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
JSM 2003
1
what is the goal of QTL study?
• uncover underlying biochemistry
–
–
–
–
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
–
–
–
–
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
Yandell © 2003
JSM 2003
2
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
Yandell © 2003
JSM 2003
3
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 )
2
h 
2
var( GQ )  
2
8
Yandell © 2003
data
histogram
qq
9
10
11
QQ
12
JSM 2003
13
14
15
16
17
18
19
20
4
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
Yandell © 2003
JSM 2003
5
Pareto diagram of QTL effects
3
(modifiers)
minor
QTL
polygenes
1
2
major
QTL
0
3
additive effect
major QTL on
linkage map
2
1
Yandell © 2003
0
4
5
10
15
20
25
30
rank order of QTL
5
JSM 2003
6
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
Yandell © 2003
JSM 2003
7
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
Yandell © 2003
JSM 2003
8
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
Yandell © 2003
JSM 2003
9
epistasis examples
3
1
effect +/- 2 se
0
4
bb
Bb
BB
a1
d1
a2
d2
iaa iad ida idd
aa
Aa
AA
Aa
AA
bb
Bb
BB
-2
0
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
Yandell © 2003
0.0
-1.0
-0.5
3.0
2.5
aa
1.5
2.0
aa
Aa
AA
1.0
2.0
1.5
1.0
bb
Bb
BB
effect +/- 2 se
3.5
genotypic value
genotypic
value 3.5
2.5
3.0
4.0
4
0.5
4.5
aa
aa
0
bb
BB
Bb
-40
20
20
-1
2
2
-20
genotypic value
2
6
6
20
0
60
aa
40
40
Bb
bb
9
genotypic4value
80
Aa
genotypic value
genotypic
value
60
80
AA
effect +/- 2 se
BB
1
100
100
(Doebley Stec Gustus 1995; Zeng pers. comm.)
bb
Bb
BB
a1
d1
a2
d2
iaa iad ida idd
JSM 2003
10
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
–
–
–
–
–
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)
Yandell © 2003
JSM 2003
11
idea of mapping microarrays
(Jansen Nap 2001)
Yandell © 2003
JSM 2003
12
goal: unravel biochemical pathways
(Jansen Nap 2001)
Yandell © 2003
JSM 2003
13
central dogma via microarrays
(Bochner 2003)
Yandell © 2003
JSM 2003
14
coordinated expression in mouse
genome (Schadt et al. 2003 Nature)
expression
pleiotropy
in yeast genome
(Brem et al.
2002 Science)
Yandell © 2003
JSM 2003
15
glucose
insulin
(courtesy AD Attie)
Yandell © 2003
JSM 2003
16
decompensation
Yandell
from
Unger©
& 2003
Orci FASEB J. (2001) 15,312
JSM 2003
17
Type 2 Diabetes Mellitus
Yandell
from
Unger©
& 2003
Orci FASEB J. (2001) 15,312
JSM 2003
18
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
•
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,…
Yandell © 2003
JSM 2003
19
LOD map for PDI: cis-regulation
Lan et al. (2003 submitted)
Yandell © 2003
JSM 2003
20
effect (add=blue, dom=red)
-0.5 0.0 0.5 1.0
0
LOD
2
4
6
8
Multiple Interval Mapping
SCD1: multiple QTL plus epistasis!
0
0
Yandell © 2003
50
chr2
100
50
chr2
100
150
200
250
chr9
300
200
250
chr9
300
chr5
150
chr5
JSM 2003
21
2-D scan: assumes only 2 QTL!
epistasis LOD
joint LOD
Yandell © 2003
JSM 2003
22
trans-acting QTL for SCD1
(no epistasis yet: see Yi, Xu, Allison 2003)
dominance?
Yandell © 2003
JSM 2003
23
1
3
Yandell © 2003
5
7
9
11
model index
13
15
JSM 2003
1
3
5
7
9
model index
11
2
3
5
6
moderate
6
6
5
5
4
4
6
5
3
4
3:1,2,3
0.15
posterior / prior
0.2
0.4
0.6 0.8
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
2:1,2
model posterior
0.05
0.10
pattern posterior
2
0.00
Bayesian model assessment:
chromosome QTL pattern for SCD1
Bayes factor ratios
weak
13
15
24
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
–
–
–
–
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
Yandell © 2003
JSM 2003
25
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
Yandell © 2003
JSM 2003
26
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
Yandell © 2003
JSM 2003
27
1
0
V2
-1
0
-2
-5
mRNA2
5
2
PC simply rotates & rescales
to find major axes of variation
-5
0
5
-2
mRNA1
Yandell © 2003
-1
0
1
V1
JSM 2003
28
multivariate screen
for gene expressing mapping
principal components
PC2 (22%)
PC1(red) and SCD(black)
PC1 (42%)
Yandell © 2003
JSM 2003
29
0.020
0.000
ch2
50
0
50
0
50
100
150
100
150
100
150
ch5
200
250
200
250
200
250
ch9
300
0.5
0
-1.0
additive
ch2,ch5,ch9
ch2
ch5
ch9
300
0.0
dominance
hong7pc.bim summaries with pattern
-2.0
loci histogram
mapping first diabetes PC as a trait
Yandell © 2003
ch2
ch5
JSM 2003
ch9
300
30
0.2
1.0
0.1
0.15
Storey pFDR(-)
0.05
0.2
0.0
0.2
0.4
0.6
0.8
relative size of HPD region
Yandell © 2003
0
0.0
0.0
0.2
pr( H=0 | p>size )
0.4
0.6
0.8
prior probability
fraction of posterior
found in tails
BH pFDR(-) and size(.)
0.4
0.6
0.8
1.0
pFDR for PC1 analysis
0.0
JSM 2003
0.2
0.4
0.6
0.8
pr( locus in HPD | m>0 )
1.0
31
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
Yandell © 2003
JSM 2003
32
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
Yandell © 2003
JSM 2003
33