Epistasis / Multi-locus Modelling

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Epistasis / Multi-locus Modelling

• Shaun Purcell, Pak Sham
• SGDP, IoP, London, UK

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I
3
Single locus model
T
QTL1
4
Multilocus model
QTL1 QTL2
T
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GENE 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)

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Nonadditive 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

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Separate analysis

• locus A shows an association with the trait
• locus B appears unrelated

Locus B
Locus A
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Joint analysis

• locus B modifies the effects of locus A

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Genotypic 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 ?

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Partitioning of effects

• Locus A
• Locus B

M
P
M
P
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4 main effects
M
Additive effects
P
M
P
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6 twoway interactions
M
P
?
Dominance
M
P
?
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6 twoway interactions
M
M
?
Additive-additive epistasis
P
P
?
M
P
?
P
M
?
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4 threeway interactions
M
P
M
?
?
Additive-dominance epistasis
P
P
M
?
?
M
P
M
?
?
M
P
P
?
?
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1 fourway interaction
Dominance-dominance epistasis
M
M
P
P
?
?
?
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One locus

• Genotypic
• means
• AA m a
• Aa m d
• aa m - a

0
d
a
-a
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Two loci

• AA Aa aa
• BB
• Bb
• bb

dd
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Research 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?

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Variance 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
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Covariance 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
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• 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

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• 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

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IBD 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
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Joint 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

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Joint IBD sharing for two linked loci
? at QTL 1
? at QTL 2
0
1/2
1
0
1/2
1
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Potential 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

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Power calculations for epistasis

• Specify
• genotypic means,
• allele frequencies
• residual variance
• Calculate
• under full model and submodels
• variance components
• expected non-centrality parameter (NCP)

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Submodels

• 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

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Detecting 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

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(No Transcript)
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- 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 - - - - - - - -
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Example 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)

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Example 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

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Exercise

• 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?

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Poor 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

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Epistasis 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

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• Probability Function Calculator
• http//statgen.iop.kcl.ac.uk/bgim/
• Genetic Power Calculator
• http//statgen.iop.kcl.ac.uk/gpc/