PBG 650 Advanced Plant Breeding

Module 6: Quantitative Genetics– Environmental variance– Heritability– Covariance among relatives

More interactions

• Interlocus interactions are important, but difficult to quantify

• Many designs for genetic experiments lump dominance and epistatic interactions into one component called “non-additive” genetic variance

For an individualG = A + D + IP = A + D + I + E

2

E

2

I

2

D

2

A

2

P For a population

2

DD

2

AD

2

AA

2

I Two-locus interactions

More than two loci….

Genetic variances from a factorial model

Bernardo, Chapt. 5

Source of Variation Degrees of Freedom Variance Component

A locus main effect 2 2locusAD

2locusAA )()(

A linear 1 2locusAA )(

A quadratic (deviations) 1 2locusAD )(

B locus main effect 2 2locusBD

2locusBA )()(

B linear 1 2locusBA )(

B quadratic (deviations) 1 2locusBD )(

A x B interaction 4 2I

A linear x B linear 1 2AA

A linear x B quadratic 1 2

AD (pooled) A quadratic x B linear 1

A quadratic x B quadratic 1 2DD

Environmental variance

GE

2

E

2

G

2

P Cov2

P = G + E• covariance would occur if better

genotypes are given better environments

• randomization should generally remove this effect from genetic experiments in plants

P = G + E + GE2GE

2E

2G

2P

• genotype by environment interactions• differences in relative performance of

genotypes across environments• experimentally, GE is part of E

DeLacey et al., 1990 – summary of results from many crops and locations

70-20-10 rule E: GE: G

For a particular crop, only 10% of variation in phenotype is due to genotype!

Repeatability

• Multiple observations on the same individuals

– May be repetitions in time or space (e.g. multiple fruit on a plant)

2Es

2

Within

2

Between

2

P 2

Within variation among observations on the same individual due to temporary environmental effects( = special environmental variance)

2

Between variation among individuals due to genetic differences and permanent environmental effects( = general environmental variance)

Falconer & Mackay, pg 136

2Es

2Eg

2G

2P )(

2Eg

Repeatability

2Es

2G

2Es

2Eg

2G

2P )(

2P

2Eg

2Gr

Repeatability

• Sets an upper limit on heritabilities

• is easy to measure

• To separate and , you must evaluate repeatability of genetically uniform individuals

2Es

2Eg

2Egr

2Eg

Gain from multiple measurements

• Multiple measurements can increase precision and increase heritability (by reducing environmental and phenotypic variation)

• Greatest benefits are obtained for measurements that have low repeatability (large )

2Esn

12Eg

2G

2nP )()(

n

)1n(r12

P

2

)n(P

fyi

2Es

Heritability

• For an individual: P = A + D + I + E

• For a population:

• Broad sense heritability

– degree of genetic determination

• Narrow sense heritability– extent to which phenotype is determined by genes transmitted from the parents

2E

2I

2D

2A

2P

2G

2P

2GH

2P

2A2h

“heritability”

Falconer & Mackay, Chapter 8

h2 = the regression of breeding value on phenotypic value

Narrow sense heritability – another view

h2=0.5

h2=0.3

+1

+2

h2 is trait specific, population specific, and greatly influenced by the choice of testing environments

Narrow sense heritability

2E

2D

2A

2A

2P

2A2h

– Can be applied to individuals in a single environment (generally the case in animal breeding)

– In plants, it is commonly expressed on a family (plot) basis, which are often replicated within and across environments

Heritability in plants - complications

• Different mating systems, including varying degrees of selfing

• Different ploidy levels

• Annuals, perennials

• For many crops, measurement of some traits is only meaningful with competition, in a full stand

– variables such as yield are measured on a plot basis

– other traits are averages of multiple plants/plot

– plot size varies from one experiment to the next

• Replicates are evaluated in different microenvironments

• Genotype x environment interaction is prevalent for many important crop traits

Nyquist, 1991; Holland et al., 2003

Heritability in plants - definition

• Fraction of the selection differential that is gained when selection is practiced on a defined reference unit (Hanson, 1963)

Selection Differential S=s-0

Selection Response R=1-0

Y=bX

R=Sbyx

R/S=h2=byx

• Main purpose for estimating heritability is to make predictions about selection response under varying scenarios, in order to design the optimum selection strategy

R=h2S

Applications in plant breeding

• Selection in a cross-breeding population

• Selection among purelines (with or without subsequent recombination)

• Selection among clones

• Selection among testcross progeny in a hybrid breeding program

• Must specify the unit of selection, the selection method, and unit on which the response is measured

Heritability of a genotype mean

2 2

22

2 GL e

G

l rlG

H

2 22 2 GL e

P G l rl

2 2

22

2 GL e

A

l rlG

h

Error varianceGXE

broad sense heritability narrow sense heritability or “heritability”

Resemblance between Relatives

• Covariance between relatives measures degree of genetic resemblance

• Variance among groups = covariance within groups

2

W

2

B

2

Bt

Intraclass correlationof phenotypic values

Strategy:

• Determine expected covariance among relatives from theory, and compare to experimental observations

• Estimate genetic variances and heritabilities

Falconer & Mackay, Chapt. 9

Covariance between offspring and one parent

Genotype Frequency Genotypic Value Breeding

Value

Mean Genotypic Value of Offspring

A1A1 p2 2q(-qd) 2q qA1A2 2pq (q-p)+2pqd (q - p) (1/2)(q - p)A2A2 q2 -2p(+pd) -2p -p

CovOP=p2*2q(-qd)q+2pq[(q-p)+2pqd](1/2)(q - p) +q2[-2p(+pd)](-p)

CovOP = pq2 = (1/2)σA2

This result is true for a single offspring and for the mean of any number of offspring

i i i X YCov(X,Y) f X Y μ μ

Resemblance between offspring and one parent

• For parents and offspring, observations occur in pairs

• Regression is more useful than the intraclass correlation as a measure of resemblance

– does not depend on the number of offspring

– does not require parents and offspring to have the same variance

2

P

2

A21

2

P

)P,O(Covb

phenotypic

variance of the parental population

Estimate

Resemblance between offspring and mid-parent

• Regression on mid-parent is twice the regression of offspring on a single parent

• Number of offspring does not affect the covariance or the regression

CovO,MP = pq2 = (1/2)σA2

2

P212

MP

2

P

2

A2

P21

2

A21

2

MP

)MP,O(Covb

Resemblance among half-sibs

Covariance of half-sibs = variance among half-sib progeny

Genotype FrequencyBreeding

Value

Mean Genotypic Value of Offspring Freq. x Value2

A1A1 p2 2q q p2q22

A1A2 2pq (q - p) (1/2)(q - p) (1/2)pq(q - p)22

A2A2 q2 -2p -p p2q22

CovHS = pq2[(1/2)(q - p)2+2pq] = pq2[(1/2)(p+q)2]

= (1/2)pq2=(1/4)σA2

2

P

2

A41

2

P

HSCovt

Resemblance among full-sibs

Progeny

Genotype of parents

Frequency of mating

A1A1

aA1A2

dA2A2

-aMean Value of Progeny

A1A1 A1A1 p4 1 a

A1A1 A1A2 4p3q 1/2 1/2 (1/2)(a+d)

A1A1 A2A2 2p2q2 1 d

A1A2 A1A2 4p2q2 1/4 1/2 1/4 (1/2)d

A1A2 A2A2 4pq3 1/2 1/2 (1/2)(d-a)

A2A2 A2A2 q4 1 -a

CovFS= σFS2 = p4a2+4p3q[(1/2)(a+d)]2….+q4(-a)2 - 2

= pq[a+d(q-p)]2 + p2q2d2

Resemblance among full-sibs

222 dqpp-qdapq 4222

G

2

D412

A212

FSFSCov

2

P

2

D412

A21

2

P

FSCovt

CovFS= σFS2 = p4a2+4p3q[(1/2)(a+d)]2….+q4(-a)2 - 2

= pq[a+d(q-p)]2 + p2q2d2

General formula for covariance of relatives

• Unilineal relatives

– Resemblance involves only

• Bilineal relatives

– Potential exist for relatives to have two common alleles that are identical by descent

2

A 2

AA 2

AAA etc.

A

C

B

D

X1X2 X3X4

(X1X3, X1X4, X2X3, or X2X4)

X1X3 X1X3

2

D 2

AD 2

AAD etc.

Resemblance will also involve:

Covariance due to breeding values

A B C D

X Y

), kikiα )Cov(AP(ACov

),

),

),

ljlj

kjkj

lili

)Cov(AP(A

)Cov(AP(A

)Cov(AP(A

(Ai Aj) (Ak Al)

2A

2AXY

2XY σσθσθ

ir24

Covariance due to dominance deviations

A B C D

X Y

), klijljki )Cov(AA,AP(ACov

(Ai Aj) (Ak Al)

2DBCADBDAC σθθθθ )(

), klijkjli )Cov(AA,AP(A

2DXY σ

General formula for covariance of relatives

2D

2ArCov

A B C D

X Y

r = 2XY

= ACBD + ADBC

Extended to include epistasis:

... 2DD

22AD

2AA

22D

2A rrrCov

Adjusting coefficients for inbreeding

Relatives r = 2XY Parent-offspring 1/2 0

Half-sibs

Common parent not inbred 1/4 0

Common parent inbred (1+F)/4

Full-sibs

Parents not inbred 1/2 1/4

Parents inbred (2+FA+FB)/4 (1+FA)(1+FB)/4

2

A 2

D