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

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Systems Biology: Mathematics for Biologists

Kirsten ten Tusscher, Theoretical Biology, UU

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Chapter 3

Equilibrium types in 2D systems

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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.

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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.

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Equilibrium types: stable node

dx

dt = −2x + y dy dt = x − 2y

x y x y

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Equilibrium types: stable node

dx

dt = −2x + y dy dt = x − 2y

Null-clines: y = 2x y = 1

2x

x y x y

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Equilibrium types: stable node

dx

dt = −2x + y dy dt = x − 2y

Null-clines: y = 2x y = 1

2x

fill in point (1, 0):

dx dt = −2 ∗ 1 + 0 = −2 < 0 so ← dy dt = 1 − 0 = 1 > 0 so ↑

x y x y

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Equilibrium types: stable node

dx

dt = −2x + y dy dt = x − 2y

Null-clines: y = 2x y = 1

2x

fill in point (1, 0):

dx dt = −2 ∗ 1 + 0 = −2 < 0 so ← dy dt = 1 − 0 = 1 > 0 so ↑

x y x y Vectorfield: all arrows point to equilibrium → stable node

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Equilibrium types: stable node

dx

dt = −2x + y dy dt = x − 2y

Null-clines: y = 2x y = 1

2x

fill in point (1, 0):

dx dt = −2 ∗ 1 + 0 = −2 < 0 so ← dy dt = 1 − 0 = 1 > 0 so ↑

x y x y Vectorfield: all arrows point to equilibrium → stable node Phase portrait gives same information as numerical solutions

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Equilibrium types: stable node (2)

dx

dt = −2x − y dy dt = −x − 2y

x y x y

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Equilibrium types: stable node (2)

dx

dt = −2x − y dy dt = −x − 2y

Null-clines: y = −2x y = −1

2x

x y x y

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Equilibrium types: stable node (2)

dx

dt = −2x − y dy dt = −x − 2y

Null-clines: y = −2x y = −1

2x

fill in point (1, 0):

dx dt = −2 ∗ 1 − 0 = −2 < 0 so ← dy dt = −1 − 2 ∗ 0 = −1 < 0 so ↓

x y x y

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Equilibrium types: stable node (2)

dx

dt = −2x − y dy dt = −x − 2y

Null-clines: y = −2x y = −1

2x

fill in point (1, 0):

dx dt = −2 ∗ 1 − 0 = −2 < 0 so ← dy dt = −1 − 2 ∗ 0 = −1 < 0 so ↓

x y x y Vectorfield: all arrows point to equilibrium → stable node

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Equilibrium types: stable node (2)

dx

dt = −2x − y dy dt = −x − 2y

Null-clines: y = −2x y = −1

2x

fill in point (1, 0):

dx dt = −2 ∗ 1 − 0 = −2 < 0 so ← dy dt = −1 − 2 ∗ 0 = −1 < 0 so ↓

x y x y Vectorfield: all arrows point to equilibrium → stable node Compare: different nullclines, similar vectorfield!

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Equilibrium types: unstable node

dx

dt = 2x + y dy dt = x + 2y

x y x y

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Equilibrium types: unstable node

dx

dt = 2x + y dy dt = x + 2y

Null-clines: y = −2x y = −1

2x

x y x y

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Equilibrium types: unstable node

dx

dt = 2x + y dy dt = x + 2y

Null-clines: y = −2x y = −1

2x

fill in (1, 0):

dx dt = 2 ∗ 1 + 0 = 2 > 0 so → dy dt = 1 + 2 ∗ 0 = 1 > 0 so ↑

x y x y

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Equilibrium types: unstable node

dx

dt = 2x + y dy dt = x + 2y

Null-clines: y = −2x y = −1

2x

fill in (1, 0):

dx dt = 2 ∗ 1 + 0 = 2 > 0 so → dy dt = 1 + 2 ∗ 0 = 1 > 0 so ↑

x y x y Vectorfield: all arrows away from equilibrium → unstable node

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Equilibrium types: unstable node

dx

dt = 2x + y dy dt = x + 2y

Null-clines: y = −2x y = −1

2x

fill in (1, 0):

dx dt = 2 ∗ 1 + 0 = 2 > 0 so → dy dt = 1 + 2 ∗ 0 = 1 > 0 so ↑

x y x y Vectorfield: all arrows away from equilibrium → unstable node Compare: same nullclines, very different vectorfield

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Equilibrium types: saddle point

dx

dt = −x − 2y dy dt = −2x − y

x y x y

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Equilibrium types: saddle point

dx

dt = −x − 2y dy dt = −2x − y

Null-clines: y = −1

2x

y = −2x x y x y

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Equilibrium types: saddle point

dx

dt = −x − 2y dy dt = −2x − y

Null-clines: y = −1

2x

y = −2x fill in (1, 0):

dx dt = −1 − 2 ∗ 0 = −1 < 0 so ← dy dt = −2 ∗ 1 − 0 = −2 < 0 s0 ↓

x y x y

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Equilibrium types: saddle point

dx

dt = −x − 2y dy dt = −2x − y

Null-clines: y = −1

2x

y = −2x fill in (1, 0):

dx dt = −1 − 2 ∗ 0 = −1 < 0 so ← dy dt = −2 ∗ 1 − 0 = −2 < 0 s0 ↓

x y x y Vectorfield:

ne vector-pair points towards, one points away from equilibrium:

stable and unstable direction → saddle point

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Equilibrium types: stable spiral

dx

dt = −x + 2y dy dt = −2x − y

x y t x, y x y

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Equilibrium types: stable spiral

dx

dt = −x + 2y dy dt = −2x − y

Null-clines: y = 1

2x

y = −2x x y t x, y x y

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Equilibrium types: stable spiral

dx

dt = −x + 2y dy dt = −2x − y

Null-clines: y = 1

2x

y = −2x fill in (1, 0):

dx dt = −1 + 2 ∗ 0 = −1 < 0 so ← dy dt = −2 ∗ 1 − 0 = −2 < 0 so ↓

x y t x, y x y

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Equilibrium types: stable spiral

dx

dt = −x + 2y dy dt = −2x − y

Null-clines: y = 1

2x

y = −2x fill in (1, 0):

dx dt = −1 + 2 ∗ 0 = −1 < 0 so ← dy dt = −2 ∗ 1 − 0 = −2 < 0 so ↓

x y t x, y x y Inward spiraling motion towards equilibrium Oscillations with decreasing amplitude

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Equilibrium types: stable spiral

dx

dt = −x + 2y dy dt = −2x − y

Null-clines: y = 1

2x

y = −2x fill in (1, 0):

dx dt = −1 + 2 ∗ 0 = −1 < 0 so ← dy dt = −2 ∗ 1 − 0 = −2 < 0 so ↓

x y t x, y x y Inward spiraling motion towards equilibrium Oscillations with decreasing amplitude Vectorfield: arrows only suggest rotation!

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Equilibrium types: stable spiral

dx

dt = −x + 2y dy dt = −2x − y

Null-clines: y = 1

2x

y = −2x fill in (1, 0):

dx dt = −1 + 2 ∗ 0 = −1 < 0 so ← dy dt = −2 ∗ 1 − 0 = −2 < 0 so ↓

x y t x, y x y Inward spiraling motion towards equilibrium Oscillations with decreasing amplitude Vectorfield: arrows only suggest rotation! Phase portrait gives less information than numerical solutions...

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Equilibrium types: unstable spiral

dx

dt = x + 2y dy dt = −2x + y

x y t x, y x y

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Equilibrium types: unstable spiral

dx

dt = x + 2y dy dt = −2x + y

Null-clines: y = −1

2x

y = 2x x y t x, y x y

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Equilibrium types: unstable spiral

dx

dt = x + 2y dy dt = −2x + y

Null-clines: y = −1

2x

y = 2x fill in (1, 0):

dx dt = 1 + 2 ∗ 0 = 1 > 0 so → dy dt = −2 ∗ 1 + 0 = −2 < 0 so ↓

x y t x, y x y

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Equilibrium types: unstable spiral

dx

dt = x + 2y dy dt = −2x + y

Null-clines: y = −1

2x

y = 2x fill in (1, 0):

dx dt = 1 + 2 ∗ 0 = 1 > 0 so → dy dt = −2 ∗ 1 + 0 = −2 < 0 so ↓

x y t x, y x y Outward spiraling motion away from equilibrium Oscillations with increasing amplitude

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Equilibrium types: unstable spiral

dx

dt = x + 2y dy dt = −2x + y

Null-clines: y = −1

2x

y = 2x fill in (1, 0):

dx dt = 1 + 2 ∗ 0 = 1 > 0 so → dy dt = −2 ∗ 1 + 0 = −2 < 0 so ↓

x y t x, y x y Outward spiraling motion away from equilibrium Oscillations with increasing amplitude Vectorfield: arrows again only suggest rotation!

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Equilibrium types: center point

dx

dt = x + 2y dy dt = −2x − y

x y t x, y x y

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Equilibrium types: center point

dx

dt = x + 2y dy dt = −2x − y

Null-clines: y = −1

2x

y = −2x x y t x, y x y

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Equilibrium types: center point

dx

dt = x + 2y dy dt = −2x − y

Null-clines: y = −1

2x

y = −2x fill in (1, 0):

dx dt = 1 + 2 ∗ 0 = 1 > 0 so → dy dt = −2 ∗ 1 − 0 = −2 < 0 so ↓

x y t x, y x y

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Equilibrium types: center point

dx

dt = x + 2y dy dt = −2x − y

Null-clines: y = −1

2x

y = −2x fill in (1, 0):

dx dt = 1 + 2 ∗ 0 = 1 > 0 so → dy dt = −2 ∗ 1 − 0 = −2 < 0 so ↓

x y t x, y x y Rotation around equilibrium at constant distance Oscillations amplitude determined by initial conditions

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Equilibrium types: center point

dx

dt = x + 2y dy dt = −2x − y

Null-clines: y = −1

2x

y = −2x fill in (1, 0):

dx dt = 1 + 2 ∗ 0 = 1 > 0 so → dy dt = −2 ∗ 1 − 0 = −2 < 0 so ↓

x y t x, y x y Rotation around equilibrium at constant distance Oscillations amplitude determined by initial conditions Vectorfield: arrows again only suggest rotation!

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Vectorfield insufficient

Sometimes the vectorfield does not give enough information:

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Vectorfield insufficient

Sometimes the vectorfield does not give enough information:

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Vectorfield insufficient

Sometimes the vectorfield does not give enough information:

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Vectorfield insufficient

Sometimes the vectorfield does not give enough information:

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Vectorfield insufficient

Sometimes the vectorfield does not give enough information: All vectorfields suggest rotation, but we may even have a node!

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Self-feedback: when and how

First look at the entire vectorfield: is it clearly a stable node, unstable node, saddle? YES: you are finished! NO: look at self-feedback

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Self-feedback: when and how

First look at the entire vectorfield: is it clearly a stable node, unstable node, saddle? YES: you are finished! NO: look at self-feedback Self-feedback: Feedback of x variable on itself add a little x (small horizontal step from eq. to the right) does x increase or decreases (horizontal vector to left or right)

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Self-feedback: when and how

First look at the entire vectorfield: is it clearly a stable node, unstable node, saddle? YES: you are finished! NO: look at self-feedback Self-feedback: Feedback of x variable on itself add a little x (small horizontal step from eq. to the right) does x increase or decreases (horizontal vector to left or right) Feedback of y variable on itself add a little y (small vertical step from eq. upwards) does y increase or decreases (vertical vector up or down)

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