A Primer in Bifurcation Theoryfor Computational Cell Biologists

John J. TysonVirginia Polytechnic Institute

& Virginia Bioinformatics Institute

tyson@vt.edu

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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