Association Mapping
LD
Definition
Causes
Haplotype Blocks
Recombination
Hotspots
Extent of LD
Marker Density
Candidate loci
or whole genome?
Regression
Multiple testing
vs. Shrinkage
Sub-population
structure
Model-based
or PCA?
Genomic selection
Methods
Signatures of
selection
Gene identification or
Marker-assisted
selection?
Breeding
System
Species
Panel diversity
Confounded
structure and
polymorphism
Germplasm
123
Outline
• Association mapping is regression
• Accounting for structure
– Estimating structure using markers
– Truly multi-factorial models
• Miscelaneous topics:
– Genomic control; TDT; Confounding with
structure; Haplotype predictors; Genetic
heterogeneity; Missing heritability; NAM;
Validation
124
Association Mapping
• It’s the same thing as linkage mapping in a biparental population but in a population that
has not been carefully designed and
generated experimentally
• Because the experiment has not been
designed, it is messy. Statistical methods are
needed to deal with the mess
125
Regression
• xi is the allelic state at a marker
• Consider the total genotypic effect of I
• qi is the allelic state at a QTL with which the
marker is (hopefully) in LD
• Now estimate β
126
Estimate of Beta
127
When is cov(x, g) non-zero?
• Differences in allele frequencies at the marker
between subpopulations AND difference in
phenotypic mean between subpopulations
– The difference in mean can be due to a single or
many loci
• Difference in the frequency of alleles between
families AND difference in family phenotypic
means within a (sub)population
128
Population structure
Structure possibilities
Familial relatedness
Yu, J., Pressoir, G., et al. 2006. Nat Genet 38:203-208
129
Controlling for structure
• Basic quantitative genetics:
– Two individuals who share many alleles should
resemble each other phenotypically
– Use markers to figure out how many alleles
individuals share and then use that to adjust
statistically for their phenotypic resemblance
130
Controlling for structure
• The “mixed model”
Yu, J., Pressoir, G., et al. 2006. Nat Genet 38:203-208
131
Controlling for structure
• Structure => large differences in allele
frequencies across many markers
Average marker Set2 score
1
0.8
First PCA axis
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
Average marker Set1 score
1
Regression coefficients of
the phenotype on the PCA
values
132
Use of PCA
• Results are not sensitive to the
number of PCA, provided you
have enough
– Price, A.L. et al. 2006. Principal
components analysis corrects
for stratification in genomewide association studies. Nat
Genet 38:904-909
• The number of significant PC
can be determined
– Patterson, N. et al. 2006.
Population Structure and
Eigenanalysis. PLoS Genetics
2:e190
• Use a “Screeplot”
133
Historical footnote
• PCA achieves what the Pritchard program
Structure does
• PCA is faster and more robust
Pritchard, J.K. et al. 2000. Genetics 155:945-959
Price A.L. et al. 2006. Nat Genet 38:904-909.
Patterson N. et al. 2006. PLoS Genetics 2:e190
134
Kinship
• We are all a little bit related:
– Two unrelated people: go back 1 generation, all
four parents must be different people.
– Go back 2 generations, all eight grand-parents
must be different people.
– Go back 30 generations, all 2.1 billion ancestors
would need to be different people: Impossible!
135
Identity by Descent
• Two alleles that are copies (through
reproduction) of the same ancestral allele
Coefficient of Coancestry
• Choose a locus
• Pick an allele from Ed and one from Peter
• Probability that the alleles are IBD = Ed and
Peter’s Coefficient of Coancestry, θEP
136
Coef. of Coancestry –> A matrix
• A is the additive relationship or kinship matrix
Winter
Six-Row
“Bison”
Two-Row
137
A constrains u
• Two individuals who share many alleles should
resemble each other phenotypically
• u is the polygenic effect
• Its covariance matrix is Var(u) = Aσ2u
• If aij has a high value, the ui and uj should have
similar values (they have high covariance)
• A constrains the values that are possible for u
138
Single locus, additive model: cov(ui, uj)
139
A matrix from the pedigree
• The cells in the A matrix are aij = 2θij, the
additive relationship coefficients between i in
the row and j in the column
• Coefficient of coancestry θij: the prob that a
random alleles from i and j are IBD
• Calculate from the pedigree by recursion:
140
A matrix from marker data
, the homozygosities over
all markers and alleles
141
With inbreeding, parental
contributions NOT 50:50
• Maize intermated
population
• Drift during intermating
and inbreeding
• Markers can give more
accurate θ than
pedigree
90
80
70
60
50
40
30
20
10
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
0
/
142
Mixed Model Example
•
•
•
•
Five individuals, a, b, c, d, and e.
a and b in subpop 1; c, d, and e in subpop2.
a, b, c, and d unrelated; e is offspring of c and d.
a and d carry the 0; b, c, and e carry the 1 allele
y=
μ
+ Xβ
+ Qv
143
Mixed Model Example
•
•
•
•
Five individuals, a, b, c, d, and e.
a and b in subpop 1; c, d, and e in subpop2.
a, b, c, and d unrelated; e is offspring of c and d.
a and d carry the 0; b, c, and e carry the 1 allele
y=
μ
+ Xβ
+ Qv
144
Mixed Model Example
•
•
•
•
Five individuals, a, b, c, d, and e.
a and b in subpop 1; c, d, and e in subpop2.
a, b, c, and d unrelated; e is offspring of c and d.
a and d carry the 0; b, c, and e carry the 1 allele
y=
μ + Xβ + Qv
+ Zu
+e
145
Mixed Model Example
•
•
•
•
Five individuals, a, b, c, d, and e.
a and b in subpop 1; c, d, and e in subpop2.
a, b, c, and d unrelated; e is offspring of c and d.
a and d carry the 0; b, c, and e carry the 1 allele
Zu
A=
var(u) =
σ2u
146
Mixed Model Example
• There is a polygenic effect u for each
individual => overdetermined model?
• NO: u is a random effect, constrained by Aσ2u
147
Mixed Model Example
μ + Xβ + Qv
y=
+ Zu
+e
–1
=
✕
148
Control false positives from structure
Flowering time
(High population structure)
Ear height
(Moderate population structure)
0.5
0.5
0.5
a.
0.4
Cumulative P
Ear diameter
(Low population structure)
b.
Simple
Simple
0.4
Q
0.3
c.
0.4
Q
Q
K
GC
Q+K
0.3
0.3
Q+K
Simple
K
0.2
K
0.2
0.2
Q+K
GC
0.1
0.1
GC
0
0.1
0
0
0.1
0.2
0.3
Observed P
0.4
0.5
Simple
Q
K
Q+K
GC
0
0
0.1
0.2
0.3
Observed P
0.4
0.5
0
0.1
0.2
0.3
Observed P
0.4
0.5
A straight diagonal line indicates an appropriate control of false
positives.
Q + K model has best Type I error control, most important when
trait is related to population structure (e.g., flowering time).
149
Statistical power
Flowering time
(High population structure)
1
Ear height
(Moderate population structure)
1
d.
1
e.
K
Q+K
Adjusted average power
Ear diameter
(Low population structure)
Q+K
0.8
K
Simple
Simple
Simple
0.6
K
GC
0.4
0.2
0.2
0
0.4
(3.3)
0.6
(7.1)
(11.9)
Genetic effect
(Phenotypic variation explained in %)
0.8
(17.4)
1
Simple
Q
K
Q+K
GC
0.2
0
0.2
(0.8)
GC
0.4
0.4
GC
0
(0)
Q+K
Q
0.6
0.6
Q
0.8
0.8
Q
f.
0
0
(0)
0.2
(0.8)
0.4
(3.3)
0.6
(7.1)
(11.9)
Genetic effect
(Phenotypic variation explained in %)
0.8
(17.4)
1
0
(0)
0.2
(0.8)
0.4
(3.3)
(7.1)
0.6
(11.9)
0.8
(17.4)
Genetic effect
(Phenotypic variation explained in %)
Q + K model had highest power to detect SNPs
with true effects.
150
1
Controlling for Structure
Original
P Matrix
K Matrix
151
FDR vs. Power for 300 lines, 10 QTL
152
Effect of line number, P-only
10 QTL, 0.75 heritability
153
Effect of Reduced Population Diversity
154
Take homes on diversity
• At equal population size
– A less diverse population can increase power
because relative to the extent of LD, the average
marker distance is lower
– Given that you are testing fewer markers, the
multiple testing problem is reduced
• Avoid as much as possible reducing
population size for the sake of obtaining a
more homogeneous population
155
Guidelines
•
•
•
•
More lines and more markers are better
For a diverse population, 800+ lines
For a narrower population, 300+ (?)
FDR is a reasonable method of determining
significance, but probably conservative
156
1680
2360
1330
2320
1290
2280
1250
Q constant K estimated
1640
1600
• Q estimated with all markers, K estimated
with varying fraction of markers available
1560
2240
1210
1520
2200
1170
1480
2160
0%
25%
50%
75%
100%
1130
0%
25%
Flowering time
Variance ratio
d
50%
75%
100%
0%
Ear height
e
0.80
f
0.80
0.60
0.40
0.40
0.40
0.20
0.20
0.20
0.00
25%
50%
75%
Marker number
100%
75%
100%
0.80
0.60
0%
50%
Ear diameter
0.60
0.00
25%
SSR
SNP
0.00
0%
25%
50%
75%
Marker number
100%
0%
25%
50%
75%
100%
Marker number
157
1680
2360
1330
2320
1290
2280
1250
Q estimated K constant
1640
1600
• Q estimated with varying fraction of markers
available, K estimated with all markers
1560
2240
1210
1520
2200
1170
1480
2160
0%
25%
50%
75%
100%
1130
0%
Flowering time
Variance ratio
d
25%
50%
75%
100%
0%
Ear height
e
0.80
f
0.80
0.60
0.40
0.40
0.40
0.20
0.20
0.20
0.00
25%
50%
75%
Marker number
100%
75%
100%
0.80
0.60
0%
50%
Ear diameter
0.60
0.00
25%
SSR
SNP
0.00
0%
25%
50%
75%
Marker number
100%
0%
25%
50%
75%
100%
Marker number
158
History / future of controlling for
structure
159
Single locus: model mis-specification
• “the problem is better thought of as model
mis-specification: when we carry out GWA
analysis using a single SNP at a time, we are in
effect modeling a multifactorial trait as if it
were due to a single locus”
– Atwell S. et al. 2010. Nature 465:627-631
160
History: Candidate locus studies
• AM started out with candidate locus studies
where the effects of few loci could be fitted
• The biotechnology was not there to type more
than a few loci
• The genetic background needed to be
accounted for somehow (see above)
• In any event, the computational power was
not there to fit all 106 loci simultaneously
161
Logsdon B. et al. 2010.
BMC Bioinformatics 11:58.
Future: GWAS fitting all loci
• These methods could displace mixed models
accounting for structure
162
Sundry topics
•
•
•
•
•
•
•
Other methods to control structure
QTL confounded with structure
Single markers or haplotypes?
Genetic heterogeneity
Missing heritability
Linkage disequilibrium / Linkage analysis
Validation
163
Genomic Control
• Calculate bias in distribution of test statistic
using “neutral” loci, then account for bias
•
Devlin, B. and Roeder, K. 1999. Genomic Control for
Association Studies. Biometrics 55:997-1004.
• Works best for candidate genes: test loci can
be distinguished from neutral control loci.
Works less well for whole genome scans
•
•
•
Marchini, J. et al. 2004. Nat. Genet. 36:512-517
Devlin, B. et al. 2004. Nat. Genet. 36:1129-1131.
Marchini, J. et al. 2004. Nat. Genet. 36:1131-1131
164
Transmission Disequilibrium Test
• Experimental rather than statistical control of
effects of structure
• Originally conceived for dichotomous (e.g.,
disease / no disease) traits
• Affected offspring and both parents, of which one
must be heterozygous
• Test whether the a putative causal allele is
transmitted more often that 50% of the time
• Spielman, R.S. et al. 1993. Am. J. Hum. Genet. 52:506-516
165
TDT
• Extensions for quantitative traits
• Allison, D.B. 1997. Am. J. Hum. Genet. 60:676-690
• Extensions for larger-than-trio pedigrees
• Monks, S.A., and N.L. Kaplan. 2000. Am J Hum Genet
66:576-92
• Using for populations under artificial selection
• Bink, M.C.A.M. et al. 2000. Genetical Res. 75:115-121
166
QTL confounded with structure
• Particularly important for QTL affecting
adaptation, e.g., flowering time
Camus-Kulandaivelu, L. et al. 2006. Genetics 172:2449–2463
167
Also in rice…
Ghd7-0a
Non-functional
Ghd7-2
Weak allele
Ghd7-0
Deleted
Ghd7-1, Ghd7-3
Functional
Given geographic
distribution and
role in adaptation,
selection using
this locus will
have marginal
utility
Xue, W. et al. 2008. Nat Genet
40:761-767
168
Confounded QTL with structure
• Association analysis will have difficulty
identifying such QTL: the QTL needs to be
polymorphic within subpopulations
• Traditional linkage studies of crosses between
members of different subpopulations should
be very effective in this case
• e.g., Xue, W. et al. 2008. Nat Genet 40:761-767
• Multi-factorial methods will have difficulty
identifying loci under strong structure
169
Dwarf8: Confounded with
structure
• Thornsberry, J.M. et al. 2001. Nat. Genet.
28:286-289
– First structured association test applied to plants
Camus-Kulandaivelu, L. et al. 2006. Genetics 172:2449–2463
170
Single markers or haplotypes?
• The jury is still out
• Infinite ways to simulate
and analyze
– Ne, QTL MAF, QTL effect,
quantitative vs. binary, age of
mutation
• Ex. 1: Dramatically more
power for haplotypes vs
single markers
• Durrant, C. et al. 2004. Am J
Hum Genet 75:35-43
171
Single markers or haplotypes?
• Ex. 2: Similar or lower power for haplotype
method relative to single marker method
• Zhao, H.H. et al. 2007. Genetics 175:1975-1986
• Process to sort out what method most
appropriate for when still has to happen
172
Exploiting Haplotype Blocks
• Objective: reduce the genotyping cost while
capturing polymorphism at all (most) loci
• Haplotype: series of alleles at adjacent
polymorphic loci
• Blocks: majority of diversity in few haplotypes
• => Strong LD between loci within blocks; weak LD
between loci across blocks
• Knowledge of the allele at one locus provides
much information on the alleles at other loci
173
What causes blocks?
• Recombination heterogeneity: coldspots
within blocks, hotspots between blocks
• Random sampling of alleles and timing of
mutation relative to recombination events
174
Evidence for mechanisms
• Humans: High marker density resources
– Observed LD structure reproduced best if
recombination hotspots every ~ 100 kbp
• Reich, D.E. et al. 2002. Nat Genet 32:135-142.
Wall, J.D., and J.K. Pritchard. 2003. Am J Hum Genet 73:50215.
– Block boundaries correspond with positions of high
current recombination
• Jeffreys, A.J. et al. 2005. Nat Genet 37:601-606
– Block boundaries consistent across different human
populations
• De La Vega, F.M. et al. 2005. Genome Res. 15:454-462
175
Hotspots exist in plants (Arabidopsis) too
70 kbp
Kim, S. et al. 2007. Nat Genet 39:1151-1155
176
Blocks also arise randomly
• No relation between historic recombination
(histogram) and block boundaries (dark bars)
• Verhoeven, K.J.F. and K.L. Simonsen. 2005. Mol. Biol. Evol. 22:735-740
177
Blocks in barley
40 ?? Mbp
178
Block cause matters
• If blocks arise from a recombination process,
they will be consistent across populations
• Markers that tag blocks identified in one
population will therefore be useful in others
• If blocks arise from random processes, tags
useful in one population will not be so in
another
179
Haplotypes for discovery in barley
• 2198 mapped SNP in 1807 lines across barley
• Five methods
– Traditional single SNP
– Four gamete: use D’ to determine boundaries
– Tree scan: single df contrasts based on parsimony
– HapBlock: group to capture diversity
– Sliding window of 3 SNP
• Simulation: mask a SNP and pretend it’s a QTL
• Real data: heading date on 1040 lines
180
Results
• Simulation: single SNP best in 5 / 8 cases and
never worse than the best haplotype method
• CAUTION: the QTL had the same properties as
the SNP: ideal for single SNP discovery
• IF QTL simulated as recent mutations on
blocks with haplotype properties THEN
haplotype methods had higher power
– Even then, single SNP did pretty well
181
Real data (heading date)
Only
Tree
Scan
All methods
Only 4gamete
1H
2H
3H
4H
5H
6H
7H
Chromosome
182
4gamete success
• Rare recombinants split off early-heading lines
001
-0.99
3
000
-0.53
64
111
-0.69
12
010
-0.44
82
110
-0.48
386
*
011
0.16
489
183
Take-homes
• Simulations don’t support use of haplotype
blocks
– But we don’t know how to simulate the true
nature of QTL
• With real data, a diversity of approaches
might produce the most useful candidate list
184
Block vs tag identification
• Blocks require position, tags do not
• General tag marker approach:
– Identify markers in high LD with each other
– Retain only one
– Aggressive: among tags, see if combinations can
be used instead of single tags
de Bakker, P.I.W. et al. 2005. Nat Genet 37:12171223
185
Reducing marker numbers
• Tag SNP and Imputation:
• Tag marker approach
– Identify markers in high LD with each other
– Retain only one
– Aggressive: among tags, see if combinations can
be used instead of single tags
186
Tagging works
• Power is maintained; genotyping is reduced
Power
•Greedy
•Best N
•Random Tags
•No LD
Average marker spacing (kbp)
de Bakker, P.I.W. et al. 2005
187
Tags serve as a base for imputation
• Model-based imputation using fastPHASE
Scheet and Stephens. 188
2006
Imputation on tag markers works
189
Jannink et al. 2007
Imputation can increase power
Chromosomal Position (Mb)
190
Marchini et al. 2007
Imputation can increase power
191
Guan and Stephens 2008
Genome scans with low LD
• Numerous species have too low LD to perform
(as of yet) whole genome scans
– You would need too many SNP on too many genos
• “Nested Association Mapping”
• Known as “Linkage disequilibrium linkage
analysis” (LDLA) in animal genetics
Meuwissen, T.H.E. et al. 2002. Genetics
161:373-379.
Yu, J. et al. 2008. Genetics 178:539-551
192
NAM Design
• B73 is the reference parent, crossed to 26
other inbred lines, representing a large part of
maize diversity
B73
CML52
26 Times
…
B73
F1
F1

RIL1
P39

RIL2
…
RIL199
RIL200
RIL1
RIL2
…
RIL199
RIL200
193
SSD
RIL






×





Tzi8
Tx303
P39
Oh7B
Oh43
NC358
NC350
MS71
Mo18W
M37W
M162W
Ky21
Ki3
Ki11
Il14H
Hp301
CML69
CML52
CML333
CML322
CML277
CML247
CML228
CML103
B97
25 DL
B73
F1s




1
2



200
194
NAM Genotyping
• Type parents at high density (2.5 M SNP…)
• Type RIL at low density (10 k SNP): know, on a
sub-cM scale, which parental allele inherited
195
NAM linear models
• P: matrix indicating which parent contributed
the allele to each offspring. α: vector of
effects of parental alleles.
•
Eq. 1
• A linear model for the parental allele effects:
•
Eq. 2
• This latter model is what Yu et al. call
“Projecting parental SNP on to the progeny.”
196
NAM / LDLA on Maize
• Consider:
– 2.5 Gbp genome with LD extending to 1000 bp
– Requires 2.5 M SNP…
• Apply Eq. 1 to identifiy QTL with, say, 3 cM C.I.
– Within interval there are ~ 3000 parental SNP
• Apply Eq. 2 to dissect the QTL to its causal SNP
– Feasible with 25 genotypes (apparently)
– Note that α will be accurately estimated
197
Advantages of NAM / LDLA
• Adds power without adding huge genotyping
burden
• Reduces / eliminates problems related to
structure: the linkage part of the analysis
removes long-distance LD
198
Sugary1: Genetic heterogeneity
Tracy, W.F. et al. 2006. Crop Sci 46:S-49-54
199
Genetic heterogeneity hinders AM
• Distinct mutations at the B locus
B2 associated with A1
B3 associated with A2
A1B1 Exists
A2B1 Exists
A1B1 Exists
A2B1 Exists
A1B2 New
A2B2
A1B3
A2B3 New
--
--
• If B2 and B3 cause a phenotype (e.g. loss of
function at isoamylase), it will be associated
with A1 in one case and A2 in the other case.
• B2 and B3 can be identified by linkage mapping
200
How prevalent is heterogeneity?
Buckler et al. 2009. Science 325:714-718
201
Multiple hits => Allelic series
Buckler et al. 2009. Science 325:714-718
202
Heterogeneity and Pop. History
• If a population has gone through a severe
bottleneck, polymorphic loci are unlikely to
have > 2 alleles…
• Heterogeneity is less likely in domesticated
populations with low Ne
203
204
Missing Heritability
• Heritabtility for height in humans is ~ 0.80.
• Very large GWAS studies find ~ 50 SNP
together accounting for 5% of that heritability
• Where’s the rest?
– Infinitesimal effects
– Low frequency SNP in same causal genes
– Epigenetics
– Genotype x environment interaction
– Epistasis
Maher, B. 2008. Nature 456:18-21
Manolio T.A. et al. 2009.Nature 461:747-753.
205
Plants are not like humans
• Atwell et al. 2010. Nature 465:627-631
– Just 192 lines!
– Some large effect variants (intermediate
frequency and explain 20% of variation…)
– Inbred lines enable noise reduction
– Extended association peaks because of low Ne
• Less evidence of missing heritability
206
Mouse composite not like humans
• Valdar et al. 2006. Nature Genetics 38:879887
• QTL account for 73% of observed heritability
207
Humans are not like humans
• Yang et al. 2010. Nat. Genet. 10.1038/ng.608
– Common SNP accounted for 45% of variation if all
SNP included in the model
• i. Many very small QTL effects
• ii. QTL generally have lower MAF than arrayed SNP
• Dickson S.P. et al. 2010. PLoS Biol 8:e1000294
– Several rare variants can combine to produce an
association with a common SNP
208
Validation
• All genome-wide studies raise the question of
validation
• In candidate studies, independent evidence
from biological reasoning for candidate choice
• In Zhao et al. 2007, used previous linkage
analyses of parents in the association panel
209
Real data (heading date)
Linkage Studies
VRN3
1H
2H
3H
4H
5H
6H
7H
Chromosome
210
Arabidopsis: Residual structure
Residual Confounding: no bi-parental QTL found despite it segregating in the cross
Low Power: No association found despite large effect in the cross
211
Recap
• Model has focused on one locus at a time
• The locus has been treated as a fixed effect
– Makes sense in the candidate locus context
• We have dealt with residual “polygenic” effects that,
through structure, wreak havoc
• Going forward, statistical models will be multi-factorial
• Linkage mapping needed to find loci associated with
structure
• LD exhibits block-like structure: what to do with that?
• Potential for genetic heterogeneity depends on population
history
• GWAS can miss substantial heritability
• If you have very low LD, nested association, or LDLA, is a
good idea
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