Epistasis / Multi-locus Modelling Shaun Purcell, Pak Sham SGDP, IoP, London, UK

TTT I MMMQTL Multiplex (larger families) I T M QTL T I M TT TT Multivariate (more traits) M M MM Multipoint (more markers) Multilocus (modelling more QTLs) QTL

Single locus model T QTL 1 QTL 3 QTL 4 QTL 2 QTL 5 E3E3 E4E4 E2E2 E1E1

Multilocus model T QTL 1 QTL 2 QTL 4 QTL 3 QTL 5 E3E3 E4E4 E2E2 E1E1

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 Locus A Locus B

Joint analysis locus B modifies the effects of locus A

Genotypic Means Locus A Locus BAAAaaa 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 MP MP

4 main effects M P M P Additive effects

6 twoway interactions MP MP   Dominance

6 twoway interactions M PM P   Additive-additive epistasis M PP M  

4 threeway interactions MPM P  M P MP   M P  MP   Additive- dominance epistasis

1 fourway interaction MMP  Dominance- dominance epistasis P

One locus Genotypic means AAm + a Aam + d aam - a 0 d +a-a

Two loci AAAaaa BB Bb bb m m m m m m m m m + a A – a A + d A + a B – a B + d B – 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) =  2 A +  2 D +  2 S +  2 N Under H 1 : Cov(P 1,P 2 ) =  2 A + z  2 D +  2 S 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 H 0 : Cov(P 1,P 2 ) = ½  2 A + ¼  2 D +  2 S where½ = proportion of alleles shared identical-by- descent (ibd) between siblings ¼ = prior probability of complete allele sharing ibd between siblings

Covariance matrix Sib 1Sib 2 Sib 1  2 A +  2 D +  2 S +  2 N  2 A + z  2 D +  2 S Sib 2  2 A + z  2 D +  2 S  2 A +  2 D +  2 S +  2 N Sib 1Sib 2 Sib 1  2 A +  2 D +  2 S +  2 N ½  2 A + ¼  2 D +  2 S Sib 2 ½  2 A + ¼  2 D +  2 S  2 A +  2 D +  2 S +  2 N

QTL linkage : two locus model P = A 1 + D 1 + A 2 + D 2 + A 1 A 1 + A 1 D 2 + D 1 A 2 + D 1 D 2 + S + N Var (P) =  2 A +  2 D +  2 A +  2 D +  2 AA +  2 AD +  2 DA +  2 DD +  2 S +  2 N

Under linkage : Cov(P 1,P 2 ) =  2 A + z  2 D +  2 A + z  2 D +  2 A +  z  2 AD + z  2 DA + zz  2 DD +  2 S Under null : Cov(P 1,P 2 ) = ½  2 A + ¼  2 D + ½  2 A + ¼  2 D + E(  )  2 A +E(  z)  2 AD +E(z  )  2 DA + E(zz)  2 DD +  2 S

IBD locus 1 2 Expected Sib Correlation 01  2 A /2 +  2 S 02  2 A +  2 D +  2 S 10  2 A /2 +  2 S 11  2 A /2 +  2 A /2 +  2 AA /4 +  2 S 12  2 A /2 +  2 A +  2 D +  2 AA /2 +  2 AD /2 +  2 S 20  2 A +  2 D +  2 S 21  2 A +  2 D +  2 A /2 +  2 AA /2 +  2 DA /2 +  2 S 22  2 A +  2 D +  2 A +  2 D +  2 AA +  2 AD +  2 DA +  2 DD +  2 S 002S002S

Joint IBD sharing for two loci For unlinked loci, Locus A 012 Locus B01/161/81/161/4 11/81/41/81/2 21/161/81/161/4 1/41/21/4

0 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 A 1 A 1 A 1 A 2 A 2 A 2 Marginal Freq B 1 B Locus B B 1 B B 2 B 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

AB Y  a b True Model A Y  a* Assumed Model a* is the apparent co-efficient a* will deviate from a to the extent that A and B are correlated

- DDV* A1 V* D1 V* A2 V* D2 V* AA V* AD V* DA - - ADV* A1 V* D1 V* A2 V* D2 V* AA AAV* A1 V* D1 V* A2 V* D DV* A1 -V* A AV* A H H FullV A1 V D1 V A2 V D2 V AA V AD V DA V DD V S and V N estimated in all models

Example 1 : epi1.mx Genotypic MeansB 1 B 1 B 1 B 2 B 2 B 2 A 1 A A 1 A A 2 A Allele frequenciesA 1 = 50% ; B 1 = 50% QTL variance 20% Shared residual variance40% Nonshared residual variance40% Sample N10, 000 unselected pairs Recombination fractionUnlinked (0.5)

Example 2 : epi2.mx Genotypic MeansB 1 B 1 B 1 B 2 B 2 B 2 A 1 A A 1 A A 2 A Allele frequenciesA 1 = 90% ; B 1 = 50% QTL variance 10% Shared residual variance20% Nonshared residual variance 70% Sample N2, 000 unselected pairs Recombination fraction0.1

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

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