Mrca

Past

Deme 1

-- Coalescence

-- Migration

-- Coalescence Coalescence

Migration Coalescence

Coalescence Migration

Deme 2

Present

Figure 4.16 A hypothetical genealogy for two demes. Initially there are three lineages in each deme. The very first event going back in time is the migration of a lineage from deme one into deme two. Immediately after this migration occurs, the chance of coalescence in deme two increases since there are more lineages and the chance of coalescence in deme one decreases since there are fewer lineages. Continuing back in time, a coalescence event occurs in deme one and then a coalescence event occurs in deme two. The lineage that migrated out of deme one migrates back into deme one by chance. Coalescence to the single most recent common ancestor (MRCA) of all lineages cannot occur until the final two lineages are brought together in a single deme by migration.

Combining coalescent and migration events

Describing genealogies with gene flow can be accomplished by adding another type of possible event that can occur working from the present to a time in the past where all lineages find their most recent common ancestor. We will assume that both coalescent and migration events are rare (or that Ne is large and the rate of migration is small), so that when an event does occur going back in time it is either coalescence or migration. In other words, we will assume that migration and coalescence events are mutually exclusive. The fact that events are mutually exclusive is an important assumption. When two independent processes are operating, the coalescence model becomes one of following lineages back in time and waiting for an event to happen. When events are independent but mutually exclusive, the probability of each event is added over all possible events to obtain the total chance that an event occurs. For example, the chance that a diploid genotype for a diallelic locus is a heterozygote under random mating is 2pq. This is the sum of the independent chance of sampling Aa and the chance of sampling aA since a heterozygote results from one of the two ways of sampling of two different alleles (the probability of a heterozygote under random mating is not (pq)2, which is the chance of sampling Aa and aA simultaneously). Therefore, if we can find an exponential approximation for the chance that a lineage migrates to a different deme each generation, we can just add this to the exponential approximation for the chance of coalescence.

In a subdivided population, each generation there is the chance that a lineage in one deme migrates to some other deme. The rate of migration, m, is the chance that a lineage migrates each generation. The chance that a lineage does not migrate is therefore 1 - m each generation. The chance that t generations pass before a migration event occurs is then the product of the chances of t - 1 generations of no migration followed by a migration, or

migration

This is in an identical form to the chances that a coalescent event occurs after t generations given in Chapter 3. Like the probability of coalescence, the probability of a migration through time is a geometric series that can be approximated by the exponential distribution (see Math box 3.2). To obtain the exponent of e (or the intensity of the migration process), we need to determine the rate at which migration is expected to occur in a population.

Now consider migration events in the context of an island model of gene flow where there are d demes and each deme contains 2Ne lineages. The total population size is the sum of the sizes of all demes or 2Ned lineages. When time is measured on a continuous scale with t = ■

, one unit of time is equivalent to 2Ned generations. If 2Ned generations elapse and m is the chance of migration per generation, then 2Nedm migration events are expected in the total population during one unit of continuous time. If we define M = 4Nem, then M/2 is equivalent to 2Nem or the chance that a lineage in one deme migrates (the per deme migration rate). The chance of migration is independent in all of the demes, so the expected number of migration events in the total population is the sum of the per deme chances M

of migration or — d. This leads to the exponential approximation for the chances that a single lineage in any of the demes migrates at generation t:

two demes (d = 2) with k1 and k2 ancestral lineages in each deme, equation 4.70 reduces to

P(Tmigration t) e

on a continuous time scale. When there is more than one lineage, each lineage has an independent chance of migrating but only one lineage will migrate. So we

Md add the e 2 chance of migration for each lineage over all k lineages to obtain the total chance of migration:

-tMik

P(Tmigration t) e for k ancestral lineages of the d demes. The chance that one of k lineages migrates at or before a certain time can then be approximated with the cumulative exponential distribution:

P(Tmigration

in exactly the same fashion that times to coalescent events are approximated.

When two independent processes are operating, the genealogical model becomes one of following lineages back in time and waiting for an event to happen. The possible events in this case are migration or coalescence, so the total chance of any event is the sum of the independent probabilities of each type of mutually exclusive event. Since lineages cannot coalesce unless they are in the same deme, the chance of a coalescent event is

when there are ki ancestral lineages in deme i, a slightly modified version of the basic coalescent model that takes into account the d demes and time scaling by 2Ned. (Note that when d = 1 the expected time to coalescence reduces to —-^ on a continuous time

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