QTL Mapping in Natural Populations

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
Linkage disequilibrium mapping natural
population

• -Marker, Prob(M)p, Prob(m)1-p
• -QTL, Prob(A)q, Prob(a)1-q
• Four haplotypes
• Prob(MA)p11pqD pp11p10
• Prob(Ma)p10p(1-q)-D qp11p01
• Prob(mA)p01(1-p)q-D Dp11p00-p10p01
• Prob(ma)p00(1-p)(1-q)D

3
Joint and conditional (?ji) genotype prob.
between marker and QTL

• AA Aa aa Obs
• MM p112 2p11p10 p102 n2
• Mm 2p11p01 2(p11p00p10p01) 2p10p00 n1
• mm p012 2p01p00 p002 n0
• MM p112 2p11p10 p102 n2
• p2 p2 p2
• Mm 2p11p01 2(p11p00p10p01) 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

4
Mixture model-based likelihoodwith marker
information
Linkage disequilibrium mapping natural
population

• L(?y,M)?i1n?2if2(yi) ?1if1(yi)
  ?0if0(yi)
• Sam- Height Marker genotype QTL genotype
• ple (cm, y) M AA Aa aa
• 1 184 MM (2) ?22i ?12i ?02i
• 2 185 MM (2) ?22i ?12i ?02i
• 3 180 Mm (1) ?21i ?11i ?01i
• 4 182 Mm (1) ?21i ?11i ?01i
• 5 167 Mm (1) ?21i ?11i ?01i
• 6 169 Mm (1) ?21i ?11i ?01i
• 7 165 mm (0) ?20i ?10i ?00i
• 8 166 mm (0) ?20i ?10i ?00i

Prior prob.
5
Conditional probabilities of the QTL genotypes
(missing) based on marker genotypes (observed)
Linkage disequilibrium mapping natural
population

• L(?y,M)
• ?i1n ?2if2(yi) ?1if1(yi) ?0if0(yi)
• ?i1n2 ?22if2(yi) ?12if1(yi)
  ?02if0(yi) Conditional on 2 (n2)
• ? ?i1n1 ?21if2(yi) ?11if1(yi)
  ?01if0(yi) Conditional on 1 (n1)
• ? ?i1n0 ?20if2(yi) ?10if1(yi)
  ?00if0(yi) Conditional on 0 (n0)

6
Normal distributions of phenotypic values for
each QTL genotype group
Linkage disequilibrium mapping natural
population

• 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

7
Differentiating L with respect to each unknown
parameter, setting derivatives equal zero and
solving the log-likelihood equations
Linkage disequilibrium mapping natural
population

• L(?y,M) ?i1n?2if2(yi) ?1if1(yi)
  ?0if0(yi)
• log L(?y,M) ?i1n log?2if2(yi) ?1if1(yi)
  ?0if0(yi)
• Define
• ?2i ?2if1(yi)/?2if2(yi) ?1if1(yi)
  ?0if0(yi) (1)
• ?1i ?1if1(yi)/?2if2(yi) ?1if1(yi)
  ?0if0(yi) (2)
• ?0i ?0if1(yi)/?2if2(yi) ?1if1(yi)
  ?0if0(yi) (3)
• ?2 ?i1n(?2iyi)/ ?i1n?2i (4)
• ?1 ?i1n(?1iyi)/ ?i1n?1i (5)
• ?0 ?i1n(?0iyi)/ ?i1n?0i (6)
• ?2 1/n?i1n?2i(yi-?2)2?1i(yi-?1)2?0i(yi-?0
  )2 (7)

8

• Complete data Prior prob
• QQ Qq qq Obs
• MM p112 2p11p10 p102 n2
• Mm 2p11p01 2(p11p00p10p01) 2p10p00 n1
• mm p012 2p01p00 p002 n0
• QQ Qq qq Obs
• MM n22 n21 n20 n2
• Mm n12 n11 n10 n1
• mm n02 n01 n00 n0
• p112n22 (n21n12) ?n11/2n,
• p102n20 (n21n10) (1-?)n11/2n,
• p012n02 (n12n01) (1-?)n11/2n,
• p112n00 (n10n01) ?n11/2n,
  ?p11p00/(p11p00p10p01)

9

• Incomplete (observed) data
• Posterior prob
• QQ Qq qq Obs
• MM ?22i ?12i ?02i n2
• Mm ?21i ?11i ?01i n1
• mm ?20i ?10i ?00i n0
• p11?i1n2(2?22i?12i)?i1n1(?21i??11i)/2n
  , (8)
• p10?i1n2(2?02i?12i)?i1n1?01i(1-?)?11i
  /2n, (9)
• p01?i1n0(2?20i?10i)?i1n1?21i(1-?)?11i
  /2n, (10)
• p00?i1n2(2?00i?10i)?i1n1(?01i??11i)/2n
  (11)

10
EM algorithm

• (1) Give initiate values ?(0) (?2,?1,?0,?2,p11,p1
  0,p01,p00)(0)
• (2) Calculate ?2i(1), ?1i(1) and ?0i(1) using
  Eqs. 1-3,
• (3) Calculate ?(1) using ?2i(1), ?1i(1) and
  ?0i(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 -2ln 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 -2ln 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
• SNP Chrom. Black White
• ADRA1A 8p21 q 0.20
• D 0.04
• a 11.40
• d -2.63
• LR 3.90 NS
• WHR
• ADRB1 10q24 q 0.83
• D -0.07
• a -0.15
• d -0.24
• LR 5.91 NS
• ADRB2 5q32-33 q 0.16
• D 0.07
• a 0.16
• d -0.20

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