8.592 - HST.452J (Fall 2025)

Statistical Physics in Biology

Lecture 22

1. Stability of Fixed Points:
   • Linear stability matrix; eigenvectors and eigenvalues
   • Loss of stability from a real eigenvalue; stability exchange, bifurcation
   • Biological applications of bifurcation theory
   • Biological switches (Genetic toggle switch by James Collins)
2. Oscillations:
   • Loss of stability from imaginary eigenvalues; Poincaré–Bendixson theorem and periodic orbits
   • Predator-Prey system (applet) (another applet)
   • (Nlogo Appletincluding stochasticity and grass)
   • Oscillating reactions
   • The Belousov-Zhabotinsky Reaction (video, video) (Explanation)
   • Excitable media: neurons, cardiac tissue (FitzHugh-Nagumo model)
3. Biochemical clocks:
   • The cogs of a biological clock
   • Animations of biological clocks (offline}
   • The Repressilator [Elowitz and Liebler, Nature 403, 305 (2000)]
4. Synchronization:
   • Examples: Firefly, fireflies, heart pacemaker cells, cicada
   • Phase synchronization model from Nlogo
   • Collective synchronization
   • The Kuramoto model: (Math Insight Applet) (with more details) (another Applet)

(Lecture Notes)

Biological Clocks

• Internet Resources on Chronobiology, Biological Clocks and Rhythms
• Biotiming Tutorial
• The Repressilator figures