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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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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)=p 11 =pq+D p=p 11 +p 10 Prob(Ma)=p 10 =p(1-q)-Dq=p 11 +p 01 Prob(mA)=p 01 =(1-p)q-DD=p 11 p 00 -p 10 p 01 Prob(ma)=p 00 =(1-p)(1-q)+D Linkage disequilibrium mapping – natural population

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AAAaaa Obs MMp p 11 p 10 p 10 2 n 2 Mm2p 11 p 01 2(p 11 p 00 +p 10 p 01 )2p 10 p 00 n 1 mmp p 01 p 00 p 00 2 n 0 MM p p 11 p 10 p 10 2 n 2 p 2 p 2 p 2 Mm2p 11 p 01 2(p 11 p 00 +p 10 p 01 )2p 10 p 00 n 1 2p(1-p) 2p(1-p) 2p(1-p) mmp p 01 p 00 p 00 2 n 0 (1-p) 2 (1-p) 2 (1-p) 2 Joint and conditional ( j|i ) genotype prob. between marker and QTL

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Mixture model-based likelihood with marker information L( |y,M)= i=1 n [ 2|i f 2 (y i ) + 1|i f 1 (y i ) + 0|i f 0 (y i )] Sam- Height Marker genotype QTL genotype ple(cm, y) M AA Aaaa 1184MM (2) 2|2i 1|2i 0|2i 2185MM (2) 2|2i 1|2i 0|2i 3180Mm (1) 2|1i 1|1i 0|1i 4182Mm (1) 2|1i 1|1i 0|1i 5167Mm (1) 2|1i 1|1i 0|1i 6169Mm (1) 2|1i 1|1i 0|1i 7165mm (0) 2|0i 1|0i 0|0i 8166mm (0) 2|0i 1|0i 0|0i Prior prob. Linkage disequilibrium mapping – natural population

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Conditional probabilities of the QTL genotypes (missing) based on marker genotypes (observed) L( |y,M) = i=1 n [ 2|i f 2 (y i ) + 1|i f 1 (y i ) + 0|i f 0 (y i )] = i=1 n2 [ 2|2i f 2 (y i ) + 1|2i f 1 (y i ) + 0|2i f 0 (y i )] Conditional on 2 (n 2 ) i=1 n1 [ 2|1i f 2 (y i ) + 1|1i f 1 (y i ) + 0|1i f 0 (y i )] Conditional on 1 (n 1 ) i=1 n0 [ 2|0i f 2 (y i ) + 1|0i f 1 (y i ) + 0|0i f 0 (y i )] Conditional on 0 (n 0 ) Linkage disequilibrium mapping – natural population

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Normal distributions of phenotypic values for each QTL genotype group f 2 (y i ) = 1/(2 2 ) 1/2 exp[-(y i - 2 ) 2 /(2 2 )], 2 = + a f 1 (y i ) = 1/(2 2 ) 1/2 exp[-(y i - 1 ) 2 /(2 2 )], 1 = + d f 0 (y i ) = 1/(2 2 ) 1/2 exp[-(y i - 0 ) 2 /(2 2 )], 0 = - a Linkage disequilibrium mapping – natural population

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Differentiating L with respect to each unknown parameter, setting derivatives equal zero and solving the log-likelihood equations L( |y,M) = i=1 n [ 2|i f 2 (y i ) + 1|i f 1 (y i ) + 0|i f 0 (y i )] log L( |y,M) = i=1 n log[ 2|i f 2 (y i ) + 1|i f 1 (y i ) + 0|i f 0 (y i )] Define 2|i = 2|i f 1 (y i )/[ 2|i f 2 (y i ) + 1|i f 1 (y i ) + 0|i f 0 (y i )](1) 1|i = 1|i f 1 (y i )/[ 2|i f 2 (y i ) + 1|i f 1 (y i ) + 0|i f 0 (y i )](2) 0|i = 0|i f 1 (y i )/[ 2|i f 2 (y i ) + 1|i f 1 (y i ) + 0|i f 0 (y i )](3) 2 = i=1 n ( 2|i y i )/ i=1 n 2|i (4) 1 = i=1 n ( 1|i y i )/ i=1 n 1|i (5) 0 = i=1 n ( 0|i y i )/ i=1 n 0|i (6) 2 = 1/n i=1 n [ 2|i (y i - 2 ) 2 + 1|i (y i - 1 ) 2 + 0|i (y i - 0 ) 2 ](7) Linkage disequilibrium mapping – natural population

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Complete dataPrior prob QQQqqqObs MMp p 11 p 10 p 10 2 n 2 Mm2p 11 p 01 2(p 11 p 00 +p 10 p 01 )2p 10 p 00 n 1 mmp p 01 p 00 p 00 2 n 0 QQQqqqObs MMn 22 n 21 n 20 n 2 Mmn 12 n 11 n 10 n 1 mmn 02 n 01 n 00 n 0 p 11 =[2n 22 + (n 21 +n 12 ) + n 11 ]/2n, p 10 =[2n 20 + (n 21 +n 10 ) + (1- )n 11 ]/2n, p 01 =[2n 02 + (n 12 +n 01 ) + (1- )n 11 ]/2n, p 11 =[2n 00 + (n 10 +n 01 ) + n 11 ]/2n, =p 11 p 00 /(p 11 p 00 +p 10 p 01 )

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Incomplete (observed) data Posterior prob QQQqqqObs MM 2|2i 1|2i 0|2i n 2 Mm 2|1i 1|1i 0|1i n 1 mm 2|0i 1|0i 0|0i n 0 p 11 =[ i=1 n2 (2 2|2i + 1|2i )+ i=1 n1 ( 2|1i + 1|1i )]/2n,(8) p 10 ={ i=1 n2 (2 0|2i + 1|2i )+ i=1 n1 [ 0|1i +(1- ) 1|1i ]}/2n,(9) p 01 ={ i=1 n0 (2 2|0i + 1|0i )+ i=1 n1 [ 2|1i +(1- ) 1|1i ]}/2n, (10) p 00 =[ i=1 n2 (2 0|0i + 1|0i )+ i=1 n1 ( 0|1i + 1|1i )]/2n (11)

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EM algorithm (1) Give initiate values (0) =( 2, 1, 0, 2,p 11,p 10,p 01,p 00 ) (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.

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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

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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)

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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)

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

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BMI SNPChrom. BlackWhite ADRA1A8p21 q 0.20 D 0.04 a11.40 d-2.63 LR 3.90* NS WHR ADRB110q24 q 0.83 D-0.07 a-0.15 d-0.24 LR 5.91* NS ADRB25q32-33q 0.16 D 0.07 a 0.16 d-0.20 LR 5.88* NS ADRB2-5/20q GNAS1D a d LR 8.42* 8.06*

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