Bayesian MCMC QTL mapping in outbred mice Andrew Morris, Binnaz Yalcin, Jan Fullerton, Angela Meesaq, Rob Deacon, Nick Rawlins and Jonathan Flint Wellcome Trust Centre for Human Genetics University of Oxford

Motivation Analysis of heterogeneous stock (HS) mice provides reasonable evidence of (at least) one QTL for anxiety related trait in ~4Mb region of chromosome 1, encompassing cluster of RGS genes. Intensive sequencing of HS founder strains has identified two sub-regions with high probability of containing QTLs. Can we replicate these findings in other samples of mice? Can we refine the location of potential QTLs? Can we distinguish between single and multiple QTL effects?

MF1 sample Sample of MF1 outbred mice obtained from Harlan. Large sibships of mice phenotyped for anxiety related trait and genotyped at more than 40 SNPs and microsatellites in 4Mb candidate region. Parental phenotype and marker information not generally available. 93 sibships, 729 phenotyped offspring.

Method outline Reconstruct marker haplotypes in sibships and estimate inheritance vectors, taking account of uncertainty in phase assignment. Approximate distribution of location of di- allelic QTL(s) (mutant/normal) in candidate region given phenotypes and inheritance vectors, allowing for uncertainty in parental QTL alleles. Compare additive vs dominant genetic effect models and single QTL vs multiple QTL models.

Inheritance vectors (1)

Inheritance vectors (2) Parental genotypes Possible offspring inheritance vectors: Offspring genotype

Inheritance vectors (2) θ recombination fraction Parental genotypes Possible offspring inheritance vectors: (1-θ) 0.5θ 0.5(1-θ) Homozygous parents Offspring genotype

Inheritance vectors (2) θ recombination fraction Parental genotypes Possible offspring inheritance vectors: p(1-θ)/T (1-p)θ/T Unknown parental phase Offspring genotype p 1-p Estimate p from offspring genotypes across sibship

Inheritance vectors (2) θ recombination fraction Parental genotypes Possible offspring inheritance vectors: Missing parental genotypes Offspring genotype p 1-p Sum probabilities over all parental genotypes!

Inheritance vectors (2) Parental genotypes Possible offspring inheritance vectors: Offspring genotype Sensitivity to genotyping errors

Inhertitance vectors (3) For MF1 sample, parental genotypes not currently available. Too many combinations of parental genotypes to consider all markers simultaneously. Use overlapping sliding window of five markers, and combine information across windows.

Bayesian framework (1) Goal is to approximate posterior distribution of location(s) of QTL(s), f(x|Y,V), given offspring phenotypes, Y, and estimated inheritance vectors V. Recovered by integration f(x|Y,V) = ∫ M ∫ Q f(x,M,Q|Y,V) dQdM over genetic effect model parameters, M, and parental QTL alleles, Q.

Bayesian framework (2) By Bayes' theorem f(x,M,Q|Y,V) = C f(Y|x,M,Q,V) f(x,M,Q) where f(Y|x,M,Q,V) is the likelihood of the phenotype data and f(x,M,Q) is the prior density of location(s), genetic effect model parameters and parental QTL alleles. Assume independent uniform priors for location(s) and genetic effect parameters, and that all assignments of parental QTL alleles are equally likely, a priori. Hence f(x,M,Q) is constant.

Likelihood calculations (1) Conditional on inheritance vectors and parental QTL alleles, the phenotype of offspring, k, in the same sibship, i, will be independent: f(Y|x,M,Q,V) = Π i Π k f(y ik |x,M,Q,V) where y ik is distributed N(μ ik,σ 2 ) and, under a single QTL model: μ ik = s i + aq ik + d[I(q ik =1)] The sibship effect, s i, is distributed N(λ,σ S 2 ). The number of mutant QTL alleles, q ik = (0,1,2), will have a distribution determined by x, Q and V, so weight likelihood according to corresponding inheritance vector probabilities…

Likelihood calculations (2) L θ L x θ R R Parents Offspring

Likelihood calculations (2) L θ L x θ R R Parents Offspring (1-θ L ) 2 (1-θ R ) 2 /T(1-θ L )θ L (1-θ R )θ R /Tθ L 2 θ R 2 /T(1-θ L )θ L (1-θ R )θ R /T q ik = 1 q ik = 2 q ik = 0 q ik = 1

MCMC algorithm Employ Metropolis-Hastings MCMC algorithm to approximate target posterior distribution f(x,M,Q|Y,V). Random initial parameter configuration, P = {x,M,Q}. Propose small change to parameter configuration, P*. Accept new parameter configuration with probability f(P*|Y,V)/f(P|Y,V), otherwise current configuration retained. On convergence, each configuration accepted (or retained) by the algorithm represents a random draw from f(P|Y,V).

MF1 analysis Comparison of four models: no QTLs in candidate region (null); one additive QTL in candidate region; one dominant QTL in candidate region; two dominant QTLs in candidate region. Assume uniform recombination rate across candidate region, a priori. 2.2 million iterations of MCMC algorithm, thinned to every 2,000 th output. Initial 200,000 iterations excluded as burn-in, resulting in 1,000 thinned sampling outputs.

MF1 analysis: 1 dominant QTL 95% credibility interval: Mb

MF1 analysis: 2 dominant QTLs 95% credibility intervals: Mb (DOM1) Mb (DOM2) DOM1 DOM2

MF1 analysis: 2 dominant QTLs 95% credibility intervals: Mb (DOM1) Mb (DOM2) DOM1 DOM2

MF1 analysis: Comparison of models Model mScaled log[f(M=m|Y,V)] Posterior probability Null additive QTL dominant QTL dominant QTLs

MF1 analysis: ongoing work Model 3 dominant QTLs in candidate region. Incorporate parental genotype information. Additional genotyping in vicinity of DOM1. Sensitivity to marker selection and genotyping error. Investigate properties of algorithm under null model by random permutation of offspring phenotypes.

Summary Bayesian MCMC method developed to approximate distribution of location of QTLs in candidate region. Designed for use with large sibships of outbred mice, but could be generalised to other pedigree structures. Analysis of MF1 sample suggests evidence of (at least) two QTLs, one in the vicinity of RGS18.