A Primer in Bifurcation Theory for Computational Cell Biologists John J. Tyson Virginia Polytechnic Institute & Virginia Bioinformatics Institute Click on icon to start audio

The Dynamical Perspective in Molecular Cell Biology Molec Genetics Biochemistry Cell Biology Kinetic Equations Molecular Mechanism

Wee1 Cdc25 MPF= Mitosis Promoting Factor

The Dynamical Perspective in Molecular Cell Biology Molec Genetics Biochemistry Cell Biology Kinetic Equations Molecular Mechanism The Curse of Parameter Space

[Cyclin] [CKI] [MPF] Kinetic Equations State Space, Vector Field Molecular Mechanism Attractors, Transients, Repellors Henri Poincare (1890)

The Dynamical Perspective in Molecular Cell Biology Molec Genetics Biochemistry Cell Biology Kinetic Equations State Space, Vector Field Attractors, Transients, Repellors Bifurcation Diagrams Molecular Mechanism Signal-Response Curves

Wee1 Cdc25 = k 1 - (k wee + k 2 ) * MPF + k 25 (cyclin - MPF) = k 1 - k 2 * cyclin d MPF dt d cyclin dt

MPF Cyclin d cyclin dt = k 1 - k 2 * cyclin = 0 k 1 / k 2 d MPF dt = … = 0

MPF Cyclin d cyclin dt = k 1 - k 2 * cyclin = 0 k 1 / k 2 d MPF dt = … = 0

MPF Cyclin d cyclin dt = k 1 - k 2 * cyclin = 0 k 1 / k 2 d MPF dt = … = 0 saddle-node

MPF Cyclin d cyclin dt = k 1 - k 2 * cyclin = 0 k 1 / k 2 d MPF dt = … = 0

One-parameter bifurcation diagram Parameter, k1 Variable, MPF stable steady state unstable steady state saddle-node Signal Response t t p x OFF ON (signal) (response) x y

Frog egg MPF Cdc25- P Cdc25 MPF- P response (MPF) signal (cyclin) interphase metaphase (inactive) CycB MPF = M-phase Promoting Factor

MPF activity depends on total cyclin concentration and on the history of the extract Cyclin concentration increasing inactivation threshold at 90 min MPF activity nM  cyclin B M I I I II I MPF activity nM  cyclin B M M M I/M I I I Cyclin concentration decreasing I M bistability Wei Sha & Jill Sible (2003) zero

Oscillations MPF cyclin MPF Cdc25- P Cdc25 MPF- P (inactive) cyclin synthesis cyclin degradation APC negative feedback loop

Pomerening, Kim & Ferrell Cell (2005) MPF activity Total Cyclin stable limit cycle

Variable, MPF Parameter, k 1 sss uss slc max min One-parameter bifurcation diagram Hopf Bifurcation stable limit cycle

The Dynamical Perspective in 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 Oscill8