Joint Linkage and Linkage Disequilibrium Mapping

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Presentation transcript:

Joint Linkage and Linkage Disequilibrium Mapping Key Reference Li, 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.

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.

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

Diagrammatic Presentation

Family Design: family number and size

Mating frequencies of families and oﬀspring genotype frequencies per family

HWE assumed Can you figure out where this assumption is needed?

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

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)

A joint two-stage log-likelihood Let unknown parameters

Upper-stage Likelihood

EM algorithm for Θ E step M step

Lower-stage Likelihood

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

E step (cont’d) Calculate the probability with which a double heterozygote oﬀspring carries recombinant haplotypes by

M step where m equals the sum of the following terms:

Hypothesis tests Linkage and Linkage disequilibrium H0: r = 0 and D = 0 H1: At least one equality does not hold LR = -2(log L0 – log L1) Critical threshold x2 (df=2)

Hypothesis tests Sex-specific difference in population structure

Hypothesis test Sex-specific difference in the recombination fraction

Simulation

Power

Power

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