1. PBG 650 Advanced Plant Breeding
Module 6: Quantitative Genetics
Environmental variance
Heritability
Covariance among relatives

2. More interactions For an individual G = A + D + I P = A + D + I + E
For a population
Two-locus interactions
More than two loci….
Interlocus interactions are important, but difficult to quantify
Many designs for genetic experiments lump dominance and epistatic interactions into one component called “non-additive” genetic variance

3. Genetic variances from a factorial model
Bernardo, Chapt. 5

4. Environmental variance
covariance would occur if better genotypes are given better environments
randomization should generally remove this effect from genetic experiments in plants
P = G + E
P = G + E + GE
genotype by environment interactions
differences in relative performance of genotypes across environments
experimentally, GE is part of E
DeLacey et al., 1990 – summary of results from many crops and locations
For a particular crop, only 10% of variation in phenotype is due to genotype!
rule E: GE: G

5. Repeatability Multiple observations on the same individuals
May be repetitions in time or space (e.g. multiple fruit on a plant)
variation among observations on the same individual due to temporary environmental effects
( = special environmental variance)
variation among individuals due to genetic differences and permanent environmental effects
( = general environmental variance)
Falconer & Mackay, pg 136

6. Repeatability Repeatability Sets an upper limit on heritabilities
is easy to measure
To separate and , you must evaluate repeatability of genetically uniform individuals

7. Gain from multiple measurements
fyi
Multiple measurements can increase precision and increase heritability (by reducing environmental and phenotypic variation)
Greatest benefits are obtained for measurements that have low repeatability (large )

8. Heritability For an individual: P = A + D + I + E For a population:
Broad sense heritability
degree of genetic determination
Narrow sense heritability
extent to which phenotype is
determined by genes
transmitted from the parents
“heritability”
Falconer & Mackay, Chapter 8

9. Narrow sense heritability – another view
h2 = the regression of breeding value on phenotypic value
h2=0.5 +1 +2
h2=0.3
h2 is trait specific, population specific, and greatly influenced by the choice of testing environments

10. Narrow sense heritability
Can be applied to individuals in a single environment (generally the case in animal breeding)
In plants, it is commonly expressed on a family (plot) basis, which are often replicated within and across environments

11. Heritability in plants - complications
Different mating systems, including varying degrees of selfing
Different ploidy levels
Annuals, perennials
For many crops, measurement of some traits is only meaningful with competition, in a full stand
variables such as yield are measured on a plot basis
other traits are averages of multiple plants/plot
plot size varies from one experiment to the next
Replicates are evaluated in different microenvironments
Genotype x environment interaction is prevalent for many important crop traits
Nyquist, 1991; Holland et al., 2003

12. Heritability in plants - definition
Fraction of the selection differential that is gained when selection is practiced on a defined reference unit (Hanson, 1963)
Selection Differential S=s-0
Selection Response R=1-0
Y=bX
R=Sbyx
R/S=h2=byx
Main purpose for estimating heritability is to make predictions about selection response under varying scenarios, in order to design the optimum selection strategy
R=h2S
High heritability – use mass selection, single environment
Low heritabiltiy – progeny testing, family selection

13. Applications in plant breeding
Selection in a cross-breeding population
Selection among purelines (with or without subsequent recombination)
Selection among clones
Selection among testcross progeny in a hybrid breeding program
Must specify the unit of selection, the selection method, and unit on which the response is measured

14. Heritability of a genotype mean
GXE
Error variance
High heritability – use mass selection, single environment
Low heritabiltiy – progeny testing, family selection
broad sense heritability narrow sense heritability
or “heritability”

15. Resemblance between Relatives
Covariance between relatives measures degree of genetic resemblance
Variance among groups = covariance within groups
Intraclass correlation
of phenotypic values
Strategy:
Determine expected covariance among relatives from theory, and compare to experimental observations
Estimate genetic variances and heritabilities
Falconer & Mackay, Chapt. 9

16. Covariance between offspring and one parent
Genotype
Frequency
Genotypic Value
Breeding Value
Mean Genotypic Value of Offspring
A1A1 p2
2q(-qd)
2q q
A1A2
2pq
(q-p)+2pqd
(q - p)
(1/2)(q - p)
A2A2 q2
-2p(+pd)
-2p
-p
CovOP=p2*2q(-qd)q+2pq[(q-p)+2pqd](1/2)(q - p) +q2[-2p(+pd)](-p)
CovOP = pq2 = (1/2)σA2
This result is true for a single offspring and for the mean of any number of offspring

17. Resemblance between offspring and one parent
For parents and offspring, observations occur in pairs
Regression is more useful than the intraclass correlation as a measure of resemblance
does not depend on the number of offspring
does not require parents and offspring to have the same variance
Get SE b from any standard stats book
phenotypic variance of the parental population
Estimate

18. Resemblance between offspring and mid-parent
CovO,MP = pq2 = (1/2)σA2
Regression on mid-parent is twice the regression of offspring on a single parent
Number of offspring does not affect the covariance or the regression

19. Resemblance among half-sibs
Genotype
Frequency
Breeding Value
Mean Genotypic Value of Offspring
Freq. x Value2
A1A1 p2
2q q
p2q22
A1A2
2pq
(q - p)
(1/2)(q - p)
(1/2)pq(q - p)22
A2A2 q2
-2p
-p
Covariance of half-sibs = variance among half-sib progeny
CovHS = pq2[(1/2)(q - p)2+2pq] = pq2[(1/2)(p+q)2]
= (1/2)pq2=(1/4)σA2

20. Resemblance among full-sibs
Progeny
Genotype of parents
Frequency of mating
A1A1 a
A1A2 d
A2A2 -a
Mean Value of Progeny p4 1
4p3q
1/2
(1/2)(a+d)
2p2q2
4p2q2
1/4
(1/2)d
4pq3
(1/2)(d-a) q4
have to subtract the population mean because we’re working with the coded values
CovFS= σFS2 = p4a2+4p3q[(1/2)(a+d)]2….+q4(-a)2 - 2
= pq[a+d(q-p)]2 + p2q2d2

21. Resemblance among full-sibs
CovFS= σFS2 = p4a2+4p3q[(1/2)(a+d)]2….+q4(-a)2 - 2
= pq[a+d(q-p)]2 + p2q2d2

22. General formula for covariance of relatives
Unilineal relatives
Resemblance involves only
Bilineal relatives
Potential exist for relatives to have two common alleles that are identical by descent
etc.
(X1X3, X1X4, X2X3, or X2X4) A B
Resemblance will also involve:
X1X2
X3X4
etc. C D
X1X3
X1X3

23. Covariance due to breeding values
A B C D X Y
(Ai Aj)
(Ak Al)

24. Covariance due to dominance deviations
A B C D X Y
(Ai Aj)
(Ak Al)

25. General formula for covariance of relatives
A B C D X Y
r = 2XY
 = ACBD + ADBC
Extended to include epistasis:

26. Adjusting coefficients for inbreeding
Relatives
r = 2XY 
Parent-offspring
1/2
Half-sibs
Common parent not inbred
1/4
Common parent inbred
(1+F)/4
Full-sibs
Parents not inbred
Parents inbred
(2+FA+FB)/4
(1+FA)(1+FB)/4