3 Negative feedback

Question 1

What model describes the predator-prey relationship?

Answer 1

Lotka-VOlterra Model

Question 2

What are the ODEs for prey A and predator B?

Answer 2

A dot = + aA -bAB
Reproduction increases A
Predation decreases A

B dot = -cB + dAB
Death decreases B
Successful hunt promotes B reproduction

Question 3

What assumptions does Lotka-Volterra model make?

Answer 3

Natural death is not shown
Lynx don’t only hunt hare
Assumes constant reproduction rate of hare

Question 4

What is feedback?

Answer 4

When the system outputs loop back onto the input

Question 5

What does feedback allow?

Answer 5

Feedback allows self-regulation

Question 6

Give 3 examples of negative feedback

Answer 6

Pendulum = relationship between velocity and angular acceleration
Negative feedback gene circuit
Lotka-Volterra Model

Question 7

What are the ODE for normal linear negative feedback loop?

Answer 7

A dot = -rB
Bdot = +sA

With these equations we can draw the phase portrait

Question 8

What are the nullclines of the simplest negative feedback?

Answer 8

When A dot=0 is B=0
On B=0 direction of B dot ONLY depends on sA

Question 9

Describe the nullclines of a simple negative feedback system?

Answer 9

There seems to be unidirectional rotation

Question 10

What is the exact shape of the trajectories of simple negative feedback on the phase plane?

Answer 10

Closed cycles and oscillations (time dynamics are waves in this case)

Think of rotating disc with the spiral elongating from it

Question 11

What determines if the closed orbits are stable?***

Answer 11

Stable fixed point = all trajectories point to one point

Question 12

Are linear oscillations structurally stable?

Answer 12

No, the origin has a 0 vector and is stable fixed point

But the orbits are not stable, given a small perturbation (A0 + m) the oscillation will pushed onto a new, closed trajectory
It will not move back to previous trajectory

The amplitude is set entirely by the initial condition = structural unstable

Question 13

What does the amplitude of the wave correlate to on the phase portrait?

Answer 13

The trajectory

The amplitude is set entirely by the initial condition = structural unstable

Question 14

How is structural stability achieved?

Answer 14

Dampening leads to inward spiral = centre is an attractor
Non-linear systems may generate limit cycle

Question 15

What shape does negative feedback with dampening make, and why?

Answer 15

Spiral
Example: Gene A is an activator of B
Gene B is a repressor of Gene A

If both gene products undergo slow degradation
A dot = -rB -mA
B dot = sA - mB

Trajectories spiral into origin = attractor (stable fixed point)
Expression of the two genes will reach homeostasis

Question 16

What does the time-dynamic graph look like of negative feedback?

Answer 16

Oscillation is not maintained = system converges to a steady state (if infinite time will always converge to this point so everything is stable)

Expression of the two genes will reach homeostasis

Question 17

What is the difference between under-damping and over-damping oscillators?

Answer 17

Under-damped = slow degradation

Over-damped = fast degradation
(like pendulum swinging in honey, will hit the steady state almost immediately)

Question 18

Explain the vector composition for oscillation of the 3 decay rates (negative decay rates)

Answer 18

Closed cycle phase portrait = sustained oscillation no decay

Spiral, underdampened = gradually decay towards centre

Spiral, over-dampened = quickly decays towards centre

Question 19

What happens if the decay rate is not negative but positive?***

Question 20

How is a stable limit cycle acheived?

Answer 20

Delay, noise, non-linearity, positive feedback & more complex circuit etc

Question 21

Describe the limit cycle

Answer 21

Centre = repeller (unstable fixed point)

Attractant = not just one point but a whole trajectory?***

Question 22

What is the vector composition for oscillation of limit cycle?

Answer 22

Delay adds an outwards vector that balances off the decay = forming sustained oscillation