Linkage analysis: basic principles
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Linkage analysis: basic principlesManuel Ferreira & Pak Sham
Boulder Advanced Course 2005
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Outline
1. Aim2. The Human Genome3. Principles of Linkage Analysis4. Parametric Linkage Analysis5. Nonparametric Linkage Analysis
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1. Aim
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For a heritable trait...
localizes region of the genome where a locus (loci) that regulates the trait is likely to be harboured
identifies a locus that regulates the trait
Linkage:
Association:
Family-specific phenomenon: Affected individuals in a family share the same ancestral predisposing DNA segment at a given trait locus
Population-specific phenomenon: Affected individuals in a population share the same ancestral predisposing DNA segment at a given trait locus
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2. Human Genome
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A DNA molecule is a linear backbone of alternating sugar residues and phosphate groupsAttached to carbon atom 1’ of each sugar is a nitrogenous base: A, C, G or TTwo DNA molecules are held together in anti-parallel fashion by hydrogen bonds between bases [Watson-Crick rules]Antiparallel double helix
Only one strand is read during gene transcription
Nucleotide: 1 phosphate group + 1 sugar + 1 base
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - GA - TC - GC - GT - AA - TG - CG - CC - GG - CA - TT - AA - TC - GT - AA - TA - TA - T
DNA structure
A gene is a segment of DNA which is transcribed to give a protein or RNA product
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C - GA - TA - TT - AG - CC - GT - AT - AT - AG - CT - AA - TC - GG - CA - TC - GA - TC - GA - T (CA)nG - CG - CC - GG - CA - TT - AA - T C - G G - C T - GC - GT - AA - TA - TA - T
DNA polymorphismsRFLPs
A
B
MinisatellitesMicrosatellites>100,000Many alleles, (CA)n, veryinformative, even, easily automatedSNPs 10,054,521 (25 Jan ‘05)Most with 2 alleles (up to 4), not veryinformative, even, easily automated
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Haploid gametes
♁
♂
♂ ♁
G1 phase
chr1
chr1
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
S phase
Diploid zygote 1 cell
M phase
Diploid zygote >1
cell
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
♁♂ ♁
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
A -
B -
A -
B -
A -
B -
A -
B -
A -
B -
A -
B -
♂ ♁C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
A -
B -
A -
B -
♂ ♁C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
- A
- B
- A
- B- A
- B
- A
- B
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
DNA organization
Mitosis
22 + 1 2 (22 + 1)
2 (22 + 1)
2 (22 + 1)
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Diploid gamete precursor cell
(♂) (♁)
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
(♂)
(♁)
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - GHaploid
gamete precursors Hap. gametes
NR
NR
R
R
♁
A -
B -
- A
- B
A -
B -
- A
- B
A -
B -
- A
- B
A -
B -
- A
- B
♂ ♁C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
A -
B -
A -
B -
- A
- B
- A
- B
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
DNA recombination
Meiosis
2 (22 + 1)
2 (22 + 1)
22 + 122 + 1
chr1 chr1 chr1 chr1
chr1
chr1
chr1
chr1
chr1
chr1
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Diploid gamete precursor
(♂) (♁)
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
(♂)
(♁)
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - GHaploid
gamete precursors Hap. gametes
NR
NR
NR
NR
♁
A -B -
- A- B
A -B -
- A- B
A -B -
- A- B
A -B -
- A- B
♂ ♁C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
A -B -
A -B -
- A- B
- A- B
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
C - GA - TA - TT - AG - CC - GT - AT - AT - AA - TA - TG - CC - GG - CA - TT - AG - CT - AA - TC - G
DNA recombination between linked loci
Meiosis2 (22 + 1)
22 + 1
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Human Genome - summary
Recombination fraction between loci A and B (θ)Proportion of gametes produced that are recombinant for A and BIf A and B are very far apart: 50%R:50%NR - θ = 0.5If A and B are very close together: <50%R - 0 ≤ θ < 0.5Recombination fraction (θ) can be converted to genetic distance (cM)Haldane: eg. θ=0.17, cM=20.8Kosambi: eg. θ=0.17, cM=17.7
21ln5.0100cM
2121ln25.0100cM
DNA is a linear sequence of nucleotides partitioned into 23 chromosomesTwo copies of each chromosome (2x22 autosomes + XY), frompaternal and maternal origins. During meiosis in gamete precursors,recombination can occur between maternal and paternal homologs
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3. Principles of Linkage Analysis
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Linkage Analysis requires genetic markers
M1
M2
Mn
M1
M2
Mn
M1M2
Mn
θ 0.5 0.5 .4 .3.15
.3 .4 0.5
Q
θ 0.50.5 .4 .3 .1
.26 .35 0.5.35 .22.3 .4
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Linkage Analysis: Parametric vs. Nonparametric
QM
Phe
A
D
C
E
Genetic factors
Environmental factors
Mode of inheritanc
e
Recombination
Correlation
ChromosomeGene
Adapted from Weiss & Terwilliger 2000
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4. Parametric Linkage Analysis
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Linkage with informative phase known meiosis
M2M5Q2Q2 M1M6Q1Q?
M1Q1/M2Q2 M3M4Q2Q2
M1Q1/M3Q2 M2Q2/M3Q2 M1Q1/M4Q2 M1Q1/M4Q2 M2Q2/M4Q2 M2Q1/M3Q2
Chromosome
M1..6 Q1,2Autosomal dominant, Q1 predisposing
allele
Gene
♁♂
NR: M1Q1 NR: M2Q2
R: M1Q2R: M2Q1
θMQ = 1/6 = 0.17
InformativePhase known
(~20.8 cM)
M1M2Q1Q2
M1 Q1
M2 Q2
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M1M2Q1Q2 M3M4Q2Q2
NR: M1Q1 NR: M2Q2
R: M1Q2R: M2Q1
Q2Q2 Q1Q?
P 1-θ 1-θ
θ
θ
M1Q1/M2Q2
R: M1Q1 R: M2Q2
NR: M1Q2NR: M2Q1
P θ θ
1-θ
1-θ
M1Q2/M2Q1N 3 2
0 1
N 3 2
0 1
|XL 51 121 15 1
21 +
5.0|XL 51 5.015.021
15 5.015.021
+ 65.0
InformativePhase unknown
Linkage with informative phase unknown meiosis
M1Q1/M3Q2 M2Q2/M3Q2 M1Q1/M4Q2 M1Q1/M4Q2 M2Q2/M4Q2 M2Q1/M3Q2
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0.1 0.2 0.3 0.4 0.5
LOD
sco
re-5
-4
-3
-2
-1
0
1
2
3
θ
Parametric LOD score calculation
)5.0|()|(log10
XL
XLLOD)5.0|(
)|(
XL
XLOD
n
i i
i
XLXLLOD
110 )5.0|(
)|(log
n
i i
i
XLXLOD
1 )5.0|()|(
n
ii
n
i i
i LODXL
XLLOD11
10 )5.0|()|(log
Overall LOD score for a given θ is the sum of all family LOD scores at θeg. LOD=3 for θ=0.28
6
1551
10 5.0
1211
21
log
LOD
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M1
M2
Mn
θ 0.5 0.5 .4 .3.1
.3 .4 0.5
Q
For each marker, estimate the θ that yields highest LOD score across all families
Markers with a significant parametric LOD score (>3) are said to be linkedto the trait locus with recombination fraction θ
This θ (and the LOD) will depend upon the mode of inheritance assumedMOI determines the genotype at the trait locus Q and thus determines thenumber of meiosis which are recombinant or nonrecombinant. Limited to Mendelian diseases.
Parametric Linkage Analysis - summary
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M1M2Q1Q1 M3M4Q1Q2
M2M3Q1Q1 M1M4Q1Q2 M1M4Q1Q1 M2M4Q1Q2
NR: M3Q1 NR: M4Q2
R: M3Q2R: M4Q1
Q1Q1 Q2Q?
P 1-θ 1-θ
θ
θ
M3Q1/M4Q2
R: M3Q1 R: M4Q2
NR: M3Q2NR: M4Q1
P θ θ
1-θ
1-θ
M3Q2/M4Q1N 1 2
0 1
N 1 2
0 1
|XL 31 121 13 1
21 +
5.0|XL 31 5.015.021
13 5.015.021
+ 45.0
Practical
1. Identify informative individual(s)2. Reconstruct possible phase(s)3. Classify gametes as R or NR4. Count R and NR gametes5. Express |XL 5.0| XL
6. Express LOD score )(f
6
1551
10 5.0
1211
21
log
LOD
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Practical II
6
1551
10 5.0
1211
21
log
LOD
Talk example
4
1331
10 5.0
1211
21
log
LOD
Practical example
Graph each…
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Outline
1. Aim2. The Human Genome3. Principles of Linkage Analysis4. Parametric Linkage Analysis5. Nonparametric Linkage Analysis
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5. Nonparametric Linkage Analysis
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Approach
Parametric: genotype marker locus & genotype trait locus(latter inferred from phenotype according to a specific disease model) Parameter of interest: θ between marker and trait lociNonparametric: genotype marker locus & phenotypeIf a trait locus truly regulates the expression of a phenotype, then tworelatives with similar phenotypes should have similar genotypes at amarker in the vicinity of the trait locus, and vice-versa.Interest: correlation between phenotypic similarity and marker genotypicsimilarity
No need to specify mode of inheritance, allele frequencies, etc...
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Phenotypic similarity between relatives
Squared trait differencesSquared trait sumsTrait cross-product
221 XX
221 XX
21 XX
Trait variance-covariance matrix
221
211
XVarXXCovXXCovXVar
Affection concordance
T2
T1
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Genotypic similarity between relativesIBS Alleles shared Identical By State “look the same”, may have the
same DNA sequence but they are not necessarily derived from a known common ancestorIBD Alleles shared
Identical By Descent are a copy of the same
ancestor allele
M1Q1
M2Q2
M3Q3
M3Q4
M1Q1
M3Q3
M1Q1
M3Q4
M1Q1
M2Q2
M3Q3
M3Q4
IBS IBD2 1
Inheritance vector (M)
0 0 0 1 1
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Genotypic similarity between relatives -
M1Q1
M3Q3
M2Q2
M3Q4
Number of alleles IBD
0
M1Q1
M3Q3
M1Q1
M3Q4
1
M1Q1
M3Q3
M1Q1
M3Q3
2
Proportion of alleles IBD -
0
0.5
1
Inheritance vector (M)
0 0 1 1
0 0 0 1
0 0 0 0
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Genotypic similarity between relatives -
21
210 222
21
20ˆ
x0/x1 x0/x1
x0/x0x0/x0x0/x0x0/x0x0/x1
x0/x1x0/x1x0/x1x1/x0x1/x0x1/x0x1/x0
x1/x1x1/x1x1/x1
x1/x1
x0/x0x0/x1
x1/x0x1/x1
x0/x0
x0/x1x1/x0x1/x1x0/x0x0/x1x1/x0
x1/x1
x0/x0x0/x1x1/x0
x1/x1
Inheritance vector
0000000100100011010001010110011110001001101010111100110111101111
Prior probability
1/161/161/161/161/161/161/161/161/161/161/161/161/161/161/161/16
IBD
2110120110210112
A1/A3 A1/A2
Posterior probability
01/400
1/4000000
1/400
1/40
A1A3 A1A2
Posterior probability
01/60
1/121/60
1/1200
1/120
1/61/12
01/60
1 2
3 4
A1A2 A3A2
A1/A2
Posterior probability
A1/A3
A1/A2 A3/A2
0100000000000000
P (IBD=0)P (IBD=1)P (IBD=2)
1/41/21/4
1/32/30
010
010
22n
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Statistics that incorporate both phenotypic and genotypic similarities
Genotypic similarity ( )
Phen
otyp
ic s
imila
rity
0 0.5 1
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Haseman-Elston regression – Quantitative traits
221 XX
|221 XXE
|2 2121 XXCovXVarXVar
|2 2122
21 XXXXE
ECAQ VVVVXVarXVar 21
CAQ VlVVXXCov 2ˆˆ|, 21
EAQQ VVVVXXE 22ˆ2ˆ|221
Phenotypic dissimilarity
Genotypic similarity
b ×= + c
0 0.5 1
X1 X2 (X1-X2)2
1 2.2 2.1 0.01 0.92 1.9 2.3 0.16 0.63 2.3 2.6 0.09 0.74 3.4 1.6 3.24 0.15 2.5 2.3 0.04 0.8
…1000 2.4 2.4 0 0.9
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VC ML – Quantitative & Categorical traits method
0 0.5 1
21, XXCov
H1: CAQ VlVVXXCov 2ˆˆ|, 21
H0: |, 21 XXCov )()(log
0
110 HL
HLLOD CA VlV 2
e.g. LOD=3
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Individual LOD scores can be expressed as P values (Pointwise)LOD Chi-sq (n-df) P value2.1 9.67 0.0009
Genome-wide linkage analysis (e.g. VC)
(x4.6)
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Statistics for selected samples
T2
T1
H0 (No linkage): Mean 5.0ˆ H1 (Linkage): Mean 5.0ˆ
H0 (No linkage): Mean H1 (Linkage): Mean
5.0ˆ 5.0ˆ
Mean IBD sharing statistics(Risch & Zhang 1995, 1996)
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Other Linkage statistics
Dependent variable: Phenotypes Independent variable:
Dependent variable: Independent variable: Phenotypes
Extensions to Haseman Elston
VC ML with mixture distribution
Pedwide-regression Analysis (“reverse HE”)Reverse VC ML
(Wright 1997, Drigalenko 1998, Elston et al. 2000, Forrest 2001, Visscher & Hopper 2001, Xu et al. 2000,
Sham & Purcell 2001)(Eaves et al. 1996)
(Sham et al. 2002)
(Sham et al. 2000)
Statistics for affection traitsBased on IBD scoring functions eg. Sall(Whittemore & Halpern 1994, Kong & Cox 1997)
Forrest & Feingold 2000 Mixed statistic
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No need to specify mode of inheritance
Nonparametric Linkage Analysis - summary
Models phenotypic and genotypic similarity of relativesExpression of phenotypic similarity, calculation of IBD
HE and VC are the most popular statistics used for linkage of quantitative traitsOther statistics available, specially for affection traits
Type I error?Power?
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Type I error
Type I error
True positive
LOD k Theoretical (Lander & Kruglyak
1995)Empirical
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Theoretical genome-wide thresholds
Genome-wide threshold for suggestive linkageLOD score that occurs by chance alone on average once per scanLOD = 2.2, Chi-sq = 10.1, Pointwise P = 0.00074
Genome-wide threshold for significant linkageLOD score that occurs by chance alone on average once per 20 scansLOD = 3.6, Chi-sq = 16.7, Pointwise P = 0.000022
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Empirical genome-wide thresholds
Genome-wide threshold for suggestive linkageLOD score that occurs by chance alone on average once per scan
Genome-wide threshold for significant linkageLOD score that occurs by chance alone on average once per 20 scans