Origins of the Coalescent: 1974-1982

Abstract

THE circle of ideas that has come to be known as the coalescent has proved to be a useful tool in a range of genetical problems, both in modeling biological phenomena and in making statistical sense of the rich data now available. In the 20 years since the concept crystallized, it has been extended in a number of directions. It is not the purpose of this note to document recent developments or to record the way in which others later arrived at similar conclusions by different routes; for that, consult, for instance, Hudson (1990), Donnelly and Tavaré (1995), and Stephens and Donnelly (2000). The technique has been widely applied in recent studies of evolution, thanks to advances in molecular biology and computer technology. By being sample based, it has provided rigorous statistical analyses of population data and has provided a rationale for designing simulations. It has led to two different estimators of the key parameter, θ = 4Nμ, where N is the effective population number and μ is the mutation rate, and, therefore, to a test of neutrality. It has provided estimates of the time to a common ancestor, and, in particular, a very long time estimate provided strong evidence for balancing selection in the ancestry of the HLA and other loci. This technique has also provided estimates of recombination and rate of selfing. It has been helpful in assessing migration patterns in human ancestry, in particular, sex differences as revealed by comparison of within- and between-group variability of Y chromosome and mitochondrial DNA. For a recent review, see Fu and Li (1999). I shall not discuss these applications either. Mine is the much simpler aim of describing the way in which the ideas first came together, in the period leading to my 1982 articles. This is inevitably a personal account, but one that I hope is accurate, being based on records from these years. I have had the benefit of comments from Warren Ewens and Peter Donnelly, for which I am most grateful, but the interpretations and emphases are mine. Three insights, in combination, comprise the essential basis of the coalescent. The first is the idea of tracing the ancestry of a gene backward in time and building up the family tree of the genes (at a particular locus) in a population sample back to the point at which they have a single common ancestor. This is just a generalization of Malécot’s “identity by descent” (Nagylaki 1989) to more than two genes. It becomes powerful because of the second insight, that for a large class of demographic models, characterized by selective neutrality and constrained population size, the stochastic structure of the genealogy does not depend on the detail of the reproductive mechanism. Finally, for such models the effect of mutation is statistically independent of the genealogy. What is surprising is that these rather simple ideas took so long to emerge. For me the story begins in 1974, when I was traveling in Australia meeting mathematicians who shared my interests in random processes and their applications. I had not worked on genetics since, as a Cambridge undergraduate, I had published juvenilia on polymorphisms maintained by single locus selection. But in Melbourne I encountered Warren Ewens, who was exploring some ideas of Ohta and Kimura (1973) on neutral evolution in finite populations (the “charge-state” model). Moving on to Canberra, I found that Pat Moran was working on a similar problem (Moran 1975), and the enthusiasm of these two Australians inspired their English visitor. They considered a locus at which the different alleles can be labeled by a single numerical quantity and in which mutation causes a random addition or subtraction. Thus, a single line of descent will have genes that perform a random walk on the line. There was little biological credibility in such a description, but it accorded with the experimental techniques of gel electrophoresis, which were then the best way of distinguishing alleles (e.g., Singh et al. 1976). A haploid population of fixed size n was supposed to evolve in discrete generations, the numbers of children of the members of one generation having a symmetric multinomial joint distribution (the Wright-Fisher model). Thus, the genes of one generation are represented by N points on the line. As we observe from generation to generation, the N points perform a “coherent random walk.” The group strays farther and farther from its starting point, but the extent of the group remains relatively limited, and the distribution of the relative positions of the points converges to a proper limit. In my 1976 article I quoted Ewens as explaining this phenomenon by noting that “the probability that two points of Gt (the tth generation) have a common ancestor in Gs is 1 - (1 - N-1)t-s, which is near unity when (t - s) is large compared with N. Thus, the whole of Gt is descended from a common ancestor in Gt-Δ, where the random integer Δ = Δ(t) remains stochastically bounded as t → ∞. The relative