Population history and infrequent mutations: How old is a rare mutation? GUCY2D as a worked example

The mosaic pattern of haplotypes observed around a single mutation results from one or several founder events. The difficulties involved in calculating the age of the variant are greatly reduced by assuming a single event, but this simplification may bias analysis of the genealogy of the mutation. However, if it is assumed that more than one founder event occurred, the number of genealogies is very large and the likelihood of every possible tree could not be realistically calculated. A multipoint approach is required, given the number of independent variables needed to describe a complex bifurcating genealogy. Starting from the observation that a limited number of parameters is needed for calculation of the simplest models of bifurcating genealogies, we show that the probability density of a two-ancestor model genealogy can be simply described as an algebraic function in a closed form, two coalescence times being calculated simultaneously without compromising accuracy. Implementation in a Bayesian framework is facilitated by the simplicity of the function, which describes the reciprocal relationship between the region of complete linkage disequilibrium and the branch length of the tree. We illustrate the use of haplotype information about allele-sharing decay around a mutation as a genetic clock, using data for two GUCY2D mutations in Mediterranean populations.