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Quantitative genetics

- Many traits that are important in agriculture,

biology and biomedicine are continuous in their

phenotypes. For example, - Crop Yield
- Stemwood Volume
- Plant Disease Resistances
- Body Weight in Animals
- Fat Content of Meat
- Time to First Flower
- IQ
- Blood Pressure

The following image demonstrates the variation

for flower diameter, number of flower parts and

the color of the flower Gaillaridia pilchella

(McClean 1997). Each trait is controlled by a

number of genes each interacting with each other

and an array of environmental factors.

- Number of Genes Number of Genotypes
- 1 3
- 2 9
- 5 243
- 10 59,049

Consider two genes, A with two alleles A and a,

and B with two alleles B and b.- Each of the

alleles will be assigned metric values- We give

the A allele 4 units and the a allele 2 units-

At the other locus, the B allele will be given 2

units and the b allele 1 unit

- Genotype Ratio Metric value
- AABB 1 12
- AABb 2 11
- AAbb 1 10
- AaBB 2 10
- AaBb 4 9
- Aabb 2 8
- aaBB 1 8
- aaBb 2 7
- aabb 1 6

A grapical format is used to present the above

results

Normal distribution of a quantitative trait may

be due to

- Many genes
- Environmental effects
- The traditional view polygenes each with small

effect and being sensitive to environments - The new view A few major gene and many

polygenes (oligogenic control), interacting with

environments

Traditional quantitative genetics research

Variance component partitioning

- The phenotypic variance of a quantitative trait

can be partitioned into genetic and environmental

variance components. - To understand the inheritance of the trait, we

need to estimate the relative contribution of

these two components. - We define the proportion of the genetic variance

to the total phenotypic variance as the

heritability (H2). - - If H2 1.0, then the trait is 100 controlled

by genetics - - If H2 0, then the trait is purely affected

by environmental factors.

- Fisher (1918) proposed a theory for partitioning

genetic variance into additive, dominant and

epistatic components - Cockerham (1954) explained these genetic variance

components in terms of experimental variances

(from ANOVA), which makes it possible to estimate

additive and dominant components (but not the

epistatic component) - I proposed a clonal design to estimate additive,

dominant and part-of-epistatic variance

components - Wu, R., 1996 Detecting epistatic genetic

variance with a clonally replicated design

Models for low- vs. high-order nonallelic

interaction. Theoretical and Applied Genetics 93

102-109.

Genetic Parameters Means and (Co)variances

- One-gene model
- Genotype aa Aa AA
- Genotypic value G0 G1 G2
- Net genotypic value -a

0 d

a -

origin(G0G1)/2 - a additive genotypic value
- d dominant genotypic value
- Environmental deviation E0 E1 E2
- Phenotype or
- Phenotypic value Y0G0E0 Y1G1E1 Y2G2E2
- Genotype frequency P0 P1 P2
- at HWE q2 2pq p2
- Deviation from population mean ? -a - ? d -

? a - ? - -2pa(q-p)d (q-p)a(q-p)d

2qa(q-p)d

- Population mean ? q2(-a) 2pqd p2a

(p-q)a2pqd - Genetic variance ?2g q2(-2p?-2p2d)2

2pq(q-p)?2pqd2 p2(2q?-2q2d)2 - 2pq?2 (2pqd)2
- ?2a (or VA) ?2d (or VD)

- Additive genetic variance, Dominant genetic

variance, - depending on both on a and d depending only on

d - Phenotypic variance ?2P q2Y02 2pqY12 p2Y22

(q2Y0 2pqY1 p2Y2)2 - Define
- H2 ?2g /?2P as the broad-sense heritability
- h2 ?2a / ?2P as the narrow-sense heritability
- These two heritabilities are important in

understanding the relative contribution of

genetic and environmental factors to the overall

phenotypic variance.

What is ? a(q-p)d?

- It is the average effect due to the substitution

of gene from one allele (A say) to the other (a). - Event A a contains two possibilities
- From Aa to aa From AA to Aa
- Frequency q p
- Value change d-(-a) a-d
- ? qd-(-a)p(a-d)
- a(q-p)d

Midparent-offspring correlation

- __________________________________________________

__________________ - Progeny
- Genotype Freq. of Midparent AA Aa aa Mean

value - of parents matings value a d -a of progeny
- __________________________________________________

__________________ - AA AA p4 a 1 - - a
- AA Aa 4p3q ½(ad) ½ ½ - ½(ad)
- AA aa 2p2q2 0 - 1 - d
- Aa Aa 4p2q2 d ¼ ½ ¼ ½d
- Aa aa 4pq3 ½(-ad) - ½ ½ ½(-ad)
- aa aa q4 -a - - 1 -a
- ________________________________________________

- Covariance between midparent and offspring
- Cov(OP)
- E(OP) E(O)E(P)
- p4a a 4p3q ½(ad) ½(ad) q4 (-a)(-a)

(p-q)a2pqd2 - pq?2
- ½?2a
- The regression of offspring on midparent values

is - b Cov(OP)/?2(P)
- ½?2a / ½?2P
- ?2a /?2P
- h2
- where ?2(P)½?2P is the variance of midparent

value.

- IMPORTANT
- The regression of offspring on midparent values

can be used to measure the heritability! - This is a fundamental contribution by R. A.

Fisher.

You can derive other relationships

- Degree of relationship Covariance
- __________________________________________________

__ - Offspring and one parent Cov(OP) ?2a/2
- Half siblings Cov(FS) ?2a/4
- Full siblings Cov(FS) ?2a/2 ?2a/4
- Monozygotic twins Cov(MT) ?2a ?2d
- Nephew and uncle Cov(NU) ?2a/4
- First cousins Cov(FC) ?2a /8
- Double first cousins Cov(DFC) ?2a/4 ?2d/16
- Offspring and midparent Cov(O) ?2a/2
- __________________________________________________

__

Cockerhams experimental and mating designs

- By estimating the covariances between relatives,

we can estimate the additive (or mixed additive

and dominant) variance and, therefore, the

heritability. - Next, I will introduce mating and experimental

designs used to estimate the covariances between

relatives.

Mating design

- Mating design is used to generate genetic

pedigrees, genetic information and materials that

can be used in a breeding program - Mating design provides genetic materials, whereas

experimental design is utilized to obtain and

analyze the data from these materials

Objectives of mating designs

- Provide information for evaluating parents
- 2) Provide estimates of genetic parameters
- 3) Provide estimates of genetic gains
- 4) Provide a base population for selection

Commonly used mating designs

- 1) Open-pollinated
- 2) Polycross
- 3) Single-pair mating
- 4) Nested mating
- 5) Factorial mating tester design
- 6) Diallel mating (full, half, partial

disconnected)

Nested mating (NC Design I)

- Each of male parents is mated to a subset of

different female parents

- Cov(HSM)1/4VA
- V(female/male) Cov(FS) Cov(HSM)
- 1/2VA1/4VD 1/4VA
- 1/4VA 1/4VD
- - Provide information for parents and full-sib

families - - Provide estimates of both additive and

dominance effects - - Provide estimates of genetic gains from both

VA and VD - - Not efficient for selection
- - Low cost for controlled mating

Example Date structure for NC Design I

- Sample Male Female Full-sib family Individual Phen

otype - 1 1 A 1 1 y1A1
- 2 1 A 1 2 y1A2
- 3 1 B 2 1 y1B1
- 4 1 B 2 2 y1B2
- 5 1 C 3 1 y1C2
- 6 1 C 3 2 y1C2
- 7 2 D 4 1 y2D1
- 8 2 D 4 2 y2D2
- 9 2 E 5 1 y2E1
- 10 2 E 5 2 y2E2
- 11 2 F 6 1 y2F1
- 12 2 F 6 2 y2F2
- 13 3 G 7 1 y3G1
- 14 3 G 7 2 y3G2
- 15 3 H 8 1 y3H1
- 16 3 H 8 2 y3H2
- 17 3 I 9 1 y3I1
- 18 3 I 9 2 y3I2

Estimates by statistical software

- VTotal 40
- VFS Cov(FS) 10
- VM Cov(HSM) 4
- VE VTotal VFS 40 10 30
- V(female/male) Cov(FS) Cov(HSM)
- 10 4 6
- VA 4Cov(HSM) 4 4 16 h2 16/40

0.x - V(female/male) 1/4VA 1/4VD 4 1/4VD 6
- VD 8, VG VA VD 16 6 22
- H2 22/40 0.x

Factorial mating (NC Design II)

- Each member of a group of males is mated to each

member of group of females

- Cov(HSM) 1/4 VA
- Cov(HSF) 1/4 VA
- V(female ? male) Cov(FS)Cov(HSM)Cov(HSF)
- 1/4 VD
- - Provide good information for parents and

full-sib families - - Provide estimates of both additive and

dominance effects - - Provide estimates of genetic gains from both

VA and VD - - Limited selection intensity
- - High cost

Tester mating design (Factorial)

- Each parent in a population is mated to each

member of the testers that are chosen for a

particular reason

- Cov(HSM)1/4VA
- Cov(HSF)1/4VA
- V(female ? male) Cov(FS)COV(HSM)-COV(HSF)
- 1/4VD
- - Provide good information for parents and

full-sib families - - Provide estimates of both additive and

dominance effects - - Provide estimates of genetic gains from both

VA and VD - - Limited selection intensity
- - High cost

Diallel mating design

- Full diallel each parent is mated with every

other parent in the population, including selfs

and reciprocal

- Half diallel each parent is mated with every

other parent in the population, excluding selfs

and reciprocal

- Partial Diallel selected subsets of full

diallels

- Disconnected half diallel selected subsets of

full diallels

- Diallel analysis
- Cov(HS) 1/4VA
- Cov(FS) 1/2VA 1/4VD
- Cov(FS) Cov(FS) 2Cov(HS) 1/4VD
- - Provide good evaluation of parents and

full-sib families - - Provide estimates of both additive and

dominance effects - - Provide estimates of genetic gains from both

VA and VD - - High cost

Genomic Imprinting or parent-of-origin effectThe

same allele is expressed differently, depending

on its parental origin

- Consider a gene A with two alleles A (in a

frequency p) and a (in a frequency q) - Genotype Frequency Value
- AA p2 a Average effect
- Aa pq di No imprinting ? a

d(q-p) - aA qp d-i Imprinting ?M a

i d(q-p) A ? a - aa q2 -a ?P a i d(q-p)

A ? a - Mean a(p-q)2pqd
- No imprinting ?g2 2pq?2 (2pqd)2
- Imprinting ?gi2 2pq?2 (2pqd)2 2pqi2
- Imprinting leads to increased genetic variance

for a quantitative trait and, therefore, is

evolutionarily favorable.

Genomic Imprinting

The callipygous animals 1 and 3 compared to

normal animals 2 and 4 (Cockett et al. Science

273 236-238, 1996)

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Predicting Response to Selection

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Population Mean, Xp - phenotypic mean of the

animals or plants of interest and expressed in

measurable units. Selection Mean, Xs - phenotypic

mean of those animals or plants chosen to be

parents for the next generation and expressed in

measurable units. Selection Differential, SD -

difference between the phenotypic means of the

entire population and its selected mean.

Genetic Gain the amount that the phenotypic

mean in the next generation change by selection.

- that change can be or -

Selection Differential

G h2 SD

How to Calculate Genetic Gain

M2 M h2 (M1 - M) M2 resulting mean

phenotype M mean of parental population M1

mean of selected population h2 heritability of

the trait ? M2 - M h2 (M1

- M) ? G h2 SD (SD/?p)h2?p ih2?p i

selection intensity h2 narrow-sense

heritability ?p standard phenotypic deviation

- Factors that influence
- the Genetic Gain
- Magnitude of selection differential
- Selection intensity
- Broad-sense heritability heritability
- Phenotypic variation

Knowing the Selection Differential, and the

response to selection, an estimate of the traits

heritability can be calculated G / SD Realized

Heritability

Realized heritability can also be calculated

as M2 M h2 (M1 - M) rearranged,

(M2 - M) (M1 - M)

h2

- Maximizing Genetic Gain
- Examples

N48, Population Mean 109.7

Goal Improve the Mean Select those in red, N

6, Mean of Selected 119.5 SD 9.8 G h2 SD

0.7 x 9.8 6.86

Goal Reduce the Mean Select those in blue, N

8, Mean of Selected 100.4

Nature 432, 630 - 635 (02 December 2004)The

role of barren stalk1 in the architecture of maize

- ANDREA GALLAVOTTI1,2, QIONG ZHAO3,

JUNKO KYOZUKA4, ROBERT B. MEELEY5,

MATTHEW K. RITTER1,, JOHN F. DOEBLEY3,

M. ENRICO PÈ2 ROBERT J. SCHMIDT1 - 1 Section of Cell and Developmental Biology,

University of California, San Diego, La Jolla,

California 92093-0116, USA2 Dipartimento di

Scienze Biomolecolari e Biotecnologie, Università

degli Studi di Milano, 20133 Milan,

Italy3 Laboratory of Genetics, University of

Wisconsin, Madison, Wisconsin 53706,

USA4 Graduate School of Agriculture and Life

Science, The University of Tokyo, Tokyo 113-8657,

Japan5 Crop Genetics Research, Pioneer-A DuPont

Company, Johnston, Iowa 50131, USA Present

address Biological Sciences Department,

California Polytechnic State University, San Luis

Obispo, California 93407, USA

Mapping Quantitative Trait Loci (QTL) in the F2

hybrids between maize and teosinte

Maize Teosinte tb-1/tb-1 mutant maize

Effects of ba1 mutations on maize development

Mutant Wild type No tassel

Tassel

Data format for a backcross

- Sample Height Marker 1 Marker 2 QTL
- (cm, y)
- 1 184 Mm (1) Nn (1) ?
- 2 185 Mm (1) Nn (1) ?
- 3 180 Mm (1) Nn (1) ?
- 4 182 Mm (1) nn (0) ?
- 5 167 mm (0) nn (0) ?
- 6 169 mm (0) nn (0) ?
- 7 165 mm (0) nn (0) ?
- 8 166 mm (0) Nn (1) ?

- Heights classified by markers (say marker 1)
- Marker Sample Sample Sample
- group size mean variance
- Mm n1 4 m1182.75 s21
- mm n0 4 m0166.75 s20

The hypothesis for the association between the

marker and QTL

- H0 m1 m0
- H1 m1 ? m0
- Calculate the test statistic
- t (m1m0)/?s2(1/n11/n0),
- where s2 (n1-1)s21(n0-1)s20/(n1n02)
- Compare t with the critical value

tdfn1n2-2(0.05) from the t-table. - If t gt tdfn1n2-2(0.05), we reject H0 at the

significance level 0.05 ? there is a QTL - If t lt tdfn1n2-2(0.05), we accept H0 at the

significance level 0.05 ? there is no QTL

Why can the t-test probe a QTL?

- Assume a backcross with two genes, one marker

(alleles M and m) and one QTL (allele Q and q). - These two genes are linked with the recombination

fraction of r. - MmQq Mmqq mmQq mmqq
- Frequency (1-r)/2 r/2 r/2 (1-r)/2
- Mean effect ma m ma m
- Mean of marker genotype Mm
- m1 (1-r)/2 (ma) r/2 m m (1-r)a
- Mean of marker genotype mm
- m0 r/2 (ma) (1-r)/2 m m ra
- The difference
- m1 m0 m (1-r)a m ra (1-2r)a

- The difference of marker genotypes can reflect

the size of the QTL, - This reflection is confounded by the

recombination fraction - Based on the t-test, we cannot distinguish

between the two cases, - - Large QTL genetic effect but loose linkage with

the marker - - Small QTL effect but tight linkage with the

marker

Example marker analysis for body weight in a

backcross of mice

- __________________________________________________

___________________ - Marker class 1 Marker class 0
- ______________________ _____________________
- Marker n1 m1 s21 n1 m1 s21 t P value
- __________________________________________________

___________________________ - 1 Hmg1-rs13 41 54.20 111.81 62 47.32 63.67 3.754

lt0.01 - 2 DXMit57 42 55.21 104.12 61 46.51 56.12 4.99

lt0.01 - 3 Rps17-rs11 43 55.30 101.98 60 46.30 54.38 5.231

lt0.000001 - __________________________________________________

___________________

Marker analysis for the F2

- In the F2 there are three marker genotypes, MM,

Mm and mm, which allow for the test of additive

and dominant genetic effects. - Genotype Mean Variance
- MM m2 s22
- Mm m1 s21
- mm m0 s20

Testing for the additive effect

- H0 m2 m0
- H1 m2 ? m0
- t1 (m2m0)/?s2(1/n21/n0),
- where s2 (n2-1)s22(n0-1)s20/(n1n02)
- Compare it with tdfn2n0-2(0.05)

Testing for the dominant effect

- H0 m1 (m2 m0)/2
- H1 m1 ? (m2 m0)/2
- t2 m1(m2 m0)/2/?s21/n11/(4n2)1/(4n0)

, - where s2 (n2-1)s22(n1-1)s21(n0-1)s20/(n2n1

n03) - Compare it with tdfn2n1n0-3(0.05)

Example Marker analysis in an F2 of maize

- __________________________________________________

____________________________________________ - Marker class 2 Marker class 1 Marker class

0 Additive Dominant - ____________ ______________ ______________
- M n2 m2 s22 n1 m1 s21 n0 m0

s20 t1 P t2 P - __________________________________________________

_____________________________________________ - 43 5.24 2.44 86 4.27 2.93 42

3.11 2.76 6.10 lt0.001 0.38 0.70 - 2 48 4.82 3.15 89 4.17 3.26 34

3.54 2.84 3.28 0.001

-0.05 0.96 - 3 42 5.01 3.23 92 4.14 3.18 37

3.57 2.68 3.71 0.0002

-0.57 0.57 - __________________________________________________

_____________________________________________

Testing for the dominant effect

- H0 m1 (m2 m0)/2
- H1 m1 ? (m2 m0)/2
- t2 m1(m2 m0)/2/?s21/n11/(4n2)1/(4n0)

, - where s2 (n2-1)s22(n1-1)s21(n0-1)s20/(n2n1

n03) - Compare it with tdfn2n1n0-3(0.05)

Example Marker analysis in an F2 of maize

- __________________________________________________

____________________________________________ - Marker class 2 Marker class 1 Marker class

0 Additive Dominant - ____________ ______________ ______________
- M n2 m2 s22 n1 m1 s21 n0 m0

s20 t1 P t2 P - __________________________________________________

_____________________________________________ - 43 5.24 2.44 86 4.27 2.93 42

3.11 2.76 6.10 lt0.001 0.38 0.70 - 2 48 4.82 3.15 89 4.17 3.26 34

3.54 2.84 3.28 0.001

-0.05 0.96 - 3 42 5.01 3.23 92 4.14 3.18 37

3.57 2.68 3.71 0.0002

-0.57 0.57 - __________________________________________________

_____________________________________________

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