D ddd

where each dt is the number of demes with i lineages and n is the total number of demes. The total number of lineages is then the product of the number of demes containing i lineages and the number of lineages i summed over all possible numbers of lineages n per deme or ^ idi. With a sample of two lineages i=1

taken from a total population composed of two demes, there are two possible ways the lineages could be sampled. The two lineages could either be sampled from different demes to give d = (2,0) or sampled from a single deme to give d = (0,1). This notation specifies what is called the sample configuration of a number of lineages drawn from some number of demes. Figure 4.18 gives several examples of sample configurations for two or three demes. With coalescence to a single ancestral lineage the sample configuration becomes (1). This sample configuration notation is useful because the mean and variance of coalescence times in a structured population depends on whether lineages are located in the same or different demes.

Figure 4.18 The possible events that can occur when two lineages are in the same deme (0,1) or when two lineages are in two different demes (2,0) along with their probabilities of occurring. The separation between demes is represented by a dotted vertical line. Two lineages can coalesce only when they are in the same deme. The probability of coalescence (a), migration of one lineage such that the two lineages are in different demes (b), and migration that places both lineages in the same deme (c) determine the overall chances that two lineages coalesce. The chance that both lineages migrate (with probability m2) is not shown in (b) and applies when there are three or more demes.

With that background on sample configurations, let's move on to derive the average coalescence time and expected total length of a genealogy in a structured population. We will focus on the simplest case of two lineages in the context of two demes. We need to determine the chances that two lineages in either of the two possible sample configurations ((2,0) or (0,1)) experience coalescence. Figure 4.18 shows these possible transitions between sample configuration states. As in the basic coalescent process, the chance of coalescence is the product of one over the population size and the number of unique pairs of lineages that can coalesce. If each deme contains

2Ne lineages the probability of coalescence is-for

2Ne two lineages in one deme. However, two lineages cannot coalesce unless they are together in the same deme and restricted gene flow will make this less likely to happen.

For two lineages that are together in the same deme, or in sample configuration (0,1), there are two possible events that eventually lead to coalescence. The first possible event is simply that the two lineages coalesce with probability-. The second possible

2Ne event is that one or both of the two lineages migrates into another deme before they can coalesce. If the proportion of migrants per generation in any deme is m then the chance that a single lineage is an emigrant is m and the chance that it is not an emigrant is 1 - m. The chance that one lineage migrates and the other lineage does not is m(1 - m) + (1 - m)m = 2m(1 - m). The chance that both lineages migrate is m2. The total chance that one or both lineages migrate is then 2m(1 - m) + m2, which is approximately 2 m if m is small and m2 terms can be ignored. For two lineages in the same deme or (0,1), the total chance that any event occurs in the previous generation, either coalescence or migration, is therefore 2 m +——.

For two lineages that are in different demes, or in sample configuration (2,0), the total chance that one lineage migrates is 2m, following the same logic as when two lineages are in a single deme. However, to transition from (2,0) to (0,1), the migration event is not into any random deme but must be into the one other deme where the second lineage is found. The chance that migration into a specific deme occurs is-where d is the number of demes. The total d -1

probability that two lineages initially in separate demes end up in a single deme with the possibility that they can later coalesce is therefore 2m-

To determine the average time to coalescence in two demes we can use the fact that the average time to an event is one over the probability of each event in a process where waiting times are exponentially distributed. Let T(0 1) represent the average time until coalescence for two lineages in the same deme and T(2 0) the average time to coalescence for two lineages in different demes. For two lineages in the same deme, the average time to coalescence is the average time to coalescence plus the average time spent in two different demes if there is a migration event. The average time either of coalescence or migration is

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