A Haploid Wright-Fisher Mutation Model with Three Alleles

Mainstream population genetics uses diffusion models with approximative solutions rather than encompassing the exact stochastic process. This practice was necessary due to practical computing limitations for large populations. The present paper develops the exact Markov chain algebra (MCA) for a discrete haploid tri-allelic Wright-Fisher model (TA-WFM) with a full mutation matrix to address this challenge. Nonzero mutations between all three alleles give a tri-allelic model irreducible to previous bi-allelic models. The gamete frequencies for asymptotic equilibrium are calculated analytically. The exact time-dependent Markov model is evaluated numerically and presented concisely in terms of diffusion variables. The convergence with increasing population size to a diffusion limit is demonstrated for the population composition distribution. There will never be exact (irreversible) extinction when there are nonzero mutation rates into each allele. Moreover, there will never be an exact (irreversible) fixation when there are nonzero mutation rates out from each allele. From our general haploid triallelic mutation model, we only present results where there is no exact extinction and no exact fixation. We present some computations for the Markov process in full detail, exposing the behavior near the boundaries for the compositional domains, which are non-singular boundaries according to diffusion theory.