Mutation in populations governed by a Galton-Watson branching process

Area of Science:

• Population Genetics
• Mathematical Biology
• Evolutionary Dynamics

Background:

• Understanding the evolutionary dynamics of populations with multiple alleles is crucial.
• Previous models often simplified the interplay between genetic drift and mutation.
• Branching processes offer a powerful framework for modeling population genetics.

Purpose of the Study:

• To develop and analyze a population genetics model based on a multitype branching process.
• To investigate the effects of neutral mutations on population structure and dynamics.
• To derive and solve the forward Kolmogorov equation in the diffusion limit.

Main Methods:

• Utilizing a multitype branching process (Galton-Watson process for multiple alleles).
• Deriving the diffusion limit forward Kolmogorov equation for neutral mutations.
• Obtaining asymptotic stationary and approximate time-dependent solutions.
• Employing numerical simulations to validate the approximate solution.

Main Results:

• The asymptotic stationary solution shows population partitioning influenced by mutation rates.
• An approximate time-dependent solution reveals a rapid transition from drift to mutation dominance.
• The transition point depends on growth factors and mutation rates.
• Numerical simulations confirm the accuracy of the approximate solution.

Conclusions:

• Neutral mutations drive population subdivision in a manner determined by mutation rates.
• A distinct phase transition occurs from drift- to mutation-dominated dynamics.
• The model provides a robust framework for studying allele frequency dynamics and population structure.