Lab of Control and Systems Biology, Department of Electrical Engineering, National Tsing Hua University, Hsinchu, 30013, Taiwan

Abstract

Background

Collective rhythms of gene regulatory networks have been a subject of considerable interest for biologists and theoreticians, in particular the synchronization of dynamic cells mediated by intercellular communication. Synchronization of a population of synthetic genetic oscillators is an important design in practical applications, because such a population distributed over different host cells needs to exploit molecular phenomena simultaneously in order to emerge a biological phenomenon. However, this synchronization may be corrupted by intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise. Therefore, robust synchronization is an important design topic in nonlinear stochastic coupled synthetic genetic oscillators with intrinsic kinetic parameter fluctuations and extrinsic molecular noise.

Results

Initially, the condition for robust synchronization of synthetic genetic oscillators was derived based on Hamilton Jacobi inequality (HJI). We found that if the synchronization robustness can confer enough intrinsic robustness to tolerate intrinsic parameter fluctuation and extrinsic robustness to filter the environmental noise, then robust synchronization of coupled synthetic genetic oscillators is guaranteed. If the synchronization robustness of a population of nonlinear stochastic coupled synthetic genetic oscillators distributed over different host cells could not be maintained, then robust synchronization could be enhanced by external control input through quorum sensing molecules. In order to simplify the analysis and design of robust synchronization of nonlinear stochastic synthetic genetic oscillators, the fuzzy interpolation method was employed to interpolate several local linear stochastic coupled systems to approximate the nonlinear stochastic coupled system so that the HJI-based synchronization design problem could be replaced by a simple linear matrix inequality (LMI)-based design problem, which could be solved with the help of LMI toolbox in MATLAB easily.

Conclusion

If the synchronization robustness criterion, i.e. the synchronization robustness ≥ intrinsic robustness + extrinsic robustness, then the stochastic coupled synthetic oscillators can be robustly synchronized in spite of intrinsic parameter fluctuation and extrinsic noise. If the synchronization robustness criterion is violated, external control scheme by adding inducer can be designed to improve synchronization robustness of coupled synthetic genetic oscillators. The investigated robust synchronization criteria and proposed external control method are useful for a population of coupled synthetic networks with emergent synchronization behavior, especially for multi-cellular, engineered networks.

Background

Many biochemically dynamical systems are controlled by intrinsic rhythms generated by specialized cellular clocks within the organism itself. These rhythm generators are composed of thousands of clock cells that are intrinsically diverse, but nevertheless manage to function in a coherent oscillatory state

Recently designed synthetic genetic oscillators can offer an alternative approach, and provide a relatively well-controlled test bed in which the function and behavior of natural genetic oscillators can be isolated and characterized in detail

In previous studies

Generally, biological systems or organisms are subject to time-varying uncertainties, assumed to be in the form of both internal noise resulting from the birth and death of biochemical molecules, and external noise resulting from environmental perturbations

Based on nonlinear stochastic equations of coupled synthetic genetic oscillators distributed in different host cells, the robust synchronization mechanism is discussed from the _{
∞
} noise-filtering perspective. The robust ability to tolerate stochastic kinetic parameter fluctuations and the filtering ability to attenuate extrinsic environmental molecular noise to maintain synchronization of nonlinear stochastic coupled synthetic genetic oscillations was measured from the nonlinear stochastic system theory point of view. In the case where robust synchronization cannot be maintained or is corrupted by intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise, some robust synchronization design methods are discussed to enhance synchronization. Both the physical insight into the robust synchronization mechanisms and the designs to improve these mechanisms require solving nonlinear HJIs, which cannot be easily achieved by any analytical or numerical method at present. In order to simplify this analysis and design, the Takagi-Sugeno (T-S) fuzzy model

If the robust synchronization of nonlinear stochastic coupled synthetic genetic oscillators could not be achieved spontaneously, an external control input was developed to synchronize the coupled synthetic genetic oscillators. External stimulation inputs are known to play an important role in the synchronization of biological rhythms. For instance, many organisms display a circadian rhythm of 24-hours periodicity entrained to the light–dark cycle

The contributions of this paper are fourfold: (1) A nonlinear stochastic system is introduced to model a population of nonlinear stochastic coupled synthetic genetic oscillators under random intrinsic kinetic parameter fluctuations and extrinsic molecule noise

Methods

Stochastic models of nonlinear stochastic synthetic genetic oscillators under intrinsic kinetic parameter fluctuations and external molecular noise

Model description

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

Synchronization scheme of

**Synchronization scheme of **
**
N
**

Based on the synchronization scheme of coupled repressilators via quorum sensing mechanism (Figure

where _{
ai
} , _{
bi
} and _{
ci
} are the concentration of mRNA transcribed from _{
Ai
} , _{
Bi
} and _{
Ci
} , respectively. The concentration of AI inside each cell is denoted by _{
Si
}. _{
a
}, _{
b
}, and _{
c
} are the dimensionless transcription rate in the absence of the repressor, and _{
S
} is the maximal contribution to _{
S
} is the activation coefficient. _{
m
} is the respective dimensionless degradation rate of mRNA for

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

where parameters _{
A
}
_{
B
} and _{
C
} are the translation rates of the proteins from their mRNA, and _{
P
} represents the dimensionless degradation rate of proteins TetR, CI, and LacI in the cell. The intercellular protein AI in cell

where _{s} measures the diffusion rate of AI across the cell membrane, _{
s
} is the synthesis rate of AI, and _{
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

where _{
i
}(_{
ai
}(_{
bi
}(_{
ci
}(_{
Ai
}(_{
Bi
}(_{
Ci
}(_{
Si
}(^{
T
} ∈ ^{
m
} is the state vector of the ^{
m
} → ^{
m
} is a smooth nonlinear function that characterizes the behavior of the synthetic oscillator; ^{
m
} → ^{
m
} is a smooth nonlinear inner-coupling function; and _{
ij
})_{
N × N
} ∈ ^{
N × N
} is the coupling configuration matrix, where _{
ij
}
_{
ij
}=0. Assume that the diagonal elements of

Synthetic genetic oscillators under intrinsic parameter fluctuations

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

where _{
j
}, _{
j
}, _{
j
}, _{
s
} denote the amplitudes of the kinetic parameter fluctuations and _{
i
}(_{
j
}, Δ_{
j
}, Δ_{
j
} and Δ_{
s
} denote the deterministic part of the stochastic kinetic parameter fluctuations Δ_{
j
}
_{
i
}(_{
j
}
_{
i
}(_{
j
}
_{
i
}(_{
s
}
_{
i
}(_{
i
}(_{
j
}
_{
i
}(_{
j
}
_{
i
}(_{
j
}
_{
i
}(_{
s
}
_{
i
}(_{
i
}(_{
i
}(^{2}
_{
t,τ
}, with _{
j
}, _{
j
}, _{
j
}, _{
s
}}, respectively, where _{
t,τ
} denotes the delta function, that is, _{
t,τ
} = 1 if _{
t,τ
} = 0 if _{
j
}, _{
j
}, _{
j
} and _{
s
} denote the corresponding standard deviations of the stochastic parameter fluctuations _{
j
}
_{
i
}(_{
j
}
_{
i
}(_{
j
}
_{
i
}(_{
s
}
_{
i
}(

where

where _{
i
}(_{
i
}(_{
i
}(

Synthetic genetic oscillators under external environmental molecular noise

In general, a synthetic genetic oscillator

where the signal vector _{
i
}(_{
i
} denotes the noise-coupling matrix in cell

Robust synchronization control design for nonlinear synthetic genetic oscillators under intrinsic kinetic parameter fluctuations and external environmental molecular noise

The nonlinear stochastic coupled synthetic genetic oscillators in (8) are said to reach synchronization asymptotically if _{1}(_{2}(_{
N
}(_{
f
} or

in which ^{
m
} is the synchronization solution satisfying

Let us denote the synchronization error signal for synthetic genetic oscillators as

According to (8), the synchronization error dynamics for cell

for

where

and

Suppose the influence of extrinsic environmental molecular noise on the synchronization error can be bounded by the following noise-filtering level

for all possible environmental noises ^{2} denotes the upper bound of the effect of _{0} of coupled synthetic genetic oscillators, and

Remark 1

(i) The inequality in (14) means that the effect of extrinsic environmental molecular noise on the synchronization error is less than _{0} is the lower bound of _{0} for all possible extrinsic noises _{0} can be given in (14) at first. Then, we will decrease the upper bound _{0}, i.e., to get _{0} by minimizing _{0} = min _{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

For some Lyapunov function

Results

Based on the synchronization error dynamic equation in (13) and the _{∞} noise filtering performance in (14) or (15), we obtain the following robust synchronization result for nonlinear stochastic coupled synthetic genetic oscillators under intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise.

If there exists a positive function

then the stochastic intrinsic noise can be tolerated (i.e. the synchronization of coupled synthetic genetic networks cannot destroyed by intrinsic noise) and the influence of extrinsic environmental molecular noise

Proof: see Additional file

Since

subject to HJI in (16) with

i.e., the noise-filtering ability _{0} on _{0} of the synchronized synthetic oscillators in (17) can be obtained by decreasing

If the noise-filtering ability _{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 _{
a
}, _{
b
}, _{
c
}, _{
A
}, _{
B
}, _{
C
}, _{
m
}, and _{
p
} in (5) to solve the constrained optimization in (17) to enhance the noise-filtering ability, i.e.,

subject to HIJ in (16) with

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.

Remark 2

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

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 _{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 _{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

Robust synchronization design of synthetic genetic oscillators via t-s fuzzy methodology

In this study, the T-S fuzzy method is employed to simplify the analysis and design procedure for robust synchronization of nonlinear stochastic coupled synthetic oscillators under intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise. The T-S fuzzy model for the synchronization error dynamics is described by fuzzy if-then rules. The

**Rule** **
k
**

for _{
g,i
} is the element of premise variables of the _{
i
} = _{1,i
}, …, _{
g,i
}
^{
T
}; _{
kg
} is the fuzzy set; _{
k
}, _{
k
}, _{
Wk
}, and _{
Wk
} are the fuzzy system matrices; _{1,i
}(_{2,i
}(_{
g,i
}(_{
k1}, _{
k2}, ⋯, _{
kg
}, then the synchronization error dynamics in (12) can be represented by interpolating the linearized synchronization error dynamics in (20) via the fuzzy basis. The fuzzy synchronization error dynamics in (20) is referred as follows

where _{
kj
}(_{
j,i
}) is the grade of membership of _{
j,i
}(_{
kj
} or the possibility function of _{
j,i
}(_{
kj
}, and _{
k
}(_{
i
}) is called fuzzy basis function for _{
k
}(_{
i
}) to approximate the nonlinear stochastic system in (13). In this situation, the nonlinear stochastic coupled oscillation systems in (13) can be represented by the fuzzy interpolation system as follows:

where _{
k
}(_{
k,1}(_{1}), …, _{
k,N
}(_{
N
})) and _{1}, …, _{
N
}
^{
T
}.

Remark 3

In _{
k
}, _{
k
}, _{
Wk
}, and _{
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 _{∞} 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.

Proposition 2

If there exists a positive definite symmetric matrix

for

Proof: See Additional file 1: Appendix B

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

subject to

This file contains appendices A, B and C.

Click here for file

Remark 4

(i) If the prescribed noise-filtering level _{
a
}, _{
b
}, _{
c
}, _{
A
}, _{
B
}, _{
C
}, _{m}, and _{
p
} of the synthetic gene oscillators in _{
k
}, so that the LMIs in (23) have a positive solution

(ii) In this study, the fuzzy approximation method in (21) or (22) is only employed to simplify the analysis and design procedure via solving _{
W
}(_{
W
}(^{
T
}

(iii) In general, the constrained optimization problems in (24) are called eigenvalue problem

(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

for _{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 _{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 _{
N
} ⊗ _{
Wk
} + _{
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.

Robust synchronization of synthetic genetic oscillators by external control input

If robust synchronizations of coupled synthetic genetic oscillators cannot be achieved spontaneously via the parameter design in the above sections, then a control strategy is needed from external stimulation inputs to improve the robust synchronization of coupled synthetic genetic oscillators. External stimulation inputs are known to play an important role in the synchronization of biological rhythms. Recently, several methods of periodic stimulation for synchronization of nonlinear oscillators have been introduced _{∞} stochastic control theory, an input control strategy is introduced to enhance the robust synchronization. If AI is injected into a common medium to increase the average concentration of AI protein in the exteracellular environment, which in turn increases the cellular communication of coupled oscillation systems, then the dynamics of the signaling molecule AI in the cellular environment, as shown in (3), should be modified as

where _{
e
} =

For the simplicity of control design, suppose that the following control input _{
e
} =

where _{
e
}(_{
eij
}(_{
N × N
} ∈ ^{
N × N
} is the coupling configuration matrix, in which _{
eii
}(_{
s
}(1 − ^{− 1})(_{
e
} + _{
eij
}(_{
s
}
^{− 1}(_{
e
} +

Corollary 3

For the nonlinear stochastic coupled synthetic genetic oscillators with an extracellular control input _{
e
}
_{
e
}(

for a prescribed filtering level

Proof: similar to the proof of proposition 1

The inequality (28) is equivalent to synchronization robustness criterion

The physical meaning of synchronization robustness criterion in (29) is that if we can specify control parameter

subject to

In general, it is still very difficult to specify the control parameter _{
e
}

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.

Corollary 4

For stochastic synchronized oscillation systems, if there exists a symmetric solution

for

Proof: similar to the proof of proposition 2

The physical meaning of Corollary 4 is that if we can select a control parameter _{
N
} ⊗ _{
k
} + _{
e
}(_{
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:

subject to

The physical meaning of the constrained optimization in (33) is that if we can select a control parameter

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

(2) Given the prescribed disturbance attenuation level

(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

Design examples

In this section, we provide a simulated example to illustrate the design procedure of robust synchronization of the nonlinear stochastic coupled synthetic oscillation systems and to confirm the performance of the robust synchronization of proposed method against intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise.

The purpose of this example is to demonstrate the effectiveness of the theoretical synchronization result of synthetic gene oscillators in mimicking real biological oscillator systems. We consider a synthetic genetic network of _{0}
^{100}
^{
T
}(_{0}
^{100}
^{
T
}(^{2} < 0.56^{2}. It can be clearly seen that based on our proposed design method, the coupled gene network cannot only tolerate kinetic parameter variations but also attenuate the extrinsic molecular noise below a desired level to achieve a robust synchronization.

Ten coupled genetic oscillators

**Ten coupled genetic oscillators.** The parameter values in (1), (2), and (3) are set as follows: _{a} = _{b} = _{c} = 216, _{S} = 20, _{S} = 1, _{S} = 1, _{S} = 2, _{S} = 0.1, _{A} = _{B} = _{C} = 1, _{m} = 6.9315, _{p} = 1.1552 and _{e} = 0.09 _{a} = _{b} = _{c} = 2.16, _{S} = 0.2, _{A} = _{B} = _{C} = 0.01, _{S} = 0.001, _{S} = 0.02, _{m} = 0.06, _{p} = 0.01, and _{S} = 0.01. For the convenience of simulation, we assume that the extrinsic molecular noise _{1}~_{10} is independent Gaussian white noise with a mean of zero and standard deviation of 0.02. It can be seen that coupled synthetic oscillators cannot achieve synchronization under these intrinsic kinetic parameter fluctuations and extrinsic molecular noise.

The robust synchronization result of ten coupled synthetic oscillators in Figure

**The robust synchronization result of ten coupled synthetic oscillators in Figure****, by external control with ****=** **0.66****.** Based on a Monte Carlo simulation with 100 runs, the noise filtering level is given by (_{0}^{100}^{T}(_{0}^{100}^{T}(^{2} < 0.56^{2}.

Discussion

The cell is the functional unit of all living things, either unicellular or multicellular. A cell can sense many different signals from the internal or external context and can respond to the constantly changing environment via appropriate cellular processes. Also, cells can interact with each other via cell-to-cell communication and achieve specific physiological functions essential for life in a cooperative manner. However, many fundamental questions remain regarding how cellular phenomena arise from the interactions between genes and proteins, what features make the cell operate reliably in diverse conditions, and how the cell is responsible for these operations. To gain insight into these questions, one can construct the underlying mechanisms that constitute the web of interactions. This idea is useful to separate a complicated network into many simpler ones, which can work independently but also cooperate with each other. It may not only enhance our understanding of collective behavior particularly via synchronization but may also establish a foundation to design robust implementation of coupled synthetic gene networks

In this paper, we consider a nonlinear stochastic coupled network with two or more coupled synthetic oscillators. By transforming nonlinear stochastic coupled network dynamics into synchronization error dynamics, we can use Lyapunov’s direct method to infer a sufficient condition required for robustness of the nonlinear synchronized network. Assuming that each synthetic oscillator suffers from intrinsic kinetic parameter fluctuations and extrinsic molecular noise, robust synchronization performance is defined as the effect of extrinsic molecular noise upon the synchronization error. Based on this definition, robust synchronization performance of a nonlinear coupled network can be calculated by solving an associated HJI-constrained optimization problem. We also show that nonlinear coupled networks with robust synchronization performance are also

Recently a simple synthetic device was engineered in a cell, and several cells were then combined, so that their connections allowed the construction of a more complex synthetic gene circuit, i.e., so-called multi-cellular engineered networks. This approach not only uses cellular consortia as an efficient way of engineering complex gene circuits, but also demonstrates the great potential for reutilization of small parts of the gene circuit. In such situations, our proposed evaluation framework may offer a possible guideline for the design or compensation of such coupled networks with a given connected topology toward a desired collective behavior.

Conclusions

In this study, several robust synchronization criteria and designs are proposed for a population of synthetic genetic oscillators in order to exploit an emergent synchronization phenomenon by quorum sensing molecules under intrinsic kinetic parameter fluctuations and extrinsic molecular noise. When the synchronization of nonlinear stochastic coupled synthetic genetic oscillators cannot be maintained, a robust _{∞} control scheme is developed to enhance synchronization by adding external control to increase the cell-to-cell communication through quorum sensing. This study enhances our understanding in this area in the following ways: (a) nonlinear stochastic systems are employed to model the coupled synthetic genetic oscillators with intrinsic kinetic parameter fluctuations, extrinsic molecular noises on the host cells and quorum sensing molecules; (b) two robust synchronization criteria (19) and (25) of coupled synthetic oscillators are developed from the nonlinear stochastic filtering point of view, so that we can gain more insight into the robust synchronization mechanism from a systematic perspective. If these robust synchronization criterion cannot be guaranteed, robust synchronization control schemes via selecting adequate kinetic parameters and inducer concentration are also developed to improve the synchronization robustness of coupled synthetic genetic oscillators; and (c) the fuzzy approximation is employed to approximate the nonlinear stochastic synchronization error model by interpolating several linear stochastic systems, so that the powerful LMI toolbox in MATLAB can be used to significantly simplify the system analysis and design procedure for robust synchronization of coupled synthetic gene networks, which is very important for the emergent phenomenon of synthetic gene networks through quorum sensing at the molecular level. In addition, the proposed robust synchronization design and control scheme for nonlinear stochastic coupled genetic oscillators can be easily applied to the robust synchronization problem of other coupled genetic networks as a cellular consortium.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

BSC formulated the research topic and carried out some robust synchronization control designs. CYH participated in some control designs and performed simulation. Both authors read and approved the final manuscript.

Acknowledgements

The work was supported by the National Science Council of Taiwan under grant NSC 100-2745-E-007-001-ASP and NSC 101-2745-E-007-001-ASP.