Major Genes, Polygenes, andQTLs

Major genes --- genes that have a significant effect on the phenotype

Polygenes --- a general term of the genes of small effect that influence a trait

QTL, quantitative trait locus --- a particular gene underlying the trait.

Usually used when a gene underlying a trait ismapped to a particular chromosomal region

Candidate gene --- a particular known gene that is of interest as being a potential candidate for contributing to the variation in a trait

Mendelizing allele. The allele has a sufficiently large effect that its impact is obvious when looking at phenotype

Major Genes

• Major morphological mutations of classical geneticsthat arose by spontaneous or induced mutation

• Genes of large effect have been found selected lines

– pygmy, obese, dwarf and hg alleles in mice

– booroola F in sheep

– halothane sensitivity in pigs

• Major genes tend to be deleterious and are at verylow frequencies in unselected populations, andcontribute little to Var(A)

Genes for Genetic modification of muscling

“Natural” mutations in the myostatin gene in cattle

“Natural” mutation in the callipyge - gene in sheep

“Booroola” gene in sheep increasing ovulation rate

Merino Sheep

Major genes for mouse body size

The mutations ob or db cause deficiencies in leptin production, or leptin receptor deficiencies

Major Genes and IsoallelesWhat is the genetic basis for quantitative variation?

Honest answer --- don’t know.

One hypothesis: isoalleles. A locus that has an allele ofmajor effect may also have alleles of much smaller effect(isoalleles) that influence the trait of interest.

Structural vs. regulatory changes

Structural: change in an amino acid sequence

Regulatory: change affecting gene regulation

General assumption: regulatory changes are likely more important

Cis vs. trans effectsCis effect --- regulatory change only affects genes (tightly) linked on the same chromosome

Cis-acting locus. The allele influencesThe regulation of a gene on the sameDNA molecule

Trans effect --- a diffusible factor that can influence regulation of unlinked genes

Trans-acting locus. This locus influences genes onother chromosomes and non-adjacent sites on the samechromosome

Genomic location of mRNA level modifiers

Geno

mic loc

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gene

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CIS-modifiers

Genomic location of mRNA level modifiers

Geno

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TRANS-modifiers

Genomic location of mRNA level modifiers

Geno

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MASTER modifiers

Polygenic Mutation

For “normal” genes (i.e., those with large effects) simply giving a mutation rate is sufficient (e.g. the rate at which an dwarfing allele appears)

For alleles contributing to quantitative variation, we must account for both the rate at which mutants appear as well as the phenotypic effect of each

Mutational variance, Vm or !2m - the amount of new additive

genetic variance introduced by mutation each generation

Typically Vm is on the order of 10-3 VE

Simple Tests for thePresence of Major Genes

• Phenotypes fall into discrete classes

• Multimodality --- distribution has several modes (peaks)

Simple Visual tests:

Simple statistical tests

• Fit to a mixture model (LR test)

p(z) = pr(QQ)p(z|QQ) +pr(Qq)p(z|Qq) + pr(qq)p(z|qq)

• Heterogeneity of within-family variances

• Select and backcross

Mixture Models

p(z) =n!

i=1

Pr(i)pi(z)

The distribution of trait value z is the weighted sum of nunderlying distributions

The probability that arandom individual is

from class i

The distribution of phenotypes zconditional of the individual

belonging to class i

The component distributions are typically assumed normal

p(z) =n!

i=1

Pr(i)!(z, µi, "2i )

The component distributions are typically assumed

p(z) =n!

i=1

Pr(i)!(z, µi, "2i )

!(z, µi,"2i ) = 1"

2#"2i

exp#

� (z µi)2

2"2i

$

Normal with mean µ and variance !2

3n-1 parameters: n-1 mixture proportions, n means, n variances

Typically assume common variances -> 2n-1 parameters

In quantitative genetics, the underlying classes aretypically different genotypes (e.g. QQ vs. Qq) althoughwe could also model different environments in the samefashion

Likelihood function for an individual under a mixture model

$(zj) = Pr(QQ) pQQ(zj) + Pr(Qq)pQq(zj) + Pr(qq)pqq(zj)= Pr(QQ) !(zj ,µQQ,"2) + Pr(Qq)!(zj, µQq, "2) +Pr(qq)!(zj ,µqq , "2)

Mixture proportions follow from Hardy-Weinberg, e.g. Pr(QQ) = pQ* pQ

$(zj) = Pr(QQ) pQQ(zj) + Pr(Qq)pQq(zj) + Pr(qq)pqq(zj)= Pr(QQ) !(zj ,µQQ,"2) + Pr(Qq)!(zj, µQq, "2) +Pr(qq)!(zj ,µqq , "2)

Likelihood function for a random sample of m individuals

$(z) = $(z1, z2, · · · , zm) =m%

j=1

$(zj)

Likelihood Ratio test for MixturesNull hypothesis: A single normal distribution is adequate tofit the data. The maximum of the likelihood function underthe null hypothesis is

!&

'

(

)max $0(z1, z2, · · · , zm) = (2#S2) m/2 exp ! 12S2

m

j=1

(zj z )2

The LR test for a significantly better fit under a mixtureis given by 2 ln (max { likelihood under mixture}/max l0 )

The LR follows a chi-square distribution with n-2 df, wheren-1 = number of fitted parameters for the mixture

!S2 =

1m

(zi ! z)2

Complex Segregation Analysis

A significant fit to a mixture only suggests the possibilityof a major gene.

A much more formal demonstration of a major gene isgiven by the likelihood-based method of ComplexSegregation Analysis (CSA)

Testing the fit of a mixture model requires a sample ofrandom individuals from the population.

CSA requires a pedigree of individuals. CSA useslikelihood to formally test for the transmission ofa major gene in the pedigree

Building the likelihood for CSAStart with a mixture modelDifference is that the mixing proportions are not the same for eachindividual, but rather are a function of its parental (presumed) genotypes

Phenotypic value ofindividual j in family i

Major-locusgenotypes of

parents

$(zij |gf , gm) =3!

go=1

Pr(go |gf , gm)!(zij, µgo , "2)

Transmission Probability of anoffspring having genotype go giventhe parental genotypes are gf, gm.

Sum is over allpossible genotypes,indexed by go =1,2,3

Mean ofgenotype go

Phenotypicvariance

conditioned onmajor-locus

genotype

$(zi·) =3!

gf=1

3!

gm=1

$(zi· |gf , gm) (gf, gm)Likelihood for family i

(go = 3 | gf = 1, gm = 2) = (qq |gf = QQ, gm = Qq) = 0(go = 2 | gf = 1, gm = 2) = (Qq | gf = QQ, gm = Qq) = 1/2(go = 1 | gf = 1, gm = 2) = (QQ |gf = QQ, gm = Qq) = 1/2

Transmission Probability example: code qq=3, Qq=2,QQ=1

$(zi· |gf, gm) =ni%

j=1

$(zij | gf , gm)Conditional family likelihood

$(zij |gf , gm) =3!

go=1

Pr(go |gf , gm)!(zij, µgo , "2)

Transmission ProbabilitiesExplicitly model the transmission probabilities

Pr(qq | gf , gm) = (1! %gf ) (1! %gm)Pr( Qq | gf , gm) = %gf (1! %gm) + %gm (1! %gf )Pr(QQ | gf , gm) = %gf %gm

Probability that thefather transmits Q

Probability that themother transmits Q

Formal CSA test of a major gene (three steps):

Formal CSA test of a major gene (three steps):

• Rejection of the hypothesis of equal transmission for allgenotypes ("QQ = "Qq = "qq )

• Failure to reject the hypothesis of Mendelian segregation : "QQ = 1, , "Qq = 1/2, "qq = 0

• Significantly better overall fit of a mixture model compared with a single normal

Pr(qq | gf , gm) = (1! %gf ) (1! %gm )Pr( Qq | gf , gm) = %gf (1! %gm ) + %gm (1! %gf )Pr(QQ | gf , gm) = %gf %gm

CSA Modification: Common Family Effects

Families can share a common environmental effect

Expected value for go genotype, family i is µgo + ci

$(zi | gf , gm, ci) =ni%

j=1

*

+3!

goj =1

Pr(goj|gf , gm)!(zij, µgoj

+ ci, "2)

,

-

Likelihood conditioned on common family effect ci

Unconditional likelihood (average over all c) --- assumedNormally distributed with mean zero and variance !c

2

$(zi | gf , gm, ci) =ni%

j=1

*

+3!

goj =1

Pr(goj|gf , gm)!(zij, µgoj

+ ci, "2)

,

-

Unconditional likelihood (average over all c) --- assumedNormal with mean zero and variance !c

2

$(zi | gf , gm) =. !

"!$(zi | gf , gm, c)!(c, 0, "2

c )dc

Likelihood function with no major gene, but family effects

$(zi ) =. !

"!$(zi | c)!(c, 0, "2

c)dc

=. !

"!

*

+ni%

j=1

!/zij ,µ +c, "2

0,

- !/c,0, "2

c

0dc