Quantitative Genetics

Most of the traits people care about, such as height, blood pressure, milk yield, grain weight and the risk of type 2 diabetes, do not sort into neat Mendelian classes. They vary smoothly across a range because many genes and many environmental factors each nudge them a little. Quantitative genetics gives you the tools to measure that variation, divide it into genetic and environmental parts, predict how a population will respond to selection, and finally track down the individual genes hiding inside the bell curve.

Section 1Continuous traits and polygenic inheritance

Discrete and continuous variation

Mendel chose traits that fall into a few clear categories: round or wrinkled seeds, purple or white flowers. Traits like these show discontinuous (discrete) variation, and a single gene with large effect explains the categories. Most traits in nature behave differently. Measure the height of 1,000 adults, the weight of 1,000 bean seeds or the systolic blood pressure of 1,000 patients and you get a smooth, roughly bell shaped distribution with no natural breaks. These are quantitative traits, also called continuous or complex traits. Their study is quantitative genetics, and its language is the language of statistics rather than of ratios.

Quantitative traits come in three broad types, and it helps to recognize which one you are dealing with:

• Continuous traits can take any value within a range: height, body mass, blood glucose, milk yield, the time a plant takes to flower.
• Meristic traits are counted in whole numbers but still show many classes and a roughly normal distribution: the number of bristles on a fruit fly, kernels on a corn ear, eggs laid by a hen in a year, vertebrae in a snake.
• Threshold traits appear as present or absent, yet the liability that underlies them is continuous. An individual develops the condition only when the combined effect of many genes and environmental factors pushes its liability past a threshold. Cleft lip with or without cleft palate, pyloric stenosis, type 2 diabetes, schizophrenia and many congenital heart defects behave this way.

The multiple factor hypothesis

An early resolution of how Mendelian genes could produce continuous variation came from experiments on wheat kernel color by the Swedish geneticist Herman Nilsson-Ehle. He crossed a red kerneled line with a white kerneled line. The F1 were an intermediate pink. In some F2 families he found 1/16 as red as the red parent and 1/16 as white as the white parent, with three intermediate shades in between in the ratio 1:4:6:4:1. In other crosses only 1/64 of the F2 were as extreme as either parent.

He explained these results with the multiple factor hypothesis (the polygene hypothesis): a quantitative trait is controlled by several genes, each segregating in a Mendelian way, and each with a small, additive effect. In the two gene case, call the alleles that add red pigment A and B (the contributing or additive alleles) and the alleles that add nothing a and b (the noncontributing alleles). Every contributing allele adds the same increment of color, so the phenotype depends only on how many contributing alleles an individual carries, not on which gene they belong to. An AaBb F1 has two of four, so it is intermediate. In the F2, the number of contributing alleles follows the binomial distribution: 0, 1, 2, 3 or 4 contributing alleles occur in the ratio 1:4:6:4:1.

Three general rules follow for n genes, each with two equally effective alleles, in a cross of two heterozygotes (or in the F2 from two homozygous extreme parents):

• The number of distinct phenotypic classes is 2n + 1.
• The fraction of offspring that matches one extreme parent is (1/4)ⁿ. For one gene this is 1/4, for two genes 1/16, for three 1/64, for four 1/256.
• The frequency of each class is given by the row of Pascal's triangle with 2n entries, so the probability of exactly k contributing alleles is C(2n, k) × (1/2)²ⁿ.

Why the bell curve appears

Even a few additive genes produce a distribution with a central peak and tapering tails. Two further factors smooth the steps into a continuous curve. First, real traits usually involve many more loci, often hundreds or thousands, so the steps become tiny. Second, the environment adds its own variation: nutrition, temperature, disease, maternal effects and chance events during development shift each individual a little up or down. A plant with exactly four contributing alleles might be 34 cm in poor soil and 38 cm in rich soil. When every genotypic class is spread out by environmental noise, neighboring classes overlap and the histogram looks like a single normal distribution. This is the central limit theorem at work: sums of many small independent effects tend toward a normal shape.

The overlap has an important practical consequence. In a continuous trait, you cannot read genotype from phenotype. Two plants of the same height may carry different numbers of contributing alleles, and two plants with identical genotypes may differ in height. That is why quantitative genetics works with averages and variances of populations rather than with individual genotypes.

Gene action in polygenic traits

The simple model assumes that alleles act additively: the heterozygote is exactly halfway between the two homozygotes, and the effects of different loci simply add. Real loci depart from this in two ways:

• Dominance is an interaction between alleles at the same locus. If one allele is dominant, the heterozygote resembles one homozygote rather than lying at the midpoint. Dominance makes the F2 distribution skewed toward the dominant class.
• Epistasis is an interaction between alleles at different loci. The effect of an allele at one gene depends on the genotype at another. Epistasis is common in metabolic pathways and developmental networks.

The distinction matters because, as we will see in Section 3, only the additive part of genetic variation is reliably passed from parent to offspring. Dominance and epistatic combinations are broken up each generation by segregation and recombination.

Threshold traits and liability

Many common diseases appear to run in families without following Mendelian ratios. The liability threshold model explains them. Each person has a continuous, normally distributed liability built from many genetic and environmental factors. Only those whose liability exceeds a threshold are affected. The model makes several predictions that fit clinical data well. The risk to relatives of an affected person is higher than the population risk and falls off sharply with distance of relationship, faster than for a single gene trait. The risk is higher when more relatives are affected, and higher when the affected relative has a severe form, because both indicate that the family carries many risk alleles. When the sexes have different thresholds, relatives of an affected person of the less often affected sex have higher risk. Pyloric stenosis is the classic example: it is about five times more common in boys, and the sons of affected women carry the highest risk because a woman who crossed the higher female threshold must carry an unusually heavy load of liability alleles.

Section 2Statistics for quantitative traits

Mean, variance and standard deviation

The mean (x̄) is the arithmetic average: add all n measurements and divide by n. It tells you where the distribution is centered, but nothing about its spread. Two populations can share a mean of 170 cm yet differ greatly in how variable they are.

The variance (s²) measures spread. For each value, find its deviation from the mean, square it so that positive and negative deviations do not cancel, add the squares, and divide by n − 1:

s² = Σ(x_i − x̄)² / (n − 1)

Dividing by n − 1 rather than n (the degrees of freedom) corrects for the fact that the sample mean was itself calculated from the same data, which makes the raw squared deviations slightly too small on average. The variance is in squared units (cm²), which is awkward to interpret but has a great advantage: independent sources of variance add. That additivity is the foundation of heritability.

The standard deviation (s) is the square root of the variance and is in the original units. In a normal distribution, about 68 percent of values fall within 1 standard deviation of the mean, about 95 percent within 2 (more exactly 1.96) and about 99.7 percent within 3. If adult height in a population has a mean of 170 cm and a standard deviation of 7 cm, about 95 percent of adults are between 156 and 184 cm.

The standard error of the mean, s/√n, is a different quantity. It describes how precisely the sample mean estimates the true population mean, and it shrinks as sample size grows. Students often mix it up with the standard deviation, which describes variation among individuals and does not shrink with more data.

Covariance and correlation

Quantitative genetics constantly asks whether two measurements vary together: the height of parents and of their children, the weight of a calf and its later milk yield, the trait values of twins. The covariance measures this joint variation. For each pair, multiply the deviation of x from its mean by the deviation of y from its mean, add the products, and divide by n − 1:

cov(x, y) = Σ(x_i − x̄)(y_i − ȳ) / (n − 1)

If large x values tend to go with large y values, most products are positive and the covariance is positive. If large x goes with small y, the covariance is negative. If there is no relationship, positive and negative products cancel and the covariance is near zero. The covariance between relatives is the raw material from which heritability is estimated, because relatives share genes in known proportions.

Covariance depends on the units of measurement, so it is hard to compare across traits. The correlation coefficient (r) standardizes it by dividing by both standard deviations:

r = cov(x, y) / (s_x s_y)

The correlation ranges from −1 (perfect negative linear relationship) through 0 (no linear relationship) to +1 (perfect positive). Two cautions apply. First, correlation does not imply causation: ice cream sales and drowning deaths are positively correlated because both rise in summer. Second, r measures only linear association. The square of r, r², gives the proportion of the variation in y that is accounted for by its linear relationship with x.

Regression

Correlation tells you how tightly points cluster around a line. Regression tells you the line itself, so that you can predict y from x. The least squares regression line y = a + bx has slope

b = cov(x, y) / s_x²

and intercept a = ȳ − b x̄. The slope says how much y changes, on average, for each one unit change in x. In quantitative genetics the regression of offspring on parents is especially useful: as Section 3 shows, the slope of offspring values regressed on the average of the two parents (the midparent value) estimates narrow sense heritability directly.

Francis Galton noticed the key pattern in the late 1800s while studying human height. Children of very tall parents were tall, but on average not as tall as their parents; children of very short parents were short, but not as short. He called this regression toward the mean. It happens because part of what made the parents extreme was environment and nonadditive gene combinations that are not transmitted, so the children drift back toward the population mean. The slope of the regression line measures how much of the parental deviation is passed on.

Section 3Heritability: broad and narrow sense

Partitioning phenotypic variance

The central idea of quantitative genetics is that the total variation we see in a trait, the phenotypic variance (V_P), can be divided into parts that have different causes. In the simplest model, an individual's phenotype is the sum of a genetic value and an environmental deviation, P = G + E. If genes and environment are independent, their variances add:

V_P = V_G + V_E

Real populations add two complications. If particular genotypes tend to find themselves in particular environments (for example, children with high musical aptitude are more likely to receive music lessons), there is a genotype environment covariance that adds 2cov(G, E). If genotypes respond differently to the same environmental change (one wheat variety yields best in drought, another in wet years), there is a genotype by environment interaction, V_G×E. A fuller version is therefore V_P = V_G + V_E + V_G×E + 2cov(G, E). Most problems assume these two terms are zero.

The genetic variance itself has three components that correspond to the kinds of gene action described in Section 1:

• Additive genetic variance (V_A) comes from the average effects of individual alleles, summed across loci. It is the part that makes offspring resemble their parents, because a parent passes on alleles, not genotypes.
• Dominance variance (V_D) comes from interactions between the two alleles at the same locus. A parent cannot transmit a dominance interaction, because each child receives only one of the parent's two alleles.
• Interaction (epistatic) variance (V_I) comes from interactions between loci, and it too is largely broken up by segregation and recombination.

Putting the pieces together: V_P = V_A + V_D + V_I + V_E. The environmental variance can also be split into a common or shared part (V_Ec, effects shared by members of a family, such as household diet) and a special or unique part (effects specific to each individual, including random developmental noise and measurement error).

Two kinds of heritability

Heritability is the proportion of phenotypic variance in a population that is due to genetic differences among individuals. There are two versions, and confusing them is a classic exam error.

Broad sense heritability (H²) includes all genetic variance:

H² = V_G / V_P = (V_A + V_D + V_I) / V_P

It tells you how much of the variation reflects genotype in general, which matters, for example, when clones or inbred lines can be propagated, because a clone transmits its entire genotype.

Narrow sense heritability (h²) includes only the additive part:

h² = V_A / V_P

It is the quantity that predicts how closely offspring resemble parents and how a sexually reproducing population will respond to selection. Plant and animal breeders care most about h². Both heritabilities range from 0 (no variation is genetic) to 1 (all variation is genetic), and h² can never exceed H².

Estimating heritability

Heritability cannot be measured directly. It is estimated by comparing groups whose genetic or environmental relationships are known. Four approaches are commonly taught.

1. Genetically uniform populations. Members of an inbred line, or the F1 from a cross of two inbred lines, are genetically identical, so all their variance is environmental: V_P = V_E. The F2 from the same cross contains both genetic and environmental variance. Grown in the same environment, V_G = V_P(F2) − V_E. This yields H².

2. Parent offspring regression. Plot offspring phenotypes against parental phenotypes. When offspring are regressed on the midparent value, the slope b estimates h² directly. When they are regressed on one parent only, the slope estimates h²/2, because a single parent contributes only half of the offspring's alleles; double the slope to get h². The method works because the covariance between a parent and its offspring equals ½V_A, while dominance and epistatic deviations are not shared.

3. Twin studies. Monozygotic (identical) twins share all their genes; dizygotic (fraternal) twins share on average half, like ordinary siblings. If both kinds of twins share their environments to a similar degree, the extra similarity of identical twins reflects genes. Falconer's formula gives a heritability estimate from the twin correlations:

h² ≈ 2(r_MZ − r_DZ)

Because the difference in genetic sharing is one half, doubling the difference in correlation recovers the additive genetic proportion. Twins reared apart provide another approach: the correlation of identical twins raised in separate homes estimates H² directly, provided the homes are not systematically similar. Twin studies have limits: identical twins may be treated more alike, and adoptive homes are not random environments, biases that tend to inflate estimates.

4. Response to selection. The amount a population changes after selection gives the realized heritability, discussed in Section 4.

What heritability does not mean

Heritability is one of the most misunderstood numbers in biology. Keep these points straight:

• It describes a population, not an individual. A heritability of 0.8 for height does not mean that 80 percent of your height came from genes and 20 percent from food. It means that 80 percent of the differences among people in that population are associated with genetic differences.
• It is specific to a population and an environment. If an environment becomes more uniform, V_E shrinks and heritability rises, even though no gene has changed. If environments become more variable, heritability falls.
• High heritability does not mean a trait cannot be changed. Height is highly heritable, yet average height in many countries rose by more than 10 cm over the 20th century as nutrition and health improved. Phenylketonuria is entirely genetic in cause, yet a low phenylalanine diet prevents its intellectual disability.
• Heritability within groups says nothing about the cause of differences between groups. Imagine seeds from one genetically variable bag planted in two fields, one fertile and one poor. Within each field, height differences are largely genetic (high heritability), but the difference between the field averages is entirely environmental. This is the classic argument against inferring genetic causes for average differences between human populations.
• A trait with zero heritability can still be under genetic control. Number of eyes in humans has a heritability near zero because almost everyone has two; differences that do occur come from accidents. The genes that build eyes are fixed, so they create no variance.

Section 4Response to selection

The breeder's equation

For thousands of years farmers improved crops and livestock by saving seed from the best plants and breeding the best animals. Quantitative genetics turns that practice into a prediction. Suppose a breeder measures a trait in a population, picks the individuals with the highest values to be parents, and measures their offspring. Three quantities are involved:

• The selection differential (S) is the difference between the mean of the selected parents and the mean of the whole population they were chosen from: S = x̄_selected − x̄_population.
• The response to selection (R) is the difference between the mean of the offspring and the mean of the original population: R = x̄_offspring − x̄_population.
• The narrow sense heritability (h²) links them.

The breeder's equation states:

R = h² S

The logic is the same as for regression toward the mean. The selected parents are superior partly because of their additive genes and partly because of good luck in environment and favorable dominance or epistatic combinations. Only the additive part is transmitted, so offspring recover only the fraction h² of their parents' advantage. If h² = 1, the offspring would match their selected parents exactly. If h² = 0, the offspring would revert completely to the original population mean, no matter how carefully the parents were chosen.

Realized heritability

Rearranging the breeder's equation gives a direct way to estimate heritability from a selection experiment:

h² = R / S

This is the realized heritability. For example, if mice average 40 g, the selected parents average 50 g (S = 10 g) and their offspring average 43 g (R = 3 g), then h² = 3/10 = 0.30. Realized heritability is valuable because it is measured from the very process it is meant to predict, rather than from correlations among relatives.

Selection intensity

The size of S depends on two things: how strictly the breeder selects and how variable the trait is. Breeders separate these by writing S = i σ_P, where σ_P is the phenotypic standard deviation and i is the selection intensity, the selection differential in standard deviation units. For a normally distributed trait, keeping the top 50 percent gives i ≈ 0.80, the top 20 percent gives i ≈ 1.40, the top 10 percent gives i ≈ 1.76, and the top 1 percent gives i ≈ 2.67. The breeder's equation then becomes R = i h² σ_P. Dividing by the generation interval gives gain per year, which is why genomic selection of young bulls has sped dairy improvement.

Correlated response

Selecting on one trait often changes others. A correlated response occurs when two traits share a genetic correlation, usually because some genes affect both (pleiotropy) or because genes for the two traits are tightly linked. Broiler chickens selected intensely for rapid growth also developed leg problems and heart failure; dairy cows selected for milk yield showed declining fertility; and in many species, selection for large body size lengthens development time. Correlated responses can also be exploited: if a trait is hard or expensive to measure, a breeder can select on an easily measured trait that is genetically correlated with it.

Long term selection and selection limits

Two famous experiments show what sustained selection can do. In the Illinois long term corn selection experiment, begun in the late 1800s, separate lines were selected for high and low kernel oil content. After more than 100 generations, the high oil line had risen from about 5 percent oil to more than 20 percent, and the low oil line had fallen to near 1 percent, values far outside the range of the starting population.

Why does response continue so long? First, a polygenic trait has many loci, and a starting population is very unlikely to contain an individual with the favorable allele at every one; selection gradually assembles combinations that never existed before. Second, new mutations continually add a small amount of additive variance each generation. Eventually, however, response may slow or stop, reaching a selection plateau. Causes include fixation of favorable alleles (V_A exhausted), opposing natural selection (very high growth lines may become infertile), and negative genetic correlations with fitness.

Section 5QTL mapping and genome wide association

From variance components to genes

Heritability tells us that genes matter for a trait, but not which genes, where they are, or how large their effects are. A quantitative trait locus (QTL) is a region of the genome containing one or more genes that influence a quantitative trait. Two families of methods find them. QTL mapping uses controlled crosses between lines that differ in the trait; genome wide association studies (GWAS) use natural populations, including humans, where controlled crosses are impossible. Both rely on the same principle: if a genetic marker sits near a gene that affects the trait, individuals carrying different marker alleles will differ, on average, in the trait.

QTL mapping in experimental crosses

The classic design starts with two inbred lines that differ strongly in the trait, for example a high yielding and a low yielding tomato. The lines also differ at thousands of molecular markers, such as single nucleotide polymorphisms (SNPs) or microsatellites, scattered across the genome. They are crossed to produce an F1, and the F1 is either intercrossed to make an F2, backcrossed to one parent, or selfed for several generations to make recombinant inbred lines (RILs), each a homozygous mosaic of the two parental genomes. Every individual in the mapping population is then genotyped at all markers and measured for the trait.

At each marker, the population is sorted by marker genotype and the trait means are compared. At a marker unlinked to any QTL, the genotype classes have the same mean, because independent assortment randomizes the trait genes relative to the marker. At a marker tightly linked to a QTL, marker genotype predicts QTL genotype, so the classes differ. The size and pattern of the differences estimate the QTL's effects:

• The additive effect a is half the difference between the two homozygous classes.
• The dominance effect d is how far the heterozygote mean departs from the midpoint of the homozygotes.

Because recombination between the marker and the QTL weakens the association, the effect is strongest at the marker closest to the QTL. Interval mapping refines this by testing positions between markers, using the known map distances to calculate the probability of each QTL genotype at each position. The evidence at each position is expressed as a LOD score, the logarithm (base 10) of the odds that the data arose with a QTL at that position versus with no QTL. A LOD of 3 means the data are 10³ = 1,000 times more likely under the QTL hypothesis. Plotting LOD along each chromosome produces peaks at likely QTL positions; thresholds of about 3 (often set more exactly by permutation testing) declare significance, and the region within about one LOD unit of the peak gives a confidence interval.

Limits of QTL mapping

QTL mapping in crosses is powerful but coarse. An F2 has had only one round of meiosis to break up parental chromosomes, so a QTL peak often spans 10 to 30 centimorgans and hundreds of genes. Only variants that differ between the two parental lines can be detected, and small effect QTL are easily missed unless the mapping population is large. Intervals are narrowed with further crosses and candidate genes tested by gene editing. Success stories include the tomato gene fw2.2, which accounts for a large share of the difference in fruit size between wild and cultivated tomatoes, and the maize gene tb1, whose altered regulation helped turn bushy teosinte into single stalked corn.

Genome wide association studies

In humans and in many natural populations, experimental crosses are not possible. Instead, a GWAS genotypes hundreds of thousands to millions of SNPs in thousands to hundreds of thousands of unrelated people and tests each SNP for association with the trait. For a continuous trait like height, the test asks whether the trait mean differs among SNP genotypes, usually by regressing the trait on the number of copies of one allele (0, 1 or 2). For a disease, it compares allele frequencies in cases and controls and expresses the effect as an odds ratio.

GWAS depends on linkage disequilibrium (LD), the nonrandom association of alleles at nearby loci. In a population, recombination has broken up most chromosome segments over many generations, but alleles very close together are still inherited as blocks called haplotypes. A genotyped SNP therefore acts as a tag for untyped variants in the same LD block. A significant SNP is usually not the causal variant itself, only a neighbor in LD with it. Because LD blocks in human populations span only tens of kilobases, GWAS localizes signals far more precisely than F2 QTL mapping, often to a handful of genes.

Multiple testing and the significance threshold

With a million tests, the usual P < 0.05 would produce about 50,000 false positives by chance alone. GWAS therefore uses a Bonferroni style correction, dividing 0.05 by the approximate number of independent common variants in the human genome, about one million: 0.05/1,000,000 = 5 × 10⁻⁸. This is the conventional genome wide significance threshold. Results are displayed on a Manhattan plot, with −log₁₀(P) on the vertical axis so that small P values become tall points. The threshold 5 × 10⁻⁸ sits at −log₁₀(5 × 10⁻⁸) = 8 − log₁₀(5) = 8 − 0.70 = 7.3. Significant associations must then be replicated in an independent sample.

Pitfalls: population stratification

If cases and controls come from populations with different ancestries, any SNP whose allele frequency differs between those ancestries will appear associated with the disease, whether or not it has anything to do with it. This population stratification produces false positives. Modern GWAS correct for stratification by including principal components of genome wide genotypes, or relatedness matrices, in the analysis, and by checking that the bulk of the P values follow the null expectation on a quantile quantile (Q-Q) plot.

What GWAS has revealed

Two decades of GWAS have transformed our picture of complex traits. Most traits are highly polygenic: adult height is associated with more than 10,000 independent SNPs, each shifting height by about 1 mm or less. Most associated variants lie in noncoding DNA, especially in enhancers active in the relevant tissue, implying that they change gene regulation rather than protein structure. Many loci are shared among traits, which is pleiotropy on a genomic scale. Some findings have pointed to unexpected biology: GWAS of Crohn disease highlighted autophagy genes, and GWAS of age related macular degeneration identified complement factor H, opening new treatment directions.

Early GWAS posed the puzzle of missing heritability. Twin studies put the heritability of height near 0.8, yet the first significant SNPs together explained only a few percent of the variance. The gap has largely closed for height as samples grew into the millions, showing that much of the missing heritability was spread across thousands of variants too small to detect individually. Remaining gaps are attributed to rare variants poorly tagged by SNP arrays, structural variants, imperfect LD tagging, gene by environment interaction and inflated twin estimates.

Polygenic scores

A polygenic score (polygenic risk score) adds up an individual's alleles across many associated SNPs, each weighted by its GWAS effect size: score = Σ (number of effect alleles × effect size). For coronary artery disease, breast cancer and type 2 diabetes, people in the top few percent of the score distribution can have risk comparable to carriers of single gene mutations. Polygenic scores also have important limits. They predict poorly in populations whose ancestry differs from the GWAS sample, because LD patterns and allele frequencies differ; most large GWAS have been done in people of European ancestry, so scores currently work best for them, an equity problem the field is working to correct. A score gives probabilities for a population of people with similar scores, not a diagnosis for an individual, and its predictive power is capped by the trait's heritability.

Words worth knowing