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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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Dynamical aspects of generalized Goodwin-Griffith oscillator.A1. Phase portraits of Goodwin-Griffith oscillator as described by Eqs (4). One needs to substitute Q  =  M (scaled concentration of mRNA) for red line, Q  =  X (promoter occupancy) for blue line and Q  =  Z (end-product) for pink line. Simulation settings are , ,  and we set  which required a critical Hill coefficient of Cna = 6 to generate oscillations. Total simulation time is 100 (measured in terms of number of lifetimes of the protein product of TF gene A) and integration step is . A2. Trajectories corresponding to the settings in A1. A3. Roots of the (biquadratic) characteristic polynomial (PI) associated with the Jacobian matrix for settings in A1. B1. Variation of critical Hill coefficient with the parameter set . Minimum of this critical value seems to be achieved at , and . B2. Variation of critical Hill coefficient with the parameter set . With the optimized settings in B1, the system seems to be robust when . B3. Variation of critical Hill coefficient with the parameter set . With the optimized settings in B1, the system seems to be robust when . B4. Variation of critical Hill coefficient with the parameter set . Default values of other parameters in B1-4 are as in A1. C1. Variation of period and amplitude of the negative-feedback-only model considered in reference [25] with respect changes in the promoter affinity parameter . Simulation settings are given in Table 1. Red solid line in the period and blue solid line is amplitude of the oscillator.
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pone-0104328-g003: Dynamical aspects of generalized Goodwin-Griffith oscillator.A1. Phase portraits of Goodwin-Griffith oscillator as described by Eqs (4). One needs to substitute Q  =  M (scaled concentration of mRNA) for red line, Q  =  X (promoter occupancy) for blue line and Q  =  Z (end-product) for pink line. Simulation settings are , , and we set which required a critical Hill coefficient of Cna = 6 to generate oscillations. Total simulation time is 100 (measured in terms of number of lifetimes of the protein product of TF gene A) and integration step is . A2. Trajectories corresponding to the settings in A1. A3. Roots of the (biquadratic) characteristic polynomial (PI) associated with the Jacobian matrix for settings in A1. B1. Variation of critical Hill coefficient with the parameter set . Minimum of this critical value seems to be achieved at , and . B2. Variation of critical Hill coefficient with the parameter set . With the optimized settings in B1, the system seems to be robust when . B3. Variation of critical Hill coefficient with the parameter set . With the optimized settings in B1, the system seems to be robust when . B4. Variation of critical Hill coefficient with the parameter set . Default values of other parameters in B1-4 are as in A1. C1. Variation of period and amplitude of the negative-feedback-only model considered in reference [25] with respect changes in the promoter affinity parameter . Simulation settings are given in Table 1. Red solid line in the period and blue solid line is amplitude of the oscillator.

Mentions: Sample trajectories and phase portraits of GG oscillator for are shown in Figures 3A1-3 and 4A1-3. Irrespective of the type of initial conditions and magnitude of the control parameters, the trajectories always start with an overshoot of protein production that is followed by asymptotic spirals towards a stable limit cycle. This seems to be an inherent property of negatively self-regulated loops [9]. Figures 3B1-4 and 4B1-4 suggest that there exists an optimum range of Group I parameters and at which the critical Hill coefficient () that is required to generate self-sustained oscillations is a minimum which is in turn strongly dependent on the promoter state occupancy parameter . This optimum range is also dependent on the values of other Group II and III parameters. The optimum range of the conversion parameter seems to be . Results suggest that strong binding conditions () and () are required to minimize the value of critical Hill coefficient with respect to changes in Group I type parameters. The minimum achievable values of critical Hill coefficients seems to be () and (). When there is an additional dimerization step as described in Eqs (4–5) corresponding to the negative-feedback-only (NFO) model considered in reference [25], the minimum achievable critical Hill coefficient seems to be . One should note that in the Lac I oscillatory system the effective Hill coefficient is since four dimers of lac I end-products involved in the overall negative feedback. Numerical analysis of this NFO model system using the physiological range of parameters as given in Table 1 suggests that the period of oscillator can be well tuned by changing the promoter state affinity of the repressor without compromising the amplitude much as shown in Figure 3C1.


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

Murugan R - PLoS ONE (2014)

Dynamical aspects of generalized Goodwin-Griffith oscillator.A1. Phase portraits of Goodwin-Griffith oscillator as described by Eqs (4). One needs to substitute Q  =  M (scaled concentration of mRNA) for red line, Q  =  X (promoter occupancy) for blue line and Q  =  Z (end-product) for pink line. Simulation settings are , ,  and we set  which required a critical Hill coefficient of Cna = 6 to generate oscillations. Total simulation time is 100 (measured in terms of number of lifetimes of the protein product of TF gene A) and integration step is . A2. Trajectories corresponding to the settings in A1. A3. Roots of the (biquadratic) characteristic polynomial (PI) associated with the Jacobian matrix for settings in A1. B1. Variation of critical Hill coefficient with the parameter set . Minimum of this critical value seems to be achieved at , and . B2. Variation of critical Hill coefficient with the parameter set . With the optimized settings in B1, the system seems to be robust when . B3. Variation of critical Hill coefficient with the parameter set . With the optimized settings in B1, the system seems to be robust when . B4. Variation of critical Hill coefficient with the parameter set . Default values of other parameters in B1-4 are as in A1. C1. Variation of period and amplitude of the negative-feedback-only model considered in reference [25] with respect changes in the promoter affinity parameter . Simulation settings are given in Table 1. Red solid line in the period and blue solid line is amplitude of the oscillator.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4128676&req=5

pone-0104328-g003: Dynamical aspects of generalized Goodwin-Griffith oscillator.A1. Phase portraits of Goodwin-Griffith oscillator as described by Eqs (4). One needs to substitute Q  =  M (scaled concentration of mRNA) for red line, Q  =  X (promoter occupancy) for blue line and Q  =  Z (end-product) for pink line. Simulation settings are , , and we set which required a critical Hill coefficient of Cna = 6 to generate oscillations. Total simulation time is 100 (measured in terms of number of lifetimes of the protein product of TF gene A) and integration step is . A2. Trajectories corresponding to the settings in A1. A3. Roots of the (biquadratic) characteristic polynomial (PI) associated with the Jacobian matrix for settings in A1. B1. Variation of critical Hill coefficient with the parameter set . Minimum of this critical value seems to be achieved at , and . B2. Variation of critical Hill coefficient with the parameter set . With the optimized settings in B1, the system seems to be robust when . B3. Variation of critical Hill coefficient with the parameter set . With the optimized settings in B1, the system seems to be robust when . B4. Variation of critical Hill coefficient with the parameter set . Default values of other parameters in B1-4 are as in A1. C1. Variation of period and amplitude of the negative-feedback-only model considered in reference [25] with respect changes in the promoter affinity parameter . Simulation settings are given in Table 1. Red solid line in the period and blue solid line is amplitude of the oscillator.
Mentions: Sample trajectories and phase portraits of GG oscillator for are shown in Figures 3A1-3 and 4A1-3. Irrespective of the type of initial conditions and magnitude of the control parameters, the trajectories always start with an overshoot of protein production that is followed by asymptotic spirals towards a stable limit cycle. This seems to be an inherent property of negatively self-regulated loops [9]. Figures 3B1-4 and 4B1-4 suggest that there exists an optimum range of Group I parameters and at which the critical Hill coefficient () that is required to generate self-sustained oscillations is a minimum which is in turn strongly dependent on the promoter state occupancy parameter . This optimum range is also dependent on the values of other Group II and III parameters. The optimum range of the conversion parameter seems to be . Results suggest that strong binding conditions () and () are required to minimize the value of critical Hill coefficient with respect to changes in Group I type parameters. The minimum achievable values of critical Hill coefficients seems to be () and (). When there is an additional dimerization step as described in Eqs (4–5) corresponding to the negative-feedback-only (NFO) model considered in reference [25], the minimum achievable critical Hill coefficient seems to be . One should note that in the Lac I oscillatory system the effective Hill coefficient is since four dimers of lac I end-products involved in the overall negative feedback. Numerical analysis of this NFO model system using the physiological range of parameters as given in Table 1 suggests that the period of oscillator can be well tuned by changing the promoter state affinity of the repressor without compromising the amplitude much as shown in Figure 3C1.

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