A Primer in Bifurcation Theory for Computational Cell Biologists Lecture 2 John J. Tyson Virginia Polytechnic Institute & Virginia Bioinformatics Institute Click on icon to start audio

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

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

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

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

Stability Analysis of Steady States …at a steady state (x o, y o ). Expand using Taylor’s Theorem: = 0 =   =  

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

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

tr(J) det(J) Saddle-Node bifurcation at det(J) = 0   2    2    2  Re(  Re(  saddle point unstable nodestable nodeunstable focusstable focus Hopf bifurcation at Tr(J) = 0

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

Numerical Bifurcation Theory Two equations in three unknowns. Fix p = p o ; solve for (x o, y o ). Expand using Taylor’s Theorem: = 0

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

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

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

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

References Strogatz, Nonlinear Dynamics and Chaos (Addison Wesley) Kuznetsov, Elements of Applied Bifurcation Theory (Springer) XPP-AUT Oscill8