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QTL Mapping in Natural Populations

QTL Mapping in Natural Populations. Basic theory for QTL mapping is derived from linkage analysis in controlled crosses There is a group of species in which it is not possible to make crosses QTL mapping in such species should be based on existing populations.

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QTL Mapping in Natural Populations

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  1. QTL Mapping in Natural Populations • Basic theory for QTL mapping is derived from linkage analysis in controlled crosses • There is a group of species in which it is not possible to make crosses • QTL mapping in such species should be based on existing populations

  2. Linkage disequilibrium mapping – natural population Association between marker and QTL -Marker, Prob(M)=p, Prob(m)=1-p -QTL, Prob(A)=q, Prob(a)=1-q Four haplotypes: Prob(MA)=p11=pq+D p=p11+p10 Prob(Ma)=p10=p(1-q)-D q=p11+p01 Prob(mA)=p01=(1-p)q-D D=p11p00-p10p01 Prob(ma)=p00=(1-p)(1-q)+D

  3. Joint and conditional (j|i) genotype prob. between marker and QTL AA Aa aa Obs MM p112 2p11p10 p102 n2 Mm 2p11p01 2(p11p00+p10p01) 2p10p00 n1 mm p012 2p01p00 p002 n0 MM p112 2p11p10 p102 n2 p2p2p2 Mm 2p11p01 2(p11p00+p10p01) 2p10p00 n1 2p(1-p) 2p(1-p) 2p(1-p) mm p012 2p01p00 p002 n0 (1-p)2 (1-p)2 (1-p)2

  4. Linkage disequilibrium mapping – natural population Mixture model-based likelihoodwith marker information L(|y,M)=i=1n[2|if2(yi) + 1|if1(yi) + 0|if0(yi)] Sam- Height Marker genotype QTL genotype ple (cm, y) MAAAa aa 1 184 MM (2) 2|2i1|2i 0|2i 2 185 MM (2) 2|2i1|2i 0|2i 3 180 Mm (1) 2|1i1|1i 0|1i 4 182 Mm (1) 2|1i1|1i 0|1i 5 167 Mm (1) 2|1i1|1i 0|1i 6 169 Mm (1) 2|1i1|1i 0|1i 7 165 mm (0) 2|0i1|0i 0|0i 8 166 mm (0) 2|0i1|0i 0|0i Prior prob.

  5. Linkage disequilibrium mapping – natural population Conditional probabilities of the QTL genotypes (missing) based on marker genotypes (observed) L(|y,M) = i=1n [2|if2(yi) + 1|if1(yi) + 0|if0(yi)] = i=1n2 [2|2if2(yi) + 1|2if1(yi) + 0|2if0(yi)] Conditional on 2 (n2) i=1n1 [2|1if2(yi) + 1|1if1(yi) + 0|1if0(yi)] Conditional on 1 (n1) i=1n0 [2|0if2(yi) + 1|0if1(yi) + 0|0if0(yi)] Conditional on 0 (n0)

  6. Linkage disequilibrium mapping – natural population Normal distributions of phenotypic values for each QTL genotype group f2(yi) = 1/(22)1/2exp[-(yi-2)2/(22)], 2 =  + a f1(yi) = 1/(22)1/2exp[-(yi-1)2/(22)], 1 =  + d f0(yi) = 1/(22)1/2exp[-(yi-0)2/(22)], 0 =  - a

  7. Linkage disequilibrium mapping – natural population Differentiating L with respect to each unknown parameter, setting derivatives equal zero and solving the log-likelihood equations L(|y,M) = i=1n[2|if2(yi) + 1|if1(yi) + 0|if0(yi)] log L(|y,M) = i=1n log[2|if2(yi) + 1|if1(yi) + 0|if0(yi)] Define 2|i= 2|if1(yi)/[2|if2(yi) + 1|if1(yi) + 0|if0(yi)] (1) 1|i= 1|if1(yi)/[2|if2(yi) + 1|if1(yi) + 0|if0(yi)] (2) 0|i= 0|if1(yi)/[2|if2(yi) + 1|if1(yi) + 0|if0(yi)] (3) 2 = i=1n(2|iyi)/ i=1n2|i (4) 1 = i=1n(1|iyi)/ i=1n1|i (5) 0 = i=1n(0|iyi)/ i=1n0|i (6) 2 = 1/ni=1n[2|i(yi-2)2+1|i(yi-1)2+0|i(yi-0)2] (7)

  8. Complete data Prior prob QQ Qq qq Obs MM p112 2p11p10 p102 n2 Mm 2p11p01 2(p11p00+p10p01) 2p10p00 n1 mm p012 2p01p00 p002 n0 QQ Qq qq Obs MM n22 n21 n20 n2 Mm n12 n11 n10n1 mm n02 n01 n00n0 p11=[2n22 + (n21+n12) + n11]/2n, p10=[2n20 + (n21+n10) + (1-)n11]/2n, p01=[2n02 + (n12+n01) + (1-)n11]/2n, p11=[2n00 + (n10+n01) + n11]/2n, =p11p00/(p11p00+p10p01)

  9. Incomplete (observed) data Posterior prob QQ Qq qq Obs MM 2|2i1|2i 0|2i n2 Mm 2|1i1|1i 0|1in1 mm 2|0i1|0i 0|0in0 p11=[i=1n2(22|2i+1|2i)+i=1n1(2|1i+1|1i)]/2n, (8) p10={i=1n2(20|2i+1|2i)+i=1n1[0|1i+(1-)1|1i]}/2n, (9) p01={i=1n0(22|0i+1|0i)+i=1n1[2|1i+(1-)1|1i]}/2n, (10) p00=[i=1n2(20|0i+1|0i)+i=1n1(0|1i+1|1i)]/2n (11)

  10. EM algorithm (1) Give initiate values (0) =(2,1,0,2,p11,p10,p01,p00)(0) (2) Calculate 2|i(1), 1|i(1)and 0|i(1)using Eqs. 1-3, (3) Calculate (1) using 2|i(1), 1|i(1)and 0|i(1)based on Eqs. 4-11, (4) Repeat (2) and (3) until convergence.

  11. Hypothesis Tests • Is there a significant QTL? H0: μ2 = μ1 = μ1 H1: Not H0 LR1 = -2[ln L0 – L1] Critical threshold determined from permutation tests

  12. Hypothesis Tests • Can this QTL be detected by the marker? H0: D = 0 H1: Not H0 LR2 = -2[ln L0 – L1] Critical threshold determined from chi-square table (df = 1)

  13. A case study from human populations • 105 black women and 538 white women; • 10 SNPs genotyped within 5 candidates for human obesity; • Two obesity traits, the amount of body fat (body mass index, BMI) and its distribution throughout the body (waist to hip circumference ratio, WHR)

  14. Objective Detect quantitative trait nucleotides (QTNs) predisposing to human obesity traits, BMI and WHR

  15. BMI SNP Chrom. Black White ADRA1A 8p21 q 0.20 D 0.04 a 11.40 d -2.63 LR 3.90* NS WHR ADRB1 10q24 q 0.83 D -0.07 a -0.15 d -0.24 LR 5.91* NS ADRB2 5q32-33 q 0.16 D 0.07 a 0.16 d -0.20 LR 5.88* NS ADRB2- 5/20 q 0.83 0.78 GNAS1 D 0.02 0.03 a -0.18 -0.15 d -0.10 -0.16 LR 8.42* 8.06*

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