From d79e4d6a2bfe118e06301e5c5e35130b1576bc58 Mon Sep 17 00:00:00 2001 From: Per Unneberg Date: Mon, 23 Oct 2023 23:01:49 +0200 Subject: [PATCH] Add missing slide to foundations --- docs/slides/foundations/index.qmd | 280 +++++++++++++++++++++++++++++- 1 file changed, 279 insertions(+), 1 deletion(-) diff --git a/docs/slides/foundations/index.qmd b/docs/slides/foundations/index.qmd index a4371e86..b1847bf1 100644 --- a/docs/slides/foundations/index.qmd +++ b/docs/slides/foundations/index.qmd @@ -2334,7 +2334,285 @@ floating up and down at random. ::: -## Allele frequency distribution for +## 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? + +:::: {.columns .fragment fragment-index=1} + +::: {.column width="10%"} + +t=0 + +::: + +::: {.column width="30%"} + +```{r, engine='tikz', fig.ext="svg"} +#| label: tikz-afd-aa +#| echo: false +#| eval: true +#| out-width: 200px +\begin{tikzpicture}[ybar interval] +\tikzstyle{tick}=[font=\scriptsize]; +\tikzstyle{label}=[font=\small]; + \draw[fill=black, draw=black] +plot coordinates{(0,1) (1,0) (2, 0) (3, 0)}; +\node[label] (aa) at (0.5, 1.8) {aa}; +\node[label] (Aa) at (1.5, 1.8) {Aa}; +\node[label] (AA) at (2.5, 1.8) {AA}; +\node[label] (0) at (0.5, 1.4) {0}; +\node[label] (1) at (1.5, 1.4) {1}; +\node[label] (2) at (2.5, 1.4) {2}; + \draw[->] (0, -0.2) -- (0, 1.2); +\draw (-0.1, 1) -- (0.1, 1); +\draw (-0.1, 0) -- (0.1, 0); +\node[tick] at (-0.4, 1) {1}; +\node[tick] at (-0.4, 0) {0}; +\end{tikzpicture} +``` + +::: + +::: {.column width="30%"} + +```{r, engine='tikz', fig.ext="svg"} +#| label: tikz-afd-aA +#| echo: false +#| eval: true +#| out-width: 200px +\begin{tikzpicture}[ybar interval] +\tikzstyle{tick}=[font=\scriptsize]; +\tikzstyle{label}=[font=\small]; + \draw[fill=black, draw=black] +plot coordinates{(0,0) (1,1) (2, 0) (3, 0)}; +\node[label] (aa) at (0.5, 1.8) {aa}; +\node[label] (aA) at (1.5, 1.8) {aA}; +\node[label] (AA) at (2.5, 1.8) {AA}; +\node[label] (0) at (0.5, 1.4) {0}; +\node[label] (1) at (1.5, 1.4) {1}; +\node[label] (2) at (2.5, 1.4) {2}; + \draw[->] (0, -0.2) -- (0, 1.2); +\draw (-0.1, 1) -- (0.1, 1); +\draw (-0.1, 0) -- (0.1, 0); +\node[tick] at (-0.4, 1) {1}; +\node[tick] at (-0.4, 0) {0}; +\end{tikzpicture} +``` + +::: + +::: {.column width="30%"} + +```{r, engine='tikz', fig.ext="svg"} +#| label: tikz-afd-AA +#| echo: false +#| eval: true +#| out-width: 200px +\begin{tikzpicture}[ybar interval] +\tikzstyle{tick}=[font=\scriptsize]; +\tikzstyle{label}=[font=\small]; + \draw[fill=black, draw=black] +plot coordinates{(0,0) (1,0) (2, 1) (3, 1)}; +\node[label] (aa) at (0.5, 1.8) {aa}; +\node[label] (aA) at (1.5, 1.8) {aA}; +\node[label] (AA) at (2.5, 1.8) {AA}; +\node[label] (0) at (0.5, 1.4) {0}; +\node[label] (1) at (1.5, 1.4) {1}; +\node[label] (2) at (2.5, 1.4) {2}; + \draw[->] (0, -0.2) -- (0, 1.2); +\draw (-0.1, 1) -- (0.1, 1); +\draw (-0.1, 0) -- (0.1, 0); +\node[tick] at (-0.4, 1) {1}; +\node[tick] at (-0.4, 0) {0}; +\end{tikzpicture} +``` + +::: + +:::: + +:::::{.translatey50} + +:::: {.columns} + +::: {.column width="10%"} + +::: {.fragment fragment-index=2} + +t=1 + +::: + +::: + +::: {.column width="30%"} + +::: {.fragment fragment-index=2} + +```{r, engine='tikz', fig.ext="svg"} +#| label: tikz-afd-aa-1 +#| echo: false +#| eval: true +#| out-width: 200px +\begin{tikzpicture}[ybar interval] +\tikzstyle{tick}=[font=\scriptsize]; +\tikzstyle{label}=[font=\small]; + \draw[fill=black, draw=black] +plot coordinates{(0,1) (1,0) (2, 0) (3, 0)}; + \draw[->] (0, -0.2) -- (0, 1.2); +\draw (-0.1, 1) -- (0.1, 1); +\draw (-0.1, 0) -- (0.1, 0); +\node[tick] at (-0.4, 1) {1}; +\node[tick] at (-0.4, 0) {0}; +\end{tikzpicture} +``` + +::: + +::: + +::: {.column width="30%"} + +::: {.fragment fragment-index=4} + +```{r, engine='tikz', fig.ext="svg"} +#| label: tikz-afd-aA-1 +#| echo: false +#| eval: true +#| out-width: 200px +\begin{tikzpicture}[ybar interval] +\tikzstyle{tick}=[font=\scriptsize]; +\tikzstyle{label}=[font=\small]; + \draw[fill=black, draw=black] +plot coordinates{(0,0.25) (1,0.5) (2, 0.25) (3, 0.25)}; + \draw[->] (0, -0.2) -- (0, 1.2); +\draw (-0.1, 1) -- (0.1, 1); +\draw (-0.1, 0) -- (0.1, 0); +\node[tick] at (-0.4, 1) {1}; +\node[tick] at (-0.4, 0) {0}; +\end{tikzpicture} +``` + +::: + +::: + +::: {.column width="30%"} + +::: {.fragment fragment-index=3} + +```{r, engine='tikz', fig.ext="svg"} +#| label: tikz-afd-AA-1 +#| echo: false +#| eval: true +#| out-width: 200px +\begin{tikzpicture}[ybar interval] +\tikzstyle{tick}=[font=\scriptsize]; +\tikzstyle{label}=[font=\small]; + \draw[fill=black, draw=black] +plot coordinates{(0,0) (1,0) (2, 1) (3, 1)}; + \draw[->] (0, -0.2) -- (0, 1.2); +\draw (-0.1, 1) -- (0.1, 1); +\draw (-0.1, 0) -- (0.1, 0); +\node[tick] at (-0.4, 1) {1}; +\node[tick] at (-0.4, 0) {0}; +\end{tikzpicture} +``` + +::: + +::: + +:::: + +::::: + +:::::{.translatey100} + +:::: {.columns} + +::: {.column width="10%"} + +::: {.fragment fragment-index=5} + +t=2 + +::: + +::: + +::: {.column width="30%"} + +::: + +::: {.column width="30%"} + +::: {.fragment fragment-index=5} + +```{r, engine='tikz', fig.ext="svg"} +#| label: tikz-afd-aA-2 +#| echo: false +#| eval: true +#| out-width: 200px +\begin{tikzpicture}[ybar interval] +\tikzstyle{tick}=[font=\scriptsize]; +\tikzstyle{label}=[font=\small]; + \draw[fill=black, draw=black] +plot coordinates{(0,0.375) (1,0.25) (2, 0.375) (3, 0.25)}; + \draw[->] (0, -0.2) -- (0, 1.2); +\draw (-0.1, 1) -- (0.1, 1); +\draw (-0.1, 0) -- (0.1, 0); +\node[tick] at (-0.4, 1) {1}; +\node[tick] at (-0.4, 0) {0}; +\end{tikzpicture} +``` + +::: + +::: + +::: {.column width="30%"} + +::: + +:::: + +::::: + +::: {.notes} + +Example from [@gillespie_PopulationGeneticsConcise_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