The Exact Stochastic Process of the Haploid Multi-Allelic Wright-Fisher Mutation Model

Diffusion models are widely applied in population genetics, but their approximate solutions may not accurately capture the exact stochastic process. Nevertheless, this practice was necessary due to computing limitations, particularly for large populations. In this article, we develop the exact Markov chain algebra (MCA) for a discrete haploid multi-allelic Wright-Fisher model (MA-WFM) with a full mutation matrix to address this challenge. A special case of nonzero mutations between multiple alleles have not been captured by previous bi-allelic models. We formulated the mean allele frequencies for asymptotic equilibrium analytically for the tri- and quad-allelic case. We also evaluated the exact time-dependent Markov model numerically, presenting it concisely in terms of diffusion variables. The convergence with increasing population size to a diffusion limit is demonstrated for the population composition distribution. Our model shows that there will never be exact irreversible extinction when there are nonzero mutation rates into each allele and never be an exact irreversible fixation when there are nonzero mutation rates out of each allele. We only present results where there is no complete extinction and no complete fixation. Finally, we present detailed computations for the full Markov process, exposing the behavior near the boundaries for the compositional domains, which are non-singular boundaries according to diffusion theory.