Mutational pattern of a sample from a critical branching population.

Area of Science:

• Population Genetics
• Stochastic Processes
• Mathematical Biology

Background:

• Understanding the evolutionary history of populations, or genealogies, is crucial for interpreting genetic data.
• Branching processes are fundamental models for population dynamics and have been extended to incorporate mutations.

Purpose of the Study:

• To develop a universal model for sample genealogy in populations with mutations.
• To analyze mutational patterns and the structure of genealogical trees under various population models.

Main Methods:

• Utilizing a critical birth-death process with Poissonian mutations, conditioned on population size.
• Investigating large population asymptotics towards the continuum random tree.
• Extending models to include random foundation times with different prior distributions (uniform, log-uniform).
• Analyzing the site frequency spectrum and convergence of sample genealogies.

Main Results:

• The critical birth-death process with mutations accurately models genealogies in large, nearly critical branching populations.
• Explicit formulas for the expected site frequency spectrum are derived for fixed and random foundation times.
• Convergence in distribution is established for sample genealogies under specific priors.
• Limiting genealogies from different priors can be embedded within a common Poisson point measure realization.

Conclusions:

• The proposed universal model provides a unified framework for studying population genealogy and mutation patterns.
• The results offer insights into the structure of evolutionary trees and the distribution of mutations in large populations.
• The mathematical framework allows for the analysis of diverse population histories, including those with random origins.