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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
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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
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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
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Reconstruct marker haplotypes in sibships
and estimate inheritance vectors, taking
account of uncertainty in phase assignment.
Approximate distribution of location of diallelic 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 (1)
Inheritance vectors (2)
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Parental
genotypes
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Possible offspring inheritance vectors:
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Offspring genotype
Inheritance vectors (2)
θ recombination fraction
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Parental
genotypes
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Offspring genotype
Possible offspring inheritance vectors:
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0.5(1-θ)
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0.5θ
0.5θ
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0.5(1-θ)
Homozygous
parents
Inheritance vectors (2)
p
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θ recombination fraction
124
1-p
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Parental
genotypes
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Offspring genotype
Possible offspring inheritance vectors:
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p(1-θ)/T
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(1-p)θ/T
Estimate p from offspring genotypes across sibship
Unknown
parental phase
Inheritance vectors (2)
p
θ recombination fraction
Parental
genotypes
1-p
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144
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144
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Offspring genotype
Possible offspring inheritance vectors:
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124
144
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144
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144
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Sum
probabilities
over all
parental
genotypes!
Missing
parental
genotypes
Inheritance vectors (2)
140
124
142
122
Parental
genotypes
140
124
148
126
144
126
148
120
Offspring genotype
Possible offspring inheritance vectors:
140
124
148
126
Sensitivity
to genotyping
errors
Inhertitance vectors (3)
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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)
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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)
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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.
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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) = Πi Πk f(yik|x,M,Q,V)
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where yik is distributed N(μik,σ2) and, under a single QTL model:
μik = si + aqik + d[I(qik=1)]
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The sibship effect, si, is distributed N(λ,σS2).
The number of mutant QTL alleles, qik = (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
θL
x
θR
R
Parents
Offspring
Likelihood calculations (2)
L
θL
x
θR
R
Parents
Offspring
(1-θL)2(1-θR)2/T (1-θL)θL(1-θR)θR/T (1-θL)θL(1-θR)θR/T
qik = 1
qik = 2
qik = 0
θL2θR2/T
qik = 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).
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MF1 analysis
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Comparison of four models:
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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 2,000th output.
Initial 200,000 iterations excluded as burn-in,
resulting in 1,000 thinned sampling outputs.
MF1 analysis: 1 dominant QTL
95% credibility interval:
2.442-3.263Mb
MF1 analysis: 2 dominant QTLs
95% credibility intervals:
2.452-3.115Mb (DOM1)
3.774-4.485Mb (DOM2)
DOM1
DOM2
MF1 analysis: 2 dominant QTLs
95% credibility intervals:
2.452-3.115Mb (DOM1)
3.774-4.485Mb (DOM2)
DOM1
DOM2
MF1 analysis:
Comparison of models
Model m
Scaled
log[f(M=m|Y,V)]
-274.66
Posterior
probability
0.000
1 additive QTL
-202.66
0.000
1 dominant QTL
-107.98
0.000
2 dominant QTLs
39.43
1.000
Null
MF1 analysis: ongoing work
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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
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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.