## S N

Past

Present

Time

Figure 3.28 The effects of a population bottleneck on gene genealogies. During the bottleneck the chance that two randomly sampled gene copies are derived from one copy in the previous generation

2Ne increases. This can also be thought of as a reduction in the overall height of a genealogical tree caused by the bottleneck since lineages that find their ancestors during the bottleneck lead to short branches. The overall effect of a bottleneck on coalescence among gene copies sampled in the present depends on the reduction in the effective population size and the duration. The arrows indicate the point in time when gene copies were sampled from the population.

After a population recovers from a bottleneck, it is possible that the lineages present will all descend from a most recent common ancestor during the bottleneck period. If this occurs, it is equivalent to saying that a single lineage among those present in the pre-bottleneck population becomes the most recent common ancestor of all lineages during the bottleneck. The chance that this occurs increases as the bottleneck exhibits a smaller population size or persists for a longer period of time. The expected height of a coalescent tree (the sum of all the time periods between coalescence of pairs of gene copies until there is a single lineage, equation 3.75) can be used to show this effect quantitatively. In the continuous coalescent the expected height in units of

2N generations is 2

k where k is the number of gene copies in the present. Figure 3.28 shows k = 10 so the expected height of each coalescent tree is (1.8)(2Ne) = 3.6Ne generations. Before the bottleneck, a sample of 10 gene copies from a population of Ne = 100 would coalesce to a single lineage in an average of 360 generations. The same sample of 10 gene copies taken from a population of Ne = 10 would coalesce to a single lineage in an average of 36 generations. At the time point closest to the present in Fig. 3.28, the population has an effective size of about 25 based on the harmonic mean as well as the probability of coalescence over three generations.

Therefore, the expected height of the coalescent tree is 90 generations.

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