)  p = , , . . . , n k= ∂lk (dk ∂lk ) – C v (t) + B u (t – τ (t), l),

⎧ ⎨ ∂u (t,l) = l k= ∂t ⎩ ∂v (t,l) = l

A = diag(a , a , . . . , an ), B = diag(b , b , . . . , bn ), C = diag(c , c , . . . , cn ), A = diag(a , a , . . . , an ), B = diag(b , b , . . . , bn ), C = diag(c , c , . . . , cn ), Dk = diag(Dk , Dk , . . . , Dn k ),   D∗k = diag D∗k , D∗k , . . . , D∗n k , dk = diag(dk , dk , . . . , dn k ),   ∗ ∗ , dk , . . . , dn∗ k , dk∗ = diag dk  T u (t, l) = u (t, l), u (t, l), . . . , un (t, l) ,  T v (t, l) = v (t, l), v (t, l), . . . , vn (t, l) ,  T u (t, l) = u (t, l), u (t, l), . . . , un (t, l) ,  T v (t, l) = v (t, l), v (t, l), . . . , vn (t, l) ,         T F(vj t – σ (t), l = (f (v t – σ (t), l , f (v t – σ (t), l , . . . , fn vn t – σ (t), l ,         T G(vq t – σ  (t), l = (g (v t – σ  (t), l , g (v t – σ  (t), l , . . . , gn vn t – σ  (t), l . fi (·) and gp (·) satisfy inequality () and inequality (), respectively, because fi (·) and gp (·) are monotonically increase functions with saturation ≤

fi (xi ) ≤ ςi , xi

≤

gp (yp ) ≤ χp , yp

∀xi = , i = , , . . . , n ∀yp = , p = , , . . . , n

() ()

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i.e.   f T (x) f (x) – K x ≤ ,   g T (y) g(y) – K y ≤ ,

() ()

where K = diag(ς , ς , . . . , ςn ) > , K = diag(χ , χ , . . . , , χn )T > , x = [x , x , . . . , xn ]T and y = [y , y , . . . , yn ]T . Lemma  Let f (v) be a real-valued function defined on [a, b] ⊂ R, with f (a) = f (b) = . If f (v) ∈ C  [a, b], then

b

(b – a) f (v) dv ≤ π





a

b

 f  (v) dv.

()

a

Lemma  If  is a bounded C  open set in Rn and η, ϕ ∈ C  (), then



ηϕ dx =



ηϕ dx + 

η ∂

∂ϕ ∂η –ϕ dS, ∂n ∂n

()

∂η where ∂n and ∂ϕ are the directional derivatives of η and ϕ in the direction of the outward ∂n  pointing normal n to the surface element dS, respectively. lk= ∂x∂ (Dk ∂x∂ ) can be regarded k k as a Laplacian operator which is formally self-adjoint and a differential; in Lemma  we have an inner product for a function with Dirichlet boundary.

Lemma  ([]) From the Green formula, under Dirichlet boundary conditions, and by using Lemma  and Lemma , we can obtain

l  ∂μ π ∂ dx ≤ – μT μ dx.  μ ∂xk ∂xk   

T

()

k=

Lemma  Let M >  ∈ Rn×n , a positive scalar ϑ > , vector function x : [, ϑ] → Rn such that the integrations concerned are well defined, and they exist:

ϑ

x(s) ds 

T M

ϑ

x(s) ds ≤ ϑ



ϑ

x(s)Mx(s) ds .

()



Lemma  For vectors X, Y ∈ Rn are any positive definite matrix, and any scalar ε, there exists the following inequality: X T Y ≤ εX T X + ε– Y T Y .

()

Lemma  For any vectors X, Y ∈ Rn , and any scalar ε >  are positive, there exists the following inequality: X T HY ≤ εX T HX + ε– Y T HY .

()

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3 Model of interactional GRNs According to GRN () and (), W or W express the interaction of genes in single GRN, we assumed that interaction of the different GRNs is like a single GRN. We have constructed a bidirectional coupling model for the type of interactional GRNs as equations () and (), and we investigate a stability criterion under Dirichlet boundary condition. ⎧ l du (t,l) ⎪ ⎪ ⎨ dt = k= ⎪ ⎪ ⎩ dv (t,l) dt

∂ (Dk ∂u∂x (t,l) ) – A u (t, l) + W F(v (t ∂lk k + W∗ G(v (t – σ  (t), l)),

=

l

∂ ∗ ∂v (t,l) k= ∂lk (Dk ∂lk ) – C v (t, l) + B u (t

– σ (t), l))

– τ (t), l),

⎧ l ∂ du (t,l) ∂u (t,l) ⎪ ⎪ ⎨ dt = k= ∂lk (dk ∂xk ) – A u (t, l) + W G(v (t – σ  (t), l)) + W∗ F(v (t – σ (t), l)), ⎪ ⎪ ⎩ dv (t,l) l ∂ ∗ ∂v (t,l) = k= ∂l (dk ∂l ) – C v (t, l) + bp u (t – τ  (t), l), dt k

k

() i = , , . . . , n

() p = , , . . . , n

where W∗ = (ωiq ) ∈ Rn ×n , W∗ = (pj∗ ) ∈ Rn ×n are described by equations () and (), ∗ and βpj∗ are the dimensionless transcriptional rates of transcriptional factor q of GRN αiq () to gene i of GRN () and transcriptional factor j of GRN () to gene p of GRN (), respectively. ⎧ ∗ ⎪ if transcription factor q is an activator of gene i, ⎪ ⎨αiq ωiq =  if there is no link from node q to i, ⎪ ⎪ ⎩ ∗ –αiq if transcription factor q is a repressor of gene i, ⎧ ∗ ⎪ if transcription factor j is an activator of gene p, ⎪ ⎨βpj ∗ pj =  if there is no link from node q to p, ⎪ ⎪ ⎩ ∗ –βpj if transcription factor j is a repressor of gene p.

()

()

Theorem For given scalars τ , σ , τ , σ , λ , λ , η and η satisfying equation (), GRN () and GRN () under a Dirichlet boundary condition are robust stable if there exist matrices PiT = Pi >  and Ti = i >  (i = , . . . , ); RTi = Ri >  (i = , . . . ); QTi = Qi >  (i = , . . . , ), such that the following linear matrices inequalities (LMIs) hold:  = –P A –

π P DL + R + σ Q , 

 = –R + R ,  = (λ – )R + ( – λ )R + BT P B ,  = – + K K + (λ – )Q ,  = –P A –

π P dL + εP W + R + σ  Q , 

 = –R + R ,  = (η – )R + ( – η )R + BT P B ,

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 = (η – )Q –  + K K +  = –P C –

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P W , ε

π P D∗L + P + R + τ Q , 

 = –R + R ,  = (λ – )R + ( – λ )R + K  K ,  = Q –  ,  = –P C –

π P dL∗ + P + R + τ  Q , 

 = –R + R ,  = (η – )R + ( – η )R + K  K ,  = Q –  , ⎡   ⎢ ⎢ ∗  ⎢ ⎢∗ ∗ ⎢ ⎢ ∗ ⎢∗ ⎢ ⎢∗ ∗ ⎢ ⎢∗ ∗ ⎢  = ⎢ ⎢∗ ∗ ⎢ ⎢∗ ∗ ⎢ ⎢ ⎢∗ ∗ ⎢ ⎢∗ ∗ ⎢ ⎢ ∗ ∗ ⎣ ∗ ∗ < , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢  = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

  –R ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

  –R ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

    ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

    ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

    –Q ∗ ∗ ∗ ∗ ∗ ∗ ∗

    –Q ∗ ∗ ∗ ∗ ∗ ∗ ∗

P W      ∗ ∗ ∗ ∗ ∗ ∗

K        ∗ ∗ ∗ ∗ ∗ ∗

     W∗ P  ∗ ∗ ∗ ∗ ∗

       ∗ ∗ ∗ ∗ ∗

        ∗ ∗ ∗ ∗

        ∗ ∗ ∗ ∗

        –R ∗ ∗ ∗

        –R ∗ ∗ ∗

          ∗ ∗

          ∗ ∗

          –Q ∗

          –Q ∗

< .

⎤ P W∗ ⎥  ⎥ ⎥  ⎥ ⎥ ⎥  ⎥ ⎥  ⎥ ⎥  ⎥ ⎥ ⎥  ⎥ ⎥  ⎥ ⎥ ⎥  ⎥ ⎥  ⎥ ⎥ ⎥  ⎦  () ⎤

 ⎥  ⎥ ⎥  ⎥ ⎥ ⎥  ⎥ ⎥  ⎥ ⎥  ⎥ ⎥ ⎥ K   ⎥ ⎥  ⎥ ⎥ ⎥  ⎥ ⎥  ⎥ ⎥ ⎥  ⎦ 

()

Proof Define a Lyapunov-Krasovskii functional candidate for GRN () as

V (t, l) =

  i=

Vi (t, l)

()

Zou et al. Advances in Difference Equations (2017) 2017:250

Page 8 of 20

where



V (t, l) = 

uT (t, l)P u (t, l) dl









t–σ

vT (s, l)R v (s, l) ds dl +





–τ (t)

t+s







–σ (t)



t

vT (s, l)Q v (s, l) dθ ds dl,

()

    F T v (s, l) Q F v (s, l) ds dl,

()





+

+ 





vT (t, l)P v (t, l) dl,

t–τ



t–τ  (t)

+

uT (s, l)R u (s, l) ds dl

uT (s, l)R u (s, l) ds dl

t

vT (s, l)R v (s, l) ds dl +



t–σ



t–σ  (t)

vT (s, l)R v (s, l) ds dl

t–σ 

vT (s, l)R v (s, l) ds dl

t–σ





–τ  (t)

 t

t–σ  (t)

t+s



()

t

uT (s, l)Q u (s, l) dθ ds dl



–σ  (t)

t

t+s

vT (s, l)Q v (s, l) dθ ds dl,

    GT v (s, l) Q G v (s, l) ds dl,

then, computing the derivatives of Vi (t, m, p) (i = , , ), we can get ∂u (t, l) ∂v (t, l) dl +  vT (t, l)P dl ∂t ∂t    l  ∂ ∂u (t, l)    T =  u (t, l)P Dk – A u (t, l)W F v t – σ (t), l ∂lk ∂lk 

∂V (t, l) = ∂t

()

t–τ  (t)

t–σ



V (t, l) =



uT (s, l)R u (s, l) ds dl

t–τ



+ σ

+

t

+

V (t, l) = τ 

t+s

uT (t, l)P u (t, l) dl

t–τ

()

t

t–σ (t)





vT (s, l)R v (s, l) ds dl

uT (s, l)Q u (s, l) dθ ds dl



V (t, m, p) =

V (t, l) =

t–σ (t)



t

+σ



t–σ

vT (s, l)R v (s, l) ds dl,

V (t, l) = τ



uT (s, l)R u (s, l) ds dl

t–σ (t) t–σ



t–τ (t)



t

+ 

t–τ

uT (s, l)R u (s, l) ds dl

t–τ

+

V (t, l) =



()

t–τ (t)

+



vT (t, l)P v (t, l) dl,

uT (s, l)R u (s, l) ds dl +

t–τ





t

V (t, l) = 

+



uT (t, l)P

k=

() ()

Zou et al. Advances in Difference Equations (2017) 2017:250

+ W∗ G

   v t – σ  (t), l

Page 9 of 20







l  ∂ ∗ ∂u (t, l) + Dk ∂lk ∂lk  k=    – C v (t, l) + B u t – τ (t), l ,

vT (t, l)P

∂V (t, l) = ∂t







uT (t, l)R u (t, l) dl –





+ 



+  – τ˙ (t)













+ 



    uT t – τ (t), l R u t – τ (t), l dl

uT (t – τ , l)R u (t – τ , l) dl vT (t, l)R v (t, l) dl

– 



+  – σ˙ (t)





≤







    vT t – σ (t), l R v t – σ (t), l dl     vT t – σ (t), l R v t – σ (t), l dl

vT (t – σ , l)R v (t – σ , l) dl

uT (t, l)R u (t, l) dl –





+ 



+ ( – λ )







+ 



uT (t – τ , l)R u (t – τ , l) dl

    uT t – τ (t), l R u t – τ (t), l dl     uT t – τ (t), l R u t – τ (t), l dl

uT (t – τ , l)R u (t – τ , l) dl vT (t, l)R v (t, l) dl –

+



uT (t – τ , l)R u (t – τ , l) dl

– ( – λ )

–

vT (t – σ , l)R v (t – σ , l) dl

vT (t – σ , l)R v (t – σ , l) dl

  –  – σ˙ (t)

–



    uT t – τ (t), l R u t – τ (t), l dl

+

uT (t – τ , l)R u (t – τ , l) dl

uT (t – τ , l)R u (t – τ , l) dl

  –  – τ˙ (t)

–



()



vT (t – σ , l)R v (t – σ , l) dl

vT (t – σ , l)R v (t – σ , l) dl

– ( – λ )



+ ( – λ ) 

    vT t – σ (t), l R v t – σ (t), l dl     vT t – σ (t), l R v t – σ (t), l dl

Zou et al. Advances in Difference Equations (2017) 2017:250

Page 10 of 20

vT (t – σ , l)R v (t – σ , l) dl,

– 





∂V (t, l) = τ  (t) ∂t

()



uT (t, l)Q u (t, l) dl – τ (t)



+ σ  (t) 

vT (t, l)Q v (t, l) dl – σ (t) 

≤ τ



uT (t, l)Q u (t, l) dl –



+ σ





t–σ (t)

vT (s, l)Q v (s, l) ds dl

t

T



t

Q

t–τ (t)



 u (s, l) ds dl

t–τ (t)

 v (s, l) ds dl,

t

Q

t–σ (t)

()

t–σ (t)

    F T v (t, l) Q F v (t, l) dl



  –  – σ˙ (t)



      F T v t – σ (t), l Q F v t – σ (t), l dl



≤

F

 T

   v (t, l) Q F v (t, l) dl





      F T v t – τ (t), l Q F v t – τ (t), l dl,

+ (λ – ) 



∂V (t, l) = ∂t

t

u (s, l) ds



T

t

v (s, l) ds



∂V (t, l) = ∂t



uT (s, l)Q u (s, l) ds dl

vT (t, l)Q v (t, l) dl

– 

t–τ (t)





t

∂u (t, l) uT (t, l)P

dl + 

vT (t, l)P

()

∂v (t, l) dl ∂t

∂t   l

 ∂ ∂u (t, l) =  uT (t, l)P Dk – A u (t, l) ∂lk ∂lk  k=        + W G v t – σ  (t), l + W∗ F v t – σ (t), l 







l  ∂ ∗ ∂u (t, l) + Dk ∂lk ∂lk  k=    – C v (t, l) + B u t – τ  (t), l ,

vT (t, l)P

∂V (t, l) = ∂t







uT (t, l)R u (t, l) dl –



+ 

    uT t – τ , l R u t – τ , l dl

  –  – τ˙  (t)   +  – τ˙  (t) –



+ 



    uT t – τ , l R u t – τ , l dl







    uT t – τ  (t), l R u t – τ  (t), l dl     uT t – τ  (t), l R u t – τ  (t), l dl

    uT t – τ , l R u t – τ , l dl vT (t, l)R v (t, l) dl

()

Zou et al. Advances in Difference Equations (2017) 2017:250



    uT t – τ , l R u t – τ , l dl

–



vT (t, l)R v (t, l) dl

+ 



    vT t – σ , l R v t – σ , l dl +

– 

  –  – σ˙  (t)   +  – σ˙  (t)



    vT t – σ  (t), l R v t – σ  (t), l dl







    uT t – τ , l R u t – τ , l dl

    uT t – τ , l R u t – τ , l dl

+ 



– ( – η ) 

+ ( – η )





    uT t – τ  (t), l R u t – τ  (t), l dl     uT t – τ  (t), l R u t – τ  (t), l dl

    uT t – τ , l R u t – τ , l dl +

– 



    vT t – σ , l R v t – σ , l dl +

– 





+ ( – η ) 









vT (t, l)R v (t, l) dl     vT t – σ , l R v t – σ , l dl

   v t – σ  (t), l R v t – σ  (t), l dl  T

– ( – η )

    vT t – σ  (t), l R v t – σ  (t), l dl

 T

   v t – σ , l R v t – σ , l dl,

– 

∂V (t, l) = τ   (t) ∂t



+ σ   (t)

uT (t, l)Q u (t, l) dl





+ σ  – 



t



t–τ 

– τ (t)







()

vT (t, l)Q v (t, l) dl – σ  (t)

uT (t, l)Q u (t, l) dl –



    vT t – σ  (t), l R v t – σ  (t), l dl

uT (t, l)R u (t, l) dl –

≤ τ 



    vT t – σ , l R v t – σ , l dl

   v t – σ , l R v t – σ , l dl





∂V (t, l) = ∂t





 T

– ≤







Page 11 of 20





uT (s, l)Q u (s, l) ds dl

t

t–σ 

vT (s, l)Q v (s, l) ds dl

t

t–τ 

u (s, l) ds



T Q

t t–τ 

 u (s, l) ds dl

vT (t, l)Q v (t, l) dl T

t

t–σ 

v (s, l) ds

 Q

t t–σ 

 v (s, l) ds dl,

    GT v (t, l) Q F v (t, l) dl



  –  – σ˙ 



      GT v t – σ  , l Q G v t – σ  , l dl

()

Zou et al. Advances in Difference Equations (2017) 2017:250



Page 12 of 20

    GT v (t, l) Q G v (t, l) dl

≤ 



+ (η – )

     GT v (t – σ , l) Q G v t – σ  , l dl.

()



According to Lemma , we have  



uT (t, l)P



vT (t, l)P

 



uT (t, l)P



π ∂ ∗ ∂v (t, l) Dk dl ≤ – vT (t, l)P D∗L v (t, l) dl, ∂lk ∂lk   



l  π ∂ ∂u (t, l) uT (t, l)P dL u (t, l) dl, dk dl ≤ – ∂lk ∂lk   

()

()

k=

l  k=

()

k=

l  k=



l  π ∂ ∂u (t, l) Dk dl ≤ – uT (t, l)P DL u (t, l) dl, ∂lk ∂lk   



vT (t, l)P



π ∂ ∗ ∂v (t, l) vT (t, l)P dL∗ v (t, l) dl, d dl ≤ – ∂lk k ∂lk   

()

where

∂u (t, l) L ∂uT (t, l) ∂uT (t, l) T T T = u (t, l)P D , . . . , u (t, l)P DL , u (t, l)P Dk ∂lk ∂l ∂lL k=

∂v (t, l) L ∂v (t, l) ∂v (t, l) , vT (t, l)P D∗k = vT (t, l)P D∗k , . . . , vT (t, l)P D∗L ∂lk ∂l ∂lL k=

∂u (t, l) L ∂uT (t, l) ∂uT (t, l) uT (t, l)P dk , = uT (t, l)P d  , . . . , uT (t, l)P dL  ∂lk ∂l ∂lL k=

∂v (t, l) L ∂v (t, l) ∂v (t, l) vT (t, l)P dk∗ . = vT (t, l)P dk∗ , . . . , vT (t, l)P dL∗ ∂lk ∂l ∂lL k= Considering inequalities () and (), for diagonal matrices  > ,  > ,  >  and  > , the following inequalities hold:       F T v (t, l)  F v (t, l) – vT (t, l)K  F v (t, l) ≤ ,       F T v t – σ (t), l  F v t – σ (t), l      – vT t – σ (t), l K  F v t – σ (t), l ≤ ,       GT v (t, l)  G v (t, l) – vT (t, l)K  G v (t, l) ≤ ,       GT v t – σ  (t), l  G v t – σ  (t), l      – vT t – σ  (t), l K  G v t – σ  (t), l ≤ .

()

() ()

()

According to Lemma , we have inequalities as follows:      vT t – σ (t), l K  F v t – σ (t), l     ≤ vT t – σ (t), l K  K v t – σ (t), l       + F T v t – σ (t), l  F v t – σ (t), l ,

()

Zou et al. Advances in Difference Equations (2017) 2017:250

Page 13 of 20

     vT t – σ  (t), l K  G v t – σ  (t), l     ≤ vT t – σ  (t), l K  K v t – σ  (t), l       + GT v t – σ  (t), l  G v t – σ  (t), l ,   vT (t, l)P B u t – τ (t), l     ≤ vT (t, l)P v (t, l) + u t – τ (t), l BT P B u t – τ (t), l ,   vT (t, l)P B u t – τ  (t), l     ≤ vT (t, l)P v (t, l) + u t – τ  (t), l BT P B u t – τ  (t), l .

()

()

()

According to Lemma , for a positive scalar ε, there exist    u (t, l)P W G v t – σ  (t), l        ≤ εuT (t, l)P W u (t, l) + GT v t – σ  (t), l P W G v t – σ  (t), l . ε

()

Taking equations ()-() into consideration, derivatives of V (t, m, p) (i = , , ) can be formed as follows: ∂V (t, l)  ∂Vi (t, l) = . ∂t ∂t i= 

()

Taking inequalities ()-() into consideration, equation () is rewritten as  ∂V (t, l)  ∂Vi (t, l) = ∂t ∂t i=

T  ≤ ξ (t, l) ξ (t, l) + ξT (t, l) ξ (t, l) dl < ,

()



for ζ (t, x) = , where   T T T T ξ = u (t, l), u (t – τ , l), u (t – τ , l), u (t – τ , l),

T

t

u (s, l) ds

,

t–τ

         F T v t – σ (t), l , uT (t, l), uT t – τ , l , uT t – τ , l , uT t – τ  , l ,  t T     T u (s, l) ds , GT v t – σ  (t), l , 

t–τ 



ξ = vT (t, l), vT (t – σ , l), vT (t – σ , l), vT (t – σ , l),       vT (t, l), vT t – σ , l , vT t – σ , l , vT t – σ  , l ,   t T  T  T v (s, l) ds , G v (t, l) .

T

t

v (s, l) ds

  , F T v (t, l) ,

t–σ

t–σ 



4 Numerical simulation Consider the reaction-diffusion-delayed GRN (), () with the following parameters: k = ,

L = ,

γ = ,

γ  = .,

K = K = .I,

Zou et al. Advances in Difference Equations (2017) 2017:250

A = diag(, , , , ),

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B = diag(., ., ., ., .),

C = diag(., ., ., ., .), D = diag(., ., ., ., .), ⎡     – ⎢ ⎢–     ⎢ W = ⎢ ⎢  –    ⎢ ⎣  –         A = diag(., ., .),

D∗ = diag(., ., ., ., .), ⎤ ⎡ ⎤ γ   ⎥ ⎢ ⎥ ⎥ ⎢γ   ⎥ ⎥ ⎢ ⎥ ⎥, ⎥ W∗ = ⎢ ⎥ ⎢   –γ ⎥ , ⎥ ⎢ ⎥ ⎦ ⎣   ⎦   

B = diag(., ., .),

d∗ = diag(., ., .), d = diag(., ., .), ⎡ ⎡ ⎤ . –.  –γ  –γ  ⎢ ⎢ ⎥ ∗ W = ⎣. –. W = ⎣    ⎦,  γ .  –. τ = . + . sin(t),

C = diag(., ., .),

 γ 

 γ 

⎤  ⎥  ⎦. –γ 

σ = . + . sin(t),

τ  = . + . sin(t),

σ  = . + . sin(t),

where fi (x) =

x ,  + x

gi (y) =

y  + y

(i = , , . . . , n).

By using the Toolbox YALMIP in MATLAB to solve the LMI () and () we can obtain feasible solutions, the processes of asymptotic stability of mRNA concentration and protein concentration under Dirichlet boundary conditions are shown by Figures -. It is obvious that the proposed theory is feasible. The topological structure of interactional GRNs is shown by Figure . We set up the rules to measure the stabilizing time of interactional GRNs as follows: γ (t) = 

       wi ui (t, l) – ui (t – δ, l) + wi vi (t, l)vi (t – δ, l) + wp up (t, l) – up (t – δ, l)

  + wp vp (t, l) – vp (t – δ, l) dl,

Figure 1 The trajectory of u11 (t, l) and v11 (t, l).

()

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Figure 2 The trajectory of u12 (t, l) and v12 (t, l).

Figure 3 The trajectory of u13 (t, l) and v13 (t, l).

Figure 4 The trajectory of u14 (t, l) and v14 (t, l).

⎧ ⎨(t) = ,

γ (t) > δ ,

⎩(t) = ,

γ (t) ≤ δ ,

()

where wi > , wi >  (i = , . . . , n ); wp > , wp >  (p = , . . . , n ) are weights of ui , vi , up and vp , respectively, δ is a positive small quantity, (t) is a quantity for stability identification. When (t) = , it indicates interactional GRNs are stable; when (t) = , it indicates interactional GRNs are unstable. When γ  ∈ {., ., ., ., ., ., ., .}, γ = ., wi = wi = wp = wp = /, δ = ., the evolutions of (t) with γ  are shown by Figure , it is found that the increase of coupling strength γ  will lengthen the stabilizing time of interactional GRNs. Further-

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Figure 5 The trajectory of u15 (t, l) and v15 (t, l).

Figure 6 The trajectory of u21 (t, l) and v21 (t, l).

Figure 7 The trajectory of u22 (t, l) and v22 (t, l).

more, we can find that the iterative number of solution of LMIs and the time of reaching a steady state have the same change with different γ  as shown in Figure , therefore we utilize the iterative number to characterize the time of reaching steady state, because of the independence of the initial condition and the step size for the iterative number. When γ ∈ {, , , ., ., ., .}, the performance of number of iterations with γ  and γ is shown in Figure , where the color represents the time of reaching a steady state.

5 Discussion Further numerical simulations show that: (I) Interactional GRNs without reaction-diffusion terms cannot reach a steady state, because of the defection of information from reaction-diffusion terms. In other

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Figure 8 The trajectory of u23 (t, l) and v23 (t, l).

Figure 9 The topological structure of interactional GRNs.

Figure 10 The evolution of (t) with γ  .

words, reaction-diffusion terms are important and indispensable for interactional GRNs. (II) γ and γ  have an absolute effect on the time of reaching steady state as shown in Figure , the increase of coupling strength γ  and γ will collectively lengthen the stabilizing time of interactional GRNs.

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Figure 11 Iterative number and required time for stable with different γ  (γ = 0.0.4).

Figure 12 Performance of number of iterations with γ  and γ .

(III) The matrices W and W∗ qualitatively affect the stability of the interactional GRNs, in other words, the topological structure of the unstable GRNs and W∗ are very important for the stability of the interactional GRNs. The out-degree and in-degree of stable GRNs as well as the coupling term of W∗ can change at will, but the out-degree and in-degree of unstable GRNs and the coupling term of W∗ cannot change freely.

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6 Conclusion In this paper, we have constructed a model for interactional GRNs with reaction-diffusion terms under a Dirichlet boundary condition, and we analyzed the robust stability of interactional GRNs. Through constructing appropriate Lyapunov-Krasovskii functions and linear matrix inequalities (LMIs), we have given stability criteria corresponding to interactional GRNs. By numerical simulations, we found three important conclusions: interactional GRNs without reaction-diffusion terms cannot reach a steady state, because of the defection of information from reaction-diffusion terms, in other words, reaction-diffusion terms are important and indispensable for interactional GRNs; due to the smaller coupling strength γ  (in a certain range) and γ , interactional GRNs tend to become stable more quickly; the topological structures of the unstable GRNs and coupling term of W∗ determine the stability of interactional GRNs. Acknowledgements This work is supported by the National Natural Science Foundation of China (Nos. 61425002, 61672121, 61572093, 61402066, 61402067, 61370005, 31370778), Program for Changjiang Scholars and Innovative Research Team in University (No. IRT_15R07), and the Program for Liaoning Key Lab of Intelligent Information Processing and Network Technology in University. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors have made equal contributions to the writing of this paper. All authors have read and approved the final version of the manuscript. Author details 1 Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian, 116024, China. 2 Key Laboratory of Advanced and Intelligent Computing, Dalian University, Ministry of Education, Dalian, 116622, China.

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