Top Banner
Linkage Analysis: Linkage Analysis: An Introduction An Introduction Pak Sham Twin Workshop 2001
33

Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Dec 18, 2015

Download

Documents

Theresa Moore
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Linkage Analysis:Linkage Analysis:An IntroductionAn Introduction

Pak Sham

Twin Workshop 2001

Page 2: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Linkage MappingLinkage Mapping

Compares inheritance pattern of trait with the inheritance pattern of chromosomal regions

First gene-mapping in 1913 (Sturtevant)

Uses naturally occurring DNA variation (polymorphisms) as genetic markers

>400 Mendelian (single gene) disorders mapped

Current challenge is to map QTLs

Page 3: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Linkage = Co-Linkage = Co-segregationsegregation

A2A4

A3A4

A1A3

A1A2

A2A3

A1A2 A1A4 A3A4 A3A2

Marker allele A1

cosegregates withdominant disease

Page 4: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

RecombinationRecombinationA1

A2

Q1

Q2

A1

A2

Q1

Q2

A1

A2 Q1

Q2

Likely gametes(Non-recombinants)

Unlikely gametes(Recombinants)

Parental genotypes

Page 5: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Recombination of three Recombination of three linked locilinked loci

(1-1)(1-2)

1 2

(1-1)2

1(1-2)

12

Page 6: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Map distanceMap distance

Map distance between two loci (Morgans)

= Expected number of crossovers per meiosis

Note: Map distances are additive

Page 7: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Recombination & map Recombination & map distancedistance

2

1 2me

Haldane mapfunction

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.2 0.4 0.6 0.8 1

Map distance (M)

Re

co

mb

ina

tio

n f

rac

tio

n

Page 8: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Methods of Linkage Methods of Linkage AnalysisAnalysis

Model-based lod scores Assumes explicit trait model

Model-free allele sharing methods Affected sib pairs Affected pedigree members

Quantitative trait loci Variance-components models

Page 9: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Double Backcross :Double Backcross :Fully Informative GametesFully Informative Gametes

AaBb aabb

AABB aabb

AaBb aabb Aabb aaBb

Non-recombinant Recombinant

Page 10: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Linkage Analysis :Linkage Analysis :Fully Informative GametesFully Informative Gametes

Count Data Recombinant Gametes: RNon-recombinant Gametes: N

Parameter Recombination Fraction:

Likelihood L() = R (1- )N

Parameter

Chi-square

)(ˆ RNR

)5log(.)(

)1log(log22

NR

NR

Page 11: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Phase Unknown MeiosesPhase Unknown Meioses

AaBb aabb

AaBb aabb Aabb aaBb

Non-recombinant Recombinant

Recombinant Non-recombinant

Either :

Or :

Page 12: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Linkage Analysis :Linkage Analysis :Phase-unknown MeiosesPhase-unknown Meioses

Count Data Recombinant Gametes: XNon-recombinant Gametes: Y

or Recombinant Gametes: YNon-recombinant Gametes: X

Likelihood L() = X (1- )Y + Y (1- )X

An example of incomplete data :

Mixture distribution likelihood function

Page 13: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Parental genotypes Parental genotypes unknownunknown

Likelihood will be a function of

allele frequencies (population parameters)

(transmission parameter)

AaBb aabb Aabb aaBb

Page 14: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Trait phenotypesTrait phenotypes

Penetrance parameters

Genotype Phenotype

f2AA

aa

Aa

Disease

Normal

f1

f0

1- f2

1- f1

1- f0

Each phenotype is compatible with multiple genotypes.

Page 15: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

General Pedigree General Pedigree LikelihoodLikelihood

Likelihood is a sum of products (mixture distribution likelihood)

n

f

imf

f

i

G

n

gggtransgpopgxpenL iiii

111

)|()()|( ,

number of terms = (m1, m2 …..mk)2n

where mj is number of alleles at locus j

Page 16: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Elston-Stewart algorithmElston-Stewart algorithmReduces computations by Peeling:

Step 1Condition likelihoods of family 1 on genotype of X.

1

2X

Step 2Joint likelihood of families 2 and 1

Page 17: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Lod Score: Morton Lod Score: Morton (1955)(1955)

5.0

log

L

LLod

Lod > 3 conclude linkage

Prior odds linkage ratio Posterior odds1:50 1000 20:1

Lod <-2 exclude linkage

Page 18: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Linkage AnalysisLinkage AnalysisAdmixture TestAdmixture Test

Model

Probabilty of linkage in family =

Likelihood

L(, ) = L() + (1- ) L(=1/2)

Page 19: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Allele sharing Allele sharing (non-parametric) (non-parametric)

methodsmethodsPenrose (1935): Sib Pair linkage

For rare disease IBDConcordant affected

Concordant normalDiscordant

Therefore Affected sib pair design

Test H0: Proportion of alleles IBD =1/2

Page 20: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Affected sib pairs: Affected sib pairs: incomplete marker incomplete marker

informationinformationParameters: IBD sharing probabilities

Z=(z0, z1, z2)

iIBDMPzzLi

i

|2

0

Marker Genotype Data M: Finite Mixture Likelihood

SPLINK, ASPEX

Page 21: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Joint distribution of Joint distribution of Pedigree IBDPedigree IBD

IBD of relative pairs are independent

e.g If IBD(1,2) = 2 and IBD (1,3) = 2

then IBD(2,3) = 2

Inheritance vector gives joint IBD distribution

Each element indicates whether

paternally inherited allele is transmitted (1)

or maternally inherited allele is transmitted (0)

Vector of 2N elements (N = # of non-founders)

Page 22: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Pedigree allele-sharing Pedigree allele-sharing methodsmethods

Problem

APM: Affected family members Uses IBS

ERPA: Extended Relative Pairs Analysis Dodgy statisticGenehunter NPL: Non-Parametric Linkage Conservative

Genehunter-PLUS: Likelihood (“tilting”)

•All these methods consider affected members only

Page 23: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Convergence of Convergence of parametric and non-parametric and non-parametric methodsparametric methods

Curtis and Sham (1995)

MFLINK: Treats penetrance as parameter

Terwilliger et al (2000)

Complex recombination fractions

Parameters with no simple biological interpretation

Page 24: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Quantitative Sib Pair LinkageQuantitative Sib Pair Linkage

X, Y standardised to mean 0, variance 1r = sib correlationVA = additive QTL variance

(X-Y)2 = 2(1-r) – 2VA(-0.5) +

Haseman-Elston Regression (1972)Haseman-Elston Regression (1972)

Haseman-Elston Revisited (2000)Haseman-Elston Revisited (2000)

XY = r + VA(-0.5) +

Page 25: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Improved Haseman-Improved Haseman-ElstonElston

Sham and Purcell (2001) Use as dependent variable

Gives equivalent power to variance components model for sib pair data

2YX

2

2

)1( r

YX

2

2

2

2

)1()1( r

YX

r

YX

Page 26: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Variance components Variance components linkagelinkage

Models trait values of pedigree members jointly Assumes multivariate normality conditional on IBD Covariance between relative pairs

= Vr + VA [-E()]

Where V = trait variance

r = correlation (depends on relationship)

VA= QTL additive variance

E() = expected proportion IBD

Page 27: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

QTL linkage model for sib-pair QTL linkage model for sib-pair datadata

PT1

QSN

PT2

Q S N

1

[0 / 0.5 / 1]

n qs nsq

Page 28: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

No linkageNo linkage

Page 29: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Under linkageUnder linkage

Page 30: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Incomplete Marker Incomplete Marker InformationInformation

IBD sharing cannot be deduced from marker genotypes with certainty

Obtain probabilities of all possible IBD values

Finite mixture likelihood

Pi-hat likelihood

Ai ViIBDXLZL ;|

AVIBDXLL ;ˆ2|

Page 31: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

QTL linkage model for sib-pair QTL linkage model for sib-pair datadata

PT1

QSN

PT2

Q S N

1

n qs nsq

Page 32: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

Conditioning on Trait Conditioning on Trait ValuesValues

Usual test

0;ln

;|lnln

A

Ai

VXL

ViIBDXLZMaxLR

Conditional test

Ai

Ai

ViIBDXLP

ViIBDXLZMaxLR

;|ln

;|lnln

Zi = IBD probability estimated from marker genotypesPi = IBD probability given relationship

Page 33: Linkage Analysis: An Introduction Pak Sham Twin Workshop 2001.

QTL linkage: some QTL linkage: some problemsproblems

Sensitivity to marker misspecification of marker allele frequencies and positions

Sensitivity to non-normality / phenotypic selection Heavy computational demand for large pedigrees or

many marker loci Sensitivity to marker genotype and relationship errors Low power and poor localisation for minor QTL