Put Markers and Trait Data into box below Linkage Disequilibrium Mapping - Natural Population OR

Linkage Disequilibrium Mapping - Natural Population Initial value of p11, p10, p01:

Linkage Disequilibrium Mapping - Natural Population

Mixture model-based likelihood Height markers Sample (cm, y) m1 m2 m3 … Linkage disequilibrium mapping – natural population

Association between marker and QTL -Marker, Prob(M)=p, Prob(m)=1-p -QTL, Prob(Q)=q, Prob(q)=1-q Four haplotypes: Prob(MQ)=p 11 =pq+D p=(p 11 +p 10 ) Prob(Mq)=p 10 =p(1-q)-Dq=(p 11 +p 01 ) Prob(mQ)=p 01 =(1-p)q-DD=p 11 p 00 -p 10 p 01 Prob(mq)=p 00 =(1-p)(1-q)+D Estimate p, q, D AND  2,  1,  0 Linkage disequilibrium mapping – natural population

Mixture model-based likelihood 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 QQQqqq 1184MM (2)  2|i  1|i  0|i 2185MM (2)  2|i  1|i  0|i 3180Mm (1)  2|i  1|i  0|i 4182Mm (1)  2|i  1|i  0|i 5167Mm (1)  2|i  1|i  0|i 6169Mm (1)  2|i  1|i  0|i 7165mm (0)  2|i  1|i  0|i 8166mm (0)  2|i  1|i  0|i Prior prob. Linkage disequilibrium mapping – natural population

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 MMp 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

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|2 f 2 (y i ) +  1|2 f 1 (y i ) +  0|2 f 0 (y i )] Conditional on 2 (n 2 )   i=1 n1 [  2|1 f 2 (y i ) +  1|1 f 1 (y i ) +  0|1 f 0 (y i )] Conditional on 1 (n 1 )   i=1 n0 [  2|0 f 2 (y i ) +  1|0 f 1 (y i ) +  0|0 f 0 (y i )] Conditional on 0 (n 0 ) Linkage disequilibrium mapping – natural population

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

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  1|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 [  1|i (y i -  1 ) 2 +  0|i (y i -  0 ) 2 ](7) Linkage disequilibrium mapping – natural population

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 =1/2n{  i=1 n2 [2  2|2i +  1|2i ]+  i=1 n1 [  2|1i +  1|1i ],(8) p 10 =1/2n{  i=1 n2 [2  0|2i +  1|2i ]+  i=1 n1 [  0|1i +(1-  )  1|1i ],(9) p 01 =1/2n{  i=1 n0 [2  2|0i +  1|0i ]+  i=1 n1 [  2|1i +(1-  )  1|1i ], (10) p 00 =1/2n{  i=1 n2 [2  0|0i +  1|0i ]+  i=1 n1 [  0|1i +  1|1i ] (11)

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.

Do While (Abs(mu(1) - omu(1)) + Abs(p00 - p00old) > ) kkk = kkk + 1 ‘cumulate the number of iteration p00old = p00 ‘keep old value of p00 prob(1, 1) = p11 ^ 2 / p ^ 2 ‘prior conditional probability  2|2 prob(1, 2) = 2 * p11 * p10 / p ^ 2 ‘  2|1 prob(1, 3) = p10 ^ 2 / p ^ 2 ‘  2|0 prob(2, 1) = 2 * p11 * p01 / (2 * p * q) ‘  1|2 prob(2, 2) = 2 * (p11 * p00 + p10 * p01) / (2 * p * q) ‘  1|1 prob(2, 3) = 2 * p10 * p00 / (2 * p * q) ‘  1|0 prob(3, 1) = p01 ^ 2 / q ^ 2 ‘  0|2 prob(3, 2) = 2 * p01 * p00 / q ^ 2 ‘  0|1 prob(3, 3) = p00 ^ 2 / q ^ 2 ‘  0|0 Given a initial  2,  1,  0,  2, p11, p10, p01, p00 mu(1), mu(2), mu(3), s2 PROGRAM:

For j = 1 To 3 omu(j) = mu(j) : cmu(j) = 0 : cpi(j) = 0 : bpi(j) = 0 For i = 1 To 3 nnn(i, j) = 0 ’3 by 3 matrix to store  2|2,  2|1,  2|0, ….  0|0 Next Next j cs2 = 0 ll = 0 For i = 1 To N sss = 0 For j = 1 To 3 ’ f2(yi), f1(yi), f0(yi) f(j) = 1 / Sqr(2 * * s2) * Exp(-(y(i) - mu(j)) ^ 2 / 2 / s2) sss = sss + prob(datas(i, mrk), j) * f(j) Next j ll = ll + Log(sss) ’calculate log-likelihood For j = 1 To 3 bpi(j) = prob(datas(i, mrk), j) * f(j) / sss ’FORMULA (1-3) cmu(j) = cmu(j) + bpi(j) * datas(i, nmrk) ’ numerator of FORMULA (4-6) cpi(j) = cpi(j) + bpi(j) ’ denominator of FORMULA (4-6) cs2 = cs2 + bpi(j) * (y(i) - mu(j)) ^ 2 ’FORMULA (7) nnn(datas(i, mrk), j) = nnn(datas(i, mrk), j) + bpi(j) ’FORMULA (8-11) Next j Next i ‘[  2|if2(yi) +  1|if1(yi) +  0|if0(yi)]

‘ Update  2,  1,  0 formula (4-6) For j = 1 To 3 mu(j) = cmu(j) / cpi(j) Next j ‘Update  2 formula 7 s2 = cs2 / N ‘Update p11, p10, p01, p00 FORMULA (8-11) phi = p11 * p00 / (p11 * p00 + p10 * p01) p11 = (2 * nnn(1, 1) + nnn(1, 2) + nnn(2, 1) + phi * nnn(2, 2)) / 2 / N p10 = (2 * nnn(1, 3) + nnn(1, 2) + nnn(2, 3) + (1 - phi) * nnn(2, 2)) / 2 / N p01 = (2 * nnn(3, 1) + nnn(2, 1) + nnn(3, 2) + (1 - phi) * nnn(2, 2)) / 2 / N p00 = (2 * nnn(3, 3) + nnn(2, 3) + nnn(3, 2) + phi * nnn(2, 2)) / 2 / N p = p11 + p10 q = 1 - p Loop LR = 2 * (ll - ll0)

Put Markers and Trait Data into box below Linkage Disequilibrium Mapping - Natural Population Binary Trait OR

Linkage Disequilibrium Mapping - Natural Population Binary Trait Initial value of p11, p10, p01: Initial value of f2, f1, f0:

Linkage Disequilibrium Mapping - Natural Population Binary Trait Initial value of f2, f1, f0:

Linkage Disequilibrium Mapping - Natural Population Binary Trait L(  |y)=  j=0 2  i=0 nj log [  2|ij Pr{y ij =1|G ij =2,  } yij Pr{y ij =0|G ij =2,  } (1-yij) +  1|ij Pr{y ij =1|G ij =1,  } yij Pr{y ij =0|G ij =1,  } (1-yij) +  0|ij Pr{y ij =1|G ij =0,  } yij Pr{y ij =0|G ij =0,  } (1-yij) ] =  j=0 2  i=0 nj log[  2|ij f 2 yij (1-f 2 ) (1-yij) +  1|ij f 1 yij (1-f 1 ) (1-yij) +  0|ij f 0 yij (1-f 0 ) (1-yij) ]  = (p 11, p 10, p 01, p 00, f 2, f 1, f 0 ) (6 parameters)

For j = 1 To 3 omu(j) = mu(j) : cmu(j) = 0 : cpi(j) = 0 : bpi(j) = 0 For i = 1 To 3 nnn(i, j) = 0 ’3 by 3 matrix to store  2|2,  2|1,  2|0, ….  0|0 Next Next j cs2 = 0 ll = 0 For i = 1 To N sss = 0 For j = 1 To 3 ’ f2(yi), f1(yi), f0(yi) f(j) = 1 / Sqr(2 * * s2) * Exp(-(y(i) - mu(j)) ^ 2 / 2 / s2) f(j)=mu(j) ^ datas(i, nmrk) * (1 - mu(j)) ^ (1 - datas(i, nmrk)) sss = sss + prob(datas(i, mrk), j) * f(j) Next j ll = ll + Log(sss) ’calculate log-likelihood For j = 1 To 3 bpi(j) = prob(datas(i, mrk), j) * f(j) / sss ’FORMULA (1-3) cmu(j) = cmu(j) + bpi(j) * datas(i, nmrk) ’ numerator of FORMULA (4-6) cpi(j) = cpi(j) + bpi(j) ’ denominator of FORMULA (4-6) cs2 = cs2 + bpi(j) * (y(i) - mu(j)) ^ 2 ’FORMULA (7) nnn(datas(i, mrk), j) = nnn(datas(i, mrk), j) + bpi(j) ’FORMULA (8-11) Next j Next i