Inferring Genetic Architecture
of Complex Biological Processes
Brian S. Yandell12, Christina Kendziorski13, Hong Lan4,
Elias Chaibub1, Alan D. Attie4
U Alabama Birmingham
1
Department of Statistics
2 Department of Horticulture
3 Department of Biostatistics & Medical Informatics
4 Department of Biochemistry
University of Wisconsin-Madison
http://www.stat.wisc.edu/~yandell/statgen
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glucose
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(courtesy AD Attie)
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insulin
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studying diabetes in an F2
• mouse model: segregating panel from inbred lines
– B6.ob x BTBR.ob  F1  F2
– selected mice with ob/ob alleles at leptin gene (Chr 6)
– sacrificed at 14 weeks, tissues preserved
• physiological study (Stoehr et al. 2000 Diabetes)
– mapped body weight, insulin, glucose at various ages
• gene expression studies
– RT-PCR for a few mRNA on 108 F2 mice liver tissues
• (Lan et al. 2003 Diabetes; Lan et al. 2003 Genetics)
– Affymetrix microarrays on 60 F2 mice liver tissues
• U47 A & B chips, RMA normalization
• design: selective phenotyping (Jin et al. 2004 Genetics)
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The intercross (from K Broman)

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mRNA expression as phenotype:
effect (add=blue, dom=red)
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LOD
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interval mapping for SCD1 is complicated
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chr2
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chr9
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taking a multiple QTL approach
• improve statistical power, precision
– increase number of QTL detected
– better estimates of loci: less bias, smaller intervals
• improve inference of complex genetic architecture
– patterns and individual elements of epistasis
– appropriate estimates of means, variances, covariances
• asymptotically unbiased, efficient
– assess relative contributions of different QTL
• improve estimates of genotypic values
– less bias (more accurate) and smaller variance (more precise)
– mean squared error = MSE = (bias)2 + variance
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Pareto diagram of QTL effects
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(modifiers)
minor
QTL
polygenes
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major
QTL
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additive effect
major QTL on
linkage map
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rank order of QTL
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Bayesian model assessment:
number of QTL for SCD1 with
R/bim
Bayes factor ratios
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posterior / prior
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QTL posterior
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QTL posterior
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moderate
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weak
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number of QTL
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number of QTL
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model index
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model index
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posterior / prior
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6:3*1,2,2*3
6:3*1,2*2,3
5:1,2*2,2*3
6:4*1,2,3
6:2*1,2*2,2*3
2:1,3
3:2*1,2
2:1,2
model posterior
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pattern posterior
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Bayesian model assessment
genetic architecture: chromosome pattern
Bayes factor ratios
weak
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trans-acting QTL for SCD1
Bayesian model averaging with R/bim
additive QTL?
dominant QTL?
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Bayesian LOD and h2 for SCD1
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LOD
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density
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(summaries from R/bim)
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LOD conditional on number of QTL
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heritability
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marginal LOD, m
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marginal heritability, m
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heritability conditional on number of QTL
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SCD mRNA expression phenotype
2-D scan for QTL (R/qtl)
epistasis
LOD
peaks
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joint
LOD
peaks
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sub-peaks can be easily overlooked
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hetereogeneity: many genes affect each trait
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(modifiers)
minor
QTL
polygenes
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major
QTL
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additive effect
major QTL on
linkage map
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rank order of QTL
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interval mapping basics
•
observed measurements
– Y = phenotypic trait
– X = markers & linkage map
observed
• i = individual index 1,…,n
•
missing
• alleles QQ, Qq, or qq at locus
Q
unknown quantities
– M = genetic architecture
–  = QT locus (or loci)
–  = phenotype model parameters
•
Y
missing data
– missing marker data
– Q = QT genotypes
•
X
unknown
pr(Q=q|X,) genotype model
after
Sen & Churchill
(2001 Genetics)

– grounded by linkage map, experimental cross
– recombination yields multinomial for Q given X
•
f(Y|q) phenotype model
– distribution shape (assumed normal here)
– unknown parameters  (could be non-parametric)
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M
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genetic architecture: heterogeneity
•
heterogeneity: many genes can affect phenotype
–
–
–
•
different allelic combinations can yield similar phenotypes
multiple genes can affecting phenotype in subtle ways
multiple genes can interact (epistasis)
genetic architecture: model for explained genetic variation
–
–
loci (genomic regions) that affect trait
genotypic effects of loci, including possible epistasis
M = {1, 2, 3, (1, 2)}
= 3 loci with epistasis between two
q = 0 + q1 + q2 + q3 + q (1,2) = linear model for genotypic mean
 = (1, 2, 3)
q = (q1, q2, q3)
Q = (Q1, Q2, Q3)
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= loci in model M
= possible genotype at loci 
= genotype for each individual at loci 
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multiple QTL interval mapping
• genotypic mean depends on model M
q = 0 + {q in M} qk
• interval mapping between flanking markers
f(Y | X, M) = q f(Y | q) f(Q = q | X, )
• model selection
– choice of distribution: f is normal
– sample many possible architectures
– compare based on Bayes factors (BIC)
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2M observations
30,000 traits
60 mice
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modern high throughput biology
• measuring the molecular dogma of biology
– DNA  RNA  protein  metabolites
– measured one at a time only a few years ago
• massive array of measurements on whole systems (“omics”)
– thousands measured per individual (experimental unit)
– all (or most) components of system measured simultaneously
•
•
•
•
whole genome of DNA: genes, promoters, etc.
all expressed RNA in a tissue or cell
all proteins
all metabolites
• systems biology: focus on network interconnections
– chains of behavior in ecological community
– underlying biochemical pathways
• genetics as one experimental tool
– perturb system by creating new experimental cross
– each individual is a unique mosaic
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finding heritable traits
(from Christina Kendziorski)
•
reduce 30,000 traits to 300-3,000 heritable traits
•
probability a trait is heritable
pr(H|Y,Q) = pr(Y|Q,H) pr(H|Q) / pr(Y|Q)
Bayes rule
pr(Y|Q) = pr(Y|Q,H) pr(H|Q) + pr(Y|Q, not H) pr(not H|Q)
•
phenotype averaged over genotypic mean 
pr(Y|Q, not H) = f0(Y) =  f(Y| ) pr() d
if not H
pr(Y|Q, H) = f1(Y|Q) = q f0(Yq )
if heritable
Yq = {Yi | Qi =q} = trait values with genotype Q=q
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hierarchical model for expression phenotypes
(EB arrays: Christina Kendziorski)

YQQ QQ ~ f  QQ


YQq  Qq ~ f   Qq
mRNA phenotype models
given genotypic mean q


Yqq qq ~ f  qq
 QQ
 qq
 Qq
common prior on q across all mRNA
(use empirical Bayes to estimate prior)
 q ~ pr 
 QQ
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
 Qq
 qq
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expression meta-traits: pleiotropy
• reduce 3,000 heritable traits to 3 meta-traits(!)
• what are expression meta-traits?
– pleiotropy: a few genes can affect many traits
• transcription factors, regulators
– weighted averages: Z = YW
• principle components, discriminant analysis
• infer genetic architecture of meta-traits
– model selection issues are subtle
• missing data, non-linear search
• what is the best criterion for model selection?
– time consuming process
• heavy computation load for many traits
• subjective judgement on what is best
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7.6
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ettf1
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PC2 (7%)
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PC for two correlated mRNA
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PC1 (93%)
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PC across microarray functional groups
Affy chips on 60 mice
~40,000 mRNA
2500+ mRNA show DE
(via EB arrays with
marker regression)
1500+ organized in
85 functional groups
2-35 mRNA / group
which are interesting?
examine PC1, PC2
circle size = # unique mRNA
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factor loadings for PC1&2
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how well does PC1 do?
lod peaks for 2 QTL at best pair of chr
data (red) vs. 500 permutations (boxplots)
blue bars at 1%, 5%; width proportional to group size
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84 PC meta-traits by functional group
focus on 2 interesting groups
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red lines: peak
for PC meta-trait
black/blue: peaks
for mRNA traits
arrows: cis-action?
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(portion of) chr 4 region
chr 15 region
?
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DA meta-traits: separate pleiotropy
from environmental correlation
pleiotropy only
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environmental
correlation only
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both
Korol et al. (2001)
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interaction plots for DA meta-traits
DA for all pairs of markers:
separate 9 genotypes based on markers
(a) same locus pair found with PC meta-traits
(b) Chr 2 region interesting from biochemistry (Jessica Byers)
(c) Chr 5 & Chr 9 identified as important for insulin, SCD
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B.H
PC ignores genotype
2
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note better
spread of circles
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correlation of
PC and DA meta-traits
B.B
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PC1 (25%)
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DA2 (18%)
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H.H
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DA creates best
separation by
genotype
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PC2 (12%)
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DA uses genotype
H.B
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PC captures
spread without
genotype
H.B
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genotypes from
Chr 4/Chr 15
locus pair
(circle=centroid)
3
comparison of PC and DA meta-traits on 1500+ mRNA traits
B.H
A.B
-5
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PC1 (25%)
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SCD trait
log2 expression
DA meta-trait
standard units
relating meta-traits to mRNA traits
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DA: a cautionary tale
(184 mRNA with |cor| > 0.5; mouse 13 drives heritability)
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building graphical models
• infer genetic architecture of meta-trait
– E(Z | Q, M) = q = 0 + {q in M} qk
• find mRNA traits correlated with meta-trait
– Z  YW for modest number of traits Y
• extend meta-trait genetic architecture
– M = genetic architecture for Y
– expect subset of QTL to affect each mRNA
– may be additional QTL for some mRNA
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posterior for graphical models
•posterior for graph given multivariate trait & architecture
pr(G | Y, Q, M) = pr(Y | Q, G) pr(G | M) / pr(Y | Q)
–pr(G | M) = prior on valid graphs given architecture
•multivariate phenotype averaged over genotypic mean 
pr(Y | Q, G) = f1(Y | Q, G) = q f0(Yq | G)
f0(Yq | G) =  f(Yq | , G) pr() d
•graphical model G implies correlation structure on Y
•genotype mean prior assumed independent across traits
pr() = t pr(t)
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from graphical models to pathways
• build graphical models
QTL  RNA1  RNA2
– class of possible models
– best model = putative biochemical pathway
• parallel biochemical investigation
– candidate genes in QTL regions
– laboratory experiments on pathway components
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graphical models (with Elias Chaibub)
f1(Y | Q, G=g) = f1(Y1 | Q) f1(Y2 | Q, Y1)
QTL
DNA
RNA
QTL
D1
R1
D2
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unobservable
protein
meta-trait
P1
observable
cis-action?
P2
observable
trans-action
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