Lecture 1 of the Quantitative Genetics series. Why quantitative traits need their own theory, and why they end up normally distributed.
Genetics as most people first meet it is the genetics of discrete traits. A discrete trait takes a few distinct values, eye colour, the ability to roll the tongue, coat colour in Labradors. Such traits are usually controlled by the genotype at one or a few loci, and for many of them the responsible genes are known and understood. You can write down a genotype, apply Mendel's rules, and predict the phenotype.
Now consider milk yield, growth rate, feed intake, disease resistance or human height. None of these falls into neat categories. They vary continuously, and no single gene determines them.
A quantitative trait is a trait that assumes values on a continuous scale. Quantitative genetics deals with the inheritance of variation in quantitative traits. Most quantitative traits are influenced by variation in many genes, each of small effect, together with the environment.
This is not a minor special case. Nearly every trait of economic importance in animal and plant breeding is quantitative. So we need a different language: one built on expectations and probabilities rather than on deterministic Mendelian ratios.
The strategy of this course is to model one locus in complete detail, and then extend that model to many loci. Almost everything that follows, breeding values, variance components, heritability, response to selection, is built on the one-locus foundation.
Quantitative genetics deals primarily with metric traits, traits we measure on a scale rather than classify into categories. Working with metric traits forces an important admission:
This constraint determines the structure of the subject. Because the effect of an individual gene cannot be observed, the analysis cannot proceed gene by gene. It instead describes the distribution of the trait in a population and partitions the variation in that distribution into components attributable to genes and to environment. Probability and expectations, the subject of the next lecture, are therefore the working tools of the subject rather than a mathematical preliminary.
Start with the model for a single locus. For a locus \(j\) with two alleles, the effect an individual carries is
where \(G_j\in\{0,1,2\}\) counts copies of one allele, \(p_j\) is that allele's frequency and \(\alpha_j\) is the allele substitution effect. The full derivation of this expression is the subject of the one-locus lecture; here we only need its shape.
Adding a second locus simply adds a second term:
and the general model for \(L\) loci is
where \(y\) is the observed phenotype, \(\mu\) the overall mean, \(g_j\) the effect of the genotype at locus \(j\), and \(e\) the residual or environmental effect. The individual's breeding value across all loci is \(g=\sum_{j=1}^{L}g_j\).
Because a parent transmits one allele of each pair, the offspring's expected genetic value is the average of the parental breeding values:
with \(s\) the sire and \(d\) the dam. This relation underlies genetic improvement by selection, and reappears as the parent average in breeding-value prediction.
Height, milk yield, body weight and feed intake are all routinely assumed to be normally distributed. However, consider the building block:
A single gene effect is emphatically not normally distributed. It is a three-point discrete variable. So where does the familiar bell curve come from?
The normal density itself is
written \(Y\sim N(\mu,\sigma^2)\).
Normality of a quantitative trait is not an arbitrary convenience. It is a consequence of many small, independent genetic contributions adding up. This also tells you when to be careful: if a trait is governed by one or two large-effect genes, the normal approximation is poor and the distribution may be visibly multi-modal.
Pushing the logic to its limit gives the model that underpins most of animal and plant breeding.
The third assumption is easily misread. The animal's actual phenotype does not change when a one-locus, two-locus or many-locus model is adopted. What changes is how the same total genetic effect is distributed among the terms:
As more loci enter the model, the total genetic effect on \(y\) becomes more diluted among the individual \(\alpha_j\). Each locus explains less, but the sum explains the same. The infinitesimal model takes this to the extreme: infinitely many loci, each of infinitesimally small effect.
The topics ahead, in the order they build on one another:
Two applied strands accompany these topics. Artificial selection aims to change the genetic properties of a population by preferentially breeding individuals whose genetics move the population in a desired direction. Breeding value prediction combines genetic, pedigree and phenotypic information into the best available prediction of which individuals carry the most desirable genetics, using linear models that pool information across many animals and several traits. Breeding schemes then organise this optimally, maximising genetic gain within physical, economic and biological constraints.
These exercises use the R functions from the course (QG_KU_functions.R,
simY()). The point is not the coding but the shape of the distributions you get.
Simulate 100 phenotypes from a single-locus model and plot the density:
rm(list=ls(all=TRUE)); source("QG_KU_functions.R")
data <- simY(100, mu=-5)
plot.density(data$y)Inspect the object with str(data). What are the components y,
G, s and e? Why is the density not bell-shaped?
str(data) returns a list of four: y the phenotypes,
G the genotype codes (0/1/2) per locus, s the simulated allele
substitution effects, and e the residuals.
With one locus the genetic contribution \(g_1=(G_1-2p_1)\alpha_1\) takes only three distinct values. The density is therefore a mixture of three narrow humps (one per genotype) smeared only by the environmental term \(e\), not a single smooth bell.
Repeat with an increasing number of loci:
data <- simY(100, n.loci=2, mu=0); plot.density(data$y)
data <- simY(100, n.loci=10, mu=0); plot.density(data$y)
data <- simY(100, n.loci=50, mu=0); plot.density(data$y)
data <- simY(100, n.loci=100, mu=0); plot.density(data$y)At roughly how many loci does the distribution become visually indistinguishable from a normal curve? Relate what you see to the Central Limit Theorem.
With 2 loci you can still often see discrete humps (up to 9 genotype combinations). By about 10 loci the humps have merged; by 50 to 100 the density is smooth and symmetric and looks normal.
This is the CLT in action: the sum \(\sum_j g_j\) of many independent, individually non-normal three-point variables converges to a normal distribution. Note that the number of loci needed depends on how equal the effects are, a few large-effect loci among many small ones will keep the distribution skewed for longer.
Conceptual, no computing. A colleague argues that because each gene effect \(g_j\) can only take three values, it is wrong to model a quantitative trait as normally distributed. What is the flaw in the argument, and under what circumstances would the colleague actually be right?
The flaw is confusing the distribution of a single term with the distribution of their sum. Normality is claimed for \(y=\mu+\sum_j g_j+e\), not for any individual \(g_j\).
The colleague is right, however, whenever the sum is dominated by a small number of large-effect loci. Then the CLT approximation fails and the phenotypic distribution can be visibly multi-modal or skewed, coat colour scored on a numeric scale, or a trait with a major gene such as double-muscling, are practical examples.
