Title: Epistasis / Multi-locus Modelling
1Epistasis / Multi-locus Modelling
- Shaun Purcell, Pak Sham
- SGDP, IoP, London, UK
2I
3Single locus model
T
QTL1
4Multilocus model
QTL1 QTL2
T
5GENE x GENE Interaction
- GENE x GENE INTERACTION Epistasis
- Additive genetic effects
- alleles at a locus and across loci independently
sum to result in a net phenotypic effect - Nonadditive genetic effects
- effects of an allele modified by the presence of
other alleles (either at the same locus or at
different loci)
6Nonadditive genetic effects
- Dominance
- an allele ? allele interaction occurring within
one locus - Epistasis
- an interaction occurring between the alleles at
two (or more) different loci - Additionally, nonadditivity may arise if the
effect of an allele is modified by the presence
of certain environments
7Separate analysis
- locus A shows an association with the trait
- locus B appears unrelated
Locus B
Locus A
8Joint analysis
- locus B modifies the effects of locus A
9Genotypic Means
- Locus A
- Locus B AA Aa aa
- BB ?AABB ?AaBB ?aaBB ?BB
- Bb ?AABb ?AaBb ?aaBb ?Bb
- bb ?Aabb ?Aabb ?aabb ?bb
- ?AA ?Aa ?aa ?
10Partitioning of effects
M
P
M
P
114 main effects
M
Additive effects
P
M
P
126 twoway interactions
M
P
?
Dominance
M
P
?
136 twoway interactions
M
M
?
Additive-additive epistasis
P
P
?
M
P
?
P
M
?
144 threeway interactions
M
P
M
?
?
Additive-dominance epistasis
P
P
M
?
?
M
P
M
?
?
M
P
P
?
?
151 fourway interaction
Dominance-dominance epistasis
M
M
P
P
?
?
?
16One locus
- Genotypic
- means
- AA m a
- Aa m d
- aa m - a
0
d
a
-a
17Two loci
dd
18Research questions
- How can epistasis be modelled under a variance
components framework? - How powerful is QTL linkage to detect epistasis?
- How does the presence of epistasis impact QTL
detection when epistasis is not modelled?
19Variance components
- QTL linkage single locus model
- P A D S N
- Var (P) ?2A ?2D ?2S ?2N
Under H1 Cov(P1,P2) ??2A z?2D
?2S where ? proportion of alleles shared
identical-by- descent (ibd) between siblings
at that locus z probability of complete allele
sharing ibd between siblings at that locus
Under H0 Cov(P1,P2) ½?2A ¼?2D
?2S where ½ proportion of alleles shared
identical-by- descent (ibd) between
siblings ¼ prior probability of complete allele
sharing ibd between siblings
20Covariance matrix
- Sib 1 Sib 2
- Sib 1 ?2A ?2D ?2S ?2N ??2A z?2D ?2S
- Sib 2 ??2A z?2D ?2S ?2A ?2D ?2S ?2N
Sib 1 Sib 2 Sib 1 ?2A ?2D ?2S
?2N ½?2A ¼?2D ?2S Sib 2 ½?2A ¼?2D
?2S ?2A ?2D ?2S ?2N
21- QTL linkage two locus model
- P A1 D1 A2 D2
- A1A1 A1D2 D1A2 D1D2
- S N
- Var (P) ?2A ?2D ?2A ?2D
- ?2AA ?2AD ?2DA ?2DD
- ?2S ?2N
22- Under linkage
- Cov(P1,P2) ??2A z?2D ??2A z?2D
- ???2A ?z?2AD z??2DA zz?2DD
- ?2S
- Under null
- Cov(P1,P2) ½?2A ¼?2D ½?2A ¼?2D
- E(??)?2AE(?z)?2AD E(z?)?2DA E(zz)?2DD
- ?2S
23IBD locus 1 2 Expected Sib
Correlation
0 0 ?2S
0 1 ?2A/2 ?2S
0 2 ?2A ?2D ?2S
1 0 ?2A/2 ?2S
1 1 ?2A/2 ?2A/2 ?2AA/4 ?2S
1 2 ?2A/2 ?2A ?2D ?2AA/2 ?2AD/2 ?2S
2 0 ?2A ?2D ?2S
2 1 ?2A ?2D ?2A/2 ?2AA/2 ?2DA/2 ?2S
2 2 ?2A ?2D ?2A ?2D ?2AA ?2AD ?2DA
?2DD ?2S
24Joint IBD sharing for two loci
- For unlinked loci,
- Locus A
- 0 1 2
- Locus B 0 1/16 1/8 1/16 1/4
- 1 1/8 1/4 1/8 1/2
- 2 1/16 1/8 1/16 1/4
- 1/4 1/2 1/4
25Joint IBD sharing for two linked loci
? at QTL 1
? at QTL 2
0
1/2
1
0
1/2
1
26Potential importance of epistasis
- a genes effect might only be detected within
a framework that accommodates epistasis - Locus A
- A1A1 A1A2 A2A2 Marginal
Freq. 0.25 0.50 0.25 - B1B1 0.25 0 0 1 0.25
- Locus B B1B2 0.50 0 0.5 0 0.25
- B2B2 0.25 1 0 0 0.25
- Marginal 0.25 0.25 0.25
27Power calculations for epistasis
- Specify
- genotypic means,
- allele frequencies
- residual variance
- Calculate
- under full model and submodels
- variance components
- expected non-centrality parameter (NCP)
28Submodels
- Apparent variance components
- - biased estimate of variance component
- - i.e. if we assumed a certain model (i.e. no
epistasis) which, in reality, is different from
the true model (i.e. epistasis) - Enables us to explore the effect of misspecifying
the model
29Detecting epistasis
- The test for epistasis is based on the difference
in fit between - - a model with single locus effects and epistatic
effects and - - a model with only single locus effects,
- Enables us to investigate the power of the
variance components method to detect epistasis
30(No Transcript)
31- DD VA1 VD1 VA2 VD2 VAA VAD VDA -
- AD VA1 VD1 VA2 VD2 VAA - - -
- AA VA1 VD1 VA2 VD2 - - - -
- D VA1 - VA2 - - - - -
- A VA1 - - - - - - -
H0 - - - - - - - -
32Example 1 epi1.mx
- Genotypic Means B1B1 B1B2 B2B2
- A1A1 0 0 1
- A1A2 0 0.5 0
- A2A2 1 0 0
- Allele frequencies A1 50 B1 50
- QTL variance 20
- Shared residual variance 40
- Nonshared residual variance 40
- Sample N 10, 000 unselected pairs
- Recombination fraction Unlinked (0.5)
33Example 2 epi2.mx
- Genotypic Means B1B1 B1B2 B2B2
- A1A1 0 1 2
- A1A2 0 1 2
- A2A2 2 1 0
- Allele frequencies A1 90 B1 50
- QTL variance 10
- Shared residual variance 20
- Nonshared residual variance 70
- Sample N 2, 000 unselected pairs
- Recombination fraction 0.1
34Exercise
- Using the module, are there any configurations of
means, allele frequencies and recombination
fraction that result in only epistatic components
of variance? - How does linkage between two epistatically
interacting loci impact on multilocus analysis?
35Poor power to detect epistasis
- Detection reduction in model fit when a term is
dropped - Apparent variance components soak up variance
attributable to the dropped term - artificially reduces the size of the reduction
36Epistasis as main effect
- Epistatic effects detected as additive effects
- Main effect versus interaction effect blurred
- for linkage, main effects and interaction effects
are partially confounded
37- Probability Function Calculator
- http//statgen.iop.kcl.ac.uk/bgim/
- Genetic Power Calculator
- http//statgen.iop.kcl.ac.uk/gpc/