Hardy-Weinberg law

Source: Gillespie, Population Genetics

Describes the equilibrium state of a single locus in a randomly-mating diploid population free of other evolutionary forces (mutation, migration, genetic drift).

• Consider two alleles, A and B, occurring in the population with frequencies p and q respectively. The frequency of AA individuals is p²; AB is 2pq; BB is q²
  • Derived from simple probability — choosing each allele is an independent event
• In hermaphrodite populations, HW-equilibrium is reached in one generation; in two-sex (dioecious) populations it can take two generations if allele frequencies differ between the sexes

Genotypes occupied by rare alleles

• Assume allele B is very rare — thus q = 1−p is very small
• Are rare alleles more likely to appear in homozygous BB or heterozygous AB genotypes?
• Ratio of AB:BB = 2pq/q² = 2p/q = 2/q → more likely to be heterozygous

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GWAS and linkage disequilibrium

Linkage disequilibrium (LD) = non-random association of alleles at different loci in a given population. Loci are "in LD" if the frequency of association of their alleles differs from what would be expected if the loci were independent and randomly associated.

LD is influenced by selection, mutation rate, genetic drift, mating system, genetic linkage (nearby DNA sequences are likely inherited together during meiosis), and population structure.

Formal definition

• Consider the set of gametes from a reproducing population: at one locus, allele A occurs with frequency $p_A$; at another locus, allele B with $p_B$. Frequency of gametes carrying both A and B = $p_{AB}$
• If completely independent: $p_A \times p_B = p_{AB}$
• Quantify the level of LD with the coefficient of LD: $D_{AB} = p_{AB} - p_A \times p_B$
  • $D_{AB}$ corresponds to LD between alleles, not loci — can differ for other alleles at the same loci
• A more normalized measure is the correlation coefficient between loci

Linkage disequilibrium score regression (LDSR)

Technique for quantifying the separate contributions of polygenic effects and confounding factors from GWAS summary stats, using regression analysis to examine the relationship between LD scores and test statistics from GWAS SNPs.

• LD score for a SNP = sum of LD $r^2$ measured with all other SNPs
• Uses: SNP-based heritability estimation; partitioning heritability into categories; calculating genetic correlations between phenotypes
• Compared to fine-mapping (which ideally uses an in-sample LD matrix), LDSR uses a representative reference LD matrix
  • LD matrix construction requires individual-level data — if only summary stats are available, you can't construct the exact in-sample matrix (common in large meta-analyses)

MPG primer: Linking SNPs with genes in GWAS (2022)

Luke O'Connor and Dan Weiner

Background

• Original stratified LDSR paper: Finucane et al. 2015, Nature Genetics — shows that non-coding variants are often cell-type specific
• Protein-coding variants are often rare variants, implicated in rare diseases and complex traits; also "low-frequency" (h² ~1%)

Linking SNPs to genes with eQTLs

The hypothesis is that GWAS SNPs are eQTLs affecting their causal gene. Two main challenges:

Co-regulation

• "Most eQTLs are not cell-type specific" → hard to identify relevant tissue; enrichment is driven by sample size
• Co-regulation of genes can be due to proximity, LD, or general promiscuity of enhancers
• Approach: explicitly model co-regulation (similar to fine-mapping)

eQTLs may not measure the right thing

• Wrong cell-type, context, or developmental stage — "Only 10% of heritability is explained by expression levels in observed cell-types and tissues"
• May be finding eQTLs for the wrong gene — genes with higher cis-heritability (more/stronger eQTLs) are actually depleted for trait heritability (possibly explained by selective constraints)
• Possible solutions: study more context-specific cell-types; use larger sample sizes to detect smaller effect sizes
• Note: "closest gene" is a surprisingly good starting point

Proximity and abstract mediation model

Two existing approaches and their limitations:

• Proximity-based enrichment (LDSC, MAGMA): SNP enrichment near gene set of interest — fails when the closest gene is not the correct one
• Integrating molecular data (eQTLs): fails due to differences between eQTL and GWAS signals

This approach — abstract mediation — models SNP→trait as proximity (SNP→gene) × gene set membership (gene→trait). Decay in heritability as a gene set gets further from SNPs allows partitioning of heritability among SNP-gene pairs.

Assumptions

• Heritability enrichment of SNPs mapped to a gene set is mediated by those genes (genes can be near disease-related SNPs without being causally relevant — careful gene set selection is needed)
• Cis-regulatory architecture doesn't vary by gene set (probably not true)
• Genes in cis mediate all heritability (ignoring trans effects)

Model is trained on "constrained genes" from 47 traits to estimate SNP→gene architecture.

Limitations

• Gene sets must be prespecified — doesn't identify causal genes
• Doesn't consider cellular context or cell-state
• Limited power for small gene sets and individual genes

Combining different SNP→gene-linking strategies

Alkes Price (+ JE!)

A combined approach can leverage the compatible strengths of different methods. The idea is to construct a SNP × gene matrix with values representing the strength of each connection.

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Quantitative trait loci (QTLs)

A QTL is a locus (section of DNA) that correlates with quantitative variation in the phenotype of a population, mapped via associated SNPs before the actual gene is identified. See Wikipedia: Quantitative trait locus.

• Most common mapping technique: interval mapping — assume one QTL in a certain range; consider each locus and calculate the logarithm of the odds (LOD) ratio that the locus is the QTL; link odds ratio to Pearson correlation coefficient between phenotype and marker genotype for each experimental subject/cross
• Composite interval mapping extends this without assuming a single QTL
• Combined with DNA microarrays → expression QTLs (eQTLs), which explain variation in mRNA expression
• sQTLs — associated with splicing variation
• cis-QTLs — associated with a nearby gene (same chromosome); trans-QTLs — associated with a gene on another chromosome, likely by first affecting a cis gene that then regulates the other

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Fine-mapping

Introduction to Bayesian fine-mapping methods

Hilary Finucane, Broad Institute

Why fine-mapping?

The goal is to identify the actual causal SNPs at a locus driving a GWAS association — disentangling the true signal from LD, noise, and sample size effects. Downstream goals: find causal coding/regulatory variants, identify variant-to-gene mechanisms, and understand genetic architecture (enrichment of certain variants, cross-trait associations).

Posterior inclusion probabilities (PIPs) and credible sets

• PIP = posterior probability that a variant is causal
• Credible set = smallest set of variants that contains a causal variant with ≥95% probability; each credible set corresponds to one putative causal variant

Methods

Single-causal-variant

The building block for more complex methods; PIPs sum to one (assuming exactly one causal variant).

• PIP = P(j causal | data)
• Applying Bayes rule with a flat prior: P(j causal | data) = P(data | j causal) / Σ_k P(data | k causal)
  • Flat prior: "before seeing GWAS data, I assume every variant has equal probability of being causal"
• Dividing numerator and denominator by the null probability P(data | no causal variant) gives the Bayes factor: BF_k = P(data | k causal) / P(data | no causal)
  • BF depends only on genotype at variant k — can be computed from summary data and is not dependent on LD
• To compute the smallest credible set, add variants in order of decreasing PIP until the cumulative probability reaches ≥0.95

Factors affecting power:

• LD: more LD between variants → less resolution (perfect LD means you can never distinguish which is causal without additional information) — combining data across populations helps by introducing LD diversity
• Sample size and effect size

Multiple-causal-variant

More common in practice, especially with larger sample sizes. Two approaches:

1. Conditional analysis — condition on the top variant; if a single causal variant exists it should be in high LD with others, so conditioning removes all signal. If multiple causal variants exist, signal remains — iterate to get a set of indexed causal variants. Limitation: top variant may not be in LD with the true causal variant.
2. Joint modeling — model sets instead of individual variants: P(set causal | data). Too many possible sets for direct enumeration → addressed by: limiting max causal variants and locus size and summing over most likely configurations (FINEMAP); or variational approaches similar to iterative conditional analysis (SuSIE).

Functionally informed

Incorporate functional data to replace the flat prior — e.g., coding variants or those in enhancers/promoters are more likely to be disease-driving. Priors can be learned from data across many loci. Methods include CAVIARBF, PolyFun, PAINTOR, and fgwas. Note: functional information cannot be used after fine-mapping for enrichment analyses.

Summary stats vs. full data

Many methods can run with just an LD matrix and summary stats vector, rather than requiring the full genotype matrix and phenotype vector. This is particularly convenient using an LD reference panel — though panel populations must match the study population well, and larger GWAS studies require larger panels.

Evaluating fine-mapping methods

• Field needs more benchmarking — applying multiple methods and checking for agreement is a useful heuristic
• Important to stress-test models; most assume:
  • All causal variants in the locus are modeled (no LD between modeled and unmodeled causal variants)
  • No imputation noise
  • A specific range for the number of causal variants
  • A specific distribution of phenotype given genotype