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. 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

3. AAAaaa Obs MMp 11 2 2p 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 01 2 2p 01 p 00 p 00 2 n 0 MM p 11 2 2p 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 01 2 2p 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

4. 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

5. 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

6. 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

7. 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

8. Complete dataPrior prob QQQqqqObs MMp 11 2 2p 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 01 2 2p 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 )

9. 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)

10. 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.

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 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 0.830.78 GNAS1D 0.02 0.03 a-0.18-0.15 d-0.10-0.16 LR 8.42* 8.06*