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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!

Agro 643 - Genetic Models for Means

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If only it were so simple

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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.


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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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Genotyp

e

Frequenc

y

Number

of B

Genotypi

c Value

Coded

Gen.

Value

BB p2 2 z + 2a a


bb q2 0 z -a

BB

bb

Bb 110 bu/a

100 bu/a

Gene action Value


d = 0



Overdominance d > a

a

-a

Complete Dominance Model


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Genotyp

e

Frequenc

y

Number

of B

Genotypi

c Value

Coded

Gen.

Value

BB p2 2 z + 2a a


bb q2 0 z -a

BB

bb

Bb

110 bu/a

107.5 bu/a

100 bu/a

Gene action Value


d = 0



Overdominance d > a

d = a

a

-a

Partial Dominance Model


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Genotyp

e

Frequenc

y

Number

of B

Genotypi

c Value

Coded

Gen.

Value

BB p2 2 z + 2a a


bb q2 0 z -a

BB

bb

Bb

110 bu/a

115 bu/a

100 bu/a

Gene action Value


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


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


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


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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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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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


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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


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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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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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?


Domestication and the Domestication Bottleneck

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Domestication and the Domestication Bottleneck


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

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