Introduction to Quantitative Genetics

Peter Sørensen

Center for Quantitative Genetics and Genomics

Aarhus University

2026-03-27

Learning Goals - Quantitative Genetics

1. Measuring Quantitative Variation

• Understand how mathematical models and statistical methods are used to study complex traits

2. A Simple Genetic Model

• Assess the relative contributions of genetic and environmental factors to phenotypic variation

3. Heritability

• Calculate and interpret broad-sense heritability
  
• Calculate and interpret narrow-sense heritability
  
• Use parental phenotypes to predict offspring phenotypes

4. Mapping Quantitative Traits

• Explain the principle behind genome-wide association studies

In quantitative genetics, the focus shifts from single-gene traits to quantitative traits.

These are traits that vary continuously, such as height, milk yield, blood pressure, or crop yield.

The first step is measuring variation.

Mathematical models and statistical methods are used to describe and analyze complex traits.

This is where genetics meets statistics.

Next, a simple genetic model is introduced.

This model separates phenotypic variation into genetic and environmental components, and quantifies their contributions.

Then, heritability is considered.

Broad-sense heritability measures the total genetic contribution to variation.

Narrow-sense heritability focuses on additive effects, which enable prediction of offspring phenotypes from parental phenotypes.

Finally, association mapping is introduced.

These methods test for statistical links between genetic variants and traits, helping identify the loci involved.

By the end of this chapter, key concepts include: how quantitative traits are analyzed, how heritability is interpreted, and how genetic loci can be identified.

What Is Quantitative Genetics?

Quantitative genetics studies complex traits influenced by many genes and environmental factors.

It extends Mendelian genetics by focusing on the combined effects of many genes and the environment.

Statistical models are central to its analysis.

A biological discipline with great applied significance for agriculture, human healthcare, and for understanding evolutionary processes.

Core concepts

• Definition of a quantitative trait
• Genetic models for quantitative traits
• Genetic parameters (heritability, variance, correlation)
• Statistical and mathematical modeling

Relevant for genetic analyses

• Estimating heritability
• Estimating genetic risk or breeding values from pedigree or genomic data
• Predicting response to selection
• Evaluating consequences of selection

Quantitative genetics is the study of complex traits — traits influenced by many genes and environmental factors.

It builds on Mendelian genetics, but instead of focusing on single genes with large effects, it considers the combined effects of many genes acting together.

Because these effects are small and distributed, statistical models are essential.

Quantitative genetics has major applications: in agriculture, in human healthcare, and in understanding evolutionary change.

The course focuses on four core elements: definition of quantitative traits, genetic modeling, estimation of key parameters such as heritability and genetic correlation, and statistical analysis of data.

These tools make it possible to estimate heritability, predict breeding values, assess genetic risk, and predict response to selection.

In summary, quantitative genetics provides a mathematical framework for understanding and predicting complex trait variation.

Natural and Artificial Selection

Natural selection and selective breeding can both cause changes in animals and plants:

Natural selection happens naturally.

Selective breeding occurs when humans intervene => artificial selection.

Selective breeding (and natural selection) takes place over many generations.

This slide contrasts natural and artificial selection.

The top image shows Darwin’s finches. Different beak shapes evolved because birds with beaks better suited to available food survived and reproduced more successfully. This is natural selection — the environment determines which traits increase in frequency.

The bottom image shows vegetables with diverse shapes and colors. This diversity is the result of artificial selection. Humans choose individuals with desirable traits and breed them intentionally.

In both cases, the underlying mechanism is the same: individuals with certain heritable traits leave more offspring.

The difference lies in who or what applies the selection pressure — nature or humans.

Both processes operate over multiple generations and lead to genetic change in populations.

What Is Quantitative Genetics?

Phenotype (P) is determined by genotype (G) and environment (E):

\[ P = G + E \]

• Parents transmit a random sample of their genes to offspring
  
• Each offspring receives a different genetic combination

Quantitative genetics asks:

How much of the phenotypic variation is due to genetic differences?

This leads to the concept of heritability.

Why does this matter?

Quantitative genetics allows us to:

• Improve traits through selective breeding
  
• Predict genetic risk in human populations
  
• Understand how natural selection changes populations

In all cases, we seek to estimate the effect of genotypes (\(G\)) on phenotypes (\(P\)).

Quantitative genetics starts from a simple model:

Phenotype equals genotype plus environment.

Because parents transmit a random sample of their genes, offspring differ genetically.

These genetic differences contribute to phenotypic differences.

The central question is:

How much of the observed variation is due to genetic variation?

This leads to the concept of heritability.

Quantitative genetics is not only theoretical.

In animal and plant breeding, it is used to improve traits over generations.

In human genetics, it is used to estimate genetic risk from genomic information.

In evolutionary biology, it helps explain how natural selection changes populations over time.

In all cases, the key task is to estimate the effect of genotypes on phenotypes.

What Are Complex (Quantitative) Traits?

Mendelian traits

• Discrete phenotypes (e.g., red/white, sick/healthy)
• Simple genotype–phenotype relationship
• Often large gene effects

Quantitative traits

• Many phenotype classes, often continuously distributed
• Influenced by many genes, each with small effects
• Individual gene effects are typically not directly observable
• Often influenced by the environment

This slide distinguishes Mendelian and quantitative traits.

Mendelian traits are simple.

They fall into clear categories, such as red versus white, or healthy versus sick.

Usually, a single gene has a large effect, so the genotype–phenotype relationship is straightforward.

Quantitative traits are different.

They show a continuous range of values, such as height or milk yield.

They are influenced by many genes, each with a small effect.

They are also affected by the environment.

So instead of clear categories, there is a smooth distribution.

The key point is:

Most traits of interest are controlled by many genes acting together.

Types of Traits and Inheritance

Individuals are described by their phenotype (e.g., appearance, performance, or disease status).

Continuous traits

• Infinite range of values (e.g., height)
  
• Typically influenced by many genes and the environment

Categorical traits

• Discrete groups (e.g., purple vs. white flowers)
  
• May involve simple or complex inheritance

Threshold traits

• Categorical outcome
  
• Underlying liability is quantitative (e.g., type 2 diabetes)

Meristic traits

• Countable values (e.g., clutch size)
  
• Quantitative but not continuous

Quantitative traits come in several forms, and it is important not to confuse the type of trait with the type of inheritance.

Continuous traits can take an infinite range of values. Height is a classic example.

Categorical traits fall into distinct groups. Some follow simple Mendelian inheritance, such as flower color in peas.

Many, especially diseases, involve complex inheritance.

Threshold traits are a special case.

The observed outcome is categorical, but the underlying liability is quantitative.

Disease occurs when the underlying risk exceeds a threshold.

Meristic traits are countable traits.

A bird can lay one, two, or three eggs — but not two and a half.

These traits are discrete, yet still quantitative.

The key message is this:

Complex traits are defined by complex inheritance, not by whether the phenotype is continuous or categorical.

Measuring Quantitative Variation

To study complex traits, we need basic statistical tools:

• Mean describes the average level of a trait
  
• Variance quantifies variation around the mean
  
• Normal distribution describes how trait values are distributed

These tools allow us to quantify and compare phenotypic variation.

Mean

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \]

Variance

\[ \mathrm{Var}(X) = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]

Normal distribution

\[ f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \]

To study quantitative traits, basic statistical tools are required.

First, the mean.

The mean describes the average level of a trait in a population.

Second, variance.

Variance describes how much individuals differ around that average.

Two populations may have the same mean, but very different levels of variability.

Many quantitative traits follow an approximately normal, bell-shaped distribution.

This distribution plays a central role in quantitative genetics, because it emerges naturally when many small effects combine.

These simple concepts — mean, variance, and the normal distribution — form the foundation of quantitative genetics.

The Mean

When studying quantitative traits, we usually sample from a population.

Sample mean:

\[ \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \]

Population mean:

\[ \mu \]

The mean describes the center of the data.

Example

Height (cm) for 5 individuals:
168, 172, 170, 174, 166

\[ \bar{X} = \frac{168 + 172 + 170 + 174 + 166}{5} \]

\[ \bar{X} = \frac{850}{5} = 170 \]

We use \(\bar{X}\) to estimate the true population mean \(\mu\).

When studying quantitative traits, the entire population is usually not observed.

Instead, a random sample is used.

If the sample is random, it provides an unbiased estimate of population parameters.

The sample mean is calculated by adding all observed values and dividing by the number of observations.

The sample mean is denoted by X-bar, and the population mean by mu.

Because traits such as height vary among individuals, the phenotype is a random variable.

Sampling therefore introduces variation due to chance.

On the left, the formal definition of the sample mean is shown.

On the right, a simple example is computed.

The same calculation applies regardless of sample size.

The key idea is this:

The sample mean is the best estimate of the unknown population mean.

The Variance

Variance describes the spread of the data.

Deviation from the mean:

\[ d_i = X_i - \bar{X} \]

Population variance:

\[ \sigma^2 = \frac{\sum (X_i - \mu)^2}{n} \]

Standard deviation:

\[ \sigma = \sqrt{\sigma^2} \]

Example

Using the sample: 168, 172, 170, 174, 166

Mean: \[ \bar{X} = 170 \]

Deviations: -2, 2, 0, 4, -4

Squared deviations: 4, 4, 0, 16, 16 (sum = 40)

Variance (n = 5): \[ \sigma^2 = \frac{40}{5} = 8 \]

Standard deviation: \[ \sigma = \sqrt{8} \approx 2.83 \text{ cm} \]

The mean describes the center of the data, but not how spread out the values are.

Variance measures the spread.

It is the average squared deviation from the mean.

Deviations are defined as each observation minus the mean.

Because positive and negative deviations cancel, they are squared before averaging.

The standard deviation is the square root of the variance, which returns the units to the original scale.

On the right, the calculation is shown for a simple example.

The key idea is this:

Variance and standard deviation describe how much individuals differ from the mean.

This becomes central in quantitative genetics, when phenotypic variance is partitioned into genetic and environmental components.

The Normal Distribution

Many biological traits follow an approximately normal distribution.

A normal distribution is fully determined by:

• Mean: \(\mu\)
  
• Variance: \(\sigma^2\)
  (or standard deviation \(\sigma\))

Probability density function:

\[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right) \]

Key Properties

• Bell-shaped curve
  
• Symmetric around the mean \(\mu\)
  
• Spread determined by the standard deviation \(\sigma\)
  
• Total area under the curve equals 1

Central role in quantitative genetics

• Describing quantitative traits
  
• Modeling phenotypic variation
  
• Partitioning genetic and environmental variance

Many biological traits follow an approximately normal, or bell-shaped, distribution.

A normal distribution is determined by two parameters:

The mean, which sets the center,

and the variance, which determines the spread.

The curve is symmetric around the mean.

Values close to the mean are more common, while extreme values are rare.

Changing the mean shifts the distribution left or right.

Changing the variance changes how spread out the values are.

The total area under the curve equals one, because it represents total probability.

This distribution plays a central role in quantitative genetics.

It is used to describe trait variation and to partition phenotypic variance into genetic and environmental components.

Normal Distribution (Effect of μ and σ)

Here are three normal distributions.

The dashed vertical lines mark the mean of each distribution.

First, focus on the two curves with the same mean, μ equals zero.

The only difference is the standard deviation.

When σ increases, the distribution becomes wider and flatter.

When σ decreases, it becomes narrower and taller.

Changing σ affects the spread, but not the location of the mean.

Now consider the curve where μ equals one.

Here, the spread is the same, but the entire distribution is shifted to the right.

Changing μ shifts the center, while changing σ changes the spread.

In all cases, the total area under the curve equals one.

In quantitative genetics, many traits follow this type of distribution.

Partitioning phenotypic variance means partitioning σ², the variance of this distribution.

A Simple Genetic Model for Quantitative Traits

Model

A quantitative trait can be expressed as:

\[ P = \mu + G + E \]

• \(\mu\) = population mean
  
• \(G\) = genetic deviation
  
• \(E\) = environmental deviation

Deviation from the mean:

\[ p = P - \mu = G + E \]

Interpretation

Differences between individuals arise from:

• Genetic effects
  
• Environmental effects
  
• Or both

This model underlies:

• Variance partitioning
  
• Heritability
  
• Prediction of response to selection

This is the fundamental model of quantitative genetics.

The phenotype equals the population mean plus a genetic deviation plus an environmental deviation.

Individuals differ from the mean because of genetic effects, environmental effects, or both.

It is often useful to subtract the mean and work with deviations, so that p equals G plus E.

This focuses on what causes individuals to differ.

The genetic component reflects how genotype shifts the phenotype.

The environmental component reflects all non-genetic influences.

This simple equation is the foundation of quantitative genetics.

It underlies variance partitioning, heritability, and prediction of response to selection.

Illustrative Example: Decomposing a Phenotype

Step 1: Observed phenotype

• Yao Ming’s height: ( P = 229 ) cm
  
• Population mean: ( = 170 ) cm

Step 2: Deviation from the mean

\[ p = P - \mu \]

\[ p = 229 - 170 = 59 \text{ cm} \]

He is 59 cm taller than average.

Step 3: Interpret using the genetic model

Recall:

\[ p = G + E \]

Therefore:

\[ 59 = G + E \]

The deviation must arise from:

• Genetic effects (\(G\))
  
• Environmental effects (\(E\))
  
• Or both

We observe only the total deviation — not its separate components.

This slide applies the genetic model to a concrete example.

Yao Ming is 229 cm tall.

If the population mean is 170 cm, the deviation is:

p = 229 − 170 = 59 cm.

According to the model:

p equals G plus E.

So this deviation must arise from genetic effects, environmental effects, or both.

The individual components G and E cannot be observed directly.

Only the total phenotype is observed.

The purpose of the model is not to separate G and E for individuals, but to analyze variation in populations.

This decomposition leads directly to variance partitioning and heritability.

Modes of Gene Action at One Locus

Why this matters

Additive gene action:

• Allele effects add linearly
• Produces predictable transmission
• Creates additive variance (\(V_A\))

Dominance:

• Depends on allele combinations
• Does not transmit predictably
• Contributes to \(V_D\), not \(V_A\)

Key insight:

Only additive effects contribute to narrow-sense heritability

This figure illustrates three modes of gene action at a single locus.

On the x-axis are the genotypes: bb, Bb, and BB.

On the y-axis is the trait value.

In the top panel, there is additive gene action.

The heterozygote lies exactly between the two homozygotes.

Each allele contributes a fixed amount, and the effects add linearly.

In the middle panel, there is dominance.

The heterozygote has the same phenotype as one of the homozygotes.

The effect depends on allele combinations, not simple addition.

In the bottom panel, there is partial dominance.

The heterozygote is intermediate, but not exactly halfway.

The key idea is this:

Only additive effects are transmitted predictably from parents to offspring.

Dominance effects depend on allele combinations and are reshuffled each generation.

Therefore, additive variance determines narrow-sense heritability and response to selection.

Genotype by Genotype Interaction (Epistasis)

Interaction between Locus 1 and Locus 2

No Interaction (Additive)

• Parallel lines: The lines do not cross or diverge.
• The effect of \(AA\) is the same regardless of whether the background is \(bb\) or \(BB\).
• Genetic effects at different loci simply add up linearly.

G\(\times\)G Interaction (Epistasis)

• Non-parallel lines: Lines cross or show different slopes.
• The effect of \(AA\) depends on the genotype at Locus 2.
• Example: \(AA\) only increases the trait value if the individual also carries \(BB\).

Key Idea:
The phenotypic expression of one gene is modified by the presence of another gene.

This figure illustrates gene-by-gene interaction, also called epistasis.

In the top panel, there is no interaction.

The lines are parallel.

The effect of locus 1 is the same regardless of locus 2.

Genetic effects simply add up.

In the bottom panel, there is epistasis.

The lines are not parallel.

The effect of one gene depends on the genotype at another gene.

For example, the AA genotype only increases the trait when combined with BB at the second locus.

The key idea is this:

The effect of a gene can depend on the genetic background.

In quantitative genetics, this complicates modeling, because allele effects are not fixed.

Additive variance remains the main driver of selection, but epistasis contributes to total genetic variation.

Genotype × Environment Interaction

No interaction

• Parallel lines
  
• Genotype difference is constant across environments

G×E interaction

• Lines cross (or diverge)
  
• Genotype ranking depends on environment

Key idea:
Genotype performance can be environment-dependent.

This figure shows reaction norms, how genotypes perform across environments.

In the top panel, there is no genotype-by-environment interaction.

The lines are parallel.

The difference between genotypes is constant across environments.

In the bottom panel, there is genotype-by-environment interaction.

The lines cross.

The performance of the genotypes depends on the environment.

A genotype that performs best in one environment may perform worse in another.

This is the key idea:

Genetic effects can depend on the environment.

This matters because selection in one environment may not give the best genotype in another.

In such cases, the model is extended to include a genotype-by-environment interaction term.

From Phenotype to Variance

Start with the model

\[ P = \mu + G + E \]

Phenotypic variance:

\[ V_P = \mathrm{Var}(P) \]

Since \(\mu\) is constant:

\[ V_P = \mathrm{Var}(G + E) \]

Basic statistical rule:

\[ V_P = V_G + V_E + 2\mathrm{Cov}(G,E) \]

If genetic and environmental effects are independent

\[ \mathrm{Cov}(G,E) = 0 \]

Then:

\[ V_P = V_G + V_E \]

Meaning:

Total variation = genetic variation + environmental variation

This is the foundation of heritability.

Start with the phenotype model:

P equals μ plus G plus E.

Phenotypic variance measures how much individuals differ from the mean.

Because μ is constant, it does not contribute to variance.

So the variance of P becomes the variance of G plus E.

A basic rule states:

The variance of a sum equals the sum of the variances plus twice the covariance.

If genetic and environmental effects are independent, the covariance is zero.

In that case:

Total phenotypic variation equals genetic variation plus environmental variation.

This relationship is the foundation of heritability.

Visualizing Variance Partitioning

This figure shows total phenotypic variance as a single bar.

That total variance can be partitioned into two components:

a genetic component and an environmental component.

In this example, genetic variance accounts for 40 percent, and environmental variance for 60 percent.

The exact values are not important.

The key idea is:

Phenotypic variation can be decomposed into genetic and environmental components.

This provides the basis for heritability, which is the proportion of total variance that is genetic.

Example: Partitioning Variance (Maize Experiment I)

Numerical results

Total phenotypic variance:

\[ V_P = 14.67 \]

Genetic variance
(variation among inbred line means):

\[ V_G = 12.00 \]

Environmental variance
(variation among environments):

\[ V_E = 2.67 \]

Check:

\[ V_P = V_G + V_E = 12.00 + 2.67 = 14.67 \]

Interpretation

• Most variation is genetic
  (12.00 of 14.67)

• Environmental variation is smaller
  (2.67)

• The decomposition holds because:

  \[ \mathrm{Cov}(G,E) = 0 \]

• Genotypes were randomized
  across environments

This illustrates that variance components add when genotype and environment are independent.

This slide applies variance decomposition to real data.

In this experiment, inbred maize lines were grown across different environments.

The total phenotypic variance is 14.67.

This is partitioned into:

genetic variance, 12.00,

and environmental variance, 2.67.

These components add exactly to the total variance.

This works because genetic and environmental effects are independent.

Genotypes were randomized across environments, so the covariance is zero.

The key idea is:

Phenotypic variance can be decomposed into genetic and environmental components.

This leads directly to heritability, which is defined as a ratio of these components.

Covariance and Correlation in Quantitative Genetics

Covariance: The Core Quantity

Covariance measures whether two variables vary together:

\[ \mathrm{Cov}(X,Y) \]

It appears everywhere in Quantitative Genetics:

• Phenotypic variance
  
• Genetic variance
  
• Genetic covariance between traits
  
• Response to selection

Variance is simply:

\[ \mathrm{Var}(X) = \mathrm{Cov}(X,X) \]

Standardized Form: Correlation

Correlation rescales covariance:

\[ r = \frac{\mathrm{Cov}(X,Y)}{\sigma_X \sigma_Y} \]

• Unitless
  
• Bounded between −1 and 1
  
• Measures strength of linear association

In the variance model:

\[ V_P = V_G + V_E + 2\mathrm{Cov}(G,E) \]

If Cov(G,E) = 0, variance components add cleanly.

Covariance measures whether two variables vary together.

In quantitative genetics, this concept is fundamental.

Even variance is a special case of covariance, the covariance of a variable with itself.

When phenotypic variance is decomposed, it equals genetic variance plus environmental variance plus twice the covariance between genetic and environmental effects.

If genotype and environment are independent, the covariance is zero, and the variance partitions cleanly.

If they are correlated, the covariance term remains, and the components cannot be separated cleanly.

In this sense, quantitative genetics is largely the study of variances and covariances.

Visualizing Correlation

This figure visualizes correlation between two variables.

Each point represents a pair of observations, here shown as twin heights.

In the top panel, there is perfect correlation.

All points lie exactly on a straight line.

Knowing one value completely determines the other.

In the middle panel, there is strong positive correlation.

The points follow a clear upward trend, but with some scatter.

As one variable increases, the other tends to increase.

In the bottom panel, there is no correlation.

The points are scattered randomly.

Knowing one value gives no information about the other.

The key idea is this:

Correlation measures the strength of a linear relationship between two variables.

Broad-Sense Heritability (\(H^2\))

Definition

Key question:

• How much of the observed variation is genetic?

Broad-sense heritability:

\[ H^2 = \frac{V_G}{V_P} \]

where:

• \(V_G\) = genetic variance
  
• \(V_P\) = total phenotypic variance

Range:

• \(0 \le H^2 \le 1\)

Interpretation

• \(H^2 = 0\) → all variation is environmental
  
• \(H^2 = 1\) → all variation is genetic

Important:

• Describes variation in a population
• Not the “% genetic” of an individual
• Population- and environment-specific

“Broad-sense” includes:

• Additive effects
  
• Dominance effects
  
• Gene–gene interactions (epistasis)

Broad-sense heritability answers a key question:

How much of the variation in a population is associated with genetic differences?

It is defined as genetic variance divided by total phenotypic variance.

Its value ranges from zero to one.

If broad sense heritability equals zero, all variation is environmental.

If broad sense heritability equals one, all variation is genetic.

But this does not apply to individuals.

Heritability always describes variation within a population.

If all individuals had identical genotypes, genetic variance would be zero, even though genes still affect the trait.

Broad-sense heritability includes all genetic effects:

additive, dominance, and interaction effects.

Heritability is not a fixed property.

It depends on the population, the environment, and the amount of genetic variation present.

Heritability Depends on the Environment (Maize Example)

Experiment I: Similar Environments

• \(V_G = 12.0\)
• \(V_E = 2.67\)
• \(V_P = 14.67\)

\[ H^2 = \frac{12.0}{14.67} = 0.82 \]

Interpretation:

• 82% of the observed variation is genetic
  
• Trait appears highly heritable

Experiment II: More Variable Environments

• \(V_G = 12.0\)
• \(V_E = 24.0\)
• \(V_P = 36.0\)

\[ H^2 = \frac{12.0}{36.0} = 0.33 \]

Interpretation:

• 33% of the observed variation is genetic
  
• Environmental variation dominates

This slide compares two experiments using the same maize lines.

Importantly, genetic variance is identical in both cases.

In Experiment I, environmental variance is small.

Most of the total variation is genetic, and heritability is high.

In Experiment II, environmental variance is much larger.

Total variation increases, and heritability drops substantially.

Even though the genetics did not change, heritability changes.

This illustrates a key concept:

Heritability is not a fixed property of a trait.

It depends on the balance between genetic and environmental variation.

If environmental variance increases, heritability decreases.

If environmental variance decreases, heritability increases.

This does not mean genes are less important.

It means their contribution to observed variation has changed.

Estimating Heritability Using Identical Twins

Theory

Identical (monozygotic) twins:

• Share 100% of their genes
  
• If reared apart → environments treated as independent

Under independence:

\[ \mathrm{Cov}(\text{Twin 1}, \text{Twin 2}) = V_G \]

Therefore:

\[ H^2 = \frac{V_G}{V_P} = \frac{\mathrm{Cov}(\text{twins})}{V_P} \]

Key assumption:

\[ \mathrm{Cov}(G,E) = 0 \]

Example: IQ in Twins Reared Apart

From data:

• Covariance between twins = 119.2
  
• Total phenotypic variance \(V_P\) = 154.3

Heritability estimate:

\[ H^2 = \frac{119.2}{154.3} = 0.77 \]

Interpretation

• 77% of IQ variation (in this population)
  is associated with genetic differences

• Refers to population variation
  — not individuals

Identical twins share the same genotype.

If they are reared apart, their environments can be treated as independent.

Under this assumption, the covariance between twin phenotypes reflects genetic variance.

So covariance between twins equals V_G.

Broad-sense heritability can therefore be estimated as:

covariance between twins divided by total phenotypic variance.

In this example, the covariance is 119.2, and total variance is 154.3.

This gives broad sense heritability equal to 0.77.

Interpretation is important:

This does not apply to individuals.

It means that 77 percent of the variation in IQ in this population is associated with genetic differences.

Twin studies rely on key assumptions:

independent environments, no strong gene–environment correlation, and accurate measurement.

Broad-Sense Heritability in Humans

Trait	\(H^2\)
Physical traits
Height	0.88
Fingerprint ridge count	0.97
Systolic blood pressure	0.64
Cognitive traits
IQ	0.69
Spatial processing speed	0.36
Personality traits
Extraversion	0.54
Neuroticism	0.48
Psychiatric disorders
Autism	0.90
Schizophrenia	0.80
Major depression	0.37
Beliefs / attitudes
Religiosity (adults)	0.30–0.45
Conservatism (adults)	0.45–0.65

This table shows broad-sense heritability (\(H^2\)) estimates from twin studies.

\[ H^2 = \frac{V_G}{V_P} \]

Key patterns:

• Physical traits → often high \(H^2\)
  (e.g., height, fingerprints)

• Cognitive & personality traits → moderate \(H^2\)

• Psychiatric disorders → variable
  (some high, some moderate)

• Even beliefs and attitudes show measurable heritability

This table shows broad-sense heritability estimates from twin studies.

Recall: broad sense heritability equals genetic variance divided by total phenotypic variance.

These values describe how much variation in a population is associated with genetic differences.

Several patterns emerge.

Physical traits often show high heritability, such as height and fingerprint patterns.

Cognitive and personality traits tend to show moderate heritability.

Psychiatric disorders vary, with some showing high values and others more moderate.

Even beliefs and attitudes can show measurable heritability.

The key interpretation is critical:

High heritability does not mean a trait is genetically determined, or that it cannot change.

Heritability is always population-specific and environment-specific.

It measures variation — not destiny.

Narrow-Sense Heritability: Predicting Phenotypes

From Broad to Narrow Heritability

Broad-sense heritability:

\[ H^2 = \frac{V_G}{V_P} \]

But genetic variance contains multiple components:

• Additive variance (\(V_A\))
  → predictably transmitted

• Dominance variance (\(V_D\))
  → depends on allele combinations

Total genetic variance:

\[ V_G = V_A + V_D + V_I \]

where \(V_I\) = interaction (epistatic) variance

Narrow-Sense Heritability

\[ h^2 = \frac{V_A}{V_P} \]

Focuses only on additive variance.

Why?

• Additive effects are predictably inherited
  
• Determine parent–offspring resemblance
  
• Determine response to selection

Key idea:

Additive genetic variation drives response to selection.

Broad-sense heritability includes all genetic variance.

But genetic variance has different components.

Additive variance reflects the average effect of alleles.

Dominance and interaction variance depend on specific allele combinations.

Because allele combinations are reshuffled each generation, these components are not transmitted predictably.

Only additive effects are passed on in a consistent way.

Narrow-sense heritability focuses on this component:

narrow sense heritability equals additive variance divided by total phenotypic variance.

This is the key quantity for prediction.

It determines resemblance between relatives and response to selection.

If narrow sense heritability is high, selection is effective.

If narrow sense heritability is low, response is limited.

The key distinction is simple:

Broad-sense heritability describes total genetic influence, while narrow-sense heritability captures what can be inherited predictably.

Narrow-Sense Heritability (\(h^2\))

Definition

\[ h^2 = \frac{V_A}{V_P} \]

• \(V_A\) = additive genetic variance
  
• \(V_P\) = total phenotypic variance

Only additive variance is predictably transmitted.

Broad-sense heritability:

\[ H^2 = \frac{V_G}{V_P} \]

But:

\[ V_G = V_A + V_D + V_I \]

Estimation

Parent–offspring covariance:

\[ \mathrm{Cov}_{\text{parent, offspring}} = \frac{1}{2} V_A \]

Therefore:

\[ h^2 = \frac{2\,\mathrm{Cov}_{\text{parent, offspring}}}{V_P} \]

Half-sibs share \(\frac{1}{4}\) of additive effects:

\[ V_A = 4\,\mathrm{Cov}_{\text{half-sibs}} \]

Example:

Human height: \(h^2 \approx 0.8\)

Left panel:

Narrow-sense heritability focuses on additive genetic variance.

Broad-sense heritability includes all genetic effects, including dominance and interactions.

But only additive effects are transmitted predictably from parents to offspring.

That is why narrow-sense heritability focuses on V_A.

Right panel:

An offspring inherits half its alleles from each parent.

Because additive effects sum across alleles, the covariance between parent and offspring equals one-half of the additive variance.

Rearranging gives:

Narrow-sense heritability equals two times the covariance between parent and offspring, divided by total phenotypic variance.

Similarly, half-siblings share one-quarter of their additive effects.

So additive variance can be estimated as four times the covariance between half-sibs.

For example, human height has high narrow-sense heritability, around 0.8.

This means most of the variation in height within a population is associated with additive genetic differences.

But this does not mean height is genetically fixed.

Environmental effects still contribute to differences between individuals.

Narrow-Sense Heritability (h²)

Parent–Offspring Regression for Height

• Each point = one parent–offspring pair
  
• Clear positive relationship
  
• Slope of regression line ≈ \(h^2\)

Interpretation:

The steeper the slope, the greater the additive genetic contribution to variation.

This figure shows parent–offspring regressions for height.

In the top panel are mother–daughter pairs. In the bottom panel are father–son pairs.

Each point represents one parent–offspring pair.

There is a clear positive relationship: taller parents tend to have taller offspring.

The red line is the regression line.

Its slope estimates narrow-sense heritability.

Why?

Because additive genetic effects are transmitted predictably from parents to offspring.

If additive genetic variance is large, offspring closely resemble their parents, and the slope is steep.

If additive genetic variance is small, the resemblance is weaker, and the slope is flatter.

In this example, the slope is close to 0.8, indicating a strong additive genetic contribution to variation in height.

But this describes variation in a population, not how genetic height is for an individual.

Environmental effects still contribute to differences between individuals.

Narrow-Sense Heritability (h²) Across Species

Trait	h² (%)
Agronomic species
Body weight (cattle)	65
Milk yield (cattle)	35
Back-fat thickness (pig)	70
Litter size (pig)	5
Body weight (chicken)	55
Egg weight (chicken)	50
Natural species
Bill length (Darwin’s finch)	65
Flight duration (milkweed bug)	20
Plant height (jewelweed)	8
Fecundity (red deer)	46
Life span (collared flycatcher)	15

These are estimates of narrow-sense heritability (h²) in different species and traits.

\[h^2 = V_A / V_P\]

• Measures additive (transmissible) variation
  
• Predicts response to selection

Patterns:

• Production traits in livestock often show moderate–high h²
  
• Reproductive traits tend to have low h²
  
• Natural populations show wide variation

Low h² does not mean genes are unimportant —
it means additive variance is limited relative to total variance.

This table shows estimates of narrow-sense heritability across different traits and species.

Narrow-sense heritability measures the proportion of variation due to additive genetic effects — the component that is transmitted from parents to offspring.

This makes it directly relevant for selection and evolutionary change.

Several patterns emerge.

In agricultural species, production traits such as body weight and milk yield often show moderate to high heritability.

These traits respond well to selection.

In contrast, reproductive traits, such as litter size, typically show low heritability.

In natural populations, heritability varies widely across traits.

Some traits, like bill size in finches, show high values, while others, such as lifespan or plant height, are much lower.

The key point is:

Heritability depends on the trait, the population, and the environment.

Low heritability does not mean genes are unimportant.

It means that additive genetic variation is small relative to total variation.

Predicting Offspring Phenotypes

What is transmitted?

Individual deviation from the mean:

\[ x = a + d + e \]

Parents:

\[ x' = a' + d' + e' \]

\[ x'' = a'' + d'' + e'' \]

Expected offspring deviation:

\[ \mathbb{E}[x_0] = \frac{a' + a''}{2} \]

Only additive effects (\(a\))
are predictably transmitted.

Dominance (\(d\)) and environment (\(e\))
are not transmitted predictably.

Prediction Using \(h^2\)

Since \(a\) is not directly observed:

\[ \widehat{x}_{\text{offspring}} = h^2 \,\bar{x}_{\text{parents}} \]

Predicted phenotype:

\[ \widehat{X}_{\text{offspring}} = \mu + \widehat{x}_{\text{offspring}} \]

Example (Sheep fleece)

Population mean = 6.0 lb
Mean parental deviation = 0.75
\(h^2 = 0.4\)

\[ 0.4 \times 0.75 = 0.3 \]

Predicted offspring:

\[ 6.0 + 0.3 = 6.3 \text{ lb} \]

Left panel:

An individual’s deviation from the mean can be decomposed into three components:

Additive genetic effects, dominance effects, and environmental effects.

When parents reproduce, only additive effects are transmitted predictably.

Dominance effects depend on allele combinations, which are reshuffled each generation.

Environmental effects are not inherited, because offspring experience their own environments.

Therefore, the expected offspring deviation equals the average of the parents’ additive effects.

Right panel:

Because additive effects are not directly observed, they are estimated using narrow-sense heritability.

The expected offspring deviation equals:

narrow-sense heritability times the mean parental deviation.

In the sheep example:

The population mean is 6.0 pounds. The parents are 0.75 pounds above the mean. Narrow-sense heritability is 0.4.

The predicted offspring deviation is 0.3.

So the predicted offspring phenotype is 6.3 pounds.

Notice that offspring are predicted to be less extreme than their parents.

This is regression toward the mean.

If narrow-sense heritability is high, offspring closely resemble their parents.

If it is low, offspring are closer to the population mean.

This principle underlies selective breeding and evolutionary response to selection.

Selection on Complex Traits

Artificial selection

Humans select individuals with desirable phenotypes for reproduction.

Selection differential \[ S = \bar{X}_{\text{selected}} - \bar{X}_{\text{population}} \]

Response to selection \[ R = \bar{X}_{\text{offspring}} - \bar{X}_{\text{population}} \]

Breeder’s equation \[ R = h^2 S \qquad h^2 = \frac{R}{S} \]

Only additive genetic variance contributes to the response.

Example: Provitamin A in maize

Base population mean = 1.25 µg/g
Selected mean = 1.63 µg/g
Offspring mean = 1.44 µg/g

Selection differential:

\[ S = 1.63 - 1.25 = 0.38 \]

Response:

\[ R = 1.44 - 1.25 = 0.19 \]

Heritability:

\[ h^2 = \frac{0.19}{0.38} = 0.50 \]

About 50% of the variation contributing to selection is additive and transmitted to the next generation.

Left panel:

Artificial selection occurs when humans choose individuals with desirable phenotypes for reproduction.

The selection differential, S, is the difference between the selected parents and the original population mean.

The response, R, is the change in the offspring generation relative to that original mean.

The breeder’s equation links these quantities:

The response to selection equals narrow-sense heritability times the selection differential.

Only additive genetic variance contributes to a predictable response.

Right panel:

In the maize example:

The base population mean is 1.25 micrograms per gram.

The selected plants have a mean of 1.63, giving a selection differential of 0.38.

The offspring mean is 1.44, so the response is 0.19.

The response is about half of the selection differential, so narrow-sense heritability is approximately 0.5.

This means that about half of the variation driving the response to selection is additive and transmitted to the next generation.

The remaining variation reflects environmental effects and non-additive genetic effects, which do not produce a consistent response.

Key idea:

The larger narrow-sense heritability is, the stronger and faster the response to selection.

Selection Shifts the Population Mean

Figure 19-9.

One generation of artificial selection for provitamin A in maize.

• Base population mean = 1.25 μg/g
  

• Selected group mean = 1.63 μg/g
  

• Offspring mean = 1.44 μg/g

• Selection differential:
  \[ S = 1.63 - 1.25 = 0.38 \]

• Selection response:
  \[ R = 1.44 - 1.25 = 0.19 \]

Only the additive (heritable) component causes the shift in the next generation.

This figure shows how artificial selection shifts the population mean.

In panel (a), the base population is centered at 1.25 micrograms per gram.

Individuals in the right tail are selected for reproduction, with a mean of 1.63.

The difference between these values is the selection differential, equal to 0.38.

In panel (b), we see the offspring of the selected individuals.

The entire distribution has shifted to the right, with a new mean of 1.44.

The change from 1.25 to 1.44 is the response to selection, equal to 0.19.

Notice that the response is smaller than the selection differential.

This is because only additive genetic effects are transmitted predictably to the next generation.

Dominance and environmental effects do not contribute to a consistent response.

This illustrates the breeder’s equation:

The response to selection equals narrow-sense heritability times the selection differential.

Long-Term Response to Artificial Selection

Figure 19-10.

Long-term selection experiments demonstrate sustained evolutionary change.

1. Fruit flies
   

• Selected for increased flight speed
  
• Mean speed increased dramatically over 100 generations

2. Mice
   

• Selected for voluntary wheel running
  
• Large increase over just 10 generations
  
• Control (unselected) lines changed very little

Key message:

Populations contain substantial additive genetic variation
and can respond to selection for many generations.

This figure shows the long-term effects of artificial selection.

In panel (a), fruit flies were selected for increased flight speed.

In each generation, the fastest individuals were chosen to reproduce.

Over many generations, the mean flight speed increased substantially.

Importantly, the response continued over time, rather than stopping after a few generations.

In panel (b), mice were selected for voluntary wheel running.

The selected lines show a rapid increase in activity, while the control lines remain relatively unchanged.

This demonstrates a clear response to selection.

Two key principles emerge.

First, complex traits often contain substantial additive genetic variation.

Second, as long as additive variation is present, populations can continue to respond to selection over many generations.

This principle underlies modern breeding and provides a clear example of evolution in action.

What Heritability Does NOT Mean

Broad-sense heritability \[ H^2 = \frac{V_G}{V_P} \]

Narrow-sense heritability \[ h^2 = \frac{V_A}{V_P} \]

Heritability is:

• Population-specific
  
• Environment-specific
  
• A measure of variation, not destiny

Heritability does NOT mean:

• Not the “% genetic” of an individual
  
• Not that a trait is genetically fixed
  
• Not that a trait cannot change
  
• Not universal across populations
  
• Not independent of environment

Heritability is one of the most misunderstood concepts in genetics.

It does not describe how genetic a trait is in an individual.

For example, if height has high narrow-sense heritability, this does not mean a person is “mostly genetic.”

It means that most of the variation in height within a specific population is associated with genetic differences.

Heritability always refers to variation in a population, under specific environmental conditions.

It does not imply that a trait is fixed.

Even highly heritable traits can change if the environment changes.

For example, average human height increased substantially over the past century due to improved nutrition, despite high heritability.

Heritability is also not universal.

It depends on the population, the genetic variation present, and the environment.

If environmental variation increases, heritability decreases.

The key idea is simple:

Heritability measures variation — not destiny, and not individuals.

From Phenotype to Genotype: The “Missing” Link

What we know so far

• Traits such as height are heritable (\(h^2 \approx 0.8\)).
• Variation is multifactorial (many genes + environment).
• We can predict response to selection from parents.

The Big Question

• What do the underlying genes actually do?
• Where are they located in the genome?
• Which DNA variants contribute to \(V_G\)?
• What is the biological basis of allele effects?
• How do genes interact within pathways?

The Solution: Association Mapping

1. Strategy: Scan the genome for statistical associations.
2. Tool: High-density SNP markers (M).
3. Principle: Linkage disequilibrium (LD) with causal variants (Q).

We move from describing how much is inherited
to identifying what is inherited.

So far, the focus has been on phenotypes.

Many traits, such as height, are highly heritable. Variation is multifactorial, with contributions from many genes and the environment.

Quantitative genetics allows prediction of response to selection.

But it does not identify the specific genes involved.

This leads to the key question:

Which DNA variants generate genetic variation?

Where are they located in the genome, and what biological mechanisms do they affect?

To address this, the focus shifts from variance components to association mapping.

The idea is to scan the genome and test whether genetic markers are associated with the trait.

This is done using dense SNP markers.

If a marker is in linkage disequilibrium with a causal variant, it will show a statistical association.

In this way, the focus shifts from how much is inherited to what is inherited.

Genome-Wide Association Study (GWAS)

Goal

Identify genomic regions associated with complex traits
in natural, random-mating populations.

Core Idea

Test statistical associations between:

• Genetic markers (SNPs)
  
• Phenotypic variation

Key Characteristics

• Scans the entire genome
  
• Tests thousands to millions of SNPs
  
• No need to specify candidate genes in advance

Compared to classical QTL mapping

• No controlled crosses required
  
• No pedigree information required
  
• Applied directly in natural populations

A genome-wide association study, or GWAS, aims to identify genomic regions associated with complex traits.

It is applied in natural, random-mating populations, rather than controlled experimental crosses.

The core idea is simple:

Test whether genetic variation at a marker, typically a SNP, is associated with phenotypic variation.

This is done across the entire genome, often using hundreds of thousands or even millions of markers.

One key advantage is that no candidate genes need to be specified in advance.

The approach is unbiased, allowing discovery of novel loci.

Compared to classical QTL mapping, GWAS does not require controlled crosses or pedigree information.

Instead, it relies on naturally occurring genetic variation and statistical association.

The next step is to understand why this works, through linkage disequilibrium, which connects markers to causal variants.

Why GWAS Works: Marker–QTL Linkage Disequilibrium

Concept

• A causal mutation (QTL) influences the trait
  
• A nearby SNP may be in linkage disequilibrium (LD) with the QTL
  
• The SNP is therefore correlated with the causal variant

Therefore, the SNP shows statistical association
even if it is not itself causal.

Key Point

GWAS detects markers that are in LD
with causal variants.

This figure explains why genome-wide association studies work.

Each horizontal line represents a chromosome segment.

In Scenario 1, the red marker represents a causal mutation, a quantitative trait locus that directly affects the trait.

The blue marker is a nearby SNP.

Because these loci are physically close, they tend to be inherited together.

This non-random association is called linkage disequilibrium.

Even if the causal mutation is not observed directly, the nearby SNP is correlated with it.

As a result, the SNP shows a statistical association with the trait.

In Scenario 2, the marker and the causal variant are unlinked.

They segregate independently, so there is no linkage disequilibrium.

The marker then carries no information about the trait, and no association is detected.

This illustrates the core idea of GWAS:

Markers do not need to be causal.

They only need to be in linkage disequilibrium with causal variants.

Identifying the true causal mutation requires further analysis, such as fine-mapping and functional studies.

From Biological Data to Statistical Model

Phenotype data

Measured trait values for \(n\) individuals.

Example (height):

Individual	Phenotype (\(y\))
Ind₁	170
Ind₂	165
Ind₃	182
…	…

Phenotypes form a vector:

\[ \mathbf{y} = \begin{pmatrix} y_1 \\ y_2 \\ y_3 \\ \vdots \end{pmatrix} \]

Genotype matrix

Genotypes organized into an \(n \times m\) matrix.

	SNP₁	SNP₂	SNP₃	…
Ind₁	0	1	2
Ind₂	1	1	0
Ind₃	2	0	1
…

Common SNP coding:

• 0 = Homozygous reference
  
• 1 = Heterozygous
  
• 2 = Homozygous alternative

\[ \mathbf{X} \]

Genome-wide association studies rely on two types of data.

First, phenotypes.

These are measured trait values, such as height or milk yield.

They are represented as a vector, y, with one value per individual.

Second, genotypes.

These are organized into a matrix, X, where rows correspond to individuals and columns correspond to SNP markers.

Each entry is coded as 0, 1, or 2, representing the number of alternative alleles.

Together, the phenotype vector and genotype matrix form the foundation of the statistical model.

The key question is:

Does variation in a SNP, that is, a column of X, predict variation in the phenotype y?

This is the core idea behind GWAS, and it leads directly to the models used for association testing.

The Principle Behind GWAS

Goal

Identify loci associated with variation in a trait.

For each SNP test whether genotype predicts phenotype.

Statistical model (single SNP):

\[ y_i = \mu + \beta x_{ij} + \varepsilon_i \]

• \(y_i\) = phenotype of individual \(i\)
  
• \(x_{ij}\) = genotype (0,1,2) at SNP \(j\)
  
• \(\beta\) = SNP effect
  
• \(\varepsilon_i\) = residual variation

Procedure

1. Test SNP₁
   
2. Test SNP₂
   
3. Test SNP₃
   
4. …
   
5. Test all \(m\) SNPs

Thousands to millions of tests.

If \(\beta\) is not equal to zero, the test is statistically significant and there is evidence of association.

This slide builds directly on the previous one.

The genotype matrix X and phenotype vector y are now connected through a statistical model.

For each SNP — that is, each column of X — a simple regression model is fitted.

The phenotype is modeled as:

an overall mean, plus the SNP effect, plus residual variation.

The key question is:

Is the SNP effect, beta, different from zero?

If beta is significantly different from zero, there is evidence that the SNP is associated with the trait.

This test is repeated for every SNP in the genome.

In other words, GWAS scans the genome one column of the genotype matrix at a time.

Each SNP is tested independently.

Later, multiple testing must be accounted for, and results are summarized in plots such as Manhattan plots.

The central idea is:

GWAS is repeated regression across the genome.

How GWAS Works: Single Marker Test

The Additive Model

We fit a linear regression: \[y = \mu + x\beta + \varepsilon\]

• \(y\): Phenotype (LDL)
• \(x\): Genotype (0, 1, or 2)
• \(\beta\): Effect size per allele

Interpretation

• \(\beta\) represents the average change in LDL for each additional T allele.
• The \(p\)-value from this model tests the null hypothesis: \(\beta = 0\).
• Here, each T allele significantly decreases LDL.

This slide shows how a single SNP is tested in GWAS.

Each point represents one individual.

The x-axis shows genotype: 0, 1, or 2 copies of the T allele.

The y-axis shows the phenotype, here LDL cholesterol.

The boxplots summarize the distribution within each genotype group.

As the number of T alleles increases, the average LDL level decreases.

To test this formally, the genotype is treated as a numeric variable, taking values 0, 1, and 2.

A linear regression model is then fitted.

The red line represents this additive model.

Its slope is the SNP effect, beta.

Beta measures the average change in LDL for each additional T allele.

The key question is:

Is beta different from zero?

If it is, the SNP is statistically associated with the trait.

In GWAS, this exact test is repeated for every SNP across the genome.

How GWAS Works: Many Marker Test

The Interpretation

• Association Peak: The highest points cluster over a specific genomic region.
• Candidate Gene (QTL): The vertical dashed line shows the “lead SNP” falls directly within Gene B.
• The “Neighborhood” Effect: Note how the shading spans the whole gene. Because of LD, many SNPs in the red-shaded region appear significant, but only one gene is likely the true functional driver.

Key point This visual demonstrates Fine-Mapping: the process of narrowing down a broad GWAS “skyscraper” to the specific gene region most likely to contain the causal variant.

This figure shows what happens after many SNPs have been tested in the same genomic region.

Each point represents one marker.

The highest points cluster over a specific region of the genome.

This creates an association peak.

The dashed vertical line marks the lead SNP, the marker with the strongest statistical signal.

In this example, that signal falls within Gene B, which becomes a candidate region.

But the key point is this:

Because of linkage disequilibrium, many nearby SNPs can appear significant at the same time.

So GWAS usually identifies a neighborhood, not a single causal mutation.

Fine-mapping is the next step.

Its goal is to narrow this broader signal to the most likely gene or causal variant.

How GWAS Works: From P-values to Manhattan Plot

Each point = one SNP

• X-axis → genomic position
  
• Y-axis → \(-\log_{10}(p)\)

Higher points = stronger evidence
against the null hypothesis.

Dashed line

• Genome-wide significance threshold
  \(\bigl(\text{e.g., } p < 5\times10^{-8}\bigr)\)

Peaks indicate genomic regions
associated with the trait.

After testing all SNPs, a p-value is obtained for each marker.

To visualize the results, genomic position is plotted on the x-axis and minus log base 10 of the p-value on the y-axis.

This transformation makes very small p-values easier to see.

Each point represents one SNP.

Most SNPs show little or no association, so they remain near the bottom.

A few SNPs show strong association and form visible peaks.

The dashed horizontal line marks the genome-wide significance threshold.

Points above this line provide strong evidence of association.

This type of figure is called a Manhattan plot, because the peaks resemble skyscrapers.

A peak identifies an associated genomic region, but not necessarily the causal mutation itself.

GWAS Identifies a Gene for Body Size in Dogs

Figure 19-18.

1. Genome-wide association study (GWAS) for body size in dogs

• Each dot = one SNP
  
• Y-axis = −log10(P value)
  
• Horizontal line = significance threshold
  
• Strong peak on chromosome 15

→ Region contains the IGF1 gene

2. Small vs. large dog breeds

IGF1 is a major contributor to size differences.

This figure shows a real example of a genome-wide association study.

Panel (a) displays a Manhattan plot.

The x-axis shows genomic position across chromosomes. The y-axis shows statistical significance, plotted as minus log base 10 of the p-value.

Higher points correspond to smaller p-values, and therefore stronger evidence of association.

The horizontal line marks the genome-wide significance threshold.

Most regions show no strong signal.

However, on chromosome 15, there is a clear peak of highly significant SNPs.

This indicates a genomic region associated with body size.

That region contains the IGF1 gene, insulin-like growth factor 1.

IGF1 is known to play a key role in growth and development.

Panel (b) shows the phenotypic effect: small and large dog breeds differ dramatically in size.

Variation in IGF1 explains a large part of this difference.

The key takeaway is:

GWAS can identify genomic regions associated with complex traits without prior knowledge of candidate genes.

However, association alone does not prove causation.

Further analysis is required to confirm the functional role of the gene.

GWAS Identifies Genes for Common Human Diseases

Figure 19-19.

Genome-wide association studies (GWAS)
for several common human diseases.

• X-axis: Chromosomes 1–22
  
• Y-axis: −log10(P value)
  
• Green dots = significant SNP associations
  
• Red labels = candidate genes identified

Examples:

• Coronary artery disease → APOE
  
• Crohn’s disease → IL23R, ATG16L1, IRGM
  
• Rheumatoid arthritis → PTPN22, HLA-DRB1
  
• Type 1 diabetes → PTPN22, HLA-DRB1

This figure shows genome-wide association results for several common human diseases.

The x-axis shows chromosomes 1 through 22.

The y-axis shows statistical significance, plotted as minus log base 10 of the p-value.

Higher points indicate stronger evidence of association.

Green dots represent SNPs that pass the genome-wide significance threshold.

Clusters of these points form peaks, highlighting genomic regions associated with disease risk.

Several candidate genes are labeled in red.

For example:

Coronary artery disease shows association near APOE, a gene involved in lipid metabolism.

Crohn’s disease shows multiple associated loci, including IL23R, ATG16L1, and IRGM.

Rheumatoid arthritis and type 1 diabetes both show strong associations with PTPN22 and HLA-DRB1, which are involved in immune function.

This illustrates an important principle:

Common diseases are typically polygenic.

They are influenced by many genetic variants, often within biologically relevant pathways.

The key takeaway is:

GWAS identifies genomic regions associated with disease, but further studies are required to identify the causal variants and mechanisms.

Quantitative Genetics

What is studied?

• Complex traits influenced by many genes and environment
  
• Traits that do not follow simple Mendelian ratios

Examples:

• Continuous (e.g., height)
  
• Threshold (e.g., disease status)
  
• Meristic (e.g., litter size)

Core model

\[ P = G + E \]

• Phenotypic variation arises from genetic and environmental effects
  
• Key question: How much variation is genetic?

Why it matters

• Estimate heritability
  
• Predict genetic risk or breeding values
  
• Predict response to selection
  
• Understand evolutionary change

Quantitative genetics studies complex traits — traits influenced by many genes and environmental factors.

These traits do not follow simple Mendelian inheritance.

They can be continuous, such as height, threshold traits, such as disease status, or meristic traits, such as litter size.

At the core is a simple model:

Phenotype equals genotype plus environment.

Differences between individuals arise from genetic effects, environmental effects, or both.

The central question is:

How much of the observed variation is due to genetic differences?

This leads to heritability and the concept of genetic architecture.

These ideas allow us to:

predict breeding values, estimate genetic risk, forecast response to selection, and understand evolutionary change.

In essence, quantitative genetics provides a mathematical framework for analyzing and predicting complex trait variation in populations.

Genetic Model and Response to Selection

Genetic Model

Genetic deviation:

\[ G = A + D \]

• \(A\) = additive deviation (breeding value)
  
• \(D\) = dominance deviation

Only additive effects are predictably transmitted
from parents to offspring.

Additive variance (\(V_A\)) is the key component for long-term genetic change.

Breeder’s Equation

Response to selection:

\[ R = h^2 S \]

• \(S\) = selection differential
  
• \(R\) = response to selection
  
• \(h^2 = \dfrac{V_A}{V_P}\)

Response depends on:

• Strength of selection
  
• Amount of additive genetic variation

High \(h^2\) → rapid response
Low \(h^2\) → limited response

We begin with the genetic model.

Genetic deviation can be decomposed into additive and dominance components.

The additive component, A, represents the breeding value.

It is the part that is transmitted predictably from parent to offspring.

Dominance effects depend on allele combinations.

They influence the phenotype, but are not transmitted consistently across generations.

For this reason, additive variance is the key driver of long-term genetic change.

Now consider selection.

The breeder’s equation states that response equals heritability times the selection differential.

Heritability reflects the proportion of variation that is additive.

If additive variance is large, selection produces a strong response.

If additive variance is small, the response is limited.

The key connection is:

The genetic model explains why the breeder’s equation works.

Only additive effects contribute to sustained evolutionary change.

Variance and Heritability

Phenotype model \[ P = \mu + G + E \]

Phenotypic variance \[ V_P = V_G + V_E \]

Broad-sense heritability \[ H^2 = \frac{V_G}{V_P} \]

Narrow-sense heritability \[ h^2 = \frac{V_A}{V_P} \]

Interpretation

Total variation can be partitioned into Genetic variance and Environmental variance

Broad-sense heritability (H²)
Proportion of total variation due to genetic effects
(additive, dominance, epistatic).

Narrow-sense heritability (h²)
Proportion due to additive (transmissible) variation.

Only additive variance predicts response to selection.

The phenotype can be modeled as:

the population mean, plus genetic effects, plus environmental effects.

Phenotypic variance can therefore be partitioned into genetic variance and environmental variance.

Broad-sense heritability measures the proportion of total variation due to all genetic effects.

This includes additive, dominance, and interaction effects.

However, not all genetic effects are transmitted predictably.

Narrow-sense heritability focuses only on additive variance — the component passed from parents to offspring.

This is the key quantity for predicting response to selection and evolutionary change.

Mapping Complex Traits

The Goal of Mapping

• Move from statistical variance (\(V_G\)) to physical loci (QTLs).
• Identify which specific regions of the genome drive the trait.

The Strategy: GWAS

• Scan the whole genome using SNPs.
• Test for statistical association between Marker (M) and Trait.
• Relies on Linkage Disequilibrium (LD).

Key Discoveries from GWAS

• Polygenicity: Most complex traits are influenced by thousands of loci.
• Small Effects: Most individual variants explain \(<1\%\) of the variance.
• Non-coding regions: Many QTLs are in regulatory regions, not just proteins.

The Result: Manhattan Plot

• Skyscrapers = Genomic regions in LD with a causal variant.
• Threshold = Corrects for millions of tests.

This slide connects quantitative genetics to the physical genome.

The goal of mapping is to move from abstract variance components to specific genomic loci, called quantitative trait loci, or QTLs.

Genome-wide association studies, or GWAS, provide the main strategy.

They scan the genome using SNP markers and test for statistical association between markers and the trait.

This approach relies on linkage disequilibrium.

If a marker is associated with a trait, it is likely located near a causal variant.

GWAS has revealed several key insights.

First, most complex traits are highly polygenic, influenced by many loci.

Second, individual variants typically have small effects.

Third, many associated variants lie in non-coding regions, highlighting the importance of gene regulation.

The results are visualized in a Manhattan plot.

Each peak represents a genomic region in linkage disequilibrium with a causal variant.

These regions provide starting points for understanding the biological basis of complex traits.