Linkage disequilibrium and the infinitesimal limit

Area of Science:

• Quantitative genetics
• Population genetics
• Statistical genetics

Background:

• The Fisher-Bulmer model assumes a Gaussian within-family distribution for additive traits.
• This model is often derived from an infinite number of unlinked loci contributing to trait variation.

Purpose of the Study:

• To investigate the precise conditions under which the Fisher-Bulmer model emerges in the infinite locus limit.
• To explore the role of linkage disequilibrium in the validity of the Fisher-Bulmer model.
• To present an alternative population genetic derivation and extend it to include linkage disequilibrium.

Main Methods:

• Analysis of the Fisher-Bulmer infinitesimal model in quantitative genetics.
• Investigation of additive traits with unlinked loci and the central limit theorem.
• Development of population genetic models with varying degrees of linkage disequilibrium, including diallelic models.

Main Results:

• A counterexample demonstrates the failure of the Fisher-Bulmer model in the infinite locus limit due to excessive linkage disequilibrium.
• A degenerate form of the Fisher-Bulmer model can arise under specific conditions.
• An alternative derivation is presented for populations at linkage equilibrium, with extensions for various linkage disequilibrium patterns.

Conclusions:

• The Fisher-Bulmer model's applicability is contingent on specific genetic architectures, particularly the degree of linkage disequilibrium.
• The findings contribute to understanding the limiting behavior of population genetic models with numerous loci.
• The general results may have implications for modeling cultural inheritance.