IntroductionNew theoretical approach
Animal Breeding Seminar
Gota Morota
November 25, 2008
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
Epistatic EffectEpistatic Variance
Outline
1 IntroductionEpistatic EffectEpistatic Variance
2 New theoretical approachWilliam G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
Epistatic EffectEpistatic Variance
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
Epistatic EffectEpistatic Variance
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
Epistatic EffectEpistatic Variance
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
Epistatic EffectEpistatic Variance
Outline
1 IntroductionEpistatic EffectEpistatic Variance
2 New theoretical approachWilliam G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
Epistatic EffectEpistatic Variance
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
Epistatic EffectEpistatic 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)
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Outline
1 IntroductionEpistatic EffectEpistatic Variance
2 New theoretical approachWilliam G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Outline
1 IntroductionEpistatic EffectEpistatic Variance
2 New theoretical approachWilliam G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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.
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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)
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Single Locus Model
Table 3: Genotypic values
B bB a d1
b d1 -a1 Arbitrary dominance
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Two Locus Additive x Additive Model without Dominance
Table 5: Genotypic values
CC Cc ccBB -a1 02 aBb 0 0 0bb a 0 -a1 double homozygote +a or -a2 single or double heterozygotes 0
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
’U’ Distribution:
E(VA )
E(VG)=
E(VA )
E(VA ) + E(VAA )= 1 −
12K − 3
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Result – Additive x Additive Model without Dominance
Table 6: Expected proportion of VG that is VA
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Duplicate Factor Model with Two Loci
Table 7: Genotypic values
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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 ]
’U’ Distribution:
E(VA )
E(VG)=
a2
2L−1L
3K (1 − 116K )L−1
a2
2L [(1 − 1K )L − (1 − 11
6K )L ]
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Result – Duplicate Factor Model with Two loci
Table 8: Expected proportion of VG that is VA
Distribution of allele frequenciesp = 1
2 Uniform ’U’ (N = 100) ’U’ (N = 1000)0.27 0.56 0.71 0.75
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Outline
1 IntroductionEpistatic EffectEpistatic Variance
2 New theoretical approachWilliam G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Stabilizing Selection
Before
After
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Directional Selection
After
Before
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
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
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Conclusion
Even in the presence of non-addtive gene action, most geneticvariance appears to be additive
⇓
Because allele frequencies are distributed towards extreme values
Gota Morota Animal Breeding Seminar
IntroductionNew theoretical approach
William G. Hill et. al.Distribution of Allele FrequenciesRelaxation of Assumptions
Thank You
Gota Morota Animal Breeding Seminar