Genealogical distances under low levels of selection

Area of Science:

• Population genetics
• Evolutionary biology
• Mathematical biology

Background:

• Kingman's coalescent theory accurately models neutral population genetics.
• Understanding genealogical trees under selection remains a challenge, with limited knowledge of key metrics like tree length.
• Previous research has not provided analytical expressions for tree length distributions under selection.

Purpose of the Study:

• To derive an analytic expression for the distribution of total tree length in a sample of size n under low levels of selection.
• To investigate the impact of selection and dominance on the length of genealogical trees.
• To compare tree lengths under selection with those under neutrality.

Main Methods:

• Development of an analytic expression for the distribution of total tree length.
• Analysis of a two-allele model with low levels of selection.
• Mathematical proof to determine the relationship between tree length and selection parameters.

Main Results:

• An analytic expression for the distribution of total tree length under low selection is derived.
• Genealogical trees are shorter than neutral trees when selection favors a beneficial mutant with dominance h < 1/2.
• Genealogical trees are longer than neutral trees when selection favors a beneficial mutant with dominance h > 1/2.
• The deviation from neutrality is of order O(α^2) for genic selection and O(α) for other dominance modes, where α is the selection intensity.

Conclusions:

• The study provides the first analytic expression for gene tree length distribution under low selection.
• Selection, dependent on the degree of dominance, can alter genealogical tree lengths compared to neutral evolution.
• These findings offer new insights into the impact of selection on population genealogy.