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A Primer in Bifurcation Theory for Computational Cell Biologists

A Primer in Bifurcation Theory for Computational Cell Biologists. John J. Tyson Virginia Polytechnic Institute & Virginia Bioinformatics Institute. Click on icon to start audio. tyson@vt.edu. Molec Genetics Biochemistry Cell Biology.

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A Primer in Bifurcation Theory for Computational Cell Biologists

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  1. A Primer in BifurcationTheoryfor Computational Cell Biologists John J. Tyson Virginia Polytechnic Institute & Virginia Bioinformatics Institute Click on icon to start audio tyson@vt.edu

  2. Molec Genetics Biochemistry Cell Biology The Dynamical Perspectivein Molecular Cell Biology Molecular Mechanism Kinetic Equations

  3. MPF = Mitosis Promoting Factor Wee1 Cdc25

  4. The Curse of Parameter Space Molec Genetics Biochemistry Cell Biology The Dynamical Perspectivein Molecular Cell Biology Molecular Mechanism Kinetic Equations

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

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

  7. Wee1 Cdc25 d MPF dt = k1 - (kwee + k2) * MPF + k25 (cyclin - MPF) = k1 - k2 * cyclin d cyclin dt

  8. d MPF dt = … = 0 k1 / k2 d cyclin dt = k1 - k2 * cyclin = 0 MPF Cyclin

  9. d MPF dt = … = 0 k1 / k2 d cyclin dt = k1 - k2 * cyclin = 0 MPF Cyclin

  10. d MPF dt = … = 0 k1 / k2 d cyclin dt = k1 - k2 * cyclin = 0 MPF saddle-node Cyclin

  11. d MPF dt = … = 0 k1 / k2 d cyclin dt = k1 - k2 * cyclin = 0 MPF Cyclin

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

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

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

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

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

  17. uss Hopf Bifurcation sss stable limit cycle One-parameter bifurcation diagram max slc Variable, MPF min Parameter, k1

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

  19. Signal-Response Curve = One-parameter Bifurcation Diagram • Saddle-Node (bistability, hysteresis) • Hopf Bifurcation (oscillations) • Subcritical Hopf • Cyclic Fold • Saddle-Loop • Saddle-Node Invariant Circle Rene Thom

  20. 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

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