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.

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 stepwhere m equals the sum of the following terms:

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)

Hypothesis tests

Sex-specific difference in population structure

Hypothesis test

• Sex-specific difference in the recombination fraction

Simulation

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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