Before discussion of synchronization of more general synthetic genetic oscillators, we here provide a design example of coupled repressilators to illustrate the interesting phenomenon of synchronization in coupled dynamic cells. Then model description, definition, and theoretical results of robust synchronization of more general coupled synthetic genetic networks will be introduced for further study in the sequel. The repressilator is a network of three genes, the products of which inhibit the transcription of each other in a cyclical manner 1 . The gene lacI (from E. coli.) codes for the protein LacI, which inhibits the transcription of the gene tetR. The product of the latter, TetR, inhibits the transcription of gene cI (from λ phage); the protein product CI in turn inhibits the expression of lacI, thus completing the cycle. Garcia-Ojalvo et al. proposed a modular addition to the repressilator, with the aim of coupling a population of cells containing this network 1 . The basic mechanism of communication among the cells is based on quorum sensing, which was first discovered in the bioluminescent bacteria, Vibrio fisceri. These bacteria exhibit collective behaviors, mainly using two proteins (see Figure 1). The first protein, LuxI, synthesizes a small molecule known as an auto-inducer (AI), which diffuses freely through the cell membrane. The second protein, LuxR, binds to the AI molecule to form a complex, which subsequently activates transcription of various genes, as shown in Figure 1 (a detailed description of this molecular pathway is provided by Garcia-Ojalvo et al. 1 ). Recently, several studies have indicated that quorum sensing can be engineered and used as an intercellular signaling system in E. coli 1 7 9 15 16 17 .

Based on the synchronization scheme of coupled repressilators via quorum sensing mechanism (Figure 1), the mRNA dynamics in the cell i = 1, 2, …, N are governed by repressible transcription of all three genes of the repressilator plus transcription activation of the additional copy of the lacI gene, and by mRNA degradation 7 .

d x ai t d t = − γ m x ai t + α a μ + x Ci n t d x bi t d t = − γ m x bi t + α b μ + x Ai n t , i = 1 , 2 , … , N d x ci t d t = − γ m x ci ( t ) + α c μ + x Bi n t + α S x Si t μ S + x Si t

where x _ai , x _bi and x _ci are the concentration of mRNA transcribed from tetR, cI and lacI in cell i, respectively; concentrations of the corresponding proteins are represented by x _Ai , x _Bi and x _Ci , respectively. The concentration of AI inside each cell is denoted by x _Si . α _a , α _b , and α _c are the dimensionless transcription rate in the absence of the repressor, and μ is the repression coefficient. α _S is the maximal contribution to lacI transcription in the presence of saturating amounts of AI, and μ _S is the activation coefficient. γ _m is the respective dimensionless degradation rate of mRNA for tetR, cI and lacI in the cell, and n is the Hill coefficient 7 .

The dynamics of proteins TetR, CI and LacI are given respectively as 7

d x Ai t d t = − γ p x Ai t + β A x ai t d x Bi t d t = − γ p x Bi t + β B x bi t , i = 1 , 2 , … , N d x Ci t d t = − γ p x Ci t + β C x ci t

where parameters β _A , β _B and β _C are the translation rates of the proteins from their mRNA, and γ _P represents the dimensionless degradation rate of proteins TetR, CI, and LacI in the cell. The intercellular protein AI in cell i is synthesized by the protein LuxI and diffuses through the cell wall where it undergoes degradation, leading to the following equation 7 .

d x Si t d t = − γ s x Si t + β s x Ai t − η s Q e x Si t − N − 1 ∑ j = 1 N x Sj t , 0 ≤ Q e ≤ 1

where η _s measures the diffusion rate of AI across the cell membrane, β _s is the synthesis rate of AI, and γ _s gives the rate of decay of AI. Consequently, the whole coupled synthetic system is expressed by (1)-(3), which can be represented by the following more generalized nonlinear dynamic equation

x ˙ i t = f x i t + ∑ j = 1 N c ij g x j t , i = 1 , 2 , … , N

where x _i (t) = (x _ai (t), x _bi (t), x _ci (t), x _Ai (t), x _Bi (t), x _Ci (t), x _Si (t)) ^T  ∈ R ^m is the state vector of the ith synthetic oscillator; f(·) : R ^m  → R ^m is a smooth nonlinear function that characterizes the behavior of the synthetic oscillator; g(·) : R ^m  → R ^m is a smooth nonlinear inner-coupling function; and C = (c _ij ) _N × N  ∈ R ^N × N is the coupling configuration matrix, where c _ij >0 means that the ith synthetic oscillator is coupled with the jth synthetic oscillator directly, otherwise c _ij =0. Assume that the diagonal elements of C satisfy c ii = − ∑ j = 1 , j ≠ i N c ij .

In general, the synthetic genetic oscillators suffer kinetic parameter fluctuations from transcription control, alternative splicing, translation, genetic mutation and diffusion to biological modification of transcription factors 17 19 31 34 44 . In this situation, dynamic equations of coupled synthetic genetic oscillators in (1)-(3) are modified as

d x ai t d t = − γ m + Δ γ m n i t x ai ( t ) + α a + Δ α a n i t μ + x Ci n t d x bi t d t = − γ m + Δ γ m n i t x bi ( t ) + α b + Δ α b n i t μ + x Ai n t d x ci t d t = − γ m + Δ γ m n i t x ci ( t ) + α c + Δ α c n i t μ + x Bi n t + α S + Δ α S n i t x Si t μ S + x Si t d x Ai t d t = − γ p + Δ γ p n i t x Ai ( t ) + β A + Δ β A n i t x ai ( t ) d x Bi t d t = − γ p + Δ γ p n i t x Bi ( t ) + β B + Δ β B n i t x bi ( t ) d x Ci t d t = − γ p + Δ γ p n i t x Ci ( t ) + β C + Δ β C n i t x ci ( t ) d x Si t d t = − γ s + Δ γ s n i t x Si ( t ) + β s + Δ β s n i t x Ai ( t ) − η s + Δ η s n i t Q e x Si t − N − 1 ∑ j = 1 N x Sj t

where Δα _j , j ∈ {a, b, c, S}, Δβ _j , j ∈ {A, B, C, s}, Δγ _j , j ∈ {m, p, s} and Δη _s denote the amplitudes of the kinetic parameter fluctuations and n _i (t) is random white noise with zero-mean and unit variance, i.e., Δα _j , Δβ _j , Δγ _j and Δη _s denote the deterministic part of the stochastic kinetic parameter fluctuations Δα _j n _i (t), Δβ _j n _i (t), Δγ _j n _i (t) and Δη _s n _i (t), respectively, and n _i (t) absorbs the stochastic property of random intrinsic kinetic parameter fluctuations. The covariances of the stochastic parameter fluctuations Δα _j n _i (t), Δβ _j n _i (t), Δγ _j n _i (t) and Δη _s n _i (t) are given as cov(ξn _i (t), ξn _i (t)) = ξ ² Δ _t,τ , with ξ ∈ {Δα _j , Δβ _j , Δγ _j , Δη _s }, respectively, where Δ _t,τ denotes the delta function, that is, Δ _t,τ  = 1 if t = τ and Δ _t,τ  = 0 if t ≠ τ, i.e., Δα _j , Δβ _j , Δγ _j and Δη _s denote the corresponding standard deviations of the stochastic parameter fluctuations Δα _j n _i (t), Δβ _j n _i (t), Δγ _j n _i (t) and Δη _s n _i (t), respectively. Thus, the nonlinear stochastic coupled synthetic genetic oscillators under intrinsic kinetic parameter fluctuations in the host cell i can be represented by

x ˙ i t = f x i t + ∑ j = 1 N c ij g x j t + f W x i t + ∑ j = 1 N c ij g W x j t n i t

where f W x i t + ∑ j = 1 N c ij g W x j t n i t denotes the intrinsic kinetic parameter fluctuations of synthetic genetic oscillator in the host cell i. For the convenience of analysis and control design of the synchronization in synthetic genetic oscillators inserted into different host cells, the nonlinear stochastic equation in (6) can be represented by the following Ito’s stochastic differential equation

d x i t = f x i t + ∑ j = 1 N c ij g x j t d t + f W x i t + ∑ j = 1 N c ij g W x j t d w i t

where w _i (t) is a standard Wiener process or Brownian motion with dw _i (t) = n _i (t)dt to represent the random kinetic fluctuations of synthetic gene circuits.

In general, a synthetic genetic oscillator in vivo also suffers from extrinsic environmental molecular noise, such as transmitted noise from upstream and global noise affecting all cells. Therefore, the nonlinear stochastic equation of coupled oscillators in (7) should be modified to mimic realistic dynamic behavior, as follows:

d x i t = f x i t + ∑ j = 1 N c ij g x j t + h i v i t d t + f W x i t + ∑ j = 1 N c ij g W x j t d w i t

where the signal vector v _i (t) denotes extrinsic environmental molecular noise from the environment and h _i denotes the noise-coupling matrix in cell i.

(i) The inequality in (14) means that the effect of extrinsic environmental molecular noise on the synchronization error is less than ρ from the mean energy point of view, i.e., the value of the noise-filtering ability ρ ₀ is the lower bound of ρ. Because the statistics of extrinsic molecular noise may be unavailable or uncertain, it is very difficult to obtain the noise filtering ability ρ ₀ for all possible extrinsic noises v(t) directly and only the upper bound ρ of the noise-filtering ability ρ ₀ can be given in (14) at first. Then, we will decrease the upper bound ρ to as small a value as possible to approach its lower bound for the noise-filtering ability ρ ₀, i.e., to get ρ ₀ by minimizing ρ (or ρ ₀ = min ρ) indirectly. If the noise filtering ability ρ ₀ is small, it means that the environmental molecular noise has less influence on synchronization and vice versa. It will be further discussed in the sequel.

(ii) If the extrinsic environmental molecular noise v is deterministic, then the expectation on v(t) in (14) should be neglected. If the initial condition e(0) is considered, then the noise-filtering level in (14) should be modified as follows 15 16 :

E ∫ 0 t f e t T R e t d t ≤ E V e 0 + ρ 2 E ∫ 0 t f v t T v t d t

For some Lyapunov function V(e(0)), i.e., the energy due to the initial condition e(0) should be considered in the influence of noise on synchronization 30 45 .

Since ρ denotes an upper bound of the effect of v(t) on synchronization, the real effect can be obtained by minimizing ρ to as small a value as possible. Therefore, the noise-filtering ability of synchronized oscillators on v(t) can be obtained by solving the following constrained optimization:

ρ 0 = min ρ

subject to HJI in (16) with V(e) > 0

i.e., the noise-filtering ability ρ ₀ on v(t) for the synchronized synthetic oscillators can be evaluated by solving the constrained optimization in (17). The noise-filtering ability ρ ₀ of the synchronized synthetic oscillators in (17) can be obtained by decreasing ρ until no positive solution V(e) > 0 exists for HJI in (16) again.

If the noise-filtering ability ρ ₀ cannot satisfy the designer’s specification, in order to enhance the noise filtering of extrinsic noise, we need to specify the design parameters of nonlinear stochastic coupled synthetic genetic oscillators, for example, the kinetic parameters α _a , α _b , α _c , β _A , β _B , β _C , γ _m , and γ _p in (5) to solve the constrained optimization in (17) to enhance the noise-filtering ability, i.e.,

ρ 0 = min α a , α b , α c , β A , β B , β C , γ m , γ p ρ

subject to HIJ in (16) with V(e) > 0

Before the discussion on the synchronization robustness criterion of coupled synthetic genetic oscillators, some definitions on synchronization robustness, intrinsic robustness and extrinsic robustness are given as follows:

(1) Synchronization robustness: The ability of coupled synthetic genetic oscillators to resist both intrinsic noise and extrinsic noise so that the synchronization can be maintained.

(2) Intrinsic robustness: The ability of coupled synthetic genetic oscillator to tolerate intrinsic parameter fluctuation to maintain synchronization.

(3) Extrinsic robustness: The filtering ability to attenuate the effect of environmental noise on the synchronization of coupled synthetic genetic network.

Substituting the noise-filtering ability ρ ₀ of (17) into (16) in Proposition 1, we get the following equivalent synchronization robustness criterion

1 2 F W x , s + C ⊗ I m G W x , s ∂ 2 V e ∂ e 2 F W x , s + C ⊗ I m G W x , s ︸ intrinsic robustness + e T R e + 1 4 ρ 0 2 ∂ V e ∂ e T H H T ∂ V e ∂ e ︸ extrinsic robustness ≤ − ∂ V e ∂ e T F x , s + C ⊗ I m G x , s ︸ synchronization robustness

The first term on the left hand side of (19) indicates the intrinsic robustness to tolerate the intrinsic parameter fluctuation in (13) because this term is induced by intrinsic noise (or random parameter fluctuation), the second and third term on the left hand side are due to the noise filtering in (14) and indicate the extrinsic robustness to filter the extrinsic noise with the noise filtering ability ρ ₀, and the term on the right hand side of (19) indicates the synchronization robustness of the coupled synthetic gene networks. The biological meaning of synchronization robustness criterion in (19) is that if the synchronization robustness can confer both the intrinsic robustness to tolerate intrinsic parameter fluctuation and extrinsic robustness to filter the environmental noise, then the coupled synthetic networks will synchronize with a noise filtering ability ρ ₀. If the synchronization robustness criterion in (19) is violated, then the synchronization of coupled synthetic gene networks may not be achieved due to the intrinsic parameter fluctuation and extrinsic noise.

In general, it is still very difficult to solve the second-order HJI in (16) with V(e) > 0 and V(0) = 0 to guarantee robust synchronization of nonlinear stochastic coupled synthetic genetic oscillators with a prescribed attenuation level ρ under intrinsic kinetic parameters fluctuations and extrinsic environmental molecular noise or to solve the constrained minimization in (18) for robust synchronization design to achieve the optimal molecular noise filtering of the synchronized coupled synthetic oscillators. Recently, the fuzzy dynamic model has been widely used to interpolate several local dynamic models to efficiently approximate a nonlinear dynamic system 32 38 39 . Hence, in this situation, we employ the T-S fuzzy model to interpolate several linear synthetic stochastic oscillators at different local operation points to efficiently and globally approximate the error dynamic in (13), so that the analysis and design procedure for robust synchronization of nonlinear stochastic coupled synthetic genetic oscillators can be simplified.

In 39 , Takagi and Sugeno have proposed the systematic method to build T-S fuzzy model for nonlinear function approximation by the system identification tool, i.e. the local system matrix A _k , B _k , A _Wk , and B _Wk in (21) or (22) can be identified by least square estimation method. On the other hand, many studies have proved that the T-S fuzzy model can approximate a continuous function with any degree of accuracy. Actually, there is still some fuzzy approximation error in (22). In the robust synchronization control design, for simplicity, the fuzzy approximation error can be merged into the external noise, which could be efficiently attenuated by the proposed H _∞ robust synchronization control design in the sequel.

After investigating the approximation of nonlinear stochastic coupled synthetic oscillators by the fuzzy interpolation method, in order to avoid solving the nonlinear constrained optimization problem in (18) for the robust synchronization design problem of coupled synthetic oscillators under intrinsic kinetic parameter fluctuation and extrinsic environmental molecular noise, the measurement procedure for the noise-filtering ability of synchronized synthetic genetic oscillators could also be simplified by the fuzzy approximation method. Then, we get the following result.

If there exists a positive definite symmetric matrix P>0 solving the following LMIs,

R + P ( I N ⊗ A k + C ⊗ B k ) + I N ⊗ A k + C ⊗ B k T P + I N ⊗ A Wk + C ⊗ B Wk T P ( I N ⊗ A Wk + C ⊗ B Wk ) P H H T P − ρ 2 I < 0

for k = 1, 2, …, L, then the noise-filtering level ρ in (14) holds, or the intrinsic kinetic parameter fluctuations are tolerated by the synchronized synthetic genetic oscillators, and the influence of extrinsic environmental molecular noise v(t) on the synchronized synthetic oscillation systems in (13) is less than or equal to a prescribed filtering level ρ.

Therefore, the optimal noise-filtering design of synchronized oscillation systems obtained by solving the HJI-constrained optimization problem in (18) could be replaced by solving the following constrained optimizations, respectively

ρ 0 = min α a , α b , α c , β A , β B , β C , γ m , γ p ρ

subject to P>0 and LMIs in (23)

(i)  If the prescribed noise-filtering level ρ is prescribed by a biological engineer, a robust synchronization design would involve specifying the design parameters α _a , α _b , α _c , β _A , β _B , β _C , γ _m, and γ _p of the synthetic gene oscillators in A _k , so that the LMIs in (23) have a positive solution P>0 in Proposition 2. If we want to achieve optimal filtering of extrinsic noise for the synchronized synthetic oscillators, some design parameters need to be specified for coupled synthetic oscillators to achieve the constrained optimization in (24).

(ii)  In this study, the fuzzy approximation method in (21) or (22) is only employed to simplify the analysis and design procedure via solving P>0 for LMIs in (23) instead of solving V(e) > 0 for HJI in (16) directly. Further, based on the fuzzy interpolation of local linear systems, i.e., replacing F(x, s), G(x, s), F _W (x, s) and G _W (x, s) by the fuzzy approximations in (22), in Proposition 2, V(e) = e ^T Pe is employed to solve the HJI (16) in Proposition 1. The HJI in Proposition 1 is replaced with a set of LMIs in Proposition 2 and we only need to solve P>0 for LMIs to guarantee the coupled synthetic genetic oscillators have a noise filtering level ρ.

(iii)  In general, the constrained optimization problems in (24) are called eigenvalue problem 41 , which can be efficiently solved by the MATLAB LMI toolbox.

(iv)  In addition to the robust oscillation synchronization, the proposed method can be applied to robust synchronization design of coupled synthetic gene networks with any kind of dynamic behavior.

(v)  In the fuzzy approximation case, the synchronization robustness criterion in (19) is equivalent to the following

I N ⊗ A Wk + C ⊗ B Wk T P I N ⊗ A Wk + C ⊗ B Wk ︸ local intrinsic robustness + R + 1 ρ 0 2 P H H T P ︸ local extrinsic robustness ≤ − P I N ⊗ A k + C ⊗ B k + I N ⊗ A k + C ⊗ B k T P ︸ local synchronization robustness

for k = 1, 2, …, L which is equivalent to (23) with ρ being replace by ρ ₀. The biological meaning of synchronization robustness criterion in (25) is that if the local synchronization robustness of local coupled synthetic genetic oscillators can confer local intrinsic robustness to tolerate local intrinsic parameter fluctuation and local extrinsic robustness to filter external noise, then the coupled synthetic genetic oscillators can be synchronized with a noise filtering ability ρ ₀. If the synchronization robustness criterion in (25) is violated, then the synchronization of coupled synthetic genetic oscillators may not be achieved due to intrinsic parameter fluctuation and extrinsic noise. In general, if the design parameters of coupled synthetic genetic oscillators are specified so that the eigenvalues of local coupled system matrix I _N  ⊗ A _Wk  + C ⊗ B _Wk are far in the left hand side of complex s-domain (i.e. with more negative real part), then the coupled synthetic genetic networks are more easy to synchronize in spite of intrinsic parameter fluctuation and extrinsic noise.

For the nonlinear stochastic coupled synthetic genetic oscillators with an extracellular control input u _e =Q in the terms of C _e (Q) in (27), if there exists a positive solution V(e) > 0 with V(0) = 0 to the following HJI

e T R e + ∂ V e ∂ e T F x , s + C e Q ⊗ I m G x , s + 1 4 ρ 2 ∂ V e ∂ e T H H T ∂ V e ∂ e + 1 2 F W x , s + C e Q ⊗ I m G W x , s ∂ 2 V e ∂ e 2 F W x , s + C e Q ⊗ I m G W x , s < 0

for a prescribed filtering level ρ, then the stochastic intrinsic kinetic noise can be robustly tolerated, and the influence of extrinsic environmental molecular noise v(t) on the synchronization of the nonlinear stochastic coupled synthetic oscillation systems in (27) is less than or equal to ρ, i.e., the inequality in (14) or (15) holds.

The inequality (28) is equivalent to synchronization robustness criterion

1 2 F W x , s + C e Q ⊗ I m G W x , s ∂ 2 V e ∂ e 2 F W x , s + C e Q ⊗ I m G W x , s ︸ intrinsic robustness + e T R e + 1 4 ρ 2 ∂ V e ∂ e T H H T ∂ V e ∂ e ︸ extrinsic robustness ≤ − ∂ V e ∂ e T F x , s + C e Q ⊗ I m G x , s ︸ synchronization robustness

The physical meaning of synchronization robustness criterion in (29) is that if we can specify control parameter Q to improve the synchronization robustness to provide more intrinsic robustness and more extrinsic robustness to tolerate more intrinsic parameter fluctuation and filter more extrinsic noise, then the robust synchronization of the nonlinear stochastic coupled synthetic genetic oscillators in (27) can be guaranteed. Similarly, the optimal noise-filtering design of synchronized oscillation systems by the extracellular control input in (27) can be achieved by solving the following constrained optimization problem:

ρ 0 = min Q ρ

subject to V(e) > 0 and HJI in (28)

In general, it is still very difficult to specify the control parameter u _e =Q to solve the HJI-constrained optimization in (30) for achieving the optimal noise filtering for synchronized synthetic genetic oscillators. Therefore, the fuzzy approximation method is again employed to simplify the control design procedure. Based on the fuzzy approximation method, the following fuzzy interpolation system is employed to approach the nonlinear stochastic coupled oscillation systems in (27):

d e = ∑ k = 1 L μ k z I N ⊗ A k + C e Q ⊗ B k e + H v d t + I N ⊗ A Wk + C e Q ⊗ B Wk e d w

Applying the fuzzy approximation method, the external signal control design can be obtained as described in the following corollary, for robust filtering of synchronized oscillation systems with intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise.

For stochastic synchronized oscillation systems, if there exists a symmetric solution P > 0 to the following LMIs for a prescribed noise-filtering level ρ

R + P ( I N ⊗ A k + C e ( Q ) ⊗ B k ) + I N ⊗ A k + C e Q ⊗ B k T P + I N ⊗ A Wk + C e Q ⊗ B Wk T P ( I N ⊗ A Wk + C e ( Q ) ⊗ B Wk ) P H H T P − ρ 2 I < 0

for k = 1, 2, ⋯, L, then intrinsic parametric noise can be tolerated and the effect of extrinsic molecular noise v(t) on the synchronization of nonlinear stochastic coupled oscillation systems is less than or equal to a prescribed filtering level ρ.

The physical meaning of Corollary 4 is that if we can select a control parameter Q, such that the LMIs in (32) have a positive definite solution P > 0, then the robust synchronization with a prescribed noise filtering level ρ on extrinsic environmental molecular noise is guaranteed for the nonlinear stochastic coupled synthetic genetic oscillators. If we specify control parameter Q so that the eigenvalues of local system matrix of I _N  ⊗ A _k  + C _e (Q) ⊗ B _k of coupled synthetic gene oscillators have more negative real part (i.e. in far left hand complex s-domain), the coupled synthetic gene oscillators are with more robust synchronization to tolerate more intrinsic parameter fluctuations and to filter more extrinsic noise.

Similarly, based on the fuzzy approximation method, an optimal noise-filtering design of synchronized oscillation systems by using the extracellular control input in (30) can be achieved by solving the following constrained optimization problem:

ρ 0 = min Q ρ

subject to V(e) > 0 and LMIs in (32)

The physical meaning of the constrained optimization in (33) is that if we can select a control parameter Q through the inducer concentration control method to solve the constrained optimization problem, we can achieve both robust synchronization against intrinsic kinetic parameter fluctuations and optimal filtering against external environmental molecular noise on the synchronization by using the external control signal in the coupled synthetic oscillation systems.

The design procedure of external inducer control for robust synchronization of the coupled network is summarized as follows.

(1) Consider a synthetic genetic network of N coupled oscillators with intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise.

(2) Given the prescribed disturbance attenuation level ρ.

(3) Represent the nonlinear stochastic synchronization error dynamic by the T-S fuzzy synchronization error dynamic model, using the interpolation of several local linear stochastic systems.

(4) Specify Q to solve LMI in (32) with the help of LMI toolbox in MATLAB so that N coupled synthetic genetic oscillators can be synchronized with a prescribed noise filtering level ρ.