1. What we’ve learned so far…
Previously in this course, we analyzed the frequency of a genetic variant in a deterministic population (deterministic means not-at-all random).
In the absence of new mutations, we showed that the frequency (x) of a genetic variant with fitness advantage s is given by the Logistic equation: x t
For example, s > 0: selective sweep

2. Intro to stochastic pop-gen
The deterministic approach was important for building intuition, but it’s unrealistic:
in real life, “accidents” happen: # offspring has a stochastic (i.e. random) component that has nothing to do with fitness.
ex: freak environmental catastrophe
ex: discretization of gene frequency in finite populations
As we will see, fixation of beneficial mutant is certain in deterministic model, but far-from-certain in stochastic model.
In stochastic model, neutral (s=0) or even deleterious (s<0) mutations can take over population (i.e. fix, i.e. achieve fixation)
moral : stochasticity is hugely important! It isn’t just “noise on top” of what we learned before– rather, it qualitatively alters the general picture.
log(n) t
exponential growth: average outcome
example you’ve already seen: “cell” with b>d. Branching process.
extinction: what usually happens

3. From x to P(x)
As in the deterministic approach, we will focus on the frequency (x) of a genetic variant. Sorry about the change in notation  (previously, p was frequency).
But, x is now a random variable– we can’t predict the exact value it will take, but we can predict its tendencies.
The precise description of “tendencies” is the probability distribution P(x). Examples:
x tends to be one of two extremes (bi-modal)
x tends to be some characteristic value x
P(x) x
P(x) x
P(x)
x has no tendency– it defies prediction

4. Ways of summarizing P(x)
P(x) is the nuanced answer to the question “how big is x?” But, as in life, sometimes we want to “cut to the chase” and avoid nuance.
the mean simplifies P(x) to just a number:
variance tells you how faithfully the mean represents P(x):
Less trivial: the so-called heterozygosity measures diversity within a population. It tells you the average probability that two randomly chosen individuals are different. <H>=average heterozygosity.
Have them write out the corresponding formulas with sums
Note: I’m using < > to mean average over realizations (not average across popultion)

5. Clicker questions: A B C x P(x) x P(x) x P(x)
Assume that these distributions all have the same mean.
Rank order their variances and heterozygosities.
For which of these is the mean an accurate representation of the whole distribution?
Can P(x) ever be greater than 1? A B C
V: c>b>a
H: a>b>c x
P(x) x
P(x) x
P(x)

6. What does “average probability” mean?
In the last slide we italicized average probability because it’s a tricky concept. Even the language we used was a little weird:
usually, a random variable (x) is described by a probabilities
usually, x is the thing that has an average. We use probabilities to compute the average. Probabilities themselves usually don’t usually have an average.
The subtlety here is that there are two levels of randomness
One can randomly sample individuals within a population.
The composition/contents of a population is itself random (b/c evolution is random)
When we say “average value of x”, we’re averaging over the second kind of randomness.
The second kind of randomness only exists in the stochastic framework. In the deterministic framework, only the first sort of randomness exists.

7. Example from simulator:
a = green, A = red
A, a have same fitness (they’re selectively neutral).
x= frequency of a
at t=0, x= Run 5 replicate populations each for fixed # generations.
x=0.75
x=0.4
x=0.45
x=0.85
If you draw an individual from the population, the chance of picking a (green) randomly varies from from 40% to 85% (among these 5 sample populations).
Apart from the randomness involved in sampling a single individual, the genetic composition of the population is also random.
notes: <x>=0.75 (why?). Also, if population were deterministic, x(t)=0.75, for all t

8. Random genetic drift x=0.75
green and red are selectively neutral, yet x changes. Why?
x=0.75
x=0.4
x=0.45
x=0.85
If selection isn’t driving these changes in x, what is? The answer is “random genetic drift,” and it has to do with the size of the population (N). We’ll devote a lot of time to this subtle concept.