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A Primer in Bifurcation Theory for Computational Cell Biologists Lecture 2

A Primer in Bifurcation Theory for Computational Cell Biologists Lecture 2. http://www.biology.vt.edu/faculty/tyson/lectures.php. John J. Tyson Virginia Polytechnic Institute & Virginia Bioinformatics Institute. Click on icon to start audio. The Curse of Parameter Space.

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A Primer in Bifurcation Theory for Computational Cell Biologists Lecture 2

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  1. A Primer in BifurcationTheoryfor Computational Cell BiologistsLecture 2 http://www.biology.vt.edu/faculty/tyson/lectures.php John J. Tyson Virginia Polytechnic Institute & Virginia Bioinformatics Institute Click on icon to start audio

  2. The Curse of Parameter Space Molec Genetics Biochemistry Cell Biology Computational Cell Biology Molecular Mechanism Kinetic Equations

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

  4. metaphase response (MPF) interphase signal (cyclin) MPF Wee1 Cdc25 SN SN

  5. 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

  6. = D1 = 0 = D2 Stability Analysis of Steady States …at a steady state (xo, yo). Expand using Taylor’s Theorem:

  7. Jacobian Matrix The solution is… where… are called the eigenvalues and eigenvectors of the Jacobian matrix.

  8. The eigenvalues are solutions of the “characteristic” equation: tr(J) det(J)

  9. Re(l) < 0 Re(l) > 0 stable focus unstable focus Hopf bifurcation at Tr(J) = 0 l1 < 0 l2< 0 l1 > 0 l2> 0 stable node unstable node l1 < 0 l2> 0 Saddle-Node bifurcation at det(J) = 0 saddle point det(J) tr(J)

  10. p < pSN node p = pSN saddle Variable, x pSN node p > pSN Parameter, p Saddle-Node Bifurcation f(x,y;p)=0 y g(x,y;p)=0 x

  11. = 0 Numerical Bifurcation Theory Two equations in three unknowns. Fix p = po; solve for (xo, yo). Expand using Taylor’s Theorem:

  12. This is perfectly generalizable to any number of variables. As long as With this equation, we can follow a steady state as p changes.

  13. SN Variable, x SN Parameter, p A problem arises when which is exactly the case at a saddle-node bifurcation point. Fix: swap x and p.

  14. one ss Parameter, q one ss three ss Parameter, p Two-parameter Bifurcation Diagram Three equations in four unknowns. Fix p = po; solve for (xo, yo, qo). Follow the solution using… D = det(J)

  15. Actually, AUTO does not try to solve det(J) = 0. That’s too hard. Instead, AUTO solves the following equations:

  16. 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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