On the genealogy of a sample of neutral rare alleles

This paper concerns the genealogical structure of a sample of chromosomes s haring a neutral rare allele. We suppose that the mutation giving rise to t he allele has only happened once in the history of the entire population, a nd that the allele is of known frequency q in the population. Within a coal escent framework C. Wiuf and P. Donnelly (1999, Theor. Popul. Biol. 56, 183 -201) derived an exact analysis of the conditional genealogy but it is inco nvenient for applications. Here, we develop an approximation to the exact d istribution of the conditional genealogy, including an approximation to the distribution of the time at which the mutation arose. The approximations a re accurate for frequencies q<5-10%. In addition, a simple and fast simulat ion scheme is constructed. We consider a demography parameterized by a d-di mensional vector alpha = (alpha(1),..., alpha(d)). It is shown that the con ditional genealogy and the age of the mutation have distributions that depe nd on a = qa and q only, and that the effect of q is a linear scaling of ti mes in the genealogy; if q is doubled, the lengths of all branches in the g enealogy are doubled. The theory is exemplified in two different demographi es of some interest in the study of human evolution: (1) a population of co nstant size and (2) a population of exponentially decreasing size (going ba ckward in time). (C) 2000 Academic Press.