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Theory on the dynamics of oscillatory loops in the transcription factor networks.

Murugan R - PLoS ONE (2014)

Bottom Line: Though there is a period-buffering, the amplitudes of oscillators coupled through -OR- type logic are more sensitive to perturbations in the parameters associated with the promoter state dynamics than -AND- type.Further analysis shows that the period of -AND- type coupled dual-feedback oscillators can be tuned without conceding on the amplitudes.Using these results we derive the basic design principles governing the robust and tunable synthetic gene oscillators without compromising on their amplitudes.

View Article: PubMed Central - PubMed

Affiliation: Department of Biotechnology, Bhupat and Jyoti Mehta School of Biosciences, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India.

ABSTRACT
We develop a detailed theoretical framework for various types of transcription factor gene oscillators. We further demonstrate that one can build genetic-oscillators which are tunable and robust against perturbations in the critical control parameters by coupling two or more independent Goodwin-Griffith oscillators through either -OR- or -AND- type logic. Most of the coupled oscillators constructed in the literature so far seem to be of -OR- type. When there are transient perturbations in one of the -OR- type coupled-oscillators, then the overall period of the system remains constant (period-buffering) whereas in case of -AND- type coupling the overall period of the system moves towards the perturbed oscillator. Though there is a period-buffering, the amplitudes of oscillators coupled through -OR- type logic are more sensitive to perturbations in the parameters associated with the promoter state dynamics than -AND- type. Further analysis shows that the period of -AND- type coupled dual-feedback oscillators can be tuned without conceding on the amplitudes. Using these results we derive the basic design principles governing the robust and tunable synthetic gene oscillators without compromising on their amplitudes.

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Perturbation responses of Goodwin-Griffith oscillator.A1. Effects of perturbations in  on the limit-cycle orbit of GG oscillator. The default simulation settings are  and , ,  which required a critical Hill coefficient of Cna = 6. Perturbation introduced in the interval from time 30 to 100 by abruptly raising the value to . Increase in  increases the period and reduce the amplitude of oscillations. Total simulation time is 200 (measured in terms of number of lifetimes of the protein product of TF gene A) and integration step is . A2. Effects of perturbations in . The default simulation settings are as in A1. Perturbation introduced in the interval from time 30 to 100 by abruptly raising the value to . Increase in  increases the period and reduce the amplitude. A3. Effects of perturbations in . The default simulation settings are as in A1. Perturbation introduced in the interval from time 30 to 100 by abruptly raising the value to . Increase in  increases the period and reduce the amplitude. B1. Effects of variation of  on the period and critical Hill coefficient that is required to generate oscillations. With the current default settings, there exists a range of  at which the critical Hill coefficient is minimum. B2. Effects of variation of  on the period and critical Hill coefficient. Both period of oscillations and critical Hill coefficient are linearly dependent on  and one cannot assume  as in cases of several earlier studies.
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pone-0104328-g005: Perturbation responses of Goodwin-Griffith oscillator.A1. Effects of perturbations in on the limit-cycle orbit of GG oscillator. The default simulation settings are and , , which required a critical Hill coefficient of Cna = 6. Perturbation introduced in the interval from time 30 to 100 by abruptly raising the value to . Increase in increases the period and reduce the amplitude of oscillations. Total simulation time is 200 (measured in terms of number of lifetimes of the protein product of TF gene A) and integration step is . A2. Effects of perturbations in . The default simulation settings are as in A1. Perturbation introduced in the interval from time 30 to 100 by abruptly raising the value to . Increase in increases the period and reduce the amplitude. A3. Effects of perturbations in . The default simulation settings are as in A1. Perturbation introduced in the interval from time 30 to 100 by abruptly raising the value to . Increase in increases the period and reduce the amplitude. B1. Effects of variation of on the period and critical Hill coefficient that is required to generate oscillations. With the current default settings, there exists a range of at which the critical Hill coefficient is minimum. B2. Effects of variation of on the period and critical Hill coefficient. Both period of oscillations and critical Hill coefficient are linearly dependent on and one cannot assume as in cases of several earlier studies.

Mentions: Figure 4B4 shows the strong influence of on the critical which means that the approximation () as in case of most of the earlier studies on various genetic oscillators is not a valid one. At the critical Hill coefficient, the period as well as amplitude of oscillations are strongly dependent on the Group I parameters as shown in Figures 5A1-3. These results also demonstrate how the oscillator responds to temporal perturbations is Group I parameters. As we have shown in the theory section, the period of oscillations increases with increase in the Group I parameters whereas the amplitude seems to decrease as the value of Group I parameters increase. One should note that square of period of oscillation is inversely proportional to the total energy of an oscillator whereas the total energy is directly proportional to the square of amplitude. This means that the total energy of a GG oscillator can be fine-tuned by perturbing the Group I parameters. The Goodwin-Griffith oscillator seems to abruptly enter into the modified limit-cycle orbit upon introducing the perturbation and relax back much faster upon removal of perturbation in the parameter rather than perturbations in other parameters . In the latter cases, as shown in Figures 5A2-3 the relaxation of oscillator to the original orbit upon removal of perturbation seems to be through slow asymptotic spirals. Figure 5B1 suggest that the period of oscillations increases monotonically with respect to increase in the value of Group I parameters as we have predicted in the theory section. When then there exists a range of at which the period of limit cycle oscillations and the required are almost independent of changes in . Figure 5B2 shows that when then the period of oscillations linearly increases as increases whereas it linearly decreases with increase in when .


Theory on the dynamics of oscillatory loops in the transcription factor networks.

Murugan R - PLoS ONE (2014)

Perturbation responses of Goodwin-Griffith oscillator.A1. Effects of perturbations in  on the limit-cycle orbit of GG oscillator. The default simulation settings are  and , ,  which required a critical Hill coefficient of Cna = 6. Perturbation introduced in the interval from time 30 to 100 by abruptly raising the value to . Increase in  increases the period and reduce the amplitude of oscillations. Total simulation time is 200 (measured in terms of number of lifetimes of the protein product of TF gene A) and integration step is . A2. Effects of perturbations in . The default simulation settings are as in A1. Perturbation introduced in the interval from time 30 to 100 by abruptly raising the value to . Increase in  increases the period and reduce the amplitude. A3. Effects of perturbations in . The default simulation settings are as in A1. Perturbation introduced in the interval from time 30 to 100 by abruptly raising the value to . Increase in  increases the period and reduce the amplitude. B1. Effects of variation of  on the period and critical Hill coefficient that is required to generate oscillations. With the current default settings, there exists a range of  at which the critical Hill coefficient is minimum. B2. Effects of variation of  on the period and critical Hill coefficient. Both period of oscillations and critical Hill coefficient are linearly dependent on  and one cannot assume  as in cases of several earlier studies.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4128676&req=5

pone-0104328-g005: Perturbation responses of Goodwin-Griffith oscillator.A1. Effects of perturbations in on the limit-cycle orbit of GG oscillator. The default simulation settings are and , , which required a critical Hill coefficient of Cna = 6. Perturbation introduced in the interval from time 30 to 100 by abruptly raising the value to . Increase in increases the period and reduce the amplitude of oscillations. Total simulation time is 200 (measured in terms of number of lifetimes of the protein product of TF gene A) and integration step is . A2. Effects of perturbations in . The default simulation settings are as in A1. Perturbation introduced in the interval from time 30 to 100 by abruptly raising the value to . Increase in increases the period and reduce the amplitude. A3. Effects of perturbations in . The default simulation settings are as in A1. Perturbation introduced in the interval from time 30 to 100 by abruptly raising the value to . Increase in increases the period and reduce the amplitude. B1. Effects of variation of on the period and critical Hill coefficient that is required to generate oscillations. With the current default settings, there exists a range of at which the critical Hill coefficient is minimum. B2. Effects of variation of on the period and critical Hill coefficient. Both period of oscillations and critical Hill coefficient are linearly dependent on and one cannot assume as in cases of several earlier studies.
Mentions: Figure 4B4 shows the strong influence of on the critical which means that the approximation () as in case of most of the earlier studies on various genetic oscillators is not a valid one. At the critical Hill coefficient, the period as well as amplitude of oscillations are strongly dependent on the Group I parameters as shown in Figures 5A1-3. These results also demonstrate how the oscillator responds to temporal perturbations is Group I parameters. As we have shown in the theory section, the period of oscillations increases with increase in the Group I parameters whereas the amplitude seems to decrease as the value of Group I parameters increase. One should note that square of period of oscillation is inversely proportional to the total energy of an oscillator whereas the total energy is directly proportional to the square of amplitude. This means that the total energy of a GG oscillator can be fine-tuned by perturbing the Group I parameters. The Goodwin-Griffith oscillator seems to abruptly enter into the modified limit-cycle orbit upon introducing the perturbation and relax back much faster upon removal of perturbation in the parameter rather than perturbations in other parameters . In the latter cases, as shown in Figures 5A2-3 the relaxation of oscillator to the original orbit upon removal of perturbation seems to be through slow asymptotic spirals. Figure 5B1 suggest that the period of oscillations increases monotonically with respect to increase in the value of Group I parameters as we have predicted in the theory section. When then there exists a range of at which the period of limit cycle oscillations and the required are almost independent of changes in . Figure 5B2 shows that when then the period of oscillations linearly increases as increases whereas it linearly decreases with increase in when .

Bottom Line: Though there is a period-buffering, the amplitudes of oscillators coupled through -OR- type logic are more sensitive to perturbations in the parameters associated with the promoter state dynamics than -AND- type.Further analysis shows that the period of -AND- type coupled dual-feedback oscillators can be tuned without conceding on the amplitudes.Using these results we derive the basic design principles governing the robust and tunable synthetic gene oscillators without compromising on their amplitudes.

View Article: PubMed Central - PubMed

Affiliation: Department of Biotechnology, Bhupat and Jyoti Mehta School of Biosciences, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India.

ABSTRACT
We develop a detailed theoretical framework for various types of transcription factor gene oscillators. We further demonstrate that one can build genetic-oscillators which are tunable and robust against perturbations in the critical control parameters by coupling two or more independent Goodwin-Griffith oscillators through either -OR- or -AND- type logic. Most of the coupled oscillators constructed in the literature so far seem to be of -OR- type. When there are transient perturbations in one of the -OR- type coupled-oscillators, then the overall period of the system remains constant (period-buffering) whereas in case of -AND- type coupling the overall period of the system moves towards the perturbed oscillator. Though there is a period-buffering, the amplitudes of oscillators coupled through -OR- type logic are more sensitive to perturbations in the parameters associated with the promoter state dynamics than -AND- type. Further analysis shows that the period of -AND- type coupled dual-feedback oscillators can be tuned without conceding on the amplitudes. Using these results we derive the basic design principles governing the robust and tunable synthetic gene oscillators without compromising on their amplitudes.

Show MeSH