Maria Speed

In population genetics, the Dirichlet (also called the Balding--Nichols) model has for 20 years been considered the key model to approximate the distribution of allele fractions within populations in a multi-allelic setting. It has often been noted that the Dirichlet assumption is approximate because positive correlations among alleles cannot be accommodated under the Dirichlet model. However, the validity of the Dirichlet distribution has never been systematically investigated in a general framework. This paper attempts to address this problem by providing a general overview of how allele fraction data under the most common multi-allelic mutational structures should be modeled. The Dirichlet and alternative models are investigated by simulating allele fractions from a diffusion approximation of the multi-allelic Wright--Fisher process with mutation, and applying a moment-based analysis method. The study shows that the optimal modeling strategy for the distribution of allele fractions depends on the specific mutation process. The Dirichlet model is only an exceptionally good approximation for the pure drift, Jukes--Cantor and parent-independent mutation processes with small mutation rates. Alternative models are required and proposed for the other mutation processes, such as a Beta--Dirichlet model for the infinite alleles mutation process, and a Hierarchical Beta model for the Kimura, Hasegawa--Kishino--Yano and Tamura--Nei processes. Finally, a novel Hierarchical Beta approximation is developed, a Pyramidal Hierarchical Beta model, for the generalized time-reversible and single-step mutation processes.