PBG 650 Advanced Plant Breeding
Module 6: Quantitative Genetics
– Environmental variance
– Heritability
– Covariance among relatives
More interactions
For an individual
G=A+D+I
P=A+D+I+E
For a population
P   A  D  I  E
Two-locus interactions
2
2
2
2
2
 I   AA   AD   DD
2
2
2
2
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
Genetic variances from a factorial model
Source of Variation
A locus main effect
Degrees of
Freedom
2
Variance Component
 2A (locusA)  D2 (locusA)
A linear
1
 2A (locusA)
A quadratic (deviations)
1
D2 (locusA)
B locus main effect
 2A (locusB)  D2 (locusB)
2
B linear
1
 2A (locusB)
B quadratic (deviations)
1
D2 (locusB)
A x B interaction
I2
4
A linear x B linear
1
 2AA
A linear x B quadratic
A quadratic x B linear
1
1
 2A D (pooled)
A quadratic x B quadratic
1
2
 DD
Bernardo, Chapt. 5
Environmental variance
• covariance would occur if better
P=G+E
 P   G   E  2CovGE •
2
2
2
• genotype by environment interactions
• differences in relative performance of
P = G + E + GE
2
P
2
G
2
E
   
genotypes are given better
environments
randomization should generally
remove this effect from genetic
experiments in plants
2
GE
•
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%
70-20-10 rule
of variation in phenotype is due
E: GE: G
to genotype!
Repeatability
•
Multiple observations on the same individuals
– May be repetitions in time or space (e.g. multiple fruit on a plant)
 P   Between   Within
2
 Within
2
 Between
2
2
2
variation among observations on the same
individual due to temporary environmental effects
2
( Es
= special environmental variance)
variation among individuals due to genetic
differences and permanent environmental effects
2
( Eg= general environmental variance)
2
P
2
 ( G
Falconer & Mackay, pg 136
2
2
 Eg )  Es
Repeatability
2
P
2
 ( G
Repeatability
2
2
 Eg )  Es
r
2
G
2
 Eg
P2
• Sets an upper limit on heritabilities
2

• Es is easy to measure
2
• To separate  G2 and  Eg
, you must evaluate repeatability of
2
genetically uniform individuals
Eg
r 2
2
Eg  Es
Gain from multiple measurements
2
P(n)
2
 (G
fyi
2
1 2
 Eg )  n Es
 P( n ) 1  r(n  1)

2
P
n
2
•
Multiple measurements can increase precision and
increase heritability (by reducing environmental and
phenotypic variation)
•
Greatest benefits are obtained for measurements that
2
have low repeatability (large  Es
)
Heritability
•
•
For an individual:
For a population:
P=A+D+I+E
     
2
P
2
A
2
D

•
Broad sense heritability
– degree of genetic determination
•
Narrow sense heritability
– extent to which phenotype is
determined by genes
transmitted from the parents
Falconer & Mackay, Chapter 8
2
I
2
E
2
G
2
G
H 2
P
h2 
2
A
2
P
“heritability”
Narrow sense heritability – another view
Breeding Value
h2 = the regression of breeding value on phenotypic value
4,5
4,0
3,5
3,0
2,5
2,0
1,5
1,0
0,5
0,0
h2=0.5
+1
+2
h2=0.3
0
2
4
6
8
10
Phenotypic Value
h2 is trait specific, population specific, and greatly influenced by
the choice of testing environments
Narrow sense heritability
h 
2
2
A
2
P

2
A
2
2
 A  D
2
 E
– 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
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
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=h2S
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
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
Heritability of a genotype mean
GXE
  
2
P
H 

2
 
2
G
2
G
2
G
2
 GL
l
2
 GL

l
h 
Error variance
 e2
rl

2

 e2
rl
broad sense heritability
 
2
G
2
A
2
GL
l

 e2
rl
narrow sense heritability
or “heritability”
Resemblance between Relatives
•
Covariance between relatives measures degree of
genetic resemblance
•
Variance among groups = covariance within groups
Intraclass correlation
of phenotypic values
B
t 2
2
B   W
2
Strategy:
•
Determine expected covariance among relatives
from theory, and compare to experimental
observations
•
Estimate genetic variances and heritabilities
Falconer & Mackay, Chapt. 9
Covariance between offspring and one parent
Cov(X,Y)   fi X iYi  μX μY
Mean
Genotypic
Value of
Offspring
Genotype
Frequency
Genotypic Value
Breeding
Value
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
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
b
Cov(O, P)
Estimate
P
2
A
 2
P
1
2
2
phenotypic
variance of
the parental
population
Resemblance between offspring and mid-parent
CovO,MP = pq2 = (1/2)σA2
 MP   P
2
b
•
•
Cov(O, MP )
 MP
2
2
1
2
A A
 1 2  2
P
2 P
1
2
2
2
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
Resemblance among half-sibs
Genotype
Breeding
Frequency
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
p2q22
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
t
CovHS
P
2
A
 2
P
1
4
2
Resemblance among full-sibs
Progeny
Genotype of
parents
Frequency
of mating
A1A1
A1A2
A2A2
a
d
-a
A1A1
A1A1
p4
1
A1A1
A1A2
4p3q
1/2
A1A1
A2A2
2p2q2
A1A2
A1A2
4p2q2
A1A2
A2A2
4pq3
A2A2
A2A2
q4
1/4
Mean Value
of Progeny
a
1/2
(1/2)(a+d)
1
d
1/2
1/4
(1/2)d
1/2
1/2
(1/2)(d-a)
1
-a
CovFS= σFS2 = p4a2+4p3q[(1/2)(a+d)]2….+q4(-a)2 - 2
= pq[a+d(q-p)]2 + p2q2d2
Resemblance among full-sibs
CovFS= σFS2 = p4a2+4p3q[(1/2)(a+d)]2….+q4(-a)2 - 2
= pq[a+d(q-p)]2 + p2q2d2
 G  2 pqa  d q - p   4 p q d
2
2
CovFS   FS 
2
t
CovFS
P
2
1
2
2
 A  D
2
1
4
2
 A  D

2
P
1
2
2
1
4
2
2
2
General formula for covariance of relatives
•
Unilineal relatives
– Resemblance involves only
•
 A  AA  AAA
2
2
2
etc.
Bilineal relatives
– Potential exist for relatives to have two common alleles
that are identical by descent (X1X3, X1X4, X2X3, or X2X4)
A
X1X2
B
X3X4
Resemblance will also involve:
 D  AD  AAD
2
C
D
X1X3
X1X3
2
2
etc.
Covariance due to breeding values
A
B
X
(Ai Aj)
C
D
Y
(Ak Al)
Covα  P(Ai  Ak )Cov(  i ,  k )
 P(Ai  Al )Cov(  i ,  l )
 P(A j  Ak )Cov(  j ,  k )
 P(A j  Al )Cov(  j ,  l )
 4θ XY σ 2 i  2θ XY σ A2  rσ A2
Covariance due to dominance deviations
A
B
C
X
(Ai Aj)
D
Y
(Ak Al)
Cov  P(Ai  Ak , A j  Al )Cov(  ij ,  kl )
 P(Ai  Al , A j  Ak )Cov(  ij ,  kl )
 (θ AC θBD  θ ADθBC )σ D2
  XY σ D2
General formula for covariance of relatives
A
B
C
X
D
Y
Cov  r2A  D2
r = 2XY
 = ACBD + ADBC
Extended to include epistasis:
2
Cov  r2A  D2  r 22AA  r2AD  2DD
 ...
Adjusting coefficients for inbreeding
A
2
Relatives
Parent-offspring
D
2
r = 2XY

1/2
0
1/4
0
Half-sibs
Common parent not inbred
Common parent inbred
(1+F)/4
Full-sibs
Parents not inbred
Parents inbred
1/2
1/4
(2+FA+FB)/4
(1+FA)(1+FB)/4