Presented at Animal Breeding & Genomics Seminar. University of Wisconsin-Madison.

1. Introduction
New theoretical approach
Animal Breeding Seminar
Gota Morota
November 25, 2008
Gota Morota
Animal Breeding Seminar

2. Introduction
New theoretical approach
Epistatic Effect
Epistatic Variance
Outline
1
Introduction
Epistatic Effect
Epistatic Variance
2
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Gota Morota
Animal Breeding Seminar

3. Introduction
New theoretical approach
Epistatic Effect
Epistatic 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

4. Introduction
New theoretical approach
Epistatic Effect
Epistatic Variance
Epistasis on Qualitative Traits (two locus)
Table 1: Some unusual segregation ratios
Interaction Type
Classical ratio
Dominant epistasis
Recessive epistasis
Duplicate genes with cumulative effect
Duplicate dominant genes
Duplicate recessive genes
Gota Morota
A-B9
A-bb aaB- aabb
3
3
1
12
3
1
9
3
4
9
6
1
15
1
9
7
Animal Breeding Seminar

5. Introduction
New theoretical approach
Epistatic Effect
Epistatic Variance
Epistasis on Quantitative Traits (two locus )
P =G+E
G = GA + GB + IAB
Table 2: Interaction effects
Interaction Type
1
2
3
locus 1
Additive
Additive
Dominance
Gota Morota
X
X
X
locus 2
Additive
Dominance
Dominance
Animal Breeding Seminar

6. Introduction
New theoretical approach
Epistatic Effect
Epistatic Variance
Outline
1
Introduction
Epistatic Effect
Epistatic Variance
2
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Gota Morota
Animal Breeding Seminar

7. Introduction
New theoretical approach
Epistatic Effect
Epistatic 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 from
additive genetic variance
Gota Morota
Animal Breeding Seminar

8. Introduction
New theoretical approach
Epistatic Effect
Epistatic 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

9. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Outline
1
Introduction
Epistatic Effect
Epistatic Variance
2
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Gota Morota
Animal Breeding Seminar

10. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Journal Paper
Data and Theory Point to Mainly Additive
Genetic Variance for Complex Traits
William 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, Australia
4 Queensland Institute of Medical Research, Brisbane, Australia
PLoS Genetics 4(2): e1000008
Gota Morota
Animal Breeding Seminar

11. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Outline
1
Introduction
Epistatic Effect
Epistatic Variance
2
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Gota Morota
Animal Breeding Seminar

12. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation 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 = 4p 2 (1 − p )2 d 2
But the allele frequencies at most genes affecting complex traits
are not known
Gota Morota
Animal Breeding Seminar

13. William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Introduction
New theoretical approach
Distribution of Allele Frequencies
Distribution of allele frequencies depends on
mutation
selection
genetic drift
Those effects (except artificial selection) on fitness of genes at
many of the loci influencing most quantitative traits are likely to be
small
⇓
Neutral alleles
Gota Morota
Animal Breeding Seminar

14. William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Introduction
New theoretical approach
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 forward
regardless of their fitness
↓
Neutral alleles
Gota Morota
Animal Breeding Seminar

15. William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Introduction
New theoretical approach
Neutral Theory
Survival of the luckiest
The vast majority of molecular differences are selectively neutral
(if selection neither favors nor disfavors the allele).
↓
Alleles that are selectively neutral have their frequencies
determined by genetic drift and mutation.
Gota Morota
Animal Breeding Seminar

16. William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Introduction
New theoretical approach
Uniform Distribution
Distribution Frequency of the Neutral Mutant
f(p) ∝ 1
m
1
1
≤p ≤1−
2N
2N
0.0
0.2
0.4
0.6
p
Gota Morota
Animal Breeding Seminar
0.8
1.0

17. William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Introduction
New theoretical approach
L-Shaped Distribution
Distribution of the Frequency of the Mutant Allele
1
p
(1/p)
f(p) ∝
mutations arising recently
0.0
0.2
0.4
0.6
p
Gota Morota
Animal Breeding Seminar
0.8
1.0

18. William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Introduction
New theoretical approach
Inverse L-Shaped Distribution
Distribution of the Frequency of the Ancestral Allele
1
1−p
(1/(1 − p))
f(p) ∝
replaced by mutations
0.0
0.2
0.4
0.6
p
Gota Morota
Animal Breeding Seminar
0.8
1.0

19. William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Introduction
New theoretical approach
U-Shaped Distribution
The Allele which Increases the Value of the Trait
1/(p * (1 − p))
f(p) ∝
1
p(1 − p)
Due to mutations
Due to ancestral alleles
0.0
0.2
0.4
0.6
p
Gota Morota
Animal Breeding Seminar
0.8
1.0

20. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Genetic Variance Components
Integration of expressions for the variance as a function of p for a
specific model of the gene frequency distribution.
N is sufficiently large
Standardization for the U distribution.
1
1− 2N
1
2N
1
p (1 − p )
dp = 2 log 1 −
1
1
− log
2N
2N
≈ 2log (2N )
f (p ) =
1
2Kp (1 − p )
where K ∼ log (2N )
Gota Morota
Animal Breeding Seminar

21. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Single Locus Model
Table 3: Genotypic values
B
b
1
B
a
d1
b
d1
-a
Arbitrary dominance
Gota Morota
Animal Breeding Seminar

22. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Single Locus Model
Arbitrary p:
2p (1 − p )(a + d (1 − 2p ))2
VA
VA
=
=
VG
VA + VD
2p (1 − p )(a + d (1 − 2p ))2 + 4p 2 (1 − p 2 )d 2
Uniform:
E (VA )
E (VA )
2d 2
=
=1− 2
E (VG )
E (VA ) + E (VD )
5a + 3d 2
’U’ Distribution:
E (VA )
E (VA )
d2
=
=1− 2
E (VG )
E (VA ) + E (VD )
3a + 2d 2
Gota Morota
Animal Breeding Seminar

23. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Result – Single Locus Model
Table 4: Expected proportion of VG that is VA
Genetic model
d = 1 a1
2
d = a2
a = 03
p= 1
2
0.89
0.67
0.00
Distribution of allele frequencies
Uniform ’U’ (N = 100) 4 ’U’ (N = 1000)
0.91
0.93
0.93
0.75
0.80
0.80
0.33
0.50
0.50
1
partial dominance
complete dominance
3
overdominance
4
population size
2
Gota Morota
Animal Breeding Seminar

24. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Two Locus Additive x Additive Model without Dominance
Table 5: Genotypic values
BB
Bb
bb
1
2
CC
-a1
0
a
Cc
02
0
0
cc
a
0
-a
double homozygote +a or -a
single or double heterozygotes 0
Gota Morota
Animal Breeding Seminar

25. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Two Locus Additive x Additive Model without Dominance
Arbitrary p :
a 2 (Hp + Hq − 4Hp Hq )
VA
VA
= 2
=
VG
VA + VAA
a (Hp + Hq − 4Hp Hq ) + a 2 Hp Hq
Uniform:
E (VA )
E (VA )
=
=
E (VG )
E (VA ) + E (VAA )
2 2
9a
2 2
1 2
9a + 9a
=
’U’ Distribution:
E (VA )
E (VA )
1
=
=1−
2K − 3
E (VG )
E (VA ) + E (VAA )
Gota Morota
Animal Breeding Seminar
2
3

26. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Result – Additive x Additive Model without Dominance
Table 6: Expected proportion of VG that is VA
1
p= 2
0.00
Distribution of allele frequencies
p = 0.99 Uniform ’U’ (N = 100) ’U’ (N = 1000)
1
0.67
0.87
0.92
Gota Morota
Animal Breeding Seminar

27. William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Introduction
New theoretical approach
Duplicate Factor Model with Two Loci
Table 7: Genotypic values
BB
Bb
bb
1
CC
a1
a
a
Cc
a
a
a
cc
a
a
0
For an arbitrary number (L) of loci, the
genotypic value is a except for the multiple recessive homozygote, and for one locus it is complete dominance
Gota Morota
Animal Breeding Seminar

28. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Duplicate Factor Model with Two loci
For pi = 0.5:
1
a 2 ( 2 )4L −1
VA
2L
= 2L
=
1 2L
1 2L
VG
2 −1
a 2 [( 2 ) − ( 4 )]
Uniform:
1 L
1 2
E (VA )
2a L(5)
=
E (VG )
a 2 [( 1 )L − ( 1 )L ]
3
5
’U’ Distribution:
E (VA )
=
E (VG )
a2 L
11
(1 − 6K )L −1
2L −1 3K
a2
1
11
[(1 − K )L − (1 − 6K )L ]
2L
Gota Morota
Animal Breeding Seminar

29. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Result – Duplicate Factor Model with Two loci
Table 8: Expected proportion of VG that is VA
1
p= 2
0.27
Distribution of allele frequencies
Uniform ’U’ (N = 100) ’U’ (N = 1000)
0.56
0.71
0.75
Gota Morota
Animal Breeding Seminar

30. William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Introduction
New theoretical approach
Summary
The fraction of the genetic variance that is additive genetic
decreases 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 the
differences found in genotypic values
The liner regression of genotypic value on number of C genes
accounts for the genotypic difference
⇓
Almost all VG is accounted for by VA
Gota Morota
Animal Breeding Seminar

31. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Outline
1
Introduction
Epistatic Effect
Epistatic Variance
2
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Gota Morota
Animal Breeding Seminar

32. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation 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

33. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Stabilizing Selection
After
Before
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Animal Breeding Seminar

34. William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Introduction
New theoretical approach
Effects of Stabilizing Selection
Mutants are at a disadvantage if they increase (decrease) trait
values
⇓
The gene frequency distribution is still U-shaped with much more
concentration near 0 or 1
⇓
Likely to increase proportions of additive variance
Gota Morota
Animal Breeding Seminar

35. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Directional Selection
Before
After
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Animal Breeding Seminar

36. William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Introduction
New theoretical approach
Effects of Directional Selection
Rapid fixation or increase to intermediate frequency of genes
affecting the trait
⇓
Theoretically, under extreme frequency distributions, net increase
in variance over generations might be expected
Gota Morota
Animal Breeding Seminar

37. William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Introduction
New theoretical approach
Conclusion
Even in the presence of non-addtive gene action, most genetic
variance appears to be additive
⇓
Because allele frequencies are distributed towards extreme values
Gota Morota
Animal Breeding Seminar

38. Introduction
New theoretical approach
William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions
Thank You
Gota Morota
Animal Breeding Seminar