hill, w. g, et al. 2008. data and theory point to mainly additive genetic variance for complex...

IntroductionNew theoretical approach

Animal Breeding Seminar

Gota Morota

November 25, 2008

Gota Morota Animal Breeding Seminar


Epistatic EffectEpistatic Variance

Outline

1 IntroductionEpistatic EffectEpistatic Variance

2 New theoretical approachWilliam G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions




Quantitative Traits

Controlled by many genes and by environmental factors

Typically,

genes do not act additively with each other within loci - dominance

genes do not act additively with each other between loci - epistasis




Epistasis on Qualitative Traits (two locus)

Table 1: Some unusual segregation ratios

Interaction Type A-B- A-bb aaB- aabbClassical ratio 9 3 3 1

Dominant epistasis 12 3 1Recessive epistasis 9 3 4

Duplicate genes with cumulative effect 9 6 1Duplicate dominant genes 15 1Duplicate recessive genes 9 7




Epistasis on Quantitative Traits (two locus )

P = G + E

G = GA + GB + IAB

Table 2: Interaction effects

Interaction Type locus 1 locus 21 Additive X Additive2 Additive X Dominance3 Dominance X Dominance




Outline






Component of Variance (two locus)

VP = VG + VE

VG = VA + VD + VI

= VA + VD + VAA + VAD + VDD

Estimate variance components using REML with the animal model.

⇓

It is difficult to differentiate non-additive genetic variance fromadditive genetic variance




Controversy

We know epistasis plays very important role on total genetic effect.

But how much do they contribute on genetic variance?

Small portion

Falconer DS, Mackay TFC (1996)

Lynch M, Walsh B (1998)

Large portion

Schadt EE, Lamb J, Yang X, Zhu J, Edwards S, et al. (2005)

Evans DM, Marchini J, Morris AP, Cardon LR (2006)

Marchini J, Donnelly P, Cardon LR (2005)

Carlborg O, Haley CS (2004)



William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions

Outline






Journal Paper

Data and Theory Point to Mainly AdditiveGenetic Variance for Complex TraitsWilliam G. Hill1, Michael E. Goddard2,3, Peter M. Visscher4 (2008)

1 Institute of Evolutionary Biology, School of Biological Sciences,University of Edinburgh, Edinburgh, UK

2 Faculty of Land and Food Resources, University of Melborne,Victoria, Australia

3 Department of Primary Industries, Victoria, Australia4 Queensland Institute of Medical Research, Brisbane, Australia

PLoS Genetics 4(2): e1000008




Outline






Allele Frequencies

Genetic variance components depend on

the mean value of each genotype

the allele frequencies at the gene affecting the trait

VA = 2p(1 − p)[a + d(1 − p)]2

VD = 4p2(1 − p)2d2

But the allele frequencies at most genes affecting complex traitsare not known




Distribution of Allele Frequencies

Distribution of allele frequencies depends on

mutation

selection

genetic drift

Those effects (except artificial selection) on fitness of genes atmany of the loci influencing most quantitative traits are likely to besmall

⇓

Neutral alleles




Neutral Alleles

Mutation:

CGA ( Arginine )→ CGG ( Arginine )GGU ( Glycine)→ GGC ( Glycine )

Single-nucleotide changes have little or no biological effect↓

Neutral substitutions create new neutral alleles

Genetic drift

Chance events determine which alleles will be carried forwardregardless of their fitness

↓

Neutral alleles




Neutral Theory

Survival of the luckiestThe vast majority of molecular differences are selectively neutral

(if selection neither favors nor disfavors the allele).

↓

Alleles that are selectively neutral have their frequenciesdetermined by genetic drift and mutation.




Uniform Distribution

Distribution Frequency of the Neutral Mutant

0.0 0.2 0.4 0.6 0.8 1.0

p

m

f((p)) ∝∝ 1

12N ≤≤ p ≤≤ 1 −− 1

2N




L-Shaped Distribution

Distribution of the Frequency of the Mutant Allele

0.0 0.2 0.4 0.6 0.8 1.0

p

(1/p

)

f((p)) ∝∝ 1p

mutations arising recently




Inverse L-Shaped Distribution

Distribution of the Frequency of the Ancestral Allele

0.0 0.2 0.4 0.6 0.8 1.0

p

(1/(

1 −

p))

f((p)) ∝∝ 11 −− p

replaced by mutations




U-Shaped Distribution

The Allele which Increases the Value of the Trait

0.0 0.2 0.4 0.6 0.8 1.0

p

1/(p

* (

1 −

p))

f((p)) ∝∝ 1p((1 −− p))

Due to mutations

Due to ancestral alleles




Genetic Variance Components

Integration of expressions for the variance as a function of p for aspecific model of the gene frequency distribution.

N is sufficiently largeStandardization for the U distribution.∫ 1− 1

2N

12N

1p(1 − p)

dp = 2[log

(1 −

12N

)− log

(1

2N

)]≈ 2log(2N)

f(p) =1

2Kp(1 − p)

where K ∼ log(2N)




Single Locus Model

Table 3: Genotypic values

B bB a d1

b d1 -a1 Arbitrary dominance




Single Locus Model

Arbitrary p:

VA

VG=

VA

VA + VD=

2p(1 − p)(a + d(1 − 2p))2

2p(1 − p)(a + d(1 − 2p))2 + 4p2(1 − p2)d2

Uniform:

E(VA )

E(VG)=

E(VA )

E(VA ) + E(VD)= 1 −

2d2

5a2 + 3d2

’U’ Distribution:

E(VA )

E(VG)=

E(VA )

E(VA ) + E(VD)= 1 −

d2

3a2 + 2d2




Result – Single Locus Model

Table 4: Expected proportion of VG that is VA

Genetic model Distribution of allele frequenciesp = 1

2 Uniform ’U’ (N = 100) 4 ’U’ (N = 1000)d = 1

2a1 0.89 0.91 0.93 0.93d = a2 0.67 0.75 0.80 0.80a = 03 0.00 0.33 0.50 0.50

1 partial dominance2 complete dominance3 overdominance4 population size




Two Locus Additive x Additive Model without Dominance


CC Cc ccBB -a1 02 aBb 0 0 0bb a 0 -a1 double homozygote +a or -a2 single or double heterozygotes 0




Two Locus Additive x Additive Model without Dominance

Arbitrary p :

VA

VG=

VA

VA + VAA=

a2(Hp + Hq − 4HpHq)

a2(Hp + Hq − 4HpHq) + a2HpHq

Uniform:

E(VA )

E(VG)=

E(VA )

E(VA ) + E(VAA )=

29a2

29a2 + 1

9a2=

23


E(VA )

E(VG)=

E(VA )

E(VA ) + E(VAA )= 1 −

12K − 3




Result – Additive x Additive Model without Dominance


Distribution of allele frequenciesp = 1

2 p = 0.99 Uniform ’U’ (N = 100) ’U’ (N = 1000)0.00 1 0.67 0.87 0.92




Duplicate Factor Model with Two Loci


CC Cc ccBB a1 a aBb a a abb a a 01 For an arbitrary number (L) of loci, the

genotypic value is ′a′ except for the multi-ple recessive homozygote, and for one lo-cus it is complete dominance




Duplicate Factor Model with Two loci

For pi = 0.5:

VA

VG=

a2(12)4L−1

a2[(12)2L − (1

42L

)]=

2L22L − 1

Uniform:

E(VA )

E(VG)=

12a2L(1

5)L

a2[(13)L − (1

5)L ]


E(VA )

E(VG)=

a2

2L−1L

3K (1 − 116K )L−1

a2

2L [(1 − 1K )L − (1 − 11

6K )L ]




Result – Duplicate Factor Model with Two loci


Distribution of allele frequenciesp = 1

2 Uniform ’U’ (N = 100) ’U’ (N = 1000)0.27 0.56 0.71 0.75




Summary

The fraction of the genetic variance that is additive geneticdecreases as the proportion of genes at extreme frequencies

decreases

When an allele is rare (say C): CC Cc cc

Average effect of C vs.c accounts for essentially all thedifferences found in genotypic values

The liner regression of genotypic value on number of C genesaccounts for the genotypic difference

⇓

Almost all VG is accounted for by VA




Outline






Relaxation of Assumptions

Expectation of a Ratio of Variance Components

Influence of Linkage Disequilibrium

Consequences of Multiple Alleles

Effects of Selection on Gene Frequency Distribution




Stabilizing Selection

Before

After




Effects of Stabilizing Selection

Mutants are at a disadvantage if they increase (decrease) traitvalues⇓

The gene frequency distribution is still U-shaped with much moreconcentration near 0 or 1

⇓

Likely to increase proportions of additive variance




Directional Selection

After

Before




Effects of Directional Selection

Rapid fixation or increase to intermediate frequency of genesaffecting the trait

⇓

Theoretically, under extreme frequency distributions, net increasein variance over generations might be expected




Conclusion

Even in the presence of non-addtive gene action, most geneticvariance appears to be additive

⇓

Because allele frequencies are distributed towards extreme values




Thank You


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