3 Negative feedback Flashcards
What model describes the predator-prey relationship?
Lotka-VOlterra Model
What are the ODEs for prey A and predator B?
A dot = + aA -bAB
Reproduction increases A
Predation decreases A
B dot = -cB + dAB
Death decreases B
Successful hunt promotes B reproduction
What assumptions does Lotka-Volterra model make?
Natural death is not shown
Lynx don’t only hunt hare
Assumes constant reproduction rate of hare
What is feedback?
When the system outputs loop back onto the input
What does feedback allow?
Feedback allows self-regulation
Give 3 examples of negative feedback
Pendulum = relationship between velocity and angular acceleration
Negative feedback gene circuit
Lotka-Volterra Model
What are the ODE for normal linear negative feedback loop?
A dot = -rB
Bdot = +sA
With these equations we can draw the phase portrait
What are the nullclines of the simplest negative feedback?
When A dot=0 is B=0
On B=0 direction of B dot ONLY depends on sA
Describe the nullclines of a simple negative feedback system?
There seems to be unidirectional rotation
What is the exact shape of the trajectories of simple negative feedback on the phase plane?
Closed cycles and oscillations (time dynamics are waves in this case)
Think of rotating disc with the spiral elongating from it
What determines if the closed orbits are stable?***
Stable fixed point = all trajectories point to one point
Are linear oscillations structurally stable?
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
What does the amplitude of the wave correlate to on the phase portrait?
The trajectory
The amplitude is set entirely by the initial condition = structural unstable
How is structural stability achieved?
Dampening leads to inward spiral = centre is an attractor
Non-linear systems may generate limit cycle
What shape does negative feedback with dampening make, and why?
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
What does the time-dynamic graph look like of negative feedback?
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
What is the difference between under-damping and over-damping oscillators?
Under-damped = slow degradation
Over-damped = fast degradation
(like pendulum swinging in honey, will hit the steady state almost immediately)
Explain the vector composition for oscillation of the 3 decay rates (negative decay rates)
Closed cycle phase portrait = sustained oscillation no decay
Spiral, underdampened = gradually decay towards centre
Spiral, over-dampened = quickly decays towards centre
What happens if the decay rate is not negative but positive?***
How is a stable limit cycle acheived?
Delay, noise, non-linearity, positive feedback & more complex circuit etc
Describe the limit cycle
Centre = repeller (unstable fixed point)
Attractant = not just one point but a whole trajectory?***
What is the vector composition for oscillation of limit cycle?
Delay adds an outwards vector that balances off the decay = forming sustained oscillation
