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---
title: "Introduction to Population Genetics"
subtitle: "2024 HAAM Summerschool Presentation"
author: Stephan Schiffels
institute: Max Planck Institute for Evolutionary Anthropology
date: 2024-06-19
format:
revealjs:
smaller: true
scrollable: true
---
## Source code
Sourcecode for this presentation and the code for many of the figures can be found at [https://github.com/stschiff/2024-haam-lecture](https://github.com/stschiff/2024-haam-lecture).
# Outline
- Human genetic variation
- Genetic Drift
- FSt and friends
- Visualising population structure through PCA
- Investigating Admixture with F-statistics
# Human Genetic variation
## Demo:[https://popgen.uchicago.edu/ggv](https://popgen.uchicago.edu/ggv)
## Variation in space (Example 1)
## Variation in space (Example 2)
## Variation in space (Example 3)
# Genetic Drift
- Random process of allele frequency change.
- Wright-Fisher model:
- Parent generation of $N$ individuals
- Exactly $N$ individuals as next generation.
- Parents drawn randomly per child
- results in binomial random draws.
## Example
- $N=100$ (haploid individuals)
- two alleles $A$ and $B$
- Generation 1: 50 / 50, so allele freuqency of $B$ is 50%.
- Generation 2: 47 / 53, so allele freuqency of $B$ is 53%.
## Simulation
Implementation in R:
```{R echo=TRUE, `code-line-numbers` = "|1|2|3|4-6|"}
wfsim <- function(n, g, x0) {
res <- numeric(g + 1)
res[1] <- x0
for (i in 2:(g + 1)) {
res[i] <- (rbinom(1, n, res[i - 1])) / n
}
return(res)
}
```
We can test it:
```{R echo=TRUE}
set.seed(1)
wfsim(100, 10, 0.5)
```
---
```{R echo=TRUE}
time_series <- wfsim(100, 100, 0.5)
plot(time_series, type = "l", ylim = c(0, 1),
xlab = "generation", ylab = "allele frequency")
```
## Population size dependency
::: {.panel-tabset}
### N = 100
```{R}
gens <- 100
sims100 <- replicate(50, wfsim(100, gens, 0.5))
matplot(sims100, type = "l", lty = 1, col = "black",
ylim = c(0, 1),
xlab = "generation", ylab = "allele frequency")
```
### N = 1000
```{R}
gens <- 100
sims1000 <- replicate(50, wfsim(1000, gens, 0.5))
matplot(sims1000, type = "l", lty = 1, col = "black",
ylim = c(0, 1),
xlab = "generation", ylab = "allele frequency")
```
### N = 10000
```{R}
gens <- 100
sims10000 <- replicate(50, wfsim(10000, gens, 0.5))
matplot(sims10000, type = "l", lty = 1, col = "black",
ylim = c(0, 1),
xlab = "generation", ylab = "allele frequency")
```
:::
## Population size dependency
::: {.panel-tabset}
### N = 100
```{R}
par(mfrow = c(1, 2))
matplot(sims100, type = "l", lty = 1, col = "black",
ylim = c(0, 1),
xlab = "generation", ylab = "allele frequency")
plot(apply(sims100, 1, var), type = "l", ylim = c(0, 0.15),
xlab = "generations", ylab = "Variance")
```
### N = 1000
```{R}
par(mfrow = c(1, 2))
matplot(sims1000, type = "l", lty = 1, col = "black",
ylim = c(0, 1),
xlab = "generation", ylab = "allele frequency")
plot(apply(sims1000, 1, var), type = "l", ylim = c(0, 0.15),
xlab = "generations", ylab = "Variance")
```
### N = 10000
```{R}
par(mfrow = c(1, 2))
matplot(sims10000, type = "l", lty = 1, col = "black",
ylim = c(0, 1),
xlab = "generation", ylab = "allele frequency")
plot(apply(sims10000, 1, var), type = "l", ylim = c(0, 0.15),
xlab = "generations", ylab = "Variance")
```
:::
## Fixation
```{R}
gens <- 1000
sim_long <- replicate(100, wfsim(100, gens, 0.5))
par(mfrow = c(1, 2))
matplot(sim_long[,1:20], type = "l", lty = 1, col = "black",
ylim = c(0, 1),
xlab = "generation", ylab = "allele frequency")
plot(apply(sim_long, 1, var), type = "l", xlab = "generations", ylab = "Variance")
```
# The fixation index $F_\text{ST}$
## Formal definition of $F_\text{ST}$
Conditional probability of allele frequency $p_i$ in population $i$, given an ancestral allele frequency $p_\text{anc}$:
$$E(p_i|p_\text{anc}) = p_\text{anc}$$
and variance $$Var(p_i|p_\text{anc}) = F_\text{ST}^i p_\text{anc}(1-p_\text{anc}).$$
---
Intuition for $Var(p_i|p_\text{anc}) = F_\text{ST}^i p_\text{anc}(1-p_\text{anc}):$
- For $F_\text{ST}^i=0$: $p_i = p_\text{anc}$, no variance
- For $F_\text{ST}^i=1$: Variance is $p_\text{anc}(1-p_\text{anc})$
```{R}
gens <- 1000
sims_x05 <- replicate(1000, wfsim(100, gens, 0.5))
sims_x03 <- replicate(1000, wfsim(100, gens, 0.3))
sims_x02 <- replicate(1000, wfsim(100, gens, 0.2))
plot_dat <- cbind(
apply(sims_x05, 1, var),
apply(sims_x03, 1, var),
apply(sims_x02, 1, var)
)
par(mfrow = c(1, 1))
cols <- c("blue", "red", "green")
matplot(plot_dat, type = "l", ylim = c(0, 0.25), lty = 1,
xlab = "generations", ylab = "Variance", col = cols)
legend(x = "bottomright",
legend = c("p_anc = 0.5", "p_anc = 0.3", "p_anc = 0.2", "FST = 1"),
lty = c(1, 1, 1, 2), col = c(cols, "black"))
theory_values <- sapply(c(0.5, 0.3, 0.2), function(x) x * (1 - x))
abline(h = theory_values, lty = 2, col = cols)
```
## Estimating $F_\text{ST}$ from genomic data
From Theory:
$$F_\text{ST}(A,B) = \frac{F_\text{ST}^A+F_\text{ST}^B}{2}$$
Hudson-Estimator:
$$F_\text{ST}=1-\frac{H_w}{H_b}$$
- $H_w$ is heterozygosity within each population
- $H_b$ is heterozygosity between two populations
Intuition for $F_\text{ST}=1-\frac{H_w}{H_b}$:
- $F_\text{ST}=0$ if and only if $H_w=H_b$, (A and B are same population)
- $F_\text{ST}=1$ if and only if $H_w=0$, all variants fully fixed in both populations.
## Estimation from allele frequencies
$$F_\text{ST}(A,B)=\frac{\left\langle(a-b)^2\right\rangle}{\left\langle a(1-b)+b(1-a)\right\rangle}$$
- $a$ and $b$ are _population allele frequencies_ (in principle unobserved), but can be approximated by _sample allele frequencies_.
- Bias-corrected version:
$$F'_\text{ST}(A,B) = \frac{\left\langle\left(a-b\right)^2-\frac{a(1-a)}{n_a-1} - \frac{b(1-a}{n_b-1}\right\rangle}{\left\langle a(1-b) + b(1-a)\right\rangle}$$
## Relation to $F_2$ Statistics
$F_\text{ST}(A,B)$ is closely related to F2-statistics, introduced in [@Patterson2012]:
$$F_2(A,B)=(a-b)^2$$
- Both statistics range technically from 0 to 1, but differently scaled!
- $F_2:$
- Theoretical value of $F_2=1$: Both populations are fixed at different alleles in all studied SNPs (impossible).
- Timescale for $F_2=1$: Mutation-time-scale $1/\mu$ (hundreds of millions of generations)
- $F_\text{ST}$:
- Theoretical value of $F_\text{ST}=1$: Both populations are fixed at all alleles, but not necessarily at different ones.
- Timescale for $F_\text{ST}=1$: Drift time-scale $1/N$ (tens of thousands of generations)
## Computing FST using xerxes
Poseidon Software xerxes [](https://www.poseidon-adna.org)
We here chose a number of populations from [@Patterson2012] with more than 10 samples per population, and prepare the following config file for xerxes:
```Yaml
fstats:
- type: FST
a: ["Adygei", "Balochi", "Basque", "BedouinA", "BedouinB", "Biaka", "Brahui", "Burusho", "Druze", "French", "Han", "Hazara", "Italian_North", "Japanese", "Kalash", "Karitiana", "Makrani", "Mandenka", "Mayan", "Mozabite", "Orcadian", "Palestinian", "Papuan", "Pathan", "Pima", "Russian", "Sardinian", "Sindhi_Pakistan", "Yakut", "Yoruba"]
b: ["Adygei", "Balochi", "Basque", "BedouinA", "BedouinB", "Biaka", "Brahui", "Burusho", "Druze", "French", "Han", "Hazara", "Italian_North", "Japanese", "Kalash", "Karitiana", "Makrani", "Mandenka", "Mayan", "Mozabite", "Orcadian", "Palestinian", "Papuan", "Pathan", "Pima", "Russian", "Sardinian", "Sindhi_Pakistan", "Yakut", "Yoruba"]
- type: F2
a: ["Adygei", "Balochi", "Basque", "BedouinA", "BedouinB", "Biaka", "Brahui", "Burusho", "Druze", "French", "Han", "Hazara", "Italian_North", "Japanese", "Kalash", "Karitiana", "Makrani", "Mandenka", "Mayan", "Mozabite", "Orcadian", "Palestinian", "Papuan", "Pathan", "Pima", "Russian", "Sardinian", "Sindhi_Pakistan", "Yakut", "Yoruba"]
b: ["Adygei", "Balochi", "Basque", "BedouinA", "BedouinB", "Biaka", "Brahui", "Burusho", "Druze", "French", "Han", "Hazara", "Italian_North", "Japanese", "Kalash", "Karitiana", "Makrani", "Mandenka", "Mayan", "Mozabite", "Orcadian", "Palestinian", "Papuan", "Pathan", "Pima", "Russian", "Sardinian", "Sindhi_Pakistan", "Yakut", "Yoruba"]
```
This will then produce all combinations of $FST(A, B)$ and $F_2(A, B)$ as indicated in the population lists.
We run this config file using the command line
```bash
REPO=/path/to/community-archive/2012_PattersonGenetics
xerxes fstats -d $REPO --statConfig fstat_world_config.yaml -f fstat_world_output.tsv > fstat_world_table.txt
```
## Xerxes output
- The standard output, is a nicely layouted ASCII Table, which looks like this (skipping thousands of lines)
- The `-f` flag outputs a tab-separated file, here named `fstat_world_output.tsv`, which is easier to read into R.
```text
.-----------.-----------------.-----------------.---.---.---------.----------------.--------------------.------------------.--------------------.
| Statistic | a | b | c | d | NrSites | Estimate_Total | Estimate_Jackknife | StdErr_Jackknife | Z_score_Jackknife |
:===========:=================:=================:===:===:=========:================:====================:==================:====================:
| FST | Adygei | Adygei | | | 593124 | 0.0000 | 0.0000 | 0.0000 | NaN |
| FST | Adygei | Balochi | | | 593124 | 1.2789e-2 | 1.2789e-2 | 3.3572e-4 | 38.09517110646904 |
| FST | Adygei | Basque | | | 593124 | 1.8790e-2 | 1.8790e-2 | 4.0141e-4 | 46.810358341103225 |
| FST | Adygei | BedouinA | | | 593124 | 1.3017e-2 | 1.3017e-2 | 2.9647e-4 | 43.90737238689979 |
| FST | Adygei | BedouinB | | | 593124 | 3.3455e-2 | 3.3454e-2 | 5.7648e-4 | 58.03217592610529 |
| FST | Adygei | Biaka | | | 593124 | 0.1716 | 0.1716 | 1.2185e-3 | 140.85275693678508 |
| FST | Adygei | Brahui | | | 593124 | 1.4644e-2 | 1.4644e-2 | 3.4481e-4 | 42.46989237781921 |
| FST | Adygei | Burusho | | | 593124 | 1.8566e-2 | 1.8566e-2 | 3.8156e-4 | 48.6573908240317 |
| FST | Adygei | Druze | | | 593124 | 1.2173e-2 | 1.2173e-2 | 2.6659e-4 | 45.65975464203526 |
| FST | Adygei | French | | | 593124 | 9.7730e-3 | 9.7730e-3 | 3.1627e-4 | 30.9006924987833 |
| FST | Adygei | Han | | | 593124 | 9.8759e-2 | 9.8759e-2 | 1.1973e-3 | 82.48660429503893 |
| FST | Adygei | Hazara | | | 593124 | 3.0725e-2 | 3.0726e-2 | 7.1478e-4 | 42.98629834431124 |
| FST | Adygei | Italian_North | | | 593124 | 8.6600e-3 | 8.6601e-3 | 2.7883e-4 | 31.058813893781032 |
...
| F2 | Yoruba | Kalash | | | 593124 | 4.5530e-2 | 4.5530e-2 | 3.1580e-4 | 144.17623831657042 |
| F2 | Yoruba | Karitiana | | | 593124 | 8.0274e-2 | 8.0274e-2 | 5.0430e-4 | 159.18045338908084 |
| F2 | Yoruba | Makrani | | | 593124 | 3.7091e-2 | 3.7091e-2 | 2.9054e-4 | 127.66205740966917 |
| F2 | Yoruba | Mandenka | | | 593124 | 2.2053e-3 | 2.2053e-3 | 5.0215e-5 | 43.91726459507651 |
| F2 | Yoruba | Mayan | | | 593124 | 5.9758e-2 | 5.9758e-2 | 4.8204e-4 | 123.968969339996 |
| F2 | Yoruba | Mozabite | | | 593124 | 2.8717e-2 | 2.8717e-2 | 2.9425e-4 | 97.59603068441464 |
| F2 | Yoruba | Orcadian | | | 593124 | 4.3925e-2 | 4.3925e-2 | 3.0915e-4 | 142.08521834047312 |
| F2 | Yoruba | Palestinian | | | 593124 | 3.5635e-2 | 3.5635e-2 | 2.6730e-4 | 133.31454891682029 |
| F2 | Yoruba | Papuan | | | 593124 | 6.8316e-2 | 6.8316e-2 | 6.1762e-4 | 110.6122879514611 |
| F2 | Yoruba | Pathan | | | 593124 | 3.8770e-2 | 3.8770e-2 | 2.9069e-4 | 133.3741386458573 |
| F2 | Yoruba | Pima | | | 593124 | 6.9994e-2 | 6.9994e-2 | 5.4248e-4 | 129.02469385208832 |
| F2 | Yoruba | Russian | | | 593124 | 4.2705e-2 | 4.2705e-2 | 3.1362e-4 | 136.16918827517011 |
| F2 | Yoruba | Sardinian | | | 593124 | 4.4227e-2 | 4.4227e-2 | 3.1663e-4 | 139.68078608155477 |
| F2 | Yoruba | Sindhi_Pakistan | | | 593124 | 3.8543e-2 | 3.8543e-2 | 2.8137e-4 | 136.9825670475205 |
| F2 | Yoruba | Yakut | | | 593124 | 5.0727e-2 | 5.0727e-2 | 3.9113e-4 | 129.69267912526192 |
| F2 | Yoruba | Yoruba | | | 593124 | 0.0000 | 0.0000 | 0.0000 | NaN |
'-----------'-----------------'-----------------'---'---'---------'----------------'--------------------'------------------'--------------------'
```
## Plotting results in R
We first load the data
```{r echo=TRUE}
dat <- subset(read.table("fst_working/fstat_world_output.tsv", sep="\t", header = TRUE),
select=-c(c, d, Z_score_Jackknife))
datFST <- dat[dat$Statistic == "FST",]
datF2 <- dat[dat$Statistic == "F2",]
head(datFST[,1:6])
```
Let's check out the largest values
```{r echo=TRUE}
head(dat[order(-dat$Estimate_Total),1:6])
```
## Histograms
::: {.panel-tabset}
### FST
```{r echo = TRUE}
avg <- mean(datFST$Estimate_Total)
hist(datFST$Estimate_Total, xlab = "FST", ylab = "Nr of pairs", main = paste0("mean=", round(avg, 2)))
```
### F2
```{r echo = TRUE}
avg <- mean(datF2$Estimate_Total)
hist(datF2$Estimate_Total, xlab = "F2", ylab = "Nr of pairs", main = paste0("mean=", round(avg, 2)))
```
:::
## Heatmaps
```{r}
fstMat <- xtabs(Estimate_Total ~ a + b, datFST)
f2Mat <- xtabs(Estimate_Total ~ a + b, datF2)
```
::: {.panel-tabset}
### FST
```{r}
#| fig-height: 7
#| fig-width: 7
heatmap(fstMat, symm = TRUE, hclustfun = function(m) hclust(m, method="ward.D2"))
```
### F2
```{r}
#| fig-height: 7
#| fig-width: 7
heatmap(f2Mat, symm = TRUE, hclustfun = function(m) hclust(m, method="ward.D2"))
```
:::
## Dendrograms
::: {.panel-tabset}
### FST
```{r}
fstDist <- as.dist(fstMat)
dendro <- hclust(fstDist, method="ward.D2")
plot(dendro, hang = -1, ylab = "FST", xlab = "", main = "")
```
### F2
```{r}
f2Dist <- as.dist(f2Mat)
dendro <- hclust(f2Dist, method="ward.D2")
plot(dendro, hang = -1, ylab = "F2", xlab = "", main = "")
```
:::
- FST affected by total drift (inversely proportional to population size)
- F2 more closely reflecting evolutionary time
# Visualising population structure through PCA
## Principal Components Analyses
Genotype matrix
```text
Test Indivdual 1 (Selected SNPs) : T G A C C G G . . .
Test Indivdual 2 (Selected SNPs) : T A A C C G G . . .
Test Indivdual 3 (Selected SNPs) : C A A C C G G . . .
Test Indivdual 4 (Selected SNPs) : T A A C C G G . . .
Test Indivdual 5 (Selected SNPs) : C G G C C G G . . .
Test Indivdual 6 (Selected SNPs) : C G G C C G G . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
```
How can we visualise hundreds of individuals with a million SNPs?
## Principal Components Analyses
- Basic problem in multivariate statistics
- highdimensional data → two-dimensional visualisation, dimension reduction
- Idea: Order entities in a two-dimensional room such that the distance between them reflects the true high-dimensional distances most accurately
- Distances can be computed straight-forward, even for many dimensions
- Example: Pixel-wise distances in flags:
## PCA Example: Flags
:::: {.columns}
::: {.column width="50%"}
:::
::: {.column width="50%"}
- pixel-wise distances
- similar flags are placed nearby.
:::
::::
::: aside
Data pulled and processed in Wolfram Mathematica
:::
## PCA Example: EU Flags
{width="50%"}
::: aside
Data pulled and processed in Wolfram Mathematica
:::
## PCA Example: Greek Letters
:::: {.columns}
::: {.column width="50%"}
:::
::: {.column width="50%"}
- letters rasterised
- again pixel-wise distances
- layout corresponds to similarities
:::
::::
::: aside
Data pulled and processed in Wolfram Mathematica
:::
## PCA Example: Van Gogh Paintings
{width="50%"}
::: aside
Data pulled and processed in Wolfram Mathematica
:::
## PCA and Genetics (World)
:::: {.columns}
::: {.column width="40%"}
:::
::: {.column width="40%"}
- First Component separates groups from African and those outside of Africa
- second component separates Asia and America from Europe
:::
::::
::: aside
Data from Patterson, Nick, Priya Moorjani, Yontao Luo, Swapan Mallick, Nadin Rohland, Yiping Zhan, Teri Genschoreck, Teresa Webster, and David Reich. 2012. “Ancient Admixture in Human History.” Genetics 192 (3): 1065–93.
:::
## PCA and Genetics (Europe)
{width="60%"}
::: aside
Novembre, John, Toby Johnson, Katarzyna Bryc, Zoltán Kutalik, Adam R. Boyko, Adam Auton, Amit Indap, et al. 2008. “Genes Mirror Geography within Europe.” Nature 456 (7218): 98–101.
:::
## Genetics PCA through space and time
:::: {.columns}
::: {.column width="50%"}
:::
::: {.column width="50%"}
:::
::::
::: aside
Created by me, using data from Patterson, Nick, Priya Moorjani, Yontao Luo, Swapan Mallick, Nadin Rohland, Yiping Zhan, Teri Genschoreck, Teresa Webster, and David Reich. 2012. “Ancient Admixture in Human History.” Genetics 192 (3): 1065–93.
:::
## Genetics PCA through space and time
:::: {.columns}
::: {.column width="60%"}
:::
::: {.column width="40%"}
:::
::::
::: aside
Schmid, Clemens, and Stephan Schiffels. 2023. “Estimating Human Mobility in Holocene Western Eurasia with Large-Scale Ancient Genomic Data.” Proceedings of the National Academy of Sciences of the United States of America 120 (9): e2218375120.
:::
# Investigating Admixture with F-statistics
## Admixture - F3 Statistics
- F3 statistics are a useful to understand population relationships.
- Test whether a target population (C) is admixed between two source populations (A and B)
$$F3(A,B;C)=\langle(c−a)(c−b)\rangle$$
- If $F3(A, B; C)$ is negative, proof that population C is admixed between populations A and B:
<img src="images/fstats/f3_phylogeny.png" alt="F3-phylogeny" style="width: 300px;"/>
## Intermediate allele frequencies
- F3 statistics becomes negative if allele frequency of target population C is on average intermediate between the allele frequencies of A and B.
- Extreme example: $a=0$, $b=1$ and $c=0.5$. Then: $(c−a)(c−b)=−0.25$, negative.
- If entire statistics is negative: many positions like this
- Proofs admixture between the two sources.
| SNP | A | B | C | $(c-a)(c-b)$ |
|---------------|-----|-----|-----|--------------|
| 1 | 1 | 0 | 0.5 | -0.25 |
| 2 | 0.8 | 0 | 0 | 0 |
| 3 | 0 | 0.7 | 0 | 0 |
| 4 | 0.1 | 0.5 | 0.3 | -0.04 |
| 5 | 0 | 0.1 | 0.2 | 0.02 |
| 6 | 1 | 0.2 | 0.9 | -0.07 |
| $F_3(A, B;C)$ | | | | **-0.057** |
:::{.callout-caution}
If an F3 statistics is *not* negative, it does *not* proof that there is no admixture!
:::
## Computing Admixture-F3 with xerxes
- We test if Finnish are admixed between East and West
- For West, we use French, Icelandic, Lithuanian and Norwegian as source
- For East: Nganasan and one of the ancient individuals analysed in [@Lamnidis2018], from the Russian site of *Bolshoy Oleni Ostrov* (3500 years before present).
### Getting the data
If you happen to have downloaded a copy of the [Poseidon Community Archive](https://www.poseidon-adna.org/#/archive_overview?id=public-poseidon-archives) already, then just use the path to that archive in the following commands. Otherwise you download the entire archive via
```bash
trident fetch -d /somewhere/to/store/the/archive --downloadAll
```
or just the relevant packages for the examples in this chapter:
```bash
trident fetch -d /somewhere/to/store/the/archive -f "*2012_PattersonGenetics*,*2014_RaghavanNature*,*2014_LazaridisNature*,*2018_Lamnidis_Fennoscandia*"
```
## Running fstats
We use the software `xerxes fstats` from the [Poseidon Framework](https://www.poseidon-adna.org/#/xerxes?id=xerxes-cli-software). Here is a command line that computes 8 statistics for us:
```bash
xerxes fstats -d ~/dev/poseidon-framework/community-archive \
--stat "F3(Nganasan,French,Finnish)" \
--stat "F3(Nganasan, Icelandic, Finnish)" \
--stat "F3(Nganasan, Lithuanian, Finnish)" \
--stat "F3(Nganasan, Norwegian, Finnish)" \
--stat "F3(Russia_Bolshoy, French, Finnish)" \
--stat "F3(Russia_Bolshoy, Icelandic, Finnish)" \
--stat "F3(Russia_Bolshoy, Lithuanian, Finnish)" \
--stat "F3(Russia_Bolshoy, Norwegian, Finnish)" \
```
Note that `xerxes fstats` will automatically find the right packages from your local archive that contain these groups. You can see in the output of the program which packages contribute:
```text
[Info] 5 relevant packages for chosen statistics identified:
[Info] *2012_PattersonGenetics-2.1.3*
[Info] *2014_LazaridisNature-4.0.2*
[Info] *2016_LazaridisNature-2.1.3*
[Info] *2018_Lamnidis_Fennoscandia-2.1.0*
[Info] *2019_Flegontov_PalaeoEskimo-2.2.1*
```
So these five packages contain the samples requested in these statistics. You can inquire about this also more manually using [`trident list`](https://www.poseidon-adna.org/#/trident?id=list-command).
---
Here is the result that you should get, nicely layouted in a Text-table:
```text
.-----------.----------------.------------.---------.---.---------.----------------.--------------------.------------------.---------------------.
| Statistic | a | b | c | d | NrSites | Estimate_Total | Estimate_Jackknife | StdErr_Jackknife | Z_score_Jackknife |
:===========:================:============:=========:===:=========:================:====================:==================:=====================:
| F3 | Nganasan | French | Finnish | | 593124 | -1.0450e-3 | -1.0451e-3 | 1.2669e-4 | -8.249133659451905 |
| F3 | Nganasan | Icelandic | Finnish | | 593124 | -1.1920e-3 | -1.1920e-3 | 1.3381e-4 | -8.908381869946188 |
| F3 | Nganasan | Lithuanian | Finnish | | 593124 | -1.1605e-3 | -1.1605e-3 | 1.5540e-4 | -7.4680182465607245 |
| F3 | Nganasan | Norwegian | Finnish | | 593124 | -1.0913e-3 | -1.0914e-3 | 1.3921e-4 | -7.83945981272796 |
| F3 | Russia_Bolshoy | French | Finnish | | 542789 | -6.1807e-4 | -6.1809e-4 | 1.0200e-4 | -6.059872900228102 |
| F3 | Russia_Bolshoy | Icelandic | Finnish | | 542789 | -6.2801e-4 | -6.2802e-4 | 1.1792e-4 | -5.325695961373772 |
| F3 | Russia_Bolshoy | Lithuanian | Finnish | | 542789 | -3.7310e-4 | -3.7310e-4 | 1.2973e-4 | -2.8760029685791637 |
| F3 | Russia_Bolshoy | Norwegian | Finnish | | 542789 | -3.7646e-4 | -3.7653e-4 | 1.1630e-4 | -3.2375830440434323 |
'-----------'----------------'------------'---------'---'---------'----------------'--------------------'------------------'---------------------'
```
:::{.callout-tip}
Use the option `-f <FILE>` to output the results additionally to a tab-separated file, or `--raw` if you prefer the standard output to be tab-separated
:::
- All cases of this statistic are negative (`Estimate_Total`).
- Columns `Estimate_Jackknife` and `StdErr_Jackknife` are computed using Jackknifing.
- Z score is key: It gives the deviation of the f3 statistic from zero in units of the standard error.
- As general rule, a Z score of -3 or more suggests a significant rejection of the Null hypothesis that the statistic is not negative.
- Conclusion: Finnish have ancestral admixture of East and West Eurasian ancestry.
## Running xerxes via a configuration file
- Problem: command line gets long, requiring a separate `--stat` option for every statistic.
- More powerful interface to xerxes: configuration files:
```Yaml
fstats:
- type: F3
a: ["Nganasan", "Russia_Bolshoy"]
b: ["French", "Icelandic", "Lithuanian", "Norwegian"]
c: ["Finnish"]
```
You can then run xerxes as
```bash
xerxes fstats -d ~/dev/poseidon-framework/community-archive --statConfig fstats_working/F3_finnish.config
```
- `xerxes` automatically runs all combinations of `a`, `b` and `c`.
:::{.callout-note}
Note that there are actually three types of F3-statistics supported by xerxes:
* `F3vanilla`: The purest form, defined literally as $\langle(c−a)(c−b)\rangle$
* `F3`: A bias-corrected version, which is only valid for groups in C that have non-zero heterozygosity
* `F3star`: This one is normalised by the heterozgygosity of the third population, C, as suggested in [@Patterson2012] and implemented in the [Admixtools package](https://github.com/DReichLab/AdmixTools).
The [white-paper]((https://github.com/poseidon-framework/poseidon-analysis-hs/blob/main/docs/xerxes_whitepaper.pdf)) explains this in detail.
:::
## More about configuration files
The configuration file format has a lot more options. Here is a bit more complex example, but see also [the documentation](https://www.poseidon-adna.org/#/xerxes?id=input-via-a-configuraton-file):
```Yaml
# You can define groups right within the configuration file.
# here we use negative selection to remove individuals from the
# newly defined groups
groupDefs:
CEU2: ["CEU.SG", "-<NA12889.SG>", "-<NA12890.SG>"]
FIN2: ["FIN.SG", "-<HG00383.SG>", "-<HG00384.SG>"]
GBR2: ["GBR.SG", "-<HG01791.SG>", "-<HG02215.SG>"]
IBS2: ["IBS.SG", "-<HG02238.SG>", "-<HG02239.SG>"]
fstats:
- type: F2 # this will create 2x2 = 4 F2-Statistics
a: ["French", "Spanish"]
b: ["Han", "CEU2"]
- type: F3vanilla # This will create 3x2x1 = 6 Statistics
a: ["French", "Spanish", "Mbuti"]
b: ["Han", "CEU2"]
c: ["<Chimp.REF>"]
- type: F4 # This will create 5x5x4x1 = 100 Statistics
a: ["<I0156.SG>", "<I0157.SG>", "<I0159.SG>", "<I0160.SG>", "<I0161.SG>"]
b: ["<I0156.SG>", "<I0157.SG>", "<I0159.SG>", "<I0160.SG>", "<I0161.SG>"]
c: ["CEU2", "FIN2", "GBR2", "IBS2"]
d: ["<Chimp.REF>"]
# Altogether: 110 statistics of different types
```
which will not just create multiple statistic using row-combinations, as described, but also uses newly defined groups and combines multiple statistic types (F2, F3 and F4) in one run.
## F4 Statistics
- a different way to test for admixture
- “F4 statistics” and “D statistics” are very similar ^[Patterson, Nick, Priya Moorjani, Yontao Luo, Swapan Mallick, Nadin Rohland, Yiping Zhan, Teri Genschoreck, Teresa Webster, and David Reich. 2012. “Ancient Admixture in Human History.” Genetics 192 (3): 1065–93.]
- Also defined as correlations of allele frequency differences, similarly to F3 statistics
- involve four populations, not three.
$$F4(A,B;C,D)=\langle(a−b)(c−d)\rangle.$$
:::: {.columns}
::: {.column width="70%"}
- Allele frequency difference between A and B independent from that between C and D.
- F4(A, B; C, D) should then not be statistically different from zero
- With gene flow from C or D into A or B, $F_4$ positive or negative.
- If significantly negative -> gene flow between either C and B, or D and A.
- If significantly positive -> gene flow between A and C, or B and D.
:::
::: {.column width="30%"}
:::
::::
## Shaping intuition - the ABBA- and BABA-sites
| SNP | A | B | C | D | $(a-b)(c-d)$ |
|------------------|---|---|---|---|--------------|
| 1 | 1 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 1 | 1 | 0 |
| 3 | 0 | 1 | 1 | 0 | -1 |
| 4 | 0 | 1 | 0 | 1 | 1 |
| 5 | 1 | 0 | 0 | 1 | -1 |
| 6 | 1 | 0 | 0 | 0 | 0 |
| $F_4(A, B;C, D)$ | | | | | **-0.0167** |
- The only SNPs that contribute positively are SNPs of type `1010` and `0101`
- The only SNPs that contribute negatively are SNPs of type `1001` and `0110`.
- Inuition: In SNPs polymorphis in both $(A,B)$ and $(C,D)$, is B is genetically more similar to C than it is to D?
- Test for "treeness": If A, B, C, D are on a perfect tree, then C should be equally close to C as to D.
- If B is closer to C than to D, or vice versa, the tree is violated: Closer connection between B and C or B and D, depending on the sign of the statistic.
## From single samples to allele frequencies
- ABBA- and BABA-categories of SNPs useful for intuition with single haploid genomes.
- What about population allele frequencies?
- A nice feature of F4-Statistics: averages over all quadruples of samples: With multiple samples in one or multiple slots A, B, C or D, total F4-statistic of the _groups_ is exactly equal to the _average_ of F4-Statistics of the _individuals_.
Let's say we have 2 individuals in each of A and B: $A=\{A_1,A_2\}$ and $B=\{B_1,B_2\}$.
$$F4(A, B; C, D) = \text{Average of}[F4(A_1, B_1; C, D), F4(A_1, B_2; C, D), F4(A_2, B_1; C, D), F4(A_2, B_2; C, D)]$$
In other words: An F4-Statistic _always_ measures the _average_ excess of pairwise BABA SNPs over ABBA SNPs.
:::{.callout-note}
F4-statistics have been famously used to show that Neanderthals are more closely related to Non-African populations than to Africans, suggesting gene-flow between Neanderthals and Non-Africans (shown in [@Green2010]). You can reproduce this famous result with
```bash
xerxes fstats -d ~/poseidon_repo --stat 'F4(<Chimp.REF>,<Altai_published.DG>,Yoruba,French)' \
--stat 'F4(<Chimp.REF>,<Altai_published.DG>,Sardinian,French)'
```
which shows that the first statistic is significantly positive with a Z-score of 7.99, while the second one is insignificantly different from zero (Z=1.01)
:::
## Running F4-Statistics with xerxes
We test for East Asian ancestry in Finns, similarly to the test using Admixture F3 statistics above. We will again use `xerxes fstats`. Configuration file:
```Yaml
fstats:
- type: F4
a: ["Mbuti"]
b: ["Nganasan", "Russia_Bolshoy"]
c: ["French", "Icelandic", "Lithuanian", "Norwegian"]
d: ["Finnish"]
```
Run via
```bash
xerxes fstats -d ~/dev/poseidon-framework/community-archive --statConfig fstats_working/F4_finnish.config
```
Result:
```text
.-----------.-------.----------------.------------.---------.---------.----------------.--------------------.------------------.--------------------.
| Statistic | a | b | c | d | NrSites | Estimate_Total | Estimate_Jackknife | StdErr_Jackknife | Z_score_Jackknife |
:===========:=======:================:============:=========:=========:================:====================:==================:====================:
| F4 | Mbuti | Nganasan | French | Finnish | 593124 | 2.3114e-3 | 2.3115e-3 | 1.2676e-4 | 18.235604067907143 |
| F4 | Mbuti | Nganasan | Icelandic | Finnish | 593124 | 1.6590e-3 | 1.6590e-3 | 1.4861e-4 | 11.163339072181776 |
| F4 | Mbuti | Nganasan | Lithuanian | Finnish | 593124 | 1.3290e-3 | 1.3290e-3 | 1.4681e-4 | 9.052979707622278 |
| F4 | Mbuti | Nganasan | Norwegian | Finnish | 593124 | 1.6503e-3 | 1.6503e-3 | 1.5358e-4 | 10.745850997260929 |
| F4 | Mbuti | Russia_Bolshoy | French | Finnish | 542789 | 1.8785e-3 | 1.8785e-3 | 1.2646e-4 | 14.854487416366263 |
| F4 | Mbuti | Russia_Bolshoy | Icelandic | Finnish | 542789 | 1.0829e-3 | 1.0828e-3 | 1.4963e-4 | 7.236818881873822 |
| F4 | Mbuti | Russia_Bolshoy | Lithuanian | Finnish | 542789 | 5.4902e-4 | 5.4907e-4 | 1.4601e-4 | 3.7605973064589096 |
| F4 | Mbuti | Russia_Bolshoy | Norwegian | Finnish | 542789 | 9.3473e-4 | 9.3475e-4 | 1.5302e-4 | 6.108881868125652 |
'-----------'-------'----------------'------------'---------'---------'----------------'--------------------'------------------'--------------------'
```
In all cases, the Z score is positive and larger than 3, indicating a significant deviation from zero, and implying gene flow between Nganasan and Finnish, and BolshoyOleniOstrov and Finnish, when compared to French, Icelandic, Lithuanian or Norwegian.
## Outgroup-F3-Statistics
Outgroup F3 statistics are a special case how to use F3 statistics. The definition is the same as for Admixture F3 statistics, but instead of a target C and two source populations A and B, one now gives an outgroup C and two test populations A and B.
To get an intuition for this statistics, consider the following tree:
In this scenario, the statistic F3(A, B; C) measures the branch length from C to the common ancestor of A and B, coloured red. So this statistic is simply a measure of how closely two population A and B are related with each other, as measured from a distant outgroup. It is thus a similarity measure: The higher the statistic, the more genetically similar A and B are to one another.
Here is again a SNP table to illustrate, using haploid individuals:
| SNP | A | B | C | $(c-a)(c-b)$ |
|---------------|---|---|---|--------------|
| 1 | 1 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 |
| 3 | 0 | 0 | 1 | 1 |
| 4 | 1 | 0 | 1 | 0 |
| 5 | 1 | 1 | 0 | 1 |
| 6 | 0 | 0 | 1 | 1 |
| $F_3(A, B;C)$ | | | | **0.5** |
You can see that each position which is similar between A and B, but different to C contributes 1, all other SNPs 0. So it directly measures similarity between A and B on alleles that differ from the outgroup C.
:::{.callout-note}
Note that the averaging-relation shown for F4 statistics above is also true for Outgroup-F3 statistics, but only for populations A and B, not for C. So if you have multiple samples in A and B, you may think of this statistic being the _average_ over all pairwise nucleotide similarities between individuals in A and B with respect to the same outgroup C.
:::
## Running Outgroup-F3
We can use this statistic to measure for example the genetic affinity to East Asia, by performing the statistic F3(Han, X; Mbuti), where Mbuti is a distant African population and acts as outgroup here, Han denote Han Chinese, and X denotes various European populations that we want to test.
You can again define a configuration file that performs looping over various populations X for you:
```Yaml
fstats:
- type: F3
a: ["Han"]
b: ["Chuvash", "Albanian", "Armenian", "Bulgarian", "Czech", "Druze", "English",
"Estonian", "Finnish", "French", "Georgian", "Greek", "Hungarian", "Icelandic",
"Italian_North", "Italian_South", "Lithuanian", "Maltese", "Mordovian", "Norwegian",
"Orcadian", "Russian", "Sardinian", "Scottish", "Sicilian", "Spanish_North",
"Spanish", "Ukrainian", "Finland_Levanluhta", "Russia_Bolshoy", "Russia_Chalmny_Varre", "Saami.DG"]
c: ["Mbuti"]
```
which cycles through many populations from Europe, including the ancient individuals from Chalmny Varre, Bolshoy Oleni Ostrov and Levänluhta (described in [@Lamnidis2018]). We store this file in a file called `fstats_working/OutgroupF3_europe.config` and run via:
```bash
xerxes fstats --statConfig fstats_working/OutgroupF3_europe.config -d ~/dev/poseidon-framework/community-archive -f fstats_working/outgroupf3_europe.tsv
```
:::{.callout-warning}
Often in Outgroup-F3-statistics you use single genomes for population C, sometimes even single haploid genomes. In this case, `F3` and `F3star` will get undefined results, because ordinary `F3` and `F3star` statistics require population C to have non-zero average heterozygosity, so you will need at least one diploid sample, or multiple haploid or diploid samples.
Use `F3vanilla` if your third population C is a single pseudo-haploid sample.
:::
## Results (Outgroup F3)
Here is the output of this run (but note that a tab-separated version was also stored in `fstats_working/outgroupf3_europe.tsv` using the option `-f`):
```text
.-----------.-----.----------------------.-------.---.---------.----------------.--------------------.------------------.--------------------.
| Statistic | a | b | c | d | NrSites | Estimate_Total | Estimate_Jackknife | StdErr_Jackknife | Z_score_Jackknife |
:===========:=====:======================:=======:===:=========:================:====================:==================:====================:
| F3 | Han | Chuvash | Mbuti | | 593124 | 5.3967e-2 | 5.3967e-2 | 5.0668e-4 | 106.51180329550319 |
| F3 | Han | Albanian | Mbuti | | 593124 | 4.9972e-2 | 4.9973e-2 | 4.9520e-4 | 100.91326321202445 |
| F3 | Han | Armenian | Mbuti | | 593124 | 4.9531e-2 | 4.9531e-2 | 4.7771e-4 | 103.68366652942314 |
| F3 | Han | Bulgarian | Mbuti | | 593124 | 5.0103e-2 | 5.0103e-2 | 4.8624e-4 | 103.04188532686614 |
| F3 | Han | Czech | Mbuti | | 593124 | 5.0536e-2 | 5.0536e-2 | 4.9261e-4 | 102.58792370749681 |
| F3 | Han | Druze | Mbuti | | 593124 | 4.8564e-2 | 4.8564e-2 | 4.6788e-4 | 103.79674299622445 |
| F3 | Han | English | Mbuti | | 593124 | 5.0280e-2 | 5.0281e-2 | 4.9183e-4 | 102.23198323949656 |
| F3 | Han | Estonian | Mbuti | | 593124 | 5.1154e-2 | 5.1155e-2 | 5.0350e-4 | 101.59882496016485 |
| F3 | Han | Finnish | Mbuti | | 593124 | 5.1784e-2 | 5.1784e-2 | 5.0603e-4 | 102.33488758899031 |
| F3 | Han | French | Mbuti | | 593124 | 5.0207e-2 | 5.0208e-2 | 4.8552e-4 | 103.40976592749682 |
| F3 | Han | Georgian | Mbuti | | 593124 | 4.9711e-2 | 4.9711e-2 | 4.8100e-4 | 103.34881140790415 |
| F3 | Han | Greek | Mbuti | | 593124 | 4.9874e-2 | 4.9874e-2 | 4.8994e-4 | 101.79554640756365 |
| F3 | Han | Hungarian | Mbuti | | 593124 | 5.0497e-2 | 5.0498e-2 | 4.9878e-4 | 101.24215699276706 |
| F3 | Han | Icelandic | Mbuti | | 593124 | 5.0680e-2 | 5.0680e-2 | 4.9729e-4 | 101.91303336514295 |
| F3 | Han | Italian_North | Mbuti | | 593124 | 4.9903e-2 | 4.9904e-2 | 4.8436e-4 | 103.03094306099203 |
| F3 | Han | Italian_South | Mbuti | | 592980 | 4.9201e-2 | 4.9201e-2 | 5.1170e-4 | 96.15239597674244 |
| F3 | Han | Lithuanian | Mbuti | | 593124 | 5.0896e-2 | 5.0896e-2 | 5.0638e-4 | 100.50984037418753 |
| F3 | Han | Maltese | Mbuti | | 593124 | 4.8751e-2 | 4.8751e-2 | 4.7500e-4 | 102.63442479673623 |
| F3 | Han | Mordovian | Mbuti | | 593124 | 5.1820e-2 | 5.1820e-2 | 4.8853e-4 | 106.07409963190884 |
| F3 | Han | Norwegian | Mbuti | | 593124 | 5.0724e-2 | 5.0724e-2 | 4.9514e-4 | 102.4454387098217 |
| F3 | Han | Orcadian | Mbuti | | 593124 | 5.0469e-2 | 5.0469e-2 | 4.9485e-4 | 101.98814656611475 |
| F3 | Han | Russian | Mbuti | | 593124 | 5.1277e-2 | 5.1277e-2 | 4.8613e-4 | 105.48070801791317 |
| F3 | Han | Sardinian | Mbuti | | 593124 | 4.9416e-2 | 4.9417e-2 | 4.8908e-4 | 101.04049389691913 |
| F3 | Han | Scottish | Mbuti | | 593124 | 5.0635e-2 | 5.0635e-2 | 5.0565e-4 | 100.13962104744425 |
| F3 | Han | Sicilian | Mbuti | | 593124 | 4.9194e-2 | 4.9194e-2 | 4.8157e-4 | 102.15353663091187 |
| F3 | Han | Spanish_North | Mbuti | | 593124 | 5.0032e-2 | 5.0032e-2 | 4.9377e-4 | 101.32594226555439 |
| F3 | Han | Spanish | Mbuti | | 593124 | 4.9693e-2 | 4.9693e-2 | 4.8551e-4 | 102.35200847948641 |
| F3 | Han | Ukrainian | Mbuti | | 593124 | 5.0731e-2 | 5.0731e-2 | 4.9506e-4 | 102.47529111692852 |
| F3 | Han | Finland_Levanluhta | Mbuti | | 303033 | 5.4488e-2 | 5.4488e-2 | 5.7681e-4 | 94.46487653920919 |
| F3 | Han | Russia_Bolshoy | Mbuti | | 542789 | 5.7273e-2 | 5.7273e-2 | 5.2875e-4 | 108.31739594898687 |
| F3 | Han | Russia_Chalmny_Varre | Mbuti | | 428215 | 5.4000e-2 | 5.4000e-2 | 5.6936e-4 | 94.84371082564112 |
| F3 | Han | Saami.DG | Mbuti | | 585193 | 5.4727e-2 | 5.4728e-2 | 5.5546e-4 | 98.5265149143263 |
'-----------'-----'----------------------'-------'---'---------'----------------'--------------------'------------------'--------------------'
```
## Plotting Outgroup F3
Now it’s time to plot these results using R. Let's first read in the table:
```{r echo=TRUE}
d <- read.csv("fstats_working/outgroupf3_europe.tsv", sep = "\t")
```
We can check that it worked:
```{r echo=TRUE}
head(d)
```
Nice, now on to plotting in Base-R:
---
```{r echo=TRUE}
#| fig-height: 8
#| fig-width: 8
#| out-width: "100%"
#| fig-align: "center"
order <- order(d$Estimate_Total) # order the estimates for visual effect
x <- d$Estimate_Jackknife[order]
xErr <- d$StdErr_Jackknife[order]
y <- seq_along(d$b)
plot(x, y, xlab = "Z Score", ylab = NA, yaxt = "n", # plot no y-axis ticks
xlim = c(0.048,0.06),
main = "F3(Han, X; Mbuti)")
# plot the labels
text(x + 0.001, y, labels = d$b, adj=0)
# plot the error bars
arrows(x - xErr, y, x + xErr, y, length=0.05, angle=90, code=3)
```
As expected, the ancient samples and modern Saami are the ones with the highest allele sharing with present-day East Asians (as represented by Han) compared to many other Europeans.
## A real Example
Admixture-F3:
::: aside
Patterson et al. 2012
:::