Interval mapping with maximum likelihood Data Files: Marker file – all markers Traits file – all traits Linkage map – built based on markers For example:

Interval mapping with maximum likelihood ID #PN RG472 RG246 K5 U10 RG532 W1 RG173 Amy1B RZ276 RG146

Interval mapping with maximum likelihood ID 10D 20D 30D 40D 50D 60D 70D 80D 90D

RG472 RG K5 U10 RG532 W1 RG173 RZ276 Amy1B RG146 RG345 RG381 RZ19 RG690 RZ730 RZ801 RG810 RG RG437 RG544 RG171 RG157 RZ318 Pall RZ58 CDO686 Amy1A/C RG95 RG654 RG256 RZ213 RZ123 RG RG104 RG348 RZ329 RZ892 RG100 RG191 RZ678 RZ574 RZ284 RZ394 pRD10A RZ403 RG179 CDO337 RZ337A RZ448 RZ519 Pgi -1 CDO87 RG910 RG418A RG218 RZ262 RG190 RG908 RG91 RG449 RG788 RZ565 RZ675 RG163 RZ590 RG214 RG143 RG chrom1chrom2chrom3chrom4

- Type of Study Interval Mapping Program - Genetic Design

- Data and Options Names of Markers (optional) Cumulative Marker Distance (cM) Interval Mapping Program Map Function Parameters Here for Simulation Study Only QTL Searching StepcM

- Data Interval Mapping Program Put Markers and Trait Data into box below OR

- Analyze Data Interval Mapping Program

- Profile Interval Mapping Program

- Permutation Test Interval Mapping Program #Tests Cut Point at Level Is Based on Tests.

Backcross Population – Two Point FreqQqqq Mm1/2(1-r)/2r/2 mm1/2r/2(1-r)/2

Backcross Population – Three Point FreqQqqq MmNn(1-r)/2(1-r 1 )(1-r 2 )r 1 *r 2 Mmnnr/2(1-r 1 )r 2 r 1 (1-r 2 ) mmNnr/2r 1 (1-r 2 )(1-r 1 )r 2 mmnn(1-r)/2r1*r2(1-r)/2 M Q N

F2 Population – Two Point FreqQQQqqq MM1/4(1-r) 2 /4(1-r)r/2r 2 /4 Mm1/2(1-r)r/2½-(1-r)r(1-r)r/2 mm1/4r 2 /4(1-r)r/2(1-r) 2 /4

F2 Population – Three Point FreqQQQqqq MMNN(1-r) 2 /4 1/4(1-a) 2 (1-b) 2 1/2a(1-a)b(1-b)1/4a 2 b 2 Nn(1-r)r/2 1/2(1-a) 2 b(1-b)1/2a(2b 2 - 2b+1)(1-a) 1/2a 2 b(1-b) nnr 2 /4 1/4(1-a) 2 b 2 1/2a(1-a)b(1-b)1/4a 2 (1-b) 2 MmNN(1-r)r/2 1/2a(1-a)(1-b) 2 1/2b(1- 2a+2a 2 )(1-b) 1/2a(1-a)b 2 Nn ½-(1-r)r a(1-a)b(1-b)1/2(2b 2 - 2b+1)(1-2a+2a 2 ) a(1-a)b(1-b) Nn(1-r)r/2 1/2a(1-a)b 2 1/2b(1- 2a+2a 2 )(1-b) 1/2a(1-a)(1-b) 2 mmNNr 2 /4 1/4a 2 (1-b) 2 1/2a(1-a)b(1-b)1/4(1-a) 2 b 2 Nn(1-r)r/2 1/2a 2 b(1-b)1/2a(2b 2 - 2b+1)(1-a) 1/2(1-a) 2 b(1-b) nn(1-r) 2 /4 1/4a 2 b 2 1/2a(1-a)b(1-b)1/4(1-a) 2 (1-b) 2 M a Q b N r=a+b-2ab

Differentiating L with respect to each unknown parameter, setting derivatives equal zero and solving the log-likelihood equations L(y,M|  ) =  i=1 n [  1|i f 1 (y i ) +  0|i f 0 (y i )] log L(y,M|  ) =  i=1 n log[  1|i f 1 (y i ) +  0|i f 0 (y i )] Define  1|i =  1|i f 1 (y i )/[  1|i f 1 (y i ) +  0|i f 0 (y i )](1)  0|i =  0|i f 1 (y i )/[  1|i f 1 (y i ) +  0|i f 0 (y i )](2)  1 =  i=1 n (  1|i y i )/  i=1 n  1|i (3)  0 =  i=1 n (  0|i y i )/  i=1 n  0|i (4)  2 = 1/n  i=1 n [  1|i (y i -  1 ) 2 +  0 |i (y i -  0 ) 2 ](5)  = (  i=1 n2  1|i +  i=1 n3  0 |i )/(n 2 +n 3 )(6)

function [mk, testres]=GenMarkerForBackcross(dist, N) %genarate N Backcross Markers from marker disttance (cM) dist. if dist(1)~=0, cm=[0 dist]/100; else cm=dist/100; end n=length(cm); rs=1/2*(exp(2*cm)-exp(-2*cm))./(exp(2*cm)+exp(-2*cm)); for j=1:N mk(j,1)=(rand>0.5); end Random Generate Markers for Backcross Population

for i=2:n for j=1:N if mk(j,i-1)==1, mk(j,i)=rand>rs(i); else mk(j,i)=rand<rs(i); end Random Generate Markers for Backcross Population, Cont’

EM algorithm for Interval Mapping function intmapbackross(Datas, mrkplace) % for example, mrkplace=[ ]; N=size(Datas,1); nmrk=size(mrkplace); mm=mean(Datas(:,size(Datas,2))); vv=var(Datas(:,size(Datas,2)),1); ll0 = N * (-log(2 * * vv) - 1) / 2; %likelihood at null res=[]; omu1=0;

for cm = 1:2:mrkplace(nmrk) for i = 1:nmrk if mrkplace(i) <= cm qtlk = i end theta = (cm - mrkplace(qtlk)) / (mrkplace(qtlk + 1) - mrkplace(qtlk)); th(1) = 1; th(2) = 1 - theta; th(3) = theta; th(4) = 0; mu1 = mm; mu0 = mm; s2=vv; EM algorithm for Interval Mapping

while (abs(mu1 - omu1) > ) omu1 = mu1; cmu1 = 0; cmu0 = 0; cs2 = 0; cpi = 0; ll = 0; for j = 1:N f1 = 1 / sqrt(2 * * s2) * exp(-(Datas(j, nmrk+1) - mu1)^2 / 2 / s2); f0 = 1 / sqrt(2 * * s2) * exp(-(Datas(j, nmrk+1) - mu0)^2 / 2 / s2); pi1i = th(4 - Datas(j, qtlk + 1) - Datas(j, qtlk) * 2); pi0i = 1 - pi1i; ll = ll + log(pi1i * f1 + pi0i * f0); BPi1i = pi1i * f1 / (pi1i * f1 + pi0i * f0); %E-Step BPi0i = 1 - BPi1i; cmu1 = cmu1 + BPi1i * Datas(j, nmrk+1); %M-STEP cmu0 = cmu0 + BPi0i * Datas(j, nmrk+1); cs2 = cs2 + BPi1i * (Datas(j, nmrk+1) - mu1) ^ 2 + BPi0i * (Datas(j, nmrk+1) - mu0) ^ 2; cpi = cpi + BPi1i; end mu1 = cmu1 / cpi; mu0 = cmu0 / (N - cpi); %M-STEP s2 = cs2 / N; end

prob=th(4 - Datas(:, qtlk + 1) - Datas(:, qtlk) * 2)'; [mmmm, likelihoodback(p, … Datas(:,nmrk+1), [prob 1-prob]), [mm mm vv]); %Simplex Local Search Method LR = 2 * (ll - ll0); res=[res; [cm mu1 mu0 s2 LR]]; end EM algorithm for Interval Mapping %Simplex Local Search Method

function A=likelihoodback(par, y, marker) mu1=par(1); mu0=par(2); s2=par(3); yy1=y-mu1; yy0=y-mu0; A=sum( log( sum([exp(-yy1.^2/2/s2) … exp(-yy0.^2/s2/2)].*marker,2))... -log(s2)/2-1/2*log(2*pi)) -10.E5*(s2<0.001); A=-A; EM algorithm for Interval Mapping