Using mathematical constraints to explain narrow ranges for allele-sharing dissimilarities

Area of Science:

• Population Genetics
• Statistical Genetics

Background:

• Allele-sharing dissimilarity (ASD) statistics measure genetic differentiation between individuals or populations.
• Understanding the expected values of ASD statistics is crucial for interpreting genetic diversity.

Purpose of the Study:

• To investigate the mathematical constraints on two specific ASD statistics ($$ and $$).
• To determine how allele frequencies, particularly the frequency of the most common allele, bound the expected ASD values.
• To explain empirical observations of limited variation in human population genetic dissimilarities.

Main Methods:

• Calculating the expected ASD by averaging pairwise dissimilarities between randomly drawn individuals from specified allele-frequency distributions.
• Deriving bounds on expected ASD for a single population and for pairs of populations based on the frequency of the most frequent allele.
• Analyzing the range of expected ASD across the unit interval for allele frequencies.

Main Results:

• The expected ASD values are constrained by allele frequencies and do not span the full [0, 1] interval.
• Bounds on expected ASD were derived in terms of the frequency of the most frequent allelic type within populations.
• The mathematical constraints provide a theoretical basis for the observed narrow range of allele-sharing dissimilarities in human populations.

Conclusions:

• Mathematical constraints imposed by allele frequencies significantly restrict the possible range of expected allele-sharing dissimilarity.
• These findings offer a quantitative explanation for the empirical observation that average allele-sharing dissimilarities in human populations typically vary within a limited range.