1. PBG 650 Advanced Plant Breeding
Module 4: Quantitative Genetics
Components of phenotypes
Genotypic values
Average effect of a gene
Breeding values

2. 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

3. 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?
What are the consequences of inbreeding and outcrossing? What are the underlying causes?
Are there genetic correlations among traits?

4. 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

5. Phenotypic Value P = G + E
Components of an individual’s Phenotypic Value
P = G + E
For individual k with
genotype AiAj
P(ij)k =  + gij + e(ij)k
P = phenotypic value
G = genotypic value
E = environmental deviation
gij = Gij-mu
For now we assume that changes in the environment will only affect mu (no gxe)
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

6. Single locus model z z+a+d z+2a -a 0 d a
Genotypic
Value
Coded
Genotypic Value
-a d a
Pbar = z + a
Bernardo doesn’t endorse calling ’a’ the additive effect of a locus, because it loses its meaning when d is not equal to zero
Think of it as genetic value of homozygotes
Partial dominance 0<d<a
The origin ( ) is midway between the two homozygotes
no dominance d = 0
partial dominance < d < +a or 0 > d > –a
complete dominance d = +a or –a
overdominance d > +a or d < –a
degree of dominance =

7. Single locus model -a 0 d a 0 (1+k)a 2a 0 au a 0 a 2a+d
Different scales have been used in the literature
A1A1
A2A2
A1A2
-a d a
Falconer
(1+k)a a
Lynch & Walsh
Comstock and Robinson (1948)
au a
a a+d
Hill (1971)
Conversions can be readily made

8. Population mean Frequency Genotypic value Frequency x value A1A1 p2 a
2pq d
2pqd
A2A2 q2 –a
–q2a
M is a function of a, d, and gene frequencies
M = p2a + 2pqd – q2a
= a(p2 – q2) + 2pqd
= a(p + q)(p - q) + 2pqd
= a(p - q) + 2pqd
Mean on coded scale
(centered around zero)
This is a weighted average
contribution from homozygotes and heterozygotes
Mean on original scale

9. Population mean  = P + a(p - q) + 2pqd M = a(p - q) + 2pqd
When there is no dominance  a(p - q)
When A1 is fixed  a
When A2 is fixed  -a
Potential range  2a
A1 fixed – p=1 and q=0
A2 fixed – q=1 and p=0
If the effects at different loci are additive (independent), then
M = Σa(p - q) + 2Σpqd
 = P + a(p - q) + 2pqd

10. 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
The mean of the two BC1 populations will only be the same as the F2 if there is no dominance;
For a single locus, means in a nonrandom-mated BC1 population will be higher than in the random-mated BC1 if dominance is present
Mean for the F2 is the same regardless of whether or not it is random mated (because genotype frequencies are the same)
Effects across loci can cancel each other out.
For ½ A1A1, ½ A1A2
 = P + ½(a + d)
BC1(A1A1) = P + (1/2)a + (3/8)d
In a BC1 crossed to the unfavorable parent, p = 0.25,
so after random mating
For ½ A1A2, ½ A2A2
 = P + ½(d - a)
BC1(A2A2) = P - (1/2)a + (3/8)d

11. Average effects We have defined the mean in terms of genotypic values
Genes (alleles), not genotypes, are passed from parent to offspring
Average effect of a gene (i)
mean deviation from the population mean of individuals who received that gene from their parents (the other gene taken at random from the population)
coded values a, d and –a are expressed in terms of genotypes
Genotypes shown in table are consequences of random combinations of A1 (or A2) with gametes in the population
subtract M = a(p - q) + 2pqd
Gamete
A1A1 a
A1A2 d
A2A2 -a
Freq x value
Average effect
of a gene A1 p q
pa + qd
1=q[a+d(q-p)] A2
pd - qa
2=-p[a+d(q-p)]

12. 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)

13. Breeding value of individual Aij = i + j
Genotype
Average effect of a gene
Average effect of a gene substitution
A1A1
21
2q
A1A2
1 + 2
(q - p)
A2A2
22
-2p
Breeding value of an individual is judged by the mean value of its progeny. Breeding value can be measured. Breeding value is 2 x GCA.
Sum across all loci affecting the trait.
For a population in H-W equilibrium, the mean breeding value = 0
The expected breeding value of an individual is the average of the 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.

14. 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
variance in breeding values
variance due to regression of genotypic values on genotype (number of alleles)
● genotypic value
○ breeding value

15. Genotypic values a d a(q-p)+d(1-2pq) (q-p)+2pqd -a
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
Scaled
Adjusted for mean
A1A1 a
2q(a-pd)
2q(-qd)
A1A2 d
a(q-p)+d(1-2pq)
(q-p)+2pqd
A2A2 -a
-2p(a+qd)
-2p(+pd)
Remember  = a + d(q - p)  Substitute a =  - d(q - p)

16. P = G + E G = A + D Dominance deviation Gij =  + i + j + ij
Components of an individual’s Phenotypic Value
P = G + E
G = A + D
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)
Gij =  + i + j + ij
If there is no intralocus variance due to D, then you can say that there is only additive gene action within a locus (may not hold true across loci)

17. Partitioning Genotypic Value
Genotype
Genotypic Value (adj. for mean)
Breeding Value
(additive effects)
Dominance Deviation
A1A1
2q(-qd)
2q
-2q2d
A1A2
(q-p)+2pqd
(q - p)
+2pqd
A2A2
-2p(+pd)
-2p
-2p2d
When p = q = 0.5 (as in a biparental cross between inbred lines)
Genotype
Genotypic Value
Breeding Value
Dominance
A1A1
-(1/2)d 
-(1/2)d
A1A2
(1/2)d
A2A2
--(1/2)d -

18. Dominance deviations from regression
Genotypic Value
A1A1
2q - 2q2d
A1A2
(q-p)+2pqd
A2A2
-2p - 2p2d
-2p2d
2pqd
-2q2d

19. Interaction deviation
Components of an individual’s Phenotypic Value
P = G + E
P = A + D + E
When more than one locus is considered, there may also be interactions between loci (epistasis)
G = A + D + I
P = A + D + I + E
‘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