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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
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
The Dynamical Perspectivein Molecular Cell Biology
Molec Genetics Biochemistry Cell Biology
Kinetic Equations
Molecular Mechanism
The Curse ofParameter Space
[Cyclin]
[CKI]
[MPF]
Kinetic Equations
State Space, Vector Field
Molecular Mechanism
Attractors, Transients, RepellorsHenri Poincare (1890)
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
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
MPF
Cyclin
d cyclindt
= k1 - k2 * cyclin = 0k1 / k2
d MPFdt
= … = 0
MPF
Cyclin
d cyclindt
= k1 - k2 * cyclin = 0k1 / k2
d MPFdt
= … = 0
MPF
Cyclin
d cyclindt
= k1 - k2 * cyclin = 0k1 / k2
d MPFdt
= … = 0
saddle-node
MPF
Cyclin
d cyclindt
= k1 - k2 * cyclin = 0k1 / k2
d MPFdt
= … = 0
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
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
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
Oscillations
0
0.5
1
0 1 2
MP
F
cyclin
MPF
Cdc25- PCdc25
MPF- P(inactive)
cyclin synthesis
cyclin degradationAPC
negative feedback loop
Pomerening, Kim & FerrellCell (2005)
MP
F a
cti
vit
y
MPF activity
Total Cyclin
Total Cyclin
stable limit cycle
Variable,MPF
Parameter, k1
sss
uss
slc max
min
One-parameter bifurcation diagram
Hopf Bifurcation
stable limit cycle
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
•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
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
