INVESTIGATION
A Test for Selection Employing Quantitative Trait
Locus and Mutation Accumulation Data
Daniel P. Rice*,1 and Jeffrey P. Townsend*,†,2
*Department of Ecology and Evolutionary Biology and †Program in Computational Biology and Bioinformatics, Yale University,
New Haven, Connecticut 06520
ABSTRACT Evolutionary biologists attribute much of the phenotypic diversity observed in nature to the action of natural selection.
However, for many phenotypic traits, especially quantitative phenotypic traits, it has been challenging to test for the historical action of
selection. An important challenge for biologists studying quantitative traits, therefore, is to distinguish between traits that have evolved
under the influence of strong selection and those that have evolved neutrally. Most existing tests for selection employ molecular data,
but selection also leaves a mark on the genetic architecture underlying a trait. In particular, the distribution of quantitative trait locus
(QTL) effect sizes and the distribution of mutational effects together provide information regarding the history of selection. Despite the
increasing availability of QTL and mutation accumulation data, such data have not yet been effectively exploited for this purpose. We
present a model of the evolution of QTL and employ it to formulate a test for historical selection. To provide a baseline for neutral
evolution of the trait, we estimate the distribution of mutational effects from mutation accumulation experiments. We then apply
a maximum-likelihood-based method of inference to estimate the range of selection strengths under which such a distribution of
mutations could generate the observed QTL. Our test thus represents the first integration of population genetic theory and QTL data to
measure the historical influence of selection.
I
DENTIFYING which quantitative traits have been subject
to strong selection and which have evolved under neutral
or nearly neutral conditions is a challenging and important
task for evolutionary biologists (Boake et al. 2002). To this
end, biologists are devoting increased attention to the genetic basis and evolutionary causes of quantitative variation
(Barton and Keightley 2002; Lai et al. 2007; Barton and De
Vladar 2009). On one hand, substantial progress has been
made in revealing the genetic architecture of quantitative
traits (Zimmerman et al. 2000; Ashton et al. 2001; Mackay
2001; Gleason et al. 2002; Verhoeven et al. 2004; Mackay
and Lyman 2005; Brem and Kruglyak 2005; Lai et al. 2007;
Gleason et al. 2009). On the other hand, it has been difficult
to connect the action of microevolutionary forces detected in
studies of contemporary populations to their macroevolutionary effects (Grant and Grant 2002). Consequently, few
Copyright © 2012 by the Genetics Society of America
doi: 10.1534/genetics.111.137075
Manuscript received November 23, 2011; accepted for publication January 17, 2012
Supporting information is available online at http://www.genetics.org/content/
suppl/2012/01/31/genetics.111.137075.DC1.
1
Present address: Department of Organismic and Evolutionary Biology, Harvard
University, 26 Oxford St., Cambridge, MA 02138.
2
Corresponding author: Department of Ecology and Evolutionary Biology, Yale
University, Osborne Memorial Labs 226, 165 Prospect St., PO Box 208106, New
Haven, CT 06520. E-mail: [email protected]
attempts to detect natural selection currently exploit the
growing body of knowledge about the sizes and directions
of quantitative trait locus (QTL) effects (for counterexamples,
see Rieseberg et al. 2002; Albertson et al. 2003; Lexer et al.
2005).
Progress toward understanding the basis of quantitative
genetic variation is likely to come from studying allelic
variation at specific QTL (Barton and Keightley 2002). Identifying the genetic architecture of quantitative traits begins
with mapping QTL to broad genomic regions and ends with
the molecular definition of quantitative trait loci alleles. This
high degree of resolution has been achieved, for instance, for
some QTL in Drosophila (Zimmerman et al. 2000; Mackay
2001; Palsson and Gibson 2004; Palsson et al. 2005). QTL
mapping may be coupled with further genetic dissection (e.g.,
fine-scale mapping, disequilibrium mapping, transgenic manipulation; Mackay 2001) to characterize the specific loci
targeted by selection. Weinig et al. (2003) demonstrated that
the QTL architecture behind herbivory tolerance was one of
many loci of small effect and simultaneously elucidated locusspecific evolutionary constraints, demonstrating that linking
molecular genetic tools to quantitative genetic analysis and
field studies in ecologically relevant settings can clarify the
role of specific loci in the evolution of quantitative traits.
Genetics, Vol. 190, 1533–1545 April 2012
1533
These QTL studies not only generate maps from genotype
to phenotype, but also yield information that may be applied
to the study of phenotypic evolution (Erickson et al. 2004).
Behind every genetically based phenotypic difference lies an
evolutionary history. Despite the evident relevance of QTL
data to the study of the evolutionary process, very little theory has been developed to link data from QTL studies to
population genetic parameters via the methods of molecular
evolution. This lack of theoretical tools has inhibited progress
in explaining the process underlying the evolution of phenotypes that are genetically dissected in empirical QTL studies.
While most tests for selection depend directly on molecular data (McDonald and Kreitman 1991; Depaulis et al.
2003; Schlenke and Begun 2003; Mousset et al. 2004;
Schlenke and Begun 2004; Nurminsky et al. 2005; Schlenke
and Begun 2005), Orr (1998a) presented two innovative
attempts to integrate QTL data and population genetic theory into tests for selection: the QTL sign test and the QTL
effect size test. These tests employ a null model of the evolutionary process that produces QTL. They have been applied multiple times to empirical data (Rieseberg et al.
2002; Albertson et al. 2003; Lexer et al. 2005; Orr 2010).
However, the QTL sign test has been deemed substantially
flawed by its high false-positive rate (Anderson and Slatkin
2003) and the QTL effect size test has been shown to provide
very low power to detect selection (Anderson and Slatkin
2003). Moreover, the tests neither model selection explicitly
nor account for the distribution of mutational effects, both of
which are important to accurately model the “filtering” action of natural selection (Orr 1998b).
Accordingly, understanding the evolutionary significance of QTL observations requires evaluation of both the
origin and fate of heritable variation. In particular, the
evolutionary forces that result in the genetic differences we
observe must be estimated (Barton and Keightley 2002).
Here we develop a test for a history of directional selection
on quantitative traits and a method for estimating the
strength of that selection. Using mutation accumulation
(MA) data and the QTL effect distribution derived from
a cross between two divergent populations, we infer the
selective history of the trait. While existing tests (Orr
1998) only model neutral evolution and assume selection
to be at work whenever data does not fit the model’s predictions under neutrality, we model selection explicitly and
consider neutrality as one potential inference on a continuum. Such a flexible model is a key first step toward unraveling the complex link between molecular evolution and
quantitative trait variation.
Theory
Overview
Our framework for inference integrates information from
mutation accumulation experiment (Figure 1, A and B),
population genetic theory on selection (Figure 1, G–J), the
consequent expected outcome of selection (Figure 1, K–N),
1534
D. P. Rice and J. P. Townsend
and the empirically obtained QTL effect size distribution
(Figure 1, O–R). In our schematic example, mutation accumulation is depicted as resulting in increased variance of the
phenotype with no directional bias (Figure 1A) or, alternatively, as resulting in an increased variance in phenotype
with a downward bias (Figure 1B). From phenotypic observations obtained along the time course of a mutation accumulation experiment (Figure 1, A and B), the distribution of
mutational effects on phenotype can be inferred (Figure 1,
C–F). Data as in Figure 1A yield an inferred distribution of
mutational effects that is symmetrical around no change
(Figure 1, C and D); in contrast, data as in Figure 1B yield
asymmetry (Figure 1, E and F).
We multiply the mutation effect distribution by the
calculated probability of fixation as a function of the
strength of selection corresponding to the phenotypic effect
(Figure 1, G–J). For a neutral phenotype, fixation probabilities are equivalent across all phenotypes (Figure 1, G and I),
whereas under selection, fixation probabilities depend upon
the phenotype conferred (Figure 1, H and J). The product of
the mutation effect distribution and the fixation probability
distribution is the probability distribution of fixed phenotypic differences resulting from substitutions at quantitative
trait loci (Figure 1, K–N). Sampling a number of substitutions from this distribution and summing their effects (e.g.
Figure 1, O–R, outlines), we compare the cumulative effects
with those of the observed QTL (Figure 1, O–R, bars). The
accord between the sampled effects and the observed effects
varies with the estimated mutation effect distribution and
selective regime considered: the accord will be better when
the level of selection is appropriate for generating fixed
effects whose cumulative effects are close to those of the
observed QTL. A likelihood function (Figure 1, S and T)
quantifies the fit of the model and provides a test of the
hypothesis of neutrality and an estimate of the strength of
selection. In our schematic example, the MA and QTL data
together could support an effectively neutral model (Figure
1, A, C, D, G, H, K, L, O, P, S). Alternatively, parameterizing
with the same QTL data (Figure 1, O–R, bars), but with
different MA data (Figure 1, A and B) could yield rejection
of neutrality and thus, a conclusion that the quantitative
trait has been subject to a history of selection.
To develop this inferential framework, we first present
a method for estimating the distribution of mutational
effects with data from mutation accumulation experiments,
which are a well-established approach to inferring the
effects of mutations (Keightley, 1994; Garciadorado 1997;
Shaw et al. 2002; Halligan and Keightley 2009). In these
experiments, one or more lines of the organism under study
are kept under unselective conditions, subjected to serial
population size bottlenecks (Lopez and Lopez-Fanjul
1993). This enforced genetic drift facilitates the accumulation of mutations with minimal interference from selection.
A quantitative trait may be assayed at the end of the experiment or at regular intervals to track its change over time in
each population. Using the change in the trait means and
Figure 1 A depiction of our approach
toward characterizing the neutral or selected evolution of phenotype based on
mutation accumulation and quantitative
trait locus data for a phenotype. (A) A
constructed example of phenotypic
change due to unbiased mutation
within seven mutation accumulation
lines. (B) A constructed example of phenotypic change due to downwardly biased mutation within seven mutation
accumulation lines. (C and D) The distribution of mutational effects inferred
from A. (E and F) The distribution of mutational effects inferred from B. (G and I)
A depiction of a probability of fixation of
novel mutations that is invariant across
potential phenotypic values. (H and J) A
depiction of probability of fixation of
novel mutations that increases with the
phenotypic value. (K) The product of the
functions depicted in C and G: in this
case, a symmetrical distribution of fixed
mutations with a zero mode. (L) The
product of D and H: in this case, an
asymmetrical distribution of fixed mutations with a positive mode and positive
skewness. (M) The product of the functions depicted in I and M: in this case,
a symmetrical distribution with a negative mode. (N) The product of the functions depicted in F and J: in this case, an
asymmetrical distribution with a zero
mode and positive skewness. (O) An example of a single sample of mutations
(outlines) drawn from the distribution
depicted in K, compared to the experimentally observed positive and negative
QTL (shaded bars): in this case, the sample matched the QTL well. (P) An example of a single sample of mutations
(outlines) drawn from the distribution
depicted in L, compared to the experimentally observed positive and negative
QTL (shaded bars): in this case, the sample matched the positive QTL fairly well,
but matched the negative QTL poorly. (Q) An example of a single sample of mutations (outlines) drawn from the distribution in M, compared to the
experimentally observed positive and negative QTL (shaded bars): in this case, the sample matched the positive QTL poorly, but matched the negative
QTL fairly well. (R) An example of a single sample of mutations (outlines) drawn from the distribution in N, compared to the experimentally observed
positive and negative QTL (shaded bars): in this case, the sample matched the positive and negative QTL well. The mutations drawn here “happen” to
exactly match the QTL effects depicted in O, but in fact sampling from the fixed mutation distribution is highly stochastic and requires numerically
integrating over many samples to accurately yield the likelihood that the fixed mutation distribution underlies the experimentally observed QTL. (S) A plot
of the likelihood across a range of values of the strength of selection: in this case, samples drawn from the distribution in K are identified as fitting better
than samples drawn from the distribution in L, indicating neutral evolution of the phenotype. Parts of the plot that lie above the dashed line correspond
to a 95% CI. The plot at the y-axis corresponds to neutrality and falls within the CI, so we cannot reject a hypothesis of selective neutrality for this
phenotype. (T) A plot of the likelihood across a range of values of the strength of selection: in this case, samples drawn from the distribution in N are
identified as fitting better than samples drawn from the distribution in M, indicating evolution of the phenotype driven by natural selection. The plot at
the y-axis corresponds to neutrality and falls outside the CI, so we can reject a hypothesis of selective neutrality for this phenotype and estimate that the
strength of selection corresponds to the level of selection identified by the peak of the plot.
the number of generations between observations, we estimate a Gaussian distribution of mutational effects (Turelli
1984; Sawyer et al. 2003; Jones et al. 2007).
We then derive the phenotypic effect fixation function
from components of classical population genetic theory
(Kimura 1962). The product of this function with the mutational effect distribution provides a distribution of QTL effect
sizes. Because these components are all probabilistic, comparison of their product to estimated QTL effect sizes and
the standard errors thereof constitutes a likelihood-based
QTL, Mutation Accumulation, and Selection
1535
approach to inferring the historical regime of genetic drift or
natural selection.
We apply this test to an example trait, sensory bristle
number in the fruit fly Drosophila melanogaster. The genetic
architecture of Drosophila sensory bristle number is well
studied (Lai et al. 1994; Long et al. 1995, 1996; Dilda and
Mackay, 2002; Robin et al. 2002; MacDonald et al. 2005;
Mackay and Lyman 2005) and the selective history on this
trait has been the subject of much debate (Mather 1941;
Robertson 1955; Robertson 1967; Kearsey and Barnes
1970; Mackay et al. 1994,).
Estimating the mutational effect distribution
To estimate the distribution of the phenotypic effects of
mutations on the trait, we analyze the time course of
a mutation accumulation experiment. Although the appropriate formal distribution of the mutational effects for any
given trait is unknown (Keightley and Lynch 2003),
moments of the distribution of mutation effects on a trait
can be inferred from the changes in the trait over time. For
analytical tractability, we follow Turelli (1984), Sawyer
et al. (2003), and Jones et al. (2007) and assume that the
effects of mutations on a trait follow a Gaussian distribution
2
2
1
f ðx; m; sÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi e2 ðx2mÞ =2s ;
2
2ps
(1)
with mean m and standard deviation s. This notation and
other notation to be used in the derivation of the model are
summarized in Table 1.
The sum of k independent normally distributed random
variables with the same mean and variance is normally distributed with mean km and variance ks2 . Assuming independence of mutational effects, the distribution of the total
effect of k mutations is
2
1
2
f ðx; m; s; kÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 ðx2kmÞ =2ks :
2p ks2
(2)
The distribution of the total effect of the mutations that
accumulate in g generations is the sum of the distributions of
the effect of k mutations, as k varies from zero to infinity,
weighted by the probability that k mutations occur. If mutations are independent, the number of mutations affecting
the trait that occur in g generations follows a Poisson
distribution,
Pðnumber of mutations ¼ kÞ ¼
ðguÞk 2gu
e ;
k!
where u is the per generation rate of mutations affecting the
trait. The sum over k of Equation 2 weighted by Equation 3 is
!
N
X
2
2
ðguÞk 2gu
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 ðx2kmÞ =2ks :
hðx; g; u; m; sÞ ¼
e
k!
2p ks2
k¼1
(4)
Equation 4 is an infinite sum and cannot readily be
calculated exactly. In practice, however, accurate approximations for small gu may be achieved by sums of only
a small number of terms normalized by the total of all terms
considered. We included terms so that, for a given gu, the
total Poisson weight of the terms was at least 0.999.
Inference of historical selection from QTL and MA data
Inference of historical selection from QTL and MA data
requires specification of the relationship between the
strength of selection and the distribution of effect sizes of
mutations that fix in the population. To calculate the
probability that a mutation with a particular effect would
fix, we apply the solutions of selection–diffusion equations.
Assuming that these equations hold, the probability of a new
mutation fixing in the population is given by Kimura (1962),
PðfixationÞ ¼
1 2 e2s
;
1 2 e22Ns
x
m
s
k
u
g
s
N
c
C
n
H ¼ (H1, H2,. . ., Hn)
q ¼ (q1, q2,. . ., qn)
e ¼ (e1, e2,. . ., en)
1536
D. P. Rice and J. P. Townsend
(5)
where s is the selection coefficient and N is the population
size.
Following Lande (1983) and Chevin and Hospital
(2008), we assume that the selective value of the mutation
is proportional to its quantitative effect on the trait. To avoid
Table 1 Notation
Parameter
(3)
Description
Effect size of a mutation or substitution
Mean effect size of spontaneous mutations
Standard deviation of the effect size of spontaneous mutations
Number of mutations occurring between MA measurements
Per-generation mutation rate affecting the trait
Number of generations between MA measurements
Selection coefficient
Effective population size
Strength of selection: slope of the relationship between s and x
Normalizing constant to make Equation 8 a probability density function
The number of loci detected
Vector of the number of substitutions responsible for the QTL, where
hi is the number of substitutions responsible for QTL i
Estimated effect sizes of the QTL
Standard errors of the effect sizes of the QTL
making assumptions about the optimal value of the trait, we
assume that the fitness gradient did not change as the value
of the trait changed. This assumption stands in contrast to
some existing models of adaptive walks (Orr 1998; Martin
and Lenormand 2006) that use a Gaussian fitness landscape.
It is also a conservative assumption for our purposes: a walk
toward a nearby optimum on a peaked fitness landscape will
fix mutations of smaller effect than in our model, diminishing the signature of selection that our method is sensitive to.
Additionally, we assume that adaptation proceeds by successive fixation of beneficial mutations (Atwood et al. 1951), as
opposed to simultaneous selection at many loci. If many loci
are selected simultaneously, the strength of selection on
a particular locus will tend to decrease over time (Chevin
and Hospital 2008), rather than remain constant. Hence, the
assumption of successive fixation is also conservative. Furthermore, we assume that the mutation has no other effects
on fitness beyond its effect on the trait in question. Thus, for
a mutation with phenotypic effect x,
s ¼ cx;
(6)
where c is a constant whose sign and magnitude determine
the direction and strength of selection, respectively. A positive value of c indicates selection for an increase in the trait
and a negative value of c indicates selection for a decrease in
the trait. When c = 0, the trait is neutral (i.e., the selection
coefficient for a mutation of any effect size is zero).
Substituting Equation 6 into Equation 5 yields
8
1 2 e2cx
>
<
; c 6¼ 0
22Ncx
;
PðfixationÞ ¼ 1 2 e
>
1
:
;
c¼0
2N
measurement in each line. We also use an independent
estimate of the per-generation mutation rate of the trait. To
calculate probabilities of fixation of mutants drawn from the
effect distribution, we also use the historical effective population size of the population in question. To construct a likelihood
function for the parameters underlying the observed change in
the quantitative trait, we use the effect size and standard error
of QTL identified by crossing individuals from two divergent
populations that differ in the trait value.
Mutation effects distribution
Moments of the distribution of mutation effects on a trait
were inferred from the changes in the trait value over the
course of a mutation accumulation experiment. Given the
change in the trait means, the number of generations
between observations, and an independent estimate of the
per-generation mutation rate for the trait, a likelihood
function was specified for the parameters m and s of the
distribution of mutation effects as
X
ln hðxi ; gi ; u; m; sÞ;
(9)
Lðm; sÞ ¼
i
where L(. . .) is the log likelihood, h(. . .) is from Equation 4,
and xi and gi are the change in the mean value of the trait
and number of generations between each observation. The
values of m and s for which this function is maximized are
the maximum-likelihood estimates of the parameters of the
Gaussian distribution of mutational effect sizes.
Estimating selection
(7)
which provides the probability that a mutation of size x will
fix in the population.
The probability distribution of the phenotypic effects of
substitutions is determined by the distribution of mutational
effects multiplied by the relative fixation probability of those
effects. Accordingly, multiplying Equation 1 by Equation 7
yields the distribution of effect sizes of substitutions, given
that they have fixed
1 2 e2cx
1
2 ðx2mÞ2 =2s2
pffiffiffiffiffiffiffiffiffiffiffiffi e
(8)
f ðxÞ ¼ C
1 2 e22Ncx
2ps2
To make f(x) a probability distribution, Equation 8 contains
a normalization constant, C, chosen to ensure that the integral of f(x) over all x is equal to one.
Methods
Overview of the data required
To infer a mutation effects distribution, we use the time course
of a mutation accumulation experiment, specifically the change
in trait value and number of generations between each
We also applied a maximum-likelihood framework to
estimate the strength of historical selection on the trait.
The phenotypic effect of each QTL is the result of one or
more underlying substitutions, so we maximize the joint
likelihood of c and the vector H = (H1, H2, . . ., Hn), where Hi
is the number of substitutions at the ith locus and can be any
positive integer, and n is the number of loci. There is no
explicit constraint on the number of mutations at each locus,
nor on the directions of the underlying mutational effects,
only the constraint of the specified total effect. Thus, our
method can properly accommodate amalgamations of several QTL that are detected as single QTL by a QTL analysis
(cf. Mackay 2001; Mackay and Lyman 2005; Studer and
Doebley 2011), even if the cryptic QTL are in opposing
directions, just as it accommodates multiple mutations of
opposing direction. The likelihood of c and H given the
QTL data are the product of the joint likelihood of c and H
given each locus individually:
lðc; H; q; eÞ ¼
n
Y
lðc; Hi ; qi ; ei Þ:
(10)
i
Here q and e are the effects and standard errors of the
observed QTL. The joint likelihood of c and H given a single
locus is
QTL, Mutation Accumulation, and Selection
1537
Z
lðc; Hi ; qi ; ei Þ ¼
N
2N
gðx; qi ; ei ÞFðx; c; Hi Þdx;
(11)
where g(. . .) is the probability density function of a Gaussian
distribution with mean qi and standard deviation ei, and
F(. . .) is the probability density function of the sum of Hi
substitutions under strength of selection c. The latter is calculated by Hi convolutions of f(x) from Equation 8 with
itself.
The integral in Equation 11 has no closed-form solution
and is difficult to evaluate by typical numerical approaches
for most values of Hi. We effectively approximated the likelihood by Monte Carlo sampling from the distribution given
by F(x) and calculating average value of g(x) evaluated for
each sampled effect:
+
*
1
2 ðxj 2qi Þ2 =2ei 2
:
lðc; Hi ; qi ; ei Þ g xj ; qi ; ei j ¼ pffiffiffiffiffiffiffiffiffiffiffiffi e
2pei 2
j
(12)
Angle brackets represent averaging over all Monte Carlo
samples xj. As the number of Monte Carlo samples increases,
the right-hand side of Equation 12 approaches the true
likelihood.
To calculate the likelihood of a given value of c, we fixed
c at a value and evaluated the right-hand side of Equation 12
for a range of vectors of H. For each locus i, we started with
Hi = 1 and then increased Hi until we found its unimodal
maximum likelihood value. We took the likelihood of c to be
the product over i of these likelihoods (Equation 10). We
calculated the likelihood of a dense grid of values of c in this
way to obtain ĉ, the maximum-likelihood value. To test the
hypothesis of neutral evolution, we performed a likelihoodratio test with a null hypothesis of c = 0 and an alternate
hypothesis of c = ĉ.
Sources of data
We applied our method to QTL and MA data for sensory
bristle number in D. melanogaster. QTL effect size data were
extracted from Dilda (2002), who had applied four different
artificial selection regimes on bristle traits. Populations HST
and LST were selected for high and low numbers of sternopleural bristles, respectively, while populations HAB and
LAB were selected for high and low numbers of abdominal
bristles. The QTL effects and standard errors from each population are presented in Figures 3A, 4A, 5A, and 6A. These
QTL were assayed by crossing the artificially selected populations with a wild-type laboratory strain. Because the QTL
for this trait were assayed using artificially selected strains
rather than phenotypically divergent strains from wild populations, our example serves to illustrate the use of the test
rather than to make an inference about the selective history
of the trait in the wild.
Mutation accumulation data for sternopleural bristle
number and abdominal bristle number were extracted from
1538
D. P. Rice and J. P. Townsend
Paxman (1957) and Lopez and Lopez-Fanjul (1993), respectively. Paxman (1957) propagated six lines of flies to accumulate mutations for 40 generations and measured the
average number of sternopleural bristles in each line at 5
time points. The distribution of trait changes in the mean
sternopleural bristle number of each line between each measurement is depicted in Figure 2A. Lopez and Lopez-Fanjul
(1993) maintained 93 MA lines for 61 generations and measured the change in the number of abdominal bristles in
each line at the end of the experiment. The distribution of
trait changes is depicted in Figure 2B.
The per-generation rate of mutation affecting sensory
bristle number in Drosophila is estimated to be 0.03 (Mackay
and Lyman 2005).
Estimating the distribution of mutational effects
To estimate the mean and standard deviation of the
distribution of mutation effects m and s, we calculated their
maximum-likelihood values for each type of bristle. For each
bristle trait, we evaluated Equation 9, letting u = 0.03
(Mackay and Lyman 2005). We let xi and gi be the trait
changes and numbers of generations between measurements for all measurements in all lines. Implementing a simple hill-climbing algorithm in MATLAB, we calculated the
joint maximum-likelihood estimates for m and s (see supporting information, File S1). We estimated the standard
deviations of our estimates by finding the values of m and
s whose log likelihood was equal to the maximum log
likelihood minus one (Hudson 1971).
Estimating selection
For each of the four QTL data sets, we calculated the
likelihood of a range of values of c using an effective populations size N = 50 (Gurganus et al. 1999) and the maximum-likelihood estimates of m and s for the appropriate
bristle trait. For each value of c and Hi, we generated 6.5 ·
104 Monte Carlo samples from the distribution of substitution effects in MATLAB and calculated the likelihood according to Equation 12 (see File S2). To calculate the
significance of the likelihood-ratio test, we computed the
test statistic
lðc ¼ 0Þ
(13)
D ¼ 2 2 log
lðc ¼ ^cÞ
D is approximately x2-distributed with 1 d.f. (Hudson 1971).
It follows that the 95% confidence interval includes values
of c with likelihoods within 2 log units of the maximum.
False-positive rate
To estimate the false-positive rate of the test, we simulated
QTL data under a neutral model of evolution. For each of
the four data sets, we found the maximum-likelihood value
of H for c = 0. For the ith observed locus, we generated
a neutral locus by summing Hi effects drawn from the distribution of mutation effects for the appropriate bristle trait.
Figure 2 Frequency distribution of changes in bristle traits
during mutation accumulation. (A) Change in sternopleural bristle number between measurements in a mutation
accumulation experiment by Paxman (1957). Six lines of D.
melanogaster were raised in unselective conditions in
which they were subjected to population bottlenecks for
40 generations. The mean sternopleural bristle number
was assayed five times over the course of the experiment.
(B) Change in abdominal bristle number in a mutation
accumulation experiment by Lopez and Lopez-Fanjul
(1993). Ninety-three mutation-accumulation lines of D.
melanogaster were maintained for 61 generations and
then assayed for mean abdominal bristle number.
Although it entailed extensive computation, we generated
100 sets of simulated neutral QTL for each data set. We
applied the likelihood-ratio test for selection to these simulated data and recorded the proportion of neutral data sets
for which the neutral hypothesis was rejected.
0.05. For abdominal bristles (Lopez and Lopez-Fanjul
1993), our estimate of m was 20.25 6 0.04 and our estimate of s was 0.34 6 0.05. We calculated the MLE ĉ for the
four QTL data sets, in each case applying the corresponding
mutation-effect distribution.
Undetected QTL
Selection in the HST population
A well-documented bias in QTL studies is the failure to
detect small-effect loci (Beavis 1998; Bost et al. 2001; Xu
2003). To assess whether undetected QTL might have influenced our estimate, we applied the analytical approach of
Otto and Jones (2000) to estimate the number and average
effect of undetected loci in each data set. We then reran our
analysis including in each data set the estimated number of
additional QTL, with their effect sizes following an exponential distribution with the estimated mean.
Based on the nine QTL detected in the HST population
(Figure 3A) and the mutation-effect distribution estimated
for sternopleural bristle number (Figure 3B), the likelihoodratio test (x2 = 38.1, P , 0.001) rejects a neutral hypothesis
for an increase in sternopleural bristle number. The MLE for
c, ĉ, was 0.018, with a 95% confidence interval (CI) ranging
from 0.013 to 0.022 (Figure 3C). None of the corresponding
one hundred simulated neutral data sets based on the HST
data yielded a false positive.
Applying Otto and Jones (2000) to assess whether undetected QTL might have influenced our HST result yielded
an estimate that four loci were not detected and that these
loci had an average effect of 0.388. Including four simulated
QTL following an exponential distribution with mean 0.388,
the MLE ĉ remained 0.018 and the CI ranged from 0.013 to
0.022 (Figure 3D), the same result as in our main analysis.
To assess the sensitivity of our test to the estimates of the
mutation-effect distribution parameters, we perturbed m by
1 SD in each direction and applied the test. For m = 20.144,
ĉ = 0.027 with a CI ranging from 0.021 to 0.031, while for
m = 20.022, ĉ = 0.012 with a CI ranging from 0.007 to
0.016 (Figure 3E). Similarly perturbing s, we found that for
s = 0.183, ĉ = 0.029, with a CI ranging from 0.025 to
0.034, and for s = 0.288, ĉ = 0.005, with a CI from
0.001 to 0.01 (Figure 3F). Because confidence intervals for
ĉ did not include zero, rejection of the hypothesis of neutrality underlying the evolution of the HST QTL data are
robust to moderate perturbations in m and s.
Sensitivity to the mutation effect distribution
To assess the sensitivity of our test to the estimates of the
mutation-effect distribution parameters, we perturbed m by
1 SD in each direction and applied the test. We then similarly perturbed s and compared the resulting estimates and
confidence intervals to our primary results.
Power
To estimate the power of the test with a diverse set of
putatively realistic QTL distributions, we applied the test to
all subsets of the QTL from each of the four populations,
using the appropriate mutation-effect distribution parameters in each case. We examined how the proportion of times
the test rejected neutrality varied with the number of loci
included in the subset.
Results
The distribution of mutation effects
We estimated the maximum-likelihood mean and standard
deviation of the Gaussian distribution of mutational effects
(Equation 9) on the basis of the MA data of each type of
sensory bristle. For sternopleural sensory bristles (Paxman
1957), the maximum-likelihood estimate (MLE) of m was
20.08 6 0.06 (61 SD) and the MLE of s was 0.23 6
Selection in the LST population
On the basis of the four QTL detected in the LST population
(Figure 4A) and the mutation-effect distribution estimated
for sternopleural bristle number (Figure 4B), a likelihoodratio test (x2 = 4.88, P = 0.027) rejects a neutral hypothesis
for a decrease in sternopleural bristle number. In this case,
QTL, Mutation Accumulation, and Selection
1539
Figure 3 Selection in the HST population. (A) The effects and standard
errors of sternopleural bristle QTL detected in the HST population selected
for high sternopleural bristle number. (B) The maximum-likelihood estimate of the distribution of mutational effects for sternopleural bristles. (C)
Log likelihood vs. c, the strength of selection, given the HST QTL data and
the maximum-likelihood distribution of mutational effects. The horizontal
line indicates the likelihood threshold for a 95% confidence interval. (D)
Log likelihood vs. c, given the HST QTL data plus four imputed unobserved loci. (E) Log likelihood vs. c, given the HST QTL data, with the
estimate of m perturbed by adding 1 SD (crosses, dashed threshold line)
and subtracting 1 SD (circles, solid threshold line). (F) Log likelihood vs. c
given the HST QTL data, with the estimate of s perturbed by adding 1 SD
(crosses, dashed threshold line) and by subtracting 1 SD (circles, solid
threshold line).
the log-likelihood curve does not have a maximum, but instead increases asymptotically as ĉ decreases (Figure 4C).
Conservatively using the largest sampled likelihood as the
maximum likelihood and applying the likelihood-ratio test,
we found that ĉ , 20.005 with 95% confidence. None of the
100 simulated neutral data sets based on the LST data
yielded a false positive.
We estimated that four loci were not detected and that
these loci had an average effect of 20.284. When we included these simulated QTL in the data, we again found
ĉ , 20.005 (Figure 4D). Perturbing m, we found that for
m = 20.144, ĉ , 0.007, and for m = 20.022, ĉ , 20.017
(Figure 4E). Perturbing s, we found that for s = 0.183, ĉ ,
0.001, and for s = 0.288, ĉ , 20.007 (Figure 4F). Because
confidence intervals for ĉ included zero, rejection of the
1540
D. P. Rice and J. P. Townsend
Figure 4 Selection in the LST population. (A) The effects and standard
errors of sternopleural bristle QTL detected in the LST population, selected
for low sternopleural bristle number. (B) The maximum-ikelihood estimate
of the distribution of mutational effects for sternopleural bristles. (C) Log
likelihood vs. c, the strength of selection, given the LST QTL data and the
maximum-likelihood distribution of mutational effects. The horizontal line
indicates the likelihood threshold for a 95% confidence interval. (D) Log
likelihood vs. c, given the LST QTL data plus four imputed unobserved loci.
(E) Log likelihood vs. c, given the LST QTL data, with the estimate of m
perturbed by adding 1 SD (crosses, dashed threshold line) and subtracting 1
SD (circles, solid threshold line). (F) Log likelihood vs. c given the LST QTL
data, with the estimate of s perturbed by adding 1 SD (crosses, dashed
threshold line) and by subtracting 1 SD (circles, solid threshold line).
hypothesis of neutrality underlying the evolution of the
LST QTL data were not robust to moderate perturbations
in m and s.
Selection in the HAB population
On the basis of the three QTL detected in the HAB
population (Figure 5A) and the mutation-effect distribution
estimated for abdominal bristle number (Figure 5B), a likelihood-ratio test (x2 = 34.7, P , 0.001) rejects the neutral
hypothesis for an increase in the number of abdominal bristles. As in the LST population, the log-likelihood curve does
not have a maximum, instead increasing asymptotically as ĉ
increases (Figure 5C). Conservatively using the largest observed likelihood as the maximum likelihood and applying
the likelihood ratio test, we found that ĉ . 0.031 with 95%
confidence. Only 1 of the 100 simulated neutral data sets
based on the HAB data yielded a false positive.
We estimated that five loci were not detected and that
these loci had an average effect of 0.245. When we included
these simulated QTL in the data, we found that ĉ . 0.037
(Figure 5D). Perturbing m, we found that for m = 20.294,
ĉ .0.037, and for m = 20.215, ĉ . 0.026 (Figure 5E).
Perturbing s, we found that for s = 0.30, ĉ . 0.034, and
for s = 0.392, ĉ . 0.029 (Figure 5F). Because confidence
intervals for ĉ did not include zero, rejection of the hypothesis of neutrality underlying the evolution of the HAB QTL
data are robust to moderate perturbations of m and s.
Selection in the LAB population
On the basis of the three QTL detected in the LAB
population (Figure 6A), and the mutation-effect distribution
estimated for abdominal bristle number (Figure 6B), a likelihood-ratio test (x2 = 4.00, P = 0.046) rejects a neutral
hypothesis for a decrease in abdominal bristle number (Figure 6C). Conservatively using the largest observed likelihood as the maximum likelihood and applying the
likelihood-ratio test, we found that ĉ , 0 with 95% confidence. Only 1 of the 100 simulated neutral data sets based
on the LAB data yielded a false positive.
We estimated that 13 loci were not detected and that
these loci had an average effect of 20.240. When we included these simulated QTL in the data, ĉ was 0 with
a 95% CI ranging from 20.032 to 0.015 (Figure 6D). Perturbing m, we found that for m = 20.294, ĉ , 0.006, and for
m = 20.215, ĉ , 20.004 (Figure 6E). Perturbing s, we
found that for s = 0.30, ĉ , 0.003, and for s = 0.39, ĉ ,
20.001 (Figure 6F). Because confidence intervals for ĉ included zero, rejection of the hypothesis of neutrality underlying the evolution of the LAB QTL data are not robust to
perturbations in m and s.
Power
We examined the power of subsets of the observed QTL to
yield a positive result. The proportion of cases in which
neutrality could be rejected increased monotonically with
the number of QTL in the subset (Figure 7). For single QTL
and the observed mutation-effect distribution, the test
rejected neutrality in only 43% of cases, while for data sets
with six or more of these loci, the test always rejected
neutrality.
Discussion
We have derived a new method for detecting the level of
historical selection on quantitative traits. First, we showed
how to estimate the parameters of a Gaussian distribution of
mutational effects from the change in the trait over the
course of a mutation-accumulation experiment. Then, we
derived the distribution of the effect sizes of mutations that
fix as a function of the regime of selection or neutrality
applied to the trait. We demonstrated how to integrate these
Figure 5 Selection in the HAB population. (A) The effects and standard
errors of abdominal bristle QTL detected in the HAB population (selected
for high abdominal bristle number. (B) The maximum-likelihood estimate
of distribution of mutational effects for abdominal bristles. (C) Log likelihood vs. c, the strength of selection, given the HAB QTL data and the
maximum-likelihood distribution of mutational effects. The horizontal line
indicates the likelihood threshold for a 95% confidence interval. (D) Log
likelihood vs. c, given the HAB QTL data plus five imputed unobserved
loci. (E) Log likelihood vs. c given the HAB QTL data, with the estimate of
m perturbed by adding 1 SD (crosses, dashed threshold line) and by
subtracting 1 SD (circles, solid threshold line). (F) Log likelihood vs. c given
the HAB QTL data, with the estimate of s perturbed by adding 1 SD
(crosses, dashed threshold line) and by subtracting 1 SD (circles, solid
threshold line).
two results to yield a maximum-likelihood estimate of the
strength of historical selection given a set of observed QTL
effect sizes and the results of a mutation-accumulation
experiment. Finally, we applied the test to a well-studied
quantitative trait, detecting a history of artificial selection on
four cases of sensory bristle numbers in D. melanogaster.
The test we have derived is the first test for selection on
phenotype that incorporates information about a trait’s genetic architecture as well as the phenotypic change induced
by mutation. It is also the first test to include an explicit
model of how neutrality and selection shape the loci that
affect a trait. Finally, our test is the first that will allow
biologists not only to distinguish between neutrally evolving
traits and selected traits but also to estimate the range of
QTL, Mutation Accumulation, and Selection
1541
Figure 7 Power vs. the number of loci detected. We applied the likelihood-ratio test to all subsets of the four QTL data sets, with the appropriate mutation-effect distribution. The y-axis represents the proportion of
subsets with a given number of loci for which the test rejected the hypothesis of neutrality.
Figure 6 Selection in the LAB population. (A) The effects and standard
errors of abdominal bristle QTL detected in the LAB population. (B) The
maximum-likelihood estimate of distribution of mutational effects for
abdominal bristles. (C) Log likelihood vs. c, the strength of selection, given
the LAB QTL data and the maximum-likelihood distribution of mutational
effects. The horizontal line indicates the likelihood threshold for a 95%
confidence interval. (D) Log likelihood vs. c, given the LAB QTL data plus
13 imputed unobserved loci. (E) Log likelihood vs. c given the LAB QTL
data, with the estimate of m perturbed by adding 1 SD (crosses, dashed
threshold line) and subtracting 1 SD (circles, solid threshold line). (F) Log
likelihood vs. c given the LAB QTL data, with the estimate of s perturbed
by adding 1 SD (crosses, dashed threshold line) and subtracting 1 SD
(circles, solid threshold line).
strengths of selection that are most likely to have produced
the current phenotypic value. As such, it represents a theoretical and practical improvement on existing methods to
detect selection on quantitative traits.
Application of the test to sensory bristle number in D.
melanogaster demonstrated its ability to detect a history of
selection. Likelihood-ratio tests demonstrated sufficient
power to reject neutrality in four out of four populations
that had been subject to artificial selection. For the HST
data, we were able to identify a MLE for a coefficient determining the strength of selection, as well as to describe its
95% CI. For the other three data sets, we were able to
estimate upper or lower bounds for the strength of selection.
Because the HST data contained the largest number of QTL
1542
D. P. Rice and J. P. Townsend
(9), this difference in precision is expected. The HST data
set was also the only one that contained some QTL with
effects opposite to the actual and inferred direction of selection. Similarly, by applying the test to subsets of the data, we
demonstrated that the test’s power increased with an increased number of QTL. We also assessed the false-positive
rate by applying the test to 100 simulated neutral data sets
based on each set of real QTL. In no case did more than 1 of
the 100 simulated data sets yield a false positive.
We assessed the sensitivity of our test to the estimates of
the mutation-effect distribution parameters by perturbing m
and s by 1 SD in each direction and reapplying the test. For
the HST and HAB data, rejection of neutrality was robust to
these perturbations. However, for the LST data and the LAB
data, the result was no longer significant in the case of
a more negative m and in the case of a smaller s. This loss
of significance speaks to a perhaps unsurprisingly higher
sensitivity of the test in cases in which selection and mutation are acting in opposite directions (as they are in HST and
HAB) than in cases in which selection and mutation are
acting in the same direction. Our low false-positive rate
across both cases in which selection and mutation are acting
in concert or in conflict indicates that this difference in sensitivity is in accord with the actual inferential difficulty and
not a problem of a biased test construction.
A well-documented bias in QTL studies is the tendency to
fail to detect small-effect loci (Beavis 1998; Bost et al. 2001;
Xu 2003). To assess whether undetected QTL might have
influenced our estimates, we applied the analytical approach
of Otto and Jones (2000) to estimate the number and effect
of undetected QTL and included simulated loci following
these parameters in our analysis. For the three data sets in
which the test rejected neutrality, this modification did not
change the significance of the result and had little effect on
the estimates of c. However, when we included the simulated loci in the LAB data set, where we had already bordered on insufficient power to detect the action of selection,
the MLE for c was shifted to 0. For this data set in particular,
the analysis indicated that there were many more undetected loci (13) than observed QTL (5). Furthermore, the
simulated loci had a mean of 20.240, almost identical to the
estimate of m for abdominal bristles, making them effectively neutral for the purposes of our test. Therefore, application of Otto and Jones (2000) imputation can have
a considerable impact when the expected number of unobserved loci is much larger than the number of detected QTL.
Wider application of this test awaits the availability of
more MA and QTL data. Unfortunately, there are few traits
for which both MA and QTL analysis have been performed.
Practical and ethical considerations limit the range of
species for which mutation-accumulation experiments may
be performed. On the other hand, numerous examples of
applications of Orr’s 1998 test to empirical data (Rieseberg
et al. 2002; Albertson et al. 2003; Lexer et al. 2005; Orr
2010) demonstrate that tests for selection on phenotypic
traits are in demand. Although both types of experiments
can be labor intensive and time consuming, there are many
cases in which either MA or QTL data are already available,
and only the corresponding partner experiment need be
completed to apply this test for historical selection.
Several expansions of our model would improve its scope
and precision. First, the model presented here implicitly
rather than explicitly incorporates time. Because the generation and fixation of mutations take time, the amount of
time a trait has had to evolve is an important factor in how
many substitutions are responsible for each QTL effect.
Explicitly incorporating time into the model should provide
more information for determining in which vector H is most
likely. Since H affects the likelihood of c, this additional
discriminatory power would yield more precise estimates
of the strength of selection. However, explicit rather than
implicit incorporation of time in our framework is complex.
Our test assumes that the probability of fixation of a new
mutation follows Equation 5, which has been derived from
various models of evolution (Kimura 1962; Moran 1960;
Gillespie 1974), and is a fairly general result, accommodating a variety of evolutionary scenarios for which the effective population size captures all relevant complications.
Examples include cyclically varying population size (Otto
and Whitlock 1997), population subdivision with symmetric
migration (Pollack 1966), and deleterious mutations at completely linked loci (Charlesworth 1994). Probability of fixation results that could be used in place of Equation 5 under
more diverse evolutionary scenarios are reviewed by Patwa
and Wahl (2008).
Because we model the sequential mutation and fixation
of alleles at different loci, our test applies to QTL identified
by crossing related populations and estimates the selective
regime responsible for the trait difference between the
populations. The method is not applicable, as currently
implemented, to QTL identified by crossing individuals from
the same population. However, since selection should shape
the standing variation within a population, a related test
that applies to QTL variation in a single population is
conceivable. Accordingly, one potential weakness of the test
is that our model assumes that QTL between the divergent
populations are constructed from novel mutations, without
allowing some proportion of fixed differences to arise from
standing variation in the ancestral population. QTL arising
from standing variation fix faster and are likely to be of smaller
effect size than QTL arising from novel mutations (Barrett and
Schluter 2008). Consequently, our model assumption that all
variation arises as new mutation is conservative for the purpose
of our test: within our framework, smaller QTL effect sizes will
be more consistent with neutrality.
Finally, distributions of mutational effect other than the
Gaussian distribution used here should be explored. Implementing the model with other distributions would allow
assessment of its sensitivity to this assumption and provide
flexibility to its use as new information comes to light about
the distributions of mutational effects found in nature. The
central challenge in this endeavor is that other distributions
lack the convenient property of the normal distribution that
the sum of normally distributed random variables is also
normally distributed. As currently implemented, our approach relies on this fact for tractability (Equation 2).
As a theoretical result, this test for selection on quantitative traits is important because it bridges population
genetic theory and genetic architecture as revealed by
QTL. Furthermore, the test for selection outlined here
provides an improvement over currently available methods
for determining the regime of historical selection or neutrality that generated observed quantitative differences in
phenotype. Results of the test applied to four data sets
known to be the result of artificial selection for sensory
bristle number in D. melanogaster are highly encouraging,
because they indicate that the test has sufficient power to be
used on moderately sized QTL data sets and also that the
false-positive rate is very low. Applying the test to QTL data
generated by crossing two individuals representative of diverged wild populations will yield conclusions about the
history of selection responsible for the observed trait differences. Finally, our model provides a novel framework that
holds promise for expansion in future work to incorporate
such factors as time, varying probabilities of fixation, and
diverse shapes of the distribution of mutational effects.
Acknowledgments
We thank Trudy Mackay and Carlos Lopez-Fanjul for
graciously and helpfully digging deep in their files to provide
key details of their published experiments. DPR was partially
funded while performing this work by a Yale College
Dean’s Research Fellowship, a Yale-Howard Hughes Medical Institute Future Scientist Summer Fellowship, and a National Science Foundation Graduate Research Fellowship
DGE-1144152. The final computations were run on the
Odyssey cluster supported by the Faculty of Arts and Sciences Science Division Research Computing Group at Harvard University.
QTL, Mutation Accumulation, and Selection
1543
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Communicating editor: C. D. Jones
QTL, Mutation Accumulation, and Selection
1545
GENETICS
Supporting Information
http://www.genetics.org/content/suppl/2012/01/31/genetics.111.137075.DC1
A Test for Selection Employing Quantitative Trait
Locus and Mutation Accumulation Data
Daniel P. Rice and Jeffrey P. Townsend
Copyright © 2012 by the Genetics Society of America
DOI: 10.1534/genetics.111.137075
Supporting Files File S1: Matlab Code 1 (Estimating mu and sigma) File S2: Matlab Code 2 (Calculating likelihood) Files S1 and S2 are available for download at http://www.genetics.org/content/suppl/2012/01/31/genetics.111.137075.DC1. 2 SI D. P. Rice and J. P. Townsend