6 Stochasticity, Viability, Chaos

Question 1

What does a 95% confidence mean?

Answer 1

There is only a 5% chance that the range +- excludes the mean of the population

Question 2

What is stochasticity?

Answer 2

Subset of randomness

Random variation of processes over time and space, of characteristics what can be described by theories of probability

Question 3

What are the two origins of stochasticity?

Answer 3

Extrinsic noise and intrinsic noise

Question 4

Give an example of extrinsic noise

Answer 4

Fluctuations that affect all gene expressions in the cell

Question 5

Give an example of intrinsic noise

Answer 5

Fluctuation in transcription/translation rate of individual genes in identical cellular systems

Question 6

Describe what extrinsic fluctuations look like on a fluorescence graph

Answer 6

Only extrinsic fluctuations cause a linear line
If one signal increases, so does the other = coupled

Question 7

Describe what intrinsic fluctuations look like on a fluorescence graph

Answer 7

Only intrinsic fluctuations cause normal distribution on x and y axis
So the dots look like a sphere with more diffusion when further away from the centre = UNCOUPLED signals

Question 8

How do biological systems deal with stochasticity?

Answer 8

1 Noise filtering by damped negative feedback
2 Spatiotemporal averaging as buffering mechanism for robustness

Question 9

Define robustness

Answer 9

The system is robust when it has the ability to reproduce the same outcome despite perturbations

Robust systems can resist perturbation

Question 10

What effect does damped negative feedback have?

Answer 10

Negative feedback with delay can filter noise and achieve homeostasis = robust system

Question 11

What is temporal averaging and give an example?

Answer 11

Wandering polarity in yeast

Cell scans env averaging the molecule concentration that is scans so it can correct for erros in a noise filled env

Question 12

What is spatial averaging and give an example?***

Question 13

Explain how spatiotemporal averaging in organ shape robustenes occurs

Answer 13

Sepal has robust shape and size = needs to be reproducible in size and shape, otherwise the 4 sepals will not meet around the flower

Spatial variability model = because there are different patches of stiffness they grow less causing shape distortion

But when integrating temporal variability as well = allows patches of random stiffness to average out across sepal.

Question 14

What causes less robustness in sepal shape?

Answer 14

Local synchronization of stiffness

Question 15

Are noises all bad?

Answer 15

No, robustness arises from stochasticity

Biological noise may reduce precision but also cause symmetry breaking
The system may be in perfect balance until triggered by noise

Question 16

Why are trajectories allowed to cross in the Lorenz system

Answer 16

Because it sis a 3D phase portrait
A 2D phase portrait would be a figure 8

Question 17

What is the Lorenz attractor?

Answer 17

The Lorenz attractor is a set of chaotic solutions of the Lorenz system

The implications of the Lorenz attractor = that tiny changes in initial conditions evolve to completely different trajectories

Question 18

Give example of one dimensional

Answer 18

Logistic model
x dot = rx (1-x)

Fixed points & bifucations

Question 19

Give example of two dimensional

Answer 19

Oscillators
x dot = -ry
y dot = sx

Attractors & closed cycle

Question 20

Give example of three dimensional

Answer 20

Chaos: Lorenz equation
3 equations

Strang attractors = never repeats itself

Question 21

Define chaos

Answer 21

Deterministic = NOT random
Very sensitive to tiny differences in initial conditions = making prediction extremely difficult

Question 22

Give physical examples of chaos

Answer 22

Chaotic waterwheel that changes directions
Double-rod pendulum
Weather forecast

These will never follow the same trajectory unless initial conditions are identical

Question 23

What is the logistic model equation and graph?

Answer 23

dN/dt = RN ( 1 - N/K)

This works well for fast-growing organisms = growth akin to a smooth curve converging to the carrying capacity K

Question 24

Why can we no longer use differential equations for animals that breed annually, what do we use instead?

Answer 24

We cannot use differential equations because the change is in a descrete manner (in steps)

It changes per given time interval = so instead of dN/dt use ΔN/Δt DIFFERENCE EQUATION

Δt = 1 then look at N_t-+1 - N_t

Question 25

What is the difference equation?

Answer 25

x_n+1 = rx_n (1 - x_n)

The population of the next time point x_t+1 depends on current time point x_t

Question 26

What procedure is used to graph DIFFERENCE equations?

Answer 26

Cobwebbing ***

Question 27

What happens when 0<r<1?***
[x_n+1 = rx_n (1 - x_n)]

Answer 27

If reproduction rate is so low then the next point will be lower than diagonal line???
The final population crashes = stable fixed point?

Question 28

What happens when 1<r<2?
[x_n+1 = rx_n (1 - x_n)]

Answer 28

Higher reproduction rate = approaches steady state, smaller than K

Fluctuates before finally hitting steady state

Question 29

What happens when 2<r<3?
[x_n+1 = rx_n (1 - x_n)]

Answer 29

Fluctuations before reaching steady state

Question 30

What happens when 3<r<3.44949?***
[x_n+1 = rx_n (1 - x_n)]

Answer 30

Population oscillates during different breeding seasons
This causes bifurcation = jumping from high to low

Question 31

What happens when r goes beyond ~3.56995?

Answer 31

Chaos ensues