STAT246 : Statistical GeneticsTerry SpeedLecture 2 - January 19, 2006Scribe: Soyeon AhnEXERCISES:Ex1. Derive the Hardy-Weinberg Equilibrium ( HWE ) frequencies for >2alleles.Ex2. Suppose that your brother has genotype {u, v} at an STR locus. Whatis the probability that you are also {u, v} at that locus? Consider the case u = vand u 6= v separately. Assume “ random mating” at this locus.More about the HWEIn the HWE derivation we ignored the possibility of• mutation (mutation rate for STRs)• selection, i.e we implicitly assumed approx all genotypes were equally vi-able• finite p opulation 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 understudy.HWE in practice: usually o.k , frequently not. In reality, we don’t knowuntil we look.Possible reasons for failure of HWE : See above, & genotyping errors (seelater).Recombination fractionsOne locus(Figure 1(1)) can be generalizing to two loci(Figure 1(2)) in thenotion of recombination and linkage. We nee d a story for the transmission fromparents to gametes of variants of two loci (Generalization of the second law ofMendel).1Figure 1: (1)One locus, gamete can be A or a with probability12(2) two loci,the first and last gamete are parental and middle ones are recombinant at thesetwo loci.Figure 1 (2) Parental : AB[12(1 − r)], ab[12(1 − r)] , recombinant : Ab[12r],aB[12r]r : recombination fraction between the two loci. (some people use θ, othersuse R), r ≤12r =12for lo c i on different chromosome (unlinked)r <12(linked).TerminologyHaplotype : allelic assignment at 2 or more linked lociex) -A-B- -A-b- -a-B- -a-b-The haplotype analogue of HWE for 2 loci is called (badly) Linkage Equi-librium (LE).The haplotype frequencies in the population are products of the associatedallele frequencies, i.e, independence across loci of allele.frequency fABof haplotyp e AB =pA× qB, where pA= allele frequencyof A , qB= allele frequency of B.Similarly under LE , fAb= pAqb, etc.Unlike the simple derivation of HWE where equilibrium is reached in 1round of random mating (and no mutation, etc), LE is only reached asymptot-ically.Let fij(t) = frequency of haplotype aibjin the population at generationt. (See Figure 2), where alleles at A are a1, a2, · · · , aiand the alleles at B areb1, b2, · · · , bjwith frequencies {pi}{qj} respectively.2Figure 2: Under Linkage Equilibrium, GM : Grand maternal, GP : Grand pa-ternalTwo cases R : recombination occurs , NR : no recombination occurs in pro-ducing the gamete.In this 2 locus context, the random mating assumption is on pairs of alleleof the loci. Let H denote the gamete’s haplotype.pr(H = aibjat generation t)=XGM,GPpr(GM, GP, R, H = aibj) +XGM,GPpr(· · · , NR, · · · )=XGM,GPfGM(t − 1) × fGP(t − 1) × r × pr(H = aibj|GM, GP, R) +XGM,GPsimilar with NR(1)=XGM,GPfGM(t − 1) × fGP(t − 1) × r × pr(· · · ) +XGM,GPfGM(t − 1) × fGP(t − 1) ×12(1 − r) × pr(· · · )(2)= r × piqj+ (1 − r) × fij(t − 1), i.efij(t) = (1 − r)fij(t − 1) + r × piqjfij(t) − piqj= (1 − r)(fij(t) − piqj)= (1 − r)t(fij(0) − piqj) (3)Equation (1) : r = pr(rec.|GM, GP )Equation (2) :first term fGM(t − 1) : −i − k − and fGP(t − 1) : −u − j− orfGM(t − 1) : −h − j − and fGP(t − 1) : −i − v−second term fGM(t − 1) : −i − j − and fGP(t − 1) : −u − v− orfGM(t − 1) : −h − k − and fGP(t − 1) : −i − j−Hence independence in the limit.3As with HWE , LE is frequently observed to be approximately true, occa-sionally not.In DNA forensics, the loci are (almost all) on the different chromosome sothis argument is used with r
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