Electronic supplementary material (ESM)
Supplemental Methods
Identification of QTLs—The ‘donor genomes’ for each NIL panel were three plants,
heterozygous for loci affecting flower size. Each was an F1 from a distinct cross of large and
small flowered parents. We sampled parents from phenotypically divergent populations created
through six generations of bi-directional artificial selection (KELLY 2008; LEE 2009). The
recipient genomes, IM767 and IM62, are lines derived by single seed descent from unrelated
Iron Mountain plants.
We generated four panels of IM62 NILs by recurrent selection and backcrossing (HILL 1998;
WRIGHT 1952), backcrossing the F1s into the recipient inbred line for four generations. We
performed selection for flower size in both directions in each generation, thereby enriching the
final set of NILs for QTLs with consistent effects on corolla width in the IM62 genetic
background. Two of the three F1s were used as donors, resulting in 50 NILs per F1 per direction
of selection (200 total). Fourth generation NILs were screened at 165-168 length polymorphic
markers and assessed for over-representation of donor alleles. DNA extraction and genotyping
were done using our standard 96-well procedures (MARRIAGE et al. 2009) and all markers were
length-polymorphic fragments scored via capillary electrophoresis (MgSTS markers are listed at
http://www.mimulusevolution.org). After 4 backcross generations, we expect 6.25% of the
genome of a NIL to remain heterozygous. Markers heterozygous in 7 or more lines (out of 50)
differ significantly from this expectation, suggesting that the genomic region may have been
retained by selection for flower size. Two BC4 NILs carried five distinct overrepresented QTLs
(Q1, Q2, Q8a, Q8b, and Q10b) and these were selected for further verification and fine-mapping.
The effects sizes in the IM62 NILs (Table 1 of main paper) were obtained in multiple growups
(cohorts) representing successive BC4Fn generations descended from the two selected NILs,
where n=3-6. In each generation, we measure corolla width on all BC4Fn individuals and
genotyped for the target locus, while segregating away background residual heterozygosity as
part of a NIL clean-up process.
We generated three panels of IM767 NILs with five successive generations of backcrossing of
each donor into IM767. We self-fertilized each 6th generation NIL and collected leaf tissue for
genotyping. We grew 12 progeny from each selfed family under standard greenhouse conditions
and measured corolla width on the first two flowers. Each panel was assayed in a distinct cohort.
We then used a selective genotyping method to identify genomic regions affecting corolla width.
NILs were ranked by mean corolla width within panels and we selected NILs from the tails of
each distribution for genotyping. In total 50 NILs (the parents of the plants measured for
phenotype) were then genotyped at 236 length-polymorphic markers, 231 from MgSTS set plus
five microsatellite loci. We identified 24 markers that were over-represented in High or Low
NILs, or in both sets. We then genotyped these markers in the selfed progeny of the 50 selected
NILs and measured genotype-phenotype association, i.e. co-segregation of putative QTLs with
flower size. This procedure identified Q5a, Q5b, Q9, and Q10a. After cleaning the background
of selected NILs (using a combination of backcrossing to IM767 and selfing to eliminate donor
genotypes from other parts of the genome), plants were grown and measured in seven distinct
cohorts over a period of 9 months in 2009-2010.
Estimation of QTL variance within the Iron Mountain population—We conducted a
Replicated F2 breeding design (Kelly 2009) to estimate VQ for each of the QTLs mapped in
NILs (Figure S1). The experimental population consists of a series of crosses between a
common Reference inbred line (IM767) and each of 138 inbred lines randomly extracted from
the same natural population by many generations of self-fertilization (the ‘Random Lines’).
Each of the 138 “Line Cross Families” consists of replicate individuals of the parental Random
line, F1s (between that Random line and the Reference) and F2s (from selfing the F1s). Plants
were grown under common greenhouse conditions along with replicates of IM767 and measured
for floral traits. A detailed description of the crossing design and plant growth conditions for the
Replicated F2 experiment is given in (Scoville et al. 2009).
Figure S1. A diagram of the crosses for the replicated F2 study with n = 138 for our study.
We genotyped a total of 37 markers in the genomic vicinities of each flower size QTL (Table
S1). We selected 2-6 markers per QTL for which Random line genotypes routinely differed
from the Reference line. Locating these markers on the 8x genome build of M. guttatus, we find
that regions span DNA sequences of 70 to 2000 kilobases. Given that the estimated total
genome size of M. guttatus is 430 megabases, each of our delimited regions is less than 0.5% of
the genome. In total, the nine regions represent about 1.75% of the genome.
Table S1. A summary of markers scored in the Replicated F2 experiment aligned to the 8x
genome build of M. guttatus. Genomic position is the location of the base pair start point
for the primers of each marker. Recombination position refers to the linkage map position
(within a QTL) used in the interval mapping algorithm (units are centimorgans). e31_710
is an aggregate of marker data from two different loci, each of which was prone to
ambiguous genotype calls.
QTL
Q1
Q1
Q1
Q1
Q1
Q2
Q2
Q2
Q2
Q2
Q2
Q5a
Q5a
Q5b
Q5b
Q8a
Q8a
Q8a
Q8a
Q8a
LG
1
1
1
1
1
2
2
2
2
2
2
5
5
5
5
8
8
8
8
8
Marker
387
436
287
781
yw55
316
yw139
787
yw51
831
343
40
7
385
641
635
736
842
1
330
Genomic
Recombination
Position
Position
1641891
0
2048958
6.9
2234960
10
2455748
20
2778566
30
1770572
0
1228532
11
1221113
12
1074433
15.2
544019
27.2
417393
30
17194166
0
18226128
19
143369
0
75069
7
18555776
0
17441857
11
17236493
13
16693396
18
16510819
20
Q8a
Q8b
Q8b
Q8b
Q9
Q9
Q9
Q10a
Q10a
Q10a
Q10a
Q10a
Q10b
Q10b
Q10b
Q10b
Q10b
8
8
8
8
9
9
9
10
10
10
10
10
10
10
10
10
10
e31_710
yw179
yw116
yw187
795
523
611
75
519
70
82
332
465
109
yw91
yw164
754
3046190
2881777
2845767
12398628
12172785
11827579
18371707
18372267
18514331
18580939
18598693
2068318
1819700
1390899
1369809
1167032
24
0
8
10
0
10
25
0
3
7
10
12
0
5.5
15
15.5
20
We used AntMap (IWATA and NINOMIYA 2006) to estimate the recombination rate between all
markers within each QTL (last column of Table S1). For this, we coded F2 genotypes as
homozygous for reference line allele, homozygous for an alternative allele, heterozygous, or
missing data if the marker was uninformative in that line cross family. We then pooled data
across line cross families. The marker data determines whether the alleles of each F2 individual
were descended from the Random line or the Reference line. Combined with an interval
mapping methodology, this allows us to estimate mean effects and covariances associated with a
QTL without having to identify specific alleles.
To analyze data from the Replicated F2 experiment, we developed a likelihood based statistical
procedure and tested alternative models for each QTL. Within an F2 family, there may be strong
co-segregation between markers and phenotype (if the Random line allele is different than the
Reference line allele) or none at all (if the alleles from Random and Reference lines are the
same). To the combination of marker and phenotype data, we applied a statistical model with
three components: the environmental effects, the model for the QTL, and the model for the
genetic background. The environmental component involves a random effect (the random
normal deviation associated with all measurements, mean = 0 and variance = VE) as well as three
fixed effects (the effect of cohorts 2, 3, and 4 on mean phenotypes, respectively). The genetic
background model includes four distinct means (fixed effects): the mean of the reference line (
R),
F2).
the mean of random inbred line parents (P), the mean of F1s (F1), and the mean of F2s (
The genetic background model also includes fixed effects accommodating two ‘known
allele’ QTLs (KELLY 2009). The first is a meiotic drive locus on linkage group 11 (FISHMAN and
SAUNDERS 2008) and the second is a polymorphic inversion on linkage group 6 (Lee 2009). We
can discern the derived ‘allele’ at each locus by direct genotyping of informative markers
(Scoville et al. 2009). For each locus, we estimated the heterozygous and homozygous effect on
the trait (four parameters total), although we found that neither had a significant effect on corolla
width.
The genetic background model includes four variance components, C11, C12, C22, and VS. In
combination, these quantities determine the variances and covariances among plants within a
line-cross family. This model for the genetic background allows an arbitrary number of loci to
affect the trait with an arbitrary numbers of alleles per locus and arbitrary dominance. It does
assume independent assortment of background loci—see Appendix Notes S1 of (Scoville et al.
2009). However, our genetic background model is in one regard more general than implied by
the assumptions. We allow the mean of F2 plants to differ from (R + P + 2 F1)/4 to allow any
effect of epistasis on the trait means. There are ten distinct contrasts involving variances and
covariances in the replicated F2 design. The variance of measurements given that an individual
is a member of the reference line is VE. The variance for plants of a random line is VE + C22, VE
+ C11 for F1s, and VE + C22 / 4 + C11 / 2 + VS for F2s. There are three intra-class covariances:
the covariance between distinct individuals of a random line equals C22, the covariance between
distinct individuals of an F1 family equals C11, and the covariance between distinct F2s within a
family equals C11 / 4 + C12 / 4 + C22 / 16. Finally, there are three inter-category but intra-family
contrasts: the covariance between random line and F1 individuals equals C12, the covariance
between random line and F2 individuals equals C12 / 2 + C22 / 4, and the covariance between F1
and F2 individuals equals C11 / 2 + C12 / 4.
The null model has two QTL parameters: the additive effect of Reference Line QTL allele (a)
and QTL position (M). These variables combine to determine a single fixed effect, the additive
effect of QTL genotype on mean phenotype. The expected value for a phenotype is incremented
by (R1 + 2 R2) a, where Rk is the probability that a plant carries k QTL alleles from the
Reference Line. For Reference Line plants, R1 = 0 and R2 = 1. For Random Line plants, R1 = 0
and R2 = 0. For F1 plants, R1 = 1 and R2 = 0. The probabilities for F2s depend on marker data
and M (algorithm specified below).
The alternative QTL Model (VQ ≥ 0) treats the additive effect of the allele from each Random
Line as a random variable. We assume that the QTL has an arbitrary number of alleles with
additive effects and population variance VQ (the new parameter to this model). Within an F2
family, there may be strong co-segregation between markers and phenotype (if the Random line
allele is different than the Reference line allele) or none at all (if the alleles from Random and
Reference lines are the same). With k as the number of QTL alleles within an individual from
the random line (k = 0, 1, or 2), the variance is incremented by VQ / 2 if k = 1 and by 2 VQ if k
= 2. The covariance of two different individuals within a line cross family with k1 and k2
alleles, respectively, is inflated by k1 k2 VQ / 2 (see Appendix 1 of (Kelly 2009)). In the
alternative model, the fixed effect characterizing QTL effects on mean phenotype is not equal to
a as in the null model. Instead, the average effect of the Reference Line QTL allele on mean
phenotype, characterized by the parameter a*, depends on the probability distribution for effects
at this locus in the Random Lines. In other words, a* depends on the distribution of allelic
effects and allele frequencies within the background population. With two alleles, a* = a (1-q)
and VQ = 2 q(1-q) a2, where q is the population frequency of the Reference Line allele. With
more than two alleles at the QTL, the estimation of a* and VQ remains valid although these
quantities cannot be related directly to allele frequencies and effects without additional
information.
An interval mapping method for each QTL region—The number of alleles from the Reference
Line is known for all individuals except F2s. For these, R1 and R2 must be inferred from marker
data. We assume the QTL lies within range of genotyped markers and M denotes the relative
position (0 – 1) within this linkage map. The identity of informative markers varies among line
cross families. For each F2, the probability of each possible QTL genotype is determined
conditional on genotype at the nearest informative markers. For example, the marker positions
for Q10a are 0, 3, 7, 10 and 12 (Supplemental Table 2). If M = 0.5, the QTL is located at
position 6 and the recombination rate is 0.03 to the nearest upstream marker (MgSTS_519) and
0.01 to the nearest downstream marker (MgSTS_70). Given recombination rates to nearest
markers, R1 and R2 are given by the standard conditional probability statements for F2
segregants (see LYNCH and WALSH 1998, ch 14). For example, if the left and right markers are
both homozygous for the Reference Line allele, then R2  (1  c1 ) 2 (1  c2 ) 2 / w
and R1  2c1c2 (1  c1 )(1  c2 ) / w , where w  R1  R2  (c1c2 ) 2 , c1 is the recombination rate between
QTL and left marker, and c2 is the recombination rate between QTL and right marker. If the left
marker is homozygous for the Reference Line allele but the right marker is heterozygous,
then R2  2(1  c1 ) 2 (1  c2 )c2 / u and R1  2c1 (1  c1 )[(1  c2 ) 2  c2 ] / u , where
2
u  R1  R2  2c2 (1  c2 )c1 . Absent an informative marker (left or right), the corresponding
2
single marker conditional probabilities were used. If there was no informative marker for a
particular QTL in a particular family, then R1 = 0.5 and R2 = 0.25.
We used maximum likelihood to fit models to data assuming that the full vector of z values
(phenotypes for all individuals in the experiment) is multi-normal (SEARLE et al. 1992). The
linear model formula is:
z i = X i + g i’ + g i* + e i
where i indexes individual. z is the phenotypic value, X is the collection of fixed effects, g’ is
random effect associated with the QTL, g* is random effect due the remainder of the genome,
and e is the environmental deviation. Both QTL and genetic background can involve fixed
effects. To maximize log-likelihoods for all models, we used a combination of stochastic hillclimbing and Markov Chain Monte Carlo (MCMC) search, as in (Scoville et al. 2009). We
wrote computer code to execute these operations within two distinct platforms. J.K.K. wrote
programs in C and C++ while A.G.S. wrote a parallel set of programs for Matlab©. Maximum
likelihoods and parameter estimates obtained from the two sets of programs were completely
congruent. The likelihood ratio test (LRT) is twice the difference between maximum loglikelihoods of alternative and null models.
Supplemental Results
For both NIL sets, QTL effect was estimated from the difference in mean phenotype between
alternative homozygotes at diagnostic marker loci (Table S2). For the IM62 QTLs, we collected
the phenotypic data to estimate QTL effects simultaneously with NIL cleaning. Cleaning
preceded effect estimation for the IM767 NILs. We used a general linear model with trait
(corolla width) as the response variable to estimate QTL effects. The predictors (factors) were
QTL genotype and cohort. The latter accommodates any effect of specific growth conditions
given that the phenotypes were collected over the span of several years in different grow-ups.
For estimation in the IM62 NILs, different QTLs were assayed in distinct cohorts and so each is
treated with a different ANOVA. For estimation in the IM767 NILs, multiple QTLs were
investigated in each cohort. Given this overlap, it is possible to estimate all effects in the same
model fit with cohort as a single factor with 7 different levels. The diagnostic markers were
MgSTS_781 and MgSTS_271 for Q1, MgSTS_787 and MgSTS_316 for Q2, MgSTS_736 and
MgSTS_842 for Q8a/b, yw116 and MgSTS_59 for Q8c, MgSTS_109 and MgSTS_754 for
Q10b. The diagnostic markers were MgSTS_40 for Q5a, MgSTS_641 and MgSTS_385 for
Q5b, MgSTS_523 for Q9, and MgSTS_70 and MgSTS_82 for Q10a. Markers with the MgSTS
prefix are length-polymorphic introns of single copy nuclear genes (primer sequences at
http://www.mimulusevolution.org/). For QTL 8a, there is no test for cohort because this QTL
was tested in a single grow-up. Calculations were conducted using JMP© v8.
Table S2. A summary of hypothesis tests (F-ratios and p-values) on effects of each QTL are
given for the IM62 loci (A) and the IM767 loci (B).
(A)
QTL 1
QTL 2
QTL 8a
QTL 8b
QTL 10b
Factor
F
p
F
p
F
p
F
p
F
p
Genotype 40.2
0.009 10.0
0.002 18.0
0.000 23.6
0.000 26.4
0.000
Cohort
6.7
0.000 32.3
0.000 ------- ------- 27.2
0.000 164.8 0.000
(B)
F
p
Q5a 19.54 0.0000
Q5b 87.61 0.0000
Q9
21.07 0.0000
Q10a 9.61 0.0019
Table 1 of the main text summarizes estimation and likelihood ratio tests from the model fitting
on corolla width and pistil length. Our procedure treats the linkage map within each QTL as
known, as is standard in QTL mapping. Our linkage mapping (Table S1) produced the same
ordering of marker loci as in the genome sequence. However, the estimated recombination
fractions are rather large; 20-30 cM for intervals on the order of a megabase (mb). In part, this is
reflects that Mimulus guttatus has a substantially longer genetic map than other model
organisms. The genome wide average is about 200 kb per cM, but in gene rich regions that are
most likely to contain QTLs, the average is about 50 kb per cM which translates to 20 cM for an
interval of 1 mb (LEE 2009).
Inflated recombination estimates might also have resulted from pooling of many distinct F2
families before using AntMap or because of genotyping error of marker loci. Even a fairly low
rate of genotyping error can upwardly bias recombination estimates. Marker loci were chosen
based high variability among the IM lines and these loci might also be most prone to genotyping
error. To determine sensitivity of the results to the estimated recombination map of each QTL,
we reran the entire model fitting process on corolla width after reducing recombination rates to
one third of their estimated values. We found this to have minimal effect on likelihood ratios (an
average change of about 0.1 in log-likelihood score per test) and no effect on statistical
significance. This insensitivity is perhaps surprising, but fully consistent with the simulation
study of DODDS et al. 2004. Considering backcross designs for QTL estimation, these authors
found that map accuracy had minimal effect on the ability to detect QTLs or significance levels
for tests.
Simulation study and p-value calculations
To determine appropriate significance levels for LRT values and to evaluate statistical power, we
conducted simulations. A priori, we expect that the sampling distribution of the LRT to be an
equal mixture of 2[1] and zero values. The parameter distinguishing alternative from null model,
VQ, is bounded to non-negative values and hence we predict a mixture distribution (SELF and
LIANG 1987). The simulation algorithm reproduces trait measurements according to the planned
experimental design of our study.
Individual plant phenotypes were generated from a stochastic model of genetic and
environmental effects. For each simulated plant, a portion of the genetic variance is generated
by the target QTL and the remainder by the genetic background. We assumed the QTL has two
equally frequent alleles with additive effects. In a simulation replicate, one QTL allele was
randomly assigned to the Reference Line. Assignment of QTL genotypes to each random line
was probabilistic given the population frequency of alleles. Given QTL genotypes, a genetic
background effect (g i*) was assigned to each line by sampling from a normal distribution with
mean zero and variance 2(1 - VQ). We considered a trait with a heritability of 0.4, our estimated
value for corolla width (SCOVILLE et al. 2009).
We generated the plants of a line cross family conditional on the Random line QTL and genetic
background values. F1 plants were heterozygous at the QTL if the Random line genotype
differed from the Reference line. g i* of each F1 was the mean of the corresponding values of
parental values. Three stochastic events were simulated to produce each F2 genotype. g i* of
each F2 was obtained by assigning an independent normal deviation (mean zero and variance (1
- VQ)/2) to the F1 value. Segregation of QTL genotype to each F2 occurred by Mendelian rules
(¼ : ½ : ¼ probabilities if the F1 was heterozygous) but we also needed to simulate marker
cosegregation. Here, we assumed that the QTL was flanked by informative markers 5 cM
upstream and downstream. Recombination was simulated with a crossover occurring with
probability 0.1 at a random position uniformly distributed over the interval between markers. F2
marker genotypes were assigned conditional on recombination (if it occurred) and on the
genotype at the QTL. Finally, an independent normal deviation (mean zero and variance one)
was assigned to all plants as an environmental effect. Marker data and phenotypes from each
simulation run were input to an automated version of our estimation programs. To fit each
model to each simulation replicate, we repeated the maximum likelihood search from 20 distinct
starting points. 500 replicate datasets were simulated per parameter set to estimate the likely
distribution of outcomes.
Figure S2. The distribution of LRT values are given from 500 replicate simulations of the
experiment under the null hypothesis (VQ = 0). In (A), a = 0 and 46% of values are
identically zero. In (B), a = 1 and 47% of LRT values are zero.
350
(A)
300
Frequency
250
200
150
100
50
0
0.0
1.5
3.0
4.5
6.0
7.5
LRT
350
(B)
300
Frequency
250
200
150
100
50
0
0.0
1.5
3.0
4.5
LRT
6.0
7.5
Two different examples of the null model are illustrated by Figure S2. In panel A, the QTL has
no effect in any line cross family (a = 0), while in B, the QTL has the same consistently large
effect in all families (a = 1). The distribution of LRT values does not differ notably between
these cases. Each is consistent with the theoretical prediction of an equal mixture of 2[1] and
zero values. We apply this mixture distribution to calculate the QTL specific p-values in Table
1. To calculate the p-values for combined (whole-genome) tests, let Y be the sum of likelihood
ratio values across t independent test. Under the null hypothesis,
t 
i c
Pr[Y  c]    0.5t R [ , ]
2 2
i 0  i 
t
where R [a, b] is the regularized gamma distribution. The quantity,  t 0.5t , is the probability
 
i
that i of t draws from the mixture distribution are non-zero. To obtain p-values at the bottom of
Table 1, we set t=9 and c equal to the sum of QTL-specific LRT. Calculations were done using
Mathematica© v4.
A series of simulations with the QTL contributing to variation are depicted in Figure S3 (here
VQ > 0 except at the y-intercept). The x-axis of this figure is the ‘QTL heritability’, the ratio of
VQ to the total phenotypic variance, VP. Power to reject VQ = 0 reaches 0.5 only if the QTL
explains only 4-5% of the variance. The point estimates for our leading QTLs for corolla width
are in this range (Table 1). For a QTL with two equally frequent additive alleles that explains 13% of the phenotypic variance, the most likely outcome of our experiment would be to fail to
reject VQ= 0. VQ / VP estimates for putatively intermediate frequency polymorphisms in fruit
flies are in this range (DWORKIN et al. 2005; MACDONALD and LONG 2007) and estimates from
corn, mice, and humans are smaller (BUCKLER et al. 2009; FLINT and MACKAY 2009; YANG et
al. 2010). The important implication is that we cannot conclude that our QTLs with low VQ are
rare-allele polymorphisms.
Figure S3. The power (fraction of LRT significant) was estimated by simulation of a biallelic QTL (both alleles equally frequent) of varying effect. An experiment of size and
composition similar to the real experiment was simulated 500 times followed by model
fitting to each replicate dataset. Power is the fraction of LRT that exceed 2.71 (p=0.05 in a
locus-specific test).
1
Statistical power
0.8
0.6
0.4
0.2
0
0.00
0.02
0.04
0.06
QTL heritability
0.08
0.10
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