Basic Concepts of Quantitative Genetics

Abstract

Quantitative genetics is a special branch of genetics, which is concerned with the inheritance of the differences between individuals that are measured in degree rather than in kind. These individual differences are referred to as quantitative differences orquantitative traits. Formally, a quantitative trait is defined as a trait whose value varies continuously across individuals (Falconer and Mackay 1996; Lynch and Walsh 1998). The phenotype of a quantitative trait measured from an individual is not determined by genes alone; it is also determined by environmental variants. The proportion of the phenotypic variance explained by the segregation of a single gene is usually small. However, the contribution of all these small-effect genes collectively is significant to the variation of the phenotype. Genes controlling the variation of a quantitative trait are calledquantitative trait loci (QTL). Note that the term QTL defined in this book is used for both the singular and plural forms, e.g., one QTL for weight and two QTL for height. In the quantitative genetics literature, QTL represents the singular form and QTLs is used as the plural form. No matter how small a QTL is, it segregates just like a regularMendelian locus. For small-effect QTL, we simply cannot observe the segregation and must resort to statistical methods to infer the segregation. Most statistical methods applied in quantitative genetics require specific genetic models, which will be the focus of this chapter.

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Quantitative genetics is a special branch of genetics, which is concerned with the inheritance of the differences between individuals that are measured in degree rather than in kind. These individual differences are referred to as quantitative differences orquantitative traits. Formally, a quantitative trait is defined as a trait whose value varies continuously across individuals (Falconer and Mackay 1996; Lynch and Walsh 1998). The phenotype of a quantitative trait measured from an individual is not determined by genes alone; it is also determined by environmental variants. The proportion of the phenotypic variance explained by the segregation of a single gene is usually small. However, the contribution of all these small-effect genes collectively is significant to the variation of the phenotype. Genes controlling the variation of a quantitative trait are calledquantitative trait loci (QTL). Note that the term QTL defined in this book is used for both the singular and plural forms, e.g., one QTL for weight and two QTL for height. In the quantitative genetics literature, QTL represents the singular form and QTLs is used as the plural form. No matter how small a QTL is, it segregates just like a regularMendelian locus. For small-effect QTL, we simply cannot observe the segregation and must resort to statistical methods to infer the segregation. Most statistical methods applied in quantitative genetics require specific genetic models, which will be the focus of this chapter.

1 Gene Frequency and Genotype Frequency

Throughout the entire book, we consider onlydiploid organisms. A diploid organism carries two copies of the homologous genome, one from thepaternal parent and the other from thematernal parent. Each copy of the genome is called ahaploid. Each locus of the genome, therefore, contains two alleles, one from each parent. Although each individual carries at most two different alleles, the entire population may have many different alleles, calledmultiallelic population. For simplicity, we only consider a population that contains two different alleles, called abiallelic population. Let A ₁ and A ₂ be the two alleles of locus A in the population of interest. In the biallelic population, there are only three possible genotypes, denoted by A ₁ A ₁, A ₁ A ₂ and A ₂ A ₂, respectively. Depending on the structure and the mating system of the population, the population may have different proportions of the three genotypes. Let P ₁₁ = Pr(A ₁ A ₁), P ₁₂ = Pr(A ₁ A ₂), and P ₂₂ = Pr(A ₂ A ₂) be the frequencies of the three genotypes. The genotypes A ₁ A ₂ and A ₁ A ₁ contain one and two copies of allele A ₁, respectively. Therefore, the frequency of allele A ₁ in the population is $p_1 = \Pr (A_1 ) = P_{11} + \frac{1}{2}P_{12}$. The allelic frequency for A ₂ is then $p_2 = \Pr (A_2 ) = P_{22} + \frac{1}{2}P_{12}$. These relationships hold regardless the population history and structure. However, to express genotypic frequencies as functions of theallele frequencies, some assumptions are required. In a large random-mating population, there is a unique relationship betweengenotype frequencies andgene frequencies, which is represented by P ₁₁ = p ₁ ², P ₁₂ = 2p ₁ p ₂, and P ₂₂ = p ₂ ². This can be interpreted as independence of the two alleles joining together to form the genotype. The frequency of the heterozygote is 2p ₁ p ₂ because it contains two configurations of the same genotype, that is, A ₁ A ₂ and A ₂ A ₁, representing two different origins of the gametes. This particular relationship is represented by the binomial expansion,

$${({p}_{1} + {p}_{2})}^{2} = {p}_{ 1}^{2} + 2{p}_{ 1}{p}_{2} + {p}_{2}^{2}$$

(5.1)

corresponding to the event of

$${({A}_{1} + {A}_{2})}^{2} = {A}_{ 1}{A}_{1} + 2({A}_{1}{A}_{2}) + {A}_{2}{A}_{2}$$

(5.2)

If such a population undergoes no selection, no mutation, and no migration, the gene frequencies and genotypic frequencies will remain constant from generation to generation. Such a population is said to be inHardy–Weinberg equilibrium (Hardy 1908, Li 1955, Weinberg 1908). If a large population is not in Hardy–Weinberg equilibrium, one generation ofrandom mating will suffice to lead the population to Hardy–Weinberg equilibrium.

TheHardy–Weinberg equilibrium for a population with k(k > 2) alleles is represented by P _ij  = 2p _i p _j for i≠j and P _ii  = p _i ² for i = j, where P _ij  = Pr(A _i A _j ) and p _i  = Pr(A _i ) for $i,j = 1, \ldots ,k.$.

Gene frequencies and genotypic frequencies are properties of a population. The genes studied are usually related to fitness and thus determine the adaption of the population to environmental changes and the evolution of the population. These are contents of population genetics. In quantitative genetics, we are interested in genes that determine the expression of quantitative traits. Therefore, we must first assign some value to a genotype and a value to an allele. These values are calledgenetic effects.

2 Genetic Effects and Genetic Variance

Each individual of a population has a phenotypic value for a particular quantitative trait. Assume that we can observe the genotypes of all individuals in the population. The genetic effect for genotype A ₁ A ₁ is defined as the average phenotypic value of all individuals bearing genotype A ₁ A ₁. This genotypic value is denoted by G ₁₁. Similar notation applies to genotypes A ₁ A ₂ and A ₂ A ₂. The reason that G ₁₁ takes the average phenotypic value is explained as follows. Let Y ₁₁ be the phenotypic value for an individual with genotype A ₁ A ₁, which can be expressed as

$${Y }_{11} = {G}_{11} + {E}_{11}.$$

(5.3)

where E ₁₁ is a randomenvironmental deviation. The environmental deviation varies from one individual to another, even though all the individuals have the same genotypic value. When we take the average value across all individuals of type A ₁ A ₁, the equation becomes

$$\bar{{Y }}_{11} = {G}_{11} +\bar{ {E}}_{11}.$$

(5.4)

For a sufficient number of individuals collected from this genotype, we have ${\bar E}$ ₁₁ ≈ 0 because positive and negative deviations tend to cancel out each other. This leads to $\bar Y_{11} = G_{11}$.

We now define three parameters as functions of the three genotypic values,

$$\begin{array}{rlrlrl} \mu & = \frac{1} {2}({G}_{11} + {G}_{22}) & & \\ a & = {G}_{11} -\frac{1} {2}({G}_{11} + {G}_{22}) & & \\ d & = {G}_{12} -\frac{1} {2}({G}_{11} + {G}_{22}), &\end{array}$$

(5.5)

where μ is called themidpoint value, a theadditive effect, and d thedominance effect. The threegenotypic values are then expressed as

$$\begin{array}{rlrlrl} {G}_{11} & = \mu + a & & \\ {G}_{12} & = \mu + d & & \\ {G}_{22} & = \mu - a &\end{array}$$

(5.6)

We then express each genotypic value as a deviation from the midpoint value

$$\begin{array}{rlrlrl} {\Phi }_{11} & = {G}_{11} - \mu = a & & \\ {\Phi }_{12} & = {G}_{12} - \mu = d & & \\ {\Phi }_{22} & = {G}_{22} - \mu = -a &\end{array}$$

(5.7)

UnderHardy–Weinberg equilibrium, thepopulation mean of the genotypic values (expressed as deviations from themidpoint value) is

$$\begin{array}{rlrlrl} {\mu }_{G} =\mathrm{ E}(\Phi ) & = {P}_{11}{\Phi }_{11} + {P}_{12}{\Phi }_{12} + {P}_{22}{\Phi }_{22} & & \\ & = {p}_{1}^{2}a + 2{p}_{ 1}{p}_{2}d + {p}_{2}^{2}(-a) & & \\ & = ({p}_{1} - {p}_{2})a + 2{p}_{1}{p}_{2}d &\end{array}$$

(5.8)

and the variance of the genotypic values is

$${\sigma }_{G}^{2} =\mathrm{ var}(\Phi ) =\mathrm{ E}({\Phi }^{2}) -\mathrm{ {E}}^{2}(\Phi ),$$

(5.9)

where

$$\begin{array}{rlrlrl} \mathrm{E}({\Phi }^{2}) & = {P}_{ 11}{{\Phi }_{11}}^{2} + {P}_{ 12}{{\Phi }_{12}}^{2} + {P}_{ 22}{{\Phi }_{22}}^{2} & & \\ & = {p}_{1}^{2}{a}^{2} + 2{p}_{ 1}{p}_{2}{d}^{2} + {p}_{ 2}^{2}{(-a)}^{2} & & \\ & = ({p}_{1}^{2} + {p}_{ 2}^{2}){a}^{2} + 2{p}_{ 1}{p}_{2}{d}^{2} &\end{array}$$

(5.10)

After some algebraic manipulations, we have

$${\sigma }_{G}^{2} = 2{p}_{ 1}{p}_{2}{[a + ({p}_{2} - {p}_{1})d]}^{2} + {(2{p}_{ 1}{p}_{2}d)}^{2}.$$

(5.11)

3 Average Effect of Allelic Substitution

A single locus of an individual consists of two alleles, one from each of the two parents. When the individual reproduces, the two alleles will go to different gametes. The gametes will reunite in the next generation to form the genotypes of the next generation. Therefore, a genotype cannot be inherited from generation to generation. It is the allele (haplotype) that is passed from one generation to another. Therefore, we need to define the effect of an allele. Let us look at the following 2 × 2 table for the definition of theallelic effect (Table 5.1).

Table 5.1 Definitions for allelic effects and dominance deviations

The effect of allele A ₁ is defined as

$${\alpha }_{1} = \frac{\left ({\Phi }_{11} - {\mu }_{G}\right ){p}_{1}^{2} + \left ({\Phi }_{12} - {\mu }_{G}\right ){p}_{2}{p}_{1}} {{p}_{1}} = {p}_{2}[a + d({p}_{2} - {p}_{1})]$$

(5.12)

and the effect of allele A ₂ is defined as

$${\alpha }_{2} = \frac{\left ({\Phi }_{12} - {\mu }_{G}\right ){p}_{1}{p}_{2} + \left ({\Phi }_{22} - {\mu }_{G}\right ){p}_{2}^{2}} {{p}_{2}} = -{p}_{1}[a + d({p}_{2} - {p}_{1})]$$

(5.13)

The difference between the twoallelic effects is called theaverage effect of allelic substitution, denoted by α,

$$\alpha = {\alpha }_{1} - {\alpha }_{2} = a + ({p}_{2} - {p}_{1})d$$

(5.14)

The sum of the two allelic effects included in a genotype is called the“breeding value,” which is the expected genotypic value of the progeny of an individual bearing this genotype. Therefore, the breeding value for genotype A ₁ A ₁ is ${A}_{11} = 2{\alpha }_{1} = 2{p}_{2}\alpha$. The breeding values for the other two genotypes are ${A}_{12} = {\alpha }_{1} + {\alpha }_{2} = ({p}_{2} - {p}_{1})\alpha$ and ${A}_{22} = 2{\alpha }_{2} = -{p}_{1}\alpha$, respectively. The deviations of the actual genotypic values from the breeding values are calleddominance deviations. The three dominance deviations are

$$\begin{array}{rlrlrl} {\delta }_{11} & = {\Phi }_{11} - {\mu }_{G} - {A}_{11} = -2{p}_{2}^{2}d & & \\ {\delta }_{12} & = {\Phi }_{12} - {\mu }_{G} - {A}_{12} = 2{p}_{1}{p}_{2}d & & \\ {\delta }_{22} & = {\Phi }_{22} - {\mu }_{G} - {A}_{22} = -2{p}_{1}^{2}d &\end{array}$$

(5.15)

The genotypic values (G), the breeding values (A), and the dominance deviations (D) for the three genotypes are listed in Table 5.2.

Table 5.2 Breeding values and dominance deviations of the three genotypes in a Hardy–Weinberg population

4 Genetic Variance Components

One can verify that the expectations of both the breeding values and the dominance deviations are zero, i.e.,

$$\begin{array}{rlrlrl} \mathrm{E}(A) & = {P}_{11}{A}_{11} + {P}_{12}{A}_{12} + {P}_{22}{A}_{22} & & \\ & = {p}_{1}^{2}(2{p}_{ 2}\alpha ) + 2{p}_{1}{p}_{2}[({p}_{2} - {p}_{1})\alpha ] + {p}_{2}^{2}(-2{p}_{ 1}\alpha ) = 0 &\end{array}$$

(5.16)

and

$$\begin{array}{rlrlrl} \mathrm{E}(D) & = {P}_{11}{\delta }_{11} + {P}_{12}{\delta }_{12} + {P}_{22}{\delta }_{22} & & \\ & = {p}_{1}^{2}(-2{p}_{ 2}^{2}d) + 2{p}_{ 1}{p}_{2}(2{p}_{1}{p}_{2}d) + {p}_{2}^{2}(-2{p}_{ 1}^{2}d) = 0. &\end{array}$$

(5.17)

This leads to

$$\begin{array}{rcl}{ \sigma }_{A}^{2} =\mathrm{ E}({A}^{2}) = {p}_{ 1}^{2}{(2{p}_{ 2}\alpha )}^{2} + 2{p}_{ 1}{p}_{2}{[({p}_{2} - {p}_{1})\alpha ]}^{2} + {p}_{ 2}^{2}{(-2{p}_{ 1}\alpha )}^{2} = 2{p}_{ 1}{p}_{2}{\alpha }^{2}& & \\ & &\end{array}$$

(5.18)

and

$$\begin{array}{rcl}{ \sigma }_{D}^{2} =\mathrm{ E}({D}^{2}) = {p}_{ 1}^{2}{(-2{p}_{ 2}^{2}d)}^{2} + 2{p}_{ 1}{p}_{2}{(2{p}_{1}{p}_{2}d)}^{2} + {p}_{ 2}^{2}{(-2{p}_{ 1}\alpha )}^{2} = {(2{p}_{ 1}{p}_{2}d)}^{2}& & \\ & &\end{array}$$

(5.19)

Looking at thegenetic variance given in  (5.11), we found that the first part is σ _A ² and the second part is σ _D ². Therefore,

$${\sigma }_{G}^{2} = {\sigma }_{ A}^{2} + {\sigma }_{ D}^{2},$$

(5.20)

i.e., the total genetic variance has been partitioned into anadditive variance component and adominance variance component.

5 Heritability

The phenotypic value Y can be expressed by the following linear model:

$$Y = G + E = A + D + E,$$

(5.21)

where E is an environmental error with mean zero and variance σ _E ². The phenotypic variance σ _P ² = var(Y ) is

$${\sigma }_{P}^{2} = {\sigma }_{ G}^{2} + {\sigma }_{ E}^{2} = {\sigma }_{ A}^{2} + {\sigma }_{ D}^{2} + {\sigma }_{ E}^{2}.$$

(5.22)

The phenotypic variance contributed by the total genetic variance is calledbroad-sense heritability, denoted by

$${H}^{2} = \frac{{\sigma }_{A}^{2} + {\sigma }_{ D}^{2}} {{\sigma }_{A}^{2} + {\sigma }_{D}^{2} + {\sigma }_{E}^{2}},$$

(5.23)

while the proportion contributed by theadditive variance is callednarrow-sense heritability, denoted by

$${h}^{2} = \frac{{\sigma }_{A}^{2}} {{\sigma }_{A}^{2} + {\sigma }_{D}^{2} + {\sigma }_{E}^{2}}.$$

(5.24)

The narrow-sense heritability reflects the proportion of variance that is heritable. Therefore, it is a very important parameter to consider in developing a selection program for genetic improvement.

6 An F₂ Family Is in Hardy–Weinberg Equilibrium

AnF ₂ family initiated from the cross of lines A ₁ A ₁ and A ₂ A ₂ is inHardy–Weinberg equilibrium, and thus, the theory developed in this chapter applies to an F₂ family. The allele frequencies are ${p}_{1} = {p}_{2} = \frac{1} {2}$, and the genotypic frequencies are ${P}_{11} = {p}_{1}^{2} = \frac{1} {4}$, ${P}_{12} = 2{p}_{1}{p}_{2} = \frac{1} {2}$, and ${P}_{22} = {p}_{2}^{2} = \frac{1} {4}$. Theaverage effect of allelic substitution is $\alpha = a + ({p}_{2} - {p}_{1})d = a.$ Therefore,

$${\sigma }_{G}^{2} = 2{p}_{ 1}{p}_{2}{\alpha }^{2} + {(2{p}_{ 1}{p}_{2}d)}^{2} = \frac{1} {2}{a}^{2} + \frac{1} {4}{d}^{2}.$$

(5.25)

The same result can be obtained from a different perspective. The genotypic value of an F₂ individual can be expressed as

$$G = \mu + Za + Wd,$$

(5.26)

where

$$Z = \left \{\begin{array}{c} + 1\\ 0 \\ - 1 \end{array} \begin{array}{c} \mathrm{for}\ {A}_{1}{A}_{1}\ \mathrm{with\ probability}\ \frac{1} {4} \\ \mathrm{for}\ {A}_{1}{A}_{2}\ \mathrm{with\ probability}\ \frac{1} {2} \\ \mathrm{for}\ {A}_{2}{A}_{2}\ \mathrm{with\ probability}\ \frac{1} {4} \end{array} \right.$$

(5.27)

and

$$W = \left \{\begin{array}{c} 0\\ 1 \\ 0 \end{array} \begin{array}{c} \mathrm{for}\ {A}_{1}{A}_{1}\ \mathrm{with\ probability}\ \frac{1} {4} \\ \mathrm{for}\ {A}_{1}{A}_{2}\ \mathrm{with\ probability}\ \frac{1} {2} \\ \mathrm{for}\ {A}_{2}{A}_{2}\ \mathrm{with\ probability}\ \frac{1} {4} \end{array} \right.$$

(5.28)

The genotypic variance is partitioned into

$${\sigma }_{G}^{2} = {\sigma }_{ Z}^{2}{a}^{2} + {\sigma }_{ W}^{2}{d}^{2} = \frac{1} {2}{a}^{2} + \frac{1} {4}{d}^{2}.$$

(5.29)

where

$$\begin{array}{rcl}{ \sigma }_{Z}^{2}& =& \mathtt{E}({Z}^{2}) -{\mathtt{E}}^{2}(Z) \\ & =& \left [\frac{1} {4}{(+1)}^{2} + \frac{1} {2}{(0)}^{2} + \frac{1} {4}{(-1)}^{2}\right ] -{\left [\frac{1} {4}(+1) + \frac{1} {2}(0) + \frac{1} {4}(-1)\right ]}^{2} \\ & =& \frac{1} {2} - 0 = \frac{1} {2} \end{array}$$

(5.30)

and

$$\begin{array}{rcl}{ \sigma }_{W}^{2}& =& \mathtt{E}({W}^{2}) -{\mathtt{E}}^{2}(W) \\ & =& \left [\frac{1} {4}{(0)}^{2} + \frac{1} {2}{(1)}^{2} + \frac{1} {4}{(0)}^{2}\right ] -{\left [\frac{1} {4}(0) + \frac{1} {2}(1) + \frac{1} {4}(0)\right ]}^{2} \\ & =& \frac{1} {2} -\frac{1} {4} = \frac{1} {4} \end{array}$$

(5.31)

Note that the covariance between Z and W is zero. Otherwise, a term 2cov(Z, W)ad should be added to the genotype variance.