Population genetics

Foundations

Per Unneberg

NBIS

15-Nov-2023

Table of contents

• DNA variation
• Models of populations
• Origin and change of variation
• Selection

Intended learning outcomes

Introduction to foundations of population genetics with an emphasis on genealogies

• Description of DNA variation data
• Wright-Fisher population model and genealogies
• Genetic drift
• Wright-Fisher model with mutation
• Mutation-drift balance
• Neutral theory
• Selection basics

Objective: introduce the foundations of population genetics, with an emphasis on genealogies. Framing population genetic concepts in trees will facilitate the introduction of the coalescent in the afternoon.

The lecture presents sequence data, the Wright-Fisher model, genetic drift, mutation, and selection. The focus is to familiarise students with concepts and basic theory.

DNA variation

Goal of section: look at the data that forms the foundation for population genomic analyses

From (Nei & Kumar, 2000, p. 231):

The main subject of population genetics is to study the generation and maintenance of genetic polymorphism and to understand the mechanisms of evolution at the population level

(Casillas & Barbadilla, 2017, p. 1026):

Big data samples of complete genome sequences of many individuals from natural populations of many species have transformed population genetics inferences on samples of loci to population genomics: the analysis of genome-wide patterns of DNA variation within and between species.

(Gillespie, 2004, p. 1)

Population geneticists spend most of their time doing one of two things: describing the genetic structure of populations or theorizing on the evolutionary forces acting on populations. On a good day, these two activities mesh and true insights emerge.

DNA variation

Sequence aligmnent of four DNA sequences (Hahn, 2019, Fig 1.1).

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
T	T	A	C	A	A	T	C	C	G	A	T	C	G	T
T	T	A	C	G	A	T	G	C	G	C	T	C	G	T
T	C	A	C	A	A	T	G	C	G	A	T	G	G	A
T	T	A	C	G	A	T	G	C	G	C	T	C	G	T

The main data for molecular population genetics are DNA sequences. The alignment above shows a sample of four DNA sequences. Each sequence has 15 nucleotides (sites) “from the same locus (location) on a chromosome” (p.2 Hahn, 2019)

Alternative names for sequence:

• chromosome
• gene

• allele (different by origin)
• sample
• cistron

We will preferentially use sequence or chromosome to refer to an entire sequence, and allele to refer to individual nucleotides that differ.

DNA variation - monomorphic sites

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
T	T	A	C	A	A	T	C	C	G	A	T	C	G	T
T	T	A	C	G	A	T	G	C	G	C	T	C	G	T
T	C	A	C	A	A	T	G	C	G	A	T	G	G	A
T	T	A	C	G	A	T	G	C	G	C	T	C	G	T
*		*	*		*	*		*	*		*		*	T

The alignment has 4 DNA sequences where each sequence has length \(L=15\). A site where all nucleotides (alleles) are identical is called a monomorphic site (indicated with asterisks above). There are 9 monomorphic sites.

DNA variation - segregating sites

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
T	T	A	C	A	A	T	C	C	G	A	T	C	G	T
T	T	A	C	G	A	T	G	C	G	C	T	C	G	T
T	C	A	C	A	A	T	G	C	G	A	T	G	G	A
T	T	A	C	G	A	T	G	C	G	C	T	C	G	T
	*			*			*			*		*		*

A site where there are different nucleotides (alleles) is called a segregating site (indicated with asterisks above), often denoted S. There are \(S=6\) segregating sites.

Alternative names for segregating site are:

• polymorphism
• mutation
• single nucleotide polymorphism (SNP)

mutation here and onwards refers to the process that generates new variation and the new variants generated by this process

In contrast to mutation which corresponds to within-species variation, a substitution refers to DNA differences between species.

DNA variation - major and minor alleles

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
T	T	A	C	A	A	T	C	C	G	A	T	C	G	T
T	T	A	C	G	A	T	G	C	G	C	T	C	G	T
T	C	A	C	A	A	T	G	C	G	A	T	G	G	A
T	T	A	C	G	A	T	G	C	G	C	T	C	G	T
	*			*			*			*		*		*

Much of the nucleotide variation we study consists of bi-allelic SNPs. The most common variant is called the major allele, and the least common the minor allele.

The set of alleles found on a single sequence is called haplotype.

Describing DNA variation - heterozygosity

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
T	T	A	C	A	A	T	C	C	G	A	T	C	G	T
T	T	A	C	G	A	T	G	C	G	C	T	C	G	T
T	C	A	C	A	A	T	G	C	G	A	T	G	G	A
T	T	A	C	G	A	T	G	C	G	C	T	C	G	T
	*			*			*			*		*		*

Once we have a sample of sequences we want to describe the observed variation. At any position the ith allele has sample frequency \(p_i\), where the sum of all allele frequencies is 1. For instance, at site 1, \(p_T=1\) (and by extension \(p_A=p_C=p_G=0\)), and at site 2 \(p_C=1/4\) and \(p_T=3/4\).

Heterozygosity

The heterozygosity at a site \(j\) is given by

\[ h_j = \frac{n}{n-1}\left(1 - \sum_i p_i^2\right) \]

where the summation is over all alleles and \(p_i\) is the frequency of the \(i\)-th allele

Exercise: calculate the heterozygosity at sites 1, 2 and 5

\[ h_1 = \frac{4}{3} \left(1 - p_T^2 \right) = 0 \\ h_2 = \frac{4}{3} \left(1 - \left(p_C^2 + p_T^2\right) \right) = \frac{4}{3} \left( 1 - \left(\frac{1}{16} + \frac{9}{16}\right)\right) = \frac{1}{2}\\ h_5 = \frac{4}{3} \left(1 - \left(p_A^2 + p_G^2\right) \right) = \frac{4}{3} \left( 1 - \left(\frac{1}{4} + \frac{1}{4}\right)\right) = \frac{2}{3} \]

In a randomly mating population, the heterozygosity is equal to the frequency of heterozygotes. Note however that the definition of heterozygosity only relies on allele frequencies, which means it can be applied to populations that are not in Hardy-Weinberg equilibrium. It can also be applied to haploid organisms, like bacteria.

Describing DNA variation - nucleotide diversity

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
T	T	A	C	A	A	T	C	C	G	A	T	C	G	T
T	T	A	C	G	A	T	G	C	G	C	T	C	G	T
T	C	A	C	A	A	T	G	C	G	A	T	G	G	A
T	T	A	C	G	A	T	G	C	G	C	T	C	G	T
	*			*			*			*		*		*

Nucleotide diversity \(\pi\)

The nucleotide diversity is the sum of site heterozygosities:

\[ \pi = \sum_{j=1}^S h_j \]

where \(S\) is the number of segregating sites

Calculate the nucleotide diversity

Observation: \(h_i\) either 1/2 or 2/3 (for sites with \(p_{major}=p_{minor}\)).

\[ \pi = \frac{1}{2} + \frac{2}{3} + \frac{1}{2} + \frac{2}{3} + \frac{1}{2} + \frac{1}{2} = 3\frac{1}{3} \]

Often we provide \(\pi\) per site:

\[ \pi = 3.33/15 = 0.222 \]

Hahn (2019) implicitly assumes we are looking at DNA polymorphism. The expression actually holds for any genetic variation at a locus, and is sometimes called the gene diversity (Nei & Kumar, 2000, p. 245).

Under the infinite sites model, \(E(\pi)=\theta=4N_e\mu\), for which reason \(\pi\) sometimes is called \(\theta_\pi\). The measure gives the average number of pairwise nucleotide differences between two sequences, so an alternative expression is

\[ \pi = \frac{\sum_{i<j}k_{ij}}{n(n-1)/2} \]

The latter expression is called the nucleotide diversity (Nei & Kumar, 2000, p. 251).

Alleles as algebraic entities

Recall: alleles refer to different variants of a sequence at a locus (genomic position).

Whatever the underlying molecular nature (gene, chromosome, nucleotide, protein), let’s represent a locus by a letter, e.g., \(A\) (\(B\) if two loci, and so on)

If locus has many alleles \(1, 2, ...\) , could use indexing \(A_1, A_2, ...\).

Will use combination \(A\), \(a\) for bi-allelic loci from now on

Example: gene coding for flower color

\(A\) red color

\(a\) white color

Punnett square

\

A

a

A

a

Genotype

aa

Aa

AA

Phenotype

Heterozygote has intermediate color phenotype (pink).

Until now the examples have been based on nucleotide sequences. However, much of population genetic theory was developed before the nature of heredity (DNA) was known. In these early days, an allele would refer to variant forms of a gene, observed as differences in phenotypes. Genes, or loci, would be denoted using alphabetic characters, such as \(A\), and allelic types could be referenced with indices, e.g., \(A_1, A_2, ..., A_n\).

To simplify calculations, we often look at one locus and we assume two alleles, whereby we skip the indices and denote the allelic pairs \(A\) and \(a\) (although note that notations differs from author to author; for instance Gillespie (2004) uses \(A_1, A_2\) for bi-allelic loci). For two-locus systems we simply denote the second allele with \(B, b\), and so on.

The example shows a hypothetical locus having two alleles \(A\) and \(a\) that have phenotypes red and white flower color, and where heterozygotes are colored pink. The Punnett square shows how gamete combinations form genotypes and their corresponding phenotypes.

Alleles and frequencies

We will be interested in looking at the dynamics of alleles, i.e., how their abundances in the population change over time. Therefore we want to measure the frequencies of alleles \(A\) and \(a\).

Example

Assume following population (\(n=10\), with \(n_{AA}=5\), \(n_{Aa}=4\), \(n_{aa}=1\)):

Let \(p\) be frequency of \(A\) alleles, \(q=1-p\) frequency of \(a\) alleles; then

5 \(AA\) individuals, 4 \(Aa\) individuals \(\Rightarrow p=\frac{5\cdot2 + 4\cdot1}{10\cdot2}=\frac{14}{20}=0.7\)

and \(q=1-p=\frac{6}{20}=0.3\)

Inserting frequencies into Punnett square gives expected frequency of offspring genotypes.

\

\(A\) (\(p=0.7\))

\(a\) (\(q=0.3\))

\(A\) (\(p=0.7\))

\(p\cdot p = 0.49\)

\(p\cdot q = 0.21\)

\(a\) (\(q=0.3\))

\(q\cdot p = 0.21\)

\(q\cdot q = 0.09\)

Expected allele frequencies after mating: \(p=p^2 + pq=0.7\), \(q=1-p=0.3\)

Given \(n\) diploid individuals, there are \(2n\) alleles in the population. The frequency of allele \(A\) is then the homozygote \(AA\) times two, plus one times the individuals carrying one \(A\).

The Punnett square shows that the expected homozygote frequencies are \(0.49\) and \(0.09\) for \(AA\) and \(aa\), and hence, the frequency of \(Aa\) is \(1 - 0.49 - 0.09 = 0.42 = 0.21 + 0.21 = 2pq\)

In absence of evolutionary forces alleles are in equilibrium

The Hardy-Weinberg equilibrium

For a locus, let \(A\) and \(a\) be two different alleles and let \(p\) be the frequency of the \(A\) allele and \(q=1-p\) the frequency of the \(a\) allele. In the absence of mutation, drift, migration, and other evolutionary processes, the equilibrium state is given by the Hardy-Weinberg equilibrium (HWE).

	\(A\) (\(p\))	\(a\) (\(q\))
\(A\) (\(p\))	\(p^2\)	\(pq\)
\(a\) (\(q\))	\(qp\)	\(q^2\)

Genotype:	\(AA\)	\(Aa\)	\(aa\)
Frequency:	\(p^2\)	\(2pq\)	\(q^2\)
	\(f_{AA}\)	\(f_{Aa}\)	\(f_{aa}\)

Under HWE assumption, neither allele nor genotype frequencies change over time.

Importantly, we can calculate allele frequencies from genotype frequencies and vice versa:

\[ p = f_{AA} + \frac{f_{Aa}}{2} = p^2 + pq\\ q = f_{aa} + \frac{f_{Aa}}{2} = q^2 + pq\\ \]

Segue: apart from describing the variation via e.g., diversity measures, we want to model how allele frequencies change in time. As (Gillespie, 2004, preface p. xi) points out, “While genotype frequencies are easily measured, their change is not”

IOW: describing variation fine, but where does it come from and how does it change?

If assumptions of HWE hold, we have no change of variation. However, we want to look at change of variation and disentangle the forces that impose change

HWE assumption gives us a way to calculate allele frequencies from genotype frequencies. How well do these assumptions hold in real data? See next slide.

Natural populations do mate randomly?

Figure 1: Hardy-Weinberg proportions in 10,000 SNPs on chromosome 22 from three populations based on 1000 genomes data. For each SNP, genotypes are given as counts (minor/heterozygote/major), converted to frequencies and plotted on the y-axis. Allele frequencies are obtained from genotype frequencies and plotted on the x-axis. Most observations follow HWE proportions. Deviations from HWE can indicate sample QC issues, or that there is population structure. Illustration inspired by cooplab (2011).

So how do the HWE assumptions hold up in real data? The figure shows three human populations from the 1000 genomes data, for which 10,000 SNPs have been selected. For each SNP, we know the genotype frequencies \(AA\), \(Aa\), \(aa\) and can therefore calculate the allele frequencies for \(A\) and \(a\) using the equations on the preceding slide (e.g., \(p=p_{AA}+p_{Aa}/2\)).

Note that 1000 genomes data have sorted genotype frequencies, so to produce nice symmetrical plots, half the genotype frequencies have been reversed (minor/het/major -> major/het/minor).

See https://stackoverflow.com/questions/26587940/ggplot2-different-legend-symbols-for-points-and-lines for legend customization.

The obsession of population genetics

Population genetics is about (Gillespie, 2004)

1. describing the genetic structure of populations
2. constructing theories on the forces that influence genetic variation

Questions to ponder:

• why does variation look the way it does?
• how is variation maintained?
• how does variation change over time (\(\Delta p\))?
• what forces shape the genetic structure of populations?

\(p=0.1\)

\(\large\rightarrow\)

\(p=0.5\)

\(\large\rightarrow\)

\(p=0.9\)

From HWE: we want to look a the creation, maintenance and loss of variation and what forces affect it

Goal: describe theory behind evolving populations. This has been an obsession for a long time, even before DNA was known. We need to get back to basics.

Models of populations

Segue: take a step back to the days before the nature of genetic inheritance was known. First focus on the population.

Goal of lecture is also to understand genetic relationships in the context of genealogies -> coalescent.

Wright-Fisher model

Model of populations that describes genealogical relationships of genes (chromosomes) in a population under the following assumptions (Hein et al., 2005):

• discrete and non-overlapping generations
• haploid individuals or two subpopulations (males and females)
• constant population size
• all individuals are equally fit
• population has no geographical or social structure
• no recombination

See Hein et al. (2004) for more assumptions. The second assumption means we can use 2N chromosome interchangeably for haploid (n=2N) and diploid (n=N) populations.

Wright-Fisher model

Model of populations that describes genealogical relationships of genes (chromosomes) in a population under the following assumptions (Hein et al., 2005):

• discrete and non-overlapping generations
• haploid individuals or two subpopulations (males and females)
• constant population size
• all individuals are equally fit
• population has no geographical or social structure
• no recombination

Algorithm

1. Setup starting population at time zero

Wright-Fisher model

Model of populations that describes genealogical relationships of genes (chromosomes) in a population under the following assumptions (Hein et al., 2005):

• discrete and non-overlapping generations
• haploid individuals or two subpopulations (males and females)
• constant population size
• all individuals are equally fit
• population has no geographical or social structure
• no recombination

Algorithm

1. Setup starting population at time zero
2. Add offspring (same size) at time one

Wright-Fisher model

Model of populations that describes genealogical relationships of genes (chromosomes) in a population under the following assumptions (Hein et al., 2005):

• discrete and non-overlapping generations
• haploid individuals or two subpopulations (males and females)
• constant population size
• all individuals are equally fit
• population has no geographical or social structure
• no recombination

Algorithm

1. Setup starting population at time zero
2. Add offspring (same size) at time one
3. Select parents to offspring at random

Wright-Fisher model

Model of populations that describes genealogical relationships of genes (chromosomes) in a population under the following assumptions (Hein et al., 2005):

• discrete and non-overlapping generations
• haploid individuals or two subpopulations (males and females)
• constant population size
• all individuals are equally fit
• population has no geographical or social structure
• no recombination

Algorithm

1. Setup starting population at time zero
2. Add offspring (same size) at time one
3. Select parents to offspring at random

Wright-Fisher model

Model of populations that describes genealogical relationships of genes (chromosomes) in a population under the following assumptions (Hein et al., 2005):

• discrete and non-overlapping generations
• haploid individuals or two subpopulations (males and females)
• constant population size
• all individuals are equally fit
• population has no geographical or social structure
• no recombination

Algorithm

1. Setup starting population at time zero
2. Add offspring (same size) at time one
3. Select parents to offspring at random

Wright-Fisher model

Model of populations that describes genealogical relationships of genes (chromosomes) in a population under the following assumptions (Hein et al., 2005):

• discrete and non-overlapping generations
• haploid individuals or two subpopulations (males and females)
• constant population size
• all individuals are equally fit
• population has no geographical or social structure
• no recombination

Algorithm

1. Setup starting population at time zero
2. Add offspring (same size) at time one
3. Select parents to offspring at random

Wright-Fisher model

Model of populations that describes genealogical relationships of genes (chromosomes) in a population under the following assumptions (Hein et al., 2005):

• discrete and non-overlapping generations
• haploid individuals or two subpopulations (males and females)
• constant population size
• all individuals are equally fit
• population has no geographical or social structure
• no recombination

Algorithm

1. Setup starting population at time zero
2. Add offspring (same size) at time one
3. Select parents to offspring at random

Wright-Fisher model

Model of populations that describes genealogical relationships of genes (chromosomes) in a population under the following assumptions (Hein et al., 2005):

• discrete and non-overlapping generations
• haploid individuals or two subpopulations (males and females)
• constant population size
• all individuals are equally fit
• population has no geographical or social structure
• no recombination

Algorithm

1. Setup starting population at time zero
2. Add offspring (same size) at time one
3. Select parents to offspring at random

Wright-Fisher model

Model of populations that describes genealogical relationships of genes (chromosomes) in a population under the following assumptions (Hein et al., 2005):

• discrete and non-overlapping generations
• haploid individuals or two subpopulations (males and females)
• constant population size
• all individuals are equally fit
• population has no geographical or social structure
• no recombination

Algorithm

1. Setup starting population at time zero
2. Add offspring (same size) at time one
3. Select parents to offspring at random

Wright-Fisher model

Figure 2: Wright-Fisher model

See https://mikeyharper.uk/animated-plots-with-r/

Wright-Fisher model

Figure 3: WF model indicating time direction from past (top) to present (bottom).

Figure 4: WF model tracing the genealogies of three extant chromosomes

The Wright-Fisher sampling model

Let’s formalise the sampling process of the Wright-Fisher model¹. We assume

1. a single locus in a haploid population of size \(2N\) (or diploid of size \(N\) when random mating)
2. no mutation and selection
3. discrete generations

Each generation we sample \(2N\) new chromosomes from the previous generation. The probability of choosing a chromosome \(v\) is \(1/2N\) (coin flip with probability of success \(1/2N\)). Since the trials are independent, and we perform \(2N\) trials, the number of offspring \(k\) of a given chromosome \(v\) is binomially distributed \(\mathrm{Bin}(m, p)\), with parameters \(m=2N\) and probability of success \(p=\frac{1}{2N}\).

We assume some familiarity with Binomial sampling. For reference, some mathematical formality follows.

Let \(p\) be the probability of heads (=1) of a coin flip, and \(q=1-p\) the probability of tails (=0). A coin flip can be modelled by a random variable (r.v.) \(X\) that follows a Bernoulli distribution, where \(\mathrm{Pr(}X=1\mathrm{)}=p\), \(\mathrm{Pr(}X=0\mathrm{)}=1-p=q\). One can show (e.g., with transforms such as the probability-generating function) that a binomially distributed variable \(Y\sim Bin(n, p)\) is the sum of \(n\) independent Bernoulli variables \(X\) (“Bernoulli trials”), or \(Y=\sum^nX\). In our case, there are \(2N\) trials, each with \(p=1/2N\).

\[ P(v=k) = {2N\choose k}\left( \frac{1}{2N} \right)^k \left(1 - \frac{1}{2N} \right)^{(2N - k)} \]

1. For a more extensive treatment, see Hermisson (2017) or Hein et al. (2005)

Properties of Wright-Fisher sampling

The expected number of offspring is one

Poisson approximation for large \(N\)

\[ P(v=k) \approx \frac{1}{k!}e^{-k} \]

Prob(pick same parent) = 1/2N

Time for two sequences to coalesce \(\sim 1/2N\)

Many of the results above follow from standard results in probability theory and are provided here in brevity for reference to the interested reader. The details are not necessary to understand in detail.

Expected number of offspring

The expected value of a \(\mathrm{Bin}(m,p)\) (binomially distributed) variable with parameters \(m=2N\) and \(p=\frac{1}{2N}\) is \(mp\), hence:

\[ E(v) = mp = 2N\frac{1}{2N} = 1 \]

Poisson approximation

When \(2N\) large it holds that the probability that sequence \(v\) has \(k\) offspring is

\[ P(v=k) \approx \frac{1}{k!}e^{-k} \]

Since \(P(v=0) = e^{-1} \approx 0.37\), a fraction 0.37 of sequences lack descendants

Probability that two sequences pick same parent

Let \(u\) and \(v\) be two sequences. Pick a parent \(p_u\) of \(u\). Then the probability that \(v\) picks the same parent is (solid lines)

\[ P(p_u=p_v) = \frac{1}{2N} \]

The probability that they pick different parents is

\[ P(p_u \neq p_v) = 1 - \frac{1}{2N} \]

Time for two sequences to coalescent

Time for two sequences \(u\) and \(v\) to find common ancestor distributed as \((1 - \frac{1}{2N})^{j-1}\frac{1}{2N}\) (\(j-1\) failures followed by success). This is the geometric distribution \(Ge(p)\), with parameter \(p=\frac{1}{2N}\), and expected value \(\frac{1}{p}={2N}\). That is, the expected number of generations for two sequences to find a common ancestor (i.e., coalesce) is \(2N\) generations.

Since a large fraction of genes lack descendants, very quickly (compared to 2N) a population will descend from a small proportion of genes.

Derivation of average time to having same parent (i.e., coalescence) requires knowledge of geometric distribution. The probability that the genes find common parent \(j\) generations ago is

\[ (1-\frac{1}{2N})^{(j-1)}\frac{1}{2N} \]

due to independence between generations. This is the geometric distribution \(Ge(p)\) with \(p=1/2N\), which has expected value \(1/p\), i.e., 2N in this case.

Origin and change of variation

 

Mutation

Selection

 

Recombination

Drift

Wright-Fisher model with alleles

Alleles can randomly fix or be lost through process called genetic drift

Wright-Fisher model showing the evolution of population of 10 genes over 16 generations. Allele variants are shown in white and black. Starting frequency black variant is 0.3.

(To avoid confusion: previously black was used to indicate parent, here black/white will refer to different alleles)

Enter allelic variants; we are looking at two alleles

Important: the fate of any allele under drift is to be lost or fixed, which ultimately means variation is lost (decays) with drift

Binomial process models allele sampling

We assume two alleles \(A\), \(a\), each with \(i\) and \(j=2N-i\) copies in generation \(t\).

\(i=8\), \(j=2\cdot 6-8=4\)

Let \(p_t=i/2N\) be the frequency of \(A\) in generation \(t\), and \(q_t=1-p_t\) the frequency of \(a\).

\(p_t = 8/12\)

\(p_{t+1} = 4/12\)

Prob(\(k\) \(A\) alleles in next generation) is \(\mathsf{Bin}(2N, \frac{i}{2N})\)

A more formal mathematical description goes as follows:

The sampling model for next generation \(t+1\), given \(i\) \(A\) alleles in generation \(t\), is

\[ p_{t+1} = P(v_A=k) = {2N\choose k}\left( \frac{i}{2N} \right)^k \left(1 - \frac{i}{2N} \right)^{(2N - k)} = \mathsf{Bin}(2N, \frac{i}{2N}) \]

NB: we here slightly change notation, with \(p_{t+1} = P(X=k)\). Also, we have omitted an index \(i\) in the expressions of \(p\). The probability of observing \(k\) \(A\) alleles in generation \(t+1\) will of course depend on the number \(i\) \(A\) alleles in the previous generation.

Genetic drift

To capture dynamics, follow allele frequency trajectory (\(p_t\)) as function of time.

##' Wright Fisher model - follow allele frequency distribution
##'
##' @param p0 Starting frequency
##' @param n Population size
##' @param generations Number of generations to simulate
##'
wright_fisher <- function(p0, n, generations) {
    x <- vector(mode = "numeric", length = generations)
    x[1] <- p0
    for (i in seq(2, length(x))) {
        x[i] <- rbinom(1, size = n, prob = x[i - 1])/n
    }
    x
}

# Example simulation and plot
set.seed(1223)
generations <- 100
n <- 100  # NB: haploid population size!
plot(1:generations, wright_fisher(0.5, n, generations), type = "l", ylab = "frequency",
    xlab = "generation", ylim = c(0, 1))

Figure 5: Genetic drift for different haploid(!) population sizes, starting frequency \(p_0\)=0.5. Note dependency of variance on population size N.

Mention neutral here as this is how alleles behave under drift: like gas particle diffusing up and down. Allele is eventually either fixed or lost.

Genetic drift

Figure 6: Genetic drift for different combinations of starting frequency and population size for n=50 repetitions per parameter combination. Note how variation and time to fixation depends on population size and starting frequency.

• fate of allele: fixation or loss \(\rightarrow\) eventually loss of variation
• probability of fixation \(\pi(p)=p\), where \(p\) is the current frequency
• rate of drift (loss of variation) \(\propto \frac{1}{2N}\)

Simplifying assumptions -> rate of drift \(\propto \frac{1}{N}\) (\(\frac{1}{2N}\) in our treatment) (Leffler et al., 2012)

In reality population fluctuates and therefore Ne is substituted for N. Larger Ne -> smaller fluctuation in allele frequency (as evidenced in plots) -> maintain larger genetic diversity

Drift can be thought of as a force acting on a buoyant gas particle floating up and down at random.

Allele frequency distribution for N=1

Instead of looking at frequencies let’s switch to distributions of alleles for one individual, one locus. Then there are three possible genotypes (states) \(aa\), \(aA\), and \(AA\). Let \(n=0,1,2\) be an integer corresponding to each genotype (i.e., it counts the number of \(A\) alleles).

Assume individual mates with itself at random(!) starting in either of the three states. How does distribution evolve?

t=0

t=1

t=2

Example from (Gillespie, 2004, p. 24). We look at a single hermaphroditic individual that mates with itself at random. For each generation, given a genotype distribution, we calculate the outcome for the next generation. For instance, starting out in state 0 (only \(aa\) genotypes, hence only \(a\) alleles), we can only produce new \(aa\) genotypes and will therefore never leave state 0. The same holds for state 2. These states are absorbing states.

Starting from state 1 (\(aA\)) we can get \(aa\) genotype with 25% probability, since the probability of picking one \(a\) is 50%, and we perform two draws. Similarly, we get \(AA\) with 25% probability, leaving 50% to \(aA\).

In the next generation, the \(aA\) genotype frequency is 0.5, to be split in fractions 0.25, 0.5, 0.25 as before, and so on.

To study the system we therefore need to enumerate the probabilistic outcomes from each state (\(aa\) -> 1, 0, 0, \(aA\) -> 0.25, 0.5, 0.25, \(AA\) <- 0, 0, 1). To get the state in next generation, we multiply the current distribution with these outcomes. The next slide gives Kimura’s example for the case where we have N=10 chromosomes.

Probability distributions of allele frequencies

Figure 7: Histogram showing the course of change of the allele frequency distribution with time (Kimura, 1983, fig. 3.4). When N large (\(\gtrsim 100\)) histogram can be approximated by continuous distribution (diffusion theory). Try recipe for different values of N.

Figure 8: Frequency distributions of the brown eye (\(bw^{75}\)) allele in replicate experimental populations (\(n\sim 100\)) of Drosophila melanogaster (8 , 8 ) (Buri, 1956)

Mathematical treatment of drift can become complicated: easier to study dynamics of heterozygosity

Kimura’s plot illustrates the allelic frequency distribution of replicate populations each consisting of N=10 sequences. There are two allelic types. The x axis corresponds to the proportion of populations in a given state (e.g., the y value for x=3/10 corresponds to the proportion of populations with 3 alleles of one type and 7 of the other). At \(t=0\) all populations are in state 5/10.

Kimura’s plot is very instructive and students are encouraged to test the recipe code, increasing the number of states incrementally. For large enough N, the histograms can be approximated by continuous distributions. This observation led Kimura (even though diffusion equations were originally introduced to genetics by Fisher in 1922) to apply diffusion theory to obtain probability densities of allele frequencies, leading in turn to compact expressions of fixation probabilities, expected ages of alleles, and more. A treatment of diffusion theory is outside the scope of this course; the interested reader can consult e.g., (Ewens, 2004).

The Buri experiment is an empirical demonstration of Kimura’s plot.

On the sampling model, (Charlesworth & Charlesworth, 2010, p. 231) says:

It is, however, impossible to write down a simple algebraic expression for P, even without selection and mutation. [The equation] is useful for obtaining numerical results for relatively small populations, but becomes computationally demanding when N becomes very large.

Heterozygosity dynamics

Figure 9: Illustration of identity by descent (IBD) and state (IBS). Alleles in generation \(n\) are IBD but not IBS.

Let \(\mathcal{H}_t\) be the probability that two alleles are different by state. One can show that the time course evolution of \(\mathcal{H}_t\) in a randomly mating population consisting of \(N\) diploid hermaphroditic individuals is

\[ \mathcal{H}_t = \mathcal{H}_0 \left( 1 - \frac{1}{2N} \right)^t \]

Important consequence: heterozygosity in WF population lost at rate \(1/2N\).

Alleles in generation \(n\) are IDB but not IBS

Motivation: mathematical description of genetic drift complicated for populations with more than one individual. Easier to study the evolution of heterozygosity.

Equation presented without details. Derivation in (Gillespie, 2004, pp. 25–27) starts from G=probability that two alleles are IBS

Heterozygosity dynamics

Figure 10: Plot of \(\mathcal{H}_t\) illustrating dependency on population size

Figure 11: Heterozygosity in black-footed ferret (Wisely et al., 2002). Example from Graham Coop (2020), Fig. 4.5

Example of how rapid decline in population size can affect heterozygosity.

Population size influences genetic diversity!

However, census population size not (always) the correct measure.

Dependency on population size: for large enough populations the decline will be very slow (drift speed ~ 1/2N)

Practical example shows loss of heterozygosity tell-tale signature of population decline; conversely, not easy to show population decline in large populations (e.g., marine species with large \(N_e\)) using heterozygosity as measure

(Barton et al., 2007, p. 369) “The relation between genetic diversity and population size is difficult to discern, in part, because it is extremely hard to estimate the numbers of most species and because the number that matters is an average back into the distant past”(!)

From (Graham Coop, 2020, p. 64)

To see how a decline in population size can affect levels of het- erozygosity, let’s consider the case of black-footed ferrets (Mustela nigripes). The black-footed ferret population has declined dramatically through the twentieth century due to destruction of their habitat and sylvatic plague. In 1979, when the last known black-footed ferret died in captivity, they were thought to be extinct. In 1981, a very small wild population was rediscovered (40 individuals), but in 1985 this population suffered a number of disease outbreaks. At that point of the 18 remaining wild individuals were brought into captivity, 7 of which reproduced. Thanks to intense captive breeding efforts and conservation work, a wild population of over 300 individuals has been established since. However, because all of these individuals are descended from those 7 individuals who survived the bottleneck, diversity levels remain low. Wisely et al. measured heterozygosity at a number of microsatellites in individuals from museum collections, showing the sharp drop in diversity as population sizes crashed (see Figure 4.5).

Segue: population size important; however, census population size is not always the measure we want when relating to genetic diversity

Effective population size

Assumptions underlying Wright-Fisher model seldom fulfilled for natural populations. In particular

• non-random mating (population structure)
• fluctuations of population census size

Therefore, magnitude of drift experienced by a population different from that predicted by population size

Technically correct definition (but see Waples (2022)):

\(N_e\) is the size of an ideal population that would experience the same rate of genetic drift as the population in question.

ideal population: discrete generations with random mating, no evolutionary forces besides drift, no selective advantages

Mutation

Two-allele

Derive popgen stats

Finite sites

Recurrent mutations

Infinite alleles

Protein electrophoresis

Inifinite sites

DNA sequences

Mutations are generated randomly as changes in DNA. In WF model will be highlighted in black. We assume mutations are described by a Poisson process with rate \(\mu\) (per generation).

The mutation models are:

• two-allele model: used to derive popgen statistics
• finite sites model: recurrent mutations may occur, an assumption that is often omitted to facilitate calculations (e.g., coalescent)
• inifinite alleles model: every mutation leads to new allele (e.g., protein electrophoresis)
• infinite sites model: model commonly used for long DNA sequences

Additional model not presented here is the stepwise mutation model (mainly used to model repeats):

Mutation and drift

Genetic drift “moves” frequencies to the point that variation is lost via allele fixation or loss. New variation is introduced through mutation. We typically assume mutations are described by a Poisson process with rate \(\mu\) (per generation).

The mutation rate is denoted \(\mu\), and the population scaled mutation rate is \(2N_e\mu\) for haploid populations, \(4N_e\mu\) for diploid, where \(N_e\) is the effective population size.

The mutation - drift balance is when the diversity lost due to drift equals the diversity gained due to mutation.

Figure 12: Variation is introduced by mutations (black) at rate \(\mu=1e^{-4}\) and is occasionally lost through genetic drift.

Segue: so if drift removes variation, what introduces it? Mutation!

Note that the mutation rate in the WF model is not meaningful; the value is simply chosen such that enough mutations show up in plot.

Tracing the evolution of mutations

Figure 13: Different mutations suffer different fates. Most mutations are lost in a couple of generations. Mutant alleles are colored black and their genealogies are highlighted with thicker edges.

Observation: most mutations are in fact lost

Recall: fixation probability \(\pi(p)=p\)

Difference to previous figure: here we highlight the genealogies

Mutation drift balance

Drift removes variation. Mutation reintroduces it. At equilibrium the change in variation by definition is 0. In terms of \(\mathcal{H}_t\) (the probability that two alleles are not identical by state), \(\Delta\mathcal{H}=0\).

One can show¹ the classical formula that the equilibrium heterozygosity value is

\[ \hat{\mathcal{H}} = \frac{4N_e\mu}{1 + 4N_e\mu} \]

\(\mu\) is often assumed known, and heterozygosity is easily calculated from data, which provides a way of estimating \(N_e\).

The compound parameter \(4N_e\mu\) is called the population scaled mutation rate and is commonly named \(\theta\) such that

\[ \hat{\mathcal{H}} = \frac{\theta}{1 + \theta} \]

Gillespie (2004), pp. 30–31 uses a difference equation approach to derive \(\mathcal{H}\). Briefly, he studies the time evolution of \(\mathcal{G}\), the probability that two alleles drawn at random without replacement from the population are identical by state. Mutations are assumed unique, i.e., the infinite-alleles model. It holds

\[ \mathcal{G}^\prime = (1-\mu)^2\left[ \frac{1}{2N} + \left( 1 - \frac{1}{2N} \right) \mathcal{G} \right] \]

where \((1-\mu)^2\) is the probability that no mutation occur in either of the two sampled alleles. Since \(\mu\) is small, \((1-\mu)^2 \approx 1-2\mu\), which after some manipulation gives the desired expression for \(\mathcal{H} = 1 - \mathcal{G}\).

On pages 46–47, he shows that the expression for can be derived in a much simpler fashion using coalescent theory. Tracing two lineages backwards in time, the probability of coalescence is 1/2N, whereas the probability of a mutation is \(1-(1-\mu)^2\approx 2\mu\); \(\mathcal{H}\) is then simply the relative probability of the two events

\[ \mathcal{H} = \frac{2\mu}{2\mu + 1/2N} = \frac{4N\mu}{4N\mu + 1} \]

1. see (Gillespie, 2004, p. 29 – 31) for a concise treatement

The neutral theory of evolution

Mutation drift balance, together with the observation during 50’s-60’s that polymorphism was more common than expected, is the foundation of the neutral theory of evolution (Kimura, 1983): allele frequencies may change and fix due to chance alone and not selection; most mutations behave as if they are neutral.

Nearly neutral theory (Ohta, 1973) was later developed to explain failure to predict scaling of polymorphism with population size: most mutations are not neutral but slightly deleterious and purged from population by natural selection.

Figure 14: Heterozygosity H= predicted by the neutral theory. Shaded region shows typical heterozygosities in animals (y-axis). The observed \(N_e\mu\) range is higher than predicted from plot. From Hurst (2009), Fig 1.

Related to rate of substitution and molecular evolution is the work of Kimura that lead to the development of the neutral theory.

Motivation: if polymorphic sites deleterious, should not expect much polymorphism.

Low levels of polymorphism expected assuming little balancing selection (Hurst, 2009, p. 87); however electrophoretic studies showed polymorphism common. Would lead to detrimental load (Kimura & Ohta, 1971) -> therefore majority of polymorphism must evolve neutrally (dynamics). Also: rate of evolution (on protein level) too high (Haldane’s dilemma)

On the shaded region: the observed range of \(N_e\mu\) is larger than that which is predicted by the plot, and since \(\mu\) is constrained within a couple of orders of magnitude, \(N_e\) must vary more than predicted by the (strictly) neutral theory.

Conversely: given a constrained \(\mu\), we observe a range of \(N_e\mu\) that predicts a heterozygosity range H, which is much larger than that which we observe. In other words, the heterozygosity range is much lower than predicted by the neutral, given the observed \(N_e\mu\) range, so some other process must reduce variation somehow.

The general idea of nearly neutral theory is that most mutations are slightly deleterious and therefore purged by natural selection, thereby reducing observed variation. The efficacy of purging depends in turn on the effective population size, such that species with small \(N_e\) will have a harder time getting rid of potentially damaging variants.

Mutation rate can be estimated from substitution rate

Mutation enters populations and may be fixed by drift. Therefore, with time there will be fixed differences, or substitions (typically in the evolution of species) between populations, or species. In molecular evolution, the substition rate, \(\rho\), is the most interesting quantity.

The total number of new mutations in every generation is \(2N\mu\) (total number of gametes times mutation rate)

New mutations fix at a rate \(1/2N\)

Therefore, the average rate of substitution, \(\rho\), is \(2N\mu\times1/2N\), or

\[ \rho=\mu \]

which is independent of population size!

Practical implication: we can estimate mutation rate from the substitution rate at neutrally evolving sites (e.g., Kumar & Subramanian (2002))

What happens when sufficient amount of time passes? Mutations fix and generate substitutions between species. This slide presents Kimura’s remarkable result regarding mutation rate and substitution rate!

On \(2N\mu\): the larger the population, the larger the number of mutations that can fix

If \(4N\mu>>0\) no new mutations can fix. Issue here is there is no explicit mention of DNA sequence -> introduction of infinite sites model

For practical note on estimation of mutation rate see e.g., (Hubisz & Siepel, 2020, p. 247)

(Kumar & Subramanian, 2002, first paragraph):

Rates of point mutation can be determined indirectly by estimating the rate at which the neutral substitutions accumulate in protein-coding genes

Selection

For a list of methods see https://methodspopgen.com/methods-to-detect-selection/

Selection and fitness

Figure 15: The life cycle used in the fundamental model of selection (Gillespie, 2004, fig. 3.2)

Much confusion exists in the literature regarding how various types of selection are defined, in particular because some of the terminology is used slightly differently within different scientific communities (Nielsen, 2005)

\[ \begin{matrix} \mathrm{Genotype} & AA & Aa & aa \\ \mathrm{Frequency\ in\ newborns} & p^2 & 2pq & q^2\\ \mathrm{Viability} & w_{AA} & w_{Aa} & w_{aa}\\ \mathrm{Frequency\ after\ selection} & p^2w_{AA} / \bar{w} & 2pqw_{Aa} / \bar{w} & q^2w_{aa} / \bar{w} \\ \mathrm{Relative\ fitness} & 1 & 1-hs & 1-s\\ \end{matrix} \]

where \(\bar{w} = p^2w_{AA} + 2pqw_{Aa} + q^2w_{aa}\) is the mean fitness.

\(h=0\)	\(A\) dominant, \(a\) recessive
\(h=1\)	\(a\) dominant, \(A\) recessive
\(0<h<1\)	incomplete dominance
\(h<0\)	overdominance (heterozygote advantage)
\(h>1\)	underdominance

Notation follows Gillespie (2004), pp. 61–64.

The life cycle figure starts with newborns carrying the \(A\) allele at frequency \(p\), like parents. To reproduce they must reach adulthood, but due to selection, the allele frequency of \(A\) may shift to \(p^\prime\) because different genotypes affect survival probabilities. This is typically summarized in genotype viabilities.

The Nielsen quote is a reminder that many different notations and parametrizations exist in the literature.

The most important equation in population genetics

\[ p^\prime - p = \Delta_sp = \frac{pq[p(w_{AA} - w_{Aa}) + q(w_{Aa} - w_{aa})]}{p^2w_{AA} + 2pqw_{Aa} + q^2w_{aa}} \]

Figure 16: Allele frequency change over time for directional, balancing, and disruptive selection, for different values of \(p_0\).

Figure 17: Rate of allele frequency change as a function of allele frequency for directional, balancing, and disruptive selection.

Given the genotype fitnesses/viabilities, we can work out the difference in allele frequency \(p-p^\prime\) between successive generations as a function of viabilities (or relative fitnesses). Equation relates allele frequency change to viabilities for different genotypes. It is instructive to study the equation and plot trajectories to gain an intuition for how allele frequencies evolve given the different selection regimes.

The left panel shows that the three modes of selection have different equilibrium points: for directional selection the favored homozygote will eventually attain 100%, balancing selection has a stable equilibrium point (attractor) for a given allelic ratio (here 25:75), and disruptive selection has an unstable equilibrium point (repeller) for a given allelic ratio.

The right panel shows the rate of change as a function of allele frequency. Note how the directionality changes at the equilibrium point for balancing and directional selection.

Selection and drift - population size matters

Figure 18: The fixation probability relative to the neutral probability of fixation (\(p=1/2N\)) under the assumption \(s<0.1\). Red highlights region where \(|N_es|<0.05\). Adapted from Lynch (2007), Fig. 4.2.

In red region (\(|N_es|<0.05\)) the probability of fixation is within 10% of neutral fixation.

Consequence: for any population size there exists range of selection coefficients where mutant alleles \(\approx\) neutral (effective neutrality).

Segue: note on the strength of s and the relation to \(N_e\). For small \(N_e\) the relation will hold for higher \(s\), meaning small populations will have difficulties purging deleterious mutations.

Even if a new mutation has, say, \(s=0.01\), it still needs to overcome effects of drift. Only has 2% change of fixation.

In figure, we assume \(s<0.1\), which is almost always the case (Lynch, 2007, p. 74).

(Hahn, 2019, p. 133)

Direct selection can be inferred from protein substitutions

For genes, the ratio of nonsynonymous to synonymous substitutions can tell us about protein evolution:

Synonymous substitution

Protein          L
DNA         --- CTT ---
                  *
DNA         --- CTC ---
Protein          L

Nonsynonymous substitution

Protein          L
DNA         --- CTT ---
                 *
DNA         --- CHT ---
Protein          H

\(\mathbf{d_N/d_S << 1}\)

negative (purifying) selection

\(\mathbf{d_N/d_S < 1}\)

majority nonsynonymous deleterious, some advantageous

\(\mathbf{d_N/d_S = 1}\)

neutral or mix neutral / advantageous / deleterious mutations

\(\mathbf{d_N/d_S > 1}\)

positive selection

Figure 19: \(d_n/d_s\) comparisons for human-rat orthologs. For most genes, \(d_n/d_s << 1\) indicating purifying selection. A handful of genes (n=9) have \(d_n/d_s > 1.0\) which could indicate positive selection.

Not all mutations fall in genes. Methods for detecting direct selection not applicable to studying selection on single mutation, or e.g., balancing. This requires looking for specific patterns of diversity surrounding locus under selection.

Segue: how do we detect selection?

Early approach: look at protein-coding sequence where nucleotide substitutions can change the amino acid (non-synonymous, and presumably non-neutral) or be silent (synonymous, assumed neutral).

Not all mutations in genes. Also want to look at single advantageous mutations, or balanced polymorphisms. Must therefore look at tell-tale signs of linked neutral variants.

Linked selection reduces diversity at neighbour loci

Figure 20: A selective sweep of an advantageous mutation (gray dot). Adapted from Charlesworth & Charlesworth (2010), Fig. 8.13

Example of a selective sweep. If a sweep completes at a locus, it will become monomorphic, as will the neighbouring sites. Mutation could reintroduce variation. Recombination could increase diversity in neighbourhood, but in a manner that depends on the distance from the locus under selection.

Alternatives to direct selection tests are those that look at linked selection. In essence, we look for characteristic patterns in the vicinity of a selected locus. A sweep leads to the selected site and neigbouring sites being monomorphic (and therefore invisible to tests based on direct selection). Without recombination, mutation could reintroduce variation. In the presence of recombination, the effect of reduced diversity will be dependent on distance from the locus under selection.

Note here that every sequence can be treated as an allele, such that we in essence are looking multi- and not a bi-allelic locus.

Recombination breaks association between loci

Miller (2020), Fig. 5.12.3

Once per chromosome! But: rates vary between loci (hotspots), sex chromosomes vs autosomes, and in some species, recombination only occurs in one sex (e.g., D.melanogaster).

Main effect: association between loci breaks up.

The mechanism that breaks up association between loci is recombination. Cartoon to right shows the possible (haploid) outcomes of meiosis. \(r\) (the recombination rate) denotes the probability that there is a recombination event between the two loci.

The farther apart two loci on the same chromosome, the larger the probability that a recombination event occurs between them. Note that this can be applied to loci on different chromosomes as well; they segregate independently.

Linkage disequilibrium and its decay

Association between loci can be written as:

\[ D_{AB} = p_{AB} - p_Ap_B \]

Similar expressions hold for other pairs; only need to know one \(D_{ij}\) (e.g., \(D_{AB}\)) so drop subscript and rewrite:

\[ p_{AB}= p_Ap_B + D \]

If \(D\neq0\) the loci are in linkage disequilibrium.

Can show that decay over time is

\[ D_t = (1-r)^tD_0 \]

Recombination decreases D (linkage)

LD between pairs of autosomal SNPs for human and mouse. From (Laurie et al., 2007, fig. 2)

Handwavy introduction of LD as quantity. There are other equations, but the main point here is to point out that D measures the degree of linkage (disequilibrium); the higher D, the stronger linkage is.

The figure shows the difference in LD decay among wild populations compared to inbred mouse strains, where LD extends over several megabases. Therefore, LD decay informs about demographic history, and as we will see, evolutionary processes (sweeps -> longer LD blocks).

The effect of a selective sweep on diversity

Code

pgip-slim --seed 42 -n 1000 -r 1e-6 -m 1e-7 --threads 12 recipes/slim/selective_sweep.slim -l 1000000 --outdir results/slim
pgip-tsstat results/slim/slim*.trees -n 10 --seed 31 -s pi -s S -s TajD -w 500 --threads 10 | gzip -v - > results/slim/selective_sweep.w500.csv.gz

Figure 21: The effect of a selective sweep on diversity. The arrow points to the site under selection. The y-axis shows Tajima’s D which is proportional to the difference between two measures of diversity, nucleotide diversity \(\pi\) and Watterson’s \(\theta_W\).

Simulation: N=1e4, mu=5e-8, pi=4Nmu=0.004, seek S=a*pi=0.01, so a=S/pi which gives n. One needs to ramp up recombination rate to see the dip in TajD clearly.

The effect of a selective sweep on diversity

Code

pgip-slim --seed 42 -n 1000 -r 1e-6 -m 1e-7 --threads 12 recipes/slim/selective_sweep.slim -l 1000000 --outdir results/slim
pgip-tsstat results/slim/slim*.trees -n 10 --seed 31 -s pi -s S -s TajD -w 500 --threads 10 | gzip -v - > results/slim/selective_sweep.w500.csv.gz

Figure 22: The effect of a selective sweep on diversity. The arrow points to the site under selection. The y-axis shows Tajima’s D which is proportional to the difference between two measures of diversity, nucleotide diversity \(\pi\) and Watterson’s \(\theta_W\).

Important: individual runs show a lot of stochasticity, but the signal is clear for the mean.

The effect of a selective sweep on diversity

Figure 23: The effect of a selective sweep on diversity. The figure shows the mean of 1000 simulations with the selected locus indicated with an arrow.

cf Nielsen (2005). Not clear how scaling of statistics to have mean=1 is done.

The phases of a selective sweep

Figure 24: Time goes from left to right. As sweep progresses, tree topology changes. Adapted from Hahn (2019), Figure 8.1

Amount of diversity depends on fixation time. A neutral locus fixes in \(4N_e\) generations; for \(s=0.0001\), it takes approximately \(0.29N_e\) generations.

Selections changes the genealogy (different topology, shorter branches), an aspect used in many linkage-based tests for selection.

A selective sweep changes the genealogy (shorter branches, different topology) compared to the neutral case, a fact that can be utilised in linkage-based tests for selection.

Note that the amount of variation surrounding a sweep depends heavily on the rate of the sweep. (Hahn, 2019, p. 167) points out that an advantageous allele with \(s=0.0001\) will take approximately \(0.29N_e\) generations to fix (compared to \(4N_e\) generations for a neutral locus)

Linked selection may constrain levels of diversity

Figure 25: Hitchhiking (left) versus background selection (right).

Hitchhiking

• alleles linked to locus under selection “hitchhike” to high frequencies (Smith & Haigh, 1974)
• evidence: positive correlation between putative neutral diversity and recombination (Corbett-Detig et al., 2015)

Background selection

• loci linked to a deleterious locus will be purged from population and thus reduce diversity (Charlesworth et al., 1993)
• similar patterns to hitchhiking

Linked selection has been proposed as solution to the low range of diversity seen between species with orders of magnitude ranges in census population sizes.

Summary

We have looked at the Wright-Fisher model as a model of populations and genealogies*

Genetic drift moves allele frequencies up and down at random and removes variation at rate \(\propto 1/2N\)

Mutation reintroduces variation. The Neutral theory posits most mutations are neutral and dynamics follow mutation drift equilibrium.

Methods to detect selection are based on direct selection or studying patterns of variation caused by linked selection.

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