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Figure 3.23 Haploid (a) and diploid (b) reproduction in the context of coalescent events. In a haploid population, the probability of coalescence is (dashed lines) whereas the probability that two lineages do not have a common ancestor in the previous generation is 1 - (solid lines). In a diploid population, the two gene or allele copies in one individual in the present time have one ancestor in the female population (Nf) and one ancestor in the male population (Nm). Coalescent events in the diploid population arise when the gene copies in males and females are identical by descent. The haploid model with 2N lineages is routinely used to approximate the diploid model with N = Nj + Nm diploid individuals.

t t the process of reproduction for diploid lineages. In diploid reproduction, each offspring is composed of one allele copy inherited from a female parent and another allele copy from a male parent (Fig. 3.23b). In a time-backward view, this can be thought of as reproduction where one allele copy finds its ancestor in the male population of the last generation while the other allele copy finds its ancestor in the female population of the last generation. For a given male or female parent, each of their two allele copies has a probability of V2 of being the ancestral copy. As long as the number of males and females in a diploid population is equal and the haploid and diploid population sizes are large, the predictions of the coalescent model are very similar for haploid and diploid populations containing an identical total number of gene copies. The haploid model is more straightforward and so it is used throughout this section. In practice, the predictions that follow from the haploid coalescent model can be applied to samples of lineages (usually DNA sequences) from diploid organisms.

Like Markov chains, the probability of coalescence displays the Markov property since it is an independent event that depends only on the state of the population at the point of time of interest. Because of this, the basic probabilities of coalescence and non-coalescence between two generations can be used to describe the probability of coalescence over an arbitrary number of generations. If two randomly sampled lineages do not coalesce for t - 1 generations, then the probability that they do coalesce to their common ancestor in generation t is:

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