Systems Biology: Mathematics for Biologists Kirsten ten Tusscher, - PowerPoint PPT Presentation

Systems Biology: Mathematics for Biologists Kirsten ten Tusscher, Theoretical Biology, UU

Chapter 3 Equilibrium types in 2D systems

Equilibrium types 1D systems : Two regions, left and right of an equilibrium. Arrows can point away or toward the equilibrium. So two equilibrium types possible: stable and unstable. At bifurcation point special case: stable and unstable sides.

Equilibrium types 1D systems : Two regions, left and right of an equilibrium. Arrows can point away or toward the equilibrium. So two equilibrium types possible: stable and unstable. At bifurcation point special case: stable and unstable sides. 2D systems : Four regions around an equilibrium point. Arrows can point away or toward the equilibrium, or both ! Six different equilibrium types possible, two of which are stable.

Equilibrium types: stable node � dx dt = − 2 x + y dy dt = x − 2 y y y x x

Equilibrium types: stable node Null-clines: � dx dt = − 2 x + y y = 2 x dy y = 1 2 x dt = x − 2 y y y x x

Equilibrium types: stable node fill in point ( 1 , 0 ) : Null-clines: � dx dx dt = − 2 x + y y = 2 x dt = − 2 ∗ 1 + 0 = − 2 < 0 so ← dy y = 1 dy 2 x dt = x − 2 y dt = 1 − 0 = 1 > 0 so ↑ y y x x

Equilibrium types: stable node fill in point ( 1 , 0 ) : Null-clines: � dx dx dt = − 2 x + y y = 2 x dt = − 2 ∗ 1 + 0 = − 2 < 0 so ← dy y = 1 dy 2 x dt = x − 2 y dt = 1 − 0 = 1 > 0 so ↑ y y x x Vectorfield: all arrows point to equilibrium → stable node

Equilibrium types: stable node fill in point ( 1 , 0 ) : Null-clines: � dx dx dt = − 2 x + y y = 2 x dt = − 2 ∗ 1 + 0 = − 2 < 0 so ← dy y = 1 dy 2 x dt = x − 2 y dt = 1 − 0 = 1 > 0 so ↑ y y x x Vectorfield: all arrows point to equilibrium → stable node Phase portrait gives same information as numerical solutions

Equilibrium types: stable node (2) � dx dt = − 2 x − y dy dt = − x − 2 y y y x x

Equilibrium types: stable node (2) Null-clines: � dx dt = − 2 x − y y = − 2 x dy y = − 1 2 x dt = − x − 2 y y y x x

Equilibrium types: stable node (2) fill in point ( 1 , 0 ) : Null-clines: � dx dx dt = − 2 x − y y = − 2 x dt = − 2 ∗ 1 − 0 = − 2 < 0 so ← dy y = − 1 dy 2 x dt = − x − 2 y dt = − 1 − 2 ∗ 0 = − 1 < 0 so ↓ y y x x

Equilibrium types: stable node (2) fill in point ( 1 , 0 ) : Null-clines: � dx dx dt = − 2 x − y y = − 2 x dt = − 2 ∗ 1 − 0 = − 2 < 0 so ← dy y = − 1 dy 2 x dt = − x − 2 y dt = − 1 − 2 ∗ 0 = − 1 < 0 so ↓ y y x x Vectorfield: all arrows point to equilibrium → stable node

Equilibrium types: stable node (2) fill in point ( 1 , 0 ) : Null-clines: � dx dx dt = − 2 x − y y = − 2 x dt = − 2 ∗ 1 − 0 = − 2 < 0 so ← dy y = − 1 dy 2 x dt = − x − 2 y dt = − 1 − 2 ∗ 0 = − 1 < 0 so ↓ y y x x Vectorfield: all arrows point to equilibrium → stable node Compare : different nullclines, similar vectorfield!

Equilibrium types: unstable node � dx dt = 2 x + y dy dt = x + 2 y y y x x

Equilibrium types: unstable node Null-clines: � dx dt = 2 x + y y = − 2 x dy y = − 1 2 x dt = x + 2 y y y x x

Equilibrium types: unstable node fill in ( 1 , 0 ) : Null-clines: � dx dx dt = 2 x + y y = − 2 x dt = 2 ∗ 1 + 0 = 2 > 0 so → dy y = − 1 dy 2 x dt = x + 2 y dt = 1 + 2 ∗ 0 = 1 > 0 so ↑ y y x x

Equilibrium types: unstable node fill in ( 1 , 0 ) : Null-clines: � dx dx dt = 2 x + y y = − 2 x dt = 2 ∗ 1 + 0 = 2 > 0 so → dy y = − 1 dy 2 x dt = x + 2 y dt = 1 + 2 ∗ 0 = 1 > 0 so ↑ y y x x Vectorfield: all arrows away from equilibrium → unstable node

Equilibrium types: unstable node fill in ( 1 , 0 ) : Null-clines: � dx dx dt = 2 x + y y = − 2 x dt = 2 ∗ 1 + 0 = 2 > 0 so → dy y = − 1 dy 2 x dt = x + 2 y dt = 1 + 2 ∗ 0 = 1 > 0 so ↑ y y x x Vectorfield: all arrows away from equilibrium → unstable node Compare : same nullclines, very different vectorfield

Equilibrium types: saddle point � dx dt = − x − 2 y dy dt = − 2 x − y y y x x

Equilibrium types: saddle point Null-clines: � dx dt = − x − 2 y y = − 1 2 x dy dt = − 2 x − y y = − 2 x y y x x

Equilibrium types: saddle point fill in ( 1 , 0 ) : Null-clines: � dx dx dt = − x − 2 y y = − 1 dt = − 1 − 2 ∗ 0 = − 1 < 0 so ← 2 x dy dy dt = − 2 x − y y = − 2 x dt = − 2 ∗ 1 − 0 = − 2 < 0 s0 ↓ y y x x

Equilibrium types: saddle point fill in ( 1 , 0 ) : Null-clines: � dx dx dt = − x − 2 y y = − 1 dt = − 1 − 2 ∗ 0 = − 1 < 0 so ← 2 x dy dy dt = − 2 x − y y = − 2 x dt = − 2 ∗ 1 − 0 = − 2 < 0 s0 ↓ y y x x Vectorfield: one vector-pair points towards, one points away from equilibrium: stable and unstable direction → saddle point

Equilibrium types: stable spiral � dx dt = − x + 2 y dy dt = − 2 x − y x , y y y x t x

Equilibrium types: stable spiral Null-clines: � dx y = 1 dt = − x + 2 y 2 x dy dt = − 2 x − y y = − 2 x x , y y y x t x

Equilibrium types: stable spiral fill in ( 1 , 0 ) : Null-clines: � dx dx y = 1 dt = − 1 + 2 ∗ 0 = − 1 < 0 so ← dt = − x + 2 y 2 x dy dy dt = − 2 x − y y = − 2 x dt = − 2 ∗ 1 − 0 = − 2 < 0 so ↓ x , y y y x t x

Equilibrium types: stable spiral fill in ( 1 , 0 ) : Null-clines: � dx dx y = 1 dt = − 1 + 2 ∗ 0 = − 1 < 0 so ← dt = − x + 2 y 2 x dy dy dt = − 2 x − y y = − 2 x dt = − 2 ∗ 1 − 0 = − 2 < 0 so ↓ x , y y y x t x Inward spiraling motion towards equilibrium Oscillations with decreasing amplitude

Equilibrium types: stable spiral fill in ( 1 , 0 ) : Null-clines: � dx dx y = 1 dt = − 1 + 2 ∗ 0 = − 1 < 0 so ← dt = − x + 2 y 2 x dy dy dt = − 2 x − y y = − 2 x dt = − 2 ∗ 1 − 0 = − 2 < 0 so ↓ x , y y y x t x Inward spiraling motion towards equilibrium Oscillations with decreasing amplitude Vectorfield: arrows only suggest rotation!

Equilibrium types: stable spiral fill in ( 1 , 0 ) : Null-clines: � dx dx y = 1 dt = − 1 + 2 ∗ 0 = − 1 < 0 so ← dt = − x + 2 y 2 x dy dy dt = − 2 x − y y = − 2 x dt = − 2 ∗ 1 − 0 = − 2 < 0 so ↓ x , y y y x t x Inward spiraling motion towards equilibrium Oscillations with decreasing amplitude Vectorfield: arrows only suggest rotation! Phase portrait gives less information than numerical solutions...

Equilibrium types: unstable spiral � dx dt = x + 2 y dy dt = − 2 x + y x , y y y x t x

Equilibrium types: unstable spiral Null-clines: � dx dt = x + 2 y y = − 1 2 x dy dt = − 2 x + y y = 2 x x , y y y x t x

Equilibrium types: unstable spiral fill in ( 1 , 0 ) : Null-clines: � dx dx dt = x + 2 y y = − 1 dt = 1 + 2 ∗ 0 = 1 > 0 so → 2 x dy dy dt = − 2 x + y y = 2 x dt = − 2 ∗ 1 + 0 = − 2 < 0 so ↓ x , y y y x t x

Equilibrium types: unstable spiral fill in ( 1 , 0 ) : Null-clines: � dx dx dt = x + 2 y y = − 1 dt = 1 + 2 ∗ 0 = 1 > 0 so → 2 x dy dy dt = − 2 x + y y = 2 x dt = − 2 ∗ 1 + 0 = − 2 < 0 so ↓ x , y y y x t x Outward spiraling motion away from equilibrium Oscillations with increasing amplitude

Equilibrium types: unstable spiral fill in ( 1 , 0 ) : Null-clines: � dx dx dt = x + 2 y y = − 1 dt = 1 + 2 ∗ 0 = 1 > 0 so → 2 x dy dy dt = − 2 x + y y = 2 x dt = − 2 ∗ 1 + 0 = − 2 < 0 so ↓ x , y y y x t x Outward spiraling motion away from equilibrium Oscillations with increasing amplitude Vectorfield: arrows again only suggest rotation!

Equilibrium types: center point � dx dt = x + 2 y dy dt = − 2 x − y x , y y y x t x

Equilibrium types: center point Null-clines: � dx dt = x + 2 y y = − 1 2 x dy dt = − 2 x − y y = − 2 x x , y y y x t x

Equilibrium types: center point fill in ( 1 , 0 ) : Null-clines: � dx dx dt = x + 2 y y = − 1 dt = 1 + 2 ∗ 0 = 1 > 0 so → 2 x dy dy dt = − 2 x − y y = − 2 x dt = − 2 ∗ 1 − 0 = − 2 < 0 so ↓ x , y y y x t x

Equilibrium types: center point fill in ( 1 , 0 ) : Null-clines: � dx dx dt = x + 2 y y = − 1 dt = 1 + 2 ∗ 0 = 1 > 0 so → 2 x dy dy dt = − 2 x − y y = − 2 x dt = − 2 ∗ 1 − 0 = − 2 < 0 so ↓ x , y y y x t x Rotation around equilibrium at constant distance Oscillations amplitude determined by initial conditions

Equilibrium types: center point fill in ( 1 , 0 ) : Null-clines: � dx dx dt = x + 2 y y = − 1 dt = 1 + 2 ∗ 0 = 1 > 0 so → 2 x dy dy dt = − 2 x − y y = − 2 x dt = − 2 ∗ 1 − 0 = − 2 < 0 so ↓ x , y y y x t x Rotation around equilibrium at constant distance Oscillations amplitude determined by initial conditions Vectorfield: arrows again only suggest rotation!

Vectorfield insufficient Sometimes the vectorfield does not give enough information:

Vectorfield insufficient Sometimes the vectorfield does not give enough information:

Vectorfield insufficient Sometimes the vectorfield does not give enough information: