Fixation at a locus with multiple alleles: Structure and solution of the Wright Fisher model

We consider the Wright Fisher model for a finite population of diploid sexual organisms where selection acts at a locus with multiple alleles. The mathematical description of a such a model requires vectors and matrices of a multidimensional nature, and hence has a considerable level of complexity. In the present work we avoid this complexity by introducing a simple mathematical transformation. This yields a description of the model in terms of ordinary vectors and ordinary matrices, thereby allowing standard linear algebra techniques to be directly employed. The new description yields a common mathematical representation of the Wright Fisher model that applies for arbitrary numbers of alleles. Within this framework, it is shown how the dynamics decomposes into component parts that are responsible for the different possible transitions of segregating and fixed populations, thereby allowing a clearer understanding of the population dynamics. This decomposition allows expressions to be directly derived for the mean time of fixation, the mean time of segregation (i.e., the sojourn time) and the probability of fixation. Numerical methods are discussed for the evaluation of these quantities.