Epistasis / Multi-locus Modelling
Shaun Purcell, Pak Sham
SGDP, IoP, London, UK
Multipoint (more markers)
M
M
M
QTL QTL QTL
M
QTL QTL
M
QTL
M
T
T
T
T
Multivariate (more traits)
M
M
QTL QTL QTL
I
I
T
M
I
T
T
T
T
Multiplex (larger families)
Multilocus (modelling more QTLs)
Single locus model
E1
E2
E3
QTL2
QTL1
QTL3
QTL4
E4
QTL5
T
Multilocus model
E1
E2
E3
QTL3
QTL1
QTL2
QTL4
E4
QTL5
T
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)
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
Separate analysis
locus A shows an association with the trait
locus B appears unrelated
AA
Aa
Locus A
aa
BB
Bb
Locus B
bb
Joint analysis
locus B modifies the effects of locus A
AA
Aa
aa
BB
Bb
bb
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

Partitioning of effects
Locus A
M
P
M
P
Locus B
4 main effects
M
P
M
P
Additive
effects
6 twoway interactions
M 
P
Dominance
M 
P
6 twoway interactions
M 
M

P
M 
P
M 
P
P
Additive-additive
epistasis
4 threeway interactions
M 
P

M
M 
P

P
M 
M 
P

M 
P
P
Additivedominance
epistasis
1 fourway interaction
M 
P

M

P
Dominancedominance
epistasis
One locus
Genotypic
means
AA
m+a
Aa
m+d
aa
m-a
0
-a
d
+a
Two loci
AA
Aa
aa
BB m + aA + aB + aa
m + dA + aB + da
m – aA + aB – aa
Bb m + aA + dB + ad
m + dA + dB + dd
m – aA + dB – ad
bb m + a – aB – aa
A
m + dA – aB – da
m – aA – aB + aa
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?
Variance components
QTL linkage : single locus model
P=A+D+S+N
Var (P) = 2A + 2D + 2S + 2N
Under H01 :
2 2 ++
22
Cov(P1,P2) = ½
22AA++z
¼
D D S S
where
½==proportion

proportionof
ofalleles
allelesshared
sharedidentical-byidentical-bydescent
descent
(ibd)
(ibd)
between
between
siblings
siblings
at that locus
z ==probability
¼
prior probability
of complete
of complete
allele sharing
allele sharing
ibd
ibd between
siblings
between
siblings
at that locus
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
QTL linkage : two locus model
P = A1 + D 1 + A2 + D2
+ A1A1 + A1D2 + D1A2 + D1D2
+S+N
Var (P) = 2A + 2D + 2A + 2D
+ 2AA + 2AD + 2DA + 2DD
+ 2S + 2N
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()2A+E(z)2AD +E(z)2DA+ E(zz)2DD
+ 2S
IBD locus
1
2
0
0
Expected Sib Correlation
 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
Joint IBD sharing for two loci
For unlinked loci,
Locus A
0
Locus B
1
2
0
1/16 1/8
1/16 1/4
1
1/8
1/8
2
1/16 1/8
1/16 1/4
1/4
1/4
1/4
1/2
1/2
Joint IBD sharing for two linked loci
 at
QTL 2
0
1/2
1
    (1   )
2
 at QTL 1
0
1/2
 /4
2
 (1   ) / 2
2
1
(1  ) / 4
2
 (1   ) / 2 (1  2(1  )) / 2 (1  ) / 2
(1  ) / 4
2
 (1   ) / 2
 /4
2
Potential importance of epistasis
“… a gene’s effect might only be detected within a
framework that accommodates epistasis…”
Locus A
Locus B
A1A1
A1A2
A2A2
Freq. 0.25
0.50
0.25
B1B1
0.25
0
0
1
0.25
B1B2
0.50
0
0.5
0
0.25
B2B2
0.25
1
0
0
0.25
0.25
0.25
0.25
Marginal
Marginal
Power calculations for epistasis
Specify
genotypic means,
allele frequencies
residual variance
Calculate
under full model and submodels
variance components
expected non-centrality parameter (NCP)
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
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
True Model
A
a
Y
Assumed Model
B
A
b
a*

Y

a* is the apparent co-efficient
a* will deviate from a to the extent that A and B are correlated
Full
VA1
VD1
VA2
VD2
VAA
VAD
VDA
VDD
- DD V*A1
V*D1
V*A2
V*D2
V*AA
V*AD
V*DA
-
- AD
V*A1
V*D1
V*A2
V*D2
V*AA
-
-
-
- AA
V*A1
V*D1
V*A2
V*D2
-
-
-
-
-D
V*A1
-
V*A2
-
-
-
-
-
-A
V*A1
-
-
-
-
-
-
-
H0
-
-
-
-
-
-
-
-
VS and VN estimated in all models
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)
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
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?
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
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
Probability Function Calculator
http://statgen.iop.kcl.ac.uk/bgim/
Genetic Power Calculator
http://statgen.iop.kcl.ac.uk/gpc/