Characteristics of a population

1
Characteristics of a population 

• Genotype frequency--the relative proportion of
  different genotypes in a population with respect
  to a given locus
•  

2
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 n5, there are 243
  genotypes).

3
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

4
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.

5
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
6
Estimating Gene Frequencies

• Calculate directly from the number of genes
• 130 A alleles/200 0.65
• 70 a alleles/200 0.35

7
Estimating Gene Frequencies

• 2. Estimate from genotype frequencies
•  
• p1(A) .5 .5(.3)0.65
•  
• p2(a) .2 .5(.3)0.35
• or p21- 0.65 0.35, since p1 p21.0
•  

8
Factors affecting gene frequency

• 1. Population size
• 2. Migration
• 3. Mutation
• 4. Selection
• 5. Mating system--progeny genotypes constituted
  by union of gametes in parental generation

9
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 

10
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

11
HW Frequencies
12
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
13
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.

14
Hardy-Weinberg Equilibrium

• 3. In the absence of mutation, migration,
  selection, and drift, this equilibrium will
  persist.

15
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 farmers field of maize
• The other classic example of an HW pop is the F2
  generation of a self pollinated crop like wheat
  or soybean

16
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

17
The F2 Generation has Hardy-Weinberg Frequencies

• This is harder to grasp, but look at the
  genotypes and their frequencies in the F2

The frequency of the A and the a alleles is
0.5 The genotype frequencies in the F2 are
exactly HW
AA Aa aa
0.25 0.50 0.25

18
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

19
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
20
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 Fxy
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
F24, 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.
21
Segregation at the A_Locus Under Self Pollination
Parents AA aa
F1 Aa
F2 0.25 AA 0.5 Aa 0.25 aa
F3 AA AA AA Aa AA Aa aa aa
AA AA aA aa aA aa aa aa
Sorted by genotype 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.
22
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!

23
Development of F2 Derived Lines Without Selection
F2 Bulk Harvest 1000 plants 0.25 AA 0.5 Aa 0.25 aa
F23 Lines 250 Lines 500 Lines 250 Lines
Homogeneous AA Like an F2 population Segregating .25 AA .5 Aa .25 aa Homogeneous aa
F2 4 Lines Homogeneous AA Like an F3 population Segregating .375 AA .25 Aa .375 aa Homogeneous aa
F25 Lines Homogeneous AA Like an F4 population Segregating .4375 AA .125 Aa .4375 aa Homogeneous aa
F26 Lines 250 Homogeneous AA Lines Like an F5 population Segregating .46875 AA .0625 Aa .46875 aa 250 Homogeneous aa Lines
Inbred AA AA and aa aa
24
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 438 Homogeneous AA 125 Segregating Aa Lines 438 Homogeneous aa Lines
F45 Lines 438 Homogeneous AA Lines Like an F2 population 125 Segregating .25 AA .50 Aa .25 aa Homogeneous aa
F46 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
25
F26 vs F46 What are the practical
consequences?

• Note that in the F46 lines 876 of 1000 lines are
  descendents of homozygous F2 plants
• Contrast that with F26 lines, where only 500 of
  1000 lines are descendents of homozygous F2 plants

26
F26 vs F46 What are the practical
consequences?

• When you walk the F26 lines, you find much more
  heterogeneity within a line
• Concern will performance be repeatable?

27
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.

28
Purifying lines for Cultivar Release

• The first generation of yield testing may be F46
  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

29
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.

30
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.

31
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

32
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

33
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 gt 500 F2 populations and
  a subset at Princeton

34
Larger Populations or More Populations?

• I dont 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

35
Larger Populations or More Populations?

• Current issue are the pop sizes we are using for
  marker assisted selection big enough??

36
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

37
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

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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)

40
How to Predict Success

• The Binomial Probability Formula 
• P(x k) (nk)pk(1 - p)n-k 
• the binomial coefficient which estimates the
  number of ways in which k successes can be chosen
  from among n trials

41
How to Predict Success

• Consider the likelihood of recovering homozygotes
  for the superior allele at 12 of 20 loci
• F2 p
• F10 p

42
Linkage

• If favorable alleles are linked in coupling (eg
  AB), it works in the breeders favor
• Repulsion linkages of favorable alleles (Ab)
  require recombination to break up
• Fehr (p 56) gives example of nematode resistance
  and seed color (r0.0035)

43
Recombination More is Better, Isnt 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

44
Maize Linkage Blocks