Download The danger of applying the breeder`s equation in observational

doi:10.1111/j.1420-9101.2010.02084.x
REVIEW
The danger of applying the breeder’s equation in observational
studies of natural populations
M. B. MORRISSEY, L. E. B. KRUUK & A. J. WILSON
Institute of Evolutionary Biology, University of Edinburgh, Edinburgh, UK
Keywords:
Abstract
natural selection;
quantitative genetics;
response to selection.
The breeder’s equation, which predicts evolutionary change when a phenotypic covariance exists between a heritable trait and fitness, has provided a key
conceptual framework for studies of adaptive microevolution in nature.
However, its application requires strong assumptions to be made about the
causation of fitness variation. In its univariate form, the breeder’s equation
assumes that the trait of interest is not correlated with other traits having
causal effects on fitness. In its multivariate form, the validity of predicted
change rests on the assumption that all such correlated traits have been
measured and incorporated into the analysis. Here, we (i) highlight why these
assumptions are likely to be seriously violated in studies of natural, rather than
artificial, selection and (ii) advocate wider use of the Robertson–Price identity
as a more robust, and less assumption-laden, alternative to the breeder’s
equation for applications in evolutionary ecology.
Two simple equations to predict adaptive
phenotypic microevolution
Evolutionary change in a phenotypic trait under selection can be predicted using the univariate breeder’s
equation:
R ¼ h2 S;
ð1Þ
where R is the per generation response of a quantitative,
or continuous, trait, to selection, h2, the heritability, is
the proportion of phenotypic variance in the trait that is
attributable to additive genetic differences among individuals, and S is the selection differential (Lush, 1937,
chapter 12). The selection differential is the difference
between the population mean before and after selection,
and more generally, the selection differential is the
covariance between the trait and the relative fitness. This
equation plays a fundamental role in how we think
about and study natural selection and its microevolutionary consequences; it features prominently in many of
the texts we read and cite (Falconer, 1981; Futuyma,
1998; Lynch & Walsh, 1998; Roff, 2002), and papers and
Correspondence: Michael B. Morrissey, Institute of Evolutionary Biology,
Ashworth Labs, King’s Buildings, University of Edinburgh, Edinburgh
EH9 3JT, UK.
Tel.: +44 (0) 131 651 3608; fax: +44 (0) 131 650 6564;
e-mail: [email protected]
lectures we write and give. The breeder’s equation, as its
name implies, traces its roots to animal breeding and
artificial selection (Lerner, 1958; Lush, 1937), where it
has had a long and successful history, at least in
univariate studies (Clayton et al., 1957; Falconer, 1953;
Roff, 2007). However, the breeder’s equation has been
less successful in predicting evolutionary change in
natural, rather than agricultural or laboratory, systems
(Merilä et al., 2001a). Evolutionary biologists have made
many estimates of both the form of selection, especially
quantitative estimates of directional selection (Endler,
1986; Kingsolver et al., 2001), and also many estimates of
heritabilities, both in the laboratory and in the field, for
ecologically important traits (Mousseau & Roff, 1987). In
general, hypotheses that any given traits are under
directional selection are very often true, and furthermore
variation in these traits often, if not generally, has a
demonstrable genetic basis. Thus, the breeder’s equation
suggests that we should expect adaptive phenotypic
microevolution of quantitative traits to be common.
However, well-supported examples of microevolution in
natural populations are very difficult to produce, and on
careful inspection, stasis, or even counterintuitive phenotypic trends (e.g. Gienapp et al., 2008; Kruuk et al.,
2003; Larson et al., 1998; Wilson et al., 2007; Merilä
et al., 2001b) are at least as common as changes
consistent with breeder’s equation-based predictions
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M. B. MORRISSEY ET AL.
(e.g. Grant & Grant, 1995). Demonstrating that phenotypic change – whether consistent with predictions or not
– has an underlying genetic component is a technically
challenging task (see Hadfield et al., 2010 for a critique of
the statistical procedures most commonly used to date).
Although this difficulty undoubtedly contributes to the
scarcity of well-documented cases of adaptive phenotypic
selection responses, we also propose that there is a more
general failure to appreciate the limitations of the
breeder’s equation when applied outside of the context
of artificial selection.
An alternative simple equation to predict evolutionary
change is the Robertson–Price identity:
Dz ¼ ra ðz; wÞ;
ð2Þ
where Dz is the expected change in mean phenotype z
between generations and w is relative fitness. ra(z,w)
represents the additive genetic covariance between z and
w, or the extent to which heritable genetic differences
among individuals co-determine the trait and fitness
(Price, 1970; Robertson, 1966). The Robertson–Price
identity is also sometimes called the ‘secondary theorem
of natural selection’ (Crow & Nagylaki, 1976; Robertson,
1968). The left-hand sides of the breeder’s equation
and the Robertson–Price identity are the same: they are
both the per generation change in mean phenotype.
However, the equivalence of the right-hand sides is not
so apparent.
Translating these equations into words, the breeder’s
equation states that the response to selection is equal to
the proportion of variance in a trait that is caused by
genetic effects, multiplied by the strength of (directional) selection on the trait. The Robertson–Price
identity states that the response to selection is equal to
the genetic covariance between a trait and (relative)
fitness. The key difference between these two statements is that the breeder’s equation implies we should,
or at least can, measure the relationship between a trait
and fitness (i.e. selection) and the relationship between
a trait and genes (i.e. heritability) separately. In
contrast, the Robertson–Price identity contains only a
single term, requiring that we consider fitness variation
and the genetics of a trait jointly to predict evolution.
Expanding the h2 term in eqn 1 and setting the righthand sides of eqns 1 and 2 to be equal, as their lefthand sides are identical, we obtain
ra ðz; wÞ ¼
r2a ðzÞ
rp ðz; wÞ;
r2p ðzÞ
ra ðz; wÞ rp ðz; wÞ
¼ 2
:
r2a ðzÞ
rp ðzÞ
ð3Þ
The quotient of a covariance and a variance is a
regression, and thus in words eqn 3 states that the
regression, or more loosely the relationship, between
fitness and genes must be the same as the regression/
relationship between fitness and phenotypes in order for
the predictions of the breeder’s equation and the
Robertson–Price identity to be the same. The result in
eqn 3 is derived (differently) in the appendix to Hadfield
(2008). Each of these parameters can in fact vary
independently, and so eqn 3 does not represent an exact
theorem, but rather a narrow condition. This narrow
condition is the situation in which the two primary tools
at our disposal for predicting evolution will give us
consistent results. Below we develop a graphical model of
the genetics of selection on a heritable trait to explore the
implications of the distinction between genetic and
phenotypic patterns of trait–fitness covariance.
Decomposition of selection
An individual’s phenotype (P) is determined by genes
and environmental effects such that
P ¼ G þ E;
ð4Þ
where G is the genetic merit (i.e. what the phenotype
would be in the absence of environmental influences)
and E is the effect of the environment.1 This simple
decomposition holds at the individual level for phenotypic traits, and at the population level for variances in
(and covariances between) traits. We can therefore apply
this decomposition to selection which is simply the
covariance between two traits in the particular case that
one of those traits is fitness. Although we donot often
conceptualize fitness as a trait, it is a measurable feature
of individuals and so can be treated in this way.
Using simulated data, we can explore different
scenarios that lead to a phenotypic covariance between
a trait and fitness (Fig. 1). The first column of plots in
Fig. 1, scenarios 1–3, shows a simple and commonly
observed occurrence in studies in evolutionary biology:
the trait appears to be under directional selection, i.e. it
covaries with fitness. In the case of the simulated data
1
This decomposition can be pursued much further. For
example, the genetic variance can be decomposed into
additive and other (e.g. dominance) variances. Note that
eqn 2 is based on the additive genetic covariance,
whereas we do not proceed in eqn 4 to decompose
genetic variances into additive and other sources of
variance. We use the term ‘genetic’ throughout this
manuscript, but in fact we refer to ‘additive genetic’
variation and covariation, which are those genetically
determined effects on phenotype that contribute to a
selection response, except in special cases such as familylevel selection. Importantly, when we see, below, that
the Robertson–Price identity correctly predicts evolution
in circumstances when apparent selection caused by
unmeasured traits biases predictions of the breeder’s
equation, the same arguments will hold for failure to
recognize and model nonadditive genetic effects on the
relationships between fitness and traits (Robertson,
1967).
ª 2010 THE AUTHORS. J. EVOL. BIOL. doi:10.1111/j.1420-9101.2010.02084.x
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The genetic analysis of natural selection
Phenotypic
Genetic
3
Environmental
Scenario 1:
Fig. 1 A graphical model of the genetic basis
of natural selection. Phenotypic relations
between trait and fitness (left column) are
decomposed into mathematically permissible
genetic and environmental components.
Each row of plots represents a genetic
scenario consistent with positive directional
selection on a heritable trait. Blue downward-pointing triangles denote the 10% of
individuals with the lowest genetic breeding
values for the trait and red upward-pointing
triangles denote the 10% of individuals with
the highest breeding values for the trait. In
scenarios 1, 3 and 4, evolution is expected
because genetic effects on individuals covary
with fitness, but this pattern of change is
only predictable in scenario 1, where the
trait–fitness relationship is the same at the
phenotypic and genetic levels. Conversely,
in scenario 2, no evolution is expected,
despite apparent selection at the phenotypic
level of this hypothetical heritable trait.
Fitness
Scenario 2:
Scenario 3:
Scenario 4:
sets in scenarios 1–3, the phenotypic covariances are all
positive and we would conclude that selection favours
individuals with greater phenotypic values. The next
two columns in the figure depict alternative ways in
which this phenotypic covariance might arise from
genetic and environmental sources of covariance. In all
these scenarios, the trait under apparent selection has
genetic variance and is heritable (i.e. there is variance
along the x-axis in all plots in the middle column).
However, the scenarios differ with respect to the
presence and sign of genetic covariance between
the trait and the fitness. Scenario 4 in Fig. 1 applies
the same principle of decomposing a phenotype–fitness
relationship into plausible genetic and environmental
components. However, in this final example, the net
phenotypic covariance between a trait and fitness is
zero. Consequently, we might conclude an absence of
selection (and expected change) in this case, despite the
Trait value
fact that trait–fitness relationships do exist at the genetic
and environmental levels.
The consequence of this exercise of plotting hypothetical decompositions of a covariance representing selection may not be immediately apparent from the scatter
plots in Fig. 1 alone. Genes do not interact with environments in ways that influence fitness, only whole
organisms do. However, whereas selection only ‘sees’
phenotypes, the colours in Fig. 1 illustrate how it is that
the relationship between trait and fitness at the genetic
level is what matters for adaptive evolution. Red points
denote those 10% of simulated individuals with the
largest genetic merits for the trait. Blue points correspond to those 10% of individuals with the smallest
genetic merit. Given the first column (and remembering
that this is the only one where selection ‘sees’ traits), we
can see that the pattern of trait–fitness covariation at the
genetic level provides the only correct description of
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what genes will be represented in the next generation.
For instance, in scenario 1, individuals with genes that
promote large phenotypic values (red upward-pointing
triangles) will be increasingly represented in future
generations, given that the scenario illustrated in the
top panels exists. In contrast, there will be no adaptive
evolutionary change in scenario 2, despite apparent
positive directional selection on the heritable trait. In
scenario 3, evolution would actually occur in the
opposite direction to that expected based on the positive
covariance between trait and fitness. That is to say that
in this scenario, individuals with genes promoting
smaller than average phenotypic values (blue downward-pointing triangles) will have increased genetic
representation in the future. Clearly, the representation
of different genotypes in the next generation is a
consequence of the genetically based relationship
between the trait and the fitness.
The patterns in Fig. 1 are intended to show the
qualitative range of genetic and environmental patterns
of covariance that might potentially underlie an observation of apparent directional selection on a heritable
trait. However, an additional axis of quantitative variation likely exists in the genetic variance of fitness. In
general, variation in fitness is expected to be low or
absent in populations at equilibrium (Fisher, 1930), and
empirical results support this, at least in a relative sense,
i.e. the heritability of fitness is generally low (Coltman
et al., 2005; Merilä and Sheldon, 1999, 2000; Mousseau
& Roff, 1987; Price & Schluter, 1991; Teplitsky et al.,
2009). In the extreme equilibrium scenario, where the
additive genetic variance of fitness is zero, it is not
possible for a genetic covariance of a trait with fitness to
exist. This has two important implications for how we
relate observations of phenotypic covariances of traits
with fitness. First, if populations generally harbour little
variance for fitness, genetic covariances of traits with
fitness may often necessarily be smaller than naı̈ve
application of the breeder’s equation might implicitly
require. Specifically, the minimum genetic variance in
relative fitness required to support a given response to
selection, h2S, or in other words the genetic variance
caused by selection on heritable phenotypic variation, is
r2a ðwÞjh2 ; S ¼
h2 S2
;
r2p ðzÞ
ð5Þ
and thus this quantity must be equal to or less than the
genetic variance in relative fitness (Walsh & Blows,
2009).2 This provides a test of the plausibility of any
given prediction of evolutionary change. As the righthand side of eqn 5 can take any nonnegative value, one
2
In the multivariate case the response to selection is
GP)1S. This is the multivariate formulation of the
breeder’s equation, and it is introduced in more detail
below. The variance in relative fitness caused by multivariate selection is (P)1S)TG(P)1S) (Walsh & Blows, 2009).
can calculate its value and determine whether or not it is
consistent with an estimated or plausible value of the
genetic variance of relative fitness. Second, this highlights the important role played not only by variation in
fitness (Orr, 2009), but in particular by the genetic
variance of fitness in adaptive evolution, and so by
extension to potential benefits to bringing its consideration more thoroughly into our approaches for studying
microevolution.
So how does this graphical model relate to the two
equations for predicting change? The breeder’s equation
predicts an increase in a trait’s mean if the trait has a
positive phenotypic covariance with fitness and is genetically variable. As these criteria are both met in scenarios
1–3 of Fig. 1, the prediction would be erroneous in two
cases. Furthermore, in scenario 4, the breeder’s equation
has a zero term (i.e. S ¼ 0) and therefore predicts no
evolutionary change, despite the fact that a genetic
covariance between the trait and the fitness exists. The
breeder’s equation would fail in scenarios 2–4 because in
these situations the environmentally based association
between trait and fitness differs from the relationship at
the genetic level. This is a fundamental violation of the
model’s assumptions (as we discuss further below). In
contrast, the Robertson–Price identity makes predictions
that agree with our graphical model in all cases because it
explicitly addresses the genetic basis of the trait–fitness
relationship. We hope this graphical presentation will
promote the appreciation of the range of true compositions of the relationship between an apparently selected
heritable trait and fitness. We do not intend to suggest,
however, that we are the first to present this phenomenon, rather we aim at presenting a means by which to
make it more accessible. More general mathematical
expressions of the principle are illustrated by Rausher
(1992)’s eqn 5 and its derivation, by Queller (1992)’s
eqn 6, and by the third equation in the appendix to
Hadfield (2008).
Several studies have obtained estimates of the form of
selection on estimated individual-level effects of genes on
phenotype for ecologically important traits. The basic
premise has been to regress fitness on predicted breeding
values from linear mixed models. Predicted breeding
values (estimates of the effect of an individual’s genes;
obtained by a technique called BLUP for Best Linear
Unbiased Prediction) are predictions of the contribution
of genes to an individual’s deviation from a phenotypic
mean, based on phenotypic data of the individual and its
relatives. In plant systems, this kind of approach has
provided evidence that environmentally induced relationships between phenotypic traits and fitness differ
from genetically induced covariances in 25–30% of
studies investigated (Rausher, 1992, Stinchcombe et al.,
2002). This type of approach has also been used several
times in animal systems to test whether the selection is
actually acting on the genotype in situations where
selection on heritable traits has not resulted in demon-
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The genetic analysis of natural selection
strable evolutionary changes as predicted (Kruuk et al.,
2000; 2002). These studies have repeatedly demonstrated
a lack of correlation between predicted breeding values
and fitness for otherwise apparently selected traits.
Recently, this application of breeding values as predicted
from mixed linear models has been criticized on statistical
grounds. This is because, despite the terms ‘best’ and
‘unbiased’ used to describe them, (i) predicted breeding
values have smaller variance than true breeding values,
limiting comparison of selection differentials assessed at
phenotypic and genetic levels, (ii) predicted breeding
values are inevitably predicted with some error, and this
error represents environmental effects on phenotype
(Postma, 2006) and (iii) the very relatedness that makes
the estimation of quantitative genetic parameters possible results in nonindependence of predicted breeding
values and can make the main methods applied to
date for statistical hypothesis testing anti-conservative
(Hadfield et al., 2010). Thus, although predicted breeding
values are more closely correlated with true breeding
values than are phenotypic trait measurements, they
nonetheless remain confounded with environmental
differences among individuals. The statistical problems
that have become apparent mean that this use of
predicted breeding values is no longer considered an
appropriate way to estimate selection on the genotype
(we advocate an alternative below). Nevertheless, on
balance, the inherent problems with the methods probably render it conservative with respect to detecting a
difference in selection at the levels of genotype and
phenotype. Therefore, it should be safe to conclude that a
range of existing empirical studies have at least supported
the strong possibility that environmentally induced
covariation between traits and fitness often drives
patterns of apparent natural selection on the phenotype.
Causation
No model can be expected to provide accurate predictions
if its assumptions are seriously violated. The simple
approaches we discuss here for predicting evolutionary
change actually make many simplifying assumptions
(e.g. constant population demography, discrete generations, constant environmental conditions; Merilä et al.,
2001a), and we are certainly not the first to highlight the
potential for predictions to fail when when these models
are applied to natural systems (Hadfield, 2008; Price
et al., 1988; van Tienderen & de Jong, 1994) or to call for
caution in this regard. However, we would argue that the
most fundamental assumption of the breeder’s equation
– that of causation (Hadfield, 2008; Kruuk et al., 2003;
Pigliucci, 2006) – is very often both poorly recognised
and seriously violated. In its univariate form, the
assumption required to generate valid predictions is one
of a causal relationship between trait and fitness. In other
words, whereas we may estimate natural selection as the
covariance between fitness and a trait of interest, our
5
prediction depends on the trait differences being the
cause of the fitness variation.
We will see that causation with respect to the breeder’s
equation needs to be considered much more closely if we
are to be able to make and interpret predictions of
evolutionary change in natural populations. Relationships between traits and fitness at the genetic level, as in
the first and third scenarios of Fig. 1, could result from
causation. Alternatively, either could arise as a result of
selection acting directly on a genetically correlated trait.
Note that whereas the pattern depicted by even the first
scenario is consistent with the trait being the cause of
fitness variation, it is also consistent with the existence of
noncausal genetic and environmental factors that happen to be similarly affecting the trait and fitness.
Similarly, the trait in the third scenario could have a
causal negative effect on fitness, but may be phenotypically correlated with an unmeasured trait that has a
positive effect on fitness. The critical feature of these
scenarios is that the environmental contributions to the
phenotype–fitness covariance, whatever they may be,
are stronger and have overwhelmed the genetic agents.
Thus, the critical assumption of the breeder’s equation is
in fact not only causation but is sole causation.
A useful alternative, or rather a reformulation, to the
condition in eqn 3 is to state that the breeder’s equation
and the Robertson–Price identity are the same when
rGW P ¼ 0;
ð6Þ
i.e. beyond the effect of the phenotype P on fitness, the
correlation of breeding value G and fitness W is zero
(Queller, 1992; van Tienderen & de Jong, 1994; see also
appendix 1 to Hadfield, 2008 for complementary formulations in terms of the conditions under which the
breeder’s equation is quantitatively predictive in the
presence of unmeasured variables). In other words,
breeder’s equation-based predictions of evolutionary
change in a heritable trait with a causative effect on
fitness can be wrong if other unidentified sources of
covariance between the trait and fitness exist. This is
particularly problematic in natural systems where
unmeasured variables commonly generate the majority
of variation (Moller & Jennions, 2002; Peek et al., 2003).
Failure to explain more than a small proportion of the
variance is not a problem in and of itself. However, if
most of the variance in a trait (or in fitness) is caused by
unknown factors, then it would seem prudent to recognise that these unknown factors may also be contributing
to the observed trait–fitness covariance.
One approach to improving the predictions of the
breeder’s equation is to extend it to multiple traits with
potential causal links to fitness. The multivariate breeder’s equation is
R ¼ GP1 S;
ð7Þ
where R is a vector of responses to selection, S is a vector of
selection differentials and G and P are the genetic
and phenotypic variance–covariance matrices of the
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phenotypic traits. Descriptions and graphical representations of how the multivariate breeder’s equation works are
provided elsewhere (Futuyma, 1998; Walsh & Lynch,
2011) but for current purposes, a simple two-trait example
is useful to illustrate how its application can alleviate the
problem of causation. Consider two traits, only one of
which (trait 1) has a causative effect on fitness. If these two
traits are phenotypically correlated, both will be correlated
with fitness and thus the selection differential associated
with trait 2 will be nonzero. However, if the traits do not
covary genetically, then we do not expect phenotypic
evolution of trait 2 because it has no causal effect on
fitness. Assuming trait 2 is heritable, then the univariate
breeder’s equation would erroneously predict an evolutionary response, but this failure is corrected by application
of the bivariate model. Alternatively, the multivariate
breeder’s equation can be formulated as
R ¼ GP1 S ¼ Gb;
ð8Þ
the Lande equation (Lande, 1976, 1979), where b is a
vector of selection gradients (Lande & Arnold, 1983).
Selection gradients are classically obtained as partial
regression coefficients of fitness on a trait, and as such
account for selection on all other measured and phenotypically covarying traits. Thus, with the multivariate
breeder’s equation, we can relax the assumption that any
one trait is the sole cause of its covariance with fitness.
Conceptually, selection gradients are very important,
because in principle they identify causal effects of traits on
fitness. However, our assumption must now be that we
have identified (and meaningfully measured and appropriately modelled) all the traits and environmental factors
that cause traits of interest to covary with fitness, or in
practical terms, all the important traits and factors. At the
phenotypic level, selection gradients will correctly reflect
causative effects of phenotypic variation of fitness if all
sources of covariance are adequately measured and
modelled. At the genetic level, the post-multiplication of
G by b will correctly predict evolution if all genetically
correlated traits have known selection gradients and
genetic parameters. Let us call this new assumption
‘‘joint-sole’’ causation.
We will never achieve, nor likely approach, sole or
joint-sole causation in descriptions of fitness covariation
in studies of natural populations. Causative agents of
fitness variation and covariation need not even be factors
we would traditionally consider phenotypic traits. For
example, in discussing this problem, Stinchcombe et al.
(2002) used a hypothetical example of variation in the
nitrogen content of soil. If soil nitrogen content independently has positive effects on fitness and the biochemical composition of plant leaves, then they will
appear to be under selection. However, this nitrogeninduced component of the relationship between leaf
composition and fitness will have no evolutionary
consequences. In principle, inclusion of soil nitrogen
measurements in a multiple regression analysis (Lande &
Arnold, 1983) or path analysis (Scheiner et al., 2002)
could alleviate this problem. For analytical purposes,
there is really no necessary distinction between an
environmental variable and an additional phenotypic
trait, provided that the environmental variable can be
assigned to the individual (Wilson & Nussey, 2010).
However, other factors of huge potential to jointly
influence phenotypic traits and fitness include, but are
by no means limited to, soil content of other nutrients,
soil moisture content, light availability and disease. The
possibility that all such factors can be identified, much
less measured meaningfully (as their ecological relevance
probably depends on variation at multiple spatial and
temporal scales), is remote in studies conducted in
natural or semi-natural settings (Kruuk et al., 2003). In
animals, the situation may generally be even more
difficult. The environments experienced by more mobile
organisms are particularly difficult to measure meaningfully. Robertson (1966) described this from the perspective of genetic improvement by example:
[S]uppose a cow eats a piece of wire. Then, assuming that
wire-eating has no genetic basis, the milk yield will be
reduced, though the cow may well be culled because of her
general loss of condition rather than because of the milk
yield itself. Here we have a positive environmental contribution to the [phenotype-fitness] covariance which will
make the phenotypic regression larger than the genetic and
the genetic gain will be less than that predicted [by the
breeder’s equation].
The wire-eating tendencies of cattle may seem trivial,
but it is this triviality multiplied by the preponderance of
such environmental influences on phenotype and fitness
that makes the breeder’s equation a poor predictor of
genetic change. In fact, a myriad of such influences must
act and interact to create the phenotypes and fitness of
most organisms, especially outside of the environmental
controls of agriculture and breeding programs. Most
importantly, though, Robertson (1966) provided the
solution: the expected response to selection is the genetic
covariance between phenotype and relative fitness.
George Price discovered this relation independently
(Price, 1970) and presented the equation in a more
general form (see Gardner, 2008 for an introduction to
the Price equation).
Expectations for genetic covariances
with fitness
Little theory exists to inform our expectations of the
genetic covariances between traits and fitness or to
suggest whether or not genetic and phenotypic covariances should differ systematically. However, by adapting
existing theory on trade-offs among traits, we can make
some predictions about the relationships between traits
and fitness. van Noordwijk & de Jong (1986) provided a
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The genetic analysis of natural selection
Decomposed phenotype
Phenotype
7
Selection
Trait 2
Fitness
Scenario 1:
Fig. 2 A graphical model of the genetic basis
of natural selection on traits involved in
trade-offs. Blue diamonds represent environmental effects on the traits, red crosses
represent breeding values for the traits, and
black circles represent phenotypic values.
Arrows on the plots in the middle column
denote the direction of increasing fitness,
such that grey lines represent fitness
isoclines. Note that as the plots in the left and
centre columns are symmetrical about the
1 : 1 line, the regressions depicted in the
right column apply equally to either trait.
Scenario 2:
general model for visualizing covariances between two
traits that are involved in a life history trade-off. This
model is easily adapted to the quantitative genetic
decomposition of trait values, in a similar way to that
which proved useful in Fig. 1.
Both scenarios in Fig. 2 show hypothetical decompositions of (co)variance for a trade-off between two traits.
In both scenarios presented, the two traits are under
positive selection (i.e. fitness is increased by higher
values of both trait 1 and trait 2). Additionally, traits 1
and 2 covary positively with each other (column 1 of
Figure 2). Note that for energetically costly traits, such as
most ecologically–important characters, it is reasonable
to assume that an individual allocating more to one trait
will have less to allocate to others, but this does not
necessarily lead to a clear expectation that the traits
should show negative covariance at the population level.
In fact, positive phenotypic correlations are common
between (positively selected) life-history traits (Kruuk
et al., 2008) and are expected if individuals vary in the
amount of resource they can acquire (de Jong & van
Noordwijk, 1992; van Noordwijk & de Jong, 1986).
In scenario 1, the genetic and environmental influences
on variation in both allocation and acquisition of
resources are similar, and so the relationships between
the trait and fitness at the phenotypic and genetic levels
are similar (i.e. both traits show positive genetic covariance with fitness, and with each other). However, such a
pattern is unlikely to be maintained for very long in any
population, as selection should erode the genetic variation in fitness. Such a post-selection pattern is depicted in
scenario 2. Here, the positive covariance between the
traits arises from environmental influences (i.e. there is
environmnetally induced variance in resource acquisition). However, the genetic variance along the axis of
fitness that we depicted in scenario 1 has been largely
eroded. As a consequence, there is little genetic variance
in the direction of phenotypic change favoured by
selection, and in fact, there is negative genetic covariance
Trait 1
Trait 1
Trait
between the two traits. When we plot the traits against
fitness, it becomes apparent that there is no genetic
covariance between traits and fitness in the second
scenario. We should note here, too, that the graphical model in Fig. 2, scenario 2 describes a situation
where the genetic variance in fitness is low, and so
the maximum potential for genetic covariances of the
traits with fitness (if any existed at all) is necessarily small.
The extent to which the kind of thinking behind Fig. 2
is applicable to expectations for genetic covariances of
traits with fitness depends on (i) whether traits that we
suspect are under directional selection and are also
involved in trade-offs and (ii) whether the genetic and
environmental sources of covariance between traits
involved in trade-offs vary (in magnitude and sign).
Many traits that interest us are likely to be resourcelimited, whether the resource in question is energy, time,
space or substrates, and thus the potential for widespread
trade-offs exists. Similarly, genetic architectures in
nature are likely to have been affected by past selection.
Thus, consequent to the past fixation of any alleles that
positively affect multiple selected traits, trade-offs are
most likely to be manifested at the genetic level. In other
words, past selection is expected to have generated
genetic constraints that will determine the current and
future evolutionary potential of traits. In fact, our
conjecture that there will be a lack of genetic covariance
between traits and fitness is a corollary of the general
theory that adaptive phenotypic evolution is constrained
by multivariate genetic correlations, as discussed by
Walsh & Blows (2009).
Testing for genetic consequences
of natural selection
Selection is a phenotypic phenomenon, but it only has
evolutionary implications when fitness differences
among individuals relate directly to genetic differences.
Clearly, we must investigate the genetic basis of variation
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JOURNAL COMPILATION ª 2010 EUROPEAN SOCIETY FOR EVOLUTIONARY BIOLOGY
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M. B. MORRISSEY ET AL.
in fitness and the covariation of fitness with traits to
predict the course of adaptation. However, this is more
easily said than done. Quantitative genetic parameters
are generally difficult to measure, and genetic covariances (or correlations) have particularly large sampling
errors. Genetic covariances are especially difficult to
estimate precisely when one or more of the traits
involved has low heritability, which is likely for fitness
(Lynch & Walsh, 1998). Nonetheless, the core assumption of causality in the breeder’s equation is likely to be
seriously violated in studies of natural systems. Consequently, it would be highly desirable to better inform our
expectations for evolutionary change by parameterising
the Robertson–Price identity. We therefore propose the
following framework as an extension of existing common
practice for predicting the response of heritable traits to
selection.
Given its elegant simplicity and the widespread empirical applicability of techniques used to estimate natural
selection on phenoptype (Lande & Arnold, 1983), we
would argue that the breeder’s equation does represent a
reasonable starting point for predicting trait evolution.
However, all applications of the breeder’s equation
should be complemented by discussion of the ability of
the study to infer causality. In a typical selection analysis
involving no experimental manipulation of phenotype,
this ability is, strictly speaking, nil. Sometimes, however,
mechanistic arguments may be useful. For example, if
the primary cause of variation in viability is known to be
predation by a gape-limited predator, arguments that size
or aspects of size are likely to have causal effects on
fitness may be convincing. Obviously, if correlative
analyses can be bolstered by in situ experimental manipulations, then more powerful insights into causal agents
of selection are possible (e.g. Parachnowitsch & Caruso,
2008).
Following the application of the breeder’s equation
and providing that heritable variation for the trait(s) in
question exists, the next step is to estimate the genetic
parameters of fitness (co)variation. At first, it might seem
odd to suggest proceeding based only on evidence for
heritability of the trait in question. What if no selection
on the phenotype has been detected? Whereas we have
primarily focussed our discussion on the conditions that
lead the breeder’s equation to predict evolutionary
change when none will happen, it is useful to recognise
that the same reasoning can cause a prediction of no
change to be erroneous. That is to say an absence of a
phenotypic trait–fitness covariance (i.e. S ¼ 0) does not
have to mean that ra(z,w) ¼ 0 (as depicted in Fig. 1,
scenario 4).
Although imprecise, the genetic variance of fitness and
the genetic covariance of traits with fitness should be
estimated, reported and checked for qualitative agreement
with the predictions of the breeder’s equation. Note that
these estimates are best made using quantitative genetic
techniques to directly infer the genetic basis of the
relationship between trait and fitness, i.e. estimation of
the genetic covariance, rather than regression of fitness on
predicted breeding values, because the latter will be
substantially biased towards environmental patterns
(Hadfield, 2008; 2010; Postma, 2006; and see above). In
studies reporting multiple estimates of selection on multiple traits through multiple components of fitness, it may be
useful to break this step of the framework into two
component steps. If point estimates of genetic variance of
fitness (or in practical terms of different proxies for fitness)
are made first, estimation of genetic covariances of traits
with relative fitness can be limited to those cases where
some indication exists that the genetic variance in fitness
may be nonzero. This two-step procedure reduces the
number of imprecise parameter estimates that have to be
generated and interpreted, while still allowing the application of analyses that are relatively assumption-free with
respect to the prediction of evolutionary change.
Detailed discussion of analytical considerations for
producing quantitative genetic estimates of fitnessrelated parameters is beyond the scope of this paper. In
general, some manifestation of a mixed model (Bolker
et al., 2009; Kruuk, 2004) will be required. In the simplest
but general cases, such models might take the form
y ¼ u þ Za þ e;
ð9Þ
where y is a matrix of observations of fitness and trait
values, u is a vector of means, Z is a design matrix
relating individuals to a, a matrix additive genetic effects,
and e is a matrix of environmental effects. In a bivariate
analysis of a trait and fitness, the variance–covariane
matrix associated with a will provide an estimate of the
genetic covariance between the trait and the fitness. In
this bivariate analytical approach, one cay bypass the use
of predicted breeding values and obtain direct estimates
of the components of the covariance between trait and
fitness. Specically, one can directly test whether
ra(z,w) „ 0. The interested reader should also consult
other resources for information on estimating variance
components in natural populations, including (Bolker
et al., 2009; Garant & Kruuk, 2005; Kruuk, 2004; Lynch
& Walsh, 1998; Wilson et al., 2010).
Additionally, one can move beyond simply obtaining
an estimate of the genetic covariance between the trait
and relative fitness and can examine the genetically
induced relationship relative to the environmentally
induced relationship. We can test a corollary of Hadfield
(2008; and eqn 3) and Queller (1992; and eqn 6) results
using mixed model analysis. We can decompose the
numerator and denominator of eqn 3 into genetic and
environmental components using eqn 4, i.e. rp(z,w) ¼
ra ðz; wÞ þ re(z,w) and r2p ðzÞ ¼ r2a ðzÞ þ r2e ðzÞ, to obtain
ra ðz; wÞ ra ðz; wÞ þ re ðz; wÞ
¼
;
r2a ðzÞ
r2a ðzÞ þ r2e ðzÞ
which with rearrangment yields
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JOURNAL COMPILATION ª 2010 EUROPEAN SOCIETY FOR EVOLUTIONARY BIOLOGY
The genetic analysis of natural selection
ra ðz; wÞ re ðz; wÞ
¼ 2
:
r2a ðzÞ
re ðzÞ
ð10Þ
Each of the terms in this formulation of the conditions
for the breeder’s equation to be predictive can be
estimated directly from the solution of the mixed model
descibed in eqn 9. Thus, in addition to allowing estimation of the Robertson–Price identity, eqn 10 provides a
tool with which to test the null hypothesis that the
assumptions of the breeder’s equation hold. This can be
accomplished by comparing constrained models in a
likelihood ratio test framework. For example, the software A S R E M L (Gilmour et al., 2002) can implement a
flexible range of constraits on covariance matrices. These
analyses could also be conducted in a Bayesian framework, where, for example, one could fit the model
specified by eqn 9 using the R package MCMCglmm
(Hadfield et al., 2010) and test equality of the genetic and
residual regressions of fitness on traits by examination of
the posterior distributions of the covariance matrices
associated with a and e.
Within the context of the framework we propose here
for providing more robust predictions of responses to
selection, one technical point is worth discussing. Transformation of fitness to the relative scale is desirable at the
phenotypic level because then the breeder’s equation
works quantitatively (assuming sole causation of course!).
Analysis of relative fitness may not always be desirable in
genetic analyses. For example, no known distribution
exists for relative fitness, but some standard statistical
distributions will be useful for approximating distributions of fitness or fitness components such as counts of
offspring, i.e. the (overdispersed) Poisson distribution.
Calculation of parameters on different scales (i.e. covariances with absolute rather than relative fitness) will make
only qualitative comparison of trait–fitness relationships
at the phenotypic and genetic levels possible.
The evolutionary parameters associated with the
Robertson–Price identity, i.e. r2a ðwÞ and ra(z,w), are
difficult to estimate and will have substantial uncertainty
associated with them. The components of the breeder’s
equation can be estimated with higher precision, and so
seem statistically more desirable as a means to predict
evolution (although it is notable that while standard
errors are typically estimated and presented for h2 and S,
the uncertainty around the prediction is more rarely
quantified). However, it is necessary to consider both the
uncertainty that can be quantified, for example by
estimating standard errors, and the uncertainties that
can only be qualified, which are broken assumptions. The
difficulty of the very strong assumption of sole causation
in the application of the breeder’s equation in natural
systems means that the total amount of uncertainty in
breeder’s equation-based estimates of responses to selection is very large. Thus, especially because both r2a ðwÞ and
ra(z,w) can be estimated in any study where estimates of
selection and heritability come from the same data set,
9
little additional effort will often be necessary to evaluate
these fundamental evolutionary parameters.
We hope it is clear at this point that the Robertson–
Price covariance should provide robust predictions of
evolutionary change. However, in arguing these benefits,
it must also be made clear that we should not expect this
approach to predicting evolutionary trajectories to solve
all of the potential problems that we can encounter. For
example, the predictions of both the breeder’s equation
and the Robertson–Price covariance are for the change in
mean breeding value over one generation. This does not
equate to the change in mean phenotype if the environment changes (Dickerson, 1955; Merilä et al., 2001a;
Pemberton, 2010). Similarly, temporal and spatial variation in selection and the expression of genetic variation
(Wilson et al., 2006), nonlinear parent–offspring phenotypic relationships (Heywood, 2005), unmeasured
changes in the age-, stage- or state-structure of populations (Ozgul et al., 2009) and nonrandom immigration or
emigration (Garant et al., 2005) can all contribute to
phenotypic change (or stasis). Thus, for predicting the
evolutionary trajectories of natural populations, continued attention to the ecological theatre in which change is
occurring will often be at least as important as estimating
the quantitative genetic parameters (Pemberton, 2010).
Distinguishing between studying
selection and predicting evolution
The breeder’s equation provides a useful framework for
conceptualizing the process of adaptive evolution by
natural selection: selection causes phenotypic changes
in a population, and genetic variation transmits these
changes to future generations. This is not wrong, but
given the assumptions we have discussed, it may
generally be very inappropriate to apply this framework
as a predictive tool in nature. There is consequently a
problem with the enthusiastic way in which evolutionary biologists, ourselves included, have transferred this
model from being a tool to conceptualise trait evolution
and analyse artificial selection experiments (a context in
which the inherent assumptions can be more readily
approximated) to a predictive model in the field.
Estimates of selection based on selection gradients are
undoubtedly a conceptual step forward, but they do not
escape reliance on the assumption that all traits and
environmental factors that jointly influence studied
traits and fitness have been identified, meaningfully
measured and adequately modelled. We simply should
not expect that estimates of phenotypic selection
differentials, multiplied by values of estimates of heritability or G (whether estimated in situ, extrapolated
from other studies, or simply assumed), will yield
generally accurate predictions of phenotypic change in
natural populations. Multivariate extensions acknowledging the importance of phenotypic and genetic
correlations among traits are entirely sufficient when
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10
M . B . M O R R I S S E Y E T A L.
we can assume that all relevant factors have been
properly handled; however, this assumption is naı̈ve in
the analysis of most data from largely unmanipulated
populations.
The two equations for predicting evolutionary change
are surprisingly different, despite their common left-hand
sides. The use of these two equations to help us conceptualise the process of microevolution should occur at
different levels. When our interest is in natural selection
and the ecological factors that cause individuals with some
traits or sets of traits to vary in their fitness, the breeder’s
equation and associated methodologies for quantifying
selection, especially in their multivariate forms, provides a
coherent framework. With the breeder’s equation, we can
begin to piece together how finite sets of traits, potentially
correlated with one another and potentially varying in the
degrees to which they cause fitness variation, might
evolve. Clearly, understanding what traits influence
fitness and how their evolution affects other aspects of
phenotype is complementary and very closely related to
the business of predicting evolutionary change. However,
the direct cause of microevolutionary change is a relationship between fitness and genetic variation, regardless of
how that relationship arises. The most robust prediction of
evolutionary change will therefore be obtained by applying theory and analysis, i.e. the Robertson–Price equation,
that matches this causal level.
Summary
1. The breeder’s equation is not necessarily consistent
with the Robertson–Price identity. The two equations
provide identical predictions of evolutionary change
under the assumption that the focal trait is the sole
cause of covariance between fitness and phenotype.
The Robertson–Price identity is not hindered by an
assumption of causation at the level of phenotypic
expression and therefore provides an appealing framework for conceptualising and predicting adaptive
phenotypic microevolution.
2. Because of the assumption of sole causality, even the
predictions of evolutionary responses of heritable
traits that do have causative effects on fitness can be
misleading when estimated with the breeder’s equation. We will generally only ever identify a minority of
the causative environmental factors that can jointly
influence traits and fitness when studying wild populations. Therefore, unexplained environmental patterns of trait–fitness covariance may have more
influence on phenotypically detectable patterns of
selection than do genes influencing traits we study in
natural systems.
3. We propose that estimates of evolutionary parameters
beyond those defined by the breeder’s equation
should be calculated, reported and interpreted whenever possible. The the additive genetic variance in a
trait and the additive genetic variance in fitness jointly
define the maximum magnitude of the per-generation
response to selection. Therefore, point estimates of the
genetic variance in fitness will help to determine
whether or not the magnitudes of predicted responses
to selection are within reasonable ranges. Ultimately,
the response to selection is defined by the genetic
covariance between a trait and relative fitness. Thus
this quantity should be calculated too, to ensure at
least qualitative agreement with predictions of the
breeder’s equation.
Acknowledgments
We are grateful to Bruce Walsh, Josephine Pemberton
and Jarrod Hadfield for discussions and many useful
comments. MBM is supported by an NSERC postdoctoral
fellowship, AJW is supported by a BBSRC David Phillips
fellowship, and LEBK is supported by a Royal Society
University Research Fellowhip.
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