Linkage Disequilibrium Dynamics

Linkage Disequilibrium (LD)

Linkage disequilibrium measures the non-random association between alleles at two genetic loci. When two loci are in linkage equilibrium, the frequency of each haplotype equals the product of its constituent allele frequencies. Deviations from this equilibrium are quantified by LD statistics.

Haplotypes and Notation

• Haplotype: A combination of alleles at linked loci on a single chromosome
• Four haplotypes: AB, Ab, aB, ab with frequencies p₁₁, p₁₂, p₂₁, p₂₂
• Allele frequencies: pₐ = p₁₁ + p₁₂, pᵦ = p₁₁ + p₂₁

LD Statistics

• D (Linkage disequilibrium coefficient): D = p₁₁ - pₐ × pᵦ Measures departure from random association. D = 0 at linkage equilibrium.
• D' (Normalized D): Scales D by its theoretical maximum D' = D / Dₘₐₓ, where Dₘₐₓ depends on allele frequencies
• r² (Correlation coefficient): r² = D² / (pₐ × p₂ × pᵦ × q₂) Measures correlation between loci; r² = 1 means perfect LD

Recombination and LD Decay

Recombination breaks down linkage disequilibrium over time. With recombination rate r, LD decays as: D(t) = D(0) × (1 - r)ᵗ. Tightly linked loci (small r) maintain LD longer, while unlinked loci (r = 0.5) rapidly approach equilibrium.

Selection and LD

Natural selection can create or maintain LD even with recombination. When selection favors specific haplotype combinations (epistasis), LD persists at a balance between selection creating associations and recombination breaking them down.

Applications

• GWAS: Association mapping exploits LD between markers and causal variants
• Population history: LD patterns reveal past bottlenecks and admixture
• Recombination mapping: LD decay estimates local recombination rates
• Haplotype blocks: Regions of strong LD define common haplotypes