QTL Mapping in Natural Populations. Basic theory for QTL mapping is derived from linkage analysis in

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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

Linkage disequilibrium mapping – natural population

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)=p11=pq+D p=p11+p10

Prob(Ma)=p10=p(1-q)-D q=p11+p01

Prob(mA)=p01=(1-p)q-D D=p11p00-p10p01

Prob(ma)=p00=(1-p)(1-q)+D

Joint and conditional (j|i) genotype prob. between marker and QTL

AA Aa aa Obs

MM p112 2p11p10 p102 n2

Mm 2p11p01 2(p11p00+p10p01) 2p10p00 n1

mm p012 2p01p00 p002 n0

MM p112 2p11p10 p102 n2

p2p2p2

Mm 2p11p01 2(p11p00+p10p01) 2p10p00 n1

2p(1-p) 2p(1-p) 2p(1-p)

mm p012 2p01p00 p002 n0

(1-p)2 (1-p)2 (1-p)2

Linkage disequilibrium mapping – natural population

Mixture model-based likelihoodwith marker information

L(|y,M)=i=1n[2|if2(yi) + 1|if1(yi) + 0|if0(yi)]

Sam- Height Marker genotype QTL genotype

ple (cm, y) MAAAa aa

1 184 MM (2) 2|2i1|2i 0|2i

2 185 MM (2) 2|2i1|2i 0|2i

3 180 Mm (1) 2|1i1|1i 0|1i

4 182 Mm (1) 2|1i1|1i 0|1i

5 167 Mm (1) 2|1i1|1i 0|1i

6 169 Mm (1) 2|1i1|1i 0|1i

7 165 mm (0) 2|0i1|0i 0|0i

8 166 mm (0) 2|0i1|0i 0|0i

Prior prob.

Linkage disequilibrium mapping – natural population

Conditional probabilities of the QTL genotypes (missing) based on marker genotypes (observed)

L(|y,M)

= i=1n [2|if2(yi) + 1|if1(yi) + 0|if0(yi)]

= i=1n2 [2|2if2(yi) + 1|2if1(yi) + 0|2if0(yi)] Conditional on 2 (n2)

i=1n1 [2|1if2(yi) + 1|1if1(yi) + 0|1if0(yi)] Conditional on 1 (n1)

i=1n0 [2|0if2(yi) + 1|0if1(yi) + 0|0if0(yi)] Conditional on 0 (n0)

Linkage disequilibrium mapping – natural population

Normal distributions of phenotypic values for each QTL genotype group

f2(yi) = 1/(22)1/2exp[-(yi-2)2/(22)],

2 =  + a

f1(yi) = 1/(22)1/2exp[-(yi-1)2/(22)],

1 =  + d

f0(yi) = 1/(22)1/2exp[-(yi-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=1n[2|if2(yi) + 1|if1(yi) + 0|if0(yi)]

log L(|y,M) = i=1n log[2|if2(yi) + 1|if1(yi) + 0|if0(yi)]

Define

2|i= 2|if1(yi)/[2|if2(yi) + 1|if1(yi) + 0|if0(yi)] (1)

1|i= 1|if1(yi)/[2|if2(yi) + 1|if1(yi) + 0|if0(yi)] (2)

0|i= 0|if1(yi)/[2|if2(yi) + 1|if1(yi) + 0|if0(yi)] (3)

2 = i=1n(2|iyi)/ i=1n2|i (4)

1 = i=1n(1|iyi)/ i=1n1|i (5)

0 = i=1n(0|iyi)/ i=1n0|i (6)

2 = 1/ni=1n[2|i(yi-2)2+1|i(yi-1)2+0|i(yi-0)2] (7)

Complete data Prior prob

QQ Qq qq Obs

MM p112 2p11p10 p102 n2

Mm 2p11p01 2(p11p00+p10p01) 2p10p00 n1

mm p012 2p01p00 p002 n0

QQ Qq qq Obs

MM n22 n21 n20 n2

Mm n12 n11 n10n1

mm n02 n01 n00n0

p11=[2n22 + (n21+n12) + n11]/2n,

p10=[2n20 + (n21+n10) + (1-)n11]/2n,

p01=[2n02 + (n12+n01) + (1-)n11]/2n,

p11=[2n00 + (n10+n01) + n11]/2n, =p11p00/(p11p00+p10p01)

Incomplete (observed) data

Posterior prob

QQ Qq qq Obs

MM 2|2i1|2i 0|2i n2

Mm 2|1i1|1i 0|1in1

mm 2|0i1|0i 0|0in0

p11=[i=1n2(22|2i+1|2i)+i=1n1(2|1i+1|1i)]/2n, (8)

p10={i=1n2(20|2i+1|2i)+i=1n1[0|1i+(1-)1|1i]}/2n, (9)

p01={i=1n0(22|0i+1|0i)+i=1n1[2|1i+(1-)1|1i]}/2n, (10)

p00=[i=1n2(20|0i+1|0i)+i=1n1(0|1i+1|1i)]/2n (11)

EM algorithm

(1) Give initiate values (0) =(2,1,0,2,p11,p10,p01,p00)(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.

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

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)

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)
Objective

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

BMI

SNP Chrom. Black White

D 0.04

a 11.40

d -2.63

LR 3.90* NS

WHR

D -0.07

a -0.15

d -0.24

LR 5.91* NS

D 0.07

a 0.16

d -0.20

LR 5.88* NS

GNAS1 D 0.02 0.03

a -0.18 -0.15

d -0.10 -0.16

LR 8.42* 8.06*