Abstract
Linkage disequilibrium (LD), often expressed in terms of the squared correlation (r2) between allelic values at two loci, is an important concept in many branches of genetics and genomics. Genetic drift and recombination have opposite effects on LD, and thus r2 will keep changing until the effects of these two forces are counterbalanced. Several approximations have been used to determine the expected value of r2 at equilibrium in the presence or absence of mutation. In this paper, we propose a probability-based approach to compute the exact distribution of allele frequencies at two loci in a finite population at any generation t conditional on the distribution at generation t − 1. As r2 is a function of this distribution of allele frequencies, this approach can be used to examine the distribution of r2 over generations as it approaches equilibrium. The exact distribution of LD from our method is used to describe, quantify, and compare LD at different equilibria, including equilibrium in the absence or presence of mutation, selection, and filtering by minor allele frequency. We also propose a deterministic formula for expected LD in the presence of mutation at equilibrium based on the exact distribution of LD.
Highlights
Linkage disequilibrium (LD), the non-random association of alleles at two or more loci, is an important concept in various areas of genetics, including evolutionary, quantitative, and statistical genetics
The objective of this paper is to present a computational approach to derive the exact distribution of LD over generations and use it to study the distribution of LD whether or not there is mutation, selection, or filtering by minor allele frequency (MAF)
A contribution of this article is to propose a computational transition-matrix approach to deriving the distribution of LD between two loci over generations in the presence of multiple genetic forces including drift, mutation, and selection. These distributions of LD are studied in the presence of filtering by MAF (i.e., MAF ≥ 0.05)
Summary
Linkage disequilibrium (LD), the non-random association of alleles at two or more loci, is an important concept in various areas of genetics, including evolutionary, quantitative, and statistical genetics. LD is used to detect the genomic locations of historical selection and to estimate the time of divergence and geographic subdivision between populations (Sved and Hill, 2018). In the field of quantitative and statistical genetics, methods for genomic prediction and genome-wide association studies (GWAS) rely on the existence of LD between the molecular markers and the unobserved quantitative trait loci (QTL). Two statistics that have been widely used to quantify the LD between allelic values at two loci are: their covariance (D); or their squared correlation (r2) (Hill and Robertson, 1968).
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