6 Stochasticity, Viability, Chaos Flashcards
What does a 95% confidence mean?
There is only a 5% chance that the range +- excludes the mean of the population
What is stochasticity?
Subset of randomness
Random variation of processes over time and space, of characteristics what can be described by theories of probability
What are the two origins of stochasticity?
Extrinsic noise and intrinsic noise
Give an example of extrinsic noise
Fluctuations that affect all gene expressions in the cell
Give an example of intrinsic noise
Fluctuation in transcription/translation rate of individual genes in identical cellular systems
Describe what extrinsic fluctuations look like on a fluorescence graph
Only extrinsic fluctuations cause a linear line
If one signal increases, so does the other = coupled
Describe what intrinsic fluctuations look like on a fluorescence graph
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
How do biological systems deal with stochasticity?
1 Noise filtering by damped negative feedback
2 Spatiotemporal averaging as buffering mechanism for robustness
Define robustness
The system is robust when it has the ability to reproduce the same outcome despite perturbations
Robust systems can resist perturbation
What effect does damped negative feedback have?
Negative feedback with delay can filter noise and achieve homeostasis = robust system
What is temporal averaging and give an example?
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
What is spatial averaging and give an example?***
Explain how spatiotemporal averaging in organ shape robustenes occurs
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.
What causes less robustness in sepal shape?
Local synchronization of stiffness
Are noises all bad?
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
Why are trajectories allowed to cross in the Lorenz system
Because it sis a 3D phase portrait
A 2D phase portrait would be a figure 8
What is the Lorenz attractor?
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
Give example of one dimensional
Logistic model
x dot = rx (1-x)
Fixed points & bifucations
Give example of two dimensional
Oscillators
x dot = -ry
y dot = sx
Attractors & closed cycle
Give example of three dimensional
Chaos: Lorenz equation
3 equations
Strang attractors = never repeats itself
Define chaos
Deterministic = NOT random
Very sensitive to tiny differences in initial conditions = making prediction extremely difficult
Give physical examples of chaos
Chaotic waterwheel that changes directions
Double-rod pendulum
Weather forecast
These will never follow the same trajectory unless initial conditions are identical
What is the logistic model equation and graph?
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
Why can we no longer use differential equations for animals that breed annually, what do we use instead?
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
What is the difference equation?
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
What procedure is used to graph DIFFERENCE equations?
Cobwebbing ***
What happens when 0<r<1?***
[x_n+1 = rx_n (1 - x_n)]
If reproduction rate is so low then the next point will be lower than diagonal line???
The final population crashes = stable fixed point?
What happens when 1<r<2?
[x_n+1 = rx_n (1 - x_n)]
Higher reproduction rate = approaches steady state, smaller than K
Fluctuates before finally hitting steady state
What happens when 2<r<3?
[x_n+1 = rx_n (1 - x_n)]
Fluctuations before reaching steady state
What happens when 3<r<3.44949?***
[x_n+1 = rx_n (1 - x_n)]
Population oscillates during different breeding seasons
This causes bifurcation = jumping from high to low
What happens when r goes beyond ~3.56995?
Chaos ensues
