persist longer than it takes one species to evolve into a different species, or to diverge into two or more species.

Evolutionary Substitutions and Coalescence Theory

Now that we have defined the conditions for the passage of trans-species polymorphisms, we shall turn the argument around and attempt to extrapolate from the existing polymorphisms to the events converting an ancestral gene pool to a new pool.

The coalescence theory examines the genealogical relationships between genes (see Griffiths, 1980; Hudson, 1990). According to this theory, all alleles present in an extant pool must have descended from a single allele (to which they coalesce). For neutral alleles in a random mating population, the mean coalescence time is given by 4 N[1 - (1/i)] generations, where i is the number of sampled genes. For any two genes (i = 2), the mean coalescence time reduces to 2N generations; for a large number of genes, the mean coalescence time approaches 4N generations. These relationships are important because, if we know the time when alleles coalesce, we can estimate N, the mean number of individuals in the species.

The coalescence theory was originally developed for neutral genes (Kingman, 1982a, b; Tajima, 1983; Tavaré, 1984; and Takahata and Nei, 1985) but has been recently extended to allelic genealogies under balancing selection (Takahata, 1990, 1993a, b; Takahata and Nei, 1990). In the latter situation, the theory has the same mathematical structure as in neutral gene genealogy except for a time-scaling factor, fs. This factor is a function of population size N, selected mutation rate u, and selection coefficient s, and is given approximately by the formula

Given i sampled genes, the coalescence theory permits one to estimate the number of distinct genes, j, that existed at a certain time, with time measured in 2Nfs generations. The allelic phylogeny of the DRB1 locus (Figure 4) shows 58 HLA distinct alleles, of which 10 lineages were already in existence 13 Myr ago. The complete set coalesces 30 Myr ago, that is, before the emergence of the hominoids.

If we assume an average generation time of 15 years, the coalescence would occur two million generations ago. In the case of neutral genes, the expected time to coalescence, given by 4N generations, yields a value of N = 500,000 individuals as the long-term mean

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