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Stochastic synchronization of genetic oscillator networks.

Li C, Chen L, Aihara K - BMC Syst Biol (2007)

Bottom Line: Therefore, it is important to study the effects of noise perturbation on the synchronous dynamics of genetic oscillators.To demonstrate the effectiveness of our theoretical results, a population of coupled repressillators is adopted as a numerical example.Besides, the results are actually applicable to general oscillator networks.

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Affiliation: Centre for Nonlinear and Complex Systems, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054, PR China. cgli@uestc.edu.cn

ABSTRACT

Background: The study of synchronization among genetic oscillators is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. Genetic networks are intrinsically noisy due to natural random intra- and inter-cellular fluctuations. Therefore, it is important to study the effects of noise perturbation on the synchronous dynamics of genetic oscillators. From the synthetic biology viewpoint, it is also important to implement biological systems that minimizing the negative influence of the perturbations.

Results: In this paper, based on systems biology approach, we provide a general theoretical result on the synchronization of genetic oscillators with stochastic perturbations. By exploiting the specific properties of many genetic oscillator models, we provide an easy-verified sufficient condition for the stochastic synchronization of coupled genetic oscillators, based on the Lur'e system approach in control theory. A design principle for minimizing the influence of noise is also presented. To demonstrate the effectiveness of our theoretical results, a population of coupled repressillators is adopted as a numerical example.

Conclusion: In summary, we present an efficient theoretical method for analyzing the synchronization of genetic oscillator networks, which is helpful for understanding and testing the synchronization phenomena in biological organisms. Besides, the results are actually applicable to general oscillator networks.

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Simulation results of the coupled repressilators with the same initial values. (a) The evolution dynamics of the mRNA concentrations of tetR (ai) of all the genetic oscillators. (b) Zooming in the range t ∈ [600, 700] of (a). (c) The evolution of the synchronization error of ai - a1 for i = 2, ⋯ ,10.
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Figure 2: Simulation results of the coupled repressilators with the same initial values. (a) The evolution dynamics of the mRNA concentrations of tetR (ai) of all the genetic oscillators. (b) Zooming in the range t ∈ [600, 700] of (a). (c) The evolution of the synchronization error of ai - a1 for i = 2, ⋯ ,10.

Mentions: The purpose of this example is to demonstrate the effectiveness and correctness of the theoretical result, instead of mimicking the real biological clock system. We consider a small size of network with N = 10 coupled oscillators. The parameters are set as m = 4, α = 1.8, d1 = d2 = d3 = 0.4, μ = 1.3, k = 5, μs = 5, d4 = d5 = d6 = 0.5, β1 = β2 = β3 = 0.2, ks0 = 0.016, ks1 = 0.018, Q0 = 0.8 and η = 0.4. Since Proposition 1 holds no matter what vi(t) is and no matter where it is introduced, and the verification of Proposition 1 is independent of noise intensity vi, for simplicity, we set vi = 0.015 as a scalar for all i, and the noise term vini(t) is added to the first equation in (9), where ni(t) is a scalar Gaussian white noise process. According to Proposition 1 (by letting U = -G, and using MATLAB LMI Toolbox), we know that the above all-to-all coupled network can achieve stochastic synchronization with disturbance attenuation γ = 6. Although γ is a large value, it is easy to show from (2) that the time average of E(∑i∑j/xi(t) - xj(t)/2) is still rather small because is very small. We omit the computational details here. In Fig. 2(a) &2(b), when starting from the same initial values, we plot the time evolution of the mRNA concentrations of tetR (ai) of all the oscillators, which behaviors are similar to the experimental results (see, e.g. [1]). Fig. 2(c) shows the synchronization error of ai - a1 for i = 2, ⋯, 10.


Stochastic synchronization of genetic oscillator networks.

Li C, Chen L, Aihara K - BMC Syst Biol (2007)

Simulation results of the coupled repressilators with the same initial values. (a) The evolution dynamics of the mRNA concentrations of tetR (ai) of all the genetic oscillators. (b) Zooming in the range t ∈ [600, 700] of (a). (c) The evolution of the synchronization error of ai - a1 for i = 2, ⋯ ,10.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC1839896&req=5

Figure 2: Simulation results of the coupled repressilators with the same initial values. (a) The evolution dynamics of the mRNA concentrations of tetR (ai) of all the genetic oscillators. (b) Zooming in the range t ∈ [600, 700] of (a). (c) The evolution of the synchronization error of ai - a1 for i = 2, ⋯ ,10.
Mentions: The purpose of this example is to demonstrate the effectiveness and correctness of the theoretical result, instead of mimicking the real biological clock system. We consider a small size of network with N = 10 coupled oscillators. The parameters are set as m = 4, α = 1.8, d1 = d2 = d3 = 0.4, μ = 1.3, k = 5, μs = 5, d4 = d5 = d6 = 0.5, β1 = β2 = β3 = 0.2, ks0 = 0.016, ks1 = 0.018, Q0 = 0.8 and η = 0.4. Since Proposition 1 holds no matter what vi(t) is and no matter where it is introduced, and the verification of Proposition 1 is independent of noise intensity vi, for simplicity, we set vi = 0.015 as a scalar for all i, and the noise term vini(t) is added to the first equation in (9), where ni(t) is a scalar Gaussian white noise process. According to Proposition 1 (by letting U = -G, and using MATLAB LMI Toolbox), we know that the above all-to-all coupled network can achieve stochastic synchronization with disturbance attenuation γ = 6. Although γ is a large value, it is easy to show from (2) that the time average of E(∑i∑j/xi(t) - xj(t)/2) is still rather small because is very small. We omit the computational details here. In Fig. 2(a) &2(b), when starting from the same initial values, we plot the time evolution of the mRNA concentrations of tetR (ai) of all the oscillators, which behaviors are similar to the experimental results (see, e.g. [1]). Fig. 2(c) shows the synchronization error of ai - a1 for i = 2, ⋯, 10.

Bottom Line: Therefore, it is important to study the effects of noise perturbation on the synchronous dynamics of genetic oscillators.To demonstrate the effectiveness of our theoretical results, a population of coupled repressillators is adopted as a numerical example.Besides, the results are actually applicable to general oscillator networks.

View Article: PubMed Central - HTML - PubMed

Affiliation: Centre for Nonlinear and Complex Systems, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054, PR China. cgli@uestc.edu.cn

ABSTRACT

Background: The study of synchronization among genetic oscillators is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. Genetic networks are intrinsically noisy due to natural random intra- and inter-cellular fluctuations. Therefore, it is important to study the effects of noise perturbation on the synchronous dynamics of genetic oscillators. From the synthetic biology viewpoint, it is also important to implement biological systems that minimizing the negative influence of the perturbations.

Results: In this paper, based on systems biology approach, we provide a general theoretical result on the synchronization of genetic oscillators with stochastic perturbations. By exploiting the specific properties of many genetic oscillator models, we provide an easy-verified sufficient condition for the stochastic synchronization of coupled genetic oscillators, based on the Lur'e system approach in control theory. A design principle for minimizing the influence of noise is also presented. To demonstrate the effectiveness of our theoretical results, a population of coupled repressillators is adopted as a numerical example.

Conclusion: In summary, we present an efficient theoretical method for analyzing the synchronization of genetic oscillator networks, which is helpful for understanding and testing the synchronization phenomena in biological organisms. Besides, the results are actually applicable to general oscillator networks.

Show MeSH
Related in: MedlinePlus