1. Gene, Allele, Genotype, and Phenotype
Basic Concepts
Gene, Allele, Genotype, and Phenotype
A pair of chromosomes
Father Mother
Phenotype
Subject Genotype Height IQ AA AA
Gene A,
with two
alleles A
and a Aa Aa aa aa

2. Genetic Mapping A gene that affects a quantitative
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
Let’s see what are QTL? QTL are specific genomic segments that affect the phenotype. QTL can be detected by linked markers. This is a diagram for detecting QTL by using linked markers. The QTL detected by this approach is hypothetical chromosome segments whose DNA structure and organization are unknown. .
Marker k
Linkage Map

3. 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

4. Human Chromosomes
Male Xy
X y
Female XX X XX Xy
Daughter Son

5. Human Difference

6. How many genes control human body height?

7. Discontinuous Distribution
due to a single dwarf gene

8. Continuous Distribution
due to many genes?

10. Continuous Variation due to
Polygenes 31=3, 32=9, …, 310=59,049
Environmental modifications
Gene-environmental interactions

11. Power statistical methods are crucial for the identification of human height genes

12. Data Structure 1 AA(2) BB(2) … y1 2|1 1|1 0|1 2 AA(2) BB(2) ... y2
Subject
Marker (M)
Conditional prob
M M … Mm
Phenotype
(y)
of QTL genotype
QQ(2) Qq(1) qq(0) 1
AA(2) BB(2) … y1
2| 1| 0|1 2
AA(2) BB(2) y2
2| 1| 0|2 3
Aa(1) Bb(1) y3
2| 1|3 0|3 4 y4
2| 1|4 0|4 5 y5
2| 1|5 0|5 6
Aa(1) bb(0) y6
2| 1|6 0|6 7
aa(0) Bb(1) y7
2| 1|7 0|7 8
aa(0) bb(0) … y8
2| 1|8 0|8

13. Association between marker and QTL
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

14. 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 2p11p p102 n2
p2 p2 p2
2p(1-p) 2p(1-p) p(1-p)
mm p012 2p01p p002 n0
(1-p)2 (1-p)2 (1-p)2

15. Mixture model-based likelihood with marker information
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)]
Sam- Height Marker genotype QTL genotype
ple (cm, y) M AA Aa aa
MM (2) 2|1 1|1 0|1
MM (2) 2|2 1|2 0|2
Mm (1) 2|3 1|3 0|3
Mm (1) 2|4 1|4 0|4
Mm (1) 2|5 1|5 0|5
Mm (1) 2|6 1|6 0|6
mm (0) 2|7 1|7 0|7
mm (0) 2|8 1|8 0|8
Prior prob.

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

17. Normal distributions of phenotypic values for each QTL genotype group
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

18. 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)

19. 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
MM n22 n21 n20 n2
Mm n12 n11 n10 n1
mm n02 n01 n00 n0
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)

20. Incomplete (observed) data Posterior prob QQ Qq qq Obs
MM 2|i 1|i 0|i n2
Mm 2|i 1|i 0|i n1
mm 2|i 1|i 0|i n0
p11=[i=1n2(22|i+1|i)+i=1n1(2|i+1|i)]/2n, (8)
p10={i=1n2(20|i+1|i)+i=1n1[0|i+(1-)1|i]}/2n, (9)
p01={i=1n0(22|i+1|i)+i=1n1[2|i+(1-)1|i]}/2n, (10)
p00=[i=1n2(20|i+1|i)+i=1n1(0|i+1|i)]/2n (11)

21. 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.

22. 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

23. 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)

24. 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)

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

28. BMI
SNP Chrom. Black White
ADRA1A 8p q D a d
LR * NS
WHR
ADRB1 10q q D a d
LR * NS
ADRB2 5q q D a d
LR * NS
ADRB2- 5/20 q
GNAS1 D a d
LR * *

29. 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