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Dimension Reduction
for Mapping mRNA Abundance as
Quantitative Traits
Brian S. Yandell
University of Wisconsin-Madison
www.stat.wisc.edu/~yandell/statgen
[Lan et al. 2003 Genetics 164(4): August 2003]
Yandell © 2003
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what is the goal of QTL study?
• uncover underlying biochemistry
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identify how networks function, break down
find useful candidates for (medical) intervention
epistasis may play key role
statistical goal: maximize number of correctly identified QTL
• basic science/evolution
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how is the genome organized?
identify units of natural selection
additive effects may be most important (Wright/Fisher debate)
statistical goal: maximize number of correctly identified QTL
• select “elite” individuals
– predict phenotype (breeding value) using suite of characteristics
(phenotypes) translated into a few QTL
– statistical goal: mimimize prediction error
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what is a QTL?
• QTL = quantitative trait locus (or loci)
– trait = phenotype = characteristic of interest
– quantitative = measured somehow
• glucose, insulin, gene expression level
• Mendelian genetics
– allelic effect + environmental variation
– locus = location in genome affecting trait
• gene or collection of tightly linked genes
• some physical feature of genome
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typical phenotype assumptions
• normal "bell-shaped" environmental variation
• genotypic value GQ is composite of m QTL
• genetic uncorrelated with environment
E (Y | Q )    GQ
Qq
var (Y | Q )  
var( GQ )
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h 
2
var( GQ )  
2
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data
histogram
qq
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QQ
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why worry about multiple QTL?
• so many possible genetic architectures!
– number and positions of loci
– gene action: additive, dominance, epistasis
– how to efficiently search the model space?
• how to select “best” or “better” model(s)?
– what criteria to use? where to draw the line?
– shades of gray: exploratory vs. confirmatory study
– how to balance false positives, false negatives?
• what are the key “features” of model?
– means, variances & covariances, confidence regions
– marginal or conditional distributions
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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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advantages of 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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epistasis in parallel pathways
(Gary Churchill)
X
• Z keeps trait value low
E1
Z
• Neither E1 nor E2 is rate
limiting
Y
E2
• Loss of function alleles are
segregating from parent A at
E1 and from parent B at E2
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epistasis in a serial pathway
(Gary Churchill)
• Z keeps trait value high
X
E1
Y
E2
Z
• Neither E1 nor E2 is rate
limiting
• Loss of function alleles are
segregating from parent B at
E1 and from parent A at E2
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epistasis examples
3
1
effect +/- 2 se
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bb
Bb
BB
a1
d1
a2
d2
iaa iad ida idd
aa
Aa
AA
Aa
AA
bb
Bb
BB
-2
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AA
a1
d1
a2
d2
iaa iad ida idd
4.5
Aa
4.0
traits 1,4,9
1: dom-dom interaction
4: add-add interaction
9: rec-rec interaction
(Fisher-Cockerham effects)
Aa
AA
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0.0
-1.0
-0.5
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aa
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bb
Bb
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effect +/- 2 se
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genotypic value
genotypic
value 3.5
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aa
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bb
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genotypic value
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genotypic4value
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genotypic value
genotypic
value
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effect +/- 2 se
BB
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(Doebley Stec Gustus 1995; Zeng pers. comm.)
bb
Bb
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a1
d1
a2
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iaa iad ida idd
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why map gene expression
as a quantitative trait?
• cis- or trans-action?
– does gene control its own expression?
– evidence for both modes (Brem et al. 2002 Science)
• mechanics of gene expression mapping
– measure gene expression in intercross (F2) population
– map expression as quantitative trait (QTL technology)
– adjust for multiple testing via false discovery rate
• research groups working on expression QTLs
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review by Cheung and Spielman (2002 Nat Gen Suppl)
Kruglyak (Brem et al. 2002 Science)
Doerge et al. (Purdue); Jansen et al. (Waginingen)
Williams et al. (U KY); Lusis et al. (UCLA)
Dumas et al. (2000 J Hypertension)
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idea of mapping microarrays
(Jansen Nap 2001)
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goal: unravel biochemical pathways
(Jansen Nap 2001)
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central dogma via microarrays
(Bochner 2003)
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coordinated expression in mouse
genome (Schadt et al. 2003 Nature)
expression
pleiotropy
in yeast genome
(Brem et al.
2002 Science)
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glucose
insulin
(courtesy AD Attie)
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decompensation
Yandell
from
Unger©
& 2003
Orci FASEB J. (2001) 15,312
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Type 2 Diabetes Mellitus
Yandell
from
Unger©
& 2003
Orci FASEB J. (2001) 15,312
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studying diabetes in an F2
• segregating cross of inbred lines
– B6.ob x BTBR.ob  F1  F2
– selected mice with ob/ob alleles at leptin gene (chr 6)
– measured and mapped body weight, insulin, glucose at various ages
• (Stoehr et al. 2000 Diabetes)
– sacrificed at 14 weeks, tissues preserved
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gene expression data
– Affymetrix microarrays on parental strains, F1
• key tissues: adipose, liver, muscle, -cells
• novel discoveries of differential expression (Nadler et al. 2000 PNAS; Lan et
al. 2002 in review; Ntambi et al. 2002 PNAS)
– RT-PCR on 108 F2 mice liver tissues
• 15 genes, selected as important in diabetes pathways
• SCD1, PEPCK, ACO, FAS, GPAT, PPARgamma, PPARalpha, G6Pase, PDI,…
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LOD map for PDI: cis-regulation
Lan et al. (2003 submitted)
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effect (add=blue, dom=red)
-0.5 0.0 0.5 1.0
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LOD
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Multiple Interval Mapping
SCD1: multiple QTL plus epistasis!
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chr2
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chr2
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chr9
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chr5
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chr5
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2-D scan: assumes only 2 QTL!
epistasis LOD
joint LOD
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trans-acting QTL for SCD1
(no epistasis yet: see Yi, Xu, Allison 2003)
dominance?
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model index
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model index
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moderate
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posterior / prior
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4:2*1,2,3
4:1,2,2*3
4:1,2*2,3
5:3*1,2,3
5:2*1,2,2*3
5:2*1,2*2,3
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
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model posterior
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pattern posterior
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Bayesian model assessment:
chromosome QTL pattern for SCD1
Bayes factor ratios
weak
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high throughput dilemma
• want to focus on gene expression network
– hundreds or thousands of genes/proteins to monitor
– ideally capture networks in a few dimensions
• multivariate summaries of multiple traits
– elicit biochemical pathways
• (Henderson et al. Hoeschele 2001; Ong Page 2002)
• may have multiple controlling loci
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allow for complicated genetic architecture
could affect many genes in coordinated fashion
could show evidence of epistasis
quick assessment via interval mapping may be misleading
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why study multiple traits together?
• environmental correlation
– non-genetic, controllable by design
– historical correlation (learned behavior)
– physiological correlation (same body)
• genetic correlation
– pleiotropy
• one gene, many functions
• common biochemical pathway, splicing variants
– close linkage
• two tightly linked genes
• genotypes Q are collinear
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high throughput:
which genes are the key players?
• one approach:
clustering of expression
seed by insulin, glucose
• advantage:
subset relevant to trait
• disadvantage:
still many genes to study
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1
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V2
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mRNA2
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PC simply rotates & rescales
to find major axes of variation
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mRNA1
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multivariate screen
for gene expressing mapping
principal components
PC2 (22%)
PC1(red) and SCD(black)
PC1 (42%)
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0.020
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ch2
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ch5
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ch9
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additive
ch2,ch5,ch9
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ch5
ch9
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dominance
hong7pc.bim summaries with pattern
-2.0
loci histogram
mapping first diabetes PC as a trait
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Storey pFDR(-)
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relative size of HPD region
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prior probability
fraction of posterior
found in tails
BH pFDR(-) and size(.)
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pr( locus in HPD | m>0 )
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false detection rates and thresholds
• multiple comparisons: test QTL across genome
– size = pr( LOD() > threshold | no QTL at  )
– threshold guards against a single false detection
• very conservative on genome-wide basis
– difficult to extend to multiple QTL
• positive false discovery rate (Storey 2001)
– pFDR = pr( no QTL at  | LOD() > threshold )
– Bayesian posterior HPD region based on threshold
•  ={ | LOD() > threshold }  { | pr( | Y,X,m ) large }
– extends naturally to multiple QTL
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pFDR and QTL posterior
• positive false detection rate
– pFDR = pr( no QTL at  | Y,X,  in  )
– pFDR =
pr(H=0)*size
pr(m=0)*size+pr(m>0)*power
– power = posterior = pr(QTL in  | Y,X, m>0 )
– size = (length of ) / (length of genome)
• extends to other model comparisons
– m = 1 vs. m = 2 or more QTL
– pattern = ch1,ch2,ch3 vs. pattern > 2*ch1,ch2,ch3
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