Linear stability analysis

1. Linear stability analysis x Transcription-translation model Eigenvectors and eigenvalues Nullclines and critical points The cribsheet of linear stability analysis f m

2. Transcription-translation model m x +1 -1 +1 -1 f

3. Nullclines and critical points x 1.0 f 0.5 m 0 0.5 1.0

4. Nullclines and critical points x 1.0 f 0.5 m 0 0.5 1.0

5. Nullclines and critical points x 1.0 f 0.5 m 0 0.5 1.0

6. Nullclines and critical points x 1.0 f 0.5 m 0 0.5 1.0

7. Nullclines and critical points x or m x 1.0 0.9 0.8 mRNA f 0.7 0.6 Protein 0.5 t 0 1 2 3 4 5 m 0 0.5 1.0

8. Linear stability analysis x Transcription-translation model Eigenvectors and eigenvalues Nullclines and critical points The cribsheet of linear stability analysis f m

9. Unbending trajectories x f m 0 0.5 1.0

10. Finding the “special” direction Dx x 1.0 0.25 f 0.5 -0.25 m Dm 0 -0.25 0.5 0.25 1.0

11. x Dx Finding the “special” direction Dm m

12. Finding the “special” direction x Dx 0.5 -0.5 m Dm -0.5 0.5

13. Finding the “special” direction x Dx 0.5 Want eigenvectors! -0.5 m Dm -0.5 0.5

14. Finding the “special” direction x Dx 0.5 Want eigenvectors! -0.5 m Dm -0.5 0.5

15. Finding the “special” direction x Dx 0.5 Want eigenvectors! -0.5 m Dm -0.5 0.5

16. Finding the “special” direction x Dx 0.5 Want eigenvectors! -0.5 m Dm -0.5 0.5

17. Finding the “special” direction x Dx 0.5 Want eigenvectors! -0.5 m Dm -0.5 0.5

18. Finding the “special” direction x Dx 0.5 Trajectories along these directions do not bend -0.5 m Dm -0.5 0.5

19. Eigenvectors and eigenvalues provide analytic solution Dx x Trajectories along these directions do not bend m Dm

20. Eigenvectors and eigenvalues provide analytic solution Dx x Trajectories along these directions do not bend m Dm

21. Eigenvectors and eigenvalues provide analytic solution

22. Eigenvectors and eigenvalues provide analytic solution

23. Eigenvectors and eigenvalues provide analytic solution x Dx 0.5 Differential equations Generalsolution Initial conditions -0.5 m Dm -0.5 0.5

24. Eigenvectors and eigenvalues provide analytic solution x Dx 0.5 Differential equations Generalsolution Initial conditions -0.5 m Dm -0.5 0.5

25. Eigenvectors and eigenvalues provide analytic solution x Dx 0.5 0.5 0.4 Differential equations Generalsolution 0.3 Dx or Dm mRNA 0.2 f 0.1 Protein Initial conditions -0.5 0.0 t 0 1 2 3 4 5 m Dm -0.5 0.5

26. Linear stability analysis x Transcription-translation model Eigenvectors and eigenvalues Nullclines and critical points The cribsheet of linear stability analysis f m

27. Distinct positive eigenvalues Differential equations Generalsolution Initial conditions

28. Distinct positive eigenvalues Differential equations Generalsolution Initial conditions

29. Distinct positive eigenvalues Differential equations Generalsolution Initial conditions Node

30. Distinct negative eigenvalues Differential equations Generalsolution Initial conditions Node Node

31. Eigenvalues of opposite signs Differential equations Generalsolution Initial conditions Node Node Saddle

32. Equal eigenvalues Differential equations Generalsolution Initial conditions Node Star Degenerate node Node Star Degenerate node Saddle

33. Complex eigenvalues Differential equations Generalsolution Initial conditions Node Star Degenerate node Node Star Degenerate node Saddle

34. Complex eigenvalues: Oscillatory and spiral solutions Differential equations Generalsolution Initial conditions Rotation Scaling

35. The big cribsheet of linear stability analysis Differential equations Generalsolution Initial conditions Node Star Degenerate node Spiral Node Center Star Degenerate node Spiral Saddle