Sanja Franic, VU University Amsterdam 2010
In contrast to classical Mendelian genetics, which deals with the inheritance of interindividual
differences in traits along which individuals can be divided into distinct categories (e.g., eye
color), quantitative or biometrical genetics is concerned with inheritance of interindividual
differences in traits which vary continuously (i.e., quantitative traits, e.g. height). The fact that
the intrinsically discontinuous variation in the type of alleles present at the genome may yield
continuous variation in observed traits is explained by a) the supposition of polygenic
inheritance (quantitative traits are assumed to be affected by genes at multiple genetic loci,
whose contribution to the variation in the phenotype is small in comparison to effects of other
sources of variation), and b) non-genetic variation, which is truly continuous, being
superimposed on the genetic effects on the phenotype. Given the intrinsic reliance of genetic
models upon the quantitative genetic theory-based predictions of genetic and environmental
covariation between individuals of differing degrees of genetic relatedness, before addressing
genetic applications of SEM, we will first review how those predictions are derived.
Values and means. The measured value of a trait, or its phenotypic value (P), is typically
conceptualized as a sum of two components, one attributable to the particular assemblage of
segregating genes relevant to the phenotype in question (the genotypic value, G), and the
other to all of the non-genetic factors affecting the phenotype (environmental deviation, E).1
The mean environmental deviation in the population is typically scaled at zero; thus the mean
phenotypic value equals the mean genotypic value. The aim of succeeding sections will be to
demonstrate the derivation of the average degree of genetic resemblance between relatives; in
this light, we focus primarily on the genotypic value.
1
In the text that follows we refer only to the component of P that varies in the population; thus the effects of
monomorphic genes, as well as the non-variable aspects of the environment, are ignored (but may be modeled by
addition of appropriate constants).
0
Sanja Franic, VU University Amsterdam 2010
Consider, for instance, a single locus with two alleles, A1 and A2. The genotypic
values of the two homozygotes (A1A1 and A2A2) and that of the heterozygote (A1A2) may be
denoted +a, -a, and d, respectively (Figure 1). The point of zero genotypic value is defined as
the midpoint between the two homozygotes. The value of d reflects the degree of genetic
dominance: in the absence of dominance, d = 0; if A1 is dominant over A2, d > 0; if A2 is
dominant over A1, d < 0; in case of overdominance, d > a or d < -a.
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Sanja Franic, VU University Amsterdam 2010
Table 2
Derivation of mean genotypic values (G), breeding values (A) and dominance deviations (D)
Genotype
A1A1
A1A2
A2A2
Genotypic value (gi)
a
d
-a
Genotype frequency(fi)
p2
2pq
q2
Mean genotypic value (μgi = gifi)
p2a
2pqd
-q2a
Parental gametes
A1
Mean gen. value across genotypes: μG = ∑μgi = a(p – q) + 2dpq
Frequencies of genotypes produced
Mean values of genotypes
Average effect of allele
A1A1
A1A2
produced (μGj)
(αj = μGj - μG)
p
q
pa + qd
α1 = q[a + d(q – p)]
pd – qa
α2 = – p[a + d(q – p)]
A2
p
A2A2
q
Average effect of allele substitution: α = α1 - α2 = a + d(q – p)
Breeding value (Ai)
2α1 = 2qα
α1 + α2 = (q – p)α
2α2 = –2pα
E[A] = ∑Aifi = 2p2qα + 2pq(q – p)α – 2pq2α = 2pqα(p + q – p – q) = 0
Genotypic value (Gi = μgi - μG)
2q(α – qd)
(q – p)α + 2pqd
–2p(α + pd)
E[G] = ∑Gifi = 2p2q(α – qd) + 2pq[(q – p)α + 2pqd] – 2pq2(α + pd) = 0
2pqd
–2p2d
E[D] = ∑Difi = –2p2q2d + 4p2q2d – 2p2q2d = 0
Dominance deviation (Di = Gi – Ai) –2q2d
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Sanja Franic, VU University Amsterdam 2010
Figure 1. Arbitrarily assigned genotypic values in a system with 2 alleles (Falconer &
Mackay, 1996).
Table 1
Genotype frequencies in the offspring generation as a function of allele frequencies in the
parental generation
Paternal gametes
and their
frequencies
Maternal gametes and their frequencies
A
(p)
a
(q)
A
(p)
AA
(p2)
Aa
(pq)
a
(q)
Aa
(pq)
aa
(q2)
If the relative frequencies with which the A1 and A2 alleles occur in the population of interest
are denoted p and q, respectively (p + q = 1), the frequencies of the genotypes arising from the
process of random mating2 between individuals within this population are given by the
binomial expansion (p + q)2 = p2 + 2pq + q2, as shown in Table 1.3 The mean genotypic value
of this locus may be obtained by multiplying the value of each genotype by its frequency and
summing over the three genotypes:
μG = p2a + 2pqd - q2a = a(p – q)(p + q) + 2dpq = a(p – q) + 2dpq.
2
Mating is random if an individual has an equal chance of mating with any other individual in the population.
For effects of non-random mating see e.g. Falconer & Mackay (1996).
3
p2, 2pq, and q2 adequately describe the proportions of genotypes in the offspring generation in populations with
no migration, mutation or selection. For effects of migration, mutation and selection see e.g. Falconer & Mackay
(1996).
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Sanja Franic, VU University Amsterdam 2010
The allelic equivalent of the genotypic value is average effect of the allele. The average effect
of an allele is the mean deviation from the population mean of individuals who received that
allele from one parent, the other allele having come at random from the population (Table 2).
For instance, if a number of gametes carrying the A1 allele unite at random with gametes from
the population (where p is the frequency of the A1 allele and q of the A2 allele), the
frequencies of the genotypes produced will be p of A1A1 and q of A1A2. Taking into account
these frequencies and the genotypic values associated with each genotype, the mean genotypic
value of the locus may be expressed as pa + qd. Subtracting the population mean from this
expression yields the expression for the average effect of the A1 allele: α1 = q[a + d(q – p)].
Correspondingly, the average effect of A2 is α2 = – p[a + d(q – p)]. The average effect may
also be expressed in terms of the average effect of gene substitution, which is simply the
difference between the average effects of the two alleles: α = α1 – α2 = a + d(q – p). The
average effects of the two alleles may be conveyed in terms of the average effect of gene
substitution: α1 = qα, and α2 = -pα (Table 2).
Summing the average effects over the two alleles at a locus yields a component of the
genotypic value termed the breeding value or the additive genotype (A). The remainder of the
genotypic value is the dominance deviation (D). Dominance deviation arises in the presence
of genetic dominance (i.e., within-locus interaction between alleles), and reflects the possible
non-additive effects arising from the allelic pairing at a locus. It may be derived as D = G – A
(Table 2). If the genotypic value refers to an aggregate value of genotypes across more than
one locus, the expression for G takes on the form: G = A + D + I, where I stands for
interaction deviation4 arising from possible non-additive gene co-action across loci (i.e.,
epistasis).
4
Alleles may interact in pairs or threes or higher numbers, in expression (x.x) aggregate interactions of all sorts
are treated together as a single interaction deviation.
4