Basic Concepts
Gene, Allele, Genotype, and Phenotype
A pair of chromosomes
Father Mother
Phenotype
Subject
Gene A,
with two
alleles A
and a
Genotype Height
IQ
1
2
AA
AA
185
182
100
104
3
4
Aa
Aa
175
171
103
102
5
6
aa
aa
155
152
101
103
Bad news: It is very hard to detect such a gene directly.
Genetic Mapping
 A gene that affects a quantitative
trait is called a quantitative trait
locus (QTL).
 A QTL can be detected by the
markers linked with it.
 A QTL detected is a chromosomal
segment.
Marker 1
QTL
Marker 2
Marker 3
.
.
.
Marker k
Linkage Map
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
Human Chromosomes
Male Xy
X
y
Female
XX
X XX
Xy
Daughter
Son
Human Difference
How many genes
control human body
height?
Discontinuous Distribution
due to a single dwarf gene
Continuous Distribution
due to many genes?
Continuous Variation due to
• Polygenes 31=3, 32=9, …, 310=59,049
• Environmental modifications
• Gene-environmental interactions
Power statistical
methods are crucial
for the identification
of human height
genes
Data Structure
Marker (M)
Subject M1
1
2
3
4
5
6
7
8
AA(2)
AA(2)
Aa(1)
Aa(1)
Aa(1)
Aa(1)
aa(0)
aa(0)
M2
BB(2)
BB(2)
Bb(1)
Bb(1)
Bb(1)
bb(0)
Bb(1)
bb(0)
Conditional prob
… Mm
Phenotype
(y)
…
...
...
...
...
...
...
…
y1
y2
y3
y4
y5
y6
y7
y8
of QTL genotype
QQ(2) Qq(1) qq(0)
2|1
2|2
2|3
2|4
2|5
2|6
2|7
2|8
1|1
1|2
1|3
1|4
1|5
1|6
1|7
1|8
0|1
0|2
0|3
0|4
0|5
0|6
0|7
0|8
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
Prob(Ma)=p10=p(1-q)-D
Prob(mA)=p01=(1-p)q-D
Prob(ma)=p00=(1-p)(1-q)+D
p=p11+p10
q=p11+p01
D=p11p00-p10p01
Joint and conditional (j|i) genotype
prob. between marker and QTL
AA
Aa
aa
Obs
MM
Mm
mm
p112
2p11p01
p012
2p11p10
2(p11p00+p10p01)
2p01p00
p102
2p10p00
p002
n2
n1
n0
MM
p112
p2
2p11p01
2p(1-p)
p012
(1-p)2
2p11p10
p2
2(p11p00+p10p01)
2p(1-p)
2p01p00
(1-p)2
p102
p2
2p10p00
2p(1-p)
p002
(1-p)2
n2
Mm
mm
n1
n0
Linkage disequilibrium mapping – natural population
Mixture model-based likelihood
with marker information
L(|y,M)=i=1n[2|if2(yi) + 1|if1(yi) + 0|if0(yi)]
Prior prob.
Sam- Height
ple (cm, y)
1
184
2
185
3
180
4
182
5
167
6
169
7
165
8
166
Marker genotype
M
MM (2)
MM (2)
Mm (1)
Mm (1)
Mm (1)
Mm (1)
mm (0)
mm (0)
QTL genotype
AA
Aa
2|1
1|1
2|2
1|2
2|3
1|3
2|4
1|4
2|5
1|5
2|6
1|6
2|7
1|7
2|8
1|8
aa
0|1
0|2
0|3
0|4
0|5
0|6
0|7
0|8
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|if2(yi) + 1|if1(yi) + 0|if0(yi)] Conditional on 2 (n2)
 i=1n1 [2|if2(yi) + 1|if1(yi) + 0|if0(yi)] Conditional on 1 (n1)
 i=1n0 [2|if2(yi) + 1|if1(yi) + 0|if0(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|i = 1|if1(yi)/[2|if2(yi) + 1|if1(yi) + 0|if0(yi)]
0|i = 0|if1(yi)/[2|if2(yi) + 1|if1(yi) + 0|if0(yi)]
(1)
(2)
(3)
2 = i=1n(2|iyi)/ i=1n2|i
1 = i=1n(1|iyi)/ i=1n1|i
0 = i=1n(0|iyi)/ i=1n0|i
2 = 1/ni=1n[2|i(yi-2)2+1|i(yi-1)2+0|i(yi-0)2]
(4)
(5)
(6)
(7)
Complete data
QQ
MM
Mm
mm
Prior prob
Qq
qq
Obs
p112
2p11p01
p012
2p11p10
2(p11p00+p10p01)
2p01p00
p102
2p10p00
p002
n2
n1
n0
QQ
Qq
qq
Obs
n20
n10
n00
n2
n1
n0
MM n22
n21
Mm n12
n11
mm n02
n01
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|i
Mm 2|i
mm 2|i
n2
n1
n0
1|i
1|i
1|i
0|i
0|i
0|i
p11=[i=1n2(22|i+1|i)+i=1n1(2|i+1|i)]/2n,
p10={i=1n2(20|i+1|i)+i=1n1[0|i+(1-)1|i]}/2n,
p01={i=1n0(22|i+1|i)+i=1n1[2|i+(1-)1|i]}/2n,
p00=[i=1n2(20|i+1|i)+i=1n1(0|i+1|i)]/2n
(8)
(9)
(10)
(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.
ADRA1A 8p21
q
D
a
d
LR
Black
0.20
0.04
11.40
-2.63
3.90*
White
NS
WHR
ADRB1
ADRB2
10q24
5q32-33
ADRB2- 5/20
GNAS1
q
D
a
d
LR
0.83
-0.07
-0.15
-0.24
5.91*
NS
q
D
a
d
LR
0.16
0.07
0.16
-0.20
5.88*
NS
q
D
a
d
LR
0.83
0.02
-0.18
-0.10
8.42*
0.78
0.03
-0.15
-0.16
8.06*
Shape mapping meets LD mapping
Mapping Body Shape Genes through Shape Mapping
Ningtao Wang, Yaqun Wang, Zhong Wang, Han
Hao and Rongling Wu*
Center for Statistical Genetics, The Pennsylvania State
University, Hershey, PA 17033, USA
J Biom Biostat 2012, 3:8