Yandell © 2003 JSM 2003 1

Dimension Reductionfor Mapping mRNA Abundance

as Quantitative Traits

Brian S. YandellUniversity of Wisconsin-Madison

www.stat.wisc.edu/~yandell/statgen[Lan et al. 2003 Genetics 164(4): August 2003]

Yandell © 2003 JSM 2003 2

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

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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

22

2

)var(

)var()|(var

)|(

Q

Q

Q

G

Gh

QY

GQYE

8 9 10 11 12 13 14 15 16 17 18 19 20

qq QQ

Qqdatahistogram

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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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0 5 10 15 20 25 30

01

23

rank order of QTL

addi

tive

eff

ect

Pareto diagram of QTL effects

54

3

2

1

major QTL onlinkage map

majorQTL

minorQTL

polygenes

(modifiers)

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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)

• Z keeps trait value low

• Neither E1nor E2 is rate limiting

• Loss of function alleles are segregating from parent A

at E1and from parent B at E2

Z

X

Y

E1

E2

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epistasis in a serial pathway(Gary Churchill)

ZX YE1 E2• Z keeps trait value high

• Neither E1 nor E2 is rate limiting

• Loss of function alleles are segregating from parent B

at E1and from parent A at E2

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epistasis examples(Doebley Stec Gustus 1995; Zeng pers. comm.)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

geno

typi

c va

lue

aa Aa AA

bb

BbBB 1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

geno

typi

c va

lue

bb Bb BB

aa

AaAA

-1.0

-0.5

0.0

0.5

effe

ct +

/- 2

se

a1 d1 a2 d2 iaa iad ida idd

20

40

60

80

10

0ge

noty

pic

valu

e

aa Aa AA

bbBb

BB2

04

06

08

01

00

geno

typi

c va

lue

bb Bb BB

aa

Aa

AA

-40

-20

02

0ef

fect

+/-

2 s

e

a1 d1 a2 d2 iaa iad ida idd

02

46

geno

typi

c va

lue

aa Aa AA

bb

BbBB 02

46

geno

typi

c va

lue

bb Bb BB

aa

AaAA -2-1

01

23

effe

ct +

/- 2

se

a1 d1 a2 d2 iaa iad ida idd

traits 1,4,91: dom-dom interaction4: add-add interaction9: rec-rec interaction(Fisher-Cockerham effects)

4

91

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why map gene expressionas 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)

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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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Insulin Requirement

from Unger & Orci FASEB J. (2001) 15,312

decompensation

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Type 2 Diabetes Mellitus

from Unger & 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

• 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-regulationLan et al. (2003 submitted)

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0 50 100 150 200 250 300

02

46

8LO

D

chr2 chr5 chr9

0 50 100 150 200 250 300

-0.5

0.0

0.5

1.0

effe

ct (

add=

blue

, do

m=

red)

chr2 chr5 chr9

Multiple Interval MappingSCD1: multiple QTL plus epistasis!

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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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Bayesian model assessment:chromosome QTL pattern for SCD1

1 3 5 7 9 11 13 15

0.00

0.05

0.10

0.15

4:1,

2,2*

34:

1,2*

2,3

5:3*

1,2,

35:

2*1,

2,2*

35:

2*1,

2*2,

36:

3*1,

2,2*

36:

3*1,

2*2,

35:

1,2*

2,2*

36:

4*1,

2,3

6:2*

1,2*

2,2*

32:

1,3

3:2*

1,2

2:1,

2

3:1,

2,3

4:2*

1,2,

3

model index

mod

el p

oste

rior

pattern posterior

0.2

0.4

0.6

0.8

model index

post

erio

r /

prio

r

Bayes factor ratios

1 3 5 7 9 11 13 15

3 44 4

55 5

6 65

66

23

2

weak

moderate

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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– 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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PC simply rotates & rescalesto find major axes of variation

-5 0 5

-50

5

mRNA1

mR

NA

2

-2 -1 0 1

-2-1

01

2

V1

V2

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multivariate screenfor gene expressing mapping

principal componentsPC1(red) and SCD(black)

PC2

(22%

)

PC1 (42%)

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mapping first diabetes PC as a trait

0 50 100 150 200 250 300

0.0

00

0.0

20

loci

his

togr

am

hong7pc.bim summaries with pattern ch2,ch5,ch9

ch2 ch5 ch9

0 50 100 150 200 250 300

-1.0

0.5

addi

tive

ch2 ch5 ch9

0 50 100 150 200 250 300

-2.0

0.0

dom

inan

ce

ch2 ch5 ch9

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pFDR for PC1 analysis

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

relative size of HPD region

pr(

H=

0 | p

>si

ze )

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

pr( locus in HPD | m>0 )

BH

pF

DR

(-)

and

size

(.)

00.

050.

10.

150.

2S

tore

y pF

DR

(-)

prior probability

fraction of posterior

found in tails

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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)*sizepr(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