Exact Markov chain and approximate diffusion solution for haploid genetic drift with one-way mutation.

Area of Science:

• Population Genetics
• Mathematical Biology
• Evolutionary Dynamics

Background:

• The Wright-Fisher model is a cornerstone of population genetics, describing genetic drift in idealized populations.
• Diffusion approximations offer computationally efficient methods to study genetic processes but have limitations.

Purpose of the Study:

• To evaluate the accuracy of the classical Kimura diffusion solution for a haploid Wright-Fisher model with one-way mutations.
• To identify conditions where the diffusion approximation is inaccurate and develop improved methods.

Main Methods:

• Exact Markov chain computations using Jordan decomposition of the transition matrix.
• Perturbation of the diffusion solution near the non-fixation boundary for low mutation rates.

Main Results:

• The one-way diffusion model generally performs well, with convergence rates influenced by initial allele frequency and mutation rate.
• The diffusion approximation is poor for very low mutation rates, necessitating a quasi-fixation approach.
• An improved approximation accounting for quasi-fixation was developed.

Conclusions:

• The diffusion approximation is a useful tool in population genetics but requires careful validation and refinement for specific parameter ranges.
• The study provides a method to quantify the loss rate of genetic variants, particularly in infinite alleles models.
• The quasi-fixation approach offers a more accurate approximation for models with small mutation rates.