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@example.com
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.
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, 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 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.
2Insights from models of resource acquisition–allocation.
3The effect of directional selection on the statistical description of trade-offs.
4What 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 trade-off, 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 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 = −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 (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 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 trade-offs 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 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 trade-offs (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 = 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 σX1X2, as:
where μ designates mean values and σ2 designates variances. For a fixed acquisition () 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 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:
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 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.
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).
†Phenotypic correlation between somatic mass and ovary mass.
Mean values, variances and covariances
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:
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:
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:
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 = ln E/ ln C, the allocation model thus being ln C = 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:
where, is the variance in ln C, is the variance in ln E, is the variance in ln N and σEN is the covariance between ln E and ln N. Rearranging the above gives:
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 = 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 = ln Y − ln X = ln Y + ln X−1. This model can now be examined in the same manner as for the egg size-number 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).
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:
where Tanaka defined M as resource acquisition and s as allocation. While we agree with the definition of M, we believe that 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 solution to the integral ): 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 . 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 = c, where c is a constant, then the total fecundity is cω where ω is the reproductive lifespan and the proportion allocated at age x is c/[c + c(ω − x)] = 1/(1 + ω − 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.
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 trade-offs 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 pseudo-replication 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 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 = 0.004, n = 442, CV = 48.31%), whereas on the low ration it was 0.089 g (SE = 0.002, n = 414, CV = 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 = 8.1335, P < 0.0001, Kruskal–Wallis test, = 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∞ = 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:
where μX and μY are the mean values of traits X and Y respectively; σPXY is the phenotypic covariance between traits X and Y, is the phenotypic variance of trait X and ɛ 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 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 μY and μX. 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:
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:
where ir is approximately equal to i0√[(1 + rP/4)(1 + rP)], and i0 is the expected selection when rP = 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:
where is mean fitness and the subscript G denotes 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 (), 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 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 (=probability 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 light was considerably diminished. For both measures of the trade-off there was a marginally significant difference (P = 0.04) in the regression slope but no difference in intercept (P = 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 trade-offs that make this a separate problem. Four mechanisms that could preserve or at least reduce the rate of erosion 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.
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.
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.
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 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 trade-off 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 (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.
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).
We have discussed four main topics that we believe are important for continuing progress in the study of trade-offs. 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 long-term. On the other hand, the lack in these areas provides a rich ground for further study and the integration of many disciplines, including the emerging field of genomics.
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.