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Bifurcation Analysis of PER, TIM, dCLOCK Model

Bifurcation Analysis of PER, TIM, dCLOCK Model. Positive feedback interacting with negative feedbacks April, 2002. Model. DBT. Pm. P1. PER-P. P1. P2. T1. Tm. P2T1. T1. ITF. T1. P2T2. PN. TF. TF. dCLK. CYC. Equations.

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Bifurcation Analysis of PER, TIM, dCLOCK Model

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

  2. Model DBT Pm P1 PER-P P1 P2 T1 Tm P2T1 T1 ITF T1 P2T2 PN TF TF dCLK CYC

  3. Equations

  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 P2 Region of Hopf P1

  6. Simple Model Bifurcation Diagrams(Biophysical Journal, 1999) Region of Hopf

  7. Multiple Steady States(One parameter cut: kp1) Hopf LP LP

  8. Multiple Steady States(Two parameter cut: Keq vs. kp1) Hopf SN 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 LP LP

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

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

  13. Two parameter bifurcation: kapp vs. kp1 at vmc=0.03 and kin=2 Hopf SN TB Cusp

  14. One parameter cut with kp1 at kin=2 and vmc=0.02 Hopf Hopf

  15. Two parameter bifurcation: kapp vs. kp1 at vmc=0.02 and kin=2 Region of Hopf

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

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

  18. Two parameter bifurcation: kin vs. kdmp at kp1=0.03 Region of Hopf

  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 LP LP

  21. Two parameter bifurcation: kp1 vs. vmc TB SN Hopf 1 SN TB Cusp Hopf 2

  22. One parameter cut with kp1 at vmc=0.03 Hopf LP LP

  23. Two parameter bifurcation: kp1 vs. vmc Hopf SN TB Cusp

  24. One parameter cut with vmc Hopf 1 Hopf 2 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=2 TB Fixed period = 563.3 SN Hopf 1 TB Hopf 2 Cusp SN

  31. Two parameter bifurcation: kp1 vs. vmc at kin=1 TB 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.

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