Question

Imagine that you have a sample of 2 chromosomes in the 2 island model of population structure. Each island has a size 10,000 diploid individuals. Assume that m=0.005. Calculate: (a) The probability of a coalescent event occurring before a migration event if the 2 chromosomes are sampled from the same population. (b) The probability of a coalescent event occurring before a migration event if the 2 chromosomes are sampled from different populations. (c) The probability of a coalescent event occurring exactly 1 generation ago if the 2 chromosomes are sampled from different populations. (d) The expected coalescent time if the 2 chromosomes are sampled from different populations.

Answer 1

The word “coalescent” is used in several ways in the literature, and it will also be used in several ways here. Hopefully, the meaning will be clear from the context. The coalescent, or perhaps more appropriately, the coalescent approach, is based on two fundamental insights, which are the topic of the next subsection. The subsection after that describes the stochastic process known as the coalescent, or sometimes Kingman’s coalescent in honor of its discoverer [42, 43, 44]. This process results from combining the two fundamental insights with a convenient limit approximation. The coalescent will be introduced in the setting of the Wright-Fisher model of neutral evolution, but it applies more generally. This is one of the main topics for the remainder of the chapter. First of all, many different neutral models can be shown to converge to Kingman’s coalescent. Second, more complex neutral models often converge to coalescent processes analogous to Kingman’s coalescent. The coalescent was described by Kingman [42, 43, 44], but it was also discovered independently by Hudson [27] and by Tajima [83]. Indeed, arguments anticipating it had been used several times in population genetics (reviewed by Tavar´e [90]). The fundamental insights The first insight is that since selectively neutral variants by definition do not affect reproductive success, it is possible to separate the neutral mutation process from the genealogical process. In classical terms, “state” can be separated from “descent”. To see how this works, consider a population of N clonal organisms that reproduce according to the neutral Wright-Fisher model, i. e., generations are discrete, and each new generation is formed by randomly sampling N parents Magnus Nordborg 3 with replacement from the current generation. The number of offspring contributed by a particular individual is thus binomially distributed with parameters N (the number of trials) and 1/N (the probability of being chosen), and the joint distribution of the numbers of offspring produced by all N individuals is symmetrically multinomial. Now consider the random genealogical relationships (i. e., “who begat whom”) that result from reproduction in this setting. These can be represented graphically, as shown in Figure 1. Going forward in time, lineages branch whenever an individual produces two or more offspring, and end when there is no offspring. Going backward in time, lineages coalesce whenever two or more individuals were produced by the same parent. They never end. If we trace the ancestry of a group of individuals back through time, the number of distinct lineages will decrease and eventually reach one, when the most recent common ancestor (MRCA) of the individuals in question is encountered. None of this is affected by neutral genetic differences between the individuals. mutation time Figure 1: The neutral mutation process can be separated from the geneal

The neutral mutation process can be separated from the genealogical process. The genealogical relationships in a particular 10-generation realization of the neutral Wright-Fisher model (with population size N = 10) are shown on the left. On the right, allelic states of have been superimposed (so-called “gene dropping”). As a consequence, the evolutionary dynamics of neutral allelic variants can be modeled through so-called “gene dropping” (“mutation dropping” would be more accurate): given a realization of the genealogical process, allelic states are assigned to the original generation in a suitable manner, and the lines of descent then simply followed forward in time, using the rule that offspring inherit the allelic state of their parent unless there is a mutation (which occurs with some probability each generation). In particular, the allelic states of any group of individuals (for instance, all the members of a given generation) can be generated by assigning an allelic state to their MRCA and then “dropping” mutations along the branches of the genealogical tree that leads to them. Most of the genealogical history of the population is then irrelevant (cf. Figures 1 and 2). The second insight is that it is possible to model the genealogy of a group of