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Single Marker Analysis and Interval Mapping
Jiankang Wang, CIMMYT China and CAAS
E-mail: jkwang@cgiar.org; wangjiankang@caas.cn
Web: http://www.isbreeding.net
1
Comparison of Estimated
Recombination Frequency in Bi-
Parental Genetic Populations
Sun Z., H. Li*, L. Zhang, J. Wang. 2012. Estimation of recombination frequency in biparental genetic populations.
Genetics Research 94: 163-177
2
Populations handled in QTL IciMapping Parent P1 Parent P2 Legends
Hybridization
F1
Selfing
1. P1BC1F1 7. F2 2. P2BC1F1
Repeated selfing
9. P1BC2F1 13. P1BC1F2 8. F3 14. P2BC1F2 10. P2BC2F1
Doubled haploids
15. P1BC2F2 16. P2BC2F2
11. P1BC2RIL 5. P1BC1RIL 4. F1RIL 6. P2BC1RIL 12. P2BC2RIL BC3F1, BC4F1 etc.
P1BC2F1 P1BC1F1 F1 P2BC1F1 P2BC2F1 Marker-assisted
selection
19. P1BC2DH 17. P1BC1DH 3. F1DH 18. P2BC1DH 20. P2BC2DH CSS lines or
Introgression lines
P1 × CP P2 × CP P3 × CP … Pn × CP CP=common parent
RIL family 1 RIL family 2 RIL family 3 RIL family i RIL family n
One NAM population
The genetic analysis can be very complicated even with biparental populations!
• F1-derived populations – F2, F3, DH, RIL: p=0.5, q=0.5 at each locus
• P1BC1-derived population – F2, F3, DH, RIL: p=0.75, q=0.25 at each locus
• P2BC1-derived population – F2, F3, DH, RIL: p=0.25, q=0.75 at each locus
• P1BC2-derived population – F2, F3, DH, RIL: p=0.875, q=0.125 at each locus
• P2BC2-derived population – F2, F3, DH, RIL: p=0.125, q=0.875 at each locus
4
Twenty biparental populations
• Allele frequencies can be different
• Genotypes and their frequencies are much different
• Are they equal good in estimating the recombination frequency between two linked loci?
5
LOD scores from different populations True r=0.05 (Upper), True r=0.2 (lower)
6
0
10
20
30
40
50
60
70
F2 F3 F1DH F1RIL BC1F1 BC1F2 BC1DH BC1RIL BC2F1 BC2F2 BC2DH BC2RIL
LOD
True r = 0.05PopSize=50 PopSize=100 PopSize=200
0
5
10
15
20
25
F2 F3 F1DH F1RIL BC1F1 BC1F2 BC1DH BC1RIL BC2F1 BC2F2 BC2DH BC2RIL
LOD
True r = 0.2PopSize=50 PopSize=100 PopSize=200
Deviations to the true value (upper) and standard errors (lower) of estimated recombination frequency
7
0
0.02
0.04
0.06
0.08
F2 F3 F1DH F1RIL BC1F1 BC1F2 BC1DH BC1RIL BC2F1 BC2F2 BC2DH BC2RIL
De
viat
ion
True r = 0.3
PopSize=50 PopSize=100 PopSize=200
0
0.05
0.1
0.15
0.2
0.25
F2 F3 F1DH F1RIL BC1F1 BC1F2 BC1DH BC1RIL BC2F1 BC2F2 BC2DH BC2RIL
Stan
dar
d e
rro
r
True r = 0.3
PopSize=50 PopSize=100 PopSize=200
Observations
• When two alleles at each locus have equal frequency of 0.5, we had a better estimation.
• When a population has more genotypes, we had a better estimation.
• For F2 and F3 to be efficient, we need co-dominant markers.
8
Minimum population size to have at least one recombinant
9
Pop. r=0.01 r=0.02 r=0.03 r=0.05 r=0.1 r=0.2 r=0.3 F2 (C, C) 150 75 50 30 15 8 5 F2 (C, D) 299 149 99 60 31 16 11 F2 (C, R) 299 149 99 60 31 16 11 F2 (D, D) 299 149 99 61 31 16 11 F2 (D, R) 149786 29956 13616 4754 1197 299 132 F2 (R, R) 299 149 99 61 31 16 11 DH 299 149 99 59 29 14 9 RIL 152 77 52 32 17 9 7
In the first column, C for co-dominant marker; D for dominant marker; R for recessive marker
Outlines
• Quantitative Traits and QTL Mapping
• Single Marker Analysis
• The Conventional (Simple) Interval Mapping
10
Quantitative Traits and QTL Mapping
11
Quantitative traits in genetics
0200400600800
1000120014001600
54 56 58 60 62 64 66 68 70 72 74
Nu
mb
er
of
wo
me
n
Midpoint group value
Distribution of height (inches) among 4995 British women
0
5
10
15
20
25
30
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Nu
mb
er
of
wo
me
n
Midpoint group value
Ear length (cm) of one maize inbred line (P1)
0
5
10
15
20
25
30
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Nu
mb
er
of
wo
me
n
Midpoint group value
Ear length (cm) of aonther maize inbred line (P2)
0
5
10
15
20
25
30
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Nu
mb
er
of
wo
me
n
Midpoint group value
Ear length (cm) of their F1 hybrids
0
20
40
60
80
100
120
140
160
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Nu
mb
er
of
wo
me
n
Midpoint group value
Ear length (cm) of their F2 hybrids
Quantitative traits
• Continuous phenotypic variation
• Affected by many genes
• Affected by environment
• Epistasis
• Polygene (or multi-factorial ) hypothesis
• Classical quantitative genetics
13
Quantitative trait does not have to be “continuous”
• Categorical traits: traits in which the phenotype corresponds to any one of a number of discrete categories – Number of skin ridges forming the fingerprints
– Number of kernels on an ear of corn
– Number of puppies in a litter
• Threshold traits: traits that have only two, or a few, phenotypic classes, but their inheritance is determined by the effects of multiple genes acting together with the environment – Liability to express the trait, which is not directly observable.
– When liability is high enough (above a “threshold”), the trait will be expressed; Otherwise, the trait is not expressed. 14
What is QTL Mapping?
• The procedure to map individual genetic factors with small effects on the quantitative traits, to specific chromosomal segments in the genome
• The key questions in QTL mapping studies are:
– How many QTL are there?
– Where are they located in the marker map?
– How large an influence does each of them have on the trait of interest?
– Are they interacting with each other?
– Are they stably expressed across environments?
15
Dataset of QTL mapping
• Mapping population
• Marker data of each individual in the mapping population
• Linkage map
• Phenotypic data
16
Example: 10 RIL of Rice (linkage map of Chr. 5 )
17
Marker C263 R830 R3166 XNpb387 R569 R1553 C128 C1402 XNpb81 C246 R2953 C1447
Grain
width
(mm)
Position
(cM) 0.0 3.5 8.5 19.5 32.0 66.6 74.1 78.6 81.8 91.9 92.7 96.8
RIL1 0 0 0 0 0 0 0 0 0 0 0 0 2.33
RIL2 2 2 2 2 2 0 0 0 0 2 2 2 1.99
RIL3 0 2 2 2 2 2 2 2 2 2 2 2 2.24
RIL4 0 0 0 0 0 0 2 2 2 2 2 2 1.94
RIL5 0 0 0 0 0 2 2 0 0 0 0 0 2.76
RIL6 0 0 0 2 2 2 2 2 2 2 2 2 2.32
RIL7 0 0 0 0 0 0 0 0 0 0 0 0 2.32
RIL8 2 2 0 2 2 0 0 0 0 2 2 2 2.08
RIL9 0 0 0 0 2 2 0 0 0 0 0 0 2.24
RIL10 0 0 0 0 2 2 0 0 0 0 0 0 2.45
Classification of mapping populations
• Bi-parental mapping populations (linkage
mapping)
– Temporary population: F2 and BC
– Permanent population: RIL, DH, CSSL
– Secondary population
• Association mapping – Natural populations: human and animals
18
Overview on QTL mapping methods • Single marker analysis (Sax 1923; Soller et al. 1976)
– The single marker analysis identifies QTLs based on the difference between the mean phenotypes for different marker groups, but cannot separate the estimates of recombination fraction and QTL effect.
• Interval mapping (IM) (Lander and Botstein 1989) – IM is based on maximum likelihood parameter estimation and provides a
likelihood ratio test for QTL position and effect. The major disadvantage of IM is that the estimates of locations and effects of QTLs may be biased when QTLs are linked.
• Regression interval mapping (RIM) (Haley and Knott 1992; Martinez and Curnow 1992 ) – RIM was proposed to approximate maximum likelihood interval mapping to
save computation time at one or multiple genomic positions.
• Composite interval mapping (CIM) (Zeng 1994) – CIM combines IM with multiple marker regression analysis, which controls the
effects of QTLs on other intervals or chromosomes onto the QTL that is being tested, and thus increases the precision of QTL detection.
19
Overview on QTL mapping methods Multiple interval mapping (MIM) (Kao et al. 1999)
– MIM is a state-of-the-art gene mapping procedure. But implementation of the multiple-QTL model is difficult, since the number of QTL defines the dimension of the model which is also an unknown parameter of interest.
Bayesian model (Sillanpää and Corander 2002) – In any Bayesian model, a prior distribution has to be considered. Based on the
prior, Bayesian statistics derives the posterior, and then conduct inference based on the posterior distribution. However, Bayesian models have not been widely used in practice, partially due to the complexity of computation and the lack of user-friendly software.
Inclusive Composite Interval Mapping (Li et al. 2006)
– In the first step, stepwise regression was applied to identify the most significant regression variables in both cases but with different probability levels of entering and removing variables. In the second step, a one-dimensional scanning or interval mapping was conducted for mapping additive and a two-dimensional scanning was conducted for mapping digenic epistasis.
20
Single Marker Analysis
21
Evidence for marker and QTL association
• Three marker types MM, Mm, and mm at one marker locus
• When marker is linked with QTL, the three marker types will have un-equal means.
22
Marker type mm Marker
type Mm
Marker type MM
Marker type mm
Marker type Mm
Marker type MM
Backcrosses (P1BC1 and P2BC1) of P1: MMQQ and P2: mmqq
23
P1BC1F1 P2BC1F1
Genotype Frequency Genotypic value
Genotype Frequency Genotypic value
MMQQ (1-r)/2 m+a MmQq (1-r)/2 m+d
MMQq r/2 m+d Mmqq r/2 m-a
MmQQ r/2 m+a mmQq r/2 m+d
MmQq (1-r)/2 m+d mmqq (1-r)/2 m-a
Difference between the two marker types (P1BC1 as example)
• Two marker types:
• Difference in phenotype between the two types
24
MMQqMMQQMM )1( rr
rdarmdmramr )1()())(1(
MmQqMmQQMm )1( rr
drramdmramr )1())(1()(
))(21(MmMM dar
A barley DH population
25
0.0
0.1
0.2
0.3
0.4
0.5
36 38 40 42 44 46 48 50
Fre
qu
en
cy
Kernel weight
Type 0
Type 2
Marker locus Act8A
0.0
0.1
0.2
0.3
0.4
0.5
36 38 40 42 44 46 48 50
Fre
qu
en
cy
Kernel weight
Type 0
Type 2
Marker locus Act8B
Significance test on phenotypic means of marker types
26
Parameter Marker Act8A Marker Act8B
Type 0 Type 2 Type 0 Type 2
Sample size 70 74 58 69
Degree of freedom 69 73 57 68
Mean 42.23 42.79 43.89 41.25
Variance 4.45 5.32 3.53 2.79
Standard error 2.11 2.31 1.88 1.67
Combined variance 4.90 3.13
T-test 1.51 (P=0.1342) 8.37 (P=1.00E-13)
A soybean F2 population
Marker locus *Satt339 Marker locus *Sat_033
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 10 20 30 40 50
Fre
qu
en
cy
Chlorophy II content
Type 0
Type 1
Type 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 40 50
Fre
qu
en
cy
Chlorophy II content
Type 0
Type 1
Type 2
Significance test on phenotypic means of marker types
Parameter Marker Act8A Marker Act8B
Type 0 Type 1 Type 2 Type 0 Type 1 Type 2
Sample size 65 111 39 56 90 62
Mean 35.16 32.76 14.22 30.72 30.47 29.20
Variance 47.71 40.42 65.52 92.13 97.24 133.28
Standard error 6.91 6.65 8.09 9.60 9.86 11.54
T-test of additive 15.06 (P=1.43E-33) 0.80 (P=0.4264)
T-test of dominance 8.47 (P=5.89E-13) 0.35 (P=0.7270)
Problems with the Single Marker Analysis
• Cannot separate QTL effect and the marker-QTL distance
• Low detection power
• Does not take the advantage of genetic linkage map
29
Conventional Interval Mapping
30
Interval mapping (IM) (Lander and Botstein 1989; Milestone in QTL
mapping methodology and applications )
• Linear model (j=1,2,…,n )
• b* represent QTL effect, is the indicator
variable (0 or 1) for QTL genotype
• Likelihood profile
• Support interval: One-LOD interval
31
*
jx
jji exbby **
0
QTL genotypes under each marker type in P1BC1 (need to consider three loci simultaneously; double crossover not
considered in this slide)
P1: P2:
F1: P1:
区间标记类型1 区间标记类型2 区间标记类型3 区间标记类型4
Mi Q Mi +1
Mi Q Mi +1
mi q mi +1
mi q mi +1
×
Mi Q Mi +1 Mi Q Mi +1
Mi Q Mi +1
×
Mi Q Mi +1 Mi Q Mi +1 Mi Q Mi +1 Mi Q Mi +1
Mi Q Mi +1 Mi Q mi +1 mi q
Mi +1
mi q mi +1
Mi Q Mi +1
Mi q mi +1
mi q Mi +1
Mi Q Mi +1
mi Q Mi +1
mi q mi +1
32 Marker class I Marker class II Marker class III Marker class IV
Marker types and QTL types in DH populations (double crossover is considered in this slide)
▼
▲ ●
▽ △ ○
▼ ▲ ● ▽
△ ○
▽ △ ○ ▼ ▲ ●
▼ ▲ ●
▼ ▲ ● ▽ △ ○
▽ △ ○ ×
▼
▲
●
▽
△
○ ▼ ▲
● ▽ △
○
No crossover
One crossover between left marker and
QTL
One crossover between QTL
and right marker
Two crossovers
between the two markers
▼ ▲ ●
▽ △ ○ ▼ △ ○
▲ ● ▽ ▼ ● △
▲ ▽ ○
● ▽ △
▼ ▲ ○
)1)(1( RL21 rr )1( RL2
1 rr RL21 )1( rr RL2
1 rr
)1)(1( RL rr )1( RL rr RL )1( rr RLrr
QTL types under each marker class
Marker class I Marker class II Marker class III Marker class IV
▼
▲ ●
▽ △ ○
▼ ▲ ● ▽
△ ○
▼
▲
●
▽
△
○ ▼ ▲
● ▽ △
○
▼ ▲ ●
▽ △ ○ ▼ △ ○
▲ ● ▽ ▼ ● △
▲ ▽ ○
● ▽ △
▼ ▲ ○
)1)(1( RL21 rr
RL21 rr
)1( RL21 rr
RL21 )1( rr )1( RL2
1 rr
RL21 )1( rr )1)(1( RL2
1 rr
RL21 rr
Two QTL genotypes in 4 marker classes in DH population
35
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
Pro
bab
lity
de
nsi
ty
Quantitative trait
AABBQQqq
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
Pro
bab
lity
de
nsi
ty
Quantitative trait
AAbbQQqq
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
Pro
bab
lity
de
nsi
ty
Quantitative trait
aaBBQQqq
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
Pro
bab
lity
de
nsi
ty
Quantitative trait
aabbQQqq
Proportion of QTL genotypes depends on QTL position and the marker interval
Three QTL genotypes in 9 marker classes in F2 population
36
Proportion of QTL genotypes depends on QTL position and the marker interval
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
Pro
bab
ility
de
nsi
ty
Quantitative trait
AABB
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8P
rob
abili
ty d
en
sity
Quantitative trait
AABbQQQqqq
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
Pro
bab
ility
de
nsi
ty
Quantitative trait
AAbbQQQqqq
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
Pro
bab
ility
de
nsi
ty
Quantitative trait
AaBB
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
Pro
bab
ility
de
nsi
ty
Quantitative trait
AaBbQQQqqq
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
Pro
bab
ility
de
nsi
ty
Quantitative trait
AabbQQQqqq
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
Pro
bab
ility
de
nsi
ty
Quantitative trait
aaBB
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
Pro
bab
ility
de
nsi
ty
Quantitative trait
aaBbQQQqqq
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
Pro
bab
ility
de
nsi
ty
Quantitative trait
aaBbQQQqqq
Frequency of QTL genotypes in each marker class in DH population
37
Marker interval Sample size
Frequency QTL genotype
Left Right QQ qq
AA BB n1 ½ (1-r) ½ (1-rL-rR+rLrR) ½ rLrR
AA bb n2 ½ r ½ (1-rL)rR ½ rL(1-rR)
aa BB n3 ½ r ½ rL(1-rR) ½ (1-rL)rR
aa bb n4 ½ (1-r) ½ rLrR ½ (1-rL-rR+rLrR)
RLRL 2 rrrrr
▼
▲ ●
▽ △ ○
▼ ▲ ● ▽
△ ○
▼
▲
●
▽
△
○ ▼ ▲
● ▽ △
○
▼ ▲ ●
▽ △ ○ ▼ △ ○
▲ ● ▽ ▼ ● △
▲ ▽ ○
● ▽ △
▼ ▲ ○
)1)(1( RL21 rr
RL21 rr
)1( RL21 rr
RL21 )1( rr )1( RL2
1 rr
RL21 )1( rr )1)(1( RL2
1 rr
RL21 rr
MLEs of means of QTL genotypes
38
qk
kikij NY,,1
2 ),(~
qk
kijik
njmi
q yfL
i
,,1
2
,,1;,,1
2
1 ),|(ln)|,,,(ln
yY
EM algorithm for calculating MLE
• The Expectation step, given initial values
• wijk measures the probability of QTL genotypes of each DH line given the marker class
39
qk
kijik
kijik
ijkyf
yfw
,,1'
)0(2)0(
''
)0(2)0(
),|(
),|(
EM algorithm for calculating MLE
40
• The Maximization step, given QTL genotypes are known from wijk in E-step
injmi qk
kijijkq yfwL
,,1;,,1 ,,1
)0(2)0(2
1 ),|(ln)|,,,(ln
xX
i
i
njmi
ijk
njmi
ijijk
kw
yw
,,1;,,1
,,1;,,1
)1(
i
i
njmi
ijk
njmi
kijijk
w
yw
,,1;,,1
,,1;,,1
2)1(
)1(2
)(
Test the existence of QTL
41
010 : qH
equalnot are , , : 1 qAH
Likelihood under H0:
injmi
ijyfL
,,1;,,1
2
00
2
00 ),|()|,(
yY
Likelihood ratio test:
Likelihood of odd (LOD):
)1(~)(max
)(maxln2 20 qdf
HL
HLLRT
A
)(max
)(maxlog
0
10HL
HLLOD A
Estimation of genetic effects
DH populations: 1 for QQ, 2 for qq
F2 populations, 1 for QQ, 2 for Qq, 3 for qq
a 1a 2
)ˆˆ(ˆ212
1 )ˆˆ(ˆ212
1 a
a 1 d 2 a 3
)ˆˆ(ˆ312
1 )ˆˆ(ˆ312
1 a )ˆˆ(ˆ312
12 d
Contribution of a QTL
No distortion
With distortion
%100P
G
V
VPVE
2
G(DH) aV 2
412
21
G(F2)ˆˆ daV
2
qqQQG(DH) 4 affV
22
QqQqqqQQQq
22
qqQQqqQQG(F2) )()(2])([ dffadfffaffffV
Interval mapping in a barley DH population
02468
101214
1111111122222222222333333334444444455555555555566666666777777777
LOD
sco
re
One dimensional scanning on the seven barley chromosomes, step=1cM
-1.5
-1
-0.5
0
0.5
1
1111111122222222222333333334444444455555555555566666666777777777
Ad
dit
ive
eff
ect
One dimensional scanning on the seven barley chromosomes, step=1cM
0
10
20
30
40
1111111122222222222333333334444444455555555555566666666777777777
PV
E (%
)
One dimensional scanning on the seven barley chromosomes, step=1cM
QTL identified in the barley DH population
Chromo. Position (cM)
Left marker
Right marker
LOD PVE (%) Additive
5 3 ABA306B Act88 13.15 34.55 -1.31
7 0 dRpg1 iPgd1A 2.55 7.79 -0.62
7 98 VAtp57A MWG571D 5.36 15.77 -0.89
Interval mapping in a soybean F2 population
0
10
20
30
40
0 20 40 60 80 100 120 140 160
LOD
Scanning on one chromosome
-15
-10
-5
0
5
10
15
0 20 40 60 80 100 120 140 160
Ge
ne
tic
eff
ect
Scanning on one chromosome
Additive Dominance
QTL identified in the soybean F2 population
Position (cM)
Left marker
Right marker
LOD PVE (%) Additive Dominance
39 *Satt285 *Sat_239 27.89 73.02 -10.96 9.48
78 *Sat_239 *Satt255 36.00 68.49 -11.09 9.39
91 *Satt255 *Satt339 38.23 58.21 -10.66 8.90
131 *Satt521 *Sat_033 18.24 69.56 -11.07 9.68
Problems with Simple Interval Mapping
• Multiple peaks when QTLs are unlinked
48
0
2
4
6
8
10
02
04
06
08
01
00 0
20
40
60
80
10
0 02
04
06
08
01
00 0
20
40
60
80
10
0 02
04
06
08
01
00 0
20
40
60
80
10
0
LOD
Scanning on 6 chromosomes, each of 120cM. Step = 1cM
Problems with Simple Interval Mapping
• Ghost QTL when two QTLs are linked
49
0
5
10
15
20
02
04
06
08
01
00 0
20
40
60
80
10
0 02
04
06
08
01
00 0
20
40
60
80
10
0 02
04
06
08
01
00 0
20
40
60
80
10
0
LOD
Scanning on 6 chromosomes, each of 120cM. Step = 1cM
Ghost QTL
Problems with Simple Interval Mapping
• Biased estimation of QTL effects
50
-0.5
0
0.5
1
1.5
02
04
06
08
01
00 0
20
40
60
80
10
0 02
04
06
08
01
00 0
20
40
60
80
10
0 02
04
06
08
01
00 0
20
40
60
80
10
0
Ad
dit
ive
eff
ect
Scanning on 6 chromosomes, each of 120cM. Step = 1cM
实习5 QTL IciMapping软件 (I) (上周)
• 连锁图谱构建功能(MAP)
– Grouping:分群、设定Anchor信息
– Ordering:标记排序
– Rippling:图谱调整
–连锁图绘制
–输入和输出
• 多个图谱的整合功能(CMP)
• 两个位点间重组率的估计工具(2pointREC)
• 方差分析 工具 (ANOVA)
实习5 QTL IciMapping软件 (II)
• 双亲群体QTL作图(BIP功能)
–选择作图方法
–设定作图参数
–结果分析
• QTL作图功效分析(BIP功能)
–设定遗传模型
–功效分析
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