Download The evolution of trade‐offs: where are we?

doi:10.1111/j.1420-9101.2006.01255.x
MINI REVIEW
The evolution of trade-offs: where are we?
D. A. ROFF & D. J. FAIRBAIRN
Department of Biology, University of California, Riverside, CA, USA
Keywords:
Abstract
acquisition;
allocation;
quantitative genetics;
trade-offs;
Y-model.
Trade-offs are a core component of many evolutionary models, particularly
those dealing with the evolution of life histories. In the present paper, we
identify four topics of key importance for studies of the evolutionary biology of
trade-offs. First, we consider the underlying concept of ‘constraint’. We
conclude that this term is typically used too vaguely and suggest that
‘constraint’ in the sense of a bias should be clearly distinguished from
‘constraint’ in the sense of proscribed combinations of traits or evolutionary
trajectories. Secondly, we address the utility of the acquisition–allocation
model (the ‘Y-model’). We find that, whereas this model and its derivatives
have provided new insights, a misunderstanding of the pivotal equation has
led to incorrect predictions and faulty tests. Thirdly, we ask how trade-offs are
expected to evolve under directional selection. A quantitative genetic model
predicts that, under weak or short-term selection, the intercept will change
but the slope will remain constant. Two empirical tests support this prediction
but these are based on comparisons of geographic populations: more direct
tests will come from artificial selection experiments. Finally, we discuss what
maintains variation in trade-offs noting that at present little attention has been
given to this question. We distinguish between phenotypic and genetic
variation and suggest that the latter is most in need of explanation. We suggest
that four factors deserving investigation are mutation-selection balance,
antagonistic pleiotropy, correlational selection and spatio-temporal variation,
but as in the other areas of research on trade-offs, empirical generalizations are
impeded by lack of data. Although this lack is discouraging, we suggest that it
provides a rich ground for further study and the integration of many
disciplines, including the emerging field of genomics.
Introduction
Evolutionary biological thought is firmly grounded upon
the assumption that trait evolution is restricted or biased
by fitness trade-offs (Stephens & Krebs, 1986; Charnov,
1989; Roff, 1992, 2002; Stearns, 1992; Futuyma, 1998;
Houston & McNamara, 1999; Reznick et al., 2000). From
the perspective of life history theory, a trade-off occurs
when an increase in fitness due to a change in one trait is
opposed by a decrease in fitness due to a concomitant
change in the second trait. The term ‘trade-off’ may be
used to describe the functional relationship between two
traits or the statistical correlation between the traits,
Correspondence: Derek A. Roff, Department of Biology, University of
California, 900 University Avenue, Riverside, CA 92507, USA.
Tel.: 951 827 2437; fax: 951 827 4285; e-mail: [email protected]
although in the latter case a functional relationship is
assumed to underlie the statistical relationship. In the
absence of other factors impinging upon the trade-off, it
can be operationally measured by the statistical relationship between the two traits, i.e. the simple bivariate
correlation. However, in the presence of interacting
factors, the correspondence between the bivariate correlation and the underlying functional trade-off may break
down. Lack of a significant bivariate correlation between
the two traits is therefore insufficient to demonstrate the
absence of a trade-off, although it still measures to some
degree the extent to which the two traits can vary
independently. Because of the importance of the bivariate correlation in operationally defining trade-offs and its
frequent use in the literature, we shall use the term
‘trade-off’ in both its functional and statistical meanings,
indicating where necessary when the discussion applies
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D. A. ROFF AND D. J. FAIRBAIRN
primarily to a single aspect. To help distinguish the two
usages, we use the subscripted symbols X1 and X2, when
a functional trade-off is assumed, whereas when the
trade-off is measured by a regression analysis we use Y
and X, although the distinction clearly becomes blurred
at times.
Trade-offs between life history traits such as between
fecundity and survival, measured as either functional or
statistical relationships (more frequently the latter) have
been demonstrated in a large number of studies and
numerous taxa in laboratory, semi-natural and natural
populations (e.g. Reznick, 1985; Partridge & Sibley, 1991;
Roff, 1992, 2002; Stearns, 1992; Gustafsson et al., 1994;
Ots & Horak, 1996; Sinervo & DeNardo, 1996; Zuk,
1996). While neither the existence of fitness trade-offs
nor their central place in shaping evolutionary trajectories is in doubt, there is still little understanding, from
either a theoretical or empirical perspective, of how such
trade-offs evolve (Houle, 1991; Chippendale et al., 1996;
Fry, 1996; Reznick et al., 2000; Roff & DeRose, 2001;
Roff, 2002). In this brief review we discuss four main
topics that we believe are important for an understanding
of how trade-offs evolve. These topics encompass major
issues that remain unresolved or misunderstood in the
current literature and are discussed under the following
headings:
1 Problems with the concept of constraint.
2 Insights from models of resource acquisition–allocation.
3 The effect of directional selection on the statistical
description of trade-offs.
4 What maintains variation in the trade-off?
Problems with the concept of constraint
Problems of definition
In referring to the effect of negative genetic correlations
on evolutionary change, the term ‘constraint’ is used in
two senses: first, it is used in the sense of impeding, but
not stopping, evolution in particular directions, and
second, it is used to mean that there are evolutionary
trajectories that are unavailable to selection, termed
‘evolutionarily forbidden trajectories’ by Kirkpatrick &
Lofsvold (1992) and ‘absolute evolutionary constraints’
by Mezey & Houle (2005). Unfortunately, it is frequently
not clear in which sense ‘constraint’ is being used. This is
not a trivial source of confusion, because under the
former meaning all character states are possible, whereas
under the latter meaning some states are proscribed. We
agree with Perrin & Travis (1992) that it is perfectly
reasonable to use the term ‘constraint’ in a mathematical
sense, as for example, in the verbal statement of
X1 + X2 < Z, meaning ‘X1 plus X2 is constrained to be
less than Z’. However, because of frequent ambiguity in
its meaning, we are also sympathetic to the call by Van
Tienderen & Antonovics (1994) for a moratorium on its
use. The ambiguity in the literature has certainly not
lessened since this call in 1994 and so, because of the
potential confusion in the meaning of ‘constraint’, we
advocate not using the term unless it is precisely defined
in the given context.
Quantitative genetics and absolute constraints
Trade-offs are specified in quantitative genetics by a
negative genetic covariance between traits, a covariance
that could be caused by antagonistic pleiotropy or linkage
disequilibrium. In both cases a causal connection is
inferred. This specification is in terms of the bivariate
relationship and hence does not take into account the
effect of interactions with other variables. However, the
sum total of all interactions with other traits is taken into
account by use of the entire genetic and phenotypic
variance–covariance matrix to predict response to selection. Given these matrices it is possible to define precisely
the circumstances under which some evolutionary trajectories are not permitted. An important conclusion of
such analyses, which we describe below, is that when
trade-offs involve more than two traits, some evolutionary trajectories may be proscribed even if all of the
bivariate correlations are greater than )1. Thus, absolute
genetic constraints may exist, at least in the short term, in
spite of imperfect genetic correlations.
Variance–covariance matrices are symmetric and
hence can be reduced, using principal components
analysis, to a set of orthogonal axes designated by the
eigenvectors. Each axis is made up of a linear combination of the individual traits (the principal component
scores) and there are as many axes as there are traits. The
variance in each principal component is given by the
eigenvalue. If an eigenvalue is zero there is no genetic
variance in the respective direction and hence evolution
cannot proceed in that direction. To illustrate this we
consider two situations, one in which there are two traits
and one in which there are three traits.
Consider the trade-off illustrated in Fig. 1: from a
quantitative genetic perspective the trade-off is described
by a bivariate normal distribution (as noted in the
Introduction, the symbols X and Y are here used to
designate that the traits are being considered within a
statistical framework). Under this model, provided there
is variation orthogonal to the axis describing the tradeoff, which means that the genetic correlation between
the two traits is greater than )1, all combinations of
values are possible, although the frequency of combinations obviously varies greatly. Thus, under this model
selection can, in principle, push a population in any
direction and the trait combinations are only constrained
in the sense that for a given selection differential the
response will be greater in some directions than others.
Long-term evolution to any combination is possible. The
only case in which evolution is constrained in the sense
that some combinations cannot evolve, is that in which
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Trait Y
Trait X
Trait X
P(
X,
Y)
Trait Y
Fig. 1 A quantitative genetic view of a
trade-off between traits X and Y shown from
two orientations. Each plot shows the
bivariate normal probability distribution of
breeding values for each trait (where P(X,Y)
denotes the probability of the XY combination), with the left plot showing it in ‘contour’ perspective and the right plot showing
it in ‘3D’ mode. The solid lines in the left plot
show the major and minor axes.
435
P(X,Y)
Evolution of trade-offs
the genetic correlation between traits is exactly )1. This
condition is also specified, as noted above, by one of the
eigenvalues of the matrix being zero. To obtain a visual
understanding of this condition consider what happens if
we rotate the axes such that they now fall along the
major and minor axes of the bivariate normal distribution (Fig. 1): the equations specifying this rotation are
given by the eigenvectors of the matrix. We now have
two uncorrelated traits made up from a linear combination of the original two correlated traits, with the
eigenvalues being the genetic variances of these two
synthesized traits (Kirkpatrick & Lofsvold, 1992). When
r ¼ )1 there is no variation in the direction of the minor
axis (the second eigenvalue) and thus selection will be
totally ineffective in producing a change in this direction.
Note, however, that there can still be genetic variation
for both of the original traits.
When the genetic correlation is greater than )1, the
above scenario presents a logical conundrum, because if
both traits covary positively with fitness there is apparently nothing stopping the population moving off to ever
increasing values of each trait. We postulate that this
does not happen because the variation about the line
does not truly reflect a bivariate normal distribution but
is an approximation built up from the interaction of
multiple traits, which together do prevent evolution in
particular directions, or at least so biases it that particular
directions are highly unlikely. In other words, the
statistical description of the trade-off is a consequence
of a functional trade-off in multivariate space being
projected onto a two-dimensional surface. To illustrate
this, consider the situation in which three traits are
functionally constrained to lie upon a plane as shown in
Fig. 2. It is immediately obvious that selection cannot
drive the population to any combination of trait values
that lie off the plane, although there can be genetic
variation for all three traits. However, if we project the
observed trait values onto the X-Y plane (i.e. rotate the
axes such that the Z-axis is perpendicular to the surface
of the page; Fig. 2), we observe a scatter of points with an
overall trade-off indicated by a statistically significant
negative correlation between traits X and Y that is greater
than )1. From this we could, incorrectly conclude that
evolution in any direction is possible. In fact, depending
on the distribution of points, the projection of points onto
the X-Y plane could produce a zero correlation, and thus
it could appear that there was no trade-off between traits
X and Y. This is what Pease & Bull (1988) referred to as
‘the problem of dimensionality’. A bivariate genetic
correlation less than )1 is clearly insufficient evidence
for the conclusion that all evolutionary trajectories are
possible. In general, statistical representations of bivariate
trade-offs permit, at best, only weak inferences about
how constrained, in the sense of being biased, evolutionary trajectories are likely to be. While it is probably
true that, in most cases, selection in the direction of the
largest eigenvalue will be the fastest, failure to include
other traits could still lead to misleading predictions.
Even if there are no eigenvalues that are exactly zero,
movement along a particular evolutionary trajectory may
be very slow if the eigenvalue in that direction is very
small relative to the other eigenvalues (Blows &
Hoffmann, 2005). If we wish to make statements about
the importance of particular trade-offs in modulating and
directing evolutionary change, it is necessary to know
how this trade-off is integrated with other traits, and thus
the extent to which variation observed on the X-Y plane
actually represents variation that is in actuality more
restrictive than implied by the simple bivariate statistical
relationship. While this is possible, in principle, by
measuring the variances and covariances of a wide suite
of traits, such an approach is time consuming, potentially
costly and not guaranteed to include the requisite suite of
traits. We suggest that a better approach is to combine a
quantitative genetic analysis with a phenotypic analysis
that focuses on the underlying functional relationships,
paying particular attention to the possible influence of
unmeasured variables (path analysis may be of considerable use in this). Charlesworth’s (1990) analysis of a
hypothetical life history with functional constraints
illustrates this approach, as does the theoretical and
empirical analyses of the evolution of growth trajectories
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D. A. ROFF AND D. J. FAIRBAIRN
1.0
1.0
0.5
Trait Y
–0.5
–1.0
0.0
0.2
–0.5
0.2
0.4
Tra
it
0.0
0.4
0.6
X
Trait Z
Trait Y
0.5
0.0
0.6
0.8
1.0
0.8
1.0
it
Tra
Z
–1.0
0.2
0.4
0.6
0.8
1.0
Trait X
Fig. 2 An illustration of how a trivariate trade-off may appear as a bivariate trade-off with variation about the trade-off line when one trait is
omitted from the analysis. The actual three trait functional trade-off is shown on the left (Y ¼ 1 ) X ) Z, or, because this is the functional form,
X2 ¼ 1 ) X1 ) X3). The figure on the right shows the plot of X and Y when Z is omitted: now the distribution appears as a bivariate distribution
with a correlation greater than )1 and hence in this projection there appears to be no absolute constraint on the direction of evolution.
Combinations of X and Y that can be achieved are contingent on the fitness of the three-fold combination of X, Y and Z which is limited by the
functional trade-off.
by Kirkpatrick et al. (1990) and Kirkpatrick & Lofsvold
(1992). The latter analyses and that of wing shape in
Drosophila melanogaster (Mezey & Houle, 2005) further
illustrate the statistical problem of demonstrating that
any given eigenvalue is exactly zero: the best that we can
achieve is the statement that a particular eigenvalue is no
greater than some positive value. On the other hand,
provided the confidence limits are reasonably small, it
should be possible to delineate likely from unlikely
evolutionary trajectories. An alternative approach for
delineating the limitations imposed by functional tradeoffs is experimental manipulation of the component
traits (reviewed in Roff, 2002, pp. 132–142, and for an
excellent discussion on allometric engineering to demonstrate trade-offs see Sinervo et al., 1992) and this can
profitably be combined with predictions made from more
classic quantitative genetic analyses.
Insights from models of resource
acquisition–allocation
The Y-model
The concept that trade-offs between fitness-related traits
are fundamental in shaping both evolutionary trajectories and equilibrium trait values has proved to be a
powerful heuristic tool, with strong theoretical and
empirical support (e.g. Roff, 1992, 2002; Stearns, 1992;
Reznick et al., 2000). Nevertheless, predicted trade-offs,
generally measured by their statistical correlation, are not
always found. In fact, positive phenotypic correlations
between traits predicted to be involved in fitness tradeoffs (and hence expected to be negatively correlated) are
uncovered not infrequently in laboratory and natural
populations (Reznick et al., 2000; Roff, 2002). One of the
most influential models explaining how such positive
correlations can arise in the presence of trade-offs is the
acquisition–allocation model of Van Noordwijk & de Jong
(1986), which was first formulated by James (1974) (his
analysis contains an arithmetic error, which, unfortunately, is reproduced in Roff, 1992 and Roff, 2002) and
also derived by Riska (1986). James called the model the
‘partitioning of resources model’, whereas Riska called it
a ‘variable parts model’ and De Jong & van Noordwijk
(1992) referred to it as the ‘Y-model’, which, for
simplicity, is the name we use in this paper. This model
posits that within an individual two traits (X1,X2) are
determined by the allocation of resources from a common pool, T: T ¼ X1 + X2 (note that we here use the
symbols X1 and X2 rather than X and Y, because we are
referring to the functional form, rather than the statistical
manifestation, of the trade-off). Given this mathematical
relationship we can state that the value of trait X2 is
restricted in its possible values by the allocation of some
proportion, P, of the acquired resource T to X1, leaving
(1 ) P)T to be allocated to X2. Using this model, and
assuming that variation exists among individuals, we can
scale up to the level of the population and calculate the
covariance between the two traits X1 and X2, denoted as
rX1X2, as:
rX1 X2 ¼ r2T lP ð1 lP Þ r2P l2T r2P ;
ð1Þ
where l designates mean values and r2 designates
variances. For a fixed acquisition (r2T ¼ 0) the covariance between X1 and X2 is negative and the correlation is
)1, indicating the functional trade-off. However, when
there is variability in acquisition it is possible for the
covariance between X1and X2 to be positive, giving the
false impression, as measured by the statistical relationship between X1 and X2, that there is no trade-off. This
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Evolution of trade-offs
σX
1X 2
>
0
μ P (1 − μ P )
σ2
P
σ
X 1X 2 < 0
2
μT
Fig. 3 An illustration of the dependence of the covariance between
traits X1 and X2 on both the mean values and variances of
allocation and acquisition. The solid lines indicate the combinations
of lP(1 ) lP) and l2T at which the covariance (rX1X2) is zero
for fixed values of the variances in acquisition and allocation
(setting rX1X2 equal to zero in eqn 1 and rearranging gives
lP ð1 lP Þ ¼ r2P ½ðl2T =r2T Þ þ 1. Combinations above each line give
positive covariances and combinations below the line give negative
covariances. Thus a change in either the mean acquisition or
allocation can lead to a change in the sign of the covariance. An
increase in the variance in acquisition (r2T ) shifts the zero isocline
line downwards (dotted line), whereas an increase in the variance of
allocation shifts the zero isocline upwards (dashed line).
result applies equally to phenotypic and genetic covariances (Houle, 1991; De Jong & van Noordwijk, 1992).
The basic Y-model has provided a valuable insight into
the pitfalls of trying to deduce evolutionarily important
functional trade-offs from patterns of variation in
contemporary populations but, unfortunately, it has
been interpreted in an oversimplified manner in much
of the literature. Numerous authors have inferred from
this model that the sign of the correlation between the
two traits depends only upon the relative variances in
acquisition and allocation (e.g. Glazier, 1999; Christians,
2000; Reznick et al., 2000; Jordan & Snell, 2002; Brown,
2003; Ernande et al., 2004). This is not mathematically
correct and the misunderstanding appears to come from
interpreting the range in the two traits given in Fig. 1 of
Van Noordwijk & de Jong (1986) as the allocation
variance rather than P. It is clear from eqn 1 and as
illustrated schematically in Fig. 3, that the sign of the
covariance depends on both the mean values and
variances of acquisition and allocation.
An alternate statement of the covariance in the Y
model that emphasizes the importance of the coefficients
of variation is:
1 lP
rX1 X2 / CV2T
CV2P CV2P ;
ð2Þ
lP
where CVT is the coefficient of variation in acquisition
and CVP is the coefficient of variation in allocation. Thus
the sign of the covariance depends upon the relative sizes
of the coefficients of variation rather than the variances.
This dependency is not simple and a mere comparison
of the relative sizes of the CVs is insufficient to determine
the sign of the covariance. We illustrate this with data
on the covariance between ovary mass and somatic mass
in the sand cricket, Gryllus firmus (Table 1). The relative
size of the covariances bears no relationship to the
observed correlation: for example, for short-winged
females aged 1–3 days the CV for acquisition is considerably less than that for allocation (19.83 vs. 121.61
respectively) but the correlation is positive. Eqn 2
correctly predicts the signs of the correlations. These
Table 1 Illustrative analysis of the trade-off between allocation to ovaries (O) vs. soma (S) in the wing dimorphic sand cricket, Gryllus firmus.
Acquisition is approximated by the total body mass, T, and P is the proportion allocated to the ovaries. For further details see Crnokrak & Roff
(2002).
Age span (days)
CVT
S
S
L
L
1–3
5–7
1–3
5–7
19.83
121.62
0.3706
17.27
16.56
0.0563
13.95
99.85
0.1016
13.33
68.90
)0.3723
Mean values, variances and covariances
T
S
S
L
L
1–3
5–7
1–3
5–7
0.7756
1.1698
0.9484
1.2132
CVP
Eqn 2*
Wing morph
VT
0.0237
0.0408
0.0175
0.0261
P
0.0202
0.2608
0.0171
0.1380
*Covariances predicted from eqns 2 and 3.
Phenotypic correlation between somatic mass and ovary mass.
àSample size.
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Eqn 3*
r nà
P
0.0005
0.0061
0.0001
)0.0035
0.1419
0.5238
0.0508
)0.2104
42
37
39
32
0.3701
0.0009
0.7588
0.2478
VP
CovOS
0.00060
0.00186
0.00029
0.00904
0.00049
0.00604
0.000118
)0.00348
438
D. A. ROFF AND D. J. FAIRBAIRN
results illustrate that the effects of variation in acquisition
and allocation on the expression of bivariate trade-offs
are complex and little can be inferred from simple ‘rules
of thumb’ for comparing variances or CVs.
An unrelated but perhaps more difficult problem with
testing predictions of the Y-model is the problem of
operationally defining and measuring ‘acquisition’. This
is not a simple task: proposed measures are size at some
given age (Biere, 1995; Mitchell-Olds, 1996; Glazier,
1998, 1999; Dudycha & Lynch, 2005), particular body
stores such as lipids (Chippindale et al., 1998) and the
rate of food acquisition (Ernande et al., 2004). Whether
these are appropriate will depend upon the organism
under study and the question addressed. Body mass may
be appropriate for a capital breeder (an organism that
meets its reproductive effort from stored reserves) but
possibly not for an income breeder (an organism that
meets its reproductive effort from energy gathered during
the course of reproduction). Even in the former case
body mass does not take into account variation in energy
content of the component tissues or the energy expended
in developing and maintaining them. Feeding rate
supposes that rate of utilization does not change, but
selection experiments on response to crowding in
D. melanogaster suggest that fast feeding larvae are less
efficient at utilizing the food (Foley & Luckinbill, 2001;
Prasad & Joshi, 2003; Mueller et al., 2005), which would
generally make feeding rate a poor index of resource
acquisition.
Expanding the scope of the Y-model
It is possible to recast the Y-model in a form that is closer
to the spirit of the statement that the sign of the
correlation between the two traits depends upon the
relative variances in acquisition and allocation. The
variance in a sum of two variables is equal to the sum of
the two variances plus twice the covariance, which can be
rearranged to give the covariance of the two variables as:
h
i
rX1 X2 ¼ 12 r2T r2X1 þ r2X2 :
ð3Þ
From this it can be seen that a negative covariance
between any two traits in an allocation-based trade-off
will occur if the sum of the variances in the allocated
components exceeds the variance in acquisition. Eqn 3
correctly predicts the signs of the correlations between
somatic mass and ovary mass in the sand cricket
(Table 1).
The above equation is a more general statement and
can be applied in those cases in which the distribution of
resources is dichotomous but does not fit the assumptions
of the Y-model. In general, eqn 3 will be the appropriate
statistical model when the components of the trade-off
are measured in different units. An example of this is the
hypothesized functional trade-off between egg size and
number:
C ¼ EN;
ð4Þ
where C is clutch mass, E is individual egg mass and N is
the number of eggs. This can be made into a simple
arithmetic model by taking log, giving:
ln C ¼ ln E þ ln N:
ð5Þ
By converting to a log scale the allocation between ln(egg
mass) and ln(egg number) can be mathematically equated to the Y-model, by writing the proportional contribution, B, to ln(egg mass) as B ¼ ln E/ ln C, the
allocation
model
thus
being
ln C ¼
B ln C + (1 ) B) ln C (Christians, 2000; also adopted by
Brown, 2003). For the reasons advanced above, the
analysis of Van Noordwijk & de Jong (1986) cannot be
used to justify the claim by Christians (2000) that given a
trade-off between egg size and number, there will be a
negative correlation between egg mass and clutch size
when the variation in ln B is large relative to variation in
ln C. A much simpler approach is available by noting, as
above, that:
r2C ¼ r2E þ r2N þ 2rEN ;
r2C
ð6Þ
r2E
is the variance in ln C,
is the variance in
where,
ln E, r2N is the variance in ln N and rEN is the covariance
between ln E and ln N. Rearranging the above gives:
rEN ¼ 12 r2C r2E þ r2N
ð7Þ
from which we can derive the statement that a negative
correlation will occur when the sum of the variances in
ln E and ln N exceed the variance in ln C. Whereas ln C
can be regarded as acquisition, there is no reasonable
index of allocation, nor is it useful in this case to assign
an index of allocation. Unfortunately, the data presented
in Christians (2000) and Brown (2003) are insufficient to
use eqn 7 to determine if the conclusions reached in
these two studies are valid.
A second example of a bivariate trade-off that does not
fit simply into the Y-model is that between development
time and adult body size (Berven, 1987; Roff, 2000).
Because increases in development time increase generation time and decrease fitness, the trade-off in this case
appears as a positive relationship between body size and
development time. However, it can be recast into a
negative relationship by using the reciprocal of development time. Now, for the purpose of illustration, suppose
that all individuals follow the same linear growth
trajectory and initial size can be ignored. Adult size, Y,
is then proportional to development time, X: Y ¼ RX,
where R is the rate of growth, which is equivalent to the
rate of acquisition of energy. Taking log and rearranging
gives ln R ¼ ln Y ) ln X ¼ ln Y + ln X)1. This model can
now be examined in the same manner as for the egg sizenumber model. The immediate result is that if there is
variation in growth rate (i.e. variation in R) then the
trade-off between body size and development time, as
measured by the correlation, can be obliterated, which
has been observed in some studies (Roff, 2000).
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Evolution of trade-offs
The hierarchical Y-model
The allocation of resources may be made at a hierarchy of
levels, with a binary split at each node. Even if the
allocation is tripartite the allocation can be described
mathematically by a pathway with two splits, the first
being between one of the traits and the other two. The
several numerical and analytical examinations of the
hierarchical model have found, perhaps not surprisingly,
that correlations at the end of the hierarchy depend
critically upon allocation patterns at the beginning of the
hierarchy (De Laguerie et al., 1991; De Jong, 1993;
Worley et al., 2003; Bjorklund, 2004).
Age-specific fecundity provides a good example of a
hierarchical pattern of allocation in which a proportion
Px of the total fecundity is realized at age x leaving
(1 ) Px) to be allocated over the remaining ages. Tanaka
(1996) analysed age-specific fecundity in the bruchid
beetle Callosobruchus chinensis using this approach. In
Tanaka’s model, fecundity at age x, mx, is given by:
mx ¼ Ms esx ;
ð8Þ
where Tanaka defined M as resource acquisition and s as
allocation. While we agree with the definition of M, we
believe that s=ðs þ 1Þ is a better interpretation of allocation. In the present model, at any given age, the amount
remaining for future allocation
is M e)sx (i.e. the
R1
sx
solution to the integral x Ms e dx): thus the partitioning at age x is Ms e)sx and M e)sx, giving a total
fecundity of M(s e)sx + e)sx), and the proportion allocated at age x is thus s esx =ðs esx þ esx Þ ¼ s=ðs þ 1Þ. The
fecundity function considered by Tanaka was unusual in
its shape, more typical shapes being uniform or, most
generally, triangular. This unusual fecundity function is
responsible for the unexpected result that allocation was
predicted to be independent of age. More generally, we
would expect allocation to change with age. For example,
if fecundity is a uniform function such that mx ¼ c, where
c is a constant, then the total fecundity is cx where x is
the reproductive lifespan and the proportion allocated at
age x is c/[c + c(x ) x)] ¼ 1/(1 + x ) x), which means
that the proportional allocation increases with age.
The Y-model as an explanation for the sign of
observed trade-offs
Glazier (1999) argued that the variance in resource
acquisition in natural populations would, in general, be
greater than in captive or domestic populations and
hence trade-offs would be more often observed in the
laboratory than the field. Because, as discussed above,
the conditions under which a trade-off will be observed
are defined by the means as well as the variances, this
hypothesis implicitly assumes no differences in the
means between lab and field, or that the coefficients of
variation change in parallel with the variances, which
need not be the case.
439
A further critical assumption of this hypothesis is that
the genetic component of the variance in resource
acquisition is small. For his analysis Glazier (1999)
selected the trade-off between a measure of reproductive
investment (clutch size, number of clutches per unit
time, clutch mass, or fat content of ovaries or clutch) and
a measure of somatic investment (various measures of
body fat content or size-corrected maternal body mass).
At least some of the indexes of acquisition, such as body
fat content, are likely to be highly correlated to total body
size for which there is an abundant evidence of considerable genetic variance under laboratory conditions
(Mousseau & Roff, 1987; for plants see Geber & Griffen,
2003). Coefficients of genetic variation for morphological
traits are typically about 6% but can exceed 30% (see
Fig. 1 of Houle, 1992; Messina, 1993; Imasheva et al.,
2000; Hermida et al., 2002; Loh & Bitner-Mathé, 2005)
and demonstrate that this source of variation cannot be
ignored. Given this, it would come as no surprise not to
find trade-offs being expressed under laboratory conditions, as illustrated by the sand cricket data (Table 1).
Indeed, Tuomi et al. (1983) argued the opposite of
Glazier, namely that field conditions will be harsh,
resource acquisition less variable in the field and tradeoffs more likely to be observed in the field than in the
laboratory.
Glazier (1999) reported a significantly higher proportion of negative correlations in laboratory studies than
field studies (52% vs. 24% respectively), concluding that
the results supported his hypothesis. Unfortunately, this
result is suspect for two reasons: first there was a
taxonomic sampling bias: mammals and birds contributed the majority (85%) of the field data, whereas
crustaceans and fish contributed the majority (75%) of
the laboratory data. Secondly, Glazier did not correct for
multiple estimates from each study, the average being
1.74 estimates per study, which could lead to pseudoreplication and inflation of the degrees of freedom.
Nevertheless, the results are certainly suggestive.
An alternative method of exploring the Y-model is to
restrict acquisition experimentally by limiting intake
(Glazier, 1999). Unless the restriction is extreme, some
individuals will be able to satisfy their genetic propensity
for acquisition whereas others will not, the result being a
right truncation of the acquisition distribution realized
under ad libitum rations. The result is a reduction in both
the variance and the mean and thus it cannot be assumed
that a covariance will become more accentuated with
reduced rations. For a right-truncated normal distribution the coefficient of variation decreases monotonically
with the severity of the truncation (i.e. if the distribution
is truncated at k then CV, the coefficient of variation,
increases with k). Therefore, under this distribution,
provided that the pattern of allocation does not change,
CVT will decline with a decrease in ration and could
convert a positive correlation to a negative (see eqn 2).
Even if the correlation is negative at ‘high’ ration, under
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the scenario just described, the strength of the correlation
should increase as ration is decreased. A critical assumption of these predictions is that the pattern of allocation
of resources among components does not change with
resource availability. Unfortunately, this assumption may
not be valid (Sgro & Hoffmann, 2004; and see the
example below).
To illustrate the effect of truncation of the resource
distribution on patterns of allocation, we present data on
the allocation to ovary mass and the main flight muscles
(the dorso-longitudinal muscles, hereafter DLM) in the
macropterous (long-winged) morph of the sand cricket,
G. firmus. This experiment is described in detail elsewhere
(Roff & Gelinas, 2003) and we note here only the
pertinent features: nymphs were raised on ad libitum food
but adults were fed either ad libitum or at a rate that had
been shown to reduce seven day fecundity (measured by
ovary mass) by approximately one half. On the ad libitum
diet the combined mass of ovaries plus DLM was 0.162 g
(SE ¼ 0.004, n ¼ 442, CV ¼ 48.31%), whereas on the
low ration it was 0.089 g (SE ¼ 0.002, n ¼ 414, CV ¼
13.3%). (These values differ slightly from those calculated from Table 1 of Roff & Gelinas, because in the present
analysis we included only females for which we had both
variables.) For the present analysis we are concerned
with how the food allocated to the combination of
ovaries plus DLM is distributed between these two
components and therefore we can consider the combined
mass as the total acquisition. The low ration reduced total
acquisition by 45% and the CV of acquisition by 72%.
The percentage of this total acquisition allocated to DLM
averaged 15.5% for females on the ad libitum diet, and
23.7% for females on the low ration, a difference that is
highly significant (t854 ¼ 8.1335, P < 0.0001, Kruskal–
Wallis test, v21 ¼ 57.80, P < 0.0001; proportions transformed using arcsine square-root). In contrast, the
correlation between ovary mass and DLM mass was
)0.591 and )0.565 on ad libitum and low rations
respectively, a difference that is not significant (t¥ ¼
0.5697, P > 0.5). Thus a severe reduction in acquisition
did not result in a change in the correlation (i.e. the
trade-off), but did change the pattern of allocation.
Phenotypic plasticity in resource allocation is a common phenomenon (e.g. Reznick, 1983; Smith & Davies,
1997; Billerbeck et al., 2000; Li et al., 2001; Jordan &
Snell, 2002; Bochdanovits & de Jong, 2003; Sgro &
Hoffmann, 2004) and invalidates predictions of the sign
of the correlation between two traits based solely upon
the observed change in the variance or coefficient of
variation in resource acquisition. The assumption that a
restriction in acquisition will either convert a positive
covariance to a negative covariance or increase the
magnitude of the negative correlation is incorrect both
on theoretical and empirical grounds. That such observations have been made in some cases (e.g. Biere, 1995;
Glazier, 1999; Messina & Slade, 1999; Donohue et al.,
2000) but not others (e.g. Glazier, 1999; Lardner &
Loman, 2003) argues for a greater need for an understanding of the functional basis of trade-offs, particularly
with respect to the adaptive significance of patterns of
allocation under different acquisition regimes. The
Y-model is a powerful conceptual and analytical tool but
a misinterpretation of its predictions has led to an unfortunate number of incomplete empirical investigations.
The effect of directional selection on
trade-offs
A quantitative genetic perspective
In general, trade-off functions are described empirically
by the simple linear regression between the two traits,
which under the quantitative genetic framework can be
written as:
rPXY
rPXY
Y ¼ lY 2 lX þ 2 X þ e;
ð9Þ
rPX
rPX
where lX and lY are the mean values of traits X and Y
respectively; rPXY is the phenotypic covariance between
traits X and Y, r2PX is the phenotypic variance of trait X
and e is a normally distributed error term (Roff et al.,
2002). The first terms in parentheses define the intercept
of the regression line and the terms in the second set of
parentheses define the slope. As the designation of which
trait as the dependent or independent variable is largely
arbitrary, a better statistical model may be to define the
trade-off as the principal axis of the bivariate normal
distribution that relates the two traits as in Fig. 1, but this
does not change the qualitative predictions.
In principle, selection will eventually change the
variances and covariances (Bohren et al., 1966; Falconer,
1989; Roff, 1997) which would thereby change both the
intercept and the slope of the trade-off function. However, for the infinitesimal model, short-term directional
selection does not change the shape of the distribution of
breeding values, except under extreme conditions, and
response is dominated by changes in trait means (Barton
& Turelli, 1987; Turelli & Barton, 1994). This prediction is
supported by the empirical observation that artificial
selection experiments over 10–15 generations generally
have little effect on heritabilities and genetic correlations
(Roff, 1997). With respect to trade-offs, an important
distinction made by De Jong (1990) is between what the
author has termed ‘structured pleiotropy’ and ‘unstructured pleiotropy’. Gene substitutions in the former
category produce correlated effects on both traits
whereas those in the second category do not, although
they might still affect both traits. Structural pleiotropy is
expected when there is ‘a developmental constraint or
functional constraint underlying genetic covariances’ (De
Jong, 1990, p. 459), and should therefore, be commonly
found in functionally based trade-offs. Theoretical analysis shows that the covariance between two traits is
more resistant to change when determined by structured
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Evolution of trade-offs
pleiotropy than when determined by unstructured pleiotropy (De Jong, 1990), and provides another reason
why the slope of a trade-off will not vary substantially
under directional selection.
Thus, for the reasons described above, the slope of the
trade-off function under short-term selection is expected
to remain constant, except under very restrictive conditions (Roff et al., 2002), whereas the intercept will
change because of changes in lY and lX. The change in
the intercept for truncation selection, which is that
typically applied in artificial selection experiments, can
be predicted using the standard response equation. For
selection on a single trait, Y, the intercept will change
according to:
atþ1 ¼ at þ RY bCRX ¼ at þ ihY ðhY rPY rA hX brPX Þ; ð10Þ
where at is the intercept at generation t, RY is the direct
response of trait Y, CRX is the correlated response of trait
X, b is the slope of the trade-off function, hX and hY are
the square-roots of the heritabilities, i is the intensity of
selection and rA is the genetic correlation between X and
Y. If truncation selection acts simultaneously on traits X
and Y, the intercept will change to:
atþ1 ¼ at þ RY bRX
ir
¼ at þ
½hY rPY ðhY þ rA hX Þ
1 þ rP
brPX hX ðhX þ rA hY Þ;
ð11Þ
where ir is approximately equal to i0[(1 + rP/4)(1 + rP)],
and i0 is the expected selection when rP ¼ 0 (Sheridan &
Barker, 1974). More generally, for any type of directional
selection (G. de Jong, personal communication) the
change in trait means is given by:
" # @ ln w dX
2
@X
dt ¼ rG;X rG;XY
ð12Þ
;
@ ln w
dY
rG;XY r2G;Y
@ Y
dt
is mean fitness and the subscript G denotes
where w
genetic components of variance and covariance.
An important caveat must be added to the predicted
response when selection is applied to both traits: if the
joint selection acts to increase acquisition (l2T ), say by
selecting on the sum Y + X (assuming this to be equivalent to selection on X1 + X2), then a change in covariance may ensue (see eqn 1). Joint selection on both traits
may affect not only mean acquisition but also mean
allocation and conceivably the two variances, which
could also result in a change in the covariance (James,
1974, in fact, described changes in covariance caused by
selection on both traits but, as noted earlier, his equation
for the covariance is flawed). The important point is that
in the Y-model the two component traits within an
individual are not mathematically independent traits and
thus it is actually incorrect to regard them as separate
traits. The separate traits are in fact acquisition and
allocation, which may be much more difficult to measure, particularly as resources will generally be allocated
441
among more than two traits (i.e. the hierarchical
Y-model may be more appropriate). As formulated, the
Y-model is completely deterministic within an individual
but in many, if not most, cases the variation in the two
component traits will also be subject to other influences
and hence may be treated as separate, although correlated, traits.
Predictions and tests
The above equations predict that, under short term or
weak selection, the trade-off function, as measured in its
statistical context, will evolve by a shift in the intercept
alone, defined either as the regression line or principal
axis (Roff et al., 2002). As an initial test of this prediction
Roff et al. (2002, 2003) compared two trade-off functions
among geographically widely separated populations of
G. firmus characterized by different degrees of wing
dimorphism. For females, Roff et al. (2002) used the
linear regression between ovary mass (fecundity) and
DLM mass and for males, Roff et al. (2003) used the
linear regression between call duration (¼probability of
attracting a mate: Crnokrak & Roff, 1995) and DLM mass.
We compared three newly collected populations from
Florida, South Carolina and Bermuda, and a population
that we had maintained in the laboratory for 19 years
(approximately 80 generations). Assuming that the
differences in proportion macropterous (assayed both in
the field and in common laboratory conditions) reflected
differences in the local selection regimes, we predicted
among-population variation in the intercept of the linear
regression. In addition, because of the likelihood that
evolution in the laboratory environment had caused
changes in the variances of the component traits and the
covariances between them, we predicted that the laboratory population might also differ with respect to the
slope of the linear regression.
For both males and females, the three recently collected populations did differ in intercept but not slope of the
linear regression, as predicted. Also as predicted, the
slope of the regression between ovary and DLM mass of
the lab females differed significantly from the field
populations (Roff et al., 2002). However, the slope of
the regression between call duration and DLM mass in
lab males did not (Roff et al., 2003). This difference
between male and female traits was reflected in the
proportion macropterous: females from the lab population had significantly reduced proportion macroptery,
while males did not. This suggests that evolution within
the lab environment had altered the characteristics of the
females but not the males.
Tucic et al. (2005) compared two populations of the
iris, Iris pumila, with respect to the linear regression
between two measures of vegetative reproduction and
somatic growth. One population was drawn from a
population growing on a dune in full sunlight and the
second from the understorey of a Pinus nigra stand where
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light was considerably diminished. For both measures of
the trade-off there was a marginally significant difference
(P ¼ 0.04) in the regression slope but no difference in
intercept (P ¼ 0.1). Although the rhizomes were grown
under controlled conditions they were sampled directly
from their natal populations, introducing the possibility
of previous environmental conditions affecting their
allocation patterns. Tucic et al. (2005, p. 21) cite a
number of other studies which ‘counter the specific
prediction of Roff et al. (2002) that a shift in the slope of
the phenotypic linear regression should be less likely
than in the intercept’. However, in all the cited cases the
comparison was between regressions observed under
different environmental conditions not different populations grown under the same environment. Such experiments measure the phenotypic plasticity of the trade-off,
not the response to selection and the above theoretical
development makes no statement about this circumstance. Indeed, as discussed earlier and shown by the
simulation analysis of Malausa et al. (2005), we would
expect phenotypic plasticity in allocation and hence that
the trade-off function, as typically measured by linear
regression, could change with respect to both slope and
intercept.
The prediction derived from eqn 11 is predicated on
the (co)variances not changing. There is abundant
evidence that genetic and phenotypic (co)variances do
change in some cases but not others (Roff, 2000; Jones
et al., 2003; Cano et al., 2004) although whether this is
due primarily to drift or selection is generally uncertain
(Roff, 2000, 2004; Steppan et al., 2002; Roff & Mousseau,
2005). If changes in allele frequencies of pleiotropic
genes affect both traits, then either selection or drift will
produce proportional changes in variances and covariances (Reeve, 2000; Roff, 2004) and hence the slope of
the trade-off will be preserved but the intercept will
change as selection acts on mean trait values. Similarly, if
as expected for trade-offs, there is strong structured
pleiotropy the slope will be resistant to change (De Jong,
1990). The same arguments apply not only to trade-offs
but also to any bivariate relationships, such as the
positive covariation between body parts.
Ultimately, the stability of the trade-off relationships is
an empirical question and we require more studies of
interpopulation variation in trade-offs, as well as controlled experiments investigating responses to selection
on the trade-off itself. Common garden comparisons of
different populations can provide valuable insights into
the extent and pattern of natural variation in trade-off
functions, but the results of such studies may be difficult
to interpret because little is known about the patterns of
selection in the natural environment. Artificial selection
on the trade-off function (or, more accurately, on the
traits comprising the trade-off) has the advantage that
the forces of selection are under strict control and often
responses can be predicted a priori, allowing tests of more
complex or counter-intuitive aspects to the conceptual
models. The examples that we have given in this section
illustrate the potential complexity of responses to selection on trade-offs and the need to understand the
functional basis of the trade-off to understand the
response to selection. The evolutionary trajectory of the
trade-off can be particularly difficult to predict when
both acquisition and allocation are allowed to vary in
response to selection, as illustrated by shifts in the
correlation between longevity and stress resistance
(Archer et al., 2003; Phelan et al., 2003; Prasad & Joshi,
2003; Prasad & Shakarad, 2004) or between larval
survival and growth rate (Chippindale et al., 2003)
observed in selection experiments in D. melanogaster.
What maintains variation in the trade-off?
Thus far, we have assumed that, if an underlying
functional trade-off exists between traits, it will be
expressed in the pattern of statistical variation and
covariation of the two traits within populations. However, many trade-offs may not be visible because no
variation exists in either trait, i.e. the trade-off is fixed.
Where trade-offs are visible, an important question is
‘why does such variation persist?’ To answer this question we must first determine to what extent the variation
that reveals the trade-off reflects phenotypic plasticity
versus genetic variation among individuals. Even under a
laboratory setting, individuals do not experience exactly
the same environment and thus some, if not all, of the
variation could be a result of phenotypic plasticity. It is
relatively easy to produce a model in which a trade-off is
expressed strictly as a phenotypically plastic response: for
example, in a beetle such as Stator limbatus in which all
larval resources come from a single seed (Fox et al.,
1997), fitness may well be maximized if females facultatively increase the size of their eggs on small seeds,
even though the increase in larval survival may be
somewhat offset by a trade-off between egg size and
number (Roff, 2002, pp. 433–438). However, it is
certainly evident from the observation of trade-offs
under highly controlled (common garden) conditions
that there is genetic variation in most trade-offs (e.g.
Billerbeck et al., 2000; Donohue et al., 2000; Roff et al.,
2002, 2003; Lee et al., 2003; Roff & Gelinas, 2003).
Pedigree experiments that examine both the genetic basis
of the trade-off and possible genetic variation in plasticity
would be useful to assess the relative importance of these
two sources of variation in generating observed trade-off
functions.
A more difficult question to answer is ‘why is there
genetic variation underlying the trade-off, whether or
not plasticity is present?’ Why does not the population
collapse to a single combination of traits? In some ways
this is the same problem of accounting for variation in
general, although there are particular features of tradeoffs that make this a separate problem. Four mechanisms
that could preserve or at least reduce the rate of erosion
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of variation resulting in an observed trade-off are (1)
mutation-selection balance, (2) antagonistic pleiotropy,
(3) correlational selection, and (4) spatio-temporal heterogeneity. Of these four, correlational selection is
particularly noteworthy because it may play a much
more central role in preserving variation in trade-offs
than variation in single traits or multiple traits that are
not functionally interconnected.
Mutation-selection balance
Empirical analyses using Daphnia pulex (Lynch et al.,
1998) and D. melanogaster (Houle, 1998) lend support to
the hypothesis that much of the standing genetic variance in life history traits is because of mutational input
(see also Charlesworth & Hughes, 2000; Roff, 2005).
Whether this is sufficient to maintain the phenotypic and
genetic covariation found in trade-offs has not been
determined, although theory shows that it could be
important (Houle, 1991). Evidence suggests that most
mutations have deleterious effects on all components of
fitness and that these tend to be purged from natural
populations (Houle et al., 1994, 1997; Estes et al., 2005).
Mutations with antagonistically pleiotropic effects would
remain segregating in the populations for longer, thereby
generating variation in the trade-off.
Antagonistic pleiotropy
The conditions for the maintenance of genetic variation
by antagonistic pleiotropy appear to be quite restrictive,
requiring the presence of nonadditive genetic effects
(reviewed in Roff, 1997), although the precise requirements have not yet been ascertained. Surveys of the
amount of dominance variance suggest that antagonistic
pleiotropy is unlikely to be an explanation for the
maintenance of genetic variance in morphological traits
but could account for that in life history traits (Roff,
1997). However, Estes et al. (2005) argue that the overall
negative effects of pleiotropic mutations suggest that
antagonistic pleiotropy is unlikely. What is needed are
quantitative genetic studies that determine not only the
additive genetic components of trade-offs but also the
dominance components. Additionally, we need theoretical studies to determine what levels of dominance
variance would be sufficient to account for the observed
covariances.
Correlational selection
If the fitness surface for the two traits of a trade-off has a
single peak then, ignoring drift and mutation, the
evolutionary trajectory will take the population eventually to the peak provided that the genetic correlation is
not )1. Stabilizing selection will maintain the population
at the fitness peak and will eventually erode variation.
But suppose that the fitness surface contains not a peak
443
but a ridge aligned in the direction of the trade-off: in this
case, multiple combinations are equally fit. Variation
along the ridge is thus effectively neutral and the erosion
of variation will be retarded. Selection favouring trait
combinations is called correlational selection and will
generate a genetic correlation between the two traits
under selection as a result of linkage disequilibrium
(Sinervo & Svensson, 2002). Correlations generated by
linkage disequilibrium are relatively unstable, particularly in small populations where drift can produce wide
fluctuations in allele frequencies. Pleiotropic mutations
that produce combinations of the type favored by
selection should spread in the population replacing the
linkage disequilibrium genetic correlation with one based
on pleiotropy.
The concepts of correlational selection and antagonistic
pleiotropy are related in the sense that both predict that
fitness will be maximized only at certain combinations of
traits. However, the two concepts differ in that pleiotropy
refers to gene action, specifically, genes that affect more
than one phenotypic trait, while correlational selection
refers to the fitness surface. Confusion arises because
antagonistic pleiotropy is defined in terms of changes in
fitness generated by the pleiotropic phenotypic effects: an
increase in fitness associated with the value of one trait is
correlated with a decrease in fitness associated with the
value of the other trait. This is indeed a form of
correlational selection and it would be expected to
generate an optimal trait combination (i.e. a single
fitness peak). However, correlational selection is broader
than this, encompassing fitness ridges or even saddles,
with regions in which parallel changes in both traits (i.e.
both increase or both decrease) have parallel rather than
antagonistic effects on fitness (Phillips & Arnold, 1989).
Most relevant for this discussion, correlational selection
can generate a suite of combinations that have equal
fitnesses rather than a single fitness peak, and it is this
that may help to maintain variation in fitness trade-offs.
Correlational selection is difficult to detect statistically
but is expected to be common (Schluter & Nychka, 1994;
Blows & Brooks, 2003). Examples of correlational
selection on trade-offs include the interaction between
color pattern and antipredator behavior in the garter
snake, Thamnophis ordinoides (Brodie, 1992) and the
pygmy grasshopper, Tetrix subulata (Forsman & Appelqvist, 1998), the trade-off between water-use efficiency
and leaf size in Cakile edentula (Dudley, 1996), the tradeoff between colour morph and immunocompetence in
the side-blotched lizard, Uta stansburiana (Svensson et al.,
2002), and the trade-off between size and timing of
sexual maturation in the grasshopper Sphenarium purpurascens (Castillo & Nunez-Farfan, 1999). Correlational
selection may slow the erosion of variation but will not,
in the absence of other factors, maintain variation.
Variation will be preserved over the longest period if
correlational selection is in the same direction as the
major axis of the trade-off, and over time the genetic
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(co)variance structure is expected to evolve such that the
major axis is aligned with the direction of selection
(Jones et al., 2003, 2004; Blows et al., 2004; Estes et al.,
2005) although we certainly do not have a sufficient
number of studies yet, to say if this is generally true.
Spatio-temporal heterogeneity
The optimum combination of trait values will typically
vary with environmental conditions: for example, in the
case of the seed beetle discussed above, seed size will vary
among host plant species and if these show spatial and
temporal variation then so also will the appropriate
combination of egg size and egg number. If such
variation can be accommodated by phenotypic plasticity
then genetic variation will not be preserved. The likelihood of preservation of genetic variation increases as the
predictability of the environment decreases, but it is
important to note that this requires either the joint
effects of spatial and temporal variation or the interaction
of temporal variation with an overlapping age structure
(Roff, 2002). The importance of overlapping age structure in preserving either phenotypic or genetic variation
in trade-offs has been little explored, but would be a
fruitful avenue for further research.
The observation that the genetic correlation underlying the trade-off can itself be environmentally sensitive
(Sgro & Hoffmann, 2004), also begs more study and
theoretical insight. It is not clear if this sensitivity could
play any role in preserving variation. This plasticity in the
correlation no doubt reflects the functional behavior of
the underlying genes, some being switched on or off, or
up- or down-regulated. Whereas quantitative genetics
can describe these changes in a statistical framework, we
lack information about the morphological, physiological
and behavioural changes that occur and the suite of
genes that are activated. In this regard, further development of genomics and its application to the present
question are crucial (Bochdanovits et al., 2003; Stearns &
Magwene, 2003; Bochdanovits & de Jong, 2004; Tonsor
et al., 2005).
Conclusions
We have discussed four main topics that we believe are
important for continuing progress in the study of tradeoffs. The first, the concept of constraint, is important
because it may lead to misconceptions about the limits of
evolutionary trajectories. We suggest that an explicit
distinction be made between the bias introduced by
negative genetic correlations and the limitation in phase
space dictated by an eigenvalue of zero. In general, the
former is most likely to be the correct interpretation.
However, the existence of absolute constraints is of very
considerable importance, and when suggested by empirical data, deserves more detailed study. Here, statistical
analyses of variance–covariance matrices reach their
limit of applicability, and progress in resolving absolute
constraints will likely require studies of the mechanistic
or functional basis of the suite of trade-offs hypothesized
to restrict evolutionary change.
In discussing the second topic, acquisition–allocation
models of resource-based trade-offs, we describe
important insights gained from this approach, but also
suggest that a misunderstanding of the equation has led
to improper predictions and tests of whether such
trade-offs exist. We show how the Y-model can be
expanded to include other types of trade-offs such as
that between egg size and number, or that between
adult size and development time. The Y-model itself has
been expanded to include more than two traits (the
hierarchical Y-model: De Laguerie et al., 1991; De Jong,
1993; Worley et al., 2003; Bjorklund, 2004) and phenotypic plasticity (Malausa et al., 2005) but more theoretical and empirical research is needed on these
expanded models.
If variances and covariances are not changed under
directional selection, the answer to the question posed as
our third topic, ‘how does directional selection affect
trade-offs, as expressed by the linear regression between
the two traits?’ is that the intercept but not the slope of
the linear regression will change. Tests of this prediction
using stocks from geographic populations grown under
common garden conditions were affirmative, but artificial selection experiments would provide a better test. It
is important to distinguish the above evolutionary
predictions from predictions concerning the purely
phenotypic change in the regression expected when the
same genotypes are reared under different environmental conditions. In this case, there is no reason to expect
that the linear regression will necessarily remain
unchanged in either its slope or intercept (Malausa et al.,
2005).
The final topic that we have considered is the
perplexing question of what maintains variation in the
trade-off. We suggest that this question is best answered
by distinguishing between phenotypic variation and
genetic variation. The more difficult question to resolve
is why genetic variation is observed and this variation is,
of course, fundamental for the evolution of trade-offs.
We suggest that four phenomena are likely to be
important: mutation-selection balance, antagonistic pleiotropy, correlational selection and spatio-temporal heterogeneity. All four may be important, but unfortunately
no data exist to determine whether one or several play a
primary role.
Given the central role that trade-offs play in evolutionary theory and the evolution of life histories in
particular, it is perhaps surprising that we still know so
little about the genetic architecture underlying trade-offs,
the mechanistic basis of practically all trade-offs, or the
evolution of trade-offs in either the short-term or longterm. On the other hand, the lack in these areas provides
a rich ground for further study and the integration of
ª 2006 THE AUTHORS 20 (2007) 433–447
JOURNAL COMPILATION ª 2006 EUROPEAN SOCIETY FOR EVOLUTIONARY BIOLOGY
Evolution of trade-offs
many disciplines, including the emerging field of
genomics.
Acknowledgements
This work was supported by grant #DEB-0445140 from
the National Science Foundation. We are grateful to
Gerdien de Jong for her insightful comments on an
earlier draft of this manuscript. The manuscript was also
improved by the constructive comments of two anonymous reviewers.
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