CHANGES IN MEAN AND VARIANCE UPON SELFING IN AN IDEALIZED BREEDING NURSERY

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CHANGES IN MEAN AND VARIANCE UPON SELFING IN AN
IDEALIZED BREEDING NURSERY
The F2 Generation  1. F2 Population Mean and
Variance (p q 0.5)  Genotype
BB Bb bb Genotypic frequency
0.25 0.5 0.25 Genotypic value (bu ac-1) 24
24 16 Population mean
(frequency x value)  0.25(24) 0.5(24)
0.25(16) 22 
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The F2 Generation  1. F2 Population Mean and
Variance (p q 0.5)  Genotype
BB Bb bb Genotypic frequency
0.25 0.5 0.25 Genotypic value (bu ac-1) 24
24 16
Population mean  0.25(24) 0.5(24) 0.25(16)
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Total population variance (frequency x value2 -
mean2)  0.25 (24)2 0.5 (24)2 0.25 (16)2 -
(22)2 12
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Genetic Variance

• ?2G ?2A ?2D
• ?2A,  or additive genetic variance is the
  variance of the effects of the genes
• ?2D or dominance variance, is the variance due
  to interaction of alleles

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The total genetic variance at a single locus
comprises additive genetic variance (?2A) plus
dominance genetic variance (?2D).    when p q
0.5, as in this example, then,  ?2G ?2A
?2D.  Thus in an F2 or S0 population where
allele frequencies are 0.5, we designate, by
convention, the total genetic variance as equal
to ?2A ?2D, in the absence of epistasis. Many
studies utilize F2 or S0 populations as a
baseline, or a starting point remember the F2
is the perfect HW pop. As we will illustrate
below, the changes in ?2A and ?2D that occur
during the inbreeding generation following the F2
are expressed relative to the values of ?2A and
?2D in the original F2 population.
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The F3 Generation   F3 Population Mean and
Variance (p q 0.5)  Genotype
BB Bb bb Genotypic
frequency 0.375 0.25 0.375 Genotypic
value (bu ac-1) 24 24
16   F3 population mean   0.375 (24) 0.25
(24) 0.375 (16) 21  Total F3 population
variance   0.375 (24)2 0.25 (24)2 0.375
(16)2 - (21)2 15. 3/2 ?2A 3/4 ?2D  
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Total F3 population variance    
3/2 ?2A 3/4 ?2D   15 Remember, Total F2
population variance ?2A ?2D 12 
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Changes in the mean
Note the change in population mean. In the F2 it
was 22. In the F3 it was 21. This reflects a
slow regression back to the homozygote midparent
value (20). This is due to the decrease in
heterozygotes with genotypic value 24. Upon
complete inbreeding the frequency of Bb is zero,
and the mean is equal to the midparent value.
The F2 Generation  1. F2 Population Mean and
Variance (p q 0.5)  Genotype
BB Bb bb Genotypic frequency
0.25 0.5 0.25 Genotypic value (bu ac-1)
24 24 16
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Inbreeding Depression

• Homozygosity increases the frequency of the loci
  with alleles identical by descent. This will
  include unfavorable recessive alleles whose
  genotypic inferiority is not masked by the
  presence of an alternate dominant type. Thus,
  the overall desirability of individuals decrease.
  This is referred to as inbreeding depression

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Changes in the Mean

• Of course, if there is no dominance, the
  population mean is equal to the midparent value
  in the F2, and the F3, and the F4, etc., until
  the population is completely inbred.

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Changes in the variance
There is one-half again the additive genetic
variance expressed in the F3 population as in the
F2 population This is a direct result of the
increased frequency of homozygotes which are
tending to polarize the population into two
opposite categories--homozygous dominants (BB at
a frequency of 0.375) and homozygous recessives
(bb at a frequency of 0.375).
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Changes in Variance

• In fact, when dominance is complete, the major
  noticeable phenotypic change from the F2 to F3
  would be the increased frequency from 25 to 38
  of recessives (bb) yielding only 16 bu ac-1.
  Notice that dominance variance has decreased as a
  result of the decrease in heterozygotes.

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The Underlying Family Structure of the F3
Population Variance Among and Within
F23-Derived Lines  Unlike the F2 or S0 which
contained no family structure, the F3 generation
in this example has a defined family structure.
F23-derived lines were developed by harvesting
F2 plants individually and keeping the seed
separate. As a result, the total population
genetic variance can be partitioned into A)
Variance among F23 lines and B) Variance within
F23 lines.
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A) Variance Among F23 Lines  F2 plant source of
F23 line BB Bb bb Frequency of F23 line
0.25 0.5 0.25 Mean value of F23
line   24 22 16 Note Pay particular
attention to the mean genotypic values of the
F23 lines. Lines derived from BB individuals
will have the mean value 24. Lines derived from
bb individuals will have the mean value 16. And
(this one stumps students--but just think about
it a minute), lines derived from Bb individuals
will have the mean value equivalent to an F2
population--i.e., 22 Why? Because Bb is
equivalent to an F1 heterozygote.
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?2G among F23 lines (frequency x value2 -
mean2)  0.25 (24)2 0.5 (22)2
0.25 (16)2 - (21)2   ?2A 1/4
?2D. 9.   In addition, pay
particular attention to the correction for the
mean--i.e., (21)2. F23 lines are in the F3
generation--thus the F3 population mean is
appropriate for the correction factor.  

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B) Variance within F23 lines  F2 plant source of
F23 line BB Bb bb Frequency
of F23 line 0.25 0.5
0.25 Variance within F23 line
0 .25(24)2 .25(24)2 .25(16)2 0   -
(22)2 Note There will be no variance within an
F23 line derived from either a BB or bb F2
plant. The variance within an F23-line derived
from a Bb plant will have within-line variance
equivalent to an F2 population.
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Variance within F23 lines  F2 plant source of
F23 line BB Bb bb Frequency
of F23 line 0.25 0.5
0.25 Variance within F23 line
0 .25(24)2 . 5(24)2 .25(16)2 0   - (22)2
We can obtain a simple mean value of the
variance within all the F23 lines. Thus, we
utilize the simple (frequency x value)
method.  Mean ?2G within F23 lines (frequency x
value)  0.25 (0) 0.5 (.25(24)2
.5(24)2 .25(16)2- (22)2) 0.25 (0) 6.
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Changes in Variance

• In developing F23-derived lines, the breeder has
  enhanced the chances of making superior
  selections.
• the total genetic variance has increased due to
  the increased frequencies of homozygotes.
• Increased additive genetic variance, decreased
  dominance variance.

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Variances Partitioned

• The variances have been partitioned.
• 60 of the total variance is among the means of
  the F23 lines, and of more importance,
  two-thirds of the additive variance in the F3
  generation is observed among the means of the
  F23 lines.

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Additive Variance

• Additive variance is the chief determinant of the
  breeding value of an individual so production of
  F23 lines should help the breeder do a superior
  job during selection.

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The F4 Generation  F4 Population Mean and
Variance (p q 0.5)   Genotype
BB Bb
bb Genotypic frequency 0.437 0.125
0.437 Genotypic value (bu ac-1) 24
24 16   F4
population mean   0.437 (24) 0.125(24)
0.437(16) 20.5
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Total F4 population variance   0.437 (24)2
0.125 (24)2 0.437 (16)2 - (20.5)2
15.75 7/4 ?2A 7/16 ?2D    
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Variance among F34 lines   F3 plant source of
F34 line BB Bb
bb Frequency of F34 line 0.375
0.25 0.375 Mean value
24 22
16   ?2G among F34 lines (frequency x
value2 - mean2)  0.375(24)2 0.25
(22) 0.375 (16)2 - (20.5)2
3/2 ?2A 3/16 ?2D   12.75
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Variance within F34 lines  F3 plant source of
F34 line BB Bb bb Frequency
of F34 line 0.375 0.25
0.375 Variance within F34 line
0 .25(24)2 .25(24)2 .25(16)2
0   - (22)212 Mean ?2G within F34 lines
(frequency x value)  0.375(0)
0.25 (12) 0.375(0).   1/4 ?2A
1/4 ?2D 3 The simplest way to
calculate variance within lines is by
subtraction, Total (15.75) - Among (12.75)
Within (3)
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General Formulas for Determining Additive and
Dominance Variance Based on F, Wrights
Coefficient of Inbreeding
F the probability that at a locus in an inbred
individual I, the two alleles are Identical By
Descent
In a population where p q 0.5, the formula
simplify to  Total ?2G (1F) ?2A (1-F2)
?2D  If we assume that F 0 in the F2 or S0,
then  ?2G (10) ?2A (1-02) ?2D ?2A ?2D.
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Additive Variance Among Derived Lines  ?2A
(1Ft) ?2A  where Ft inbreeding coefficient of
the generation from which individual plants were
selected. For example, in the case of F23
lines, Ft 0, but for F34 lines, then Ft (1
1/2) 3/2. Dominance Variance Among Derived
Lines  ?2D (1Ft)/(1-Ft)(1-F)2 ?2D.
The simplest way to calculate ?2A and ?2D within
lines is by subtraction, as follows  Variance
(within lines) total variance - variance (among
lines)  For example, for F23 lines,  Variance
(within lines) (3/2 ?2A 3/4 ?2D) - (?2A 1/4
?2D) 1/2 ?2A 1/2 ?2D.
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Distribution of variances among and within lines
under continuous selfing when p q 0.5 (after
Hallauer and Miranda, 1981 Table 211).  
Among lines Within lines
Total Generation F ?2A ?2D ?2A ?2D ?2A ?2D  F2,
S0 0 1 1 F23, S01 1/2 1 1/4 1/2 1/2 3/2 3
/4 F34, S12 3/4 3/2 3/16 1/4 1/4 7/4 7/16 F45,
S23 7/8 7/4 7/64 1/8 1/8 15/8 15/64 F56,
S34 15/16 15/8 15/256 1/16 1/16 31/16 31/256 F??
1, S??1 1 2 0 0 0 2 0
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Impact of Inbreeding on Mean
At the completely inbred generation we
have  Genotype BB Bb bb Genotypi
c frequency 0.5 0 0.5 Genotypic value
24 24 16  Population mean
0.5 (24) 0.5 (16) 20 i.e., the population
mean has regressed to the midparent value.
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Impact of Inbreeding on Genetic Variance
Total population variance 0.5 (24)2 0.0
(24)2 0.5 (16)2 - (20)2 16
Recall the population variance in the F2 was 12
therefore the genetic variance has increased upon
inbreeding. It has not doubled though? Why not?
Because some of the initial genetic variance was
due to dominance and it has disappeared as
heterozygote frequency diminished.
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Other Effects of Inbreeding 

• Inbred lines, exhibit a greater sensitivity to
  environmental sources of variation than noninbred
  lines. This can interfere with experimental
  studies on changes in variation upon inbreeding.
  This may vary over successive inbreeding
  generations. 
•  

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Random mate the derived lines in all
combinations. Provided there have been no changes
in allele frequencies during inbreeding, variance
among the random F1 hybrids will be equal to the
variance in the original base population (F2 or
S0). There will be no hybrid in the resulting
population that could not have been found in an
infinitely large base population.
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Can you Calculate the Inbreeding Coefficient Of
the Famous Shorthorn Bull Roan Gauntlet?
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