Linkage Analysis: An Introduction

1
Linkage AnalysisAn Introduction

• Pak Sham
• Twin Workshop 2001

2
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
• gt400 Mendelian (single gene) disorders mapped
• Current challenge is to map QTLs

3
Linkage Co-segregation

A3A4
A1A2
A2A4
A1A3
A2A3
Marker allele A1 cosegregates with dominant
disease
A1A2
A1A4
A3A4
A3A2
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Recombination
A1
Q1
Parental genotypes
A1
Q1
A2
Q2
Likely gametes (Non-recombinants)
A2
Q2
A1
Q2
Unlikely gametes (Recombinants)
A2
Q1
5
Recombination of three linked loci
?1 ?2
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Map distance

• Map distance between two loci (Morgans)
• Expected number of crossovers per meiosis
• Note Map distances are additive

7
Recombination map distance
Haldane map function
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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

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Double Backcross Fully Informative Gametes
AABB
aabb
aabb
AaBb
Aabb
AaBb
aabb
aaBb
Non-recombinant
Recombinant
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Linkage Analysis Fully Informative Gametes
Count Data Recombinant Gametes
R Non-recombinant Gametes N Parameter Recombi
nation Fraction ? Likelihood L(?) ?R (1-
?)N Parameter Chi-square
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Phase Unknown Meioses
aabb
AaBb
Aabb
AaBb
aabb
aaBb
Either
Non-recombinant
Recombinant
Or
Recombinant
Non-recombinant
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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
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Parental genotypes unknown
Aabb
AaBb
aabb
aaBb
Likelihood will be a function of allele
frequencies (population parameters) ?
(transmission parameter)
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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.
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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
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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
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Lod Score Morton (1955)
Lod gt 3 ? conclude linkage
Prior odds linkage ratio Posterior
odds 150 1000 201
Lod lt-2 ? exclude linkage
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Linkage AnalysisAdmixture Test
Model Probabilty of linkage in family
? Likelihood L(?, ?) ? L(?) (1- ?)
L(?1/2)
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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
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Affected sib pairs incomplete marker information
Parameters IBD sharing probabilities Z(z0, z1,
z2)
Marker Genotype Data M Finite Mixture Likelihood
SPLINK, ASPEX
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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)

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Pedigree allele-sharing methods

• Problem
• APM Affected family members Uses IBS
• ERPA Extended Relative Pairs Analysis Dodgy
  statistic
• Genehunter NPL Non-Parametric Linkage Conservati
  ve
• Genehunter-PLUS Likelihood (tilting)
• All these methods consider affected members only

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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

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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) ?
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Improved Haseman-Elston

• Sham and Purcell (2001)
• Use as dependent variable
• Gives equivalent power to variance components
  model for sib pair data

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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

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QTL linkage model for sib-pair data

1
0 / 0.5 / 1
N
Q
S
Q
S
N
n
q
s
n
s
q
PT1
PT2
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No linkage
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Under linkage
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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

31

QTL linkage model for sib-pair data

1
N
Q
S
Q
S
N
n
q
s
n
s
q
PT1
PT2
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Conditioning on Trait Values
Usual test
Conditional test
Zi IBD probability estimated from marker
genotypes Pi IBD probability given relationship

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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