variance heterogeneity in genetic mapping · variance heterogeneity in genetic mapping robert corty...

Variance Heterogeneity in Genetic Mapping

Robert Corty

Curriculum of Bioinformatics and Computational BiologyValdar lab

UNC Chapel Hill

May 29, 2018

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Table of Contents

1. Introduction to Genetic Mapping (10 minutes)

2. Genetic mapping with the double generalized linear model (30 minutes)

3. Genetic Mapping with the weighted linear mixed model (10 minutes)

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Introduction to Genetic Mapping Background

Table of Contents

1. Introduction to Genetic Mapping (10 minutes)BackgroundMotivating Concept



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The Need for Genetic Mapping

I’ve tried everything in the book,but the patient is still suffering.What else can I try?

I’m a world-renowned expert inthe field, but I can’t answer yourquestion.

Would be very helpful to know which genes and pathways are involvedin disease pathophysiology.

shutterstock.com4 / 64


Approaches to Genetic Mapping

QTL = “quantitative trait locus”A genomic region containing factors that influence a quantitativetrait of interest

First step toward finding causal genes and regulatory factors, whichteach us about (patho)physiology and can provide drug targets.

Human health and disease:human studies: find QTL for the relevant trait directlymodel organisms: find QTL that influence a model trait

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Genetic Mapping is Ubiquitous and Productive

relevant for every organism(first page of PubMed results include cow, wheat, maize, barley,soybean, petunia, pundamilia, horse, peanut, eucalyptus, catfish)relevant for (nearly) every disease and characteristicmouse

the Jackson lab lists 12,397 QTL identifiedR/qtl has been cited 2,558 times

humanNHGRI lists 5,168 QTL identified in humanGCTA has been cited 2,008 times

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Genetic Mapping Procedure (Generic)

1 Measure the trait of interest in a population of genetically-diverseorganisms

2 Measure genetic variants an an appropriate density3 Apply an appropriate statistical test to each locus to quantify the

evidence that it is a QTL

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Introduction to Genetic Mapping Motivating Concept

Table of Contents




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Example QTL

A Ballele

trai

t

A Ballele

trai

t

9 / 64


Variance Heterogeneity

A Ballele

trai

t

may be interesting as a QTL itselfmay be useful to “accommodate”

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Genetic mapping with the double generalized linear model

Table of Contents


2. Genetic mapping with the double generalized linear model (30 minutes)Introduction to Linkage Disequilibrium MappingStandard Approach to LD MappingDGLM-based Approch to LD MappingBailey Reanalysis Identifies vQTLKumar Reanalysis Identifies “mQTL”Leamy Reanalysis Identifies mQTL


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Genetic mapping with the double generalized linear model Introduction to Linkage Disequilibrium Mapping

Table of Contents




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Terms

SLM = standard linear modela linear model with a Gaussian error term

DGLM = double generalized linear modela combination of two linear models, where one models patternsof mean heterogeneity and the other models patterns ofvariance heterogeneity (Smyth, 1989, JRSSB)

Both of these models assume that, conditional on the locus andcovariates, the phenotypes are independent.

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Relevant Mapping Populations

F2 intercross backcross

}}inbreds

studypop. }

}inbredsstudypop.

Collaborative CrossDrosophila Synthetic Population ResourceRequires careful and intentional breeding

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Genetic mapping with the double generalized linear model Standard Approach to LD Mapping

Table of Contents




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Standard Approach to Linkage Disequilibrium (LD)Mapping

1 Breed a mapping population2 Measure the trait of interest in each organism3 Measure genotypes at a relevant set of markers (often sparse)4 Infer haplotype probabilities at a dense set of loci5 At each locus, test whether haplotype influences trait mean

16 / 64


Example QTL Mapping Result

Suto, 2013, J. Vet. Med. Sci.

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What do the Data look like?

typical locus

A Ballele

trai

t

QTL

A Ballele

trai

t

LOD score = logarithm of the oddslogarithm of the ratio of the likelihood of the alternative model tothat of the null model

LOD = log( p(y |locus is QTL)

p(y |locus is not QTL)

)

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Standard Linear Model (SLM)

covariate locus

traitmean

yi ∼ N(mi , σ2)

with

mi = xTi β + qTi α

Legendre, 1805

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Standard Linear Model (SLM)

β α

covariate locus

traitmean

yi ∼ N(mi , σ2)

with


Legendre, 1805

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SLM Test for mQTL

covariate locus

traitmean

yi ∼ N(mi , σ2)

null model{

mi = xTi β

alternative model{


alternative modelon permuted data

{mi = xTi β + qTπ(i)α

Churchill and Doerge, 1994, Genetics

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Genetic mapping with the double generalized linear model DGLM-based Approch to LD Mapping

Table of Contents




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Types of QTL

mQTL

A Ballele

trai

t

22 / 64


Types of QTL

vQTL

A Ballele

trai

t

22 / 64


Types of QTL

gene-by-gene interaction

A Ballele

trai

t

22 / 64


Types of QTL

mvQTL

A Ballele

trai

t

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Double Generalized Linear Model (DGLM)

covariate locus

traitmean

traitvariance

yi ∼ N(mi , exp(vi ))

with

mi = xTi β + qTi αvi = zTi γ + qTi θ

Smyth, 1989

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Double Generalized Linear Model (DGLM)

β αγ θ

covariate locus

traitmean

traitvariance


with


Smyth, 1989

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Three DGLM-based Tests

mQTL test

covariate locus

traitmean

traitvariance

vQTL test

covariate locus

traitmean

traitvariance

mvQTL test

covariate locus

traitmean

traitvariance

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DGLM Test for mQTL

covariate locus

traitmean

traitvariance


null model{

mi = xTi βvi = zTi γ + qTi θ

alternative model{



mi = xTi β + qTπ(i)αvi = zTi γ + qTi θ25 / 64


DGLM Test for vQTL

covariate locus

traitmean

traitvariance


null model{

mi = xTi β + qTi αvi = zTi γ

alternative model{



mi = xTi β + qTi αvi = zTi γ + qTπ(i)θ26 / 64


DGLM Test for mvQTL

covariate locus

traitmean

traitvariance


null model{

mi = xTi βvi = zTi γ

alternative model{



mi = xTi β + qTπ(i)αvi = zTi γ + qTπ(i)θ27 / 64


Results

1 vQTL in Bailey et al. 2008 — pattern is not typically sought2 mQTL in Kumar et al. 2013 — correct for variance heterogeneity

across QTL alleles3 mQTL in Leamy et al. 2000 — correct for variance heterogeneity

across a background factor, F1 father

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Genetic mapping with the double generalized linear model Bailey Reanalysis Identifies vQTL

Table of Contents



3. Genetic Mapping with the weighted linear mixed model (10 minutes)BackgroundHeteroskedastic LMMSimulation Results

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Original Study

Intercrossed C57BL/6J and C58/J, yielding 362 F2’s.Measured six behavioral traits that model aspects of anxiety andexploratory behavior.Reported 7 QTL, but none for rearing behavior.

Bailey et al., GB&B, 2008

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New vQTL

Rearing Events

1 2 3 4 5 6 7 8 9 10

0

1

2

3

LOD

sco

re

Chromosome

Corty et al., 2018 under revision at G3

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New vQTL

Rearing Events

1 2 3 4 5 6 7 8 9 10

1

0.1

0.01

−lo

g10(

p)

Chromosome


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New vQTL

α = 0.05

α = 0.01

1 2 3 4 5 6 7 8 9 10

−lo

g10(

p)

mQTL

vQTL

mvQTL

SLM

Rearing Events


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Understanding the vQTL

● ●●

2.5

3.0

3.5

B6 Het C58

Chr 2, 65Mb marker

Rea

ring

Beh

avio

r

sex

female

male

Rearing Behavior (counts)


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Understanding the vQTL (2)

● ●

0.20

0.25

0.30

0.35

0.40

3.0 3.1

mean estimate +/− 1 SE

SD

est

imat

e +

/− 1

SE

chr 2,65Mb marker

● B6

Het

C58

Sex

●

●

female

male

Mean and Variance Effect Estimatesfor Rearing Events


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This vQTL is Robust...

...to log transform:

α = 0.05

α = 0.01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 X

−lo

g10(

p) mQTL

vQTL

mvQTL

...to inverse normal transform:α = 0.05

α = 0.01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 X

−lo

g10(

p) mQTL

vQTL

mvQTL

...to square root transform:

α = 0.05

α = 0.01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 X

−lo

g10(

p) mQTL

vQTL

mvQTL

...to Poisson regression:

α = 0.05

α = 0.01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 X

−lo

g10(

p) mQTL

vQTL

mvQTL


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Genetic mapping with the double generalized linear model Kumar Reanalysis Identifies “mQTL”

Table of Contents




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Original Study

Intercrossed C57BL/6J and C56NL/6N, two “sister strains”Measured cocaine response and circadian behavior traitsReported 1 QTL for cocaine response, identified QTNNo QTL for circadian behavior traits by standard analysis

Kumar et al., Science, 2013

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Replicate Published QTL

α = 0.05

α = 0.01

11 12 13 14 15 16 17 18 19 20

−lo

g10(

p)

mQTL

vQTL

mvQTL

SLM

30 Minute Cocaine Response

Kumar et al., Science, 2013

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New “mQTL”

α = 0.05

α = 0.01

1 2 3 4 5 6 7 8 9 10

−lo

g10(

p)

mQTL

vQTL

mvQTL

SLM

Circadian Wheel Running Activity (revolutions/minute)


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Understanding the new QTL

●●

●

0

10

20

30

40

C57BL/6J Het C57BL/6N

chr6, rs30314218

aver

age

whe

el s

peed

sex

female

male

Circadian Wheel Running Activity (revolutions/minute)


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Understanding the new QTL (2)

●

●

5.0

7.5

10.0

20 25 30


SD

est

imat

e +

/− 1

SE

rs30314218

● C57BL/6J

Het

C57BL/6N

sex

●

●

female

male

Mean and Variance Effect Estimates for Circadian Wheel Running Activity (revolutions/minute)


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Domain-specific view of the Activity Trait

0h 12h 24h 36h 48h 0h 12h 24h 36h 48h 0h 12h 24h 36h 48h


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Genetic mapping with the double generalized linear model Leamy Reanalysis Identifies mQTL

Table of Contents




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Original Study

Backcrossed CAST/Ei into M16i350 mice in mapping population, 92 markersSkull morphometrics, limb bone lengths, organ and body weightPublished many QTL, but none for body weight

Leamy et al., Genet. Res., 2000, Physiol. Genom., 2002, Yi et al., Genetics, 2005

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Body Weight at Three Weeks

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−10

−5

0

5

10

1 2 3 4 5 6 7 8 9

father

SLM

res

idua

l

0.5

1.0

1.5

2.0

DGLM weight

Residuals from Standard Linear Modelof Bodyweight at Three Weeks


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New mQTL

α = 0.05

α = 0.01

10 11 12 13 14 15 16 17 18 19

−lo

g10(

p)

mQTL

vQTL

mvQTL

traditional

Bodyweight at Three Weeks


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Understanding the new QTL

●●

5

10

15

20

AA AB

D11MIT11

bw3w

k SEX

female

male


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Understanding the new QTL (2)

●

●●

●

● ●●●

●

1

2

3

4

11 12 13 14


SD

est

imat

e +

/− 1

SE

father

●

●

●

●

●

●

●

●

●

1

2

3

4

5

6

7

8

9

D11MIT11

● AA

AB

Effects of father and D11MIT11on Bodyweight at Three Weeks


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Summary — What is a QTL and what’s not?

*BVH

non-QTL QTL

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Table of Contents




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Genetic Mapping with the weighted linear mixed model

Table of Contents




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Genetic Mapping with the weighted linear mixed model Background

Table of Contents




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Motivating Observation

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52 / 64


Relevant Populations

SLM and DGLM are only appropriate when individuals areequally-related, rare in populations not intentionally bred for linkagedisequilibrium mappingThe linear mixed model (LMM) accommodates this “differentialrelatedness” with a random effect term that has covariance patternedafter the genomic similarity.

53 / 64


Linear Mixed Model

Consider an association mapping study with T = 4 strains. Lety be the vector of estimated strain means.

y = xβ + α + �

with

α ∼ N(

0,Kτ2)

� ∼ N(

0, Iσ2)

where K is the genomic similarity matrix and I is the residualvariance matrix.

V = Kτ2 + Iσ2

h2 = τ2

τ2 + σ2

54 / 64


Fitting the LMM

For each genetic locus, use Brent’s method to find the ML value ofh2. For each fixed value of h2, the LMM can be fit by generalizedleast squares (GLS). For each h2...

1 Find a matrix M such that MTM = V−12 Use M to fit the model by GLS.3 Add in log |V| to “back out” the LMM solution

Very easy and fast if you know M!

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Genetic Mapping with the weighted linear mixed model Heteroskedastic LMM

Table of Contents




56 / 64


Considering Heteroskedastic Residuals

Homoskedastic LMM, M is known (Kang, Genetics, 2008):

α ∼ N

0, τ2 � ∼ N

0, σ2

Heteroskedastic LMM, previously no M was known:

α ∼ N

0, τ2 � ∼ N

0, σ2

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Mhom and Mhet

Mhom =(h2ΛK + (1− h2)I

)− 12 UKTMhet = (h2ΛL + (1− h2)I)−

12 ULTD−

12

where ΛX and UX are the eigen values and vectors of X and:

L = D−12 KD−

12

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Genetic Mapping with the weighted linear mixed model Simulation Results

Table of Contents




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Example QQ Plot

Null QQ plot for GWAS with100 strains and h2 = 0.5

theoretical

observed

0.000 0.002 0.004 0.006 0.008 0.010

0.000

0.002

0.004

0.006

0.008

0.010LMEMMAISAMwISAM

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QQ Plots for a GWAS with 50 Strains

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Example ROC Plot

0.00

0.25

0.50

0.75

1.00

0.00 0.25 0.50 0.75 1.00

False Positive Rate

True

Pos

itive

Rat

e

test●

●

●

●

SLM

EMMA

ISAM

wISAM

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AcknowledgmentsValdar lab: Will Valdar, Greg Keel, Paul Maurizio, Dan Oreper, Wes Crouse,Yanwei Cai, Kathie SunTarantino lab: Lisa Tarantino, Sarah Schoenrockdissertation committee: Fernando, Will, Lisa, Yun, JimMDPhD program: Gene Orringer, Mohanish Deshmukh, Toni Darville, AlisonRegan, Carol HerionBCB program: Tim Elston, Cara Marlow, John Cornettfriends and family: mom, dad, grandparents, Edward, Maithri, Mike, Brooke,Marni, Chris, Kelly, Patrick, Lee, Sarahfunders: NIMH RC: F30; NIGMS WV: R01, MIRA;

NIGMS TE, WV: T32; NIGMS GO, MD, TD: T32

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Acknowledgment

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Introduction to Genetic Mapping (10 minutes)BackgroundMotivating Concept

Genetic mapping with the double generalized linear model (30 minutes)Introduction to Linkage Disequilibrium MappingStandard Approach to LD MappingDGLM-based Approch to LD MappingBailey Reanalysis Identifies vQTLKumar Reanalysis Identifies ``mQTL''Leamy Reanalysis Identifies mQTL

Genetic Mapping with the weighted linear mixed model (10 minutes)BackgroundHeteroskedastic LMMSimulation Results

variance heterogeneity in genetic mapping · variance heterogeneity in genetic mapping robert corty...

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