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Joint Linkage and Linkage Disequilibrium Mapping
Key ReferenceLi, Q., and R. L. Wu, 2009 A multilocus model for constructing a linkage disequilibrium map in human populations. Statistical Applications in Genetics and Molecular Biology 8 (1): Article 18.
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Genetic Designs for Mapping• Controlled crosses – Backcross, F2, full-sib family, …
(linkage)• Unrelated (random) individuals from a natural
population (linkage disequilibrium)• Cases and controls from a natural population• Unrelated (random) families from a natural population
(linkage and LD)• Related (non-random) families from a natural
population (linkage, LD and identical-by-descent)
Family designs are increasingly used for genetic studies because of much information contained.
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Natural Population• Consider two SNPs 1 (with two allele A and a)
and 2 (with two alleles B and b)• The two SNPs are linked with recom. frac. r • The two SNPs form four haplotypes, AB, Ab, aB,
and ab• Prob(A) = p, Prob(B) = q, linkage disequilibrium =
D. We have haplotype frequencies as
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Diagrammatic Presentation
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Family Design: family number and size
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Mating frequencies of families and offspring genotype frequencies per family
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HWE assumed
Can you figure out where this assumption is needed?
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Segregation of double heterozygote
• Overall haplotype frequencies produced by this parent are calculated as 1/2ω1 for AB or ab and 1/2ω2 for Ab or aB
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A Joint Probability
• Mother genotypes (Mm)
• Father genotypes (Mf )
• Offspring genotypes (Mo)
P(Mm,Mf,Mo) = P(Mm,Mf)P(Mo|Mm.Mf)=
P(Mm)P(Mf)P(Mo|Mm,Mf)
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A joint two-stage log-likelihoodLet unknown parameters
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Upper-stage Likelihood
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EM algorithm for Θ• E step
• M step
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Lower-stage Likelihood
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EM algorithm for r
• E step - calculate the probability with which a considered haplotype produced by a double heterozygote parent is the recombinant type using
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E step (cont’d)
• Calculate the probability with which a double heterozygote offspring carries recombinant haplotypes by
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M stepwhere m equals the sum of the following terms:
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Hypothesis tests
Linkage and Linkage disequilibriumH0: r = 0 and D = 0H1: At least one equality does not hold
LR = -2(log L0 – log L1)
Critical threshold x2 (df=2)
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Hypothesis tests
Sex-specific difference in population structure
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Hypothesis test
• Sex-specific difference in the recombination fraction
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Simulation
€
ˆ r
€
r
€
r
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€
r
€
r
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Power
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Power
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Conclusions
The model can jointly estimate the linkage and linkage disequilibrium between two markers- LD from parents- Linkage from offspring
The model can draw a LD map to study the evolution of populations and high-resolution mapping of traits