Firstly, a nonlinear stochastic time-delayed model is developed to mimic the realistic dynamic behavior of a gene network under process delays, random intrinsic and extrinsic molecular noises. Then the robust stability and the filtering ability of nonlinear stochastic time-delayed gene network are discussed from the H∞ signal processing perspective. Further, in order to improve the robust stability and filtering ability to tolerate more parameter fluctuations, process delays and to attenuate much environmental molecular noises, based on fuzzy approximation and LMI technique, a systematic design method is proposed for nonlinear stochastic time-delayed gene networks. Finally, a simple design procedure is developed for the proposed robust gene circuit design method. These results are described in the following subsections in detail.
Robust Stability and Filtering Ability Analysis for Nonlinear Stochastic Time-Delayed Gene Networks
For the convenience of illustration, the following linear gene network with process delay is introduced first,
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqWGKbazaeaacqWGKbazcqWG4baEaaGccqWG4baEcqGGOaakcqWG0baDcqGGPaqkcqGH9aqpcqWGbbqqdaWgaaWcbaGaeGimaadabeaakiabdIha4jabcIcaOiabdsha0jabcMcaPiabgUcaRmaaqahabaGaemyqae0aaSbaaSqaaiabdUgaRbqabaGccqWG4baEcqGGOaakcqWG0baDcqGHsislcqaHepaDdaWgaaWcbaGaem4AaSgabeaakiabcMcaPaWcbaGaem4AaSMaeyypa0JaeGymaedabaGaemyBa0ganiabggHiLdGccqGGSaalaaa@519C@
x(t) = x0(t)∀t ∈ [-τ, 0], where x(t) = [x1(t),...,xN(t)]T denotes the vector of mRNA concentrations of N genes; the maximum time delay τ = max{τk, k ∈ [1, m]}; A0 and Ak denote the real-time and delay-time kinetic interaction matrices among these genes, respectively. For example, x(t) is the mRNA expression vector of N genes; the diagonal component a0,ii = -λi of A0 denotes the degradation of mRNA of i-th gene with decay rate λi; the i, j component ak,ij of Ak denotes the regulatory interaction from gene j to gene i to activate or repress gene i with time delay τk. The delay time τk may be due to the process time of gene transcription, translation, posttranslation protein modification and diffusion, which are needed for regulatory genes to product proteins (e.g. transcription factors TFs) and then convey them to their target genes (see Fig. 1). In general, in real biochemical systems, these biochemical process delays may be very long and can not be neglected in biochemical dynamic models, especially for practical gene network design applications.
<p>Figure 1</p>
A linear gene regulatory network of N genes under process delays, intrinsic molecular fluctuation Δak,ijnk(t) and extrinsic molecular noise v(t).
A linear gene regulatory network of N genes under process delays, intrinsic molecular fluctuation Δak,ijnk(t) and extrinsic molecular noise v(t).
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Suppose the linear gene network in (1) suffers from intrinsic molecular fluctuations so that ak,ij → ak,ij + Δak,ijnk(t), k = 0,1,...,m, where Δak,ij denotes the amplitude of the stochastic kinetic fluctuation and nk(t) is a white noise with zero mean and unit variance, i.e., Δak,ij denotes the deterministic part of fluctuations and nk(t) absorbs the stochastic property of intrinsic molecular fluctuations (see Fig. 1). Intrinsic molecular fluctuations are mainly attributed to thermal fluctuation and random molecular events in the transcription, splicing, and translation processes of gene expression 123. The covariances of stochastic intrinsic molecular fluctuations Δak,ijnk(t), k = 0, 1,...,m, are as follows
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for k = 0, 1,...,m, where δt1,t2
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According to the above analysis, the gene network under stochastic molecular fluctuations can be represented as
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where the fluctuation matrices Bk are given by
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If ak,ij is free of fluctuation, then Δak,ij = 0 in Bk. By the Itô stochastic differential equation, the stochastic gene network equation in (3) is equivalent to the following stochastic process 272931
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where Wk(t) is a standard Wiener process or Brownian motion with dWk(t) = nk(t)dt.
Actually, in real gene networks, the dynamic gene regulatory equations are always nonlinear. In this situation, the linear dynamic gene network regulatory equation in (4) should be modified as the following Langevin nonlinear stochastic equation with multiple time delays 1310
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where the first two terms on the right-hand side denote the nominal nonlinear interactions of gene network and the last two terms denote the effect of intrinsic molecular fluctuations on the gene network, which are state-dependent and will influence the stability of nominal gene network.
Remark 1 If the Langevin equation in (5) is linearized at an operation point, it should be a linear stochastic gene network as (4).
For simplicity, the linear stochastic gene network in (4) and nonlinear stochastic gene network in (5) will be reformulated, respectively, as follows
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and
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where τ0 = 0 and τk > 0, k = 1,...,m, denote the corresponding time delays.
We say that the stochastic gene network is asymptotically stable in probability at the equilibrium point x = 0 if there exists a Lyapunov (power-like) function V(x(t)) > 0, with V(0) = 0, in the neighborhood of the equilibrium point such that the expectation of the derivative of V(x(t)), EddtV(x(t))<0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyrauucfa4aaSaaaeaacqWGKbazaeaacqWGKbazcqWG0baDaaGccqWGwbGvcqGGOaakcqWG4baEcqGGOaakcqWGubavcqGGPaqkcqGGPaqkcqGH8aapcqaIWaamaaa@3AD8@, i.e., the power of the gene network is decreasing. In general, there exist many equilibrium points for nonlinear gene networks in (7). Gene networks perform their biochemical function within some local region of an equilibrium point, which is called the phenotype of the gene network. For the convenience of design, only the robust stability of the equilibrium point at x = 0 is discussed for the nonlinear stochastic gene network in (7). If the interested equilibrium point (steady state) of the nonlinear time-delayed gene network is not at x = 0, the origin should be shifted to the equilibrium point or the steady state. In this situation, the robust stabilization and filtering ability at the origin is equivalent to the stabilization and filtering ability at the interested equilibrium point of the gene network. According to the stochastic gene network model in (7) and the nonlinear stochastic stability theory in 273031, the robust stability and molecular noise filtering ability of stochastic time-delayed gene networks are analyzed in the following propositions.
Proposition 1 Assume there exists a positive Lyapunov function V(x(t)) > 0 and V (0) = 0 satisfying the following second-order Hamilton-Jacobi inequality (HJI)
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<
0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8774@
for all nonzero x(t) ∈ RN, then the equilibrium point x = 0 of the nonlinear stochastic gene network in (7) is asymptotically stable in probability, i.e., the effects of the time delays and intrinsic molecular fluctuations ∑k=0mhk(x(t−τk))dWk(t)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWGObaAdaWgaaWcbaGaem4AaSgabeaakiabcIcaOiabdIha4jabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKIaeiykaKIaemizaqMaem4vaC1aaSbaaSqaaiabdUgaRbqabaGccqGGOaakcqWG0baDcqGGPaqkaSqaaiabdUgaRjabg2da9iabicdaWaqaaiabd2gaTbqdcqGHris5aaaa@4750@ could be tolerated by the gene network.
Proof. See Appendix A. ■
Remark 2 The last diffusion term in (8) is from the intrinsic molecular noises in (7). If the gene network is free of intrinsic molecular fluctuations, then the asymptotic stability of the equilibrium point is only to check the existence of V(x(t)) > 0 in the following inequality without diffusion terms
(
∂
V
(
x
(
t
)
)
∂
x
)
T
∑
k
=
0
m
f
k
(
x
(
t
−
τ
k
)
)
<
0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaeWaaKqbagaadaWcaaqaaiabgkGi2kabdAfawjabcIcaOiabdIha4jabcIcaOiabdsha0jabcMcaPiabcMcaPaqaaiabgkGi2kabdIha4baaaOGaayjkaiaawMcaamaaCaaaleqabaGaemivaqfaaOWaaabCaeaacqWGMbGzdaWgaaWcbaGaem4AaSgabeaakiabcIcaOiabdIha4jabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKIaeiykaKcaleaacqWGRbWAcqGH9aqpcqaIWaamaeaacqWGTbqBa0GaeyyeIuoakiabgYda8iabicdaWaaa@51F7@
Proposition 2
For the linear stochastic gene network with process delays in (6), we could choose the Lyapunov function as
35
V
(
x
(
t
)
)
=
x
T
(
t
)
P
x
(
t
)
+
∑
k
=
1
m
∫
t
−
τ
k
t
x
T
(
s
)
Q
k
x
(
s
)
d
s
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@61C1@
for P = PT > 0 and Qk=QkT>0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyuae1aaSbaaSqaaiabdUgaRbqabaGccqGH9aqpcqWGrbqudaqhaaWcbaGaem4AaSgabaGaemivaqfaaOGaeyOpa4JaeGimaadaaa@3583@ > 0, i = 1,...,m. Then the robust stability problem in Proposition 1 is reduced to check whether or not the existence of symmetric matrices P > 0 and Qk > 0, k = 1,...,m, in the following LMI
[
Ξ
P
A
1
⋯
P
A
m
A
1
T
P
B
1
T
P
B
1
−
Q
1
0
0
⋮
0
⋱
0
A
m
T
P
0
0
B
m
T
P
B
m
−
Q
m
]
<
0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaamWaaeaafaqabeabeaaaaaqaaiabf65aybqaaiabdcfaqjabdgeabnaaBaaaleaacqaIXaqmaeqaaaGcbaGaeS47IWeabaGaemiuaaLaemyqae0aaSbaaSqaaiabd2gaTbqabaaakeaacqWGbbqqdaqhaaWcbaGaeGymaedabaGaemivaqfaaOGaemiuaafabaGaemOqai0aa0baaSqaaiabigdaXaqaaiabdsfaubaakiabdcfaqjabdkeacnaaBaaaleaacqaIXaqmaeqaaOGaeyOeI0Iaemyuae1aaSbaaSqaaiabigdaXaqabaaakeaacqaIWaamaeaacqaIWaamaeaacqWIUlstaeaacqaIWaamaeaacqWIXlYtaeaacqaIWaamaeaacqWGbbqqdaqhaaWcbaGaemyBa0gabaGaemivaqfaaOGaemiuaafabaGaeGimaadabaGaeGimaadabaGaemOqai0aa0baaSqaaiabd2gaTbqaaiabdsfaubaakiabdcfaqjabdkeacnaaBaaaleaacqWGTbqBaeqaaOGaeyOeI0Iaemyuae1aaSbaaSqaaiabd2gaTbqabaaaaaGccaGLBbGaayzxaaGaeyipaWJaeGimaadaaa@6317@
where Ξ=A0TP+PA0+B0TPB0+∑k=1mQk
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuONdGLaeyypa0Jaemyqae0aa0baaSqaaiabicdaWaqaaiabdsfaubaakiabdcfaqjabgUcaRiabdcfaqjabdgeabnaaBaaaleaacqaIWaamaeqaaOGaey4kaSIaemOqai0aa0baaSqaaiabicdaWaqaaiabdsfaubaakiabdcfaqjabdkeacnaaBaaaleaacqaIWaamaeqaaOGaey4kaSYaaabmaeaacqWGrbqudaWgaaWcbaGaem4AaSgabeaaaeaacqWGRbWAcqGH9aqpcqaIXaqmaeaacqWGTbqBa0GaeyyeIuoaaaa@4909@.
Proof. See Appendix B. ■
If the linear gene network in (6) is free of intrinsic molecular fluctuations, then the asymptotic stability is only to check whether or not P > 0 and Qk > 0, k = 1,..., m, exist in the following LMI
[
A
0
T
P
+
P
A
0
+
∑
k
=
1
m
Q
k
P
A
1
⋯
P
A
m
A
1
T
P
−
Q
1
0
0
⋮
0
⋱
0
A
m
T
P
0
0
−
Q
m
]
<
0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6640@
Obviously, the condition for the existence of P > 0 and Qk > 0, k = 1,...,m, in inequality (10) is stricter than that in inequality (11) because A0 should be more negative than that in inequality (11), i.e., the eigenvalues of A0 should be in the farther left-hand side of the complex domain (the real parts of all eigenvalues of A0 should be more negative) or more robustly stable in order to tolerate time delays and intrinsic molecular fluctuations.
In general, a gene network in vivo also suffers from extrinsic molecular noises such as the transmitted noise from upstream genes and the global noise affecting all genes. Therefore, the nonlinear stochastic gene regulatory equation with process delays in (7) should be modified in the following to mimic the realistic dynamic behavior in its host cell.
d
x
(
t
)
=
∑
k
=
0
m
f
k
(
x
(
t
−
τ
k
)
)
d
t
+
G
v
(
t
)
d
t
+
∑
k
=
0
m
h
k
(
x
(
t
−
τ
k
)
)
d
W
k
(
t
)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7049@
where signal vector v(t)=[v1(t),...,vnv(t)]T
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemODayNaeiikaGIaemiDaqNaeiykaKIaeyypa0Jaei4waSLaemODay3aaSbaaSqaaiabigdaXaqabaGccqGGOaakcqWG0baDcqGGPaqkcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWG2bGDdaWgaaWcbaGaemOBa42aaSbaaWqaaiabdAha2bqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKIaeiyxa01aaWbaaSqabeaacqWGubavaaaaaa@475B@ denotes the extrinsic molecular noises and G denotes the coupling matrix between the extrinsic molecular noises and the gene network. In general, the intrinsic molecular fluctuations are state-dependent as shown in (6)–(7). Therefore, they will influence the stability of gene network and provide an important topic for discussing the robust stability of gene network at an equilibrium point of interest, i.e. to discuss how much the gene network could tolerate the intrinsic molecular fluctuation ∑k=0mhk(x(t−τk))dWk(t)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWGObaAdaWgaaWcbaGaem4AaSgabeaakiabcIcaOiabdIha4jabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKIaeiykaKIaemizaqMaem4vaC1aaSbaaSqaaiabdUgaRbqabaGccqGGOaakcqWG0baDcqGGPaqkaSqaaiabdUgaRjabg2da9iabicdaWaqaaiabd2gaTbqdcqGHris5aaaa@4750@ without blowing away from the equilibrium point. Another important topic is how to quantify the ability to filter extrinsic molecular noises v(t) in the gene network, i.e. the attenuation analysis of v(t) on x(t) by the gene network. Such information is crucial for understanding the signal-processing scheme in gene networks, from which the design of molecular noise-tolerant and time delay-compensation gene circuits can be developed to achieve a robust H∞ noise filtering design for biotechnological purposes.
Therefore, we want to investigate the influence of extrinsic molecular noises v(t) on the gene i, i.e. xi(t), which is crucial to the design of noise-tolerant gene circuit in synthetic biology 7. Let us denote the choice of a gene of interest as
zo(t) = Cx(t)
where the row vector C = [0, 0,...0, 1, 0, 0, 0, 0], i.e., all elements of C are zero except the i-th element being 1, if the i-th gene xi(t) is the gene of interest in molecular noise filtering. Let us denote the effect of extrinsic molecular noises v(t) on a gene of interest as follows
(
E
∫
0
∞
z
o
T
(
t
)
z
o
(
t
)
d
t
)
1
/
2
(
E
∫
0
∞
v
T
(
t
)
v
(
t
)
d
t
)
1
/
2
≤
γ
,
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqGGOaakcqWGfbqrdaWdXaqaaiabdQha6naaDaaabaGaem4Ba8gabaGaemivaqfaaiabcIcaOiabdsha0jabcMcaPiabdQha6naaBaaabaGaem4Ba8gabeaacqGGOaakcqWG0baDcqGGPaqkcqWGKbazcqWG0baDaeaacqaIWaamaeaacqGHEisPaiabgUIiYdGaeiykaKYaaWbaaeqabaGaeGymaeJaei4la8IaeGOmaidaaaqaaiabcIcaOiabdweafnaapedabaGaemODay3aaWbaaeqabaGaemivaqfaaiabcIcaOiabdsha0jabcMcaPiabdAha2jabcIcaOiabdsha0jabcMcaPiabdsgaKjabdsha0bqaaiabicdaWaqaaiabg6HiLcGaey4kIipacqGGPaqkdaahaaqabeaacqaIXaqmcqGGVaWlcqaIYaGmaaaaaOGaeyizImQaeq4SdCMaeiilaWcaaa@62CE@
or
E
∫
0
∞
z
o
T
(
t
)
z
o
(
t
)
d
t
≤
γ
2
E
∫
0
∞
v
T
(
t
)
v
(
t
)
d
t
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyrau0aa8qmaeaacqWG6bGEdaqhaaWcbaGaem4Ba8gabaGaemivaqfaaOGaeiikaGIaemiDaqNaeiykaKcaleaacqaIWaamaeaacqGHEisPa0Gaey4kIipakiabdQha6naaBaaaleaacqWGVbWBaeqaaOGaeiikaGIaemiDaqNaeiykaKIaemizaqMaemiDaqNaeyizImQaeq4SdC2aaWbaaSqabeaacqaIYaGmaaGccqWGfbqrdaWdXaqaaiabdAha2naaCaaaleqabaGaemivaqfaaOGaeiikaGIaemiDaqNaeiykaKIaemODayNaeiikaGIaemiDaqNaeiykaKIaemizaqMaemiDaqhaleaacqaIWaamaeaacqGHEisPa0Gaey4kIipaaaa@59B0@
for all bounded energy noises v(t). This is the so-called H∞ noise filtering problem 27, which addresses the effect of extrinsic molecular noises on the gene of interest from the system gain (L2-gain) point of view. If the extrinsic molecular noises v(t) are deterministic signals, then the expectation E on v(t) in (14) could be neglected.
Remark 3
(i) The physical meaning of (14) is that the effect of extrinsic molecular noises v(t) on zo(t) should be less than or equal to γ. If γ < 1, then extrinsic molecular noises v(t) are filtered at gene i by the gene network. If γ > 1, then it means that extrinsic molecular noises v(t) are amplified at gene i, i.e., gene i is much influenced by external molecular noises v(t) and may be a weak site of the gene network.
(ii) In the conventional control system design and signal processing 272829303132, the H∞ noise filtering criterion in (14) has been employed to design a filter to robustly estimate the state variables from the noisy measurement output, i.e., to specify the filter design parameters to efficiently attenuate the effect of uncertain external disturbance on the state estimation error of the filter 2728. In this study, we only discuss the effect of external disturbance on the genes of nonlinear time-delayed gene network and then engineer some gene circuits to improve the noise filtering ability of the nonlinear time-delayed gene network to attenuate the effect of external disturbance. Therefore, there are some differences between the proposed noise filtering circuit design in gene network and the conventional H∞ filter design to robustly estimate state variables in a signal processing system under external disturbance.
(iii) If we want to discuss the effect of extrinsic molecular noises v(t) on the whole gene network, then C in (13) should be an identity matrix, i.e. C = I and zo(t) = x(t).
(iv) If the initial condition x(0) ≠ 0, then the filtering ability in (14) should be modified as follows 2728
E
∫
0
∞
z
0
T
(
t
)
z
o
(
t
)
d
t
≤
E
V
(
x
(
0
)
)
+
γ
2
E
∫
0
∞
v
T
(
t
)
v
(
t
)
d
t
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@622C@
for some Lyapunov function V (x(0)), i.e., the energy due to initial condition x(0) should be considered in the H∞ filtering performance.
Then we get the following molecular noise filtering result for the nonlinear stochastic gene network with process delays, intrinsic molecular fluctuations, and extrinsic molecular noises in (12).
Proposition 3 If there exists a positive function V (x(t)) > 0 with V (0) = 0 solving the following HJI
(
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(
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+
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(
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(
t
)
+
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4
γ
2
(
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@BBC9@
then the influence of molecular noises v(t) on the gene network is less than γ or equivalently, the molecular noise filtering ability γ in (14) or (15) is achieved in the nonlinear stochastic gene network under process delays, extrinsic molecular noises, and intrinsic molecular fluctuations.
Proof. See Appendix C. ■
Therefore, the optimal extrinsic molecular noise filtering ability of gene network at the gene of interest is obtained by solving the following constrained optimization
γ
0
=
min
γ
subject to
(
16
)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabiqaaaqaaiabeo7aNnaaBaaaleaacqaIWaamaeqaaOGaeyypa0JagiyBa0MaeiyAaKMaeiOBa4Maeq4SdCgabaGaee4CamNaeeyDauNaeeOyaiMaeeOAaOMaeeyzauMaee4yamMaeeiDaqNaeeiiaaIaeeiDaqNaee4Ba8MaeeiiaaIaeiikaGIaeGymaeJaeGOnayJaeiykaKcaaaaa@4759@
i.e., the molecular noise filtering ability γ0 of v(t) on gene i could be evaluated by the optimal H∞ filtering ability of gene network via solving the constrained optimization in (17).
Remark 4
(i) From Propositions 1 and 3, we gain more insight into robust stability and molecular noise filtering of nonlinear stochastic time-delayed gene network from the nonlinear stochastic H∞ filtering perspective. The solution for HJI in (16) is stricter than that for HJI in (8), because of two extra terms for the robust filtering of extrinsic molecular noises in (16). i.e., Proposition 3 not only guarantees robust stability against process delay and intrinsic molecular fluctuations, but also achieves a prescribed noise filtering ability γ against extrinsic molecular noises.
(ii) In general, a nonlinear stochastic gene network has many equilibrium points (i.e. phenotypes). We are only concerned about the robust stability of one equilibrium point of interest, i.e. the equilibrium point at the steady state of gene networks. Propositions 1 and 3 are true only at the equilibrium point (one phenotype) of the origin, i.e. xe = 0. If we are interested in an equilibrium point xe ≠ 0 (i.e. another phenotype), for the convenience of investigation, the origin should be shifted to xe, i.e., the stochastic system in (12) should be modified as
d
x
′
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0
m
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(
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′
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+
x
e
)
d
t
+
G
v
(
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)
+
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k
=
0
m
h
k
(
x
′
(
t
−
τ
k
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+
x
e
)
d
W
k
(
t
)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemizaqMafmiEaGNbauaacqGGOaakcqWG0baDcqGGPaqkcqGH9aqpdaaeWbqaaiabdAgaMnaaBaaaleaacqWGRbWAaeqaaOGaeiikaGIafmiEaGNbauaacqGGOaakcqWG0baDcqGHsislcqaHepaDdaWgaaWcbaGaem4AaSgabeaakiabcMcaPiabgUcaRiabdIha4naaBaaaleaacqWGLbqzaeqaaOGaeiykaKIaemizaqMaemiDaqhaleaacqWGRbWAcqGH9aqpcqaIWaamaeaacqWGTbqBa0GaeyyeIuoakiabgUcaRiabdEeahjabdAha2jabcIcaOiabdsha0jabcMcaPiabgUcaRmaaqahabaGaemiAaG2aaSbaaSqaaiabdUgaRbqabaGccqGGOaakcuWG4baEgaqbaiabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKcaleaacqWGRbWAcqGH9aqpcqaIWaamaeaacqWGTbqBa0GaeyyeIuoakiabgUcaRiabdIha4naaBaaaleaacqWGLbqzaeqaaOGaeiykaKIaemizaqMaem4vaC1aaSbaaSqaaiabdUgaRbqabaGccqGGOaakcqWG0baDcqGGPaqkaaa@757D@
where x'(t) = x(t) - xe. Therefore, when we are interested in an equilibrium point xe ≠ 0 of gene network, the variable x(t) in HJIs (8) and (16) should be replaced by x'(t) + xe so that the origin of the coordinate should be shifted to xe. This shift is a one-to-one correspondence 43. Therefore, instead of studying the behavior of gene network in the neighborhood of xe, one can equivalently study the behavior in the neighborhood of the origin x'(t) = 0 in (18).
In general, it is still very difficult to solve the second-order HJI in (8) or (16) with V (x(t)) > 0 and V (0) = 0 to guarantee the robust stability of a nonlinear stochastic gene network under process delay and intrinsic molecular fluctuations or to solve the constrained minimization in (17) to get the noise filtering ability γ0 on extrinsic molecular noises at any gene of the gene network, especially for complex gene networks. In this situation, the T-S fuzzy model is employed to interpolate several linear gene regulatory networks at different operation points to efficiently and globally approximate the stochastic gene networks in (7) and (12). According to the T-S fuzzy model, both the robust stability of a gene network to tolerate process delays and intrinsic molecular fluctuations and the filtering ability of attenuating extrinsic molecular noises of the gene network could be efficiently investigated and calculated via fuzzy interpolation of several local linear gene networks. Finally, if the gene regulatory network cannot tolerate process delays and intrinsic parameter variations and achieve a prescribed noise filtering level γ <γ0 for some biotechnological or drug design purpose, a robust gene circuit control design is developed from the fuzzy robust stabilization and H∞ filtering theory to achieve the robust stability and to improve the filtering ability of the gene network. The proposed robust H∞ filtering design method for gene networks is also useful for synthetic biology in the near future, if a synthetic gene network wants to perform reliably under process delays, intrinsic molecular fluctuations, and extrinsic molecular noises.
Robust H∞ Gene Circuit Design of Nonlinear Gene Network Under Process Delays and Molecular Noises via Fuzzy Methodology
If the linear stochastic gene network in (6) is not robust enough and cannot tolerate the molecular fluctuations ∑k=0mBkx(t−τk)dWk(t)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWGcbGqdaWgaaWcbaGaem4AaSgabeaakiabdIha4jabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKIaemizaqMaem4vaC1aaSbaaSqaaiabdUgaRbqabaGccqGGOaakcqWG0baDcqGGPaqkaSqaaiabdUgaRjabg2da9iabicdaWaqaaiabd2gaTbqdcqGHris5aaaa@4552@, which are state-dependent and will affect the stability of gene network. i.e., the perturbative gene network in equation (6) becomes unstable, then a gene circuit design is necessary for the gene network. We want to engineer some feedback gene circuits ∑k=0mKkx(t−τk)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWGlbWsdaWgaaWcbaGaem4AaSgabeaakiabdIha4jabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKcaleaacqWGRbWAcqGH9aqpcqaIWaamaeaacqWGTbqBa0GaeyyeIuoaaaa@3E24@ to robustly stabilize the perturbative gene network as follows
d
x
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)
=
∑
k
=
0
m
(
A
k
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−
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+
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k
x
(
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)
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+
∑
k
=
0
m
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)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeWabeqaaaqaaiabdsgaKjabdIha4jabcIcaOiabdsha0jabcMcaPiabg2da9maaqahabaGaeiikaGIaemyqae0aaSbaaSqaaiabdUgaRbqabaGccqWG4baEcqGGOaakcqWG0baDcqGHsislcqaHepaDdaWgaaWcbaGaem4AaSgabeaakiabcMcaPiabgUcaRiabdUealnaaBaaaleaacqWGRbWAaeqaaOGaemiEaGNaeiikaGIaemiDaqNaeyOeI0IaeqiXdq3aaSbaaSqaaiabdUgaRbqabaGccqGGPaqkcqGGPaqkcqWGKbazcqWG0baDaSqaaiabdUgaRjabg2da9iabicdaWaqaaiabd2gaTbqdcqGHris5aOGaey4kaSYaaabCaeaacqWGcbGqdaWgaaWcbaGaem4AaSgabeaakiabdIha4jabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKcaleaacqWGRbWAcqGH9aqpcqaIWaamaeaacqWGTbqBa0GaeyyeIuoakiabdsgaKjabdEfaxnaaBaaaleaacqWGRbWAaeqaaOGaeiikaGIaemiDaqNaeiykaKcaaaaa@713E@
=
∑
k
=
0
m
A
¯
k
x
(
t
−
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k
)
d
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+
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k
=
0
m
B
k
x
(
t
−
τ
k
)
d
W
k
(
t
)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeyypa0ZaaabCaeaacuWGbbqqgaqeamaaBaaaleaacqWGRbWAaeqaaOGaemiEaGhaleaacqWGRbWAcqGH9aqpcqaIWaamaeaacqWGTbqBa0GaeyyeIuoakiabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKIaemizaqMaemiDaqNaey4kaSYaaabCaeaacqWGcbGqdaWgaaWcbaGaem4AaSgabeaakiabdIha4jabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKIaemizaqMaem4vaC1aaSbaaSqaaiabdUgaRbqabaGccqGGOaakcqWG0baDcqGGPaqkaSqaaiabdUgaRjabg2da9iabicdaWaqaaiabd2gaTbqdcqGHris5aaaa@5D27@
where A¯k≜Ak+Kk;Kk,k=0,1,...,m
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyqaeKbaebadaWgaaWcbaGaem4AaSgabeaakiablYLiajabdgeabnaaBaaaleaacqWGRbWAaeqaaOGaey4kaSIaem4saS0aaSbaaSqaaiabdUgaRbqabaGccqGG7aWocqWGlbWsdaWgaaWcbaGaem4AaSgabeaakiabcYcaSiabdUgaRjabg2da9iabicdaWiabcYcaSiabigdaXiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabd2gaTbaa@4596@, are the circuit kinetic parameter matrices of gene circuits to be designed. The element k0,ij of K0 denotes the circuit parameter to be specified for the engineered gene circuit between gene j and gene i via transfection and transformation biotechnologies (see Fig. 1). The gene circuit from gene j to gene i can be implemented by inserting the binding site of gene product j (i.e. the protein of gene j) into the promoter region of gene i so that the protein of gene j could bind this inserted binding site to act as a transcription factor (TF) to regulate the gene expression of gene i. By inserting strong (long) or weak (short) binding site, we can get a large or small circuit kinetic parameter k0,ij. The inserting of a TF binding site into promoter region or the deleting of a TF binding site from the promoter region can be easily done by using a highly efficient phage-based homologous recombination system, called recombineering 3738. Furthermore, the change of decay rates in the diagonal terms of A0 by the diagonal terms of K0 can be implemented by the mechanisms and controls of mRNA degradation through elongating or shortening the 3' polyadenylate tail of mRNA 3940444546. These powerful biotechnologies have been employed to engineer large segments of genomic DNA to generate transgenic and knockout constructs. However, for simplicity and feasibility, only a few feasible gene circuits in K0x(t) are considered for gene circuit design in gene network, i.e., the circuit kinetic parameter matrix K0 has with only a few elements k0,ij. The design principle of Kk, k = 1,...,m, is similar. A more detailed design procedure will be given in the design example in the sequel. Therefore, in the gene circuit design procedure, we engineer some feasible gene circuits kk,ijxj(t - τk) for a gene network with adequate kk,ij to achieve robust stabilization and filtering design. In the situation, the specification of kk,ij for the embedded gene circuits is different from the convenient control or filter design method, which is used to calculate control input or state estimation from state variables or output signals. However, the closed-form solutions for Kk, k = 0, 1,...,m, are diffcult to be obtained. Therefore, some searching algorithms for components of Kk, k = 0, 1,...,m, are needed.
Following the robust stability inequality in (10), we could get the following result.
Proposition 4 The designed gene network in (20) is robustly stable if we could specify Kk, k = 0, 1,...,m, for the designed gene circuits such that there exist the positive definite symmetric matrices P > 0 and Qk > 0, k = 1,...,m, solving the following inequality
[
Ξ
P
A
¯
1
⋯
P
A
¯
m
A
¯
1
T
P
B
1
T
P
B
1
−
Q
1
0
0
⋮
0
⋱
0
A
¯
m
T
P
0
0
B
m
T
P
B
m
−
Q
m
]
<
0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6377@
where Ξ=A¯0TP+PA¯0+B0TPB0+∑k=1mQk
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuONdGLaeyypa0JafmyqaeKbaebadaqhaaWcbaGaeGimaadabaGaemivaqfaaOGaemiuaaLaey4kaSIaemiuaaLafmyqaeKbaebadaWgaaWcbaGaeGimaadabeaakiabgUcaRiabdkeacnaaDaaaleaacqaIWaamaeaacqWGubavaaGccqWGqbaucqWGcbGqdaWgaaWcbaGaeGimaadabeaakiabgUcaRmaaqadabaGaemyuae1aaSbaaSqaaiabdUgaRbqabaaabaGaem4AaSMaeyypa0JaeGymaedabaGaemyBa0ganiabggHiLdaaaa@4939@and A¯k
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyqaeKbaebadaWgaaWcbaGaem4AaSgabeaaaaa@2E83@ = Ak + Kk, i.e., the robust gene circuit design becomes how to specify Kk, k = 0, 1,...,m, such that there exist the positive symmetric matrices P and Qk, k = 1,...,m, in LMI (21).
Proof. The proof is trivial. ■
Remark 5 In order to make sure the positive concentrations of gene regulation network in (19) or (20), the choice of designed kinetic parameter Kk to solve LMI in (21) should guarantee the positive concentrations of gene networks. In general, according to the positive orthant stabilization theory 35, the guaranty for the positive concentration of x(t) is that the off-diagonal entries of A¯k
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyqaeKbaebadaWgaaWcbaGaem4AaSgabeaaaaa@2E83@ = Ak + Kk are all nonnegative. Therefore, the specification of Kk to solve LMI should be subjected to this constraint. If these concentration constraint can not be achieved, changes of gene circuits through other loop should be tried.
If the nonlinear stochastic time-delayed gene network in equation (7) cannot tolerate molecular fluctuations ∑k=0mhk(x(t−τk))dWk(t)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWGObaAdaWgaaWcbaGaem4AaSgabeaakiabcIcaOiabdIha4jabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKIaeiykaKIaemizaqMaem4vaC1aaSbaaSqaaiabdUgaRbqabaGccqGGOaakcqWG0baDcqGGPaqkaSqaaiabdUgaRjabg2da9iabicdaWaqaaiabd2gaTbqdcqGHris5aaaa@4750@, then we want to design the gene circuits ∑k=0mKkgk(x(t−τk))
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWGlbWsdaWgaaWcbaGaem4AaSgabeaakiabdEgaNnaaBaaaleaacqWGRbWAaeqaaOGaeiikaGIaemiEaGNaeiikaGIaemiDaqNaeyOeI0IaeqiXdq3aaSbaaSqaaiabdUgaRbqabaGccqGGPaqkcqGGPaqkaSqaaiabdUgaRjabg2da9iabicdaWaqaaiabd2gaTbqdcqGHris5aaaa@42C2@ in the following to robustly stabilize the perturbative gene network
d
x
(
t
)
=
∑
k
=
1
m
(
f
k
(
x
(
t
−
τ
k
)
)
+
K
k
g
k
(
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d
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+
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h
k
(
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(
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−
τ
k
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d
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@78E9@
where gk(x(t - τk)), k = 0, 1,...,m, denote the nonlinear feedback circuits and the matrices Kk, k = 0, 1,...,m, denote their corresponding circuit kinetic parameters to be specified to robustly stabilize the perturbative gene network. In general, for the nonlinear stochastic gene network with time delays in (22), it is not easy to obtain a systematic design method for Kk, k = 0,1,...,m, to achieve the robust gene circuit design.
The fuzzy dynamic model with time delay has been widely employed to interpolate several local linear time-delayed dynamic models to efficiently approximate a nonlinear time-delayed dynamic system. The T-S fuzzy dynamic model is described by fuzzy If-Then rules and employed here to deal efficiently with the robust H∞ filtering problem for gene networks under process delay, intrinsic molecular fluctuations and extrinsic molecular noises in (22). The i-th rule of the fuzzy model for nonlinear stochastic time-delayed gene network in (22) is proposed as the following form 3334:
Rule
i
:
If
z
1
(
t
)
is
F
i
1
and
z
2
(
t
)
is
F
i
2
…
and
z
g
(
t
)
is
F
i
g
,
then
d
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=
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d
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@C350@
for i = 1, 2,...,L. Fij is the fuzzy set; Ak,i, Bk,i, and Gk,i (k = 0, 1,...,m) are known constant matrices; L is the number of If-Then rules; z(t) = [z1(t),...,zg(t)]T are the premise variables; g is the number of premise variables. If all state variables x(t) are used as premise variables, then z(t) = x(t) and g = N. The physical meaning of the fuzzy rule i is that if premise variables z1(t), z2(t),...,zg(t) are with the fuzzy sets Fi1, Fi2,...,Fig, then the nonlinear stochastic gene network in (22) can be represented by interpolating the linearized systems in (23). The fuzzy stochastic system in (23) is inferred as follows 3334
d
x
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t
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]
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@89B4@
where
μ
i
(
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)
≜
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=
1
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j
(
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j
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)
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1
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j
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5F70@
Fij(zj(t)) is the grade of membership of zj(t) in Fij or the possibility function of zj(t) in Fij and μi(z), for i = 1, 2,...,L, are called the fuzzy bases. The denominator ∑l=1L∏j=1gFlj(zj(t))
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaadaqeWaqaaiabdAeagnaaBaaaleaacqWGSbaBcqWGQbGAaeqaaOGaeiikaGIaemOEaO3aaSbaaSqaaiabdQgaQbqabaGccqGGOaakcqWG0baDcqGGPaqkcqGGPaqkaSqaaiabdQgaQjabg2da9iabigdaXaqaaiabdEgaNbqdcqGHpis1aaWcbaGaemiBaWMaeyypa0JaeGymaedabaGaemitaWeaniabggHiLdaaaa@44D8@ is only for normalization so that the total sum of fuzzy bases ∑i=1Lμi(z)=1
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqaH8oqBdaWgaaWcbaGaemyAaKgabeaakiabcIcaOiabdQha6jabcMcaPiabg2da9iabigdaXaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaemitaWeaniabggHiLdaaaa@3AB5@. The physical meaning of (24) is that the fuzzy stochastic system interpolates L local linear stochastic systems through nonlinear bases μi(z) to approximate the nonlinear stochastic gene network in (22). In this situation, the Langevin stochastic equation in (22) can be represented by the fuzzy interpolatory gene network as follows
d
x
(
t
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=
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k
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeWabuqaaaaabaGaemizaqMaemiEaGNaeiikaGIaemiDaqNaeiykaKIaeyypa0ZaaabCaeaacqGGOaakcqWGMbGzdaWgaaWcbaGaem4AaSgabeaakiabcIcaOiabdIha4jabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKIaeiykaKIaey4kaSIaem4saS0aaSbaaSqaaiabdUgaRbqabaGccqWGNbWzdaWgaaWcbaGaem4AaSgabeaakiabcIcaOiabdIha4jabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKIaeiykaKIaeiykaKIaemizaqMaemiDaqhaleaacqWGRbWAcqGH9aqpcqaIWaamaeaacqWGTbqBa0GaeyyeIuoakiabgUcaRmaaqahabaGaemiAaG2aaSbaaSqaaiabdUgaRbqabaGccqGGOaakcqWG4baEcqGGOaakcqWG0baDcqGHsislcqaHepaDdaWgaaWcbaGaem4AaSgabeaakiabcMcaPiabcMcaPiabdsgaKjabdEfaxnaaBaaaleaacqWGRbWAaeqaaOGaeiikaGIaemiDaqNaeiykaKcaleaacqWGRbWAcqGH9aqpcqaIWaamaeaacqWGTbqBa0GaeyyeIuoaaOqaaiabg2da9maaqahabaGaeqiVd02aaSbaaSqaaiabdMgaPbqabaGccqGGOaakcqWG6bGEcqGGPaqkaSqaaiabdMgaPjabg2da9iabigdaXaqaaiabdYeambqdcqGHris5aOWaamWaaeaadaaeWbqaaiabcIcaOiabdgeabnaaBaaaleaacqWGRbWAcqGGSaalcqWGPbqAaeqaaOGaemiEaGNaeiikaGIaemiDaqNaeyOeI0IaeqiXdq3aaSbaaSqaaiabdUgaRbqabaGccqGGPaqkcqGHRaWkcqWGlbWsdaWgaaWcbaGaem4AaSgabeaakiabdEeahnaaBaaaleaacqWGRbWAcqGGSaalcqWGPbqAaeqaaOGaemiEaGNaeiikaGIaemiDaqNaeyOeI0IaeqiXdq3aaSbaaSqaaiabdUgaRbqabaGccqGGPaqkcqGGPaqkcqWGKbazcqWG0baDaSqaaiabdUgaRjabg2da9iabicdaWaqaaiabd2gaTbqdcqGHris5aOGaey4kaSYaaabCaeaacqWGcbGqdaWgaaWcbaGaem4AaSMaeiilaWIaemyAaKgabeaakiabdIha4jabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKIaemizaqMaem4vaC1aaSbaaSqaaiabdUgaRbqabaGccqGGOaakcqWG0baDcqGGPaqkaSqaaiabdUgaRjabg2da9iabicdaWaqaaiabd2gaTbqdcqGHris5aaGccaGLBbGaayzxaaaabaGaey4kaSYaaabCaeaacqqHuoarcqWGMbGzdaWgaaWcbaGaem4AaSgabeaakiabcIcaOiabdIha4jabcMcaPiabdsgaKjabdsha0bWcbaGaem4AaSMaeyypa0JaeGimaadabaGaemyBa0ganiabggHiLdGccqGHRaWkdaaeWbqaaiabdUealnaaBaaaleaacqWGRbWAaeqaaOGaeuiLdqKaem4zaC2aaSbaaSqaaiabdUgaRbqabaGccqGGOaakcqWG4baEcqGGPaqkcqWGKbazcqWG0baDaSqaaiabdUgaRjabg2da9iabicdaWaqaaiabd2gaTbqdcqGHris5aOGaey4kaSYaaabCaeaacqqHuoarcqWGObaAdaWgaaWcbaGaem4AaSgabeaakiabcIcaOiabdIha4jabcMcaPiabdsgaKjabdEfaxnaaBaaaleaacqWGRbWAaeqaaOGaeiikaGIaemiDaqNaeiykaKcaleaacqWGRbWAcqGH9aqpcqaIWaamaeaacqWGTbqBa0GaeyyeIuoaaOqaaiabg2da9maaqahabaGaeqiVd02aaSbaaSqaaiabdMgaPbqabaGccqGGOaakcqWG6bGEcqGGPaqkdaWadaqaamaaqahabaGafmyqaeKbaebadaWgaaWcbaGaem4AaSMaeiilaWIaemyAaKgabeaakiabdIha4jabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKIaemizaqMaemiDaqhaleaacqWGRbWAcqGH9aqpcqaIWaamaeaacqWGTbqBa0GaeyyeIuoakiabgUcaRmaaqahabaGaemOqai0aaSbaaSqaaiabdUgaRjabcYcaSiabdMgaPbqabaGccqWG4baEcqGGOaakcqWG0baDcqGHsislcqaHepaDdaWgaaWcbaGaem4AaSgabeaakiabcMcaPiabdsgaKjabdEfaxnaaBaaaleaacqWGRbWAaeqaaOGaeiikaGIaemiDaqNaeiykaKcaleaacqWGRbWAcqGH9aqpcqaIWaamaeaacqWGTbqBa0GaeyyeIuoaaOGaay5waiaaw2faaaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaemitaWeaniabggHiLdaakeaacqGHRaWkdaaeWbqaaiabfs5aejabdAgaMnaaBaaaleaacqWGRbWAaeqaaOGaeiikaGIaemiEaGNaeiykaKIaemizaqMaemiDaqhaleaacqWGRbWAcqGH9aqpcqaIWaamaeaacqWGTbqBa0GaeyyeIuoakiabgUcaRmaaqahabaGaem4saS0aaSbaaSqaaiabdUgaRbqabaGccqqHuoarcqWGNbWzdaWgaaWcbaGaem4AaSgabeaakiabcIcaOiabdIha4jabcMcaPiabdsgaKjabdsha0bWcbaGaem4AaSMaeyypa0JaeGimaadabaGaemyBa0ganiabggHiLdGccqGHRaWkdaaeWbqaaiabfs5aejabdIgaOnaaBaaaleaacqWGRbWAaeqaaOGaeiikaGIaemiEaGNaeiykaKIaemizaqMaem4vaC1aaSbaaSqaaiabdUgaRbqabaGccqGGOaakcqWG0baDcqGGPaqkaSqaaiabdUgaRjabg2da9iabicdaWaqaaiabd2gaTbqdcqGHris5aaaaaaa@9158@
where A¯k,i≜Ak,i+KkGk,i
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyqaeKbaebadaWgaaWcbaGaem4AaSMaeiilaWIaemyAaKgabeaakiablYLiajabdgeabnaaBaaaleaacqWGRbWAcqGGSaalcqWGPbqAaeqaaOGaey4kaSIaem4saS0aaSbaaSqaaiabdUgaRbqabaGccqWGhbWrdaWgaaWcbaGaem4AaSMaeiilaWIaemyAaKgabeaaaaa@3F65@; Δfk(x), Δgk(x), and Δhk(x) denote fuzzy approximation errors as follows
Δ
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@F1C4@
There are many methods for finding Ak,i and Bk,i, k = 0, 1,...,m, i = 1, 2,...,L, for fuzzy model identification 33. We could use fuzzy toolbox in Matlab to find Ak,i, Bk,i, and Gk,i easily. After finding Ak,i, Bk,i, and Gk,i, k = 0, 1,...,m, i = 1, 2,...,L, we could easily find the bounds of fuzzy approximation errors Δfk(x), Δhk(x) and Δgk(x) as follows
|
|
Δ
f
k
(
x
)
|
|
2
≤
a
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|
|
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|
|
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|
|
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h
k
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)
|
|
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|
|
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|
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|
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|
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|
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@95FD@
for some positive constants ak, bk, ck, k = 0, 1,...,m, where ||x(t−τk)||2≜(x12(t−τk)+x22(t−τk)+⋯+xN2(t−tk))1/2
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6741@, e.g. if k = 0 then ||x(t−τ0)||2=||x(t)||2=(x12(t)+x22(t)+⋯+xN2(t))1/2
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6741@ with τ0 = 0 in (7).
According to the above fuzzy approximation method, we get the following result.
Proposition 5 The nonlinear stochastic gene network with time delays in (22) is robustly stabilizable by gene circuits Kkgk(x(t - τk)), k = 0, 1,...,m, if there exist the positive definite symmetric matrices P > 0 and Qk > 0, k = 1,...,m, solving the following LMIs
[
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7304@
for all i = 1,⋯,L, and P <βI, where
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@BF31@
Therefore, the robust stabilization problem for the nonlinear stochastic time-delayed gene network in (22) becomes how to specify Kk, k = 0, 1,...,m, in LMIs (27) such that there exist the positive definite symmetric common matrices P > 0 and Qk > 0, k = 1,...,m, i.e., the gene circuits Kkgk(x(t - τk)), k = 0, 1,...,m, could robustly stabilize the nonlinear stochastic time-delayed gene network, and the equilibrium point x = 0 of gene network (22) is asymptotically stable in probability under process delays and intrinsic molecular fluctuations.
Proof. See Appendix D. ■
Remark 6 In the conventional fuzzy design 3334, we need to design a complicated fuzzy controller or fuzzy filter for engineering systems. In this paper, the fuzzy approximation in (25) is only to simplify the gene circuit design procedure by solving Kk, k = 0, 1,...,m, via LMIs in (27) instead of solving Kk, k = 0, 1,...,m, via HJI directly. So there are some differences between the proposed fuzzy gene circuit design method and the conventional fuzzy control and filter design methods.
Remark 7 For the nonlinear stochastic gene network with time delays, it is diffcult to construct a Lyapunov function V (x(t)) to satisfy the HJI in (8) or (16). However, based on fuzzy approximation method, the nonlinear stochastic gene network with time delays is approximated by interpolating several local linear systems with time delays at different operation points. The advantage of the proposed fuzzy approach is that the circuit design procedure could be simplified from the linear time-delayed system design point of view. Thus the following standard Lyapunov function for linear system with multiple time-delays can be employed for the robust stabilization of nonlinear stochastic gene network 3235
V
(
x
(
t
)
)
=
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(
t
)
P
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@61C1@
Therefore, we can avoid the diffculty of constructing a complex Lyapunov function for (16) and can employ the above Lyapunov function to obtain a systematic design procedure in the molecular circuit design.
Remark 8 If the gene circuit terms cannot be separated as (22) in gene circuit design and are merged into the terms fk(x(t - τk)), k = 0, 1,...,m, as follows
d
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(
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6A8A@
then (27) should be changed as follows
[
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5CBF@
for all i = 1,⋯,L, and P <βI, where
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@E40B@
and other matrices are same as in (27). Ak,i(Kk), k = 0, 1,...,m, denote the system matrices Ak,i, k = 0, 1,...,m, containing the designed circuit parameters Kk, k = 0, 1,...,m, as elements, respectively.
After investigating the robust stabilization design of nonlinear stochastic gene network under process delays and intrinsic molecular fluctuations by the fuzzy approximation method, in order to avoid solving the nonlinear constrained optimization for molecular noise filtering problem in (17), the extrinsic molecular noise filtering ability improvement problem of gene network could be also treated by the fuzzy interpolation method. Therefore, the robust filtering circuit design problem becomes how to engineer the gene circuits Kkgk(x(t - τk)), k = 0, 1,..., m, not only to tolerate intrinsic molecular fluctuations ∑k=0mhk(x(t−τk))dWk(t)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWGObaAdaWgaaWcbaGaem4AaSgabeaakiabcIcaOiabdIha4jabcIcaOiabdsha0jabgkHiTiabes8a0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKIaemizaqMaem4vaC1aaSbaaSqaaiabdUgaRbqabaGccqGGOaakcqWG0baDcqGGPaqkaSqaaiabdUgaRjabg2da9iabicdaWaqaaiabd2gaTbqdcqGHris5aaaa@4676@ but also to attenuate the effect of extrinsic molecular noises v(t) on zo(t) to a prescribed noise filtering level γ as (14) or (15). Especially, when the prescribed noise filtering level γ is below the optimal filtering ability γ0 in (17) of the gene network, the robust filtering circuit design is needed to improve the molecular noise filtering ability. Since it is not easy to specify Kk, k = 0, 1,...,m, to achieve robust filtering design directly for the nonlinear stochastic gene network in (22), the T-S fuzzy approximation in the following equation is employed to treat the robust filtering design problem as follows
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)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@BDA7@
zo(t) = Cx(t)
Then we get the following result.
Proposition 6 For the stochastic gene network with time delay in (29), if we could specify Kk, i = 1,...,m for the designed gene circuits such that there exist the positive definite symmetric matrices P and Qk, k = 1,...,m, solving the following LMIs
[
Ξ
11
,
i
+
Φ
Ξ
12
,
i
Ξ
13
,
i
Ξ
14
,
i
Ξ
15
,
i
Ξ
21
,
i
−
Ξ
22
,
i
0
0
0
Ξ
31
,
i
0
−
Ξ
33
,
i
0
0
Ξ
41
,
i
0
0
−
Ξ
44
,
i
0
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51
,
i
0
0
0
−
γ
I
]
<
0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8A68@
for i = 1, 2,⋯,L, and P <βI where
Φ
=
[
C
T
C
0
⋯
0
0
⋯
⋮
⋮
⋱
]
,
Ξ
15
,
i
=
[
P
G
0
⋮
]
,
Ξ
51
,
i
=
Ξ
15
,
i
T
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabeWaaaqaaiabfA6agjabg2da9maadmaabaqbaeqabmWaaaqaaiabdoeadnaaCaaaleqabaGaemivaqfaaOGaem4qameabaGaeGimaadabaGaeS47IWeabaGaeGimaadabaGaeGimaadabaGaeS47IWeabaGaeSO7I0eabaGaeSO7I0eabaGaeSy8I8eaaaGaay5waiaaw2faaiabcYcaSaqaaiabf65aynaaBaaaleaacqaIXaqmcqaI1aqncqGGSaalcqWGPbqAaeqaaOGaeyypa0ZaamWaaeaafaqabeWabaaabaGaemiuaaLaem4raCeabaGaeGimaadabaGaeSO7I0eaaaGaay5waiaaw2faaiabcYcaSaqaaiabf65aynaaBaaaleaacqaI1aqncqaIXaqmcqGGSaalcqWGPbqAaeqaaOGaeyypa0JaeuONdG1aa0baaSqaaiabigdaXiabiwda1iabcYcaSiabdMgaPbqaaiabdsfaubaaaaaaaa@5E5E@
and the other matrices are defined in (27), then the effect of extrinsic molecular noises v(t) on the gene of interest is less than γ.
Proof. See Appendix E. ■
Therefore, according to fuzzy approximation method, the optimal H∞ noise filtering design problem for nonlinear stochastic time-delayed gene network in (29) could be solved by the following constrained optimization problem
γ
0
=
min
K
k
γ
subject to
0
<
P
<
β
I
,
Q
k
>
0
and
(
31
)
.
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeaabiqaaaqaaiabeo7aNnaaBaaaleaacqaIWaamaeqaaOGaeyypa0ZaaCbeaeaacyGGTbqBcqGGPbqAcqGGUbGBaSqaaiabdUealnaaBaaameaacqWGRbWAaeqaaaWcbeaakiabeo7aNbqaaiabbohaZjabbwha1jabbkgaIjabbQgaQjabbwgaLjabbogaJjabbsha0jabbccaGiabbsha0jabb+gaVjabbccaGiabicdaWiabgYda8iabdcfaqjabgYda8iabek7aIjabdMeajjabcYcaSiabbccaGiabdgfarnaaBaaaleaacqWGRbWAaeqaaOGaeyOpa4JaeGimaaJaeeiiaaIaeeyyaeMaeeOBa4MaeeizaqMaeeiiaaIaeiikaGIaeG4mamJaeGymaeJaeiykaKIaeiOla4caaaaa@5DF0@
After solving Kk, k = 0, 1,...,m, from LMIs in (31) for a prescribed noise filtering level γ or solving Kk, k = 0, 1,...,m, from (32) for the optimal H∞ noise filtering ability γ0, we can design gene circuits Kkgk(x(t - τk)), k = 0, 1,...,m, in the nonlinear stochastic time-delayed gene network (22) to achieve a desired molecular noise filtering ability or an optimal H∞ noise filtering ability of gene network, respectively. Since the gene circuit design could improve the noise filtering ability, the γ0 in (32) should be less than the γO in (17) without gene circuit design.
Remark 9 In general, the optimization problem in (32) is called the eigenvalues problem, which can be efficiently solved by the Matlab LMI toolbox 35.
Remark 10 If the gene control circuit terms cannot be separated as (22) and are merged into fk(x(t - τk)), k = 0, 1,...,m, as follows
d
x
(
t
)
=
∑
k
=
0
m
f
k
(
x
(
t
−
τ
)
,
K
k
)
d
t
+
G
v
(
t
)
d
t
+
∑
k
=
0
m
h
k
(
x
(
t
−
τ
k
)
)
d
W
k
(
t
)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7255@
zo(t) = Cx(t)
then LMIs in (31) should be modified as follows
[
Ξ
11
,
i
+
Φ
Ξ
12
,
i
Ξ
13
,
i
Ξ
15
,
i
Ξ
21
,
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−
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22
,
i
0
0
Ξ
31
,
i
0
−
Ξ
33
,
i
0
Ξ
51
,
i
0
0
−
γ
I
]
<
0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaamWaaeaafaqabeabeaaaaaqaaiabf65aynaaBaaaleaacqaIXaqmcqaIXaqmcqGGSaalcqWGPbqAaeqaaOGaey4kaSIaeuOPdyeabaGaeuONdG1aaSbaaSqaaiabigdaXiabikdaYiabcYcaSiabdMgaPbqabaaakeaacqqHEoawdaWgaaWcbaGaeGymaeJaeG4mamJaeiilaWIaemyAaKgabeaaaOqaaiabf65aynaaBaaaleaacqaIXaqmcqaI1aqncqGGSaalcqWGPbqAaeqaaaGcbaGaeuONdG1aaSbaaSqaaiabikdaYiabigdaXiabcYcaSiabdMgaPbqabaaakeaacqGHsislcqqHEoawdaWgaaWcbaGaeGOmaiJaeGOmaiJaeiilaWIaemyAaKgabeaaaOqaaiabicdaWaqaaiabicdaWaqaaiabf65aynaaBaaaleaacqaIZaWmcqaIXaqmcqGGSaalcqWGPbqAaeqaaaGcbaGaeGimaadabaGaeyOeI0IaeuONdG1aaSbaaSqaaiabiodaZiabiodaZiabcYcaSiabdMgaPbqabaaakeaacqaIWaamaeaacqqHEoawdaWgaaWcbaGaeGynauJaeGymaeJaeiilaWIaemyAaKgabeaaaOqaaiabicdaWaqaaiabicdaWaqaaiabgkHiTiabeo7aNjabdMeajbaaaiaawUfacaGLDbaacqGH8aapcqaIWaamaaa@7245@
for i = 1, 2,⋯,L, and P <βI, where
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11
,
i
=
P
A
0
,
i
(
K
0
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+
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0
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i
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22
,
i
=
[
B
1
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+
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]