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Introduction
New theoretical approach

Animal Breeding Seminar
Gota Morota

November 25, 2008

Gota Morota

Animal Breeding Seminar
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Introduction
New theoretical approach

William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions

Stabilizing Selection

After

Before

Gota Morota

Animal Breeding Seminar
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
Introduction
New theoretical approach

William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions

Directional Selection

Before

After

Gota Morota

Animal Breeding Seminar
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
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
Introduction
New theoretical approach

William G. Hill et. al.
Distribution of Allele Frequencies
Relaxation of Assumptions

Thank You

Gota Morota

Animal Breeding Seminar

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Hill, W. G, et al. 2008. Data and Theory Point to Mainly Additive Genetic Variance for Complex Traits. PLoS Genetics 4(2): e1000008.

  • 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 Gota Morota 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 Gota Morota 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