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Analytical study of robustness of a negative feedback oscillator by multiparameter sensitivity.

Maeda K, Kurata H - BMC Syst Biol (2014)

Bottom Line: The MPS decreases with an increase in the feedback loop length (i.e., the number of molecular species constituting the feedback loop).Since a general model of negative feedback oscillators was employed, the results shown in this paper are expected to be true for many of biological oscillators.The analytical solutions give synthetic biologists some clues to design gene oscillators with robust and desired period.

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ABSTRACT

Background: One of the distinctive features of biological oscillators such as circadian clocks and cell cycles is robustness which is the ability to resume reliable operation in the face of different types of perturbations. In the previous study, we proposed multiparameter sensitivity (MPS) as an intelligible measure for robustness to fluctuations in kinetic parameters. Analytical solutions directly connect the mechanisms and kinetic parameters to dynamic properties such as period, amplitude and their associated MPSs. Although negative feedback loops are known as common structures to biological oscillators, the analytical solutions have not been presented for a general model of negative feedback oscillators.

Results: We present the analytical expressions for the period, amplitude and their associated MPSs for a general model of negative feedback oscillators. The analytical solutions are validated by comparing them with numerical solutions. The analytical solutions explicitly show how the dynamic properties depend on the kinetic parameters. The ratio of a threshold to the amplitude has a strong impact on the period MPS. As the ratio approaches to one, the MPS increases, indicating that the period becomes more sensitive to changes in kinetic parameters. We present the first mathematical proof that the distributed time-delay mechanism contributes to making the oscillation period robust to parameter fluctuations. The MPS decreases with an increase in the feedback loop length (i.e., the number of molecular species constituting the feedback loop).

Conclusions: Since a general model of negative feedback oscillators was employed, the results shown in this paper are expected to be true for many of biological oscillators. This study strongly supports that the hypothesis that phosphorylations of clock proteins contribute to the robustness of circadian rhythms. The analytical solutions give synthetic biologists some clues to design gene oscillators with robust and desired period.

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Example of dynamics of the negative feedback oscillator model. , ,  and  ().  are the intervals used to derive the analytical solutions for the period and amplitude (see Text).
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Figure 2: Example of dynamics of the negative feedback oscillator model. , , and (). are the intervals used to derive the analytical solutions for the period and amplitude (see Text).

Mentions: where and are arbitrary positive values. Introducing the step function makes dynamic models tractable [27]. This on/off behavior arises from a sigmoidal Hill function. corresponds to the concentration of a regulator molecule at which the production rate of a target molecule reaches a half maximum in Hill function. takes a positive value in the range of (0, ). The negative feedback oscillator model can produce oscillations when both and () are satisfied. It has been proved that this type of negative feedback models cannot generate a sustained oscillation when [20]. An example of the dynamics is shown in Figure 2.


Analytical study of robustness of a negative feedback oscillator by multiparameter sensitivity.

Maeda K, Kurata H - BMC Syst Biol (2014)

Example of dynamics of the negative feedback oscillator model. , ,  and  ().  are the intervals used to derive the analytical solutions for the period and amplitude (see Text).
© Copyright Policy - open-access
Related In: Results  -  Collection

License 1 - License 2
Show All Figures
getmorefigures.php?uid=PMC4305980&req=5

Figure 2: Example of dynamics of the negative feedback oscillator model. , , and (). are the intervals used to derive the analytical solutions for the period and amplitude (see Text).
Mentions: where and are arbitrary positive values. Introducing the step function makes dynamic models tractable [27]. This on/off behavior arises from a sigmoidal Hill function. corresponds to the concentration of a regulator molecule at which the production rate of a target molecule reaches a half maximum in Hill function. takes a positive value in the range of (0, ). The negative feedback oscillator model can produce oscillations when both and () are satisfied. It has been proved that this type of negative feedback models cannot generate a sustained oscillation when [20]. An example of the dynamics is shown in Figure 2.

Bottom Line: The MPS decreases with an increase in the feedback loop length (i.e., the number of molecular species constituting the feedback loop).Since a general model of negative feedback oscillators was employed, the results shown in this paper are expected to be true for many of biological oscillators.The analytical solutions give synthetic biologists some clues to design gene oscillators with robust and desired period.

View Article: PubMed Central - HTML - PubMed

ABSTRACT

Background: One of the distinctive features of biological oscillators such as circadian clocks and cell cycles is robustness which is the ability to resume reliable operation in the face of different types of perturbations. In the previous study, we proposed multiparameter sensitivity (MPS) as an intelligible measure for robustness to fluctuations in kinetic parameters. Analytical solutions directly connect the mechanisms and kinetic parameters to dynamic properties such as period, amplitude and their associated MPSs. Although negative feedback loops are known as common structures to biological oscillators, the analytical solutions have not been presented for a general model of negative feedback oscillators.

Results: We present the analytical expressions for the period, amplitude and their associated MPSs for a general model of negative feedback oscillators. The analytical solutions are validated by comparing them with numerical solutions. The analytical solutions explicitly show how the dynamic properties depend on the kinetic parameters. The ratio of a threshold to the amplitude has a strong impact on the period MPS. As the ratio approaches to one, the MPS increases, indicating that the period becomes more sensitive to changes in kinetic parameters. We present the first mathematical proof that the distributed time-delay mechanism contributes to making the oscillation period robust to parameter fluctuations. The MPS decreases with an increase in the feedback loop length (i.e., the number of molecular species constituting the feedback loop).

Conclusions: Since a general model of negative feedback oscillators was employed, the results shown in this paper are expected to be true for many of biological oscillators. This study strongly supports that the hypothesis that phosphorylations of clock proteins contribute to the robustness of circadian rhythms. The analytical solutions give synthetic biologists some clues to design gene oscillators with robust and desired period.

Show MeSH
Related in: MedlinePlus