Mathematical Biosciences 232 (2011) 116–134
Contents lists available at ScienceDirect
Mathematical Biosciences
journal homepage: www.elsevier.com/locate/mbs
Robust synchronization analysis in nonlinear stochastic cellular networks
with time-varying delays, intracellular perturbations and intercellular noise
Po-Wei Chen, Bor-Sen Chen ⇑
Lab. of Control and Systems Biology, National Tsing-Hua University, 101 Section 2, Kuang Fu Rd., Hsin-chu 300, Taiwan
a r t i c l e
i n f o
Article history:
Received 14 July 2010
Received in revised form 3 May 2011
Accepted 7 May 2011
Available online 23 May 2011
Keywords:
Robust synchronization
Cellular network
Process delay
Intercellular noise
Environmental disturbances
Robustness
a b s t r a c t
Naturally, a cellular network consisted of a large amount of interacting cells is complex. These cells have
to be synchronized in order to emerge their phenomena for some biological purposes. However, the
inherently stochastic intra and intercellular interactions are noisy and delayed from biochemical processes. In this study, a robust synchronization scheme is proposed for a nonlinear stochastic time-delay
coupled cellular network (TdCCN) in spite of the time-varying process delay and intracellular parameter
perturbations. Furthermore, a nonlinear stochastic noise filtering ability is also investigated for this synchronized TdCCN against stochastic intercellular and environmental disturbances. Since it is very difficult
to solve a robust synchronization problem with the Hamilton–Jacobi inequality (HJI) matrix, a linear
matrix inequality (LMI) is employed to solve this problem via the help of a global linearization method.
Through this robust synchronization analysis, we can gain a more systemic insight into not only the
robust synchronizability but also the noise filtering ability of TdCCN under time-varying process delays,
intracellular perturbations and intercellular disturbances. The measures of robustness and noise filtering
ability of a synchronized TdCCN have potential application to the designs of neuron transmitters, on-time
mass production of biochemical molecules, and synthetic biology. Finally, a benchmark of robust synchronization design in Escherichia coli repressilators is given to confirm the effectiveness of the proposed
methods.
2011 Elsevier Inc. All rights reserved.
1. Introduction
Since the stochastic gene expressions in a cellular population
contain a large copy number of molecules, their measurements
are difficult to observe exactly without synchronization among
cells. To collect these diverse phenomena, living organisms often
produce, secret, and detect the intercellular signal molecules,
named auto-inducers (AIs), for intra-species communication [1,2].
In recent decades, a new wave of research on cellular communication and synchronization, known as ‘quorum sensing’, has been
studied widely. Quorum sensing is a cell-to-cell community process
through AIs, and it is believed to play a key role in such synchronization characteristic as bio-luminescence, biofilm, sporulation, and
the suprachiasmatic nucleus (SCN) in the mammalian brain [3–5].
Application of quorum sensing started on the bioluminescent bacterium Vibrio fischeri [6]. Since revealing synchronization mechanism in a population of bacteria is naturally omnipresent, the
synchronization criteria of a coupled cellular network (CCN) need
further investigation at both molecular and cellular levels, e.g. circadian clocks and plant growth promoting microorganisms [7–9].
⇑ Corresponding author. Tel.: +886 3 5731155; fax: +886 3 5715971.
E-mail address: [email protected] (B.-S. Chen).
0025-5564/$ - see front matter 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.mbs.2011.05.002
Synchronization has been the highlight of CCN in various fields
of systems biology and biochemical engineering. However, the
complexity of these systems has obstructed a complete understanding for the picture of a natural CCN. To comprehend this complexity, the topological structures within a regular CCN has been
studied at first as the connection graph stability (GGS)-based
method [10]. This method makes sense and can be applied directly,
although it cannot easily be extended generally. In recent years,
more theoretical analyses of a stochastic CCN have been explored
and the topics of uncertainty, stability and controllability have also
been investigated broadly [8,11–16]. For instance, Lu et al. checked
the second-maximal eigenvalue of the coupling function to confirm the synchronization criteria [17]; Yu et al. introduced an efficient method via the linear hybrid constant-delayed coupled
network [13]; Wang et al. studied the synchronization criteria for
time-delay system under Lipschitz continuous condition [18]; Liu
et al. and Yu et al. investigated the analysis of unknown topological
structure network with Lipschitz continuous condition, too
[19,20]; Li et al. and Qiu et al. studied the stochastic synchronization with the Lur’e system [7,9]. Additionally, Chen developed a
M-synchronizable method with less conservative canonization
conditions for a complex coupled network [12]. Generally, the
analyses of a synchronized nonlinear time-varying-delay coupled
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P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
cellular network (TdCCN) are still limited for some special cases or
by some heuristic methods via dense simulations.
Furthermore, a TdCCN in real world may suffer not only timevarying delays from the biochemical processes but also intracellular perturbations and intercellular disturbances from natural uncertainties and environmental noise. Robustness is an essential and
ubiquitous property not only for a cellular function inside living
cells but also for synchronizability in a TdCCN [21–23]. For example, a population of V. fischeri need the robust synchronization for
the bioluminescence reaction [24]. For another example, the coral
spawning must be robustly synchronized for the propagation purpose. The process delay, intracellular perturbation and intercellular
disturbance will influence the synchronization of a coupled cellular
network [9,15,19,25]. In this study, a robust synchronization problem of a more general nonlinear stochastic TdCCN is considered
with nonlinear couplings, time-varying delays, intrinsic random
parameter perturbations and stochastic environmental disturbances. If the TdCCN does not have enough robustness and noise filtering ability for synchronization, then the synchronization will
decay or be destroyed [8]. Therefore, two important topics of cellular network synchronization are to investigate the general robust
synchronizability and to estimate the noise filtering ability under
time-varying process delays, intracellular parameter perturbations
and intercellular and environmental noises. Although there are a
number of open issues to analyze this robust synchronization problem, the impact of robust synchronization measurement is enormous and could be a bridge between the fundamental principles
of chaotic biochemical networks, medical practice, bioengineering,
physics and chemistry [23].
In this study, a new global estimation method of robust synchronization and noise filtering ability is proposed for a TdCCN.
After transferring the dynamic system model to a synchronization
tracking error dynamic system, we employ the robust tracking theory to efficiently estimate the robust synchronizability under timevarying process delays and intracellular perturbations. The robust
filtering theory is also employed to measure how much a synchronized TdCCN can attenuate the effect of intercellular disturbances
on the synchronization of TdCCN. To mimic a TdCCN, the intracellular perturbations (due to the intra-species biodiversity and natural random fluctuations) and the intercellular disturbances (due to
extracellular/ environmental noises) are both modeled into the
nonlinear TdCCN with stochastic noises. Based on the synchronization error dynamic system, robust synchronization to tolerate the
time-varying process delay and intracellular perturbations is transformed to an equivalent robust stabilization problem and analyzed
by the Lyapunov (energy-like) stability theory [26], while the noise
filtering ability to attenuate the effect of intercellular disturbances
on a synchronized TdCCN is examined by nonlinear robust filtering
theory. A general nonlinear time-varying-delay stochastic system
is discussed in this study. The techniques of nonlinear stabilization,
nonlinear filtering and constrained optimization are employed to
efficiently measure the robust synchronizability and the noise filtering ability of the nonlinear stochastic TdCCN.
In order to solve the robust synchronization problem efficiently,
globally and generally, we need to solve a second order Hamilton–
Jacobi inequality (HJI) to guarantee the robust synchronization
problem of a nonlinear stochastic TdCCN. However, at present,
there is no efficient method to solve the second order HJI analytically and numerically. In this situation, the global linearization
[27] technique has been employed to interpolate several linearized
stochastic systems at different operation points to approximate the
nonlinear TdCCN via some smooth interpolation functions. Hence,
the HJI for solving the robust synchronization problem could be
simplified as a set of linear matrix inequality (LMI) problems,
which is much easier to be solved via the help of LMI tool box in
MATLAB. The proposed method has potential application to the
measures of robust synchronization and noise filtering ability of
a TdCCN as designs of neuron transmitters [28], on-time processes
of cellular networks [29], and synthetic biology networks [7,30].
Finally, for convenience and easy illustration, a previous benchmark example [21] is given to illustrate the measure procedure and
to confirm the proposed criteria of robust synchronization and
noise filtering ability of a TdCCN under time-varying delays, intracellular random parameter perturbations and intercellular stochastic disturbances.
2. Robust synchronization for a nonlinear stochastic
time-varying-delay coupled cellular network
In this section, we propose a new measure of robust synchronization for a nonlinear stochastic TdCCN.
2.1. Stochastic system model of nonlinear coupled cellular network
with time-varying delay
First, for convenience of study, let us consider a modified stochastic TdCCN for a synthetic Escherichia coli multi-cellular clock
consisted of N repressilators from [21] (see Fig. 1). The ith perturbed repressilator can be represented with time-varying delay
s(t) and intracellular random perturbations due to the naturally
noisy transcription, translation, post-translation, signal transduction or molecular diffusion as follows:
dai ðtÞ ¼ c0 ai ðtÞ þ
a
0
C ni ðt dt
1þ
sðtÞÞ
Da0;1 dws
þ
þ Dc0 ai ðtÞdw;
1 þ C ni ðt sðtÞÞ
a0
Da0;1 dws
dt þ
;
dbi ðtÞ ¼ c0 bi ðtÞ þ
1 þ Ani ðt sðtÞÞ
1 þ Ani ðt sðtÞÞ
dci ðtÞ ¼ c0 ci ðtÞ þ
ð1Þ
jSi ðt sðtÞÞ
dt
1 þ Bni ðt sðtÞÞ 1 þ Si ðt sðtÞÞ
Da0;2 dws
DjSi ðt sðtÞÞdws
þ
þ
;
1 þ Si ðt sðtÞÞ
1 þ Bni ðt sðtÞÞ
a0
þ
dAi ðtÞ ¼ b0 ðai ðt sðtÞÞ Ai ðtÞÞdt þ Db0;1 Ai ðtÞdw þ Db0;2 ai ðt sðtÞÞdws ;
dBi ðtÞ ¼ b0 ðbi ðt sðtÞÞ Bi ðtÞÞdt þ Db0;1 Bi ðtÞdw;
dC i ðtÞ ¼ b0 ðci ðt sðtÞÞ C i ðtÞÞdt þ Db0;1 C i ðtÞdw;
dSi ðtÞ ¼
½ks0 þ ð1 Q 0 ÞgSi ðtÞ þ ks1 Ai ðt sðtÞÞ þ
þ Dks1 Ai ðt sðtÞÞdws þ
N
gQ 0 X
N
Sj ðtÞ Si ðtÞ
!
dt
j¼1
N
DgQ 0 X
Sj ðtÞ Si ðtÞ dw
N j¼1
for i = 1, . . . , N where ai, bi, and ci are the mRNA concentrations of
tetR, cI, and lacI, respectively, and their corresponding protein concentrations are represented by Ai, Bi and Ci in the ith bacterium; Si is
denoted as the concentrations of AI inside and outside the ith bacterium, respectively. w(t) and ws(t) are standard Wiener processes
with dw(t) = n(t)dt and dws(t) = n(t s(t))dt where n(t) and
n(t s(t)) are the Gaussian white noises due to thermal fluctuation,
alternative splicing, DNA mutation, molecular diffusion, the biodiversity, etc. [21,31,32]. In (1), the perturbed parameters, e.g. Dc0
and Dj, denote the deterministic parts of perturbations for corresponding processes, and n(t) and n(t s(t)) absorb the related stochastic properties of intracellular perturbations, respectively. Both
the covariances of n(t) and n(t s(t)) could be obtained as
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P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
Fig. 1. The LuxR-AHL quorum sensing system in an ith E. coli repressilator. The AI (AHL) can diffuse freely across the cell membrane. When AI spreads into a cell, it will bind
the promoter of luxI to regenerate itself and to activate LuxR. Then the dimer of activated LuxR (the complex) will regulate the downstream gene lacI to function the cellular
process of the repressilator. The repressilator module is located to the right of the dashed line, and the coupling module (quorum sensing) is at the left.
cov ðnðt1 Þ; nðt 2 ÞÞ ¼ r2 dt1 ;t2 where dt1 ;t2 denotes the delta function as
dt1 ;t2 ¼ 0 for t 1 – t2 or dt1 ;t2 ¼ 1 for t 1 ¼ t2 . We only assume some
parameter perturbations in this benchmark (1). Of course, the more
information of intracellular perturbation we have, the more detailed noise terms we need to recruit in.
In order to generate an emergent behavior, a large amount synthetic cellular networks have to be synchronized together to
emerge the molecular behavior of a simple synthetic network.
However, the intrinsic parameter fluctuations and environmental
disturbances will destroy the synchronization among the coupled
synthetic cellular networks. In this situation, we need to measure
the synchronizability of the TdCCN and their filtering ability to
reject the effect of environmental disturbances on the synchronization. Further, the method of how to improve the synchronizability
and filtering ability will be also discussed in the sequel.
For convenience, we simply note these molecular expression
levels as follows
xi ¼ ½ xi;1
xi;M T , ½ ai ðtÞ bi ðtÞ ci ðtÞ Ai ðtÞ Bi ðtÞ C i ðtÞ Si ðtÞ T ;
T
xs;i;M , ½ ai ðt sðtÞÞ Si ðt sðtÞÞ T :
xTs;i ¼ ½ xs;i;1
Generally, N nonlinear repressilators can be linked as the following augmented system with simple notations X, Xs, xi and xs,i
for the states X(t) [33], X(t s(t)), xi(t) and xi(t s(t)), respectively
dX ¼ ðFðX; X s Þ þ GðX; X s ÞÞdt þ ðDFðXÞ þ DGðXÞÞdw
þ ðDF s ðX s Þ þ DGs ðX s ÞÞdws , ðF þ GÞdt þ ðDF þ DGÞdw
þ ðDF s þ DGs Þdws ;
ð2Þ
where
X ¼ xT1
xTN
T
;
X s ¼ xTs;1
xTs;N
T
;
T
F ¼ f1T ðx1 ; xs;1 Þ fNT ðxN ; xs;N Þ ;
T
x1 ; ; x N ;
x1 ; ; xN ;
g TN
;
G ¼ g T1
xs;1 ; ; xs;N
xs;1 ; ; xs;N
T
DF ¼ Df1T ðx1 Þ DfNT ðxN Þ ;
T
DF s ¼ DfsT;1 ðxs;1 Þ DfsT;N ðxs;N Þ ;
h
iT
Dfi ðxi Þ ¼ Dc0 aTi ðtÞ 0 0 Db0;1 ATi ðtÞ Db0;1 BTi ðtÞ Db0;1 C Ti ðtÞ 0 ;
The initial condition of this repressilator could be noted as xi(t) =
xi(t0) "t0 2 [s(t) 0]. The upper bound of the process delay and
Dfs;i ðxs;i Þ ¼
Da0;1
Da0;1
Da0;2
DjSi ðt sðtÞÞ
þ
D
b
a
ðt
s
ðtÞÞ
0
0
D
k
A
ðt
s
ðtÞÞ
;
s1 i
0;2 i
1 þ C ni ðt sðtÞÞ 1 þ Ani ðt sðtÞÞ 1 þ Bni ðt sðtÞÞ 1 þ Si ðt sðtÞÞ
the gradient of process delay are a and b such that 0 < s(t) < a and
s_ ðtÞ < b. The values and physical meanings of parameters in (1)
can be seen in Table 1 and further details of this synthetic biochemical repressilator can be found in [21].
T
DG ¼ Dg T1 ðx1 Þ Dg TN ðxN Þ ;
T
DGs ¼ Dg Ts;1 ðxs;1 Þ Dg Ts;N ðxs;N Þ ;
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P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
Table 1
Biochemical coefficients for a repressilator [21].
a0
c0
j
Dimensionless transcription rate
The degradation rate of mRNA
Maximal contribution to lacI transcription in the presence of saturating amounts of AI
Hill coefficient
the dimensionless decay rate of AI
The dimensionless diffusion rate of AI across the cell membrane
The dimensionless binding affinity that LUXI combine lacI
The ration between the mRNA and protein lifetime
The time-varying process delay
The maximum delay time of the time-varying genetic process
The maximum of the rate of the time-varying delay process
The deterministic part of intracellular perturbation for parameter c0
The first kind of the deterministic part of intracellular perturbation for parameter a0
The second kind of the deterministic part of intracellular perturbation for parameter a0
The deterministic part of intracellular perturbation for parameter j
The first kind of the deterministic part of intracellular perturbation for parameter b0
The second kind of the deterministic part of intracellular perturbation for parameter b0
The deterministic part of intracellular perturbation for parameter ks1
The deterministic part of intracellular perturbation for parameter g
n
ks0
g
ks1
b0
s(t)
a
b
Dc 0
Da0,1
Da0,2
Dj
Db0,1
Db0,2
Dks1
Dg
"
N P
DgQ 0
N
Dg i ðxi Þ ¼ 0 0 0 0 0 0
Sj ðtÞ Si ðtÞ
#T
j¼1
6
6
6
6
6
6
6
fi ðxi ; xs;i Þ ¼ 6
6
6
6
6
6
4
0
c0 ai ðtÞ þ 1þC n aðt
sðtÞÞ
i
0
c0 bi ðtÞ þ 1þAn aðt
sðtÞÞ
i
jSi ðtsðtÞÞ
a0
c0 ci ðtÞ þ 1þBn ðt
sðtÞÞ þ 1þS ðtsðtÞÞ
i
i
b0 ðai ðt sðtÞÞ Ai ðtÞÞ
b0 ðbi ðt sðtÞÞ Bi ðtÞÞ
b0 ðci ðt sðtÞÞ C i ðtÞÞ
(i) Suppose the coupling functions of the TdCCN for this benchmark are clearly separable by the additional form of intracellular AI in each cell. If the coupling functions are more
complex, then it should be checked with the above definition beforehand.
(ii) If the real-time cellular communication from cell j to cell i
exists, then gi,j(xj) – 0; otherwise gi,j(xj) = 0. It is similar for
the delay-time communication function gi,s,j (xs,j).
(iii) If the coupling function in (2) is separable in the class S for
all xi and xs,i, i = 1, . . . , N, then when the coupled cells are synPN @gi;j ðxj Þ
chronized i.e. x1 = = xN, we have that
¼ 0 and
j¼1 @xj
PN @gi;s;j ðxs;j Þ
¼
0;
i.e.
the
coupling
functions
will
not
affect
i¼1
@xs;j
the cells.
(iv) Because of the random molecular diffusion and the quasisteady state assumption in [21], we can neglect the delaytime coupling function and say that the cellular communication is fully coupling.
3
7
7
7
7
7
7
7
7;
7
7
7
7
7
5
½ks0 þ ð1 Q 0 ÞgSi ðtÞ þ ks1 Ai ðt sðtÞÞ
3
2
0
7
6
0
7
6
7
6
7
6
0
7
6
7
6
x1 ; . . . ; x N ;
0
7
6
gi
¼6
7
7
6
xs;1 ; . . . ; xs;N
0
7
6
7
6
7
6
0
7
6
N 5
4 gQ 0 P
S
ðtÞ
S
ðtÞ
j
i
N
j¼1
in which xi is the molecular expression of cell i; fi(xi, xs,i) is the
nonlinear intracellular biochemical function and g i ðx1 ; . . . ; xN ;
xs;1 ; . . . ; xs;N Þ is the nonlinear coupling function which describes
the coupling strength and the coupling configuration between cell
i and other cells through the communication of AI. Dfi(xi), Dfs,i(xs,i),
Dgi(xi) and Dgs,i(xs,i) are parameter fluctuations related to the corresponding real-time and delay-time function. Some properties of the
nonlinear coupling function are discussed as follows:
Definition 1 [33]. The coupling function g i x1 ; . . . ; xN ; xs;1 ; . . . ; xs;N
is separable with respect to x1, . . ., xN, xs,1, . . ., xs,N if the coupling
function is differentiable and can be written as
g i ðx1 ; . . . ; xN xs;1 ; . . . ; xs;N Þ ¼ g i;1 ðx1 Þ þ þ g i;N ðxN Þ þ g i;s;1 ðxs;1 Þ
þ þ g i;s;N ðxs;N Þ:
1
0.5
0.2
0.2
0.5
0.3
0.5
0.3
0.4
0.5
With above definitions, some properties could be described as
follows.
;
Dg s;i ðxs;i Þ ¼ 0M1
2
216
1
20
2.0
1
2.0
0.01
1
ð3Þ
Definition 2 [33]. A coupling function g ðx1 ; . . . ; xN ; xs;1 ; . . . ; xs;N Þ
belongs to class S if g ðxc ; . . . ; xc xs;c ; . . . ; xs;c Þ ¼ 0 for all xc and all
xs,c where xc and xs,c are the synchronized real-time and delay-time
system states (molecular expressions), respectively.
Remark 1. In conventional studies [12,13,34], the coupling function is linear and could be described as G = C D(t)X + Cs D(t s(t))Xs where the matrices C and Cs are defined by
PN
PN
j¼1 C s;ij ¼ 0 for i = 1, . . . , N; ‘’ is the Kronecker
j¼1 C ij ¼ 0 and
j–i
product; the D(t) and D(t s(t)) are respectively the inner-coupling matrices of the network at times t and t s(t). The linear coupling is a special case in (2). In a nonlinear TdCCN, the separable
properties in Definition 1 and the class S in Definition 2 are necessary for the further study.
Before further analysis of robust synchronization and noise filtering ability, some definitions for ‘synchronization’ and ‘robust
synchronization’ should be given.
Definition 3. A coupled system (2) is called synchronization in
probability if for any e > 0 there exists d(e) > 0 such that
Ekei(t0)k , Ekxi(t0) xi+1(t0)k 6 d(e), then Ekei(t)k 6 e for all t P t0
and all i 2 {1, . . . , N 1}.
Definition 4. A coupled system is called robust synchronization if
the coupled system still has synchronization in probability under
intracellular stochastic parameter perturbations.
To analyze the robust synchronization easily and efficiently, we
employ the following error dynamic system,
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P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
e es ; X; X s Þ dt
de ¼ e
F ðe; es ; X; X s Þ þ Gðe;
e XÞ dw
þ De
F ðe; XÞ þ D Gðe;
e s ðes ; X s Þ dws
F s ðes ; X s Þ þ D G
þ De
chronize the TdCCN. If intrinsic parameter fluctuations
e De
e s exist, in order to override the last two posiD Fe ; D G;
F s and D G
tive terms in (6) due to parameter fluctuations, more negative
e is necessary.
feedback coupling G
e
e s Þdws ;
e
þ ðD e
, ðe
F þ GÞdt
þ ðD e
F þ D GÞdw
F s þ DG
ð4Þ
where
e , ½ x1 x2
xk xkþ1
es , ½ xs;1 xs;2
2
with
I
60
J¼6
4 ...
xN1 xN T ¼ JX;
xs;k xs;kþ1
I
I
..
.
0
I
..
.
..
.
xs;N1 xs;N T ¼ JX s
3
0
0 7
2 RMðN1ÞMN ; e
.. 7
F , JF
. 5
represents
0 0
I I
the synchronization error dynamic of intracellular function; and
so do D e
F , JDF and D e
F s , JDF (the errors of real-time and delaye , JG (the error of coupling
time intracellular perturbation), G
e
e
function), and D G , JDG and D G s , J DG (the errors of real-time
and delay-time coupling perturbation). Then, the concept of robust
synchronization could be taken as a robust stability problem with
respect to the single equilibrium point e = 0 of synchronization error dynamic system from system and signal processing viewpoints,
i.e., e(t) ? 0 in probability for the synchronization error dynamic
system in (4).
Remark 2. By the Mean-Value Theorem [35], we could connect the
synchronization error dynamic (4) with the augmented system (2)
generally as the following difference equation
f ðx1 Þ f ðx2 Þ ¼
Z
Z
h
@
F
@e
i
F 2 X for all e and es
02
1
ð5Þ
0
where; JðxÞ , @f@xðxÞ is the Jacobian of the continuously differentiable
vector function f(x) for x 2 RN and s is an arbitrary weighting scalar
for 0 6 s 6 1.
De
FT
A0;1 ðXÞ
De
F Ts
eT
G
eT
DG
As;1 ðX s Þ
3
2
A0;k ðXÞ
As;k ðX s Þ
3
B6
7
7
6
B6 DA0;1 ðXÞ
7
7
6 DA0;k ðXÞ
0
0
B6
7
7
6
B6
7
7
6
B6
7
6
D
A
ðX
Þ
0
D
A
ðX
Þ
0
s;1 s 7
s;k s 7
B6
7
6
X # C o B6
7; . . . ;
7; . . . ; 6
B6 B0;1 ðXÞ
7
7
6 B0;k ðXÞ
B
ðX
Þ
B
ðX
Þ
s
s
;1
s
s
;k
B6
7
7
6
B6
7
7
6
B6 DB0;1 ðXÞ
7
7
6
D
B
ðXÞ
0
0
0;k
@4
5
5
4
DBs;1 ðX s Þ
31
A0;K ðXÞ
As;K ðX s Þ
7C
6
7C
6 DA0;K ðXÞ
0
7C
6
7C
6
C
6
DAs;K ðX s Þ 7
0
7C
6
7C
6
C
6 B0;K ðXÞ
Bs;K ðX s Þ 7
7C
6
7C
6
7C
6 DB0;K ðXÞ
0
5A
4
0
0
DBs;k ðX s Þ
2
2.2. Robust synchronization measurement
Based on Lyapunov stability theory [26], we could obtain the
following robust synchronization proposition for a nonlinear perturbed TdCCN under time-varying process delays and intracellular
parameter perturbations.
Proposition 1. For a nonlinear TdCCN with time-varying-delay and
intracellular parametric uncertainty, if the following Hamilton–Jacobi
inequality (HJI) holds for a Lyapunov function V(e) > 0,
T
T 2
@VðeÞ
e þ 1 De
e @ VðeÞ D e
e
F þ DG
F þ DG
ðe
F þ GÞ
2
@e
2
@e
T @ 2 VðeÞ 1
e
es
es 6 0
F s þ DG
þ
DF s þ DG
De
2
2
@e
ð7Þ
iT
eT
and the polytope
DG
s
6MðN1Þ2MðN1Þ
X#R
.For example, if e
F ¼ A0 ðXÞeþ As ðX s Þes ;
e ¼ B0 ðXÞe þ Bs ðX s Þes ; D G
e¼
F ¼ DA0 ðXÞe; D e
F s ¼ DAs ðX s Þes ; G
De
e s ¼ DBs ðX s Þes ,
DB0 ðXÞe; D G
then
"
#T
T
T
T
T
0
B0 ðXÞ DB0 ðXÞ
0
A0 ðXÞ DA0 ðXÞ
2 X. Sup0
DATs ðX s Þ BTs ðX s Þ
0
DBTs ðX s Þ
ATs ðX s Þ
pose A0,k(X), As,k(Xs), DA0,k(X), DAs,k(Xs), B0,k(X), Bs,k(Xs),
DB0,k(X),DBs,k(Xs) for k = 1, . . . , K denote the vertices of the convex
hull of X, i.e. [37],
Jðx2 þ sðx1 x2 ÞÞðx1 x2 Þds
Jðx2 þ sðe1 ÞÞe1 ds
@
@es
h
where F ¼ Fe T
1
0
¼
In order to guarantee the synchronization character in (4), we
need to find a suitable Lyapunov function V(e) > 0 to satisfy
inequality (6). In general, it is not easy to solve the HJI and at present there is no systematic method to find a Lyapunov function for a
nonlinear TdCCN, such that (6) holds to guarantee the robust synchronization. Many conventional studies have proposed related
methods on special systems such as the Lur’e system, Chua’s circuits, linear hybrid systems and the Lipschitz continuous system
[7,13,18]. To analyze robust synchronization and noise filtering
ability properties of nonlinear stochastic TdCCN more generally
and efficiently, the global linearization method [27,36] is employed
for the error dynamic system (4). Consider the following global linearization of nonlinear system (4), suppose
ð6Þ
then the TdCCN in (2) has robust synchronization in probability in spite
of parameter fluctuations.
Proof. See Appendix A.1. h
e De
F ; D G;
F s and
If the TdCCN is free of parameter fluctuation D e
e
D G s , then the synchronization condition in (6) becomes
T
@VðeÞ
e 6 0. If the coupling G
e is negative, it is easy to synðe
F þ GÞ
@e
0
DBs;K ðX s Þ
ð8Þ
where (8) denotes a convex hull of X consisting of K vertices via the
real-time and the delay-time linearization matrices. Then the
synchronization property of the trajectory of the error dynamic
(4) could be represented by the interpolation of the following
linearized systems at vertices [37]
de ¼ A0;k ðXÞ þ B0;k ðXÞ edt þ DA0;k ðXÞ þ DB0;k ðXÞ edw þ As;k ðX s Þ
þ Bs;k ðX s Þ es dt þ DAs;k ðX s Þ þ DBs;k ðX s Þ es dws
, ðA0;k þ B0;k Þedt þ DA0;k þ DB0;k edw þ ðAs;k þ Bs;k Þes dt
þ DAs;k þ DBs;k es dws
ð9Þ
for k = 1, . . . , K. The synchronization error dynamic of TdCCN in (4)
can be interpolated by the linearized synchronization error dynamics of coupled systems at Kvertices in (9) as follows
121
P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
de ¼
K
X
Mk ðA0;k þ B0;k Þe dt þ DA0;k þ DB0;k e dw
k¼1
þ ðAs;k þ Bs;k Þes dt þ DAs;k þ DBs;k es dws þ ee þ ee dt
F
G
ð10Þ
þ e e þ e e dw þ e e þ e e dws
DF
DF s
DG
DGs
where
Mk ¼ diagðlk;1 ðe; es ; X; X s Þ; . . . ; lk;N1
ðe; es ; X; X s ÞÞT 2
RMðN1ÞMðN1Þ are some normalized interpolation functions with
PK
k¼1 lk;i ðe; es ; X; X s Þ ¼ I and 0 6 lk,i(e,es,X,Xs) 6 I [38]; the linearized
system matrices are shown as follows
where U is a symmetric matrix defined as
3
2
N1
N2
N3
2
1
7
6
7
6 N 2 2 ðN 2Þ 2 ðN 3Þ 22
21
7
6
7
6
7
6 N 3 2 ðN 3Þ 3 ðN 3Þ
3
2
3
1
7
6
7:
U,6
7
6 ..
..
..
..
..
7
6 .
.
.
.
.
7
6
7
6
7
6 ðN
2Þ
2
ðN
2Þ
1
5
4
1
ðN 2Þ 1 ðN 1Þ 1
A0;k ¼ diag A0;k;1 ðXÞ; . . . ; A0;k;N1 ðXÞ ;
As;k ¼ diag As;k;1 ðX s Þ; . . . ; As;k;N1 ðX s Þ ;
For example,
DA0;k ¼ diag DA0;k;1 ðXÞ; . . . ; DA0;k;N1 ðXÞ ;
DAs;k ¼ diag DAs;k;1 ðX s Þ; . . . ; DAs;k;N1 ðX s Þ ;
6
68
6
6
67
6
6
66
6
6
U ¼ 65
6
64
6
6
63
6
6
62
4
2
B0;k ¼ diag B0;k;1 ðXÞ; . . . ; B0;k;N1 ðXÞ ;
Bs;k ¼ diag Bs;k;1 ðX s Þ; . . . ; Bs;k;N1 ðX s Þ ;
DB0;k ¼ diag DB0;k;1 ðXÞ; . . . ; DB0;k;N1 ðXÞ ;
DBs;k ¼ diag DBs;k;1 ðX s Þ; . . . ; DBs;k;N1 ðX s Þ :
eeF , eF K
X
eDeF , D eF Mk ðDA0;k eÞ;
k¼1
eDeG , D Ge K
X
K
X
eeG , Ge Mk ðA0;k e þ As;k es Þ;
k¼1
eDeF , D eF s K
X
eDeG , D Ge s s
k¼1
Mk ðDAs;k es Þ;
k¼1
K
X
Mk ðDBs;k es Þ
k¼1
If the approximation errors are bounded and small enough, then we
can represent the nonlinear TdCCN by the interpolated system in
(10) via the global linearization method at K vertices. After finding
the system matrices via global linearization with suitable finite K
vertices, we could easily find the bounds on the approximation
errors for k = 1, . . . , K as follows
F
ke ek22 6 e25 kek22 ;
DF
k2
De
Fs 2
ke
2
2
7 kes k2 ;
6e
kee k22 6 e23 kek22 þ e24 kes k22 ;
G
2
2
6 kek2 ;
ke e k22 6 e
DG
Z
0
a
þ
Z
0
a
Z
ke e k2 6 e
DGs
tsðtÞ
t
e_ T ðsÞðU QÞe_ ðsÞds dr , eT Pe þ
tþr
3
2
16 14 12 10
8
6
4
14 21 18 15 12
9
6
12 18 24 20 16 12
8
1
10 15 20 25 20 15 10
2
6
6
6
6
6
6
6
6
6
6
6
6
4
8
12 16 20 24 18 12
6
9
12 15 18 21 14
4
6
8
10 12 14 16
2
3
4
5
ð1Þ
ð1Þ
ð1Þ
N1;1
N1;2
N1;3
6
7
8
3
7
27
7
7
37
7
7
47
7
7
5 7 for N ¼ 10
7
67
7
7
77
7
7
87
5
9
t
tsðtÞ
Nð1Þ
1;5
0
0
ð1Þ
N3;3
Nð1Þ
3;4
0
1a Q
0
c1 I
0
3
7
7
7
7
0 7
760
7
0 7
7
7
0 7
5
Nð1Þ
2;6 7
ð14Þ
c2 I
2
N
Z
Nð1Þ
1;4
ð1Þ
ð1Þ
N2;2
N2;3
eT ðsÞP s eðsÞds
ð1Þ
1;1
P s þ CT3 þ 2ðc1 þ c2 Þðe21 þ e23 ÞI
6
¼ 4 þ4c3 e25 þ e26 I
AT0
ð1Þ
N1;2
¼ C1 þ PT
2
N
ð1Þ
1;3
¼4
ð13Þ
AT0 C2
0MðN1ÞKMðN1Þ
N
¼ C1
T T
h
MðN1ÞKMðN1ÞK C3
þ1
;
ATs C1
3
T
5;
T
;
T
0MðN1ÞMðN1ÞK ;
MðN1ÞMðN1ÞK
Nð1Þ
2;3 ¼ 0
T
Nð1Þ
2;6 ¼ C2 ;
2DAT0 P DA0
0MðN1ÞKMðN1ÞK
ð1Þ
N1;4
¼ C3 0MðN1ÞMðN1ÞK
ð1Þ
1;5
ðAT0 C1 ÞT
C1
CT3 þ C4
t
_
e_ T ðsÞQ eðsÞds
dr
4
where
2
8 kes k2
tþr
Z
5
via the global linearization method, we can get the following
proposition.
ð12Þ
Then the robust synchronization and noise filtering ability measurement for a TdCCN could be discussed generally as the robust
stability problem of interpolated error system in (10) through the
help of a global linearization technique.
Choosing a general Lyapunov function V(e) > 0 for an interpolated TdCCN in (10) with positive definite matrices P; P s ; Q 2
RMM , and an irreducible symmetric semi-positive definite matrix
U 2 RN1N1 as the following quadratic function from the energy
point of view
Z t
VðeÞ , eT ðU PÞe þ
eT ðsÞðU P s ÞeðsÞds
þ
6
Proposition 2. For the nonlinear perturbative TdCCN, the robust
synchronization can be achieved if the following inequality holds with
symmetric positive definite solutions P ¼ PT > 0; P s ¼ PTs > 0;
Q ¼ Q T > 0, suitable matrices C1, C2, C3, C4, suitable positive scalars
c1, c2, c3, and P 6 c3 I
ð11Þ
keek22 6 e21 kek22 þ e22 kes k22 ;
7
Mk ðB0;k e þ Bs;k es Þ;
k¼1
s
Mk ðDB0;k eÞ;
8
1
The approximation errors of the global linearization in (10) with
finite vertices selection (for k = 1, . . . , K) are denoted as follows
K
X
9
T
Nð1Þ
2;2 ¼ C2 C2 þ aQ ;
T ;
ATs C2
iT
ð1Þ
N3;4
¼ C4 0MðN1ÞMðN1ÞK ;
3
7
5;
122
P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
2
ð1Þ
3;3
N
CT4 ð1 bÞPs þ 2ðc1 þ c2 Þ
6
¼ 4 þ4c3 e27 þ e28 I
2
e22 þ e4 I
0MðN1ÞKMðN1Þ
MðN1ÞKMðN1ÞK C4
2DATs PDAs 1
0MðN1ÞKMðN1Þ
3
7
5;
A0 ¼ A0;k þ B0;k 2 RMðN1ÞMðN1ÞK ;
As ¼ As;k þ Bs;k 2 RMðN1ÞMðN1ÞK ;
the help of global linearization method [27,36], we can
extend the robust synchronization to a more general biological network through the quadratic robust tracking ability of
a set of globally linearized error dynamic systems. Therefore,
the HJI in (6) of N-coupled nonlinear perturbed systems can
be replaced by an LMI in (14) at K vertices.
If the nonlinear TdCCN (2) is under different conditions, we
could have the following corollaries to simplify the global synchronization criterion in (14).
DA0 ¼ DA0;k þ DB0;k 2 RMðN1ÞMðN1ÞK ;
DAs ¼ DAs;k þ DBs;k 2 RMðN1ÞMðN1ÞK ;
1
is a matrix with suitable rank in which all elements are 1
Corollary 1. If the nonlinear cellular network (2) is free of any
perturbations and process delays, then the global synchronization
criterion in (14) can be easily reduced to the following inequality
Proof. see Appendix A.2. h
T
2ðc1 þ c2 Þðe21 þ e23 ÞI þ ðP CT1 ÞðC2 þ CT2 Þ1 ðP CT1 ÞT þ 2c1
1 C1 C1
The physical meaning of the results in (14) is that if the interpolated time-delay coupled system among the vertices of the globally
linearized synchronization error systems in (10) is asymptotically
stable in probability, then the nonlinear perturbative TdCCN (2)
is globally robust synchronization. If more negative couplings B0,k
are given to overcome parameter fluctuations, a more robust synchronization of TdCCN will be achieved. In general, using the LMI
toolbox in Matlab [39], it is easier to check the LMIs in (14) than
to solve the nonlinear HJI in (6) directly.
By solving the symmetric positive definite matrices P, Ps, Q, the
constraint on the LMI (14) is strict. With adequate choice of these
matrices C1, C2, C3, C4 and scalars c1,c2,c3, the conservative of the
robust synchronization solution will be released, and the synchronization robustness could be enough to tolerate more intracellular
perturbations such as DA0,k, DAs,k, DB0,k and DBs,k, together with
time-varying delays such as As,k and Bs,k. Then the robust synchronizability problem in (14) will be solved efficiently.
Remark 4
(i) U in (13) is an irreducible symmetric semi-positive definite
matrix with special properties about the augmented cou2
3
g 11 g 1N
6
.. 7, then
..
pling function G. For example, if @G
¼ 4 ...
. 5
.
@X
g N1
g NN
usually [7]
2
6
6
U¼6
6
4
g 12 þ þ g 1N
..
.
g 13 þ þ g 1N
g 13 þ þ g 1N þ g 23 þ þ g 2N
3
g 1N
7
g 1N þ g 2N
7
7:
..
..
7
5
.
.
g 1N þ þ g N1N
Choosing a suitable U in a Lyapunov function would simplify
the complicated solution of the LMI in (14).
(ii) Besides diffusing AI for communication, the other molecules
work only inside individual bacteria. This implies that the
system function e
F is also separable with respect to e1, . . . ,
eN1, es,1, . . . , es,N1, and that the corresponding linearized
matrices A0,k and As,k for k = 1, . . . , K are diagonal indeed. Further, the linearized coupling matrices B0,k and Bs,k for
k = 1, . . . , K are also diagonal due to the fully coupling network. The matrices DA0,k, DAs,k, DB0,k, and DBs,k are also
diagonal. If a cellular network is not fully coupling, e.g. the
restricted molecular transport by specific ion channel [40],
then our method still holds with only a simple modification.
(iii) In conventional studies, the synchronization analysis is limited to some special cellular network, such as the linear
hybrid constant delayed coupled networks [13], the Lipschitz continuous conditional system [18–20], the Lur’e system [7,9], and the V-decreasing property system [12]. With
1
2
T
2
þ 2 c C C2 6 C
T
1
K
X
K
X
Mk ðA0;k þ B0;k Þ þ
ðA0;k þ B0;k ÞT MTk C1
k¼1
!
k¼1
!
K
K
X
X
ðA0;k þ B0;k ÞT MTk C2 ðC2 þ CT2 Þ1
ðA0;k þ B0;k ÞT MTk C2
k¼1
!T
:
k¼1
ð15Þ
Proof. see Appendix A.3. h
From (15), it can be seen that if the two terms in the right-handside of (15) are more negative, then more robust synchronizability
will be achieved and can tolerate larger perturbations. In general, if
the couplings B0,k, k = 1, . . . , K are more negative, then the synchronization of TdCCN will be achieved.
Since the robust synchronizations in (14) and (15) are only sufficient conditions, any measure of robust synchronization derived
from these conditions might underestimate robust synchronizability, i.e. a TdCCN that violates these conditions may be still robustly
synchronized.
Remark 5
(i) Synchronization of TdCCN is related to the characteristic of
e In
system property F and coupling function G (or e
F and G).
(15), it is indicated that even if these cells are not identical,
a TdCCN is still easily synchronized with sufficiently strong
negative coupling [9]. Although the cellular coupling could
enhance the synchronization behavior, it is not sufficient
to achieve synchronization by a weak one within the cellular
network. Therefore, to synchronize a TdCCN, not only the
coupling strength with the suitable linkage but also some
appropriate system characteristics are both needed. With
suitable system characteristics and strong cellular communications (i.e. the eigenvalues of A0,k + B0,k for k = 1, . . . , K
are negative enough in the left-hand-side of the s-domain),
the TdCCN will be synchronized more easily [12,21].
(ii) Delay processes could influence the synchronization
significantly. If the delay effects in an individual cell and
between the cellular coupling are large, i.e. the eigenvalues
of delay-dependent matrices As,k and Bs,k are near the jw-axis
or at the right-hand-side of the s-domain, then the system
trajectory may diverge and destroy this synchronization [41].
(iii) According to the intra-species biodiversity and the noisy cellular processes, we model stochastic parameter perturbations.
When the influences of these perturbations are strong in
(14), i.e. DF,DFs, DG and DGs are large, they will violate the
LMIs such that the molecular trajectories in the TdCCN (e.g.
the bioluminescence protein) may diverge from each other.
123
P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
3. Noise filtering ability of a synchronized network under timevarying delays, intracellular perturbations and intercellular
disturbances
From the previous experimental implementations, although a
TdCCN could meet the synchronization criterion under the process
delays and intracellular parameter perturbations, noticeable irregularity in cellular behavior is still found in practical experiments.
This is caused by the intercellular disturbances [42]. Consider a
perturbative TdCCN suffering from intercellular disturbances as
the following general form
dX ¼ ðF þ G þ Hv Þdt þ ðDF þ DGÞdw þ ðDF s þ DGs Þdws ;
T
ð16Þ
T
where v ¼ v T1 ðtÞ v TN ðtÞ
and v i ¼ v Ti;1 v Ti;m 2 Rm1
for i = 1, . . . , N denote the intercellular disturbances on the ith cell
from the m environmental and extracellular sources through the
nonlinear weighting matrix H = H(X) , diag(h1(x1), . . . , hN(xN)).
Based on (4), we obtain the following equivalent error dynamics
with the intercellular disturbance
e þ He
e dw þ D e
e s dws ;
e v dt þ D e
F þ DG
F s þ DG
de ¼ Fe þ G
ð17Þ
where ev , Jv denotes the intercellular disturbances on the error dye
namic through the nonlinear weighting matrix H,
i.e.,
e XÞev , JHv . In the benchmark example of (16), we ase v ¼ Hðe;
He
sume that there are three kinds of intercellular disturbances
2
3T
1 1 1 0
0
0
0
(m = 3), and hi ¼ 4 0 0 0 1:5 1:5 1:5 0 5 for i = 1, . . . , N.
0 0 0 0
0
0 0:5
The intercellular disturbances include the environmental disturbances and the extracellular disturbances, which consist of the disturbances from external input (e.g. externally additional AHL),
plasmid copy-number variability, and other extra-cellular effects
[18]. The day-night cycle is a well-known example of environmental disturbance.
Before further study of the noise filtering ability of a synchronized network, let us denote a L2 measure of synchronization error
R1
e(t) as kekL2 , ð 0 eT ðtÞeðtÞdtÞ1=2 . We say that e 2 L2 if kekL2 < 1.
Then the effect of intercellular disturbances v on synchronization
error kekL2 is said to be less than a positive value q if the following
inequality holds [43]
Ekek2L2
Ekv k2L2
6 q2
or Ekek2L2 6 q2 Ekv k2L2
and the noise filtering ability is inversely proportional to q0. The
measure of filtering ability q0 on intercellular disturbances can provide more insight into the effect of environmental and extracellular
noise on the synchronization behavior of a perturbed TdCCN. The
measurement of noise filtering ability for the synchronized cellular
network has potential application to neuron transmitters, on-time
processes of cellular networks, and synthetic biology. For instance,
before in vivo experiments, bacterial population control could be
easily designed through the following proposed noise filtering ability measurement of a coupled network.
Based on the analysis above, the following propositions can be
obtained to measure the noise filtering ability of a time-varyingdelay coupled cellular network.
Proposition 3. Suppose the error dynamic of the TdCCN in (17)
suffers from intracellular perturbations and intercellular disturbances.
If the following HJI holds for a prescribed filtering value q
T
T 2
@VðeÞ
e þ 1 De
e @ VðeÞ D e
e
F þ DG
F þ DG
ðe
F þ GÞ
2
@e
2
@e
T @ 2 VðeÞ 1
e
es
es
F s þ DG
þ
DF s þ DG
De
2
2
@e
T
1 @VðeÞ e e T @VðeÞ
þ eT e 6 0
HJJ H
þ 2
4q
@e
@e
then the nonlinear TdCCN is robustly synchronized and the effect of
intercellular disturbances v on the synchronization error e is less than
q; i.e. the robust filtering with a desired attenuation value q in (18)
or (19) is achieved.
Proof. see Appendix A.4. h
Remark 6
(i) Since the HJI in (21) implies the HJI in (6), the robust synchronization and the desired disturbance attenuation value
q are both achieved in Proposition 3. Furthermore, the HJI
in (21) is more constrained than the HJI in (6) because of
the needs of filtering intercellular disturbances under a prescribed value q.
(ii) In conventional studies, Gaussian white noises are favored to
approximate the intercellular disturbance [7]. However,
there are different kinds of disturbances that should be considered, e.g. jump or sinusoidal environmental noise. In our
study, all possible kinds of noise with finite energy could
be included in our intercellular disturbance. If the intercellular disturbances are deterministic, then the expectation of E
in (18) could be neglected.
(iii) According to the definition of filtering ability q0 in (20), a
measure of noise filtering ability of intercellular disturbance
on synchronization of a TdCCN in (16) can be obtained by
solving the following constrained optimization problem
ð18Þ
for all v 2 L2, v – 0, and e(T) = 0 for T 6 0; i.e. the effect of intercellular disturbances v on the synchronization does not exceed a prescribed attenuation value q or the disturbance attenuation is
below q for the TdCCN. Then q in (18) can be considered as an
upper-bound for the filtering ability of the synchronized network.
If e(T) – 0 for T 6 0, then inequality (18) should be modified as
[43]
Ekek2L2 6 EfVðeð0ÞÞg þ q2 Ekv k2L2
ð19Þ
q0 ¼ min q
subject to q > 0; and ð21Þ
for some positive function V(e(0)) > 0. The filtering ability q0 is defined as the smallest q in (18) and is denoted as follows
q0 , min q
ð20Þ
i.e. the effect of all possible intercellular disturbances on the synchronizability should be less than q0. In other words, q0 is the lowest upper bound of q. If q0 < 1, then the effect of disturbance v on
the synchronization is attenuated by the TdCCN; if q0 > 1, then
the disturbance is amplified to influence the synchronization. A largerq0 means that synchronization of the TdCCN is more sensitive to
intercellular disturbances where the sensitivity is proportional to q0
ð21Þ
ð22Þ
However, it is not easy to solve the HJI in (21) for the robust
synchronization filtering problem. With a similar procedure, we
employ the global linearization method to simplify the measurement of noise filtering ability. For the synchronization error dynamics of a nonlinear TdCCN in (17), we can have the global
linearization method to be
@
F
@e
@
F
@es
h
@
F 2 X for all e; es ; v with F ¼ F T
@ev
eT
eTv H
iT
124
P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
and denote the vertices of convex hull of X as
02
As;1
0
DAs;1
A0;1
B6 D A
B6 0;1
B6
B6 0
B6
6
X # CoB
B6 B0;1
B6
B6 DB0;1
B6
B6
@4 0
0
Bs;1
0
DBs;1
0
0
0
0
2
3
Proof. See Appendix A.5. h
3 2
0
A0;K
6
0 7
7 6 DA0;K
7 6
6
0 7
7 6 0
7 6
0 7 6 B0;K
7 6
6
0 7
7 6 DB0;K
7 6
0 5 4 0
C 0;k
0
As;k
0
DAs;k
Bs;k
0
DBs;k
0
A0;k
7 6 DA
7 6 0;k
7 6
7 6 0
7 6
7 6
0 7 6 B0;k
7 6
6
0 7
7 6 DB0;k
7 6
0 5 4 0
0
C 0;1
As;K
0
DAs;K
Bs;K
0
DBs;K
0
0
0
0
31
7C
7C
7C
7C
7C
7C
0 7C
7C
C
0 7
7C
7C
0 5A
C 0;K
ð23Þ
where C 0;k ¼ C 0;k ðXÞ ¼ diag C 0;k;1 ðXÞ; ; C 0;k;N1 ðXÞ and C 0;k;i ðXÞ 2
Mm
R
for k = 1, . . . , K
In this situation, the error dynamics in (17) can be interpolated
by the following linearized systems at Kvertices as (10) with the
additional approximation error ee
H
de ¼
K
X
Mk ðA0;k þ B0;k Þedt þ DA0;k þ DB0;k edw þ ðAs;k þ Bs;k Þes dt
k¼1
þ DAs;k þ DBs;k es dws þ C 0;k ev dt þ ee þ ee þ ee dt
F
H
G
þ e e þ e e dw þ e e þ e e dws ;
DF
DF s
DG
DGs
ð24Þ
In Proposition 4, we need to solve an LMI in (26) instead of solving the HJI in (21).
Remark 7.
(i) There are different communication strategies for synchronization of TdCCN through low and high cell densities. With
high cell density, the TdCCN could be synchronized by its coue but by
pling communication (i.e. the characteristic of G or G)
its nominal interaction (the characteristic of F or e
F ) at low cell
density [44]. In order to maintain the robust synchronization
and noise filtering ability in (26), high cell density is a common strategy to achieve robust synchronization and noise filtering ability for a coupled cellular network [21,44].
(ii) Similar to solving the HJI-constrained optimization in (20),
the filtering ability q0 in a nonlinear TdCCN could be
obtained by minimizing q via the following constrained
optimization problem
q0 ¼
where
e v
eeH , He
K
X
2
and ee 6 e27 kev k22 :
Mk C 0;k dws ev
H
k¼1
2
ð25Þ
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
N
ð3Þ
N2;2
Nð3Þ
2;3
N
ð3Þ
1;2
N
ð3Þ
1;3
N
ð3Þ
1;4
N
ð3Þ
1;5
0
0
0
ð3Þ
N2;6
Nð3Þ
3;3
ð3Þ
N3;4
0
0
1a Q
0
0
c1 I
c22 I
4. Numerical simulation example
In this section, we simulate a TdCCN consisting of a synthetic
multi-cellular clock as our example to verify the effectiveness of
the proposed analysis methods.
3
7
7
Nð3Þ
2;7 7
0
2
N
ð3Þ
1;7
7
0 7
7
7
0 7
7 6 0;
7
0 7
7
7
0 7
5
Example. Consider a TdCCN consisting of a population of repressilators with ten individual cells (N = 10) under time-varying delay,
s(t) = 0.5 sin(pt/24 + 8) + 0.5, intracellular perturbations, and
intercellular disturbances in (16), where the parameters are given
in Table 1 and the intercellular disturbances are assumed as follows
ð26Þ
Nð3Þ
7;7
v i ðtÞ ¼
ð3Þ
ð1Þ
N1;2
¼ N1;2 ;
ð1Þ
Nð3Þ
1;5 ¼ N1;5 ;
ð3Þ
ð1Þ
N2;2
¼ N2;2 ;
ð1Þ
Nð3Þ
2;3 ¼ N2;3 ;
ð3Þ
ð1Þ
N2;6
¼ N2;6 ;
"
¼
0
0MðN1ÞKmðN1Þ
"
ð3Þ
N7;7
¼
T
ð28Þ
ð1Þ
Nð3Þ
1;1 ¼ N1;1 þ I;
N
2e0:005t sinð0:03ptÞ 5 sinð0:017ptÞ þ 5 3 cosðpt þ 5Þ þ 3
for i ¼ 1; . . . ; 10
where
ð3Þ
1;7
ð27Þ
(iii) The constrained optimization in (27) can be easily solved
with the LMI toolbox in MATLAB [45], SeDuMi in MATLAB
[46], LMI-tool in SCILAB [47] through the Interior-point
Methods[48], or SDPA implemented in C++ [49] through the
Generalized Augmented Lagrangian Method [50]. These
two methods can decrease q to q0 until no positive definite
P, Ps, and Q are solved in(26).
Proposition 4. Suppose the synchronized error dynamic in (24)
suffers from intracellular parameter perturbations and intercellular
disturbances. If there exists symmetric positive definite solutions P, Ps,
and Q; suitable matrices C1, C2, C3, and C4; suitable positive scalars
c1, c2, c3 and P 6 c3 I, such that the following inequality holds for a
prescribed filtering value q
ð3Þ
1;1
q
subject to P > 0; Ps > 0; Q > 0; c1 > 0;
c2 > 0; c3 > 0; U P 6 c3 I and ð26Þ
Therefore, the following results can be obtained to measure the
noise filtering ability of a TdCCN
2
min
P;P s ;Q ;C1 ;C2 ;C3 ;C4 ;c1 ;c2 ;c3
ð1Þ
Nð3Þ
1;3 ¼ N1;3 ;
ð3Þ
ð1Þ
N1;4
¼ N1;4 ;
ð1Þ
Nð3Þ
3;3 ¼ N3;3 ;
ð3Þ
ð1Þ
N3;4
¼ N3;4 ;
#
CT0 C1
T
0MðN1ÞKmðN1ÞK
;
h
i
ð3Þ
N2;7
¼ 0 ðCT0 C2 ÞT ;
ðc1 þ c2 Þe29 I q2 ðJ þ Þ J þ
0MðN1ÞmðN1ÞK
0MðN1ÞKmðN1Þ
0MðN1ÞKmðN1ÞK
#
;
C0 ¼ ½C 0;k 2 RMðN1ÞMðN1ÞK ;
then the robust synchronization to tolerate intracellular parameter perturbations as well as the robust filtering with a prescribed attenuation
value q are both achieved in (16).
Since the protein expressions of a repressilator are almost phase
locked, for convenience, we can focus on CI protein expression by
inserting a sequence of green fluorescent protein (gfp) in each cell
to easily observe the molecular trajectory. The simulated stochastic
gene expressions are given in Figs. 2–5 for different cell densities Q0
due to the different extracellular volumes in practice. In Fig. 2, it can
be seen that there are large intracellular perturbations and highly
random phase drifts between these individual oscillations due to
time-varying process delays, intracellular molecular perturbations,
and intercellular disturbances when the cellular network is
uncoupled (Q0 = 0). As the coupling ability increases (due to the increase of Q0), Figs. 3 and 4 are given as Q0 = 0.63 and Q0 = 0.8,
respectively.
To measure the robust synchronizability and the noise filtering
ability in this example, we take 81 vertices for the convex hull to
derive to a compromise and to interpolate the nonlinear TdCCN
through the following interpolation functions
125
P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
a
The Time Response of Uncoupled cellular network
250
CI1
CI
2
CI3
CI4
CI5
CI
6
200
CI7
CI
8
CI9
protein level (arb. units)
CI10
150
100
50
0
b
0
100
200
300
400
500
time (min)
600
700
800
900
1000
The Error Dynamic of Uncoupled Cellular Network
100
e1,5
e2,5
e
3,5
80
60
protein level (arb. units)
40
20
0
−20
−40
−60
−80
−100
0
100
200
300
400
500
600
700
800
900
1000
time (min)
Fig. 2. The time response in a multi-cellular clock consists of ten E. coli repressilators with Q0 = 0. (a) The dynamic response of protein CI: The subscript i denotes the ith cell.
(b) The corresponding error dynamics: ei,5 for i = 1, 2, 3 denote the errors of CI1–CI2, CI2–CI3 and CI3–CI4, respectively. Here we show only parts of the error dynamics to clarify.
Under this condition, the TdCCN cannot synchronize with the noise filtering ability q = 6.5017 due to the large influence of intra- and intercellular noises.
lk;i ðX; X s Þ ¼ ,
1
ek;i ei ðtÞ2
2
81
X
k¼1
1
ek;i ei ðtÞ2
2
for k ¼ 1; . . . ; 81 and i ¼ 1; . . . ; 9;
ð29Þ
where ek,i is the kth vertex for the ith error system. For Q0 = 0.63,
after bringing all possible ek,i into (12), (25) and (29), the least
approximation error bounds in (12) and (25) could be obtained as
follows
126
P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
The Time Response of coupled cellular network with Q =0.63
a
0
250
CI1
CI2
CI3
CI
4
CI5
CI
6
200
CI7
CI8
CI
9
protein level (arb. units)
CI10
150
100
50
0
0
100
200
300
400
500
time (min)
600
700
800
900
1000
The Error Dynamic of Coupled Cellular Network with Q =0.63
b
0
100
e1,5
e
2,5
e3,5
80
60
protein level (arb. unnits)
40
20
0
−20
−40
−60
−80
−100
0
100
200
300
400
500
time (min)
600
700
800
900
1000
Fig. 3. The time response in a multi-cellular clock consists of 10 E. coli repressilators with Q0 = 0.63. (a) The dynamic response of protein CI. (b) Parts of corresponding error
dynamics. Under this condition, the TdCCN could robustly synchronize with the noise filtering ability q = 0.5487, where the theoretical noise filtering ability q0 = 0.6085.
e1 ¼ 3 1015 ; e2 ¼ 3 102 ; e3 ¼ 2 1017 ; e4 ¼ 0;
e5 ¼ 4 1012 ; e6 ¼ 2:8 102 ; e7 ¼ 3:1 102 ;
e8 ¼ 0; e9 ¼ 5 1015
ð30Þ
By solving (27) with suitable matrices C1, C2, C3, C4 and positive
scales c1, c2, c3, we could get the noise filtering ability of the synchronized TdCCN as q0 = 0.6085 with
127
P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
a
The Time Response of coupled cellular network with Q0=0.8
250
CI1
CI
CI
2
3
CI4
CI
5
CI6
CI7
200
CI8
CI9
protein level (arb. units)
CI10
150
100
50
0
b
0
100
200
300
400
500
time (min)
600
700
800
900
1000
The Error Dynamic of Coupled Cellular Network with Q0=0.8
100
e1,5
e2,5
e
3,5
80
60
protein level (arb. units)
40
20
0
−20
−40
−60
−80
−100
0
100
200
300
400
500
time (min)
600
700
800
900
1000
Fig. 4. The time response in a multi-cellular clock consists of ten E. coli repressilators with Q0 = 0.8. (a) The dynamic response of protein CI. (b) Parts of corresponding error
dynamics. Under this condition, the TdCCN could robustly synchronize with the noise filtering ability q = 0.2909, where the theoretical noise filtering ability q0 = 0.4162.
128
P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
The Time Response of Couled Cellular Network with Q0=1.0
a 250
CI1
CI2
CI3
CI4
CI5
CI6
200
CI
7
CI8
CI9
protein level (arb. units)
CI10
150
100
50
0
0
100
200
300
400
500
time (min)
600
700
800
900
1000
The Error Dynamic of Coupled Cellular Network with Q =1.0
0
b 100
e1,5
e2,5
e3,5
80
60
protein level (arb. units)
40
20
0
−20
−40
−60
−80
−100
0
100
200
300
400
500
time (min)
600
700
800
900
1000
Fig. 5. The time response in a multi-cellular clock consists of ten E. coli repressilators with Q0 = 1. (a) The dynamic response of protein CI. (b) Parts of corresponding error
dynamics. As our proposed method has pointed out, sufficiently strong cell density would make the noise filtering ability worse, q = 0.4572, where the theoretical noise
filtering ability q0 = 0.5122.
P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
2
6
6
6
6
6
6
P¼6
6
6
6
6
4
1:8211 0:0483 0:0477 0:0030
0:0005
0:0005
2:3652 0:0499 0:0005
0:0033
0:0003
2:5773 0:0031 0:0028 0:0049
12
0:2598 0:3035
12
0:3140
2
6
6
6
6
6
6
Ps ¼ 6
6
6
6
6
4
2
6
6
6
6
6
6
Q ¼6
6
6
6
6
4
0:45 0:0153 0:0404 0:0059
0:48
0:0094
0:51
0:1165
0:0023
11
0:0023
0:0008
7
0:0013 7
7
0:0018 7
7
7
0:0003 7 102 ;
7
0:0003 7
7
7
0:0107 5
17
0:0062
3
0:0037 7
7
7
0:0042 0:0051 0:0201 0:0054 7
7
7
2:7
0:504
0:424
0:0001 7 102 ;
7
2:7
0:2904 0:0011 7
7
7
2:7
0:0645 5
2
0:0032
0:0906 0:0058 0:0051
3
0:0000
0:0054
0:0003
0:0000
0:0000
0:0012
3
0:0049 0:0000 0:0000 0:0000 0:0000 7
7
7
0:1068 0:0000 0:0000 0:0001 0:0000 7
7
7
0:1113 0:0348 0:0417 0:0000 7 102 :
7
0:1094 0:0425 0:0000 7
7
7
0:1472 0:0008 5
0:1373
One possible reason for the small matrices P, Ps, and Q is that the
inequality (26) is semi-negative definite. With similar procedures
for Q0 = 0 and Q0 = 0.8, the results confirm the intuitive expectation
that the perturbative coupled cellular network desynchronizes
when Q0 = 0 (i.e., q0 could not be solved), but has better noise filtering ability when Q0 = 0.8 (q0 = 0.4162). The reason for this trend
could be the higher concentration of AI among the cells. Higher cell
density implies more generation and spread for AI, which could improve the cellular communication and synchronize the TdCCN [44].
However, the noise filtering ability is disrupted when the cellular
density is strong enough ðq0;Q 0 ¼1 ¼ 0:5122 > q0;Q 0 ¼0:8 ¼ 0:4162Þ. The
computational simulation result in Fig. 3 and Fig. 4 also show this
phenomenon ðqQ 0 ¼1 ¼ 0:4572 > qQ 0 ¼0:8 ¼ 0:2909Þ. This finding is
consistent with previous studies and one possible reason is that
the increasing cell density will make the maximum catalytic fraction
of AI decrease the period of oscillation, which may finally lead to
desynchronizing the cellular oscillation [44]. Another possible
explanation is that since there is broad biodiversity from cell to cell,
sufficiently strong cell density implies very large intracellular
perturbations that may desynchronize the TdCCN [51].
Through these computational simulations for different situations, the noise filtering abilities of a robustly synchronized cellular
network are calculated as q 0.5487 < q0 = 0.6085 (Q0 = 0.63),
q 0.2909 < q0 = 0.4162 (Q0 = 0.8), and q 0.4572 < q0 = 0.5122
(Q0 = 1), respectively. By comparison between the computed noise
filtering ability q, and the theoretical one q0, the conservative nature of our proposed method is obvious, i.e., a noise filtering ability
less than 0.6085 cannot be solved (or guaranteed) theoretically by
our method, but the noise filtering ability of 0.5487 can be
achieved in practice when Q0 = 0.63. This is mainly due to the conservative nature of the global linearization method, Lyapunov synchronizability and the solution of LMIs [27,36,52,53].
The benchmark in silico example of E. coli repressilators-TdCCN
illustrates that although our proposed method is only a sufficient
condition, it is obvious that our method could not only provide
judgment of the robust synchronizability for a stochastic nonlinear
TdCCN under time-varying process delays and intracellular
(parameter) perturbations but also efficiently estimate the noise
filtering ability under intercellular disturbances. A living organism
may contract a fatal illness without a collective metabolic rhythm
from a genetic (intracellular) perturbation and/or pathological
(intercellular) disturbance such as environmental changes, infectious agents or chemical carcinogens. Based on our proposed meth-
129
ods, the robust synchronizability and noise filtering ability of a
synchronized TdCCN could be employed for a population synchronization analysis and for the prerequisite design of synthetic biology. Through our method, biologists could design a synthetic
coupled cellular network of E. coli population to simultaneously
generate the alcohol or other molecules robustly for the biomass
energy before in vivo experiments. If a population of coupled synthetic cellular networks could not synchronize robustly under the
time-varying process delay, intracellular perturbations and intercellular disturbances, biologists could improve their synchronization by increasing the cell density via the proposed method.
Furthermore, if biologists want to synthesize a larger TdCCN then
the analysis of noise filtering ability is contributive to multiple
networks.
5. Discussion and conclusions
Synchronization is an important topic for understanding and
predicting collective cellular behavior. However, the innate timevarying delays from biochemical processes and natural stochastic
noises from intracellular perturbations and intercellular disturbances will disrupt the united cellular phenomena. Robust synchronization is an essential property, which permits a population
of cells to function simultaneously and routinely under process delays as well as intracellular and intercellular disturbances. From a
systematic point of view, maintaining robust synchronizability is
not only an individual level phenomenon but also a populationand system-level one. Therefore, to study the characteristic of robust synchronization is a consideration of why and how a TdCCN
could be synchronized or not.
In this study, a newly global measurement of synchronization
for general nonlinear stochastic coupled cellular network has been
proposed based on robust tracking theory and the systems biology
approach. To efficiently estimate the robust synchronizability and
noise filtering ability without destroying the synchronization of a
general nonlinear TdCCN, we employ the global linearization
method to avoid solving the HJI in (6) and (21) but to solve the
LMI in (14) and (26). The robust synchronizability and noise filtering ability of a synchronized cellular network could provide more
insight into the effects of time-varying process delays, intracellular
parameter perturbations, and intercellular disturbances on the
synchronization of individual cells in the coupled cellular network.
Furthermore, our method could be applied to discuss other biological synchronization phenomena. The robust synchronizability
and noise filtering ability increase along with the cell density,
but a sufficiently strong cell density will desynchronize the TdCCN.
For the molecular basis of a quorum sensing system (in our benchmark, this is a LuxR-AHL quorum sensing system), since AI usually
has a low concentration among the functional cells, the quorum
sensing system is ultra sensitive to the variation of AI [5]. Thus,
the increasing concentration of AI generated by each cell will improve the cellular communication due to the linear proportion between Q0 and the cell number [21]. For example, in a micro-fluidic
system, if the flow rate in a micro-trapping chamber has low velocity, i.e. there is a high survival cell density through scrubbing, then
the cells will be synchronized more easily and the bioluminescence
will re-burst faster [54]. For another example, without sufficient V.
fischeri in the light organs of the symbiotic squid Euprymna scolopes, the V. fischeri will not have a robust bioluminescence
reaction [24]. In vitro, the cell density can be modulated by controlling the pH or temperature of the culture medium [55]. However,
when the cell density is increased beyond a threshold value, the robust synchronizability and noise filtering ability will decrease due
to the synthesizing fraction of AI and broad biodiversity [44].
Clearly, we can observe the decreasing of noise filtering ability in
our proposed method.
130
P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
In its efforts to understand and develop artificial biological systems, the synthetic biology has encountered numerous obstacles.
The design of a ‘synthetic circuit’ involves creating new biochemical parts, mass-producing these parts, fabricating these parts to be
modules, and applying these artificial modules [56]. The analysis of
a synthetic TdCCN subject to natively biochemical time-varying
process delays and the noisy intracellular and intercellular processes is an important step for constructing a population of synchronizable synthetic circuits to emerge a desired biological
behavior. Here, our main contribution is that our proposed method
can efficiently find the corresponding robust synchronizability and
noise filtering ability for the most coupled cellular networks
through solving the LMI in (14) and (26). For synthetic biology, it
is important that under process delays, intracellular perturbations
and intercellular disturbances, the synchronization of a synthetic
TdCCN may not occur or may be disrupted such that the coupled
networks could not function properly with suitable collective (or
emergent) behavior. Clearly, our method gives a promise as a useful tool in a population of synthetic biological circuits before
in vivo experiments.
There is a trade-off between overcoming the effect of intracellular perturbations and attenuating the effect of intercellular disturbances in a TdCCN i.e. we cannot significantly overcome these
two effects simultaneously. When the individual cells are quite
different or the thermal fluctuations of molecules are violent
(large intracellular perturbations), inequality (26) does not be easily held under a small q0, and vice versa. Therefore, there are
remarkable effects upon the robust synchronization by intracellular perturbations. Further, through our proposed method, biologists can analyze the cellular collective (or emergent) behavior
under all possible intercellular disturbances with finite energy,
i.e. not only Gaussian white noise or Brownian motion but also
jump motion, such as the heartbeats or impulse noise, could be
considered.
However, the assumptions of quasi-steady state and free diffusion that we employed may not be always held for a TdCCN. Smaller AIs indeed diffuse freely across the bacterial cell membranes
but larger AIs as peptides appear to be actively transported by
pumps [57,58]. From different strategies between active transport
and passive diffusion, the biochemical function and the coupling
function would be so distinct that our method should have a simple modification. Furthermore, the conservative of global linearization method and LMI are also an inherent defect of our methods.
Although the mechanism of quorum sensing system has not
previously been clear, cell-based studies have begun to reveal
some common propositions. This study is not only applicable to
estimating the synchronizability of stochastic biochemical systems
but also useful in the future for designing a population of robust
synchronized synthetic cellular networks with prescribed function.
In future research, we expect that this study could motivate the
investigations in synthetic biology to coordinate complex group
behaviors and analyze ‘multiple communicating populations’ (i.e.
multi-quorum sensing systems) [59] to mimic more realistic biology strategies.
A.1. Proof of Proposition 1
For the nonlinear error dynamic of a perturbative TdCCN (4), the
robust stability theory based on Lyapunov function will be employed to discuss its robust synchronization property. For Lyapunov function via chain rule and the Ito’s formula [60] for all
nonzero e and es, the following equation will be held with the last
two additional diffusive terms.
_ ¼E
EðVÞ
(
)
T dw dw @VðeÞ
s
e
e
e
e
e
e
F þ G þ DF þ DG
þ DF s þ DGs
@e
dt
dt
þ
)
T 2 T 2 1
e
e @ V De
e þ 1 De
e s @ V De
es :
DF þ DG
F þ DG
F s þ DG
F s þ DG
2
2
2
@e
2
@e
Therefore
8
T
T
9
>
e þ 1 De
e @ 2 VðeÞ
e >
< @VðeÞ
=
F þ DG
F þ DG
ðe
F þ GÞ
De
@e
2
@e2
_ ¼E
EðVÞ
:
T
>
>
: þ 1 De
;
e s @2 VðeÞ
es 6 0
F s þ DG
F s þ DG
De
2
2
ðA:1Þ
Then the nonlinear perturbative TdCCN is robust synchronization in
probability, i.e. the time-varying process delays and intracellular
parameter perturbations could be tolerated by the synchronized
coupled cellular network if Eq. (6) holds.
A.2
Before the proof of Proposition 1, the following lemmas are
necessary.
Lemma 2. [27]:
1 T
T
aT b þ b a 6 caT Ca þ b C1 b
This work was supported by National Science Council, R.O.C,
under Grant NSC 99-2745-E-007-001-ASP.
Appendix A
For simplifying the following notation, we represent X(t),
X(t s(t)), v(t), e(t), e(t s(t)) and ev(t) as X, Xs, v, e, es, ev
respectively.
ðA:2Þ
c
for any vector or matrix a, b, scalar c > 0, and any C = CT > 0.
Lemma 3 [61]. Given the matrices N0, Mk, N1,k and N2,k,l with suitable dimensions for k = 1, . . . , K and l = 1, . . . , K, we have the following
lemma
N0 þ
K
X
Mk N1;k þ
K
X
k¼1
3T 2
N0
6M 7 6
N
1;1
6 17 6
7 6
¼6
6 .. 7 6
.
6
4 . 5 4 ..
2
MK
NT1;l MTl þ
K X
K
X
l¼1
I
N1;K
Mk N2;k;l MTl
k¼1 l¼1
NT1;1
N2;1;1
..
.
..
N2;K;1
32
3
I
NT1;K
76
7
. . . N2;1;K 76 M1 7
7
..
.
.
N2;K;K
ðA:3Þ
76
.. 7
7
76
54 . 5
MK
Proof of Proposition 2
Take the following Lyapunov quadratic function for a nonlinear
coupled cellular network (4) as (13)
VðeÞ ¼ eT Pe þ
Acknowledgment
@e
Z
t
tsðtÞ
eT ðsÞPs eðsÞds þ
Z
0
a
Z
t
_
dr
e_ T ðsÞQ eðsÞds
ðA:4Þ
tþr
with 0 < sðtÞ < a; s_ ðtÞ < b. Then we have
n
dVðeÞ
¼ E eT Pe_ þ e_ T Pe þ eT P s e ð1 s_ ÞeTs Ps es þ ae_ T Q e_
E
dt
Z t
T 2 1
e
e @ V De
e
_
F þ DG
þ
DF þ DG
e_ T ðsÞQ eðsÞds
2
2
@e
ta
)
T 2 1
e
e s @ V D Fe s þ D G
es
þ
DF s þ DG
ðA:5Þ
2
@e2
131
P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
Along the trajectory of (10) and Leibniz–Newton formula, we have
the following two equations
0 ¼ e_ þ
K
X
Mk A0;k þ DA0;k dw=dt þ B0;k þ DB0;k dw=dt e
k¼1
With the condition P 6 c3 I, the bounded approximation errors
in (12) and Lemma 2, the effect of approximation error and
stochastic perturbation in (A.8) could be simplified with c1, c2,
c3 > 0 as follows
n
o
E 2eT CT1 ee þ ee þ 2e_ CT2 ee þ ee
þ As;k þ DAs;k dws =dt þ Bs;k þ DBs;k dws =dt es þ ee
F
þ ee þ e e þ e e dw=dt þ e e þ e e dws =dt
DF
DF s
G
DG
DGs
F
Z
t
_
eðsÞds
þ c2
ðA:7Þ
tsðtÞ
G G
tsðtÞ
DF s
DGs
9
8 " #T " #
" #
>
>
e
e
CT3 1
>
>
>
>
>
>
a
Q ½C 3 C 4 >
>
>
>
>
>
es
=
< es
CT4
;
þE
2" #T "
3
2
3
T
#
"
#
"
#
T
>
>
>
>
e
e
>
>
CT3
CT3
>
> Rt
T
1
T
>
>
4
5
4
5
>
tsðtÞ
þ e_ ðsÞQ Q
þ e_ ðsÞQ ds >
>
>
;
:
es
es
CT4
CT4
ðA:8Þ
where
DF
DA0 ðMÞ ¼
DF
DG
DG
T 6 2eT DAT0 ðMÞP DA0 ðMÞe þ 2 e e þ e e P e e þ e e
DF
DF
DG
DG
6 2eT DAT0 ðMÞP DA0 ðMÞe þ 4eT ePe e þ 4eT e P e e
DF
DG
DF
DG
2
ð
T
T
T
2
6 2e DA0 ðMÞP DA0 MÞe þ e 4c3 e5 þ e6 e
ðA:10Þ
and
T E DAs ðMÞes þ e e þ e e P DAs ðMÞes þ e e þ e e
DF s
DF s
DGs
DGs
2
T
T
T
2
6 2es DAs ðMÞes PDAs ðMÞes þ es 4c3 e7 þ e8 es :
ðA:11Þ
Since Q > 0 i.e. Q 1 is also positive, the last term in (A.8) can be neglected by the following fact
Z
2"
t
4
e
es
#T "
CT3
CT4
#
3
2" #T " #
3T
T
e
C
3
þ e_ T ðsÞQ 5Q 1 4
þ e_ T ðsÞQ 5 ds 6 0:
es
CT4
ðA:12Þ
ðA0;k þ B0;k ÞMk ;
Then
K
X
Mk ðAs;k þ Bs;k Þ ¼
ðAs;k þ Bs;k ÞMk ;
k¼1
K
X
Mk DA0;k þ DB0;k ¼
DA0;k þ DB0;k Mk ;
k¼1
G G
ðA:9Þ
K
X
k¼1
K
X
F
k¼1
G F
G
T E DA0 ðMÞe þ e e þ e e P DA0 ðMÞe þ e e þ e e
DF
DF
DG
DG
T T 6 E DA0 ðMÞe P DA0 ðMÞe þ e e þ e e P e e þ e e
DF
DF
DG
DG
T T þ DA0 ðMÞe P e e þ e e þ e e þ e e P DA0 ðMÞe
tsðtÞ
As ðMÞ ¼
o
þ eTs 2ðc1 þ c2 Þ e22 þ e24 es ;
82 3T 2 T
3
2C A0 ðMÞ þ Ps þ 2CT3 2CT1 þ P
2CT1 As ðMÞ 2CT3
e
>
>
<6 7 6 1
7
7 6
7
6E 6
2CT2 A0 ðMÞ þ P
2CT2 þ aQ
2CT2 As ðMÞ
4 e_ 5 4
5
>
>
:
T
T
es
0
2C4 ð1 bÞPs
2C4
2 39
e >
>
n
o
6 7=
T
T T
7
6
4 e_ 5> þ E 2e C1 eeF þ eeG þ 2e_ C2 eeF þ eeG
>
;
es
T þ E DA0 ðMÞe þ e e þ e e P DA0 ðMÞe þ e e þ e e
DF
DF
DG
DG
T þ E DAs ðMÞes þ e e þ e e P DAs ðMÞes þ e e þ e e
K
X
F
F
k¼1
k¼1
F
n
T
T
2
2
_ T 1 T _
6 E eT c1
1 C1 C1 e þ e c2 C2 C2 e þ e 2ðc1 þ c2 Þðe1 þ e3 Þe
þ As;k þ DAs;k dws =dt þ Bs;k þ DBs;k dws =dt es þee þ ee
F
G
i
T T
þ e e þ e e dw=dt þ e e þ e e dws =dt þ2 e C3 þ eTs CT4
DF
DF s
DG
DGs
"
#)
Z t
_
e es eðsÞds
Mk ðA0;k þ B0;k Þ ¼
G
n
o
T
T
T
_ T 1 T _
6 E eT c1
1 C1 C1 e þ e c2 C2 C2 e þ 2ðc1 þ c2 Þeeee þ 2ðc1 þ c2 Þee ee
T e s P De
e s þ 2 eT CT þ e_ T CT
F s þ DG
F s þ DG
þ De
1
2
"
K
X
Mk A0;k þ DA0;k dw=dt þ B0;k þ DB0;k dw=dt e
e_ þ
A0 ðMÞ ¼
eeF þ eeG
F
G
T
ee þ ðc1 þ c2 ÞeeT ee þ ðc1 þ c2 ÞeeT ee
þ ðc1 þ c2 Þee
ta
K
X
eeF þ eeG
T F
n
dVðeÞ
E
6 E eT Pe_ þ e_ T Pe þ eT P s e ð1 bÞeTs P s es þ ae_ T Q e_
dt
Z t
T e P De
e
_
F þ DG
F þ DG
þ De
e_ T ðsÞQ eðsÞds
DGs
n
T
T
_ T 1 T _
6 E eT c1
1 C1 C1 e þ e c2 C2 C2 e þ ðc1 þ c2 Þee ee
Substituting (A.6) and (A.7) into the derivative of (A.5) with suitable
matrices C1, C2, C3, C4, and s_ 6 b, we have
DF s
G
F
and
0 ¼ e es F
G
T T
T
_
6 E eT c1
ee þ ee þ e_ T c1
1 C1 C1 e þ c1 ee þ ee
2 C2 C2 e
ðA:6Þ
82 3 2 0
>
e T N1;1
>
>
<
7 6
6
dVðeÞ
6
6E 6
e_ 7
E
5 6
4
>
4
dt
>
>
: es
9
3
N01;2 N01;3 2 e 3>
>
=
76 7>
0
0 76
N2;2 N2;3 74 e_ 7
5 ;
N03;3
5
>
>
>
es ;
ðA:13Þ
k¼1
where
DAs ðMÞ ¼
K
X
k¼1
Mk DAs;k þ DBs;k ¼
K
X
DAs;k þ DBs;k Mk
k¼1
by the diagonal matrices A0,k, B0,k, C0,k, DA0,k, DB0,k,As,k, DAs,k, and
Mk.
N01;1 ¼ CT1 A0 ðMÞ þ AT0 ðMÞC1 þ P s þ C3 þ CT3 þ 2DAT0 ðMÞPDA0 ðMÞ
T
2
2
2
2
þ c1
1 C1 C1 þ 2ðc1 þ c2 Þðe1 þ e3 ÞI þ 4c3 ðe5 þ e6 ÞI
þ aCT3 Q 1 C3 ;
132
P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
N01;2 ¼ AT0 ðMÞC2 CT1 þ P;
T
1 T
2ðc1 þ c2 Þðe21 þ e23 ÞI þ ðP CT1 ÞðC2 þ CT2 Þ1 ðP CT1 ÞT þ 2c1
1 C1 C1 þ 2c2 C2 C2
T
6 CT1 A0 ðMÞ þ AT0 ðMÞC1 AT0 ðMÞC2 ðC2 þ CT2 Þ1 AT0 ðMÞC2
N01;3
¼ CT1 As ðMÞ CT3 þ C4 þ aCT3 Q 1 C4 ;
T
¼ C2 CT2 þ aQ þ c1
2 C2 C2 ;
N02;2
N02;3 ¼ CT2 As ðMÞ;
N03;3 ¼ C4 CT4 ð1 bÞPs þ 2DATs ðMÞPDAs ðMÞ þ 2ðc1 þ c2 Þðe22
þ e24 ÞI þ 4c3 ðe27 þ e28 Þ þ aCT4 Q 1 C4 :
By Lemma 3 and the Schur Complement [27], (A.13) could be
described as follows
2
2
6
6
6
3
6
0
N1;3
6
6
7
6
7
0 7
T6
N2;3 7 ¼ M 6
6
5
6
6
0
N3;3
6
6
6
4
N01;1 N01;2
6
6
6 6
4
0
2;2
N
Nð1Þ
Nð1Þ
Nð1Þ
1;1
1;2
1;3
Nð1Þ
Nð1Þ
2;2
2;3
Nð1Þ
1;4
ð1Þ
N1;5
0
0
3
0
Nð1Þ
3;3
Nð1Þ
3;4
0
a1 Q
0
c1 I
7
7
7
Nð1Þ
2;6 7
7
7
0 7
7
7M:
7
0 7
7
7
7
0 7
5
i.e. if the inequality (15) holds, then the TdCCN which is free of
intracellular perturbations and time-delay processes can be robustly synchronized.
A.4. Proof of Proposition 3
Consider a Lyapunov function V(e) > 0 for the cellular network
(17), with the similar procedure in Appendix A.1, we have
(
T @VðeÞ
e dw þ D e
e s dws
e þ De
e
FþG
F þ DG
F s þ DG
@e
dt
dt
_ ¼E
EfVg
þ
c2 I
T 2 1
e
e s @ V De
es
F s þ DG
DF s þ DG
2
2
@e
T
)
1 @V e
1 T e T @V
Hev þ ev H
þ
2 @e
2
@e
þ
ðA:14Þ
ð1Þ
where Ni;j are defined in (14); M ¼ diag M; I; M; I; I; I and M ¼
½I M1 Mk MK , i.e., if the inequality (14) holds, then
the nonlinear coupled system will be robustly synchronized.
6E
(
A.3. Proof of Corollary 1
By the similar proof procedure as Proposition 2, it is easy
to prove that we have the following inequality of derivative
of V(e)
E
n
o
dVðeÞ
¼ E eT Pe_ þ e_ T Pe þ 2 eT CT1 þ e_ T CT2 e_ þ A0 ðMÞ þ ee þ ee
F
G
dt
82 3T 2 T
3
T
C1 A0 ðMÞ þ AT0 ðMÞC1 þ 2c1
P CT1 þ AT0 ðMÞC2
< e
1 C1 C1
5
¼E 4 5 4
: _
T
e
C2 CT2 þ 2c1
2 C2 C2
9
2 3
e
=
T
4 5 þ 2ðc1 þ c2 Þ ee
eeF þ eeT eeG
;
F
G
e_
8
2
T T
T
>
>
>
2 3T 6 C1 A0 ðMÞ þ A0 ðMÞC1
>
>
< e 6
6
T
2
2
6 E 4 5 6 þ2c1
1 C1 C1 þ 2ðc1 þ c2 Þðe1 þ e3 ÞI
>
6
>
e_ 4
>
>
>
:
P CT1 þ AT0 ðMÞC2
T
C2 CT2 þ 2c1
2 C2 C2
9
3
>
>
>
72 e 3>
>
=
7
74 5
7
7 _ >
>
5 e >
>
>
;
6 0:
By the Schur complement, we could have the following inequality
þ
T
T 2 @VðeÞ
e þ 1 De
e @ V De
e
ðe
F þ GÞ
F þ DG
F þ DG
@e
2
@e2
T
T 2 1
e
e s @ V De
e s þ 1 @V HJJ
e H
e T @V
F
DF s þ DG
þ
D
G
s
2
@e2
4q2 @e
@e
)
þ q2 eTv ðJ Þþ Jþ ev þ ðeT e eT eÞ
(
T
T 2 @VðeÞ
e þ 1 De
e @ V De
e
ðe
F þ GÞ
¼E
F þ DG
F þ DG
@e
2
@e2
T
T 2 1
e
e s @ V De
e s þ 1 @V HJJ
e H
e T ðeÞ @V
G
DF s þ DG
þ
D
F
s
2
@e2
4q2 @e
@e
)
þ eT e eT e q2 v T v
þ
where J⁄ is the Hermitian transpose of matrix J, J+ is the pseudo-inverse of matrix J. Therefore,
if the HJI (21) holds, then we have
_ 6 E ðeT e q2 v T v Þ . After integrating the above HJI from 0
EfVg
toTas Appendix A.1, the robust synchronized coupled cellular network can filter intercellular disturbances under a desired attenuation value q in probability if (21) holds for the nonlinear TdCCN in
(17).
3
2
CT1 AT0 ðMÞ þ AT0 ðMÞC1
6
6
6 þ2ðc1 þ c2 Þðe21 þ e23 ÞI
6
6
6
6
6
6
6
6
4
T 2 1
e
e @ V De
e
F þ DG
DF þ DG
2
2
@e
T
1
PC þ
AT0 ðMÞ
T
1
C
0
ðC2 þ CT2 Þ
0
CT2
12 c1
0
C2
7
7
7
7
7
7
7 6 0:
7
7
7
7
5
12 c2
A.5. Proof of Proposition 4
With a similar procedure in Appendix A.2, consider a Lyapunov
function with the following quadratic form
VðeÞ ¼ eT Pe þ
Z
t
tsðtÞ
eT ðsÞPs eðsÞds þ
Z
0
a
Z
t
_
e_ T ðsÞQ eðsÞds
dr
tþr
ðA:15Þ
By Schur complement, the above inequality (A.15) is equivalent to
via the help of Lemma 2 and the bounded approximation error (25),
we have the differential equation of V(e) as follows
P.-W. Chen, B.-S. Chen / Mathematical Biosciences 232 (2011) 116–134
82 3 2 0
9
32 3
T
>
N1;1 N01;2 N01;3
!>
e
e
>
>
K
<
=
X
7
dVðeÞ
6 7 6
7
0
0 76
T T
T T
_
_
e
þ
2
e
6 E 4 e_ 5 6
E
C
þ
e
C
M
C
e
þ
e
N
N
5
k 0;k v
1
2
2;2
2;3 54
4
e
H >
>
dt
>
>
k¼1
: es
;
es
N03;3
3
8 2 3T 2 0
9
2 3
0
0
>
>
> e 6 N1;1 N1;2 N1;3 7 e
>
>
>
K
K
P
P
>
>
7 6
7
0
0 76
>
>
T T
T T
<6
_
_
_
þ
2e
C
M
C
e
þ
2
e
C
M
C
e
e
e
N
N
4 5 4
5
k 0;k v
k 0;k v =
1
2
2;2
2;3 54
k¼1
k¼1
6E
0
es
es
>
>
N3;3
>
>
>
>
>
>
>
>
:
;
1 T T
1 _ T T
T
þc1 e C1 C1 e þ c2 e C2 C2 e_ þ ðc1 þ c2 Þe ee
e
H H
82 3 2
9
3
2 3
T
>
>
N01;1 N01;2 N01;3
e
e
>
>
>
>
K
K
>
>
7
6
>
>
7
7
T P
T P
0
0 76
T
T
<6
6
Mk C 0;k ev þ 2e_ C2
Mk C 0;k ev =
N2;2 N2;3 54 e_ 5 þ 2e C1
4 e_ 5 4 6E
k¼1
k¼1
>
>
es
es
N03;3
>
>
>
>
>
>
>
>
: 1 T T
;
1 _ T T
T
2
þc1 e C1 C1 e þ c2 e C2 C2 e_ þ ev ðc1 þ c2 Þe9 ev
8
9
3
2
2 3T N0 þ c1 CT C
2 3>
>
N01;2
N01;3
N01;4
>
e >
e
1 1
1
>
>
>
>
7
6 1;1
>
T
0
0
0
<6 e_ 7 6
=
76 e_ 7>
1
N
þ
c
C
C
N
N
76 7
6 7 6
2 2
2;2
2;3
2;4
2
¼E 6 7 6
76 7
0
>
>
5
5
7
4
4
6
es 4
>
N3;3
0
>
>
5 es >
>
>
> e
>
:
ev ;
v
ðc1 þ c2 Þe29 I
P
P
where N01;4 ¼ CT1 Kk¼1 Mk C 0;k ; N02;4 ¼ CT2 Kk¼1 Mk C 0;k . Through the
similar procedure of Lemma3 in Appendix A.2, the interpolation
function Mk for k = 1, . . . , Kcould be separated from the linear matrix
ð3Þ
inequality. If the inequality (26) holds with Ni;j which is defined in
T
2 T
_
_ es , and ev, i.e. when the
(26), then V 6 ðe e q v v Þ for every e; e;
inequality (26) holds, the robustly synchronized TdCCN could filter
the intercellular disturbance under the disturbance filtering ability
q.
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ðA:16Þ
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