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Genetic Principles of Plant Breeding
Quantitative Genetics
Models to compose and decompose
the ‘genotypic value’
Wolfgang Link, University of Göttingen, Germany
Dedicated to H.H. Geiger, Hohenheim
1
MODELS FOR THE GENOTYPIC VALUE
In chapter 2 we did focus on genotypic effects, gi being
the sum of many single effects, effects on the expression
of a quantitative trait. In this chapter we shall study
different linear models, describing the combined action
of genes. These models belong to two categories:
 metric models
 statistical models.
In both cases we mostly assume that the genes at
different loci act independently (no epistasis) and that
their effects are not correlated (no linkage). This implies
that the effects (and contributions to the variance) of the
several loci and combinations of loci behave purely
additive and can be summarized to a genotypic value.
Hence, it is valid to construct the models with only one of
the pertinent loci, it being an valid example for all loci.
Interactions between genes at different loci (epistasis)
may involve 2, 3, 4, … loci. Any generalization of 1locus-models to these situations requires a great number
of additional parameters. We therefore restrict ourselves
to the easiest case, i.e. 2-loci-interactions (digenic
epistasis). This is the most important case of epistatic
effects and the principle of the epistatic influence on the
genotypic value is quite well elucidated with this case.
Linkage between loci is mostly neglected, because the
models would else become too cumbersome and
because only for a small part of pairs of loci linkage is to
be expected (this part being < 1/x, where x equals the
number of chromosomes in the haploid genome).
2
Genotypic
value
c+a
c+d
d = metric
dominance
effect
c
a =
metric
additive
effect
a = metric additive effect
c-a
0
aa
aa
Aa
Aa
AA
AA
Genotype
Metric model of the genotypic values
3
Metric model
The parameters of the metric model describe the
differences between the genotypic value of the genotypes under study independently from the gene- and
genotype frequencies of the respective populations.
They characterize the gene action; e.g. the degree of
dominance or type and size of epistatic effects of genes.
General asssumption:
2 alleles per locus
a) 1 locus:
Genotype
Genotypic value
AA
Aa
Aa
c + a
c + d
c - a
Interpretation of the parameters:
c
=
(AA + aa)/2
Mean of the homozygotes
a
=
(AA – aa)/2
d =
Aa – (AA +
aa)/2
Half the difference between
homozygotes
Deviation of the heterozygote
from the mean of the
homozygotes
AA, Aa & aa are taken here to specify gentypic values:
instead of the more cumbersome expressions GAA, GAa
and Gaa; (mostly we use AA etc. for both, genotype &
genotypic value) .
4
Degree of dominance = d/a
No dominance: d/a = 0
Partial dominance: 0 < d/a < 1
Complete dominance:d/a = 1
Overdominance:d/a > 1
The reference level in the ‘metric model’ is the level “c”. This c
is the average genotypic value of all possible homozygotes
(nota bene: not the population mean  !).
Full homozygosity is reached only after a number of
n = ∞ generations of selfing, hence, the models were termed
F∞-model or better, F∞-metric.
(In some texts, the reference level is chosen as the average
genotypic value of a F2-equilibrium population, leading to a
somewhat different metric. In this case the metric is analogously
5
termed F2-metric.)
Schön, C.C., 1993
-a
+a
AA-genotypes
aa-genotypes
High ............Resistance............Low
6
Schön, C.C., 1993
a
d
7
Schön, C.C., 1993
8
Schön, C.C., 1993
9
Schön, C.C., 1993
10
UMC33
UMC128
Genetic Principles of Plant Breeding
27.08.2009
Examples of the concept of
“Effect = (Main Effect)i + (Main Effect)j + (Interaction)ij”
A
Δ
a
Dominant – recessive gene action
A
a
20
20
0
≠
6
20
14
AA
Δ
aa
Epistasis
BB
20
0
≠
20
bb
20
20
0
Effect of genotype + environment on phenotypic value
Env. 1
Env. 2
Gentoype .1
20
20
Δ
0
≠
6
Genotype 2
20
14
11
Genetic Principles of Plant Breeding
27.08.2009
It makes a difference whether there is epistasis involved or not!
Example: Start with a HWE- population! Three genotypes:
P(AA/BB) = 36%, H(Aa/Bb) = 48%, Q(aa/bb) = 16%.
As the population continues to propagate by random mating (no
selection …), “nothing should change”!
Genotypic frequencies
AA
Aa
aa
BB
0.36
Genotypic values
First generation (epistasis ?)
Bb
bb
BB Bb bb
AA 20
0.48
Aa
20
0.16
aa
10
Population
performance
18.40
No epistasis: Generation ∞
BB
Bb
bb
AA 0.1296 0.1728 0.0576
Aa 0.1728 0.2304 0.0768
aa 0.0576 0.0768 0.0256
0.48
0.16
Σ 0.36
Σ
0.36
0.48
0.16
BB
AA 20
Aa 20
aa 14
Bb
20
20
14
bb
16
16
10
18.40
With epistasis*: Generation ∞
BB
Bb
bb
BB Bb bb
Σ
AA 0.1296 0.1728 0.0576 0.36
AA 20 20 20
Aa 0.1728 0.2304 0.0768 0.48
Aa 20 20 20
aa 0.0576 0.0768 0.0256 0.16
aa 20 20 10
0.48
0.16
Σ 0.36
*(aa)12=(ad)12=(da(12)=(dd)12=-2,5, see further below!
It is of great
importance whether a
population (the performance of which is
e.g. 18.40) changes
its performance
without (selection, mutation, drift etc., it
means without) any
‘reason’. This is
contradictory to the
DUS critera (distinctness, uniformity, stability; the ‘reason’
here is EPISTASIS)
19.744
12
Albina-Locus
Xantha-Locus
Albina-Locus
Xantha-Locus
13
Modes of Gene action
Supplementary gene action: different alleles of one locus code
for different effect that may compensate each other
Complementary gene action: for a full expression of the trait,
two loci have to be favourably equipped with pertinent alleles.
Example: Resistance
Genotype of
host
1 locus
Race ‘R’
Race ‘Q’
Races ‘R’ & ‘Q’
R1R1
Yes
No
No
R1R2
Yes
Yes
Yes
R2R2
Mode of gene
action
No
Dominant
R1>R2
Yes
Dominant
R2>R1
No
Supplementary
Gene Action
2 loci*
Race ‘R’
Race ‘Q’
Races ‘R’ & ‘Q’
R1R1/Q1Q1
Yes
No
No
R1R2/Q1Q2
Yes
Yes
Yes
Resistance of host when attacked by
R2R2/Q2Q2
No
Yes
No
Mode of gene
Dominant
Dominant
Complementary
action
R1>R2
Q1>Q2
Epistasis
*if linked, then only these genotypes may occur
14
2 loci:
To fully describe the genotypic value based on 2 loci,
we (additionally to the 1-locus-parameters a & d)
need the following digenic epistasis parameters:
(aa)12 Additive x additive-interaction
between
the genes
(ad)12 Additive x dominance-interaction
at locus
(da)12 Dominance x additive-interaction
(dd)12 Dominance x dominance-interaction 1 and 2
Genotypic value (c  0):
Locus 2
Locus
1
BB
Bb
bb
AA
a1 + a2 + (aa)12
a1 + d2 + (ad)12
a1 - a2 - (aa)12
Aa
d1 + a2 + (da)12
d1 + d2 + (dd)12
d1 - a2 - (da)12
aa
-a1 + a2 - (aa)12 - a1 + d2 - (ad)12 - a1 - a2 + (aa)12
Interpretation:
c =
1/4 (AABB + AAbb + aaBB + aabb)
½ [½ (AABB – aaBB) + ½ (AAbb – aabb)]
a1 =
Half difference between the two homozygotes at
the first locus, averaged across the homozygous
phases (situations) at the second locus.
½ [½ (AABB – AAbb) + ½ (aaBB – aabb)]
a2 =
Half difference between the two homozygotes at
the second locus, averaged across the homozygous phases (situations) at the first locus.
15
d1
= ½{[AaBB–½(AABB+aaBB)] + [Aabb–½(AAbb+aabb)]}
Deviation of the heterozygote from the mean of the
homozygotes at the first locus averaged across the
homozygous phases at the second locus. ½
d2
= ½{[AABb–½(AABB+AAbb)] + [aaBb–½(aaBB+aabb)]}
(aa)12 = ½ [½(AABB–aaBB) – ½(AAbb–aabb)]
= ½ [½(AABB–AAbb) – ½(aaBB–aabb)]
Dependency of the difference between the homozygotes at
one locus from the homozygous phase at the other locus;
(ad)12 = ½{[AABb–(AABB+AAbb)/2] – [aaBb–(aaBB+aabb)/2]}
Dependency of the dominance effects at the 2nd locus from
the homozygous phase at the first locus
= ½{[AABb–aaBb] – ½[(AABB–aaBB) + (AAbb–aabb)]}
Dependency of the homozygotes’ difference at 1st locus
from the hetero/homozygosity at 2nd locus.
(da)12 = Analogous to (ad)12
(dd)12 = AaBb–½(AABb+aaBb) – ½{[AaBB–½(AABB+aaBB)] +
[Aabb–½(AAbb+aabb)]}
Dependency of the dominance effect at the first locus from
the hetero/homozygosity an the second locus and vice
versa.
16
Contributions of the parameters of the metric 2-loci-model
to the genotypic values of the 9 genotypes that are
possible with 2 alleles per locus (cf. Tab. at page 5)
Genotype
c
a1
a2
Parameter
d1 d2 (aa)12 (ad)12 (da)12 (dd)12
AABB
1
1
1
0
0
1
0
0
0
AABb
1
1
0
0
1
0
1
0
0
AAbb
1
1
-1
0
0
-1
0
0
0
AaBB
1
0
1
1
0
0
0
1
0
AaBb
1
0
0
1
1
0
0
0
1
Aabb
1
0
-1
1
0
0
0
-1
0
aaBB
1
-1
1
0
0
-1
0
0
0
aaBb
1
-1
0
0
1
0
-1
0
0
aabb
1
-1
-1
0
0
1
0
0
0
Y = vector of genotypic values,
X = matrix of coefficients,
 = vector of parameters.
Solution of the linear system of equations for  by matrix
inversion leads to:
  X-1Y
17
How to “compose” parameters from genotypes
(coefficients of the inverted system of equations1))
Genotype
ParaAABB AABb AAbb AaBB AaBb Aabb aaBB aaBb aabb
meter
c
1
0
1
0
0
0
1
0
1
a1
1
0
1
0
0
0
-1
0
-1
a2
1
0
-1
0
0
0
1
0
-1
d1
-1
0
-1
2
0
2
-1
0
-1
d2
-1
2
-1
0
0
0
-1
2
-1
(aa)12
1
0
-1
0
0
0
-1
0
1
(ad)12
-1
2
-1
0
0
0
1
-2
1
(da)12
-1
0
1
2
0
-2
-1
0
1
(dd)12
1
-2
1
-2
4
-2
1
-2
1
1)
All matrix values must be multiplied by ¼, e.g.:
c = ¼ [1 x G(AABB) + 1 x G(Aabb) + 1 x G(aaBB) + 1 x G(aabb)]
etc.
18
2–loci-Model für n loci:
Any genotype
Locus
AA
Bb
CC
dd
Ee
....
1
2
3
4
5
....
n
Genotypic value:
Gi =
c
+ a1 + d2 + a3 – a4 + d5 + ... + (ad)12 + (aa)13
- (aa)14 + (ad)15 + ... + (da)23 - (da)24 + (dd)25 + ...
From the single parameters a, d, (aa), (ad), (da) and (dd), a summation parameter
can be built by simple addition. Here, we will elucidate the parameter system,
the metric, based on several numerical examples and by experimental data sets.
The genotypic values are ordered in the standard matrix form:
AABB
AABb
AAbb
AaBB
AaBb
Aabb
aaBB
aaBb
aabb
19
Examples for trait expression with 2 loci and 2 alleles
each, allowing for different modes of gene action.
1)
Intermediate
20
18
16
17
15
13
14
12
10
2)
Partial dominance
20
19
16
19
18
15
14
13
10
3)
Complete domin.
20
20
16
20
20
16
14
14
10
4)
Overdominance
20
21
16
22
23
18
14
15
10
5)
Complementary
20
20
10
20
20
10
10
10
10
6) Duplicate gene action
20
20
20
20
20
20
20
20
10
7)
38,7
6,6
4,0
2,7
2,0
3,4
“Complex“ epistasis
Callus induction rate
3,2
(%)
1,3
Corn anther culture
1,6
(Cowen et al. 1992)
Exercise:
Calculate the parameters a1, a2, d1, d2, (aa)11, (ad)12,
(da)12, (dd)12 and the degree of dominance (a/d) for the
two loci. Check whether you can „re-compose“ the
genotypic values from the calculated parameters.
20
Solutions
IntermePartial
Complete Over- CompleDuplicateeComplex
diate domininance dominance domin. mentary
c
15
15
15
15
12,5
17,5
11,55
a1
3
3
3
3
2,5
2,5
9,40
a2
2
2
2
2
2,5
2,5
9,15
d1
2
3
5
2,5
2,5
-8,90
d2
1
2
3
2,5
2,5
-6,55
(aa)12
2,5
-2,5
8,60
(ad)12
2,5
-2,5
-7,80
(da)12
2,5
-2,5
-7,80
(dd)12
2,5
-2,5
5,90
d1/a1
0,67
1
1,67
1
1
-0,95
d2/a2
0,5
1
1,5
1
1
-0,72
21
Example for epistasis:
Inheritance of restoration of pollen fertility in a F2population of winter rye (Secale cereale)
Two way tableau of marker class means for pollen fertility
(1-9)a; marker loci are two unlinked RFLP-marker loci Ma
(MWG59) und Mb (PSR371), originating from the female and
male parent;
Scores large enough for use in breeding are shaded in grey.
Marker-genotype
MWG59
Mean
MaMa
Mama
mama
MbMb
7.9
7.1
1.5
5.5
PSR371 Mbmb
7.8
8.0
2.0
5.9
mbmb
4.6
5.6
3.5
4.6
Mean
6.8
6.9
2.3
5.3
a)
1 = totally male sterile, 9 = fully male fertile.
(from: Dreyer, Ph.D. 2000)
Exercise:
model:
c =
a1 =
a2 =
d1 =
d2 =
Estimation of the parameters of the metric
(aa)12
(ad)12
(da)12
(dd)12
=
=
=
=
22
Frequency [%]
70
71,1
60
P1
96,5
83,8
Plant height
of
winter rye
P2
P
50
(from: Oetzel, 1977)
40
30
20
10
60
65 70
75 80
85
90 95 100 105 110 115 120 125 130
Plant height [cm]
Frequency [%]
60
Heterosis
50
38.0
40
F1 121,8
30
20
10
Frequency [%]
60
60
65 70 75
80
85 90
95 100 105 110 115 120 125 130
18,7
50
Plant height [cm]
F2 102,5
40
30
20
10
Frequency [%]
60
60 65
70 75
80
50
85 90
95 100 105 110 115 120 125 130
95,7 B1
Plant height [cm]
B 101,7
B2 106,8
40
30
20
10
24
60 65
70 75
80
85
90 95 100 105 110 115 120 125 130
Plant height [cm]
Schierholt, Antje, 2000:
Hoher Ölsäuregehalt (C18:1) im Samenöl:
genetische Charakterisierung von Mutanten im Winterraps (Brassica napus L.).
Dissertaion, Universität Göttingen.
25
Example
F2- ½(B1+B2)= ¼(aa)12
i.e.,
70.5 – ½ (65.1+72.7) = 1.6
thus,
Schierholt, Antje, 2000:
Hoher Ölsäuregehalt (C18:1) im
Samenöl: genetische
Charakterisierung von Mutan-ten
im Winterraps (Brassica napus L.).
Dissertaion, Universität Göttingen.
26
- 16 –
COV 3.1.2
Genetic interpretation of generation means,
(F∞-metric),
no linkage
linkage
2-loci-model (F
-metric), no
b) P1 = AAbb, P2 = aaBB
Gen.
c
a1
a2
d1
P1
P2
P
1
1
1
1
-1
-1
1
F1
F2

F∞
F
1
1

1
B1
B2
B
1
1
1
1
½

0
½ -½ ½
-½ ½ ½
½
(Repulsion phase)
d2 (aa)12 (ad)12 (da)12 (dd)12
-1
-1
-1
1
½

0
½
½
½
-¼
-¼
-¼
¼
-¼
-¼
¼
1
¼

0
¼
¼
¼
Compare the cases „coupling“ and „repulsion“; the
parameters a1, a2, (aa)12, (ad)12 and (da)12 enter with
varyiing sign into the generation means. The sign depends
on the type of association of the genes in the parents. This
fact governs the above studied contrasts between mean.
Heterosis:
F2 -
F1 – P = d1 + d2 + (aa)12 + (dd)12
(F1 + P)/2
F2 - B
=
½ (aa)12 – ¼(dd)12
=
¼(aa)12
28
10
9
7
5
4
3
2
1
0
1.00
Parental
mean;
F∞
0.75
F2- mean; BC1-mean
6
F3-generation mean
Ertragsleistung (t/ha)
Yield performance (t/ha)
Any deviation from this linearity
is indicative for epistasis. The type(s)
of epistasis depend(s) on the actual
non-linearity.
F1-hybrid
8
Any difference of F∞
and the parental mean
shows additiv-additivepistatic effects
0.50
0.25
0.00
Inbreeding coefficient, Inzuchtkoeffizient
29
Statistical models
The parameters of the statistical models characterize the
average effects of genes and gene combinations in
populations with given, in reality mostly unknown
genotype frequencies like p(A) and q(a).
In the following we will restrict ourselves to populations in
the generalized HWE (this implies: N   , random
mating, no selection, no migration, no mutation).
Parameters of the statistical models are
- The population mean µ,
- The additive or average effect of an allele, α,
- The dominance effect (interaction of allelic
genes), δ,
- The epistatic effects (interaction of non-allelic
genes), (αα) etc.
Different from the parameters of the metric models, here
the parameters are written in Greek letters
- µ, α, δ, (αα), … and the sums of effects of a given type are written in
capital Latin letters (A, D, AA, ...).
30
Genotypic trait value of offspring families
Random mating
Gentoypic variance; 100 loci; a=d=0.5; p(A)=0.634
Genotypic variance; 100 loci; a=0.5; d=0; p(A)=0.634
100
80
„Value“ of AA = 1
„Value“ of aa = 0
2
s =1.554
s=1.247
60
2
40
s =2.901
s=1.703 = 3.406/2
20
s²G=11.603
s²A= 6.218
sA = 2.494
s²D= 5.385
2
s =11.603
s=3.406
0
0
20
40
60
80
Genotypic trait value of parents
100
31
Population mean
Definition:  = E {Pij} = E {Gi}
Assumption: Random mating, generalized HWE
a) 1 locus, 2 alleles:
Runner (i) Genotype
1
2
3
Frequency (zi)
Gi
p2
2pq
q2
c+a
c+d
c–a
AA
Aa
aa
3

=
 zG =
i
i
(p2 + 2pq + q2) c + (p2 – q2) a + 2pqd
i=1

=
c + (p - q)a + 2pqd
As clear from definition, homozygotes AND heterozygotes contribute to the population mean.
b) n loci, 2 alleles each, no epistasis:
n

=
c+

n
(pl – ql)al + 2
l1

plqldl
l1
c = Mean of all 2n homozygotes
32
COV 3.2
 = c + (p - q)a + 2pqd
- 25 -
μ
c+a
d=0
intermediate
gene effects
c+a
c
c-a
0
1
p
c+a
c+a
d=½a
partial
dominance
c
c-a
0
p
1
c+a
d=a
complete
dominance
c
p
c-a
1
0
c+a
d=
c+a
3
a
2
overdominance
c
d+ a
2d
for d>a
p* 
p*=5/6
c-a
0
p
1
Dependency of the population mean (  ) on the
frequency (p) of the favourable allele when allowing for
different degrees of dominance.
33
Analysis of genotypic value: AVERAGE EFFECT OF AN ALLELE
refer to Falconer, D.S., 1981: Introdution to quantitave genetics. Longman.
Example:
Trait:
Corn (panmictic species), imagine an improvement of a population
via mass selection.
Grain yield in saline soil
Mean yield of genotypes with allelic
equipment of locus A: A1A1 = 30 dt/ha
Frequency of A1 = 10%
Frequency of A2 = 90%
A1A2 = 30 dt/ha
A2A2 = 26 dt/ha
Population mean: M = 0,12 x 30 + 2 x 0,1 x 0,9 x 30 +
0,92 x 26 = 26,76 dt/ha
The favourable allele A1 is rather rare, and it is dominant.
Mating of the rare gametes with the allele A1 to the gametes, as delivered by the
population (imagine this as pollen pool), gives:
A1 with A1 = 10%, A1 with A2 = 90%
with a resulting mean of the offspring : 0,1 x 30 dt/ha + 0,9 x 30 dt/ha = 30 dt/ha.
A2 with A1 = 10%, A2 with A2 = 90%
With a resulting mean of the offspring : 0,1 x 30 dt/ha + 0,9 x 26 dt/ha = 26,4 dt/ha.
Difference of these offsprings to population mean = average effect(341)
A1 with A1 = 10%, A1 with A2 = 90%
with a resulting mean of the offspring : 0,1 x 30 dt/ha + 0,9 x 30 dt/ha = 30 dt/ha.
A2 with A1 = 10%, A2 with A2 = 90%
With a resulting mean of the offspring : 0,1 x 30 dt/ha + 0,9 x 26 dt/ha = 26,4 dt/ha.
Difference of these offsprings to population mean = average effect(1)
The average effect of allele A1 ( 1) is:
Mean of the offspring - population mean =  1
30 dt/ha
-
26,76 dt/ha
= 3,24 dt/ha
The average effect of allele A2 ( 2) is:
26,4 dt/ha
-
26,76 dt/ha
= - 0,36 dt/ha
Accordingly the breeding value A of the genotypes amounts to:
A1A1
A1A2
A2A2
21
6,48 dt/ha
1 + 2
2,88 dt/ha
22
- 0,72 dt/ha
Average effects and breeding values depend on the frequency of the alleles and their
effect. An individual has a high breeding value, if it is carrying many favourable
and/or dominant alleles and if these alleles are rare in the population that has to be
improved.
35
The variance of the breeding values of genotypes is called additive variance,
σ²A .
COV 3.2.2
- 26 -
3.2.2 Additive and dominance effects
Definitions:
The additive effect (or average effect) of an allele Ai,
i, is defined as the average deviation in performance
(plus or minus) of all those genotypes arising from
mating Ai with the gametic array of the population. The
deviation is measured from the population’s mean ():
α1- α 2= α =[a- (p-q)d]
α is sometimes
called „average
effect of a gene
substitution“
1 =
=
=
2 =
=
=
p · G(A1A1) + q · G(A1A2)
-
p(c + a)
+ q(c + d) - c + (p – q)a + 2pqd
q a – (p – q)d
p · G(A2A1) + q · G(A2A2)
-
p(c + d)
+ q(c – a) - c + (p – q)a + 2pqd
-p a – (p-q)d
The dominance effect of the allelic gene combination
AiAj, ij, is defined as the deviation of the genotypic
value of genotype AiAj from the sum of the population
mean  plus the two additive effects i & j of the
corresponding two alleles Ai and Aj.
11 = G(A1A1) – { + 1 + 1}
= c + a – {c + (p – q)a + 2pqd + 2q a – (p-q)d }
12
22
=
=
=
=
=
-2q2d
G(A1A2) – { + 1 + 2}
+2pqd
G(A2A2) – { + 2 + 2}
- 2p2d
36
38
39
COV 3.3
- 38 –
3.3 Summary
The genotypic values of the individuals of a
population can be described by

metric and

statistical models.
Metric models characterize the gene effects, the
type of action of genes, and hence are useful to
analyse the trait differences between generation:
especially if the generations trace back to
homozygous lines and hence we know the allele
frequencies. Important phenomena of gene action
for application in breeding are:

heterosis and inbreding depression,

deviation from the linearity between performance and heterozygosity.
The parameters of the statistical models characterize the effects of genes and of combinations of
genes on average, across all genotypes of a population, with unknown allele frequencies in the population. Based on statistical models, we can describe and forecast (predict):

the suitability of a genotype for use in further
breeding,

the effects of shifts in gene frequencies caused
by selection.
40
COV 3.4
3.4 Exercises
- 39 (cf. FALCONER Exercise 7.1-7.8)
1 FOR a maize single hybrid cultivar and its parental
homozygous inbred lines, the following yields were
measured [t ha-1]:
F1: 6.80
P1: 1.65
P2: 2.35

What is the amount of heterosis of the hybrid ?

Imagine we harvest the (open pollinated) F1-plants
and took the harvested seed to sow the resulting
generation (= F2); neglect epistasis; what is the
expected yield in F2?

Discuss the potential effects of epistasis on the
performance of the performance of this F2.
2 ACHIEVE a thorough understanding of the following
equation: it describes average genotypic value of an
inbred population (F; 0 < F  1; epistasis is neglected. Which factor(s) determine the difference in performance between a non inbred (F=0) and an inbred (F>0)
population, i.e., O - F ?
Population mean under inbreeding, F; no epistasis:
F = c + (p² + Fpq)a + 2pq(1 – F)d + (q² + Fpq)(-a)
F = c + (p – q)a + 2pq (1 – F)d
41
COV 3.4
3.4 Exercises (ff)
- 40 (cf. FALCONER Exercise 7.1-7.8)
3 IN THE following table you find data on yield and height
for corn, for progenies with different degrees of
inbreeding. Do graphically show the relationship
between performance (y-axis) and inbreeding coefficient
(x-axis). Choose differenct sympols for selfing and fullsib-mating like  and . Do comment the graphic !
Inbreeding depression (yield, height) under selfing &
fullsib mating in corn (Cornelius & Dudley 1974, Tab. 2)
Selfing
Fullsib Mating
Inbreeding
Yield Height
Yield Height
F
genera- coeffi(t/ha) (cm)
(t/ha) (cm)
tion
cient F
0
0,00
5,43
229
0,00 5,43
229
1
0,50
3,29
201
0,25 4,44
215
2
0,75
2,97
185
0,375 4,02
210
3
0,50 3,59
202
4 A GRASS breeder develops a synthetic cultivar from 5
inbres lines (= starting generation: Syn-0). The first Syngeneration (Syn-1) is created by diallel crossing. The
seed of the (5·4)/2 = 10 crosses are mixed with equal
proportions, and the second generation, Syn-2, is
created by open pollination (random mating) of the crop
stand comprising this equal-dose mixture. All further
multiplications are as well by open pollination. Do make
a scheme of the expected performance across the
generations Syn-0 to Syn 4! Do neglect epistasis!
42
COV 3.4
- 42 -
Solutions of the exercises to chapter 3
Heterosis = 6,80 – (1,65 + 2,35) / 2
1
= 4,80 t ha-1
F2 = F1 –½ heterosis
= 6,80 – 2,40
= 4,40 t ha-1
Positive aa and/or dd effects would decrease the
performance of the F2, negative effects would
increase it.
2
Change of population mean due to inbreeding;
no epistasis:
c + (p - q)a + 2pqd - F
O - F =
2pqFd
O - F =
We find a linear dependency of the population
performance on its inbreeding coefficient F !
43
-1
Y ie ld (t h)a
3 Graphic
6
5
4
3
2
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
P la n t h e ig h t (c m )
In b r e e d in g c o e ffic ie n t F
240
230
220
210
200
190
180
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
In b r e e d in g c o e ffic ie n t ( F )
Dependency of grain yield and plant height on the inbreeding coefficient in corn (Cornelius & Dudley 1974)
Basic population
Selfing,
Fullsib mating
Comment: Nearly linear relation of performance & inbreeding
coefficient F. In case of grain yield there is a hint on a deviation from
linearity with the last inbreeding generation. This could be the effect
of selection when choosing the parental individuals. In case of height,
there is no such deviations. This trait most probably is not or nearly
not affected by selection (natural or artificial), hence, a nearly-atrandom-choice of parents in the experiment could be realized.
44
4
t
0
1
2
3
4
Syn generation (t)
Expected performance (t) of a synthetic population in
the first generations of multiplicaitons
45