Selection and mutation at a diallelic X-linked locus.

Area of Science:

• Population Genetics
• Evolutionary Biology
• Mathematical Biology

Background:

• Understanding gene frequency dynamics is crucial in population genetics.
• X-linked loci present unique inheritance patterns influencing allele frequencies.
• Selection and mutation are key evolutionary forces shaping genetic variation.

Purpose of the Study:

• To analyze the equilibria and convergence of gene frequencies.
• To investigate a diallelic X-linked locus under selection and mutation.
• To determine the phase portraits of the dynamical system governing allele frequency changes.

Main Methods:

• Modeling an infinite diploid population with discrete generations and random mating.
• Applying mathematical analysis to study gene frequency dynamics.
• Proving global convergence under specific conditions of mutation rates and fitnesses.

Main Results:

• Global convergence of gene frequencies to equilibria is proven when mutation rates and fitnesses are constant and mutation rates are below one-third.
• Phase portraits of the dynamical system were determined.
• Monotone convergence of gene frequencies is established when every other generation is skipped.

Conclusions:

• Constant selection and mutation rates (below 1/3) ensure global convergence of gene frequencies for X-linked loci.
• The study provides a simplified proof for monotone convergence in the absence of mutation.
• The findings contribute to the understanding of genetic drift and evolutionary trajectories in populations.