STAT246 Statistical Genetics Terry Speed Lecture 2 January 19 2006 Scribe Soyeon Ahn EXERCISES Ex1 Derive the Hardy Weinberg Equilibrium HWE frequencies for 2 alleles Ex2 Suppose that your brother has genotype u v at an STR locus What is the probability that you are also u v at that locus Consider the case u v and u 6 v separately Assume random mating at this locus More about the HWE In the HWE derivation we ignored the possibility of mutation mutation rate for STRs selection i e we implicitly assumed approx all genotypes were equally viable finite population size ignoring genetic drift population structure e g ethnic racial geographic subdivision realities of human populations overlapping generations family structure dependence etc We explicitly ruled out assortative mating in relation to the locus under study HWE in practice usually o k frequently not In reality we don t know until we look Possible reasons for failure of HWE See above genotyping errors see later Recombination fractions One locus Figure 1 1 can be generalizing to two loci Figure 1 2 in the notion of recombination and linkage We need a story for the transmission from parents to gametes of variants of two loci Generalization of the second law of Mendel 1 Figure 1 1 One locus gamete can be A or a with probability 12 2 two loci the first and last gamete are parental and middle ones are recombinant at these two loci Figure 1 2 Parental AB 12 1 r ab 21 1 r recombinant Ab 12 r aB 12 r r recombination fraction between the two loci some people use others use R r 12 r 12 for loci on different chromosome unlinked r 12 linked Terminology Haplotype allelic assignment at 2 or more linked loci ex A B A b a B a bThe haplotype analogue of HWE for 2 loci is called badly Linkage Equilibrium LE The haplotype frequencies in the population are products of the associated allele frequencies i e independence across loci of allele frequency fAB of haplotype AB pA qB where pA allele frequency of A qB allele frequency of B Similarly under LE fAb pA qb etc Unlike the simple derivation of HWE where equilibrium is reached in 1 round of random mating and no mutation etc LE is only reached asymptotically Let fij t frequency of haplotype ai bj in the population at generation t See Figure 2 where alleles at A are a1 a2 ai and the alleles at B are b1 b2 bj with frequencies pi qj respectively 2 Figure 2 Under Linkage Equilibrium GM Grand maternal GP Grand paternal Two cases R recombination occurs NR no recombination occurs in producing the gamete In this 2 locus context the random mating assumption is on pairs of allele of the loci Let H denote the gamete s haplotype pr H ai bj at generation t X X pr GM GP R H ai bj pr N R GM GP X GM GP fGM t 1 fGP t 1 r pr H ai bj GM GP R GM GP X similar with NR GM GP 1 X fGM t 1 fGP t 1 r pr GM GP X GM GP 1 fGM t 1 fGP t 1 1 r pr 2 2 r pi qj 1 r fij t 1 i e fij t 1 r fij t 1 r pi qj fij t pi qj 1 r fij t pi qj 1 r t fij 0 pi qj Equation 1 r pr rec GM GP Equation 2 first term fGM t 1 i k and fGP t 1 u j or fGM t 1 h j and fGP t 1 i v second term fGM t 1 i j and fGP t 1 u v or fGM t 1 h k and fGP t 1 i j Hence independence in the limit 3 3 As with HWE LE is frequently observed to be approximately true occasionally not In DNA forensics the loci are almost all on the different chromosome so this argument is used with r 12 4
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