Bifurcation Analysis of PER, TIM, dCLOCK Model

1
Bifurcation Analysis of PER, TIM, dCLOCK Model

• Positive feedback interacting with negative
  feedbacks
• April, 2002

2
Model
DBT
Pm
P1
PER-P
P2
P1
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 k0, vmt0,
  and vmtb1, 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 kin2
TB
Fixed period 563.3
SN
Hopf 1
TB
Hopf 2
Cusp
SN
12
One parameter cut with kp1 at kin2 and vmc0.03
Hopf
LP
LP
13
Two parameter bifurcation kapp vs. kp1 at
vmc0.03 and kin2
Hopf
SN
TB
Cusp
14
One parameter cut with kp1 at kin2 and vmc0.02
Hopf
Hopf
15
Two parameter bifurcation kapp vs. kp1 at
vmc0.02 and kin2
Region of Hopf
16
One parameter cut with vmc at kp10.03
Hopf
17
One parameter cut with vmc at kp10.03
Hopf
Hopf
18
Two parameter bifurcation kin vs. kdmp at
kp10.03
Region of Hopf
19
Bifurcation Analysis of Comprehensive Model Part 2

• For the follwing figures, both negative (kin1),
  and positive feedbacks are active (kp110). But
  the feedbacks on TIM and dCLK are still inactive
  (vmt0, vmtb1, k0).

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 vmc0.03
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 kp10.03
Hopf
27
Simple Model - Multiple Steady States(One
parameter cut kp1)
Hopf
LP
LP
28
One parameter cut with kp1 at kin2 and vmc0.03
Hopf
LP
LP
29
One parameter cut with kp1 at kin1, vmc0.03
Hopf
LP
LP
30
Two parameter bifurcation kp1 vs. vmc at kin2
TB
Fixed period 563.3
SN
Hopf 1
TB
Hopf 2
Cusp
SN
31
Two parameter bifurcation kp1 vs. vmc at kin1
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.