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What Type of Variation Cause the Diversity We
Breed for in Crops?
.Quantitative Variation!
http://www.ars.usda.gov/images/docs/6652_6836/tomato%20colors.jpg
http://agronomyday.cropsci.illinois.edu/2003/exhibits/peregrine-illo---seeds.gif
https://reader009.{domain}/reader009/html5/0408/5ac9a2e2ab983/5ac9a2e439985.jpg
Qualitative variation (mutants) are rarely useful
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Agro 643 - Review: statistics concepts
Quantitative Genetics = (Genetics + Phenotype + Statistics) for a population
MolecularQuantitative Genetics =
(Genetics + Phenotype + Genotype + Statistics) for a population
Basic Probabil ity and Statist ics Concepts Review Terms
Binary ~ two state distribution [1,0]; [black, white], etc. Qualitative random variable ~ finite and small number of possible outcomes
(usually binary) Quanti tative random variable ~ any number of possible outcomes
Discrete distribut ion ~ observations on a quantitative random variable canonly assume countable (whole) number values. Continuous distribution ~ observations on a quantitative random variable can
assume any of the uncountable number values in a line interval. Mean (arithmetic) ~ the sum of measurements divided by the total numberof measurementsVariance ~ ~ the spread of observations around the mean
where there are n measurements y1, y1, ynwith arithmetic mean1
)(
n
yyi
y
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Genetic Models
Additive model for height
Mean Base Height = 50cm Mean Allele a value = 0cm Mean Allele A value = 5cm
Individual Additive
Height
aa 50cm
aA or Aa 55cm
AA 60cm
Agro 643 - Review: genetics and statistics concepts
AA = 60cmAa = 55cmAa = 50cm
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Quanti tative Genetic Models for Means and Variances
What is a genetic model?
Concrete, mathematical way to discuss the variation in apopulation
Why do we use these mathematical genetic models?Teaching tools, understanding is needed for later concepts Calculate gain from selection Useful for people who are designing new breeding methods andanalyses Glossy Papers (Hallaur, 2006 Cornell University)
Even though we talk about allele frequencies it is abstract and based onphenotypic information not molecular markers though it can and is appliedto markers too!
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Agro 643 - Review: statistics concepts
If only it were so simple
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Agro 643 - Review: statistics concepts
Normal Distribution
First introduced by French mathematician A. DeMoivre in 1733 who called it the exponentialbell shaped curve. German mathematician K.F. Gauss made it famous so it is called aGaussian distribution. Because we believe (often incorrectly to simplify things) that it isfound everywhere (thanks Central Limit Theorem) we now call it a normal distribution.
),(~2
NX
60cm55cm50cm
Could be caused by:
Other genesEnvironmental effectsGenetic by environmental
interaction (G x E)Random error
- disease- drought- soil differences- other biotic / abiotic- random chance
Num
berofindividuals
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Basic Probability and Statistics Concepts Review
Central Limit Theorem ~
1.
2.
3. When n is large, the sampling distribution of Y will be approximatelynormal, with approximation becoming more precise as n increases.
4. When the population distribution is normal, the sampling distribution of Y isexactly normal for any sample size n
Where Y can equal either the mean:or the sum of all y1, y1, yn observations: i yn
And y is the mean of the sample
And y is the standard deviation of the sample (the standard error)
uy=
ny / =
y
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Basic Genetics and Statistics Concepts Review
Genetic Models 100 F2 individuals measured for height. m genes at 50%frequency have value of 40cm/ m ... our expectation
One Additive Gene Two Additive Genes Three Additive Genes
Four Additive Genes Five Additive Genes Six Additive Genes
Height
Frequency
50 60 70 80 90
0
50000
100000
1
50000
200000
Height
Frequency
50 60 70 80 90
0
50000
100000
150000
200000
Height
Frequency
50 60 70 80 90
0e+00
1e+05
2e+05
3e+05
4e+05
5
Height
Frequency
50 60 70 80 90
0e+00
1e+05
2e+05
3e+05
Height
Frequency
50 60 70 80 90
0
50000
100000
150000
200000
250000
30
Height
Frequency
50 60 70 80 90
0
50000
100000
150000
200000
250000
1 1
2
1 1
64 4
1 1
1 11 1 1 1
7056 56
29 29
20 1515
6 6
8 810 10
45 45
120 120
210 210
252924
792 792
495 495
220220
66 6612 12
R: #Genetic Ratios Based on Calculation
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Basic Genetics and Statistics Concepts Review
Central Limit Therom As the number of independent random variables (genesinvolved in a phenotype) approaches infinity, the sum of these approachesnormality
Height
Density
55 60 65 70 75 80 85
0.0
0
0.0
2
0.0
4
0.0
6
0.0
8
Ten Additive Genes
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Height
Frequency
60 65 70 75 80
0
5
10
15
20
25
Genetic Models 100 individuals (n) measured for height. m genes at 50%frequency have value of 40cm/ m ... Reality...
(one simulated draw of 100 individuals for each scenario)
One Additive Gene Two Additive Genes Three Additive Genes
Four Additive Genes Five Additive Genes Six Additive Genes
Height
Frequency
50 55 60 65 70 75 80
0
5
10
15
20
2
11
3
58
13
2123
25
Height
Frequen
cy
50 60 70 80 90
0
5
10
15
20
2
1 114
13
2125
13
21
Height
Frequency
50 60 70 80 90
0
5
10
15
20
25
30
11
7
14
31
2521
Height
Frequency
50 60 70 80 90
0
5
10
15
20
25
30
35
5 6
25
3529
Height
Frequency
50 60 70 80 90
0
10
20
30
40
50
27
52
21
2145
1112
2225
18
Agro 643 - Review: genetics and statistics concepts R: #Based on probability - Possible Sample Observations
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Genetic Models Something similar can also be observed if using limiteddraws using a normal distribution function.
Agro 643 - Review: genetics and statistics concepts
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Agro 643 - Phenotypic Quantitative Genetics - Review: statistics concepts
Basic Genetics Models
Individual Additive
Height
Dominance
Height
Overdominance
Height
aa 50cm 50cm 50cm
aA or Aa 55cm 55cm 55cm
AA 60cm 55cm 50cm
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Quanti tative Genetic Models for Means and Variances
ASSUMPTIONS
Hardy Weinberg Equilibrium (Hartl and Clark, 1997)- The organism is diploid.
- Reproduction is sexual.- Generations are non-overlapping.- The gene under consideration has two alleles.- The allele frequencies are identical in males and females.- Mating is random.- Population size is very large (in theory, infinite no genetic drift).- Migration is negligible.- Mutation can be ignored.- Natural selection does not affect the alleles under consideration.
Additionally:No selection (e.g. human based, or flowering time difference to pollen
competition)Single Locus- No Epistasis- No Linkage
Genetic Effects ONLY- No E or G*E
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Genetic models for means
Simple model ( one locus, two alleles)
Genotyp
e
Frequenc
y
Number
of B
Genotypi
c Value
Coded
Gen.
Value
BB p2 2 z + 2a a
Bb 2pq 1 z + a + d d
bb q2 0 z -a
BB
bb
Bb
110 bu/a
105 bu/a
100 bu/a
Gene action Value
Additive (nodominance)
d = 0
Complete dominance d = a
Partial dominance a > d > 0
Overdominance d > a
a
-a
Additive Model
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Genetic models for means
Simple model ( one locus, two alleles)
Genotyp
e
Frequenc
y
Number
of B
Genotypi
c Value
Coded
Gen.
Value
BB p2 2 z + 2a a
Bb 2pq 1 z + a + d d
bb q2 0 z -a
BB
bb
Bb 110 bu/a
100 bu/a
Gene action Value
Additive (nodominance)
d = 0
Complete dominance d = a
Partial dominance a > d > 0
Overdominance d > a
a
-a
Complete Dominance Model
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Genetic models for means
Simple model ( one locus, two alleles)
Genotyp
e
Frequenc
y
Number
of B
Genotypi
c Value
Coded
Gen.
Value
BB p2 2 z + 2a a
Bb 2pq 1 z + a + d d
bb q2 0 z -a
BB
bb
Bb
110 bu/a
107.5 bu/a
100 bu/a
Gene action Value
Additive (nodominance)
d = 0
Complete dominance d = a
Partial dominance a > d > 0
Overdominance d > a
d = a
a
-a
Partial Dominance Model
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Genetic models for means
Simple model ( one locus, two alleles)
Genotyp
e
Frequenc
y
Number
of B
Genotypi
c Value
Coded
Gen.
Value
BB p2 2 z + 2a a
Bb 2pq 1 z + a + d d
bb q2 0 z -a
BB
bb
Bb
110 bu/a
115 bu/a
100 bu/a
Gene action Value
Additive (nodominance)
d = 0
Complete dominance d = a
Partial dominance a > d > 0
Overdominance d > a
d = 2a
a
-a
Overdominance Model
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Genotyp
e
Frequenc
y
Number
of B
Genotypi
c Value
Coded
Gen.
Value
BB p2
2 z + 2a aBb 2pq 1 z + a + d d
bb q2 0 z -a
Extended to a population the mean of the population reflects the proportional
value of its individuals. Thus, it depends on both allele frequency and level ofdominance.
aqpqdapX 22 2 +=Population
Mean =
p (B) q (b) a d
0.5 0.5 2 2 1
0.5 0.5 2 1 0.5
0.7 0.3 2 2 1.64
0.7 0.3 2 1 1.22
0.3 0.7 2 2 0.44
0.3 0.7 2 1 -0.38
X
pqdq)a(pX 2+=
Which reduces to:pqd)aq(pX 222 +=
Which reduces to:
Population Genetic Mean
Agro 643 - Genetic Models for Means
R:
#Calculatepopulation
means
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An alleles average effectis dependent on population allele frequency and alleleeffect.
Additive (no dominance) - d=0, a=2
Complete dominance - d=2, a=2Partial dominance - d=1, a=2
Overdominance - d=2, a=1
(*Note a value is shown lower to f it the same
scale)
Genetic Models for Means
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Breeding value uses the mean value of progeny to calculate an individualsvalue. Unlike average effect it can be measured directly in a diploid population.Breeding value is additive genetic variation
Theoretically, breeding value is the sum of the average effects of the individualsgametes.
The reason to go through average effects to determine breeding value is so thatyou can see that the breeding value of an individual is directly connected to
frequency and effects of the alleles in the population.Genotype Breeding Value from
average effects
BB q[a+d(q-p)] + q[a+d(q-p)]=2(qa+q2d-pdq)
Bb q[a+d(q-p)] - p[a+d(q-p)]=qa+q2d-pdq - pa-pdq+dp2=q2d+dp2-2dpq +qa -pa
bb -p[a+d(q-p)] - p[a+d(q-p)]=2(- pa-pdq+dp2)
Breeding value is additive genetic variation
Agro 643 - Genetic Models for Means
Alternative explanat ion:
Basically this shows mathematicallythat if all the individuals are very goodthen the difference between the bestindividual and the population is small.
However if the population mostly poorwith a few very good individuals thenthe breeding value will be very highon the elite individuals.
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Dominance deviation is the difference between the genotypic value (what weobserve) and the breeding value (which we must calculate)
Dominance deviation = Genotypic value Breeding Value
d = G aOr G = a + dGenotypic value = Breeding Value + Dominance deviation
Genotype Genotypic (G)
value
G - population
mean
G - population
mean ( insert
= a + d (q-p)
BB a = a - a(p-q) +2dpq= 2q(a-pd)
= 2q(a-pd)
= 2q(-qd)
Bb d = d - a(p-q) +2dpq
= a(q-p) + d(1-2pq)
= a(q-p) + d(1-
2pq)= (q-p) + 2pqd
bb -a = -a - a(p-q)+2dpq
= 2p(a+qd)
= 2p(a+qd)
= 2p(+pd)
The Dominance Deviation is Caused by Dominance Effects
Agro 643 - Genetic Models for Means
Has Dominance Heterosis Increased in Maize?
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Has Dominance Heterosis Increased in Maize?
Few studies have looked at if heterosis has increased in maize over time.
The one study I am aware of:DUVICK, D. N., 1999 Heterosis: feeding people and protecting natural resources, pp. 1929 inTheGenetics and Exploitation of Heterosis in Crops, edited by J . G. COORS and S. PANDEY. ASA-CSSA-SSSA Societies, Madison, WI
Shows:
Agro 643 Heterosis in maize
Single Cross Yield
Mid Parent Value
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Genetic Models for Means Summary
- Dominance causes differences between genotypic values and breeding values
When intermating in a diverse HWE population:
- Recessive homozygotes (-/-) give progenies that appear much better thanthemselves with most progeny (+/-)
- Heterozygotes (+/-) look better than their progeny as they produce (-/-)
-Dominant homozygotes (+/+) give progenies that can appear slightly better than
themselves with (+/+) progeny and (+/-).
- Slope of the regression line is the average effect of a gene substitution =12
- Breeding values and population mean are on the regression line
- Regression depends on gene frequency and effects
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For Inbreeding, A Way to Think About Genetic Means Across
Different Generations
P1 = m + a
P2 = m aF1 = m + dF2 = m + dFn = m +()n-1dBC11 = m + a + dBC
n1= m + [1-(1/2)n]a +()n d
BC12 = m - a + dBCn2= m - [1-(1/2)
n]a +()n d
P1
P2
d
-a
a
m
d/2
d/4d/8
F1
F2
F3F4
Agro 643 - Genetic Models for Variances
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Agro 643 - Genetic Models for Variances
Genetic Models for Variances
The expected population mean of the next generation indicates how muchvariance is in the population for breeding improvement through selection.
While the red and the blue populations below have the same mean and are thesame size, they have different variances for height.
If we select the top 10% of plants to produce a new generation (assumingadditive effects only).
= 5, = 10
R: Population Mean and
Variance Normal Distribu tion
G ti M d l f V i
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Agro 643 - Genetic Models for Variances
Genetic Models for VariancesThe expected population variance indicates how much variance is in thepopulation for breeding improvement (selection) . This is a function of allelefrequency as well as additive and dominant effects.
2222222
]2)[(2 pqdaqpaqpqdap ++=Population
Variance =
])21()(2[2 222 dpqadpqapq ++=
Thus, whenp=q= then our expected2 = a2 + d2
]))5.0)(5.0(21()5.05.0(2)[5.0)(5.0(2 222 dada ++=
])5.1()[5.0( 222 da +=
The total genetic variance can be broken down into additive and dominancedeviation 222
DAG +=
222224])([2 dqpdpqapqG ++=
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The Genetic Variance for All Models is a Function of Allele FrequencyThe expected population mean indicates how much variance is in thepopulation for breeding improvement (selection) .
G2
d2
=a2
=2
C
ompletedomina
nce
G2
d2 = 0
a2 = 2
Add
itive(nodomina
nce)
Extre
me
Overdom
inance
G2
d2
= 1000a
2 = 1
Overdom
inance
G2
d2 = 2
a2 = 1
Agro 643 - Genetic Models for Variances
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Sources of Genetic Variance in Different Inbred Generations
a2
Total
a2 Between
a2
Within
d2Total
d2 Between
d2 Within
F - inbreedingCoefficient
*Geneticvariance
between inbredfamiliesincrease andwithin familiesdecreases withinbreeding
*Total geneticvariancedoubles fromF=0 to F=1
*When F=1 allvariance isadditive
Agro 643 - Genetic Models for Variances
NOTE: Should
be discrete(points) not acontinuous linegraphonlyshown forclarity
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Genetic Models for Variances
Back to Reality Why does this not work for your population?- Random Mating ( Impossible)
- Epitasis effect ( Gets very complex )
- Genes Independent / no linkage ( Impossible)
- No selection ( Near impossible - no matter how hard you try)
- Environmental effects
- Population size
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Population Genetics: Genetic Drif t
This is much easier to observe using simulations
N = 10 N = 100
N = 1000
Ten simulations
each with p=.5
Agro 643 - Review: population genetics
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Agro 643 - Review: statistics concepts
What Do You Need to Perform Quanti tative Genetic Analysis?
1. A Control led Population ( or at least one you understand)
2. Genetic Diversi ty
3. Lots and Lots of Phenotypic Observations ( data points)
4. A Genetic Model5. Good Statistical Analysis
Correlation ~ measures strength of the linear relationship of X and Y
Usually reported as r =In writing people prefer Pearson correlation coefficient
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Agro 643 - Review: statistics concepts
Heritability - Parent Offspring Regression
Correlation ~ measures strength of the linear relationship of X and YUsually reported as r =In writing people prefer Pearson correlation coefficient
http://www.biology.duke.edu/rausher/heritability.JPGHeritability ~The amountof phenotypic variationattributable to genetics
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Citing the well-known correlation betweenobese dogs and their owners, Marc kept his
New-Years resolution to get fit
Causation can not be deduced from correlation!
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Agro 643 - Review: probability theory and statistics
What is Random?
What is Significant?Meeting an acquaintance in the backstreets of Venice?
iPod shuffle playing three J ay-Z songs in a row?
Height and flowering time being correlated?
Winning at slots?
H th i T ti
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Agro 643 - Phenotypic Quantitative Genetics - Review: statistics concepts
Hypothesis Testing
Nullhypothesisis
True
Nullhypothesis is
False
Reject theNull
Hypothesis
Type 1
Error!
Fail to Rejectthe Null
Hypothesis
Type 2
Error!
Type III error: provides the right answer to the wrong question(discrepancy between the research focus and the research question )
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All Breeders Need Genetic Diversity
What Does Genetic Diversity Mean to
You?
Agro 643 - Review: population genetics
Domestication and the Domestication Bottleneck
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Domestication and the Domestication Bottleneck
Agro 643 - Review: population genetics
From: DoebleyJ F, GautBS, Smith BD(2006) The molecular genetics of cropdomestication. Cell 127: 13091321
From: Tanksley, S.D., and S.R.McCouch. 1997. Seed banksand molecular maps:Unlocking genetic potentialfrom the wild. Science277:10631066.
From: Doebleyet al. 2006
From: Doebleyet al. 2006
Genetic Diversity Whatdoes it mean?
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Genetic Diversity What does it mean?
Diversity is a relative term:
In this class we will use genetic diversity when referring to many diversity levels.Similarly geneticists and breeders mean different things when talking about
diversity.Make sure it is clear at what level we are interested in!
Genetic diversity (GD) ofall wild and cultivatedwheat
GD of allcultivated wheat
GD in Elite TAMU
material
GD in KSUprogram
GD in TAMUprogram
GD in MSUprogram
GD in a specificbi-parentalderived population