1. Bifurcation Analysis of PER, TIM, dCLOCK Model
Positive feedback interacting with negative feedbacks
April, 2002

2. Model DBT P1 PER-P P1 P2 T1 P2T1 T1 P2T2 Pm Tm ITF T1 PN TF TF dCLK
There are three feedbacks in the model.
Positive feedback loop based on the stabilization of PER upon dimerization.
Negative feedback loop inhibiting both per and tim transcriptions
Negative feedback loop on transcription of dClk. TF
dCLK
CYC

3. Equations
Parameters
vmpb=0, vmtb=1, vmp=1, vmt=0, vmc=1, kdmp=0.1, kdmt=0.1, kdmc=0.1, vp=0.5, vt=0.5, vc=0.5, kp1=10, kp2=0.03, kp3=0.1, kt3=0.1, kdc=0.1, Jp=0.05, kapp=10, kdpp=0.1, kapt=10, kdpt=0.1, kacc=10, kdcc=0.1 , kaitf=10 , kditf=0.1, k=1, kin=1, kout=0.1, K1=1, K2=1, m=7, n=2, CYCtot=10
When k=0 -> turns off feedback on dClk.
When vmtb=1 and vmt=0 -> turns off feedback on tim.
When kin=2 -> turns of negative feedback on per and tim.
When kp1=0.03 -> turns off positive feedback.

4. Typical two parameter bifurcation diagram of positive feedback
Region of Multiple steady states SN P2 SN TB
Cusp TB
Hopf P1

5. Typical two parameter bifurcation diagram of negative feedback
Region of Hopf P1

6. Simple Model Bifurcation Diagrams (Biophysical Journal, 1999)
Region of Hopf
p041802a.ps
Default parameter values as in Bioph. J., 1999.
Showing a region of Hopf where the rhythm is robust.

7. Multiple Steady States (One parameter cut: kp1)
Hopf LP
p041802b.ps
Changed parameters vm=1.2 from 1, Keq=1 from 200, and Pcrit=0.6 from 0.1 in order to create multiple steady states.
One parameter cut with kp1, showing two SNs and Hopf. LP

8. Multiple Steady States (Two parameter cut: Keq vs. kp1)
Hopf SN
p041802c.ps
Followed above SNs and Hopf in two parameters: Keq and kp1.
Hopf meets SN with infinite period (TB).
SN shows a region of multiple steady states with cusp. TB SN
Cusp

9. Bifurcation Analysis of Comprehensive Model Part 1
All of the analysis were done with k=0, vmt=0, and vmtb=1, unless otherwise indicated. This eliminated feedbacks on dClk and TIM, which were not present in our simple model. In other words, this was done to compare the dynamics between simple and comprehensive model, and explore the parameter space of comprehensive model where we could get similar dynamics as in simple model.
Unless indicated, parameters are at their default value. (see notes in page 3)

10. One parameter cut with vmc at kin = 2
Hopf
ptd042202g2.ps
When kin=2, it creates a rapid transport of P2T2 -> Pn, which eliminates the delay. In turn, this will deactivate negative feedback.
There is a Hopf bifurcation, which generates stable limit cycle.
There are two limit points indicating saddle node bifurcations.
There were no other Hopf bifurcation at greater value of vmc. (Investigated up to vmc=0.5) LP LP

11. Two parameter bifurcation: kp1 vs. vmc at kin=2TB
Fixed period = 563.3 SN
Hopf 1
ptd042202h.ps
Followed Hopf 1 and SN bifurcations in two parameters: kp1 and vmc.
Hopf 2 is calculated by getting a one parameter cut of vmc at kp1=3. This is a Hopf generated by negative feedback, and pushed down to a small region by increased kin.
A typical bifurcation diagram from positive feedback system.
There is a region of multiple steady states.
Both ends of the Hopf meets SN (Takens-Bogdanov) and disappears with infinite period.
The fixed period of h is there to indicate possible picture of SL, which also ends in TB. TB
Hopf 2
Cusp SN

12. One parameter cut with kp1 at kin=2 and vmc=0.03
Hopf LP
ptd042802c.ps
In order to generate two parameter bifurcation diagram of kp1 and keq as in simple model, one parameter cut with kp1 at kin=2 and difference vmc values are evaluated. (see page 5)
It shows a hystersis with two SNs, and a Hopf bifurcations. LP

13. Two parameter bifurcation: kapp vs. kp1 at vmc=0.03 and kin=2
Hopf SN
ptd042802d.ps
There is a multiple steady state region with cusp.
Hopf meets with SN creating TB bifurcation, and disappears.
Compare this picture with a similar figure of simple model. (page 6) TB
Cusp

14. One parameter cut with kp1 at kin=2 and vmc=0.02
Hopf
ptd042302d.ps
At vmc=0.02, there is no region of multiple steady states. When you cut with vmc=0.02, we are getting a cut before cusp. (see page 9)
Hopf

15. Two parameter bifurcation: kapp vs. kp1 at vmc=0.02 and kin=2
Region of Hopf
ptd042302e.ps
There is a region of Hopf with stable limit cycle.
Compare this with two parameter bifurcation of simple model in page 4. At kin=2 and vmc=0.02, we were able to generate a similar dynamics as in our simple model.

16. One parameter cut with vmc at kp1=0.03
ptd042202a.ps
When kp1=0.03, the positive feedback due to PER stabilization upon dimerization is disappeared. At kp1=0.03, the degradation machinery of PER is no longer effective.
There is a Hopf bifurcation generated by delayed negative feedback system.
There is no region of multiple steady states at lower values of vmc.
Hopf

17. One parameter cut with vmc at kp1=0.03
Hopf
Hopf
ptd042902f2.ps
With inactivated positive feedback (kp1=0.03), followed one parameter cut with kdmp, which is the degradation rate of per mRNA. kdmp is related with the delay of the negative feedback.

18. Two parameter bifurcation: kin vs. kdmp at kp1=0.03
Region of Hopf
ptd042902g.ps
Followed Hopf in two parameter space: kin vs. kdmp. Indicates typical region of Hopf by negative feedback.

19. Bifurcation Analysis of Comprehensive Model Part 2
For the follwing figures, both negative (kin=1), and positive feedbacks are active (kp1=10). But the feedbacks on TIM and dCLK are still inactive (vmt=0, vmtb=1, k=0).

20. One parameter cut with vmc
Hopf 1
Hopf 2
ptd041602a.ps
Both negative and positive feedbacks are active, and we have two Hopf bifurcations and two SN bifurcations.
Hopf 1 seems to be generated by delayed negative feedback, and Hopf 2 and hysterisis seem to be generated by positive feedback mechanism. LP LP

21. Two parameter bifurcation: kp1 vs. vmcTB SN
Hopf 1 SN
ptd042202b2.ps, ptd042202d.ps
Hopf 1 seems to connect both Hopf bifurcations of negative and positive feedbacks, and ends in TB.
Hopf 2 was revealed by calculating one parameter cut with vmc at kp1= Hopf 2 also ends in TB.
Multiple steady state region with cusp is present. TB
Cusp
Hopf 2

22. One parameter cut with kp1 at vmc=0.03
Hopf LP
ptd042802a.ps
In order to compare the dynamics of positive feedback in the presence of negative feedback (kin=1), similar two parameter bifurcation analysis was done. (see page10)
The qualitative aspects of the system is unchanged even when kin=1. LP

23. Two parameter bifurcation: kp1 vs. vmc
Hopf SN
ptd042802b.ps
Qualitatively similar picture as in kp1 vs. vmc at kin=2. (see page 11) TB
Cusp

24. One parameter cut with vmc
Hopf 1
Hopf 2
Here we start side by side comparison of one parameter cuts of vmc, kp1, and two parameter cuts of kp1 vs. vmc. LP LP

25. One parameter cut with vmc at kin = 2
Hopf LP LP

26. One parameter cut with vmc at kp1=0.03
Hopf

27. Simple Model - Multiple Steady States (One parameter cut: kp1)
Hopf LP LP

28. One parameter cut with kp1 at kin=2 and vmc=0.03
Hopf LP LP

29. One parameter cut with kp1 at kin=1, vmc=0.03
Hopf LP LP

30. Two parameter bifurcation: kp1 vs. vmc at kin=2TB
Fixed period = 563.3 SN
Hopf 1 TB
Hopf 2
Cusp SN

31. Two parameter bifurcation: kp1 vs. vmc at kin=1TB SN
Hopf 1 SN TB
Cusp
Hopf 2

32. Conclusion
The dynamics of the comprehensive model reveals that it is identical with our simple model, when we have either positive feedback alone, or both positive and negative feedbacks together at low values of vmc.
In the presence of both positive and negative feedbacks, they seem to interact with each other, and generates different regions of stable limit cycles.
Our next step is to analyze how other feedbacks (TIM and dCLK) interact with existing feedbacks by chaging vmt, vmtb, and k.