Gene, Allele, Genotype, and Phenotype A pair of chromosomes Father Mother Gene A, with two alleles A...
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Transcript of Gene, Allele, Genotype, and Phenotype A pair of chromosomes Father Mother Gene A, with two alleles A...
Gene, Allele, Genotype, and Phenotype
A pair of chromosomes Father Mother
Gene A,with twoalleles Aand a
Phenotype
1 AA 185 1002 AA 182 104
3 Aa 175 1034 Aa 171 102
5 aa 155 1016 aa 152 103
Subject Genotype Height IQ
Basic Concepts
Bad news: It is very hard to detect such a gene directly.
Genetic Mapping
Marker 1
Marker 2Marker 3
Marker k
QTL
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.
.
.
.
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 yFemaleXX 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
Subject
Marker (M) Conditional prob M1 M2 … Mm
Phenoty
pe (y)
of QTL genotypeQQ(2) Qq(1) qq(0)
1 AA(2) BB(2) … y1
2|1
1|1
0|1
2 AA(2) BB(2) ... y2
2|2
1|2
0|2
3 Aa(1) Bb(1) ... y3
2|3
1|3
0|3
4 Aa(1) Bb(1) ... y4
2|4
1|4
0|4
5 Aa(1) Bb(1) ... y5
2|5
1|5
0|5
6 Aa(1) bb(0) ... y6
2|6
1|6
0|6
7 aa(0) Bb(1) ... y7
2|7
1|7
0|7
8 aa(0) bb(0) … y8
2|8
1|8
0|8
Data Structure
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
Linkage disequilibrium mapping – natural population
AA Aa aa ObsMM p11
2 2p11p10 p102 n2
Mm 2p11p01 2(p11p00+p10p01) 2p10p00 n1
mm p012 2p01p00 p00
2 n0
MM p112 2p11p10 p10
2 n2
p2 p2
p2
Mm 2p11p01 2(p11p00+p10p01) 2p10p00 n1
2p(1-p) 2p(1-p) 2p(1-p)
mm p012 2p01p00 p00
2 n0
(1-p)2 (1-p)2 (1-p)2
Joint and conditional (j|i) genotype prob. between marker and QTL
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 genotypeple (cm, y) M AA Aa aa1 184 MM (2) 2|1 1|1 0|1
2 185 MM (2) 2|2 1|2 0|2 3 180 Mm (1) 2|3 1|3 0|3 4 182 Mm (1) 2|4 1|4 0|4 5 167 Mm (1) 2|5 1|5 0|5 6 169 Mm (1) 2|6 1|6 0|6 7 165 mm (0) 2|7 1|7 0|7 8 166 mm (0) 2|8 1|8 0|8
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|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/(22)1/2exp[-(yi-2)2/(22)],
2 = + a
f1(yi) = 1/(22)1/2exp[-(yi-1)2/(22)],
1 = + d
f0(yi) = 1/(22)1/2exp[-(yi-0)2/(22)],
0 = - a
Linkage disequilibrium mapping – natural population
Differentiating L with respect to each unknown parameter, setting derivatives equal zero and
solving the log-likelihood equationsL(|y,M) = i=1
n[2|if2(yi) + 1|if1(yi) + 0|if0(yi)]log L(|y,M) = i=1
n log[2|if2(yi) + 1|if1(yi) + 0|if0(yi)]
Define2|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=1
n2|i (4)1 = i=1
n(1|iyi)/ i=1n1|i (5)
0 = i=1n(0|iyi)/ i=1
n0|i (6)2 = 1/ni=1
n[2|i(yi-2)2+1|i(yi-1)2+0|i(yi-0)2] (7)
Linkage disequilibrium mapping – natural population
Complete data Prior prob
QQ Qq qq ObsMM p11
2 2p11p10 p102 n2
Mm 2p11p01 2(p11p00+p10p01) 2p10p00 n1
mm p012 2p01p00 p00
2 n0
QQ Qq qq ObsMM 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)
Incomplete (observed) data
Posterior prob
QQ Qq qq ObsMM 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(22|i+1|i)+i=1
n1(2|i+1|i)]/2n, (8)
p10={i=1n2(20|i+1|i)+i=1
n1[0|i+(1-)1|i]}/2n, (9)
p01={i=1n0(22|i+1|i)+i=1
n1[2|i+(1-)1|i]}/2n, (10)
p00=[i=1n2(20|i+1|i)+i=1
n1(0|i+1|i)]/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
BMISNP Chrom. Black WhiteADRA1A 8p21 q 0.20
D 0.04a 11.40d -2.63LR 3.90* NS
WHRADRB1 10q24 q 0.83
D -0.07a -0.15d -0.24LR 5.91* NS
ADRB2 5q32-33 q 0.16D 0.07a 0.16d -0.20LR 5.88* NS
ADRB2- 5/20 q 0.83 0.78GNAS1 D 0.02 0.03
a -0.18 -0.15d -0.10 -0.16LR 8.42* 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