# Fundamentals of Evolution

## Today's topics

1. Recap (assignment, HW)

2. Wright's F-statistics

3. Theta

4. The coalescent

5. Neutral theory (debate?)

### Cis-regulatory versus structural evolution

How has the debate over the importance of these two types of evolution advanced over the last decade? How has new data (a lot of it!) contributed new insights?

### Hardy Weinberg Equilibrium

The Hardy–Weinberg principle (HW) provides the solution to one of Darwin's biggest mysteries: how variation is maintained in a population. Under Mendelian inheritance (independent assortment) the frequencies of alleles (variants at a gene) will remain constant in the absence of selection, mutation, migration, and/or genetic drift.

### Hardy Weinberg Equilibrium

The Hardy–Weinberg principle (HW) provides the solution to one of Darwin's biggest mysteries: how variation is maintained in a population. Under Mendelian inheritance (independent assortment) the frequencies of alleles (variants at a gene) will remain constant in the absence of selection, mutation, migration, and/or genetic drift.

In modeling this idealized population we make the following assumptions, and then we can also ask how allele frequencies would change if one or more of these assumptions is relaxed:

selection=0
mutation=0
migration=0 (i.e., random mating, no population structure)
Ne=infinite (infinite population size = no effect of drift)

### Mendelian segregation and recombination

In the absence of evolutionary processes causing allele frequency changes, the frequency of diploid genotypes in a population will change only according to the probability of mating (random sampling).

At any single locus the two copies can be passed on with equal probability (segregation) and so Hardy-Weinberg equilibrium can be reached within a single generation:

$p^{2} + 2pq + q^{2} = 1$.

### Hardy Weinberg Equilibrium

The three genotypes for a locus with two alleles will occur at: $p^{2} + 2pq + q^{2} = 1$.

### Wright-Fisher Model

By incorporating a finite effective population size (N) into our sampling probabilities we can estimate the expected change in allele frequencies over time (generations).

A discrete time model in which each generation is composed of 2N copies of each gene. Each subsequent generation 2N new copies are randomly drawn from the previous generation. The probability of obtaining k copies the next generation is:

$${{2N \choose k}p^{k}(1-p)^{2N-k}}$$

https://en.wikipedia.org/wiki/Genetic_drift#Wright.E2.80.93Fisher_model

### Wright-Fisher Model

A neutral evolutionary process (no selection) can be modeled using the WF model in which allele frequencies change over time by genetic drift.

Source: Alexei Drummond

### Wright-Fisher Model

In addition to estimating the effects of genetic drift on allele frequencies in a single population, Sewall Wright developed a series of estimators for predicting how differences between populations could evolve by genetic drift. Wright's F-statistics (fixation indices) describe these probabilities in terms of sampling probabilities (i.e., bean-bag genetics).

A central measurement of F-statistics is the inbreeding coefficient (F).

$$F = 1 - \frac{Observed(Aa)}{Expected(Aa)}$$

### Wright's F-statistics

Under HW the genotype frequencies in a populations are:

$$AA = p^{2}$$ $$Aa = 2pq$$ $$aa = q^{2}$$
In the presence of inbreeding (non-random mating) these deviate from HW as:

$$AA = p^{2}(1-F) + pF$$ $$Aa = 2pq(1 - F)$$ $$aa = q^{2}(1-F) + qF$$

### Wright's F-statistics

These statistics become interesting when we start to define what we mean by a population. For example, comparing sampled individuals from two locations.

Different F-statistics look at different levels of population structure. $F_{IT}$ is the inbreeding coefficient ($F$) of an individual relative to the total population. $F_{IS}$ is the inbreeding coefficient of an individual relative to a subpopulation ($S$), and $F_{ST}$ is the effect of subpopulations relative to total.

### Wright's F-statistics

$F_{ST}$ can be interpreted as a measure of population differentiation. If $F_{ST}$ between two populations is 1 then they are fixed for different alleles (no variation is segregating in both populations).

### Wright's F-statistics

For example, large populations among which there is much migration tend to show little differentiation, whereas small populations among which there is little migration tend to be highly differentiated. FST is a convenient measure of this differentiation, and as a result FST and related statistics are among the most widely used descriptive statistics in population and evolutionary genetics.

### $F_{ST}$ applied

The most widely used of these statistics is $F_{ST}$, as we are often interested in the effect of population subdivision, and in comparing genetic diversity of subpopulations. There are several definitions of $F_{ST}$:

$$F_{ST} = \frac{\pi_{between}-\pi_{within}}{\pi_{between}}$$
Or, the variance in p across populations over the max variance that could exist among subpopulations.)
$$F_{ST} = \frac{var(p)}{p(1-p)}$$

### $F_{ST}$ applied

The most widely used estimator of $F_{ST}$ from DNA sequence data is:

$$F_{ST} = \frac{\pi_{between}-\pi_{within}}{\pi_{between}}$$
Where $\pi_{between}$ and $\pi_{within}$ are the average number of pairwise differences between two individuals sampled from different subpopulations (between) or the same subpop (within). This estimator is also biased and so we typically use another estimator, Watterson's $\Theta$, to estimate the average pairwise distance as population genetic diversity (more on this next time).

### $F_{ST}$ in humans

Genetic diversity in humans is low compared to our close relatives, and most variation is shared among human populations.

Based on three million SNPs, ~12% of the genetic variation is found to occur between continental populations, and ~1% between populations within continents.

### The neutral theory of molecular evolution

Neutral theory proposes that most genetic variation within species and genetic differences among species are a result of genetic drift acting on selectively neutral alleles.

Most new mutations are deleterious and lost, or weakly deleterious and thus effectively neutral. Beneficial mutations are rare and fix quickly and thus will not be observed as within-population variation.