Introduction to Linkage Analysis

1
Introduction to Linkage Analysis

• Pak Sham
• Twin Workshop 2003

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Human Genome

• 22 autosomes, XY
• ?3 ?109 base-pairs (2 metres long)
• ? 2 coding sequences, rest regulatory junk
• ? 30,000 - 40,000 genes
• Much communality with other species

3
Genetic Variation

• Chromosomal abnormalities
• Duplication (e.g. Downs)
• Deletion (e.g. Velo-cardio-facial syndrome)
• Major deleterious mutations
• Usually Rare (e.g. Huntingtons)
• Polymorphisms
• Single nucleotide polymorphisms (SNPs)
• Variable length repeats (e.g. microsatellites)
• Some are functional (normal variation)
• Most are non-functional (neutral markers)

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Genetic Mapping of Disease

• Levels of Genetic Analysis
• Estimate heritability (family, twins, adoption)
• Find chromosomal locations (linkage)
• Identify risk variants (association)
• Understand mechanisms (cell biology, etc)
• Applications
• Prediction of genetic risk
• More accurate prediction of genetic risk
• Even more accurate prediction of genetic risk
  prediction of prognosis and treatment response
• Development of new drug targets

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Strategies of Gene Mapping

• Functional
• Uses knowledge of disease to identify candidate
  genes
• Finds variants in candidate genes
• Looks for association between variants and
  disease
• Positional
• Systematic screen of whole genome
• Uses a set of ? 400 evenly-spaced markers
• Looks for markers which con-segregate with disease

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Co-segregation

A3A4
A1A2
A2A4
A1A3
A2A3
Marker allele A1 cosegregates with dominant
disease
A1A2
A1A4
A3A4
A3A2
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Linkage ?Co-segregation
Gametes
Parent
Alleles on the same chromosome tend to be
stay together in meiosis therefore they tend be
co-transmitted.
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Crossing over between homologous chromosomes
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Map Distance

• Map distance between two loci (Morgans)
• Expected number of crossovers per meiosis
• (1 Morgan 100 centiMorgans)
• Note Map distances are additive
• Heterogeneity in recombination frequencies
• Total map length ? 33
• (1 cM ? 106 base pairs)

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Recombination
A1
Q1
Parental genotypes
A1
Q1
A2
Q2
Non-recombinants 1-?
A2
Q2
A1
Q2
Recombinants ?
A2
Q1
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Recombination Fraction

• Recombination fraction (?) between two loci
• Proportion of gametes that are recombinant
  with respect to the two loci

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Recombination map distance
Haldane map function
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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 Estimation 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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Mixture distribution likelihood
The probability of observed data X depend on
the status of descrete variable G P(XG) The
status of G is not observed but the
probability distribution of G is
available P(G) Then the likelihood of the
observed data X is
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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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Complex 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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Lod Score Curves
lod
0
0.5
?
Lod score curves are additive over pedigrees
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Lods, chi-squares p-values
In large samples
2 ? loge(10) ? Max lod ?21
In small samples
P ? 10 -Max lod
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Problems with parametric linkage

• Requires parameters of the disease model to be
  specified
• Allele frequency
• Penetrances
• These are generally unknown for a complex trait
• Disease model assumes that a single locus is the
  only source of familial resemblance
• This is generally unrealistic

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Linkage AnalysisAdmixture Test (CAB Smith)
Model Probability of linkage in family
? Likelihood L(?, ?) ? L(?) (1- ?)
L(?1/2) Note Another example of mixture
likelihood
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Linkage Analysis MOD

• Maximise lod score over several sets of disease
  models, e.g. dominant, recessive, additive
• Make correction for multiple (k) models
• Adjusted lod lod log10(k)

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Allele sharing (non-parametric) methods
Penrose (1935) Sib Pair linkage For rare
disease IBD Concordant
affected Concordant normal Discordant Theref
ore affected sib pair (ASP) design
efficient Test H0 Proportion of alleles IBD
1/2 HA Proportion of alleles IBD gt1/2
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Correlation between IBD of two loci

• For sib pairs
• Corr(?A, ?B) (1-2?AB)2
• ? attenuation of linkage signal with increasing
  genetic distance from disease locus

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Joint distribution of Pedigree IBD

• IBD of relative pairs are not 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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Inheritance Vector An Example
Ordered genotype notation 1st allele
paternally inherited 2nd allele maternally
inherited
1/2
3/4
2/3
1/3
1/4
2/4
Inheritance vector (1, 1, 1, 0, 1, 0)
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Pedigree allele-sharing methods

• APM Affected Pedigree Members Uses IBS
• very sensitive to allele frequency
  mis-specification
• less powerful than IBD-based methods
• NPL Non-Parametric Linkage (Genehunter)
• Conservative at positions between markers
• LRT Delta parameter (Genehunter, Allegro)
• All these methods consider affected members only

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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 VQ ?-E(?)
• Where V trait variance
• r correlation (depends on relationship)
• VQ QTL additive variance
• E(?) expected proportion IBD

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Path Diagram for Sib-Pair QTL model

1
0 / 0.5 / 1
N
Q
S
Q
S
N
n
q
s
n
s
q
PT1
PT2
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Incomplete Marker Information

• IBD sharing cannot always be deduced from marker
  genotypes with certainty
• Obtain probabilities of IBD values (Z0, Z1, Z2)
• Finite mixture likelihood
• Pi-hat likelihood

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Pi-hat Model
1
N
Q
S
Q
S
N
n
q
s
n
s
q
PT1
PT2
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Parametric / Allele Sharing
Parametric
Trait Data
Marker Data
Allele sharing
IBD sharing