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

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

RG24619.2

16.1K5U10RG532

W1RG173

RZ276

Amy1B

RG146

RG345

RG381

RZ19

RG690

RZ730

RZ801

RG810

RG331

4.84.7

15.315.5

15.03.8

3.3

34.3

2.5

23.5

8.2

13.2

33.1

2.6

9.2

RG437

RG544

RG171

RG157

RZ318

Pall

RZ58

CDO686

Amy1A/C

RG95

RG654

RG256

RZ213

RZ123

RG520

13.0

5.3

22.2

27.4

6.3

29.3

10.2

8.8

12.8

8.4

5.110.0

5.4

13.1

RG104RG348

RZ329RZ892

RG100

RG191RZ678

RZ574

RZ284

RZ394

pRD10A

RZ403

RG179

CDO337

RZ337A

RZ448

RZ519

Pgi -1

CDO87

RG910

RG418A

7.7

13.26.99.82.8

17.5

41.6

37.1

15.6

18.5

2.5

5.028.6

1.9

22.5

15.0

32.1

7.1

9.217.9

RG218

RZ262

RG190

RG908RG91RG449

RG788RZ565

RZ675

RG163

RZ590

RG214

RG143

RG620

8.18.6

12.6

13.73.2

16.18.4

16.8

21.4

28.2

2.7

12.2

5.9

chrom1 chrom2 chrom3 chrom4

Simulation

- Type of Study

Back

Interval Mapping Program

F2

Backcross

- Genetic Design

- Data and Options

Names of Markers (optional)

Cumulative Marker Distance (cM)

Means sigma^2

QTL position


N

Haldane

KosambiMap Function

Parameters Here for Simulation Study Only

QTL Searching Step cM

H2 Calculate H2 or Sigma2

Back

- Data

Back


Put Markers and Trait Data into box below OR Simulate Data

- Analyze Data

Back


Analyze

Regression ModelMixture Model

- Profile

Back


(0.50)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0 20 40 60 80 100

QTL at 91 (091 1.756 1.455 0.182 3.649)

- Permutation Test

Back


(1.00)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0 20 40 60 80 100

QTL at 91 (091 1.756 1.455 0.182 3.649)

#Tests

Test

Cut Point atLevel Is Based on Tests.

Reset

Backcross Population – Two Point

Freq Qq qq

Mm 1/2 (1-r)/2 r/2

mm 1/2 r/2 (1-r)/2

Backcross Population – Three Point

Freq Qq qq

Mm Nn (1-r)/2 (1-r1)(1-r2) r1*r2

Mm nn r/2 (1-r1)r2 r1 (1-r2)

mm Nn r/2 r1 (1-r2) (1-r1)r2

mm nn (1-r)/2 r1*r2 (1-r)/2

M Q N

F2 Population – Two Point

Freq QQ Qq qq

MM 1/4 (1-r)2/4 (1-r)r/2 r2/4

Mm 1/2 (1-r)r/2 ½-(1-r)r (1-r)r/2

mm 1/4 r2/4 (1-r)r/2 (1-r)2/4

F2 Population – Three PointFreq QQ Qq qq

MM NN (1-r)2/4 1/4(1-a)2(1-b)2 1/2a(1-a)b(1-b) 1/4a2b2

Nn (1-r)r/2 1/2(1-a)2b(1-b) 1/2a(2b2-2b+1)(1-a)

1/2a2b(1-b)

nn r2/4 1/4(1-a)2b2 1/2a(1-a)b(1-b) 1/4a2(1-b)2

Mm NN (1-r)r/2 1/2a(1-a)(1-b)2 1/2b(1-2a+2a2)(1-b)

1/2a(1-a)b2

Nn ½-(1-r)r a(1-a)b(1-b) 1/2(2b2-2b+1)(1-2a+2a2)

a(1-a)b(1-b)

Nn (1-r)r/2 1/2a(1-a)b2 1/2b(1-2a+2a2)(1-b)

1/2a(1-a)(1-b)2

mm NN r2/4 1/4a2(1-b)2 1/2a(1-a)b(1-b) 1/4(1-a)2b2

Nn (1-r)r/2 1/2a2b(1-b) 1/2a(2b2-2b+1)(1-a)

1/2(1-a)2b(1-b)

nn (1-r)2/4 1/4a2b2 1/2a(1-a)b(1-b) 1/4(1-a)2(1-b)2

M a Q b Nr=a+b-2ab

Differentiating L with respect to each unknown parameter, setting derivatives equal zero and

solving the log-likelihood equationsL(y,M|) = i=1

n[1|if1(yi) + 0|if0(yi)]log L(y,M|) = i=1

n log[1|if1(yi) + 0|if0(yi)]

Define

1|i = 1|if1(yi)/[1|if1(yi) + 0|if0(yi)] (1)0|i = 0|if1(yi)/[1|if1(yi) + 0|if0(yi)] (2)

1 = i=1n(1|iyi)/ i=1

n1|i (3)0 = i=1

n(0|iyi)/ i=1n0|i (4)

2 = 1/ni=1n[1|i(yi-1)2+0|i(yi-0)2] (5)

= (i=1n21|i +i=1

n30|i)/(n2+n3) (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 end end

Random Generate Markers for Backcross Population, Cont’

EM algorithm for Interval Mapping

function intmapbackross(Datas, mrkplace)%% for example, mrkplace=[0 20 40 60 80];N=size(Datas,1);nmrk=size(mrkplace);

mm=mean(Datas(:,size(Datas,2)));vv=var(Datas(:,size(Datas,2)),1); ll0 = N * (-log(2 * 3.1415926 * 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 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) > 0.00000001) omu1 = mu1; cmu1 = 0; cmu0 = 0; cs2 = 0; cpi = 0; ll = 0; for j = 1:N f1 = 1 / sqrt(2 * 3.1415926 * s2) * exp(-(Datas(j, nmrk+1) - mu1)^2 / 2 / s2); f0 = 1 / sqrt(2 * 3.1415926 * 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, llll]=fminsearch(@(p) 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


%%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;


Documents

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