2 Genetic variation

Any two people differ at a few million positions in their genomes, and almost all of those differences are shared with many other people. Most analyses in this book rest on three properties of each variant: how common it is, what it does to a gene, and which neighboring variants travel with it. Learn to read those three and a GWAS result stops looking like a list of random positions.

2.1 Variants come in sizes

The first question about a variant is how many bases it touches. Size decides which technology can detect it and which file format holds it.

Class	Size	Example	Detected well by
SNV (single-nucleotide variant)	1 bp	`A>G`	Arrays, exome, genome
MNV (multi-nucleotide variant)	2+ adjacent bp	`AC>GT`	Sequencing
Indel	1 bp to about 50 bp	`ATG>A` (2-bp deletion)	Sequencing; arrays only for known ones
Structural variant (SV)	over 50 bp	Inversion, translocation	Long reads, genome sequencing
Copy-number variant (CNV)	about 1 kb to several Mb	Deletion or duplication of a gene	Arrays (intensity), genome sequencing

A SNP (single-nucleotide polymorphism) is an SNV common enough to be called a polymorphism, traditionally with a frequency of at least 1%. In practice the two terms are used loosely. Most variants carry an rs identifier from dbSNP, which stays stable across genome builds while positions do not.

2.2 Allele frequency and MAF

The allele frequency is the share of all copies of a site, in a population, that carry a given allele. With \(n\) diploid people there are \(2n\) copies:

\[p_{\text{ALT}} = \frac{2 \cdot n_{\text{hom-alt}} + n_{\text{het}}}{2n}\]

The minor allele frequency (MAF) is the frequency of the less common allele, so it is always at most 0.5.

A worked number: in 1,000 people, 640 are AA, 320 are AG, and 40 are GG. Then \(p_G = (2 \cdot 40 + 320) / 2000 = 0.20\). G is the minor allele and the MAF is 0.20.

Category	MAF	Typical analysis
Common	≥ 5%	Single-variant GWAS (Chapter 11)
Low-frequency	1–5%	Single-variant tests with large samples
Rare	< 1%	Gene-based burden and SKAT tests (Chapter 13)
Ultra-rare, singletons	seen once or a few times	Burden tests, clinical interpretation

Allele frequencies differ between populations. A variant that is rare in one ancestry group can be common in another, and the minor allele itself can switch. Check frequencies in a large reference such as gnomAD before calling a variant rare.

2.3 Hardy–Weinberg equilibrium

In a large, randomly mating population with no selection, migration, or mutation, genotype frequencies follow directly from allele frequencies. With allele frequencies \(p\) and \(q = 1 - p\):

\[P(AA) = p^2, \quad P(Aa) = 2pq, \quad P(aa) = q^2\]

This is Hardy–Weinberg equilibrium (HWE). A worked test: 1,000 people with 400 AA, 400 Aa, and 200 aa. Then \(p = 0.6\), and the expected counts are 360, 480, and 160. The chi-square statistic is

\[\chi^2 = \frac{40^2}{360} + \frac{80^2}{480} + \frac{40^2}{160} \approx 27.8\]

With 1 degree of freedom, \(p \approx 1.4 \times 10^{-7}\). There are too few heterozygotes.

Real populations violate HWE a little, and selection on a disease variant violates it in cases. A strong deviation in controls, however, usually means a genotyping problem: a bad cluster, a paralog, or a CNV. That is why HWE is mainly a quality-control filter, used with thresholds you will see in Chapter 9.

2.4 Copy-number variants

A CNV is a deletion or duplication of a stretch of DNA, so a person carries zero, one, three, or more copies instead of two. CNVs can cover whole genes and change their dosage directly.

Some are common and benign; others cause disease. The 22q11.2 deletion, for example, causes a well-known developmental syndrome.
Some genes vary in copy number as part of normal variation. CYP2D6 can be deleted or present in three or more copies, which matters for drug metabolism (Chapter 16).
Arrays detect CNVs from probe intensity (log R ratio and B allele frequency), with tools such as PennCNV. Sequencing detects them from read depth and split reads.

SNP-based pipelines ignore CNVs, so a deletion can masquerade as an apparent homozygote. This is one source of HWE failure.

2.5 Predicted loss-of-function variants

A predicted loss-of-function (pLoF) variant is one expected to abolish the function of the gene it falls in. The main classes are:

stop-gained (nonsense): creates a premature stop codon;
frameshift: an indel whose length is not a multiple of three, which scrambles the downstream reading frame;
essential splice-site: changes the two bases at either end of an intron (the donor or acceptor site);
in some definitions, start-lost and large deletions of coding exons.

“Predicted” is the key word. A premature stop near the end of the last exon may escape nonsense-mediated decay and leave a working protein. Annotation tools such as VEP with the LOFTEE plugin flag these cases and keep only high-confidence pLoFs (Karczewski et al., 2020, Nature). gnomAD also reports gene-level constraint, such as LOEUF: genes with far fewer pLoFs than expected in the general population are likely intolerant to losing a copy.

pLoF variants are mostly rare. They are usually analyzed together, per gene, in burden tests (Chapter 13).

2.6 Haplotypes and phasing

A haplotype is the set of alleles on one physical chromosome across a region. You inherit one haplotype for each region from your mother and one from your father. Nearby alleles tend to be inherited together, because recombination rarely separates them in a few generations.

A genotype call tells you which alleles you carry at each site. It does not tell you which alleles sit on the same chromosome. Recovering that is called phasing.

flowchart LR
    subgraph unphased["Unphased: 0/1 at two sites"]
        direction TB
        U["Site 1: A/G<br/>Site 2: C/T"]
    end
    subgraph optA["Possibility 1: cis"]
        direction TB
        A1["Chromosome 1: G - T"]
        A2["Chromosome 2: A - C"]
    end
    subgraph optB["Possibility 2: trans"]
        direction TB
        B1["Chromosome 1: G - C"]
        B2["Chromosome 2: A - T"]
    end
    U --> optA
    U --> optB
    style unphased fill:transparent,stroke:#adb5bd
    style optA fill:transparent,stroke:#adb5bd
    style optB fill:transparent,stroke:#adb5bd

The difference can be biological. If both ALT alleles damage the same gene, cis leaves one intact copy, while trans (compound heterozygosity) leaves none. In pharmacogenomics, two variants on the same chromosome can define a different star allele than the same two variants apart.

	Unphased	Phased
VCF separator	Slash, as in `0/1`	Vertical bar (pipe)
Known	Alleles at each site	Alleles at each site and which chromosome carries them
Produced by	Variant calling	Statistical phasing (SHAPEIT, Eagle, Beagle), read-backed or trio phasing
Needed for	Single-variant association	Imputation, haplotype analysis, star-allele calling, compound heterozygosity

Statistical phasing is very accurate for common variants in large samples and less reliable for rare variants. Imputation always phases first, which is why imputed VCFs use pipes.

2.7 Linkage disequilibrium

Linkage disequilibrium (LD) is the non-random association of alleles at nearby sites in a population. If allele G at one SNP almost always appears with allele T at the next one, the two are in strong LD. LD arises mostly because recombination has not yet separated them; drift, bottlenecks, admixture, and selection also shape it.

Take two biallelic sites with alleles A/a and B/b. The basic measure compares the observed haplotype frequency with what independence predicts:

\[D = p_{AB} - p_A p_B\]

Two normalized versions are used in practice:

\[D' = \frac{D}{D_{\max}} \qquad r^2 = \frac{D^2}{p_A(1-p_A)\,p_B(1-p_B)}\]

A worked example: \(p_A = 0.6\), \(p_B = 0.7\), \(p_{AB} = 0.5\). Then \(D = 0.5 - 0.42 = 0.08\). Since \(D > 0\), \(D_{\max} = \min(p_A(1-p_B),\,(1-p_A)p_B) = 0.18\), so \(D' = 0.44\). And \(r^2 = 0.0064 / 0.0504 \approx 0.13\).

Measure	Range	Means	Use it for
\(D'\)	0 to 1	1 = no recombination seen between the sites	Defining haplotype blocks, history of recombination
\(r^2\)	0 to 1	How well one SNP predicts the other	Tagging, imputation, GWAS interpretation, pruning

\(D'\) can be 1 even when the two SNPs have very different frequencies and predict each other poorly. For statistical work, \(r^2\) is usually the number you want. LD decays with distance, varies along the genome with recombination hot spots, and differs between populations. African-ancestry genomes generally show shorter LD blocks than European or East Asian genomes. Online tools such as LDlink report population-specific LD.

LD matters at every step of a GWAS:

Arrays work because of LD. A few hundred thousand genotyped SNPs tag millions of others, and imputation fills them in.
A GWAS hit is a region, not a variant. Every SNP in high LD with the causal one shows a signal. The top SNP is often not causal.
Fine-mapping uses LD to assign each SNP in a region a probability of being causal (Chapter 15).
Clumping groups significant SNPs by LD to count independent loci (Chapter 15).
LD pruning thins SNPs so that no remaining pair exceeds an \(r^2\) threshold. It is used before PCA, relatedness, and heterozygosity checks, where correlated SNPs would otherwise distort the result (Chapter 9, Chapter 10). It is not used to choose SNPs for the association test itself.

# Prune: 50-SNP windows, step 5, drop one of each pair with r2 > 0.2
plink2 --pfile chr22 --indep-pairwise 50 5 0.2 --out chr22_pruned

# Pairwise r2 for SNPs within 500 kb of a lead SNP (PLINK 1.9)
LEAD=rs0000000   # replace with an rsID present in your .bim file
plink --bfile chr22 --r2 --ld-snp "$LEAD" --ld-window-kb 500 \
      --ld-window 99999 --ld-window-r2 0.2 --out lead_ld

Pitfall

LD is population-specific. Using a European LD reference to clump, fine-map, or impute a cohort of another ancestry gives wrong answers without any error message. Match the LD reference to your sample.

2.8 Star alleles: haplotypes with names

Pharmacogenomics names whole haplotypes rather than single variants. A star allele such as CYP2C19*2 is a defined combination of variants on one chromosome of a gene. The catalog of definitions is maintained by PharmVar.

CYP2C19*1 is the reference haplotype, with normal function.
CYP2C19*2 is defined by rs4244285 (c.681G>A). The change creates an aberrant splice site, and the enzyme is not produced. It is a no-function allele.
CYP2C19*17 carries a promoter variant that increases expression.

A person’s two star alleles form a diplotype, such as *1/*2. The diplotype maps to a phenotype: *1/*1 is a normal metabolizer, *1/*2 an intermediate metabolizer, and *2/*2 a poor metabolizer. A single *2 copy does not make someone a poor metabolizer.

In practice

Star-allele calling needs phased data, and genes like CYP2D6 also need copy-number calls. Use dedicated callers rather than reading single SNPs off a VCF. Chapter 16 covers the tools and how diplotypes map to guidance.

Try it

Download chromosome 22 for the 1000 Genomes Project phase 3 samples (the PLINK 2 resources page links ready-made files). Using only the EUR samples and then only the AFR samples, compute allele frequencies with plink2 --freq and the \(r^2\) between one common SNP and its neighbors within 500 kb. Compare how far LD extends in each group, and find one SNP whose minor allele differs between the two.