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Characteristics of a population
• Genotype frequency--the relative
proportion of different genotypes in a
population with respect to a given locus
Genotype Frequencies Define the
Population
• For example, consider the A_ locus
• We can define a population in terms of the
frequencies of the AA, Aa, and aa
genotypes. (We often illustrate these
concepts with just one locus, since for 2
alleles per locus, there are 3n possible
genotypes--when n=5, there are 243
genotypes).
Gene Frequencies Also Define
the Population
• economy--there are many more genotypes
than genes (see previous slide)
• gene is the stable unit of inheritance-genes, rather than genotypes, are
transmitted from parents to progeny
Gene Frequencies
• a change in gene frequency is the goal of
selection. This is the goal of the breeder,
and one of the ways in which we measure
progress from selection.
Genotypic Array
Genotype
No.
Freq.
A
a
AA
50
.5
100
0
Aa
30
.3
30
30
aa
20
.2
0
40
100
1.0
130
70
Estimating Gene Frequencies
Calculate directly from the number of genes:
130 A alleles/200 =0.65
70 a alleles/200 = 0.35
Estimating Gene Frequencies
2. Estimate from genotype frequencies
p1(A)= .5 + .5(.3)=0.65
p2(a)= .2 + .5(.3)=0.35
or p2=1- 0.65 = 0.35, since p1+ p2=1.0
Factors affecting gene frequency
1.
2.
3.
4.
5.
Population size
Migration
Mutation
Selection
Mating system--progeny genotypes
constituted by union of gametes in parental
generation
Hardy-Weinberg
• In 1908 British mathematician Hardy and
German physician Weinberg showed
independently that:
• For any gene frequency, the following holds
true in a large random mating population in
the absence of factors which affect gene
frequency:
Hardy-Weinberg
• Given locus A_ in which the frequency of A
is p1 and the frequency of a is p2, after 1
cycle of random mating, the genotype
frequencies of the progeny will be:
• P12 AA : 2p1p2 Aa : p22 aa
HW Frequencies
Random Mating = Random
Union of Gametes
M
F
0.5
A
0.5
a
0.5
A
0.25
AA
0.25
Aa
0.5
a
0.25
Aa
0.25
aa
Hardy-Weinberg Equilibrium
1. Large RM population will reach an
equilibrium with respect to gene and
genotypic frequencies after only one
cycle of RM, regardless of gene freq. in
parents.
2. Genotype frequencies of progeny
determined solely by gene frequencies
of parents.
Hardy-Weinberg Equilibrium
3. In the absence of mutation, migration,
selection, and drift, this equilibrium will
persist.
HW: Does it Matter to the Plant
Breeder?
• All selection theory that is the underpinning
of plant breeding based on HW populations
• One classic example of an HW population
is the Aztec farmer’s field of maize
• The other classic example of an HW pop is
the F2 generation of a self pollinated crop
like wheat or soybean
HW- The OP Maize Field
• Each plant has the
opportunity to mate
with any other plant
• OP varieties were
commonly grown
before hybrids took
hold
• OP landraces are still
common in
developing countries
The F2 Generation has HardyWeinberg Frequencies
• This is harder to grasp,
but look at the
genotypes and their
frequencies in the F2
AA
0.25
Aa
0.50
aa
0.25
The frequency of the A
and the a alleles is 0.5
The genotype
frequencies in the F2
are exactly HW
Departure from HW
• Once we begin to inbreed (non random
mating) or select (change gene frequency)
we disrupt HW equilibrium
• But it is useful to know that each breeding
population starts off in HW equilibrium
Designation of Inbred Lines
The commonly used systems for describing the generations that
follow the mating of two parents are the “F” and “S” systems. F
refers to Filial, and is most commonly used with self pollinated
species and S refers to generations of selfing and is used with
cross pollinated species.
We will use the F system almost exclusively in this course, but you
should be aware of the equivalence of the two systems:
F system
S system
F2
S0
F3
S1
F4
S2
F5
S3
Designation of Inbred Lines
One of the conventions we will use in this course is the 2 tier
system described by Fehr (p. 31-32) in which a line is designated as
follows: Fx:y in which x is the generation of derivation and y is the
current generation.
Why is this important?
Because there is considerable difference between an F2-derived line
in the F6 and an F4-derived line in the F6, yet both could be classified
as ‘an F6 line’.
So if we write F2:4, we are referring to a line that is now in the F4
generation, which came about (was derived) when we harvested a
head in the F2 generation.
Segregation at the A_Locus
Under Self Pollination
Parents
AA
aa
F1
Aa
F2
0.25 AA
F3
AA AA
AA Aa
AA Aa
aa aa
AA AA
aA
aA
aa aa
Sorted by
genotype
0.5 Aa
aa
0.25 aa
aa
6/16 AA
2/16 Aa
2/16 aA
6/16 aa
Which
equals
0.375 AA
0.25 Aa
0.375 aa
F4
0.4375 AA
0.125 Aa
0.4375 aa
The frequency of heterozygotes is estimated by (1/2)G, where G is
the number of generations of selfing, i.e., F2 = (1/2)1 = 1/2; F3 = (1/2)2
= 1/4; F4 = (1/2)3 = 1/8.
F2 – F8 Generations are Critical
• If you understand the next two slides, you
will have mastered a critical component of
plant breeding
• So Pay Attention!
Development of F2 Derived Lines Without Selection
F2 Bulk
Harvest 1000
plants
F2:3 Lines
0.25 AA
250 Lines
Homogeneous
AA
F2: 4 Lines
0.5 Aa
Homogeneous
500 Lines
Like an F2
population
Homogeneous
250
Homogeneous
AA Lines
Inbred
AA
Homogeneous
aa
Segregating
.375 AA
.25 Aa
.375 aa
Homogeneous
aa
Like an F4
population
Segregating
.4375 AA
.125 Aa
.4375 aa
Homogeneous
aa
Like an F5
population
Segregating
.46875 AA
.0625 Aa
.46875 aa
250
Homogeneous
aa Lines
AA and aa
aa
AA
F2:6 Lines
250 Lines
Like an F3
population
AA
F2:5 Lines
Segregating
.25 AA
.5 Aa
.25 aa
0.25 aa
Development of F4 Derived Lines Without Selection
F2 Bulk
0.25 AA
0.5 Aa
0.25 aa
F3 Bulk
0.375 AA
0.25 Aa
0.375 aa
F4 Bulk
0.4375 AA
0.125 Aa
0.4375 aa
Harvest 1000
Individual
plants from F4
Bulk
125 Segregating
Aa Lines
438
Homogeneous
AA
438
Homogeneous
aa Lines
F4:5 Lines
438
Homogeneous
AA Lines
Like an F2
population
125 Segregating
.25 AA
.50 Aa
.25 aa
Homogeneous
aa
F4:6 Lines
438
Homogeneous
AA Lines
Like an F3
population
125 Segregating
.375 AA
. 25 Aa
.375 aa
438
Homogeneous
aa Lines
Inbred
AA
AA and aa
aa
F2:6 vs F4:6 : What are the
practical consequences?
• Note that in the F4:6 lines 876 of 1000 lines
are descendents of homozygous F2 plants
• Contrast that with F2:6 lines, where only 500
of 1000 lines are descendents of
homozygous F2 plants
F2:6 vs F4:6 : What are the
practical consequences?
• When you walk the F2:6 lines, you find
much more heterogeneity within a line
• Concern: will performance be repeatable?
Alleviate this Concern by
Delaying Selection
(1) Higher proportion of homozygotes in
later generations
(2) if one observes a superior line in later
generations, subsequent performance is
more predictable.
Purifying lines for Cultivar
Release
• The first generation of yield testing may be
F4:6 lines which will be somewhat variable
• If they make the cut for further testing, then
purification begins
• The line that is eventually released as a
cultivar may be F8 derived
Segregation at Multiple Loci
with Self-Pollination
• All F2 populations will be segregating
at more than one locus.
• This reduces the proportion of
completely homozygous individuals
compared to the case of single locus
segregation.
Segregation at Multiple Loci
with Self-Pollination
• If the probability
of homozygosity
at the A or B
locus is 0.5. The
probability of
homozygosity at
both loci is 0.5 x
0.5 = 0.25.
The Potential Number of
Genotypes is Huge
• Assume parents in a cross have
different alleles at n loci : 2n inbred
genotypes can be derived from the
cross.
• If parents differ at 20 loci, then 220
or 1,048,576 genotypes are possible
Number of Genotypes in the F2 is
even Bigger
• Consider though, that 320 different
genotypes are possible in the F2
generation--i.e., 3,486,784,401
• This fact makes inbreds seem
attractive
Larger Populations or More
Populations?
• One of the most vexing plant breeding
questions
• Our F2 populations typically contain 1800 2000 individuals; not 3,486,784,401!
• This year at Lex we have > 500 F2
populations and a subset at Princeton
Larger Populations or More
Populations?
• I don’t believe in growing enormous
populations in hopes of finding the perfect
plant
• Instead, we try to sample as many
populations as possible in hopes of finding a
few that meet our many selection criteria
Larger Populations or More
Populations?
• Current issue: are
the pop sizes we
are using for
marker assisted
selection big
enough??
What are Transgressive
Segregates?
• Progeny that exceed the high parent
• For example, assume a trait is governed by
20 loci
– Inbred Parent 1 contains the superior allele at
loci 1 to 11,
– Inbred Parent 2 contains the superior allele at
loci 12 to 20
What are Transgressive
Segregates?
• Plant breeding progress to date has been
built on incremental gains in recovering
progeny with additional superior alleles in
the homozygous state
How to Predict Success
• The Binomial Probability Formula:
•
P(x = k) = (nk)pk(1 - p)n-k
• Where n = number of loci controlling the trait
• k = number of loci homozygous for the superior
allele
• p= probability of fixing the superior allele in the
homozygous state in the selfing generation Fi
(i.e., 0.25 in F2; 0.375 in F3; 0.438 in F4)
How to Predict Success
• The Binomial Probability Formula:
•
P(x = k) = (nk)pk(1 - p)n-k
n 
n!
 
k  k!(n  k)!

the binomial coefficient which estimates the number
of ways in which k successes can be chosen
from among n trials
How to Predict Success
• Consider the likelihood of recovering
homozygotes for the superior allele at 12 of
20 loci
• F2: p = 12 0.2512 (0.75)8  0.0008
20 
• F10: p =


12 
12
 0.5 (0.5)8  .1210
20 
Linkage
• If favorable alleles are linked in coupling
(eg AB), it works in the breeder’s favor
• Repulsion linkages of favorable alleles (Ab)
require recombination to break up
• Fehr (p 56) gives example of nematode
resistance and seed color (r=0.0035)
Recombination – More is Better,
Isn’t It?
• Linkage is a conservative influence which
maintains parental arrangements
The Debate:
• Favorable linkage blocks should be
maintained vs
• More genetic variation should be unleashed
through intermating
Maize Linkage Blocks