Limits...
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.

Show MeSH

Related in: MedlinePlus

Dynamics of dual feedback Goodwin-Griffith oscillators coupled through OR gate.A1-2. Trajectories of protein-products of TF genes A and B which are two independent GG oscillators coupled through A-OR-B type logic as given in Figure 2B2. Simulation settings are , , and which required a critical Hill coefficient of Cna = 8. Total simulation time is 200 (number of lifetimes of the protein product of TF gene A) and integration step is . For each promoter A/B the parameter  will be split into  where  corresponds to self-regulation and  corresponds to cross regulation. Under identical values of all the parameters the system generates synchronized oscillations with a period of . Upon introduction of perturbation in  from scaled time 0 to 100 (where ), the amplitude of TF gene A is reduced with a phase shift and the period of entire system that includes both TF genes A and B remains the same. A2 is a magnification of certain range of A1. A3-4. Effect of perturbation in only one of the split parameters  associated with TF gens A/B. Here  is perturbed to . The system seems to be unstable and generates beats. A4 is a magnification of certain range of A3. A5-6. Here  is perturbed to . Period of beats seems to increase as the disproportion among the split parameters increases. A6 is a magnification of certain range of A5. B1-2. Effect of perturbation in the parameter  which is raised to  from the default value in the interval from 0 to 100. Increase in  reduces the amplitude of both the TF genes A and B with a phase shift and the period of oscillations of the entire system remains the same as in A1-2. B2 is a magnification of certain range of B1. C1-2. Effect of perturbations in the parameter  which is raised to  from the default value in the time interval from 0 to 100. Increase in  reduces the amplitude of both the TF genes A and B without a phase shift and the period of oscillations of the entire system remains the same as in A1-2. C2 is a magnification of certain range of C1.
© Copyright Policy
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4128676&req=5

pone-0104328-g007: Dynamics of dual feedback Goodwin-Griffith oscillators coupled through OR gate.A1-2. Trajectories of protein-products of TF genes A and B which are two independent GG oscillators coupled through A-OR-B type logic as given in Figure 2B2. Simulation settings are , , and which required a critical Hill coefficient of Cna = 8. Total simulation time is 200 (number of lifetimes of the protein product of TF gene A) and integration step is . For each promoter A/B the parameter will be split into where corresponds to self-regulation and corresponds to cross regulation. Under identical values of all the parameters the system generates synchronized oscillations with a period of . Upon introduction of perturbation in from scaled time 0 to 100 (where ), the amplitude of TF gene A is reduced with a phase shift and the period of entire system that includes both TF genes A and B remains the same. A2 is a magnification of certain range of A1. A3-4. Effect of perturbation in only one of the split parameters associated with TF gens A/B. Here is perturbed to . The system seems to be unstable and generates beats. A4 is a magnification of certain range of A3. A5-6. Here is perturbed to . Period of beats seems to increase as the disproportion among the split parameters increases. A6 is a magnification of certain range of A5. B1-2. Effect of perturbation in the parameter which is raised to from the default value in the interval from 0 to 100. Increase in reduces the amplitude of both the TF genes A and B with a phase shift and the period of oscillations of the entire system remains the same as in A1-2. B2 is a magnification of certain range of B1. C1-2. Effect of perturbations in the parameter which is raised to from the default value in the time interval from 0 to 100. Increase in reduces the amplitude of both the TF genes A and B without a phase shift and the period of oscillations of the entire system remains the same as in A1-2. C2 is a magnification of certain range of C1.

Mentions: Contrasting from the configuration given in Figure 2B1, the limit-cycle orbits of the coupled oscillators depicted in Figures 2B2-3 are robust against transient imbalances in the control parameters. The minimum achievable value of the critical Hill coefficient seems to be Cnh = 4 for the oscillator with A-OR-B type logic (Figure 2B2) whereas Cnh = 2 for the coupled oscillators with A-AND-B type logic (Figure 2B3). Results suggest that the limit cycle orbit of coupled oscillators with A-AND-B and A-OR-B type logics are stable one. When there are temporal perturbations in Group I parameters associated with one of the Goodwin oscillators (TF gene A/B) then the other unperturbed oscillator responds to the changes in the behavior of the perturbed oscillator depending on the type of logical coupling between them. As shown in Figures 7A1-2, B1-2 and C1-2 in case of A-OR-B coupling an increase in the magnitude of Group I parameters associated with one of the oscillators A/B does not change the period of the entire system of oscillators (period-buffering) though there is a decrease in the amplitude of the oscillator that is perturbed in . The decrease in the amplitude might be partially owing to the period-buffering effect. In case of A-AND-B type logical coupling, increase in the magnitude of Group I parameters increases the period of oscillations and decreases the amplitude of the entire system of oscillators that includes both TF genes A/B. Figures 8A1-4 suggest that an increase in the parameter of one of the oscillators initially increases the amplitude of other oscillator to a maximum which then decreases later. Perturbations in Group I parameters associated with one of the oscillators A/B also results in a phase-shift in cases of both A-AND-B and A-OR-B type logical couplings as shown in Figures 7-8A1-2 and B1-2. Whereas perturbation in affects only the amplitude and does not affect the phases of the coupled oscillators A and B as shown in Figures 7-8C1-2. Here one should note that in case of A-OR-B type coupling the parameter will be split into where we have the indices as given in Eqs (21). Above results corresponding to A-OR-B type logical coupling with respect to changes in the parameter are valid only when the temporal perturbations are the same for a given promoter of TF gene A/B. This means that for TF gene A (here we have subscript ) the extent of perturbation should be the same for both and while the set of parameters associated with the TF gene B () remains unperturbed. Here one should note that controls the dynamics associated with the binding of end-product of TF gene ‘H’ at its own promoter whereas controls the dynamics associated with the binding of end-product of TF gene ‘K’ at the promoter of TF gene ‘H’ as we have shown in Eqs (21). When there are perturbations in only one of these two split parameters (as ) then the coupled system of oscillators seems to be dynamically unstable and also produces modulated beats as shown in Figures 7A3-4. The period of such beats increases as the imbalance in the set of split parameters increases as shown in Figures 7A5-6. These dynamical instabilities as well as beats abruptly disappear once the perturbations in are removed. Whereas the system of coupled oscillators relaxes back to the initial unperturbed limit-cycle orbit through asymptotic spirals upon removal of perturbations in case of Group I control parameters . Results from Figures 7 and 8 suggest that coupled oscillators with -AND- type logic are more robust against promoter state perturbations than the -OR- type coupling. Period of a network of oscillators can be easily fine-tuned by manipulating merely one of the oscillators when the mode of coupling is via -AND- type.


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

Murugan R - PLoS ONE (2014)

Dynamics of dual feedback Goodwin-Griffith oscillators coupled through OR gate.A1-2. Trajectories of protein-products of TF genes A and B which are two independent GG oscillators coupled through A-OR-B type logic as given in Figure 2B2. Simulation settings are , , and which required a critical Hill coefficient of Cna = 8. Total simulation time is 200 (number of lifetimes of the protein product of TF gene A) and integration step is . For each promoter A/B the parameter  will be split into  where  corresponds to self-regulation and  corresponds to cross regulation. Under identical values of all the parameters the system generates synchronized oscillations with a period of . Upon introduction of perturbation in  from scaled time 0 to 100 (where ), the amplitude of TF gene A is reduced with a phase shift and the period of entire system that includes both TF genes A and B remains the same. A2 is a magnification of certain range of A1. A3-4. Effect of perturbation in only one of the split parameters  associated with TF gens A/B. Here  is perturbed to . The system seems to be unstable and generates beats. A4 is a magnification of certain range of A3. A5-6. Here  is perturbed to . Period of beats seems to increase as the disproportion among the split parameters increases. A6 is a magnification of certain range of A5. B1-2. Effect of perturbation in the parameter  which is raised to  from the default value in the interval from 0 to 100. Increase in  reduces the amplitude of both the TF genes A and B with a phase shift and the period of oscillations of the entire system remains the same as in A1-2. B2 is a magnification of certain range of B1. C1-2. Effect of perturbations in the parameter  which is raised to  from the default value in the time interval from 0 to 100. Increase in  reduces the amplitude of both the TF genes A and B without a phase shift and the period of oscillations of the entire system remains the same as in A1-2. C2 is a magnification of certain range of C1.
© Copyright Policy
Related In: Results  -  Collection

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

pone-0104328-g007: Dynamics of dual feedback Goodwin-Griffith oscillators coupled through OR gate.A1-2. Trajectories of protein-products of TF genes A and B which are two independent GG oscillators coupled through A-OR-B type logic as given in Figure 2B2. Simulation settings are , , and which required a critical Hill coefficient of Cna = 8. Total simulation time is 200 (number of lifetimes of the protein product of TF gene A) and integration step is . For each promoter A/B the parameter will be split into where corresponds to self-regulation and corresponds to cross regulation. Under identical values of all the parameters the system generates synchronized oscillations with a period of . Upon introduction of perturbation in from scaled time 0 to 100 (where ), the amplitude of TF gene A is reduced with a phase shift and the period of entire system that includes both TF genes A and B remains the same. A2 is a magnification of certain range of A1. A3-4. Effect of perturbation in only one of the split parameters associated with TF gens A/B. Here is perturbed to . The system seems to be unstable and generates beats. A4 is a magnification of certain range of A3. A5-6. Here is perturbed to . Period of beats seems to increase as the disproportion among the split parameters increases. A6 is a magnification of certain range of A5. B1-2. Effect of perturbation in the parameter which is raised to from the default value in the interval from 0 to 100. Increase in reduces the amplitude of both the TF genes A and B with a phase shift and the period of oscillations of the entire system remains the same as in A1-2. B2 is a magnification of certain range of B1. C1-2. Effect of perturbations in the parameter which is raised to from the default value in the time interval from 0 to 100. Increase in reduces the amplitude of both the TF genes A and B without a phase shift and the period of oscillations of the entire system remains the same as in A1-2. C2 is a magnification of certain range of C1.
Mentions: Contrasting from the configuration given in Figure 2B1, the limit-cycle orbits of the coupled oscillators depicted in Figures 2B2-3 are robust against transient imbalances in the control parameters. The minimum achievable value of the critical Hill coefficient seems to be Cnh = 4 for the oscillator with A-OR-B type logic (Figure 2B2) whereas Cnh = 2 for the coupled oscillators with A-AND-B type logic (Figure 2B3). Results suggest that the limit cycle orbit of coupled oscillators with A-AND-B and A-OR-B type logics are stable one. When there are temporal perturbations in Group I parameters associated with one of the Goodwin oscillators (TF gene A/B) then the other unperturbed oscillator responds to the changes in the behavior of the perturbed oscillator depending on the type of logical coupling between them. As shown in Figures 7A1-2, B1-2 and C1-2 in case of A-OR-B coupling an increase in the magnitude of Group I parameters associated with one of the oscillators A/B does not change the period of the entire system of oscillators (period-buffering) though there is a decrease in the amplitude of the oscillator that is perturbed in . The decrease in the amplitude might be partially owing to the period-buffering effect. In case of A-AND-B type logical coupling, increase in the magnitude of Group I parameters increases the period of oscillations and decreases the amplitude of the entire system of oscillators that includes both TF genes A/B. Figures 8A1-4 suggest that an increase in the parameter of one of the oscillators initially increases the amplitude of other oscillator to a maximum which then decreases later. Perturbations in Group I parameters associated with one of the oscillators A/B also results in a phase-shift in cases of both A-AND-B and A-OR-B type logical couplings as shown in Figures 7-8A1-2 and B1-2. Whereas perturbation in affects only the amplitude and does not affect the phases of the coupled oscillators A and B as shown in Figures 7-8C1-2. Here one should note that in case of A-OR-B type coupling the parameter will be split into where we have the indices as given in Eqs (21). Above results corresponding to A-OR-B type logical coupling with respect to changes in the parameter are valid only when the temporal perturbations are the same for a given promoter of TF gene A/B. This means that for TF gene A (here we have subscript ) the extent of perturbation should be the same for both and while the set of parameters associated with the TF gene B () remains unperturbed. Here one should note that controls the dynamics associated with the binding of end-product of TF gene ‘H’ at its own promoter whereas controls the dynamics associated with the binding of end-product of TF gene ‘K’ at the promoter of TF gene ‘H’ as we have shown in Eqs (21). When there are perturbations in only one of these two split parameters (as ) then the coupled system of oscillators seems to be dynamically unstable and also produces modulated beats as shown in Figures 7A3-4. The period of such beats increases as the imbalance in the set of split parameters increases as shown in Figures 7A5-6. These dynamical instabilities as well as beats abruptly disappear once the perturbations in are removed. Whereas the system of coupled oscillators relaxes back to the initial unperturbed limit-cycle orbit through asymptotic spirals upon removal of perturbations in case of Group I control parameters . Results from Figures 7 and 8 suggest that coupled oscillators with -AND- type logic are more robust against promoter state perturbations than the -OR- type coupling. Period of a network of oscillators can be easily fine-tuned by manipulating merely one of the oscillators when the mode of coupling is via -AND- type.

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
Related in: MedlinePlus