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

Area of Science:

• Population Genetics
• Theoretical Biology
• Mathematical Biology

Background:

• Diffusion models are common in population genetics but offer approximate solutions.
• Computing limitations historically necessitated approximations for large populations.
• Previous models often overlooked complex multi-allelic mutation dynamics.

Purpose of the Study:

• To develop an exact Markov chain algebra (MCA) for a discrete haploid multi-allelic Wright-Fisher model (MA-WFM).
• To address limitations of diffusion approximations in capturing exact stochastic processes.
• To incorporate a full mutation matrix, including nonzero mutations between multiple alleles.

Main Methods:

• Formulation of exact Markov chain algebra for the MA-WFM.
• Analytical derivation of mean allele frequencies at asymptotic equilibrium for tri- and quad-allelic cases.
• Numerical evaluation of the exact time-dependent Markov model, presented using diffusion variables.

Main Results:

• Demonstrated convergence of population composition distribution to a diffusion limit with increasing population size.
• Showed that nonzero mutation rates prevent exact irreversible extinction or fixation.
• Provided detailed computations of the Markov process, revealing non-singular boundary behaviors.

Conclusions:

• The developed MCA provides an exact framework for multi-allelic Wright-Fisher models.
• The model accurately captures allele frequency dynamics and boundary behaviors, surpassing diffusion approximations.
• Nonzero mutation rates are crucial for preventing complete allele extinction or fixation in finite populations.