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When approximating the probability of coalescent events with the exponential distribution, it is standard practice to put coalescence times on a continuous scale of units of 2N generations. To see how this continuous time scale operates, let j be time measured as a real number (e.g. 1.0, 1.1, 1.2, 1.3 . . . j) in generations. The time to coalescent events t can then be expressed as t = j/(2N). As an example, imagine that a coalescence event occurred at t = 1.4 on the continuous time scale. That coalescence event could also be thought of as occurring (1.4)(2N) = 2.8N generations in the past (see Fig. 3.26). If the population size was 2N = 100 lineages, then that coalescent event was (1.4)(100) = 140 generations in the past. However, if the population size was 2N = 20 lineages, then that coalescent event was

(1.4)(20) = 28 generations in the past. Aside from the practical matter of interpreting a specific time value, the use of a continuous time scale makes an important biological point about the effects of population size on the coalescence process. The basic nature of the coalescent process is identical for all populations no matter what their size. For example, in populations of any size a single coalescent event will occur faster on average when there is a larger number of lineages sampled than when just a pair of lineages is sampled. Population size just serves to scale the time required for coalescent events to occur. Coalescent events occur more rapidly in small populations compared to bigger populations, a conclusion analogous to that reached for genetic drift earlier in the chapter.

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