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Linkage Analysis: An IntroductionPowerPoint Presentation

Linkage Analysis: An Introduction

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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

A3A4

A1A2

A1A3

A2A4

A2A3

Marker allele A1

cosegregates with

dominant disease

A1A2

A1A4

A3A4

A3A2

Recombination

A1

Q1

Parental genotypes

A1

Q1

A2

Q2

Likely gametes

(Non-recombinants)

A2

Q2

A1

Q2

Unlikely gametes

(Recombinants)

Q1

A2

Map distance

Map distance between two loci (Morgans)

= Expected number of crossovers per meiosis

Note: Map distances are additive

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

aabb

AABB

aabb

AaBb

Aabb

AaBb

aabb

aaBb

Non-recombinant

Recombinant

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

aabb

AaBb

Aabb

AaBb

aabb

aaBb

Either :

Non-recombinant

Recombinant

Or :

Recombinant

Non-recombinant

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

Aabb

AaBb

aabb

aaBb

Likelihood will be a function of

allele frequencies (population parameters)

(transmission parameter)

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

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

2

X

Elston-Stewart algorithmReduces 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 > 3 conclude linkage

Prior odds linkage ratio Posterior odds

1:50 1000 20:1

Lod <-2 exclude linkage

Linkage AnalysisAdmixture Test

Model

Probabilty of linkage in family =

Likelihood

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

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

Parameters: IBD sharing probabilities

Z=(z0, z1,z2)

Marker Genotype Data M: Finite Mixture Likelihood

SPLINK, ASPEX

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

- 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

- 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

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

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

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

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

Conditioning on Trait Values

Usual test

Conditional test

Zi = IBD probability estimated from marker genotypes

Pi = IBD probability given relationship

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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