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coalescent.html
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coalescent.html
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<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>The coalescent</title>
<meta name="author" content="Trevor Bedford">
<meta name="apple-mobile-web-app-capable" content="yes" />
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<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no">
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<body>
<div style="position: absolute; top:10px; left:10px; z-index:100;">
<a href="http://bedford.io/projects/sismid/sequences/">
<i class="fa fa-times-circle" style="color: #bbb;"></i>
</a>
</div>
<div class="reveal">
<!-- Any section element inside of this container is displayed as a slide -->
<div class="slides">
<section data-background="#8474D1">
<h2 class="title">The coalescent</h2>
</section>
<section data-transition="none">
<h3>Epidemic process</h3>
<img class="stretch" src="images/infection_tree_1.png">
</section>
<section data-transition="none">
<h3>Sample some individuals</h3>
<img class="stretch" src="images/infection_tree_2.png">
</section>
<section data-transition="none">
<h3>Sequence and determine phylogeny</h3>
<img class="stretch" src="images/infection_tree_3.png">
</section>
<section data-transition="none">
<h3>Sequence and determine phylogeny</h3>
<img class="stretch" src="images/infection_tree_4.png">
</section>
<section data-transition="none">
<h3>Wright-Fisher process</h3>
<img class="stretch" src="images/wright_fisher_1.png">
</section>
<section data-transition="none">
<h3>Sample some individuals</h3>
<img class="stretch" src="images/wright_fisher_2.png">
</section>
<section data-transition="none">
<h3>Each generation coalescence may occur</h3>
<img class="stretch" src="images/wright_fisher_3.png">
</section>
<section data-transition="none">
<h3>Each generation coalescence may occur</h3>
<img class="stretch" src="images/wright_fisher_4.png">
</section>
<section data-transition="none">
<h3>Each generation coalescence may occur</h3>
<img class="stretch" src="images/wright_fisher_5.png">
</section>
<section data-transition="none">
<h3>Each generation coalescence may occur</h3>
<img class="stretch" src="images/wright_fisher_6.png">
</section>
<section>
<img class="stretch" src="images/xkcd.png">
</section>
<section>
<h3>Set of coalescent intervals</h3>
<img class="stretch" src="images/coalescent_intervals_1.png">
</section>
<section>
<h3>Per-generation probability of coalescence</h3>
<br>
<p>$$\mathrm{Pr}(\mathrm{coal} \, | \, i=2) = \frac{1}{N}$$</p>
<br>
<p>Probability of first lineage picking an arbitrary parent is 1, while the
probability of the 2nd lineage picking the same parent is $\frac{1}{N}$.</p>
<p>Probability of coalescence scales inversely with population size.</p>
</section>
<section>
<h3>Per-generation probability of coalescence</h3>
<br>
<p>$$\mathrm{Pr}(\mathrm{coal}) = \binom{i}{2} \frac{1}{N} = \frac{i(i-1)}{2N}$$</p>
<br>
<p>There are $\binom{i}{2}$ ways pairs of lineages can pick the same parent.</p>
<p>Probability of coalescence scales quadratically with lineage count.</p>
</section>
<section>
<h3>Expected waiting time to coalescence</h3>
<br>
<p>$$\mathrm{E}[T_i] = \frac{2N}{i(i-1)}$$</p>
<br>
<p>
This is a <a href="https://en.wikipedia.org/wiki/Geometric_distribution">geometric distribution</a>.
If each generation there is a $\frac{1}{x}$ probability of an event occurring, we expect to
wait $x$ generations for the event to occur.
</p>
</section>
<section>
<h3>Continuous time limit</h3>
<br>
<p>With per-generation probability of an event $\frac{1}{x}$ small, but many generations, then the
discrete time geometric distribution approximates to a continuous time
<a href="https://en.wikipedia.org/wiki/Exponential_distribution">exponential distribution</a>.
Thus, we assume $T_i$ to be exponentially distributed with mean
<br>
<p>$$\mathrm{E}[T_i] = \frac{2N}{i(i-1)}.$$</p>
</section>
<section>
<h3>Set of coalescent intervals with waiting times</h3>
<img class="stretch" src="images/coalescent_intervals_2.png">
</section>
<section>
<h3>Gives coalescent trees their distinctive shape</h3>
<img class="stretch" src="images/coalescent_tree_shape.png">
</section>
<section>
<h3>Rate of coalescence scales with population size <i>N</i></h3>
<img class="stretch" src="images/coalescent_pop_size.png">
</section>
<section>
<h2><a href="http://bedford.io/projects/coaltrace/">
Visualization of the coalescent process
</a></h2>
</section>
<section>
<h3>Pairwise genetic diversity</h3>
<img class="stretch" src="images/diversity.png">
</section>
<section>
<h3>Time to the most recent common ancestor (TMRCA)</h3>
<img class="stretch" src="images/tmrca.png">
</section>
<section data-background="#7F85DB">
<h2 class="title">Coalescence patterns can estimate population growth/decline</h2>
</section>
<section>
<h3>Changing population size alters coalescent rate</h3>
<img class="stretch" src="images/changing_pop_size_1.png">
</section>
<section>
<h3>Changing population size alters coalescent rate</h3>
<img class="stretch" src="images/changing_pop_size_2.png">
</section>
<section>
<h3>'Skyline' is flexible demographic model that estimates windows of coalescent rate</h3>
<img class="stretch" src="images/skyline.png">
<div class="citation">
<a href="http://mbe.oxfordjournals.org/content/22/5/1185.short">
Drummond et al. 2005
</a>
</div>
</section>
<section>
<h3>Skyline model shows population growth in HCV</h3>
<img class="stretch" src="images/skyline_hcv.png">
<div class="citation">
<a href="http://www.sciencemag.org/content/292/5525/2323">
Pybus et al. 2001
</a>
</div>
</section>
<section>
<h3>Skyline model shows seasonality in flu</h3>
<img class="stretch" src="images/skyline_flu.png">
<div class="citation">
<a href="http://www.nature.com/nature/journal/v453/n7195/full/nature06945.html">
Rambaut et al. 2008
</a>
</div>
</section>
<section>
<h3>Interpreting rate of coalescence</h3>
<br>
<p>These approaches directly estimate the pairwise rate of coalescence $\lambda$, which is measured
in terms of events per year. Thus, the timescale of coalescence $\frac{1}{\lambda}$ is measured as
the expected waiting time in years for two lineages to find a common ancestor.</p>
</section>
<section>
<h3>Interpreting rate of coalescence</h3>
<br>
<p>The timescale of coalescence $\frac{1}{\lambda}$ is equal to $N_e\tau$, where $N_e$ is
measured in generations and $\tau$ is measured in years per generation. $\tau$ acts to rescale time
from generations to years.
</p>
</section>
<section>
<h3>Example tree</h3>
<img class="stretch" src="images/skyline_exercise_example_tree.png">
</section>
<section>
<h3>Example skyline plot</h3>
<img class="stretch" src="images/skyline_exercise_example_plot.png">
</section>
<section>
<h3><i>Exercise: plot skyline from tree</i></h3>
<img class="stretch" src="images/skyline_exercise_tree.png">
</section>
<section data-background="#7F97DF">
<h2 class="title">Selection distorts coalescence patterns</h2>
</section>
<section>
<h3>Neutral population dynamics</h3>
<img class="stretch" src="images/tree_neutrality.png">
</section>
<section>
<h3>Purifying selection reduces genetic diversity and leads to population stasis</h3>
<img class="stretch" src="images/tree_negative_selection.png">
</section>
<section>
<h3>Positive selection reduces genetic diversity leads to population turnover</h3>
<img class="stretch" src="images/tree_positive_selection.png">
</section>
<section>
<h3>Episodic positive selection shows selective sweeps</h3>
<img class="stretch" src="images/tree_episodic_selection.png">
</section>
</div>
</div>
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