The nonlinear structure of linkage disequilibrium

Area of Science:

• Genetics
• Population Genetics
• Bioinformatics

Background:

• Allele frequency dependence affects linkage disequilibrium (LD) measures.
• Common LD parameters (r², D) have maximum values influenced by allele frequencies.
• Existing mathematical frameworks for LD limits are often heuristic.

Purpose of the Study:

• To provide a comprehensive mathematical framework for linkage disequilibrium.
• To elucidate the shift between linear and nonlinear dependence in LD structure.
• To demonstrate how allele frequencies impact LD measure interpretation.

Main Methods:

• Mathematical analysis of linkage disequilibrium structure.
• Estimation of linear and nonlinear dependence proportions.
• Derivation of theoretical limits for LD measures.

Main Results:

• A fundamental shift between linear and nonlinear dependence underlies LD measure behavior.
• The proportion of linear/nonlinear dependence can be estimated.
• Same r² values can imply different dependence natures.
• D' (as a positive value) has a minimum of |r|.

Conclusions:

• Understanding allele frequency dependence is crucial for accurate LD inference.
• The study provides a robust mathematical basis for LD measure interpretation.
• Distinguishing linear and nonlinear dependence clarifies locus relationships.