PBG 650 Advanced Plant Breeding
Module 4: Quantitative Genetics
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Components of phenotypes
Genotypic values
Average effect of a gene
Breeding values
What is Quantitative Genetics?
Definition:
“Statistical branch of genetics based upon
fundamental Mendelian principles extended to
polygenic characters”
Primary goal:
To provide us with a mechanistic understanding of
the evolutionary process
Lynch and Walsh, Chapter 1
Questions of relevance to breeders
•
How much of the observed phenotypic variation is
due to genetic vs environmental factors?
•
How much of the genetic variation is additive (can
be passed on from parent to offspring)?
•
What is the breeding value of the available
germplasm?
•
•
Are there genotype by environment interactions?
•
Are there genetic correlations among traits?
What are the consequences of inbreeding and
outcrossing? What are the underlying causes?
Questions of relevance to breeders
• Answers to these questions will influence
– response to selection
– choice of breeding methods
– choice of parents
– optimal type of variety (pureline, hybrid,
synthetic, etc.)
– strategies for developing varieties adapted
to target environments
Phenotypic Value
Components of an individual’s Phenotypic Value
P=G+E
P = phenotypic value
G = genotypic value
For individual k with
genotype AiAj
P(ij)k =  + gij + e(ij)k
E = environmental deviation
For the population as a whole:
E(E) = 0
 = E(P) = E(G)
Cov(G, E) = 0
Bernardo, Chapt. 3; Falconer & Mackay, Chapt. 7; Lynch & Walsh, Chapt. 4
Single locus model
Genotypic
Value
Coded
Genotypic Value
A2A2
A1A2
A1A1
z
z+a+d
z+2a
P -a
P
P d
P a
-a
0
d
a
The origin ( P ) is midway between the two homozygotes
no dominance
partial dominance
complete dominance
overdominance
d=0
0 < d < +a or 0 > d > –a
d = +a or –a
d > +a or d < –a
d
degree of dominance = a
Single locus model
Different scales have been used in the literature
A2A2
-a
0
A1A2
0
A1A1
d
a
Falconer
(1+k)a
2a
Lynch & Walsh
0
au
a
0
a
2a+d
Conversions can be readily made
Comstock and
Robinson (1948)
Hill (1971)
Population mean
A1A1
Frequency
p2
A1A2
A2A2
2pq
q2
Genotypic Frequency
value
x value
a
p2 a
d
–a
M = p2a + 2pqd – q2a
= a(p2 – q2) + 2pqd
= a(p + q)(p - q) + 2pqd
= a(p - q) + 2pqd
2pqd
–q2a
Mean on coded scale
(centered around zero)
This is a weighted average
contribution from homozygotes and heterozygotes
Mean on original scale
Population mean
M = a(p - q) + 2pqd
When there is no dominance
When A1 is fixed
When A2 is fixed
Potential range
 a(p - q)
a
 -a
 2a
If the effects at different loci are additive (independent), then
M = Σa(p - q) + 2Σpqd
 = P + a(p - q) + 2pqd
Means of breeding populations
 = P + a(p - q) + 2pqd
In an F2 population, p = q = 0.5
F2 = P + (1/2)d
In a BC1 crossed to the favorable parent, p = 0.75,
so after random mating
BC1(A1A1) = P + (1/2)a + (3/8)d
For ½ A1A1, ½ A1A2
 = P + ½(a + d)
In a BC1 crossed to the unfavorable parent, p = 0.25,
so after random mating
For ½ A1A2, ½ A2A2
BC1(A2A2) = P - (1/2)a + (3/8)d
 = P + ½(d - a)
Average effects
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•
We have defined the mean in terms of genotypic values
•
Average effect of a gene (i)
Genes (alleles), not genotypes, are passed from parent to
offspring
– mean deviation from the population mean of individuals who received
that gene from their parents (the other gene taken at random from the
population)
subtract M = a(p - q) + 2pqd
Gamete
A1
A2
A1A1
A1A2
A2A2
a
p
d
q
-a
p
q
Freq x value
Average effect
of a gene
pa + qd
1=q[a+d(q-p)]
pd - qa
2=-p[a+d(q-p)]
Average effect of a gene substitution
Average effect of changing from A2 to A1
 = 1 - 2
q[a+d(q-p)] – (-p)[a+d(q-p)]
= a+d(q-p)
Average effect of changing from A1 to A2 = -
Relating this to the average effects of alleles:
1 = q
2 = -p
• a and d are intrinsic properties of genotypes
• 1, 2, and  are joint properties of alleles and the
populations in which they occur (they vary with gene
frequencies)
Breeding Value
Breeding value of individual Aij = i + j
Genotype
A1A1
A1A2
A2A2
Breeding Value
Average effect of a Average effect of a
gene
gene substitution
21
 1 + 2
22
2q
(q - p)
-2p
• For a population in H-W equilibrium, the mean breeding value = 0
• The expected breeding value of an individual is the average of the
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breeding value of its two parents
For an individual mated at random to a number of individuals in a
population, its breeding value is 2 x the mean deviation of its progeny from
the population mean.
Regression of breeding value on genotype
Breeding values
• can be measured
• provide information about
•
genetic values
lead to predictions about
genotypic and phenotypic
values of progeny
Additive genetic variance
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•
variance in breeding values
variance due to regression
of genotypic values on
genotype (number of
alleles)
● genotypic value
○ breeding value
Genotypic values
•
•
Genotypic values have been expressed as deviations from a
midparent
To calculate genetic variances and covariances, they must
be expressed as a deviation from the population mean, which
depends on gene frequencies
subtract M = a(p - q) + 2pqd
Genotypic values
Genotype
A1A1
A1A2
A2A2
Scaled
a
d
-a
Adjusted for mean
2q(a-pd)
2q(-qd)
a(q-p)+d(1-2pq)
-2p(a+qd)
(q-p)+2pqd
-2p(+pd)
Remember  = a + d(q - p)  Substitute a =  - d(q - p)
Dominance deviation
Components of an individual’s Phenotypic Value
P=G+E
G=A+D
Gij =  + i + j + ij
• In terms of statistics, D represents
– within-locus interactions
– deviations from additive effects of genes
• Arises from dominance between alleles at a locus
– dependent on gene frequencies
– not solely a function of degree of dominance
– (a locus with completely dominant gene action contributes
substantially to additive genetic variance)
Partitioning Genotypic Value
Genotypic Value Breeding Value
Genotype (adj. for mean) (additive effects)
Dominance
Deviation
A1A1
2q(-qd)
2q
-2q2d
A1A2
(q-p)+2pqd
-2p(+pd)
(q - p)
-2p
+2pqd
-2p2d
A2A2
When p = q = 0.5 (as in a biparental cross between inbred lines)
Genotype Genotypic Value Breeding Value
A1A1
-(1/2)d

A1A2
A2A2
(1/2)d
--(1/2)d
0
-
Dominance
-(1/2)d
(1/2)d
-(1/2)d
Dominance deviations from regression
Genotypic Value
A1A1
A1A2
2q (q-p)+2pqd
-2q2d
2q2d
2pqd
-2p2d
A2A2
-2p - 2p2d
Interaction deviation
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Components of an individual’s Phenotypic Value
P=G+E
P=A+D+E
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When more than one locus is considered, there may also be
interactions between loci (epistasis)
G=A+D+I
P=A+D+I+E
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‘I’ is expressed as a deviation from the population mean and
depends on gene frequencies
For a population in H-W equilibrium, the mean ‘I’ = 0