Epistasis / Multi-locus Modelling

Shaun Purcell, Pak Sham

SGDP, IoP, London, UK

T T T

I

M MM

QTL

Multiplex (larger families)

I

T

M

QTL

T

I

M

QTL

T T T T

Multivariate (more traits)

M M M M

Multipoint (more markers)

Multilocus (modelling more QTLs)

QTL QTLQTL QTLQTL QTL

Single locus model

TQTL1

QTL3

QTL4

QTL2

QTL5

E3

E4

E2

E1

Multilocus model

TQTL1

QTL2

QTL4

QTL3

QTL5

E3

E4

E2

E1

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 aa BB Bb bb

Locus A Locus B

Joint analysis

locus B modifies the effects of locus A

BB Bb bb

AA

Aa

aa

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

Locus B

M P

M P

4 main effects

M

P

M

P

Additiveeffects

6 twoway interactions

M P

M P

Dominance

6 twoway interactions

M

PM

P

Additive-additive epistasis

M

PP

M

4 threeway interactions

M P M

P

M

P

M P

M P

M P

Additive-dominance epistasis

1 fourway interaction

M M P Dominance-dominance epistasis

P

One locus

Genotypic

means

AA m + a

Aa m + d

aa m - a

0

d +a-a

Two loci

AA Aa aa

BB

Bb

bb

m

m

m

m

m

m

m

m

m

+ aA

+ aA

+ aA

– aA

– aA

– aA

+ dA

+ dA

+ dA

+ aB + aB+ aB

– aB – aB – aB

+ dB + dB + dB

– aa

– aa

+ aa

+ aa

+ dd+ ad

– da

+ da

– ad

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 + 2

D + 2S + 2

N

Under H1 :

Cov(P1,P2) = 2A + z2

D + 2S

where = proportion of alleles shared identical-by-descent (ibd) between siblings at that locus

z = probability of complete allele sharing ibdbetween siblings at that locus

Under H0 :

Cov(P1,P2) = ½2A + ¼2

D + 2S

where ½ = proportion of alleles shared identical-by- descent (ibd) between siblings

¼ = prior probability of complete allele sharing ibd between siblings

Covariance matrix

Sib 1 Sib 2

Sib 1 2A + 2

D + 2S + 2

N 2A + z2

D + 2S

Sib 2 2A + z2

D + 2S 2

A + 2D + 2

S + 2N

Sib 1 Sib 2

Sib 1 2A + 2

D + 2S + 2

N ½2A + ¼2

D + 2S

Sib 2 ½2A + ¼2

D + 2S 2

A + 2D + 2

S + 2N

QTL linkage : two locus model

P = A1 + D1 + A2 + D2

+ A1A1 + A1D2 + D1A2 + D1D2

+ S + N

Var (P) = 2A + 2

D + 2A + 2

D

+ 2AA + 2

AD + 2DA + 2

DD

+ 2S + 2

N

Under linkage :

Cov(P1,P2) = 2A + z2

D + 2A + z2

D

+ 2A + z2

AD + z2DA + zz2

DD

+ 2S

Under null :

Cov(P1,P2) = ½2A + ¼2

D + ½2A + ¼2

D

+ E()2A+E(z)2

AD +E(z)2DA+ E(zz)2

DD

+ 2S

IBD locus1 2 Expected Sib Correlation

0 1 2A/2 + 2

S

0 2 2A + 2

D + 2S

1 0 2A/2 + 2

S

1 1 2A/2 + 2

A/2 + 2AA/4 + 2

S

1 2 2A/2 + 2

A + 2D + 2

AA/2 + 2AD/2 + 2

S

2 0 2A + 2

D + 2S

2 1 2A + 2

D + 2A/2 + 2

AA/2 + 2DA/2 + 2

S

2 2 2A + 2

D + 2A + 2

D+ 2AA + 2

AD + 2DA + 2

DD + 2S

0 0 2S

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

22 )1(

4/2 2/)1( 4/)1( 2

4/22/)1( 4/)1( 2

2/)1( 2/)1( 2/))1(21(

0

1/2

1

0 1/2 1

at QTL 1 at QTL 2

Joint IBD sharing for two linked loci

Potential importance of epistasis

“… a gene’s 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

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

A B

Y

a b

True Model

A

Y

a*

Assumed Model

a* is the apparent co-efficienta* will deviate from a to the extent that A and B are correlated

- 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 - - - - - - - -

Full VA1 VD1 VA2 VD2 VAA VAD VDA VDD

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 variance40%

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 variance20%

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/