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

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Linkage analysis an introduction

Linkage Analysis:An Introduction

Pak Sham

Twin Workshop 2001


Linkage mapping
Linkage 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


Linkage co segregation
Linkage = Co-segregation

A3A4

A1A2

A1A3

A2A4

A2A3

Marker allele A1

cosegregates with

dominant disease

A1A2

A1A4

A3A4

A3A2


Recombination
Recombination

A1

Q1

Parental genotypes

A1

Q1

A2

Q2

Likely gametes

(Non-recombinants)

A2

Q2

A1

Q2

Unlikely gametes

(Recombinants)

Q1

A2


Recombination of three linked loci

(1-1)(1-2)

(1-1)2

1(1-2)

12

Recombination of three linked loci

1 2


Map distance
Map distance

Map distance between two loci (Morgans)

= Expected number of crossovers per meiosis

Note: Map distances are additive


Recombination map distance
Recombination & map distance

Haldane map

function


Methods of linkage analysis
Methods of Linkage Analysis

  • 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


Double backcross fully informative gametes
Double Backcross :Fully Informative Gametes

aabb

AABB

aabb

AaBb

Aabb

AaBb

aabb

aaBb

Non-recombinant

Recombinant


Linkage analysis fully informative gametes
Linkage Analysis :Fully Informative Gametes

Count Data Recombinant Gametes: R

Non-recombinant Gametes: N

Parameter Recombination Fraction: 

Likelihood L() = R (1- )N

Parameter

Chi-square


Phase unknown meioses
Phase Unknown Meioses

aabb

AaBb

Aabb

AaBb

aabb

aaBb

Either :

Non-recombinant

Recombinant

Or :

Recombinant

Non-recombinant


Linkage analysis phase unknown meioses
Linkage Analysis :Phase-unknown Meioses

Count Data Recombinant Gametes: X

Non-recombinant Gametes: Y

or Recombinant Gametes: Y

Non-recombinant Gametes: X

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

An example of incomplete data :

Mixture distribution likelihood function


Parental genotypes unknown
Parental genotypes unknown

Aabb

AaBb

aabb

aaBb

Likelihood will be a function of

allele frequencies (population parameters)

 (transmission parameter)


Trait phenotypes
Trait phenotypes

Penetrance parameters

Phenotype

Genotype

f2

AA

Disease

f1

1- f2

f0

Aa

1- f1

1- f0

aa

Normal

Each phenotype is compatible with multiple genotypes.


General pedigree likelihood
General Pedigree Likelihood

Likelihood is a sum of products

(mixture distribution likelihood)

number of terms = (m1, m2 …..mk)2n

where mj is number of alleles at locus j


Elston stewart algorithm

1

2

X

Elston-Stewart algorithm

Reduces computations by Peeling:

Step 1

Condition likelihoods of family 1 on genotype of X.

Step 2

Joint likelihood of

families 2 and 1


Lod score morton 1955
Lod Score: Morton (1955)

Lod > 3  conclude linkage

Prior odds linkage ratio Posterior odds

1:50 1000 20:1

Lod <-2  exclude linkage


Linkage analysis admixture test
Linkage AnalysisAdmixture Test

Model

Probabilty of linkage in family = 

Likelihood

L(, ) =  L()+ (1- )L(=1/2)


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

Penrose (1935): Sib Pair linkage

For rare disease IBD Concordant affected

Concordant normal

Discordant

Therefore Affected sib pair design

Test H0: Proportion of alleles IBD =1/2


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

Parameters: IBD sharing probabilities

Z=(z0, z1,z2)

Marker Genotype Data M: Finite Mixture Likelihood

SPLINK, ASPEX


Joint distribution of pedigree ibd
Joint distribution of Pedigree 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)


Pedigree allele sharing methods
Pedigree allele-sharing methods

  • Problem

  • APM: Affected family members Uses IBS

  • ERPA: Extended Relative Pairs Analysis Dodgy statistic

  • Genehunter NPL: Non-Parametric Linkage Conservative

  • Genehunter-PLUS: Likelihood (“tilting”)

    • All these methods consider affected members only


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

  • Curtis and Sham (1995)

    MFLINK: Treats penetrance as parameter

    Terwilliger et al (2000)

    Complex recombination fractions

    Parameters with no simple biological interpretation


Quantitative sib pair linkage
Quantitative Sib Pair Linkage

X, Y standardised to mean 0, variance 1

r = sib correlation

VA = additive QTL variance

Haseman-Elston Regression (1972)

(X-Y)2 = 2(1-r) – 2VA(-0.5) +

Haseman-Elston Revisited (2000)

XY = r + VA(-0.5) +


Improved haseman elston
Improved Haseman-Elston

  • Sham and Purcell (2001)

  • Use as dependent variable

    Gives equivalent power to variance components model for sib pair data


Variance components linkage
Variance components linkage

  • 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


QTL linkage model for sib-pair data

1

[0 / 0.5 / 1]

N

S

Q

Q

S

N

n

s

q

q

s

n

PT1

PT2




Incomplete marker information
Incomplete Marker Information

  • IBD sharing cannot be deduced from marker genotypes with certainty

  • Obtain probabilities of all possible IBD values

    Finite mixture likelihood

    Pi-hat likelihood


QTL linkage model for sib-pair data

1

N

S

Q

Q

S

N

n

s

q

q

s

n

PT1

PT2


Conditioning on trait values
Conditioning on Trait Values

Usual test

Conditional test

Zi = IBD probability estimated from marker genotypes

Pi = IBD probability given relationship


Qtl linkage some problems
QTL linkage: some problems

  • 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


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