1.5.1 ESTIMATION OF HAPLOTYPE FREQUENCIES:

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1 .5. ESTIMATION OF HAPLOTYPE FREQUENCIES: Chapter - 8 For SNPs, alleles A j,b j at locus j there are 4 haplotypes: A A, A B, B A and B B frequencies q,q,q 3,q 4. Assume HWE at haplotype level. Only the double heterozygote A B,A B cannot be phased, Loc- Locus A A A B B B A A q q q 3 q 3 (q + q 3 ) A B q q (q q 4 q 3 q 4 (q + q 3 ) + q q 3 ) (q + q 4 ) B B q q q 4 q 4 (q + q 4 ) (q + (q + q ) (q 3 + q ) (q 3 + q 4 ) q 4 ) There are 0 two-locus genotypes, but only 9 phenotypes. Haplotype frequencies can be estimated from phenotype frequencies via the EM algorithm: Pr(A A /B B A B,A B ) = q q 4 /(q q 4 + q q 3 ) Chapter - 9 Given a set of current haplotype frequency estimates q i,i=,...,4 and the phenotypic counts, the conditional expected genotypic counts are easily obtained. New haplotype estimates then are the expected multinomial proportions of each haplotype. For consistency with PHASE and other software we will temporarily label the two alleles at each SNP 0 and. At two loci there are 4 haplotypes: 00, 0, 0,, freqs q 00,q 0... There are 0 phased two-locus genotypes: eg; 0/00, /00, 0/0 There are 9 observable two-locus genotypes: (00,00), (0, ),... Only the double-heterozygotes (0,0) is ambiguous: /00 or 0/0. For other genotype pairs we just count. Suppose there are H double-heterozygotes. Then E(#(/00) H) = Hq q 00 /(q q 00 + q 0 q 0 ) So EM is easily implemented. For two (or a very few) loci it works well.

2 Chapter ESTIMATING PHASE AND HAPLOTYPE FREQUENCIES: Consider diallelic loci (e.g. SNPs): Label alleles 0 and. Then the genotypes are a set of pairs e.g. 00, 0,, 0, 0, and haplotypes a string such as 000. Determining phase is determining which of the 4 possibilities 0 and 0000, 00 and 000, 00 and 000, 000 and 00 holds. For convenience we may write an unphased genotype as (0,0000) and the phased version as (0/0000). For large samples and/or small numbers of SNPs we may use EM algorithm to estimate haplotype frequencies, and this also provides probabilities of phasings, given the estimated frequencies. However, for large numbers of SNPs this does not work well: many sample haplotype freqs are 0, and likelihood surface is high-dimensional and multimodal..5.3 HAPLOTYPING: TWO (OTHER) ALGORITHMS: Chapter - 3 Clark (990) s algorithm: note where individuals are homozygous, haplotyping is trivial. Also trivial if heterozygous at just locus. Use individuals heterozygous at at most locus to identify haplotypes that must be present. Assuming these, see which other individuals can be explained by one of these haplotypes, plus a new one add these new ones to the collection, and continue for as long as possible. Problems: May not be able to start. May not be able to finish. Final guess may depend on order one adds haplotypes to the pool. Stephens algorithm (PHASE): use a model that summarizes similarities of haplotypes in a population the idea is that haplotypes should look like each other in chunks. Use Monte Carlo to simulate alternative phasings under the model. Produces probable phasings with estimates probabilities. (Now also fastphase. Scheet and Stephens (006).

3 Chapter ALLELIC ASSOCIATION; LINKAGE DISEQULIBRIUM: A measure of allelic association between the two loci is = Pr(A A ) Pr(A ) Pr(A ) = q (q + q )(q + q 3 ) = (q q 4 q q 3 ) since q +q +q 3 +q 4 =. This measure is known as the coefficient of linkage disequilibrium. Allelic associations between loci arise from population structure, admixture and history, or from selection.(see.6. and.6.3). Example of mixture/subdivision the nuisance case. Vs. case of interest original mutation on some genetic background. Associations are, however, maintained by tight linkage (ρ 0). LD blocks are the remnants of recombination; they are not caused by linkage, but they survive because of linkage. (See.6.5). Contrast with HWE: Even for unlinked loci equilibrium ( =0) is not achieved in one generation..6. LD ARISES BY MUTATION OR STRUCTURE: Chapter - 33!"#$%&'()*+(,$-**(*+.$-,,/.+0/&$3#4$ -,,/.+0/&,$0.)/,,$*/.+$ $$$3#4$)(5(.6$7/7%*0/&$$ $$$*(8(*$./0&.(,6)9:$ ;(<$80)+0&6$0**(*($$ $$$)(=0+&,$+&$3#$<+6>$ $$$?0.@A)/%&':$ B/7%*0/&$,%?'+8+,+/&$$ $$$$0*,/$)(,%*6,$+&$3#$ $ $ high low high low

4 Chapter LD and POPULATION STRUCTURE: Population stratification creates LD, even if there is no LD within subpopulations. If allele frequencies differ so will the frequency of haplotypes. Consider populations i in proportions α i and allele A alleles frequencies p i, p i at two loci. As before Pr(A )=p = i α ip i, and Pr(A )=p = i α ip i. Assume there is NO LD within each subpopulation, so the overall frequency of the A A haplotype is Pr(A A )= i α ip i p i Then = Pr(A A ) Pr(A )Pr(A ) = α i p i p i ( α i p i )( i i j α j p j ) = i α i (p i p )(p i p )..6.4 SEGREGATION OF HAPLOTYPES: Chapter - 35 Homozygous individuals (both loci): for example an A A,B B individual segregates only A B haplotypes. Homozygote/Heterozygote: for example, an A A,A B individual passes on A A or A B each with probability / regardless of recombinaton probability ρ. A double-heterozygote individual passes each of the four haplotypes A A, A B, B A and B B, with probabilities: ( ρ)/, ρ/, ρ/ and ( ρ)/ if his genotype is A A /B B, ρ/, ( ρ)/, ( ρ)/, and ρ/ if his genotype is A B /B A. Case: A A /B B. A B A ( ρ)/ ρ/ B ρ/ ( ρ)/ Recombination breaks up chromosomes, but we only see this directly if genotypes are heterozygous at both loci.

5 Chapter DECAY OF LD: Suppose current haplotype freqs of A A, A B, B A and B B are q, q, q 3 and q 4, and at next generation are q, q, q 3 and q 4. Now, for example, an offspring A A haplotype arises from a A A, A A parent with prob. from a A A, A B or A B, A A, with prob /, from a A B,A B who is A A /B B with prob ( ρ)/ from a A B,A B who is A B /B A with prob ρ/. Thus q = q +q (q + q 3 )/ + q q 4 ( ρ)/+q q 3 ρ/ = q (q + q + q 3 + q 4 ) ρ(q q 4 q q 3 ) = q ρ. Analogously, q = q +ρ, q 3 = q 3+ρ and q 4 = q 4 ρ. Thus, in expectation, allele frequencies are unchanged (q +q = q +q ): = q q 4 q q 3 = (q ρ )(q 4 ρ ) (q + ρ )(q 3 + ρ ) = ρ (q + q + q 3 + q 4 )+ρ ( ) = ( ρ). blank slide: Chapter - 37