A Primer in Bifurcation Theory for Computational Cell Biologists
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A Primer in Bifurcation Theoryfor Computational Cell Biologists
John J. TysonVirginia Polytechnic Institute
& Virginia Bioinformatics Institute
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The Dynamical Perspectivein Molecular Cell Biology
Molec Genetics Biochemistry Cell Biology
Kinetic Equations
Molecular Mechanism
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Cdk
C K
I
Cdk
Cyclin
C K
I
Cdk
Cyclin
Cdk
Cyclin
P
Cyclin
Cdk
Wee1
Cdc25
1 2 3 4
3 4 5
6 7
8 9
d[Cyclin][Cyclin] [Cyclin][Cdk] [MPF]
dd[MPF]
[Cyclin][Cdk] [MPF] [MPF]d
[Wee1][MPF] [Cdc25][preMPF]
[MPF][CKI] [MPF:CKI]
k k k kt
k k kt
k k
k k
MPF = Mitosis Promoting Factor
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The Dynamical Perspectivein Molecular Cell Biology
Molec Genetics Biochemistry Cell Biology
Kinetic Equations
Molecular Mechanism
The Curse ofParameter Space
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[Cyclin]
[CKI]
[MPF]
Kinetic Equations
State Space, Vector Field
Molecular Mechanism
Attractors, Transients, RepellorsHenri Poincare (1890)
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The Dynamical Perspectivein Molecular Cell Biology
Molec Genetics Biochemistry Cell Biology
Kinetic Equations
State Space, Vector Field
Attractors, Transients, Repellors
Bifurcation Diagrams
Molecular Mechanism
Signal-Response Curves
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Cdk
C K
I
Cdk
Cyclin
C K
I
Cdk
Cyclin
Cdk
Cyclin
P
Cyclin
Cdk
Wee1
Cdc25
= k1 - (kwee + k2) * MPF + k25 (cyclin - MPF)
= k1 - k2 * cyclin
d MPFdt
d cyclindt
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MPF
Cyclin
d cyclindt
= k1 - k2 * cyclin = 0k1 / k2
d MPFdt
= … = 0
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MPF
Cyclin
d cyclindt
= k1 - k2 * cyclin = 0k1 / k2
d MPFdt
= … = 0
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MPF
Cyclin
d cyclindt
= k1 - k2 * cyclin = 0k1 / k2
d MPFdt
= … = 0
saddle-node
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MPF
Cyclin
d cyclindt
= k1 - k2 * cyclin = 0k1 / k2
d MPFdt
= … = 0
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One-parameter bifurcation diagram
Parameter, k1
Variable, MPF
stable steady state
unstable steady state
saddle-nodesaddle-node
Signal Response
t t
p x
OFF
ON
(signal)
(response)
x
y
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Frog egg
MPF
Cdc25- PCdc25
MPF- P
0
0.5
1
0 1 2
resp
on
se (
MP
F)
signal (cyclin)
interphase
met
apha
se
(inactive)CycBMPF =
M-phase Promoting Factor
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02468
101214
0 6 12 18 24 30 60
MPF activity depends on total cyclin concentration
and on the history of the extract
Cyclin concentration increasing
inactivation threshold at 90 min
MP
F a
ctiv
ity
nM cyclin B
M
IIIIII
02468
101214
0 6 12 18 24 30 60
MP
F a
ctiv
ity
nM cyclin B
M
M
MI/MIII
Cyclin concentration decreasing
I M
bistabilityWei Sha & Jill Sible (2003)
zero
zero
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Oscillations
0
0.5
1
0 1 2
MP
F
cyclin
MPF
Cdc25- PCdc25
MPF- P(inactive)
cyclin synthesis
cyclin degradationAPC
negative feedback loop
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Pomerening, Kim & FerrellCell (2005)
MP
F a
cti
vit
y
MPF activity
Total Cyclin
Total Cyclin
stable limit cycle
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Variable,MPF
Parameter, k1
sss
uss
slc max
min
One-parameter bifurcation diagram
Hopf Bifurcation
stable limit cycle
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The Dynamical Perspectivein Molecular Cell Biology
Molec Genetics Biochemistry Cell Biology
Kinetic Equations
State Space, Vector Field
Attractors, Transients, Repellors
Bifurcation Diagrams
Molecular Mechanism
Signal-Response Curves
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•Saddle-Node (bistability, hysteresis)•Hopf Bifurcation (oscillations)•Subcritical Hopf•Cyclic Fold•Saddle-Loop•Saddle-Node Invariant Circle
Signal-Response Curve = One-parameter Bifurcation Diagram
Rene Thom
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References
• Strogatz, Nonlinear Dynamics and Chaos (Addison Wesley)
• Kuznetsov, Elements of Applied Bifurcation Theory (Springer)
• XPP-AUT www.math.pitt.edu/~bard/xpp
• Oscill8 http://oscill8.sourceforge.net