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PCB6555 Spring 2009 PCB-6555 Introduction to Quantitative Genetics Population Genetics Allele and Genotypic Frequencies

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Page 1: Intro to Quant Gen

PCB6555Spring 2009

PCB-6555 Introduction to Quantitative Genetics

Population Genetics

Allele and Genotypic Frequencies

Page 2: Intro to Quant Gen

PCB6555Spring 2009

Population Genetics

• Hardy-Weinburg Equilibrium– In a large random-mating population with no selection,

migration or mutation, the allele frequencies and the genotypic frequencies are constant from one generation to the next.

– Assumes• Allele frequencies of the parents are known

• Population of infinite size (no sampling error)

• Random mating

• All genotypes equally viable and fertile

• Normal segregation of alleles at gametogenesis

• Allele frequencies are equal in males and females

Page 3: Intro to Quant Gen

PCB6555Spring 2009

Population Genetics

• Hardy-Weinburg– Results in a simple relationship between allelic frequencies

and genotypic frequencies – a binomial expansion• p = f(A) and q = f(a) then (p+q)2 = p2 + 2pq + q2

• where p2 = f(AA), 2pq = f(Aa) and q2 = f(aa)

• Since for two alleles at a locus p + q = 1 then p2 + 2pq + q2 = 1

– For any starting frequencies (p and q), genotypic equilibrium is reached in one generation of random mating for a single loci

Page 4: Intro to Quant Gen

PCB6555Spring 2009

Population Genetics

• Hardy-Weinburg– Arbitrary starting genotypic frequencies – f(AA) = 0.5, f(Aa) =

0.2, and f(aa) = 0.3– Gametes produced

• f(A) from genotype AA = p(A)*f(AA) = 1*0.5 = 0.5• f(A) from genotype Aa = p(A)*f(Aa) = 0.5*0.2 = 0.1• Total f(A) = 0.5 + 0.1 = 0.6• f(a) from genotype Aa = p(a)*f(Aa) = 0.5*0.2 = 0.1• f(a) from genotype aa = p(a)*f(aa) = 1*0.3 = 0.3• Total f(a) = 0.1 + 0.3 = 0.4

– General equation f(A)=f(AA)+0.5*f(Aa) and f(a)=f(aa)+0.5*f(Aa)

Page 5: Intro to Quant Gen

PCB6555Spring 2009

Population Genetics

• Hardy-Weinburg– Idealized population – genotypes of offspring

• f(AA) = f(A)*f(A) = 0.36 = p2

• f(Aa) = 2*f(A)*f(a) = 0.48 = 2pq

• f(aa) = f(a)*f(a) = 0.16 = q2

Page 6: Intro to Quant Gen

PCB6555Spring 2009

Population Genetics

Female Gametes

p(A)=0.6 p(a)=0.4

Male

Gametes

p(A)=0.6 f(AA)=0.36 f(Aa)=0.24

p(a)=0.4 f(Aa)=0.24 f(aa)=0.16

Page 7: Intro to Quant Gen

PCB6555Spring 2009

Population Genetics Test for Hardy-Weinberg Equilibrium

Genotypes Observed Expected

AA 21 23.75

Aa 36 30.46

aa 7 9.79

Calculations

(21-23.75)2/23.75 +

(36-30.46)2/30.46+

(7-9.79)2/9.79 ~ 2 with 1 degree of freedom

Page 8: Intro to Quant Gen

PCB6555Spring 2009

Population Genetics

Page 9: Intro to Quant Gen

PCB6555Spring 2009

Population Genetics

• Genetic load – rare deleterious alleles• Example Falconer page 9 – Phenylketonuria in

Birmingham England• PKU occurred in 5 of 55,715 babies born so

q=sqrt(5/55715) or 0.0095• Carriers = 2(.9905)(.0095) or 0.019 or one person in

fifty• Percentage of deleterious allele carried in

heterozygotes is 100*.5*0.019/(0.0095+.0000903) = 99%

Page 10: Intro to Quant Gen

PCB6555Spring 2009

Population Genetics

• Causes of departure from Hardy-Weinberg equilibrium– Changes in allele frequency and genotypic frequency due to

• Selection

• Mutation

• Migration

• Sampling or genetic drift

– Non-random Mating• Inbreeding

• Positive Assortative mating and Disassortative mating

Page 11: Intro to Quant Gen

PCB6555Spring 2009

Inbreeding and Assortative Mating

Page 12: Intro to Quant Gen

PCB6555Spring 2009

Inbreeding

• Mating between relatives– Mating system used by breeders (selfing in corn)– Inevitable in small populations – natural or selected– Occurs in nature because of proximity of relatives – example

natural stands of tree whose relatives are proximal because of seed dispersion

• Without selection there is a increase in the frequency of homozygotes without a change in allele frequency.

Page 13: Intro to Quant Gen

PCB6555Spring 2009

Consequences of Inbreeding on Genotypic Frequencies (HW)

Female Genotype and Frequencies

AA p2 Aa 2pq aa q2

Male Genotype and Frequencies

AA p2 Homo AA

p4

Both AA Aa 2p3q 0.5:0.5

Hetero Aa

p2q2

Aa 2pq Both AA Aa 2p3q 0.5:0.5

All .25:.5:.25

4p2q2

Both Aa aa

2pq3 0.5:0.5

aa q2 Hetero Aa p2q2

Both Aa aa

2pq3 0.5:0.5

Homo aa

q4

Page 14: Intro to Quant Gen

PCB6555Spring 2009

Consequences of Inbreeding on Genotypic Frequencies (Selfing)

Female Genotype and Frequencies

AA p2 Aa 2pq aa q2

Male Genotype and Frequencies

AA p2 Homo AA

p2

Aa 2pq All .25:.5:.25

2pq

aa q2 Homo aa

q2

Page 15: Intro to Quant Gen

PCB6555Spring 2009

Genotypic Frequency and Allele Frequency After One Generation of Selfing

• AA = p2 + .5pq• Aa = pq• aa = q2 + .5pq• f(A) = f(AA) + .5 * f(Aa) = p2 + .5pq + .5 pq = p2 + pq =

p • f(a) = f(aa) + .5 * f(Aa) = q2 + .5pq + .5 pq = q2 + pq =

q

Page 16: Intro to Quant Gen

PCB6555Spring 2009

Assortative Mating

• Usually considered as Positive Assortative Mating or mating between similar phenotypes– Mating system used by breeders (southern pines)– Can occur in nature– The effect of PAM is dependent on the correlation between

phenotype and genotype

Page 17: Intro to Quant Gen

PCB6555Spring 2009

Consequences of Positive Assortative Mating on Genotypic Frequencies with

Corr(pheno,geno) = 1 with Complete DominanceFemale Genotype and Frequencies

AA p2

P

Aa 2pq

H

aa q2

Q

Male Genotype and Frequencies

AA p2

P

Homo AA

P2

Both AA Aa PQ

0.5:0.5

Aa 2pq

H

Both AA Aa PQ

0.5:0.5

All .25:.5:.25

H2

aa q2

Q

Homo aa

Q2

Page 18: Intro to Quant Gen

PCB6555Spring 2009

Genetic Values

Population Mean

Page 19: Intro to Quant Gen

PCB6555Spring 2009

Phenotypic Value

• P = E + G• Assume genotypic values can be measured without

error then P = G

Page 20: Intro to Quant Gen

PCB6555Spring 2009

Genotypic Value = Phenotypic Value

Genotype Phenotypic

Value

Scaled Genetic Value

A1A1 G11 a

A1A2 G12 d

A2A2 G22 -a

Page 21: Intro to Quant Gen

PCB6555Spring 2009

Rescaling

• This scale is independent of allele frequencies and genotypic frequencies.

• Zero is assigned as central point between the two homozygotes

• Then better homozygote = a and poorer homozygote = -a

• d is the departure of the value of the heterozygote from the mean of the homozygotes

Page 22: Intro to Quant Gen

PCB6555Spring 2009

Estimating Scaled Values

• a = (G11 - G22)/2

• d = G12 – (G11 + G22)/2a

-a

0 d?

Page 23: Intro to Quant Gen

PCB6555Spring 2009

Population Mean for a Hardy-Weinberg Population

Genotype Frequency Unscaled Scaled Giif(Gii) giif(gii)

A1A1 p2 G11 a p2G11 p2a

A1A2 2pq G12 d 2pqG12 2pqd

A2A2 q2 G22 -a q2G22 -q2a

Page 24: Intro to Quant Gen

PCB6555Spring 2009

Population Mean - HW

• E(G) = Giif(Gii) = p2G11+2pqG12+q2G22

• E(g) = = giif(gii) = p2a+2pqd -q2a= a(p-q)+2pqd

• Or the location of the mean is due to the difference in frequency between the two alleles and the value of a; and the heterozygote frequency and the value of d

Page 25: Intro to Quant Gen

PCB6555Spring 2009

Average Effect of an Allele

• Motivation – Parents pass alleles on to their offspring and not their genotypes. Genotypes are formed anew each generation.

• Average effect value associated with an allele in a random mating population. That is the deviation from the population mean of the individuals that received a particular allele from a parent with the other parent coming from the population at random.

Page 26: Intro to Quant Gen

PCB6555Spring 2009

Average Effect of an Allele

• So let many A1 gametes unite with gametes coming from the population at random and express the mean value of the resulting individuals as a deviation from the population mean.

Page 27: Intro to Quant Gen

PCB6555Spring 2009

Average Effect of an Allele

Allele Population Frequencies

Mean Average Effect or 1

A1 A1 f(p) A2 f(q)

Offspring A1 A1 f(p) A1 A2 f(q)

Values pa qd a(p-q)+2pqd q[a+d(q-p)]

Page 28: Intro to Quant Gen

PCB6555Spring 2009

Average Effect of Allele Substitution

• Derive 2 and then average effect of an allele substitution is 1-2 or the increase in the population mean if A1 replaced all the A2 alleles in the population

• Average effect = = a+d(q-p) then 1 = q and

2 = -p

Page 29: Intro to Quant Gen

PCB6555Spring 2009

Breeding Values

• The average effects of the parental alleles passed to the offspring determine the mean genotypic value of its offspring

• The breeding value of an individual the value of an individual judged by mean value of its progeny

• Two concepts– Sum of the average effects across loci - theoretical– Mean value of offspring – practical

• The two concepts are not equivalent when interaction between loci occurs or mating is not at random

Page 30: Intro to Quant Gen

PCB6555Spring 2009

One Locus Theoretical Breeding Value

Genotype Breeding Value

A1A1 2 1 = 2q

A1A2 1+ 2 = (q-p)

A2A2 2 2 = -2p

Page 31: Intro to Quant Gen

PCB6555Spring 2009

Example of Theoretical Value Effect of allele frequency and genetic effects on

= a +d(q-p)

Example f(p) a d

1 0.2 5 0 5

2 0.2 5 5 8

3 0.8 5 0 5

4 0.8 5 5 2

Page 32: Intro to Quant Gen

PCB6555Spring 2009

Example of Theoretical Value Calculation of BV for Each Example

Example f(p) BV

A1A1

BV

A1A2

BV

A2A2

1 0.2 8 3 -2

2 0.2 12.8 4.8 -3.2

3 0.8 2 -3 -8

4 0.8 3.2 1.2 -0.8

Page 33: Intro to Quant Gen

PCB6555Spring 2009

Dominance Deviation

• The difference between the breeding value for a genotype and the genotypic value is called the dominance deviation

• Dominance deviation is not entirely determined by the degree of dominance at the loci (d), but is also a property of the allele frequencies in the population

• G = A + D

Page 34: Intro to Quant Gen

PCB6555Spring 2009

Calculating Dominance Deviation

• Adjust the genotypic value for the population mean

for A1A1 this is a-[a(p-q)+2pqd or 2q(-qd)

• Subtract the breeding value for A1A1 from this result

2q(-qd)-2q = -2q2d

• For A1A2 this is 2pqd and for A2A2 the result is –2p2 d

• So the expected value for dominance deviation across the three genotypes is the sum of the frequencies times the values

-2p2q2d + 4p2q2d - 2p2q2d = 0

Page 35: Intro to Quant Gen

PCB6555Spring 2009

Dominance Deviation for Example 1 d = 0 then Dominance Deviation = 0

-4

-2

0

2

4

6

8

10

A2A2 A1A2 A1A1

Genotypic Value -Mean

Breeding Value

-2p

2q

(q-p)

Mean=-3

a

-a

d

Page 36: Intro to Quant Gen

PCB6555Spring 2009

Dominance Deviation for Example 2 d = 5

-6

-4

-2

0

2

4

6

8

10

12

14

A2A2 A1A2 A1A1

Genotypic Value -Mean

Breeding Value

-2p

2q

(q-p)

Mean=-1.4

a

-a

dDominance deviation

Page 37: Intro to Quant Gen

PCB6555Spring 2009

Epistatic Deviation

• G = A + D + I• I is epistasis which is interlocus interaction or non-

additivity when more than one locus is considered• If epistasis is not present, the genetic value of an

individual can be calculated as the sum of the genotypic values for each loci concerning the trait

• If epistasis is present, this simple sum is no longer correct

Page 38: Intro to Quant Gen

PCB6555Spring 2009

Genetic Variances

Page 39: Intro to Quant Gen

PCB6555Spring 2009

Genetic Variance

• The study of and partitioning of the variance of a metric character is central to quantitative genetic

• Simply stated VP = VG + VE

where VG = VA + VD + VI

– Ignoring possible correlation between the environment and genotype

– Ignoring interactions of genotype with environment

• We will assign variance estimates from analyses of data to the causal components listed above

Page 40: Intro to Quant Gen

PCB6555Spring 2009

Genetic Variance

• Important ratios can be calculated from the causal components– VG/ VP the degree of genetic determination or broad sense

heritability, important when dealing with clones

– VA/ VP narrow sense heritability or heritability, this quantity determines the degree of resemblance among relatives and is most important to breeding programs

Page 41: Intro to Quant Gen

PCB6555Spring 2009

Estimating Degree of Genetic Determination

• Must separate all environmental variance from the total genetic variance

• This requires the use of clonally propagated material or genetically uniform material such as inbred lines or the F1 hybrid of such lines

• Why? Identical genotypes must be duplicated across environments in order to separate the environmental variation– Either by differencing the phenotypic variance across environments

with and without inclusion of different genotypes

– Or by partitioning variances when many clonal genotypes are planted across environments, difficult because a portion of the environmental variance may be transmitted to clones from their origin – strictly this is clonal repeatability

Page 42: Intro to Quant Gen

PCB6555Spring 2009

Additive Genetic Variance

• To be useful for breeding the genetic variance must be partitioned into its components and the additive variance estimated

• In order to perform this partitioning you must have a dataset where the resemblance among relatives can be estimated

Page 43: Intro to Quant Gen

PCB6555Spring 2009

Theoretical Genetic VariancesOne Locus No Epistasis

• Theoretical breeding values and dominance deviations are already adjusted for the mean So:– Square the theoretical breeding values or dominance

deviations– Multiply by the genotypic frequency and – Sum the results

Page 44: Intro to Quant Gen

PCB6555Spring 2009

Theoretical Genetic VariancesOne Locus, No Epistasis (Calculations)

• VA = 4p2q22 + 2pq(q-p)2 2 + 4p2q22

= 2pq2 (p2 + 2pq + q2) = 2pq2

• VD = 4p2q4d2 + 8p3q3d2 + 4p4q2d2

= 4p2q2d2 (p2 + 2pq + q2) = (2pqd)2

• Cov(VA , VD) = 0

• So VG = VA + VD = 2pq2 + (2pqd)2

Page 45: Intro to Quant Gen

PCB6555Spring 2009

What Theoretical Variances Illustrate

• Because of the effect of allele frequency on genetic variance, all estimates are population specific

• Additive genetic variance does not imply additive gene action but may arise from additive, dominant or epistatic gene action

• Asymmetry of total genetic variance with allele frequency implies both additive and non-additive gene action

Page 46: Intro to Quant Gen

PCB6555Spring 2009

Theoretical Epistatic Variance

• Looking at two-way interactions epistatic variance may arise from– The interaction of two breeding values VAA

– The interaction between two dominance deviations = VDD

– The interaction between a breeding value at one locus and the dominance deviation at another VAD

• Epistatic variance is difficult to estimate; however, sources of epistatic variance may be estimated given an appropriate population

Page 47: Intro to Quant Gen

PCB6555Spring 2009

Other Sources of Variation

• Disequilibrium – Genotypic frequencies at two loci are not equal to the expectations given the allele frequencies– Produces a covariance between the genetic variances of two loci

(VG’ and VG’’ )

– VG = VG’ + VG’’ + 2Cov(VG’ , VG’’ ), since covariances can be positive or negative this effect can either increase of decrease the apparent variance

• Disequilibrium arising from non-random mating– Selection of parents and random mating produces gametic phase

(linkage) disequlibrium (not a random sample of population)

– Assortative mating produces linkage disequilibrium as well as a correlation between alleles in uniting pairs of gametes

Page 48: Intro to Quant Gen

PCB6555Spring 2009

Correlation Between Genotype and Environment

• Genotypes are not randomly allocated to environments – that is usually the better genotypes are allocated to the better environments producing a positive correlation between genotype and environment

VP = VG + VE + 2Cov(G,E)

Example: Southern pines are deployed so that the best genotypes (fastest growing) are placed in the best environments

Page 49: Intro to Quant Gen

PCB6555Spring 2009

Genotype by Environment Interaction

• When numerous genotypes are planted across a suite of environments, it may be that some of the genotypes are more sensitive to the differences in environments than others. This is the cause of GxE.

VP = VG + VE + VGE

• The mean of all genotypes in an environment is called the environmental mean

• The regression of means in each environment for a genotype versus the environmental means is an indication of the sensitivity of that genotype to environment

Page 50: Intro to Quant Gen

PCB6555Spring 2009

Environmental Variance

• All non-genetic variance is considered environmental– Depends on trait and organism– Experiments to estimate genetic variation are planned so as

to reduce environmental variation– Can be caused by variability in climate or nutrition or scale of

measurement or common environment (maternal effects)– Usually part of the environmental variation is from unknown

causes

Page 51: Intro to Quant Gen

PCB6555Spring 2009

Repeatability

• Assumes that the same trait is being measured at each point and the variances are the same

• VP = VG + VEg + VEs

where VE is partitioned into VEg or environmental variance contributing to the between individual variance and VEs contributing to the within individual variance

• r = (VG + VEg )/ VP and 1-r = VEs / VP

• r is always great than or equal to the degree of genetic determination

Page 52: Intro to Quant Gen

PCB6555Spring 2009

Repeatability

• To increase the repeatability of a trait you decrease VP by taking multiple measurements usually across time or space such that

VP = VG + VEg + VEs / n

Page 53: Intro to Quant Gen

PCB6555Spring 2009

Gain in Accuracy from Number of Measurements on an Individual

0

20

40

60

80

100

1 2 3 4 5 6 7 8 9 10

Measurements

Vp

n /

Vp

% r = 0.75

r = 0.5

r = 0.25

r = 0.1

Page 54: Intro to Quant Gen

PCB6555Spring 2009

Prediction of Future Performance (y) from Past Performance (x)

• Using regression

)( xxbyy

Page 55: Intro to Quant Gen

PCB6555Spring 2009

Prediction of Future Performance (y) from Past Performance (x)

• Heuristically, part of the past performance (x) is due to specific environmental influence with the remainder due to repeatable factors

• The correlation between performance in the past environment with that in the future environment quantifies the relative proportion of repeatable factors to specific environmental influence

)()(/),( yVarxVaryxCovr

Page 56: Intro to Quant Gen

PCB6555Spring 2009

Prediction of Future Performance (y) from Past Performance (x)

• Since

)(/),( xVaryxCovb )(/),( xVaryxCovb

Page 57: Intro to Quant Gen

PCB6555Spring 2009

Prediction of Future Performance (y) from Past Performance (x)

• Then if you know the means of the two measures and their variances and covariance, the equation to predict future performance becomes

xy xxryy /)(

Page 58: Intro to Quant Gen

PCB6555Spring 2009

Resemblance Between Relatives and Heritability

Page 59: Intro to Quant Gen

PCB6555Spring 2009

Estimating Causal Components from Variance Component Estimates

• Relating what can be observed (estimated) to the causal components of variation– Need a genetic model for the experiment– Apply genetic model to observed components of variation

• The genetic model hinges on understanding the level of resemblance among different types of relatives

Page 60: Intro to Quant Gen

PCB6555Spring 2009

Intraclass Correlation, t

• This concept is used to define the relative proportion of variation among groups (2

b , in particular family groups) to the within group variation (2

w)

• The notion being that the larger the intraclass correlation the greater the similarity within groups (causing differences among groups) is when compared to the dissimilarity within groups

)/( 222wbbt

Page 61: Intro to Quant Gen

PCB6555Spring 2009

Intraclass Correlation, t

• Theoretically, t ranges from 1 to 0 and the variance component estimates for the formula could be derived from this linear model for half-sib families

where yij is the phenotypic observation on offspring ‘j’ within family ‘i’ ; fi is the random variable family effect ~ (0, 2

b); and wij is the random variable for offspring within family ’i’~ (0, 2

w)

ijiij wfy

Page 62: Intro to Quant Gen

PCB6555Spring 2009

Genetic Covariance (Parent-Offspring)

• Simplifying assumptions:– Hardy-Weinburg population with equal variances for parents

and offspring– No epistasis– Genetic components only – no environmental effects

• Genetic Model – one parent and mean of offspring– Parent GP = AP + DP

– Mean of offspring oPnPooo DAADAG 2/12/1

Page 63: Intro to Quant Gen

PCB6555Spring 2009

Genetic Covariance (One Parent-Offspring)

• Then

),()( , ooPPoP DADACovGGCov

)2/12/1,()( , oPnPPPoP DAADACovGGCov

),(

)2/1,()2/1,(),(

)2/1,()2/1,()( ,

oP

PnPPPoP

PnPPPoP

DDCov

ADCovADCovDACov

AACovAACovGGCov

Page 64: Intro to Quant Gen

PCB6555Spring 2009

Genetic Covariance (One Parent-Offspring)

• Then

0),(

0)2/1,(

0)2/1,(

0),(

0)2/1,(

2/1),(2/1)2/1,(

oP

PnP

PP

oP

PnP

APPPP

DDCov

ADCov

ADCov

DACov

AACov

VAACovAACov

Unrelated parents

Hardy-Weinburg

Hardy-Weinburg

Hardy-Weinburg

Unrelated parents

Page 65: Intro to Quant Gen

PCB6555Spring 2009

Genetic Covariance (One Parent-Offspring)

• So

AoP VGGCov 2/1)( ,

Page 66: Intro to Quant Gen

PCB6555Spring 2009

Genetic Covariance (One Parent-Offspring)

• Another method: This derivation can be accomplished considering a single locus in a Hardy-Weinburg population using the theoretical BV’s in terms of ’s for the parent and offspring and the frequency of the genotypes. Left as a homework exercise for you.

Page 67: Intro to Quant Gen

PCB6555Spring 2009

Empirical Estimates of the Covariance of One Parent with Offspring Using Phenotypic or

Genotypic Values – no enviromental covariance

Parental Values One Offspring Many Offspring

P1 O1j

P2 O2j

. . .

. . .

. . .

Pn Onj

1.O

2.O

n.O

Page 68: Intro to Quant Gen

PCB6555Spring 2009

Empirical Estimates of the Covariance of One Parent with Offspring Using Phenotypic or

Genotypic Values – no enviromental covariance

• Is the covariance of a parent with one offspring the same as the covariance of the parent with the mean of many offspring?

n

j

ijiii OnPCovOPCov1

. )/1,(),(

)},(...),(),({/1),( 21. iniiiiiii OPCovOPCovOPCovnOPCov

)},({/1),( . ijiii OPnCovnOPCov

),(),( . ijiii OPCovOPCov

Page 69: Intro to Quant Gen

PCB6555Spring 2009

Regression (One Parent-Offspring)

• Assuming that the parental and offspring variances are equal

pA

p

ii

VV

VOPCovb 2/1)( .,

Page 70: Intro to Quant Gen

PCB6555Spring 2009

Covariance of an Offspring with the Mean of Its Parents

• Also, assuming equal variances for parents

)2/12/1,2(),( 212211

. opppppp

ij DAADADACovOPCov

)2/1,2/1()2/1,2/1(),( 2211. ppppij AACovAACovOPCov

AAAij VVVOPCov 2/14/14/1),( .

Page 71: Intro to Quant Gen

PCB6555Spring 2009

Regression of an Offspring on the Mean of Its Parents

• Then

)(),( PVarOPCovb ij

PVPPVarPPVarPVar 2/1)(4/1}2){()( 2121

PAPA VVVVb /2/12/1

Page 72: Intro to Quant Gen

PCB6555Spring 2009

Covariance Within Half-sib Families

• Using offspring related only by the recurrent parent

)2/12/1,2/12/1()/,( ''''' oPiPioPiPiiijij DAADAACovPOOCov

APiPiiijij VAACovPOOCov 4/1)2/1,2/1()/,( '

Page 73: Intro to Quant Gen

PCB6555Spring 2009

Variance Among Half-sib Families

• Using infinite number of unrelated offspring

)2/1(()(3

1

j

j

jj gVarffamVar

22222222 )(4/12)( pqpqpqqpfamVar j

Aj VpqfamVar 4/12/1)( 2

PA VVtSo 41

Page 74: Intro to Quant Gen

PCB6555Spring 2009

Covariance Within Full-sib Families

),()|,( '21

21

21

21

' ijkPjPiijkPjPijiijkijk DAADAACovPandPOOCov

),(),(),()|,( '21

21

21

21

' ijkijkPjPjPiPijiijkijk DDCovAACovAACovPandPOOCov

),()|,( '41

41

' ijkijkAAjiijkijk DDCovVVPandPOOCov

Page 75: Intro to Quant Gen

PCB6555Spring 2009

Cov(Dijk,Dijk’)

FEMALE

A1A2

M

A

L

E

A3A4

A1A3 A2A3

A1A4 A2A4

Page 76: Intro to Quant Gen

PCB6555Spring 2009

Covariance Within Full-sib Families

• So, based on identical dominance deviations by descent

),()|,( '41

41

' ijkijkAAjiijkijk DDCovVVPandPOOCov

DAjiijkijk VVPandPOOCov 41

21

' )|,(

PDA VVVt )( 41

21

Page 77: Intro to Quant Gen

PCB6555Spring 2009

General Formula for Covariance of Related Individuals

• Cov = rVA+ uVD

where r and u are derived from coancestry values generally using a pedigree file

• Coancestry (f) equals the inbreeding coefficient of the offspring of two individuals if they were mated

• r = 2fPQ

• u = fACfBD + fADfBC

Page 78: Intro to Quant Gen

PCB6555Spring 2009

General Formula for Covariance of Related Individuals

A B C D

P Q

X

Page 79: Intro to Quant Gen

PCB6555Spring 2009

Formula for Covariance of Full-sibs

A B A B

P Q

X

Page 80: Intro to Quant Gen

PCB6555Spring 2009

Covariance of Full-sibs Using the Formulae

21

21

21 )(2 33 r

41

21

21 )0)(0())(( u

Page 81: Intro to Quant Gen

PCB6555Spring 2009

Including Epistatic Covariance Among Relatives: No LD, Covariance May Still Be

Increased If Interacting Loci Are Linked

• General formula

...322322 DDDADDAADAAADDAADA VuVruuVrVrVuVruVrVCov

Page 82: Intro to Quant Gen

PCB6555Spring 2009

Non-genetic Sources of Resemblance Among Relatives

• VE = VEC + VES

Where VEC is environmental covariance among relatives (common environment) and inflates the variance among groups of relatives

• Examples– Animals

• Maternal effects• Sibs reared in the same pen

– Plants• Maternal effects (seed size)• Clonal cuttings taken from the same ortet• Clonal laboratory propagules where each clone is propagated

separately

Page 83: Intro to Quant Gen

PCB6555Spring 2009

Common Environment Can Cause Dissimilarity Among Members of a Family

• Competition for limited resources as– Sibling calves in a pen with limited feed– Sibling plants in nutrient or water limited circumstances

Page 84: Intro to Quant Gen

PCB6555Spring 2009

Heritability

• Heritability of a metric character is important for understanding– The degree to which relatives resemble one another– The effects of selection on a population

P

Ai

V

Vh 2

Page 85: Intro to Quant Gen

PCB6555Spring 2009

Heritability as a Regression of Breeding Value on Phenotype

• Denote P = A + R

where R contains non-additive genetic effects and environmental effects

• Then

2

)(),( h

V

VPVar

PACovbP

AAP

hbrA

PAPAP

Page 86: Intro to Quant Gen

PCB6555Spring 2009

Heritability as a Regression of Breeding Value on Phenotype

• Treating the heritability as bAP you can predict the breeding value of an individual as the heritability times the individual’s phenotypic value with A and P expressed as deviations from the mean

PhA 2

^

Page 87: Intro to Quant Gen

PCB6555Spring 2009

Facts Concerning Heritability

• Heritability is a property of– The population because of allele frequencies and history– The trait – In general, trait associated with reproductive

fitness have lower heritabilities– The environment in which the phenotypes were observed –

experiments are usually planned to increase heritability by decreasing environmental noise

• So besides error in estimation there are many reasons for differences in heritability estimates

Page 88: Intro to Quant Gen

PCB6555Spring 2009

Facts Concerning Heritability

• All estimates of heritability have error associated with them

• Experiments to estimate heritability– Appropriate environments– Experimental control of environmental noise– Large sample of parents, i.e. 100 for more parents for a

large population– Generate appropriate sibships for your purpose

• h2 = b/r or t/r where r is the coefficient of relatedness,

t is the intraclass correlation or b is a regression coefficient (usually offspring-parent)

Page 89: Intro to Quant Gen

PCB6555Spring 2009

Linear Model for Progeny within Dams within Sires

))(()( ijkijiijk wdsy

Where

is a general meansi is the random variable sire ~ (0, 2

i)dj(I) is the random variable dam within sire ~ (0, 2

d)wk(j9i)) is the random variable progeny within dam within sire ~ (0, 2

w)

Page 90: Intro to Quant Gen

PCB6555Spring 2009

Estimation of Heritability – ANOVA of Nested Model Dams Within Sires

Source df Mean Square Expected Mean Square

Between sires s-1 MSs 2w + k 2

d + dk 2s

Between dams within sires

s(d-1) MSD 2w + k 2

d

Progeny within dams

sd(k-1) MSW 2w

Page 91: Intro to Quant Gen

PCB6555Spring 2009

Estimation of Heritability – ANOVA of Nested Model Dams Within Sires

Observed Component

Covariance Causal Components Estimated

Sires 2s Cov(HS) ¼ VA

Dams 2d Cov(FS) – Cov(HS) ¼ VA+ ¼ VD + VEc

Progeny 2W 2

T – Cov(FS) ½ VA+ ¾ VD + Vew

Phenotypic 2T = 2

w + 2d + 2

s = VP VA+ VD + Vec + VEw

Sires + Dams

2d + 2

s

Cov(FS) ½ VA+ ¼ VD + VEc

Page 92: Intro to Quant Gen

PCB6555Spring 2009

Heritability Estimates Available h2 = t/r

Sire component

Dam component

Sire + Dam component

2

22 4

T

sh

2

22 4

T

dh

2

222 )(2

T

sdh

Page 93: Intro to Quant Gen

PCB6555Spring 2009

Which Estimates Are Appropriate

• Depends on whether VD and/or Vec are sources of variation that bias the latter two estimates upward

• If neither source of bias is operating (2d is

approximately equal to 2s ) then the pooled estimate

is best (lowest error of estimation)

Page 94: Intro to Quant Gen

PCB6555Spring 2009

Linear Model for Half-sib Progeny within Females

)(ijiij wfy

Where

is a general meanfj(I) is the random variable female ~ (0, 2

f)w(j9i) is the random variable progeny within female ~ (0, 2

w)

Page 95: Intro to Quant Gen

PCB6555Spring 2009

Estimation of Heritability – ANOVA for Half-sib Families

Source df Mean Square Expected Mean Square

Between females

f-1 MSf 2w + k 2

f

Progeny within females

d(k-1) MSW 2w

Page 96: Intro to Quant Gen

PCB6555Spring 2009

Estimation of Heritability – Half-sib Families

Observed Component

Covariance Causal Components Estimated

Females 2f Cov(HS) ¼ VA + VEc

Progeny 2W 2

T – Cov(HS) ¾ VA+ VD + Vew

Phenotypic 2T = 2

w + 2d + 2

s = VP VA+ VD + Vec + VEw

Page 97: Intro to Quant Gen

PCB6555Spring 2009

Heritability Estimates Available h2 = t/r

Female component

2

22 4

T

fh

Page 98: Intro to Quant Gen

PCB6555Spring 2009

Linear Model for Full-sib Families in a Half-Diallel Mating Design for ‘p’ Parents

)(ijkijjiijk wsmfy

Where

is a general meanfi is the random variable female (i=1 to p-1)~ (0, 2

f)mj is the random variable male (j=2 to p)~ (0, 2

m)Usual half-diallel assumption 2

f = 2m then estimate 2

gca pooling the variance due to females and malessij is the random variable sca (j>i) ~ (0, 2

sca)wk(ij) is the random variable progeny within female and male ~ (0, 2

w)

Page 99: Intro to Quant Gen

PCB6555Spring 2009

Estimation of Heritability – ANOVA of Half-Diallel

Source df Mean Square

Expected Mean Square

GCA p-1 MSgca 2w + k 2

sca + k(p-2) 2gca

SCA (p2-3p)/2 MSsca 2w + k2

sca

Progeny within female and male

((p2-p)/2)(k-1) MSW 2w

Page 100: Intro to Quant Gen

PCB6555Spring 2009

Estimation of Heritability – ANOVA of Half Diallel

Observed Component

Covariance Causal Components Estimated

Parents 2gca Cov(HS) ¼ VA

Crosses 2sca Cov(FS) – 2Cov(HS) ¼ VD + VEc

Progeny 2W 2

T – Cov(FS) ½ VA+ ¾ VD + Vew

Phenotypic 2T = 2

w + 2sca + 22

gca = VP

VA+ VD + Vec + VEw

Parents + Crosses

2sca + 22

gca

Cov(FS) ½ VA+ ¼ VD + VEc

Page 101: Intro to Quant Gen

PCB6555Spring 2009

Heritability Estimates Available h2 = t/r

Parent component

2

22 4

T

gcah

Page 102: Intro to Quant Gen

PCB6555Spring 2009

Linear Model for Full-sib Families in a Half-Diallel Mating Design for ‘p’ Inbred Parents

)(ijkijjiijk wsmfy

Where

is a general meanfi is the random variable female (i=1 to p-1)~ (0, 2

f)mj is the random variable male (j=2 to p)~ (0, 2

m)Usual half-diallel assumption 2

f = 2m then estimate 2

gca pooling the variance due to females and malessij is the random variable sca (j>i) ~ (0, 2

sca)wk(ij) is the random variable progeny within female and male ~ (0, 2

w)

Page 103: Intro to Quant Gen

PCB6555Spring 2009

Assumptions Concerning the ‘p’ Inbred Parents

• All inbred to the same degree• One parent per inbred line• No selection• All progeny produced are non-inbred• The reference population is the non-inbred population

from which the inbred parents are drawn at random

Page 104: Intro to Quant Gen

PCB6555Spring 2009

Estimation of Heritability – ANOVA of Half-Diallel with p Inbred Parents

Source df Mean Square

Expected Mean Square

GCA p-1 MSgca 2w + k 2

sca + k(p-2) 2gca

SCA (p2-3p)/2 MSsca 2w + k2

sca

Progeny within female and male

((p2-p)/2)(k-1) MSW 2w

Page 105: Intro to Quant Gen

PCB6555Spring 2009

Estimation of Heritability – ANOVA of Half Diallel with p Inbred Parents

Observed Component

Covariance Causal Components Estimated

Parents 2gca Cov(HS)

((1+F)/4) VA

Crosses 2sca Cov(FS) – 2Cov(HS)

((1+F)2/4) VD + Vec

Progeny 2W 2

T – Cov(FS)((1-F)/2) VA + ((3-2F-F2)/4) VD + Vec

Phenotypic 2T = 2

w + 2sca + 22

gca = VP

VA+ VD + Vec + VEw

Parents + Crosses

2sca + 22

gca

Cov(FS)((1+F)/2) VA + ((1+F)2/4) VD + Vec

Page 106: Intro to Quant Gen

PCB6555Spring 2009

Heritability Estimates Available h2 = t/rfor p Inbred Parents

Parent component

)1(

42

22

Fh

T

gca

Page 107: Intro to Quant Gen

PCB6555Spring 2009

How Good Is Your Estimate of h2

• Approximate 95% ci = • Since the stderr is the square root of the variance of

, there is a problem• The estimate of h2 is a random variable resulting from

a ratio of random variables which are correlated• Two methods are in general use to estimate the

standard error of

)ˆ(2ˆ 22 hseh

2h

2h

Page 108: Intro to Quant Gen

PCB6555Spring 2009

Mean Square or REML EstimatorsDickerson’s Method

• Treat the phenotypic variance as if it were a constant• Then

• The needed variance can be estimated in two ways– Asymptotic variance of variance component estimates

REML

SAS ASYCOV option output– Variance of combinations of mean squares used to estimate

the additive variance component

)ˆ()ˆ1

()ˆ( 22

2A

T

VVarhVar

Page 109: Intro to Quant Gen

PCB6555Spring 2009

Variance of Combinations of Mean Squares Assuming Independence of Mean Squares

• Var (MSi)= 2MSi2/(dfi+2)

• Var((MSi – MSj)/k) = )ˆ( 2iVar

222

2 /})2(2

)2(2{)ˆ( kdf

MSdf

MSVarj

j

i

ii

)ˆ()ˆ(

16)ˆˆ4()ˆ( 2

22222

iT

Ti VarVarhVar

]/})2(2

)2(2{[

)ˆ(

16)ˆ( 2

22

222 kdf

MSdf

MShVarj

j

i

i

T

Page 110: Intro to Quant Gen

PCB6555Spring 2009

Asymptotic Covariance of Variance Component Estimates and Taylor Series Approximation of

the Variance of a Ratio – REML Estimation

• Standard output available from ASREML• Let V = the covariance matrix for the variance

components (nxn) where n equals the number of variance components

• Let l be the matrix containing the weights for the numerator and denominator of h2 (2xn)

• Then the variance of the numerator (1,1) and denominator (2,2) and their covariance (1,2 or 2,1) is contained in l`Vl (2x2)

Page 111: Intro to Quant Gen

PCB6555Spring 2009

Taylor Series Approximation of Var(h2)where N = numerator and D = denominator

)()(),()(2)()1

()ˆ(4

2

322 DVar

D

NDNCov

D

NNVar

DhVar

Page 112: Intro to Quant Gen

PCB6555Spring 2009

Correlated Traits

Page 113: Intro to Quant Gen

PCB6555Spring 2009

Observed Correlation Between Traits

• The observed (phenotypic) correlation between two traits is primarily determined by the correlation between the genetic effects (genetic correlation) and the correlation between the environmental effects (environmental correlation)

• If Px = Gx + Ex and Py = Gy + Ey

• Then )()(/),(),( yxyxyxP PVarPVarPPCovPPCorrr

)()(/),(),( yxyyxxyxP PVarPVarEGEGCovPPCorrr

)()(/)},(),({),( yxyxyxyxP PVarPVarEECovGGCovPPCorrr

Page 114: Intro to Quant Gen

PCB6555Spring 2009

Observed Correlation Between Traits

• Multiplication by the square root of the product of the phenotypic variances yields

• Replacing the covariances by their solution from correlations

),(),(),( yxyxyx EECovGGCovPPCov

EYEXEAYAXAPYPXP rrr

Page 115: Intro to Quant Gen

PCB6555Spring 2009

Observed Correlation Between Traits

• Given that e2 = 1 – h2

• Multiplying by the inverse of the square root of the product of the phenotypic variances yields

EYXAYXP reerhhr

PYYPXXEPYYPXXAPYPXP eerhhrr

Page 116: Intro to Quant Gen

PCB6555Spring 2009

Causes of Genetic Correlation Between TraitsPleiotrophy

• Pleiotrophy is the property of a gene having an effect on more than one trait.

• Pleiotrophic loci are the primary cause of genetic correlations and the sum of the pleiotropic effects across all loci provides the genetic similarity between traits

• If the sum of the effects for both traits is positive then the genetic correlation is positive

• If the sum of the effects for one trait is positive and negative for the other trait then the genetic correlation is negative

• It is possible for the sum of effects for one or both traits to be near zero so no genetic correlation despite some loci involved with the traits acting pleiotrophically

Page 117: Intro to Quant Gen

PCB6555Spring 2009

Causes of Genetic Correlation Between TraitsLinkage

• Linkage can cause transient correlations particularly when genetically distinct population are crossed (that is a series of linked loci acting as a single loci where independently the loci are not pleiotrophic for the two traits but there are loci on the DNA segment which affect both traits)

• May still be useful for practical breeding until the linkage is broken

Page 118: Intro to Quant Gen

PCB6555Spring 2009

What Does a Genetic Correlation Tell Us?

• Informs concerning the biological relationships among traits

• Helps in understanding the effects of selection on one trait on another which may not be considered

Page 119: Intro to Quant Gen

PCB6555Spring 2009

Problems with Genetic Correlations

• Difficult to estimate• Maybe transient due to linkage or selection altering

allele frequencies and causing fixation of alleles