Path analysis of the genetic integration of traits in the sand cricket: a novel use of BLUPs

Authors


Derek A. Roff, Department of Biology, University of California, Riverside, CA 92521, USA.
Tel.: +1 951 827 2437; fax: +1 951 827 5903; e-mail: Derek.roff@ucr.edu

Abstract

This study combines path analysis with quantitative genetics to analyse a key life history trade-off in the cricket, Gryllus firmus. We develop a path model connecting five traits associated with the trade-off between flight capability and reproduction and test this model using phenotypic data and estimates of breeding values (best linear unbiased predictors) from a half-sibling experiment. Strong support by both types of data validates our causal model and indicates concordance between the phenotypic and genetic expression of the trade-off. Comparisons of the trade-off between sexes and wing morphs reveal that these discrete phenotypes are not genetically independent and that the evolutionary trajectories of the two wing morphs are more tightly constrained to covary than those of the two sexes. Our results illustrate the benefits of combining a quantitative genetic analysis, which examines statistical correlations between traits, with a path model that focuses upon the causal components of variation.

Introduction

Genetically based trade-offs involving reproductive traits are fundamental components of the reproductive strategies of both males and females. Differences between the sexes in these trade-off functions clearly contribute to the evolution of sex-specific reproductive strategies and their associated morphology, physiology and life history (Clutton-Brock, 1988; Andersson, 1994; Fairbairn, 1997; Blanckenhorn, 2005; Fairbairn et al., 2007). In most species, positive fecundity selection is a major determinant of female lifetime fitness whereas the analogous selection in males is sexual selection through mating and fertilization success. This sexual disparity in selection often results in marked sexual dimorphisms in behaviour, morphology, physiology and life history. However, each sex inherits more or less the same genome, and the typically high genetic correlations between the same traits in males and females are expected to constrain the evolutionary trajectories of traits common to both sexes (Lande, 1980; Reeve & Fairbairn, 1996, 1999, 2001; Fairbairn, 1997; Badyaev, 2002). The two sexes can be analysed from a quantitative genetic perspective as if they were separate ‘environments’ (Falconer, 1952). Using this approach, we can estimate the genetic correlations between the sexes and predict the extent to which males and females can respond independently to sex-specific selection regimes. Except for life history traits, between-sex genetic correlations are typically high (e.g. 0.8 ± 0.03 for morphological traits but 0.4 ± 0.07 for life history traits, see meta-analysis of Poissant et al., 2010), and thus, rate of evolution of sexual dimorphisms is expected to be slow.

In addition to sexual dimorphisms, many polygenic traits show dimorphic variation within sexes that is not associated with chromosomal variation. This type of dimorphic variation can be modelled as a continuously distributed underlying trait known as the liability that interacts with a threshold of determination, such that individuals above the threshold exhibit one morph whereas individuals below the threshold exhibit the alternate morph (Wright, 1934a,b; Falconer, 1965). The threshold model can be used to explain disease incidence (Hartl, 1980), protective dimorphisms (e.g. cyclomorphosis), trophic dimorphisms (e.g. dental dimorphism in cichlids), mating dimorphisms (e.g. horned and hornless males in some beetles), life cycle dimorphisms (e.g. paedomorphosis) as well as sexual dimorphisms in species where sex is environmentally determined (for a review of threshold traits see Roff, 1996).

In this paper, we examine the genetic architecture of a key life history trade-off that varies dramatically across two axes of dimorphic variation, sex and wing length. We incorporate both models of dimorphic variation to determine the extent to which an a priori causal model of the phenotypic relations among traits central to the evolution of the trade-off is reflected in the quantitative genetic architecture of those traits. The trade-off of interest occurs between flight capability and reproduction in wing dimorphic insects, and our test species is the sand cricket, Gryllus firmus. Gryllus have an XO (male), XX (female) sex-determining system (Lim et al., 1973; Zefa, 1999; note that in Lim et al., G. firmus is given as G. bermudiensis). In contrast, the dimorphism in wing length is a consequence of a developmental switch that can occur as late as the last nymphal stadium and is best explained by the threshold model (Roff & Shannon, 1993; Fairbairn & Yadlowski, 1997). In G. firmus, this has been confirmed by both a pedigree analysis (Roff, 1986) and a selection experiment (Roff, 1990a). We begin by using structural equation modelling to test a functional model for the phenotypic relationships among five traits associated with the trade-off: wing morph, sex, head width, mass of the principal flight muscles and gonad mass. Then, we estimate the genetic variance and covariance structure underlying the sexual and wing dimorphisms and their correlations with the other measured traits to determine the extent to which the trade-off can evolve independently in the four discrete morphs. Finally, we assess the concordance between the phenotypic and genetic architecture of the trade-off by comparing the path model established from the phenotypic relationships with the same model using best linear unbiased predictors (BLUPs), which estimate the genetic value of each individual.

Wing dimorphism and correlated traits

Wing dimorphism is a common phenomenon in many species of insects, and its evolution is intimately connected with reproduction in both sexes (Roff, 1990b; Zera & Denno, 1997; Roff & Fairbairn, 2007a; Guerra, 2011). The flightless morph, while suffering from an inability to colonize new habitats, has a reproductive advantage in that females have a higher fecundity and males a higher mating success than the flight-capable morph (Roff & Fairbairn, 1991; Dingle, 1996; Fairbairn & Preziosi, 1996; Guerra, 2011). This trade-off appears to be functionally determined by competition between the energy allocated to flight capability (primarily developing and maintaining flight muscles and flight fuels) and the energy allocated to reproduction. The latter includes energy devoted to primary reproductive components such as the ovaries and testes as well as sexually selected armaments and display traits (Roff et al., 1997; Crnokrak & Roff, 2000; Zera & Larsen, 2001). The frequency of the flightless and flight-capable morphs often differs between the sexes (Roff & Fairbairn, 1993), suggesting that selection on this trade-off is often sex specific. Nevertheless, the two sexes possess the same general morphology and physiology with respect to their flight apparatus, and thus, the evolutionary trajectory of these will be influenced by the strength of the genetic correlations both between the sexes and among the traits contributing to flight capability within each sex. Furthermore, traits genetically correlated with these traits, such as gonad development, will also be impacted, the extent of joint evolution being determined by the genetic correlations between the sexes and between the wing morphs.

The trade-off between flight capability and reproduction is expressed not only in the differences between the short-winged, flightless (SW) and the long-winged, flight-capable (LW) morphs but also within the flight-capable morph itself. This morph may undergo an ontogenetic switch from a form primed for flight to one primed for reproduction, a phenomenon that has been termed the oogenesis flight syndrome (Johnson, 1969). Males and females that make this transition retain their flight capability during the early phase of their adult life and may migrate or disperse by flight during this time. However, after this initial mobile phase, they lose their flight capability. The switch between forms involves histolysis of the flight muscles, reduction in flight fuels and increased allocation to reproductive functions. As with wing dimorphism, the proportion of winged adults making this transition and the age at which they are likely to do so depend upon reproductive strategies and hence will be sex specific.

We report the results of a half-sib breeding experiment designed to investigate the genetic correlations among traits previously demonstrated to be important in the trade-off between flight capability and reproduction within and between the sexes of G. firmus. Previous studies have revealed that genetic correlations within and between wing morphs constrain the independent evolution of the trade-off in LW and SW G. firmus (e.g. Roff & Fairbairn, 2007b and references therein). We have also found a strong between-sex genetic correlation for proportion winged (Roff & Fairbairn, 1993). A generally lower proportion LW in males than in females (Fairbairn & Roff, 1990) suggests that the optimal trade-off function differs between the sexes, but the extent to which the trade-off can evolve independently in the two sexes remains unknown. Nor have we specifically compared the genetic architecture of the trade-off in males and females. The experiment reported herein was designed to fill in these gaps and specifically to develop causal model underlying the trade-off in males and females and test this using both phenotypic and genetic data.

Background for the path model

We focus upon three adult traits measured on the seventh day after eclosion into the adult stage: head width (as a measure of structural body size), gonad mass and mass of the main flight muscle, the dorso-longitudinal flight muscle (hereafter DLM mass). In pterygote insects, with the exception of the Ephemeroptera, there are no further moults once an individual has entered the adult stage, and thus, head width cannot be affected by developmental events after final eclosion. Head width is a typical morphological trait, and based on the analysis of Poissant et al. (2010), we would expect a high between-sex genetic correlation.

Because it is directly connected with reproduction and hence fitness, we classify gonad mass as a life history trait. Inter-sex genetic correlations between life history traits are typically much lower than those between morphological traits (0.4 vs. 0.8, Poissant et al., 2010). Sex-specific timing of ovarian and testicular development also suggests a lower correlation between sexes. The ovaries of G. firmus are small and immature at the time of adult eclosion and grow enormously during the first week of adult life (Stirling et al., 2001), increasing in mass 39-fold in SW females and 20-fold in LW females (unpublished analysis of data from Crnokrak & Roff, 2002). In contrast, the testes are substantially developed at eclosion in males and increase in mass during the first week only 1.3- and 1.1-fold in SW and LW males, respectively (unpublished analysis of data from Crnokrak & Roff, 2002). Thus, testis size at day 7 is likely to be more closely determined by events immediately prior to adult eclosion whereas ovary mass is more strongly influenced by conditions in early adult life (particularly food availability). Because gonad maturation is so ontogenetically distinct in males and females, we expect relatively low phenotypic and genetic correlations between sexes for gonad mass at 7 days post-eclosion.

The third trait, DLM mass, is directly connected with the ability to fly, and previous research indicates that allocation to flight capability ‘competes’ with allocation to reproduction. SW individuals cannot fly and thus, development of flight machinery carries costs but no benefits. For this reason, development of the DLM is strongly suppressed in SW crickets of both sexes, and many LW individuals reduce the ‘cost’ of allocation to flight capability by histolysing the major flight muscles once migration is accomplished or abandoned. The reproductive costs of DLM maintenance differ between females and males (i.e. allocation to fecundity in females but allocation to mate attraction in males), and hence, the quantitative nature of the physiological and genetic basis of the trade-off may be sex specific. The extent to which DLM mass at 7 days in LW crickets (our measure of histolysis) is genetically correlated between sexes is unknown.

In general, body components covary positively with overall size, and thus, we expect that whereas the trade-off between gonad mass and DLM mass will generate negative correlations, scale effects owing to size will generate positive correlations. The final correlations between traits will result from the interaction of these opposing factors.

Materials and methods

General

The sand cricket is a fairly large North American cricket (adult mass approximately 0.7 g) found along the beaches and other sandy areas from Florida to New Hampshire and on the island of Bermuda. Both SW and LW morphs are found in this species, with < 2% of individuals having intermediate wing lengths. Not all LW individuals are capable of flight, as many histolyse their DLM early in adult life. On average, fecundity over the first 7 days post-eclosion is substantially lower in LW females with functional DLM than in SW females whereas the fecundity of LW females without DLM is similar to that of SW females (Roff et al., 2002 and references therein). In males, the duration of calling, an essential element for female attraction, is greatly reduced in the LW morph with a consequent reduction in the number of females attracted (Crnokrak & Roff, 1995; Roff et al., 2002). Most significantly, within LW males, there is a trade-off between reproductive components (testis size and call duration), and the degree of histolysis of the DLM (Saglam et al., 2008).

Experimental protocol

For our half-sib experiment, we used crickets from a laboratory stock descended from approximately 40 adults (equal sex ratio) collected in northern Florida in 1981 and maintained with a standing adult population of approximately 100–500 adults. The stock is kept under diapause averting temperatures (25 °C) that also induce almost all SW individuals, resulting in no selection for a change in proportion of LW individuals (Saglam et al., 2008). Nymphs and adults are fed ad libitum with Purina rabbit chow. The adults (parents) and offspring (F1) for our experiment were reared and maintained in cages (4 L buckets) at a constant temperature of 28 °C and a 15 : 9 h light/dark photoperiod. Adult pairs for the crosses were provided with an earth dish for oviposition, and their offspring were reared at a density of 40 nymphs per cage. Water and ad libitum food in the form of crushed Purina rabbit chow were available in each cage throughout the experiment. One hundred and thirty sires were each mated to three dams, and the offspring from each full sib family were distributed among two or three rearing cages to allow estimation of cage effects (thus, the full statistical design is Sire/Dam/Cage). Our final data consisted of 4040 individuals distributed among 130 sires mated to an average of 2.8 dams with 11.5 offspring and 2.6 cages per dam.

After eclosing as adults, F1 males and females were separated to prevent mating, maintained as above for 7 days and then preserved in 70% ethanol. They were later dissected to determine the mass of the ovaries, testes and DLM. Virgin female G. firmus produce eggs but do not lay them, and ovary mass at 7 days post-eclosion is highly correlated (= 0.98, n = 46) with the sum of the number of eggs laid and still within the body of mated females (Roff, 1994). Dissected gonads and muscles were dried for 24 h at 60 °C and weighed to an accuracy of 0.1 μg. Prior to dissection, we measured head width (100% of males, 98% of females) and scored wing morph for all individuals. Within the sexes, head width is positively correlated with other body size measures, such as femur and prothorax length (Begin et al., 2004) and body mass (Saglam et al., 2008), and we use it here as an index of overall body size.

Statistical methods

Analysis of the path models was carried out using the software package AMOS© with parameters estimated by maximum likelihood (a free version of AMOS 5, which is the version used here, is available at http://www.amosdevelopment.com/download/). Models were fit using the phenotypic values, the estimated individual BLUPs and the mean BLUPs for each Dam family. Fit was determined by χ2 with support from the alternate fit indices, normal fit index (NFI), comparative fit index (CFI) and root mean square error of approximation (RMSEA). Adequacy of fit is demonstrated by a nonsignificant value of χ2, but in large samples, trivial differences can lead to significant values (Tabachnick & Fidell, 2007). A good fit is demonstrated by NFI and CFI values > 0.95 and RMSEA values close to zero (Tabachnick & Fidell, 2007). Simpler models were constructed by deletion of a single path and compared to the proposed model with a χ2 difference test (Tabachnick & Fidell, 2007). Additionally, we tested the causal pathway DLM to Gonad by reversing the direction of causality.

Genetic parameters were estimated using the ‘animal’ model, because this approach has the advantage of easily incorporating fixed effects, testing for dam effects (Roff, 2008), estimating genetic correlations between discrete states such as sex and wing morph (Wilson et al., 2010; see also the WAMWiki site at http://www.wildanimalmodels.org/tiki-index.php) and estimating BLUPs. DLM mass is bimodally distributed (Roff & Fairbairn, 2007b; Saglam et al., 2008) and was made approximately normal by squaring the values. Because this transformation still left the distribution of residuals visually different from normal, we also used a threshold transformation (Roff, 2001) in which the data are divided at the median and assigned the value 0 (below) and 1 (above). Because of the robustness of the estimation of variance components to large deviations from normality (Sahai & Ageel, 2000; Roff, 2001), this transformation had virtually no effect on our estimates and did not alter our conclusions. We therefore present results only for squared transformation.

Wing morph was treated as a threshold trait, and the genetic parameters are thus estimated for a theoretical, underlying, normally distributed trait liability, rather than for the dichotomous trait itself (Roff, 1986, 1997). Heritability on the underlying scale can be estimated using the approach of Dempster & Lerner (1950) or using a mixed model approach (Gianola, 2000). We use the former approach because it works well for standard breeding designs as used in the present experiment (Van Vleck, 1972; Mantysaari et al., 1991) and can readily be used when both categorical and continuously distributed data are analysed together (Mercer & Hill, 1984). Using the Dempster & Lerner (1950) method, heritability is estimated by first estimating the heritability on the 0, 1 scale and then transforming to the underlying liability scale using the proportion of either morph in the population (for details see Roff, 1997, p. 52–61). Short-winged individuals were coded as 0, and LW individuals were coded as 1; thus, a trade-off between wing morph and gonad mass will be signalled by a negative correlation. Females were coded as 0 and males as 1, and thus, a positive correlation signifies that the trait is larger in males than in females. Heritability estimates for the same trait in different morphs (wing and sex) were calculated separately and with morph as a fixed effect.

Genetic correlations between the sexes or wing morphs were estimated by treating each morph as a separate environment (Falconer, 1952). Genetic correlations between wing morph liability and other traits were estimated by assigning trait values of 0 and 1 to the two wing morphs and then proceeding using the usual quantitative genetic methods, no correction to the underlying liability scale being required (Mercer & Hill, 1984). We compared genetic correlations between morphs and between traits within morphs with a log-likelihood ratio test. Differences from 0 to 1 were also tested using the log-likelihood ratio test.

The heritabilities and BLUPs were calculated separately for each trait. Because wing morph is an integral component of the path model, BLUPs were estimated using both morphs combined. To be useful for path analysis of genetic effects, the BLUPs should be reasonable estimates of the true breeding values (e.g. see cautionary notes in Postma, 2006; Hadfield et al., 2010). A low correspondence between the true breeding values and the estimated BLUPs would mean that the path coefficients were being determined largely by the phenotypic values. We used an individual-based simulation to determine the expected correlation between the true breeding values and BLUPs estimated from the animal model for our breeding design. Our model mimicked the design used in the present experiment, namely 130 sires, three dams per sire and six offspring of each sex per dam (for a description of this approach and appropriate coding, see chapter 4 of Roff, 2010). Heritability was varied from 0.1 to 0.9 in steps of 0.1 plus a final value of 0.99. For each value, we ran five replicates (Fig. 1). Variability among the five replicates is minor for heritabilities > 0.3, and the correlation between the true and estimated breeding values (BLUPs) increases from 0.7 approximately linearly above this value. A higher correlation can be obtained using the mean of the dam families, which gives a correlation of 0.85 at a heritability of 0.3.

Figure 1.

 Correlations between the ‘true’ (i.e. simulated) and estimated breeding values (BLUBs) for traits with heritabilities ranging from 0.1 to 1.0. See text for details of the simulation. Lines connect the means of the five replicates.

The calculation of the BLUPs for a threshold trait is potentially complicated by the fact that heritability is first calculated on the 0, 1 scale and then converted to the underlying scale by a transformation based on the proportion of one of the morphs in the population (Robertson, 1951). In asreml©, the software package used in the present analysis, this means that the BLUPs are estimated relative to the 0, 1 scale not to the underlying scale. To verify that this generates appropriate values, we used the above-mentioned simulation model applying a threshold condition generating 0, 1 phenotypes in accord with the proportion observed in our data. The correlation between predicted and true values is somewhat less than that obtained using the continuous data but is still reasonable for heritabilities within the range estimated for wing morph liability in previous analyses of this stock (0.60–0.67; Roff, 1986, 1990a). Dam means considerably improve the estimate (Fig. 1).

As an additional validation of the genetic path model, we compared genetic correlations predicted from the path model for each sex with the genetic correlations estimated in standard fashion using the animal model. We used genetic correlations in preference to covariances to avoid possible undue leverage of traits with relatively large variances. The necessary formulae are

image

where the subscripts refer to traits (W = wing morph, H = head width, D = DLM mass), r to correlations and p to path coefficients. Because the genetic correlation rWH is taken directly from the observed value, it was omitted in the comparison.

Results

Phenotypic comparisons between sexes and wing morphs

All analyses use the offspring values only. Differences between sexes and wing morphs were tested using a two-way anova with sex and wing morph as independent variables. Except for the interaction between sex and wing morph in head width (= 0.06), all effects were highly significant (< 0.002; Fig. 2). Males are significantly larger than females (as indicated by head width, 3.6% for LW and 2.8% for SW), and LW crickets are significantly larger than SW crickets (1.6% for females and 2.4% for males). The proportion LW was significantly higher in females than in males (51.8% vs. 38.8%, inline image = 67.9, < 0.0001). As found in earlier studies, SW females had significantly larger ovaries than LW females (14.6%). Testes mass was much smaller than ovary mass and did not differ between wing morphs (= 1.01, d.f. = 1961.16, = 0.315, Welch modified t test). In contrast to gonad mass, DLM mass differed little between the sexes within wing morphs. There was a significant interaction between sex and wing morph in DLM mass, reflecting a slightly larger DLM in SW males than in SW females, but this difference is trivial relative to the main effect of wing morph (Fig. 2).

Figure 2.

 Mean trait values as a function of wing morph and sex for F1 individuals. Error bars indicate ± SE. (Δ) males, (•) females. For ease of plotting, ovary masses have been divided by 10. The denominator degrees of freedom are 4035, 4036 and 3953 for gonad mass, DLM mass and head width, respectively.

Except for three cases, all correlations between traits within sexes are significant (Table 1). The pattern of phenotypic correlations between traits is generally similar in males and females, although correlations involving DLM mass tend to be higher (more positive) in males (Table 1). The largest discrepancy between sexes is in correlations of gonad mass with DLM mass (bottom row in Table 1), which are negative in females but positive or not different from zero in males. This suggests that the phenotypic trade-off between gonad and DLM mass that occurs in females is not present in males. However, a trade-off between two traits may be obscured by the effect of another correlated trait (Pease & Bull, 1988), and hence, such a conclusion may be premature. To address this problem, we turn to the analysis of the path model.

Table 1.   Phenotypic correlations between traits within sexes and wing morphs. Nonsignificant correlations are shown in italics. Letters indicate correlations that do not differ across a row (for methods of statistical comparison of correlation coefficients see Zar, 1999).
Trait 1Trait 2FemaleMale
  1. SW, short-winged; LW, long-winged.

  2. †Combined estimate = 0.16.

  3. ‡Combined estimate = 0.46.

  4. §Although the pairwise tests separate LW females from LW males, a test of the three similar correlations was nonsignificant and the combined estimate for all three is 0.33.

Head widthWM liability†0.15a0.16a
Gonad massWM liability−0.24a−0.02b
DLM massWM liability0.57a0.74b
 LWSWLWSW
Head widthGonad mass‡0.10a0.43b0.43b0.49b
Head widthDLM mass§0.27a0.03b0.39c0.32a,c
DLM massGonad mass−0.66a−0.15b0.06c0.27d

Path analysis using phenotypic values

Using the information outlined in Background for the path model’ and associated inferences, we developed the causal model diagrammed in Fig. 3. Wing morph, sex and head width are correlated but assigned no causal pathways. However, wing morph is expected to be functionally correlated with both DLM mass and gonad mass. Similarly, the concept of the oogenesis flight syndrome predicts that gonad mass will be determined by changes in DLM mass. Because of scaling effects, body size, as indexed by head width, should also partially determine DLM mass and gonad mass. Finally, testis mass and ovary mass are expected to differ in overall size between the sexes, and thus, we postulate a causal connection between sex and gonad mass. There is no a priori reason to suppose that DLM mass will vary as a result of sex per se, and thus, no causal path is postulated. Because previous analyses have suggested that males may allocate differently from females, we also fitted the path model separately to each sex, after dropping the paths associated with sex.

Figure 3.

 Proposed path model with paths numbered (top) and with fitted coefficients (bottom). Short-winged individuals coded as 0 and long-winged individuals as 1. Females coded as 0 and males as 1. All coefficients are highly significant (< 0.0001). The chi-square for the full model is 0.022 (d.f. = 1, = 0.883). Chi-square values for models with deletion of numbered paths are as follows (d.f. for each test = 1): 1 = 59.4, 2 = 73.6, 3 = 188.0, 4 = 1959.5, 5 = 128.8, 6 = 6453.0, 7 = 201.1, 8 = 360.4, 9 = 1140.5, direction of path 9 reversed = 894.0.

The fit of the data to the proposed model is excellent (Fig. 3). The alternate measures of fit are in agreement with the chi-square analysis, NFI and CFI being equal to 1–3 decimal places and RMSEA being equal to 0–3 decimal places. Deletion of any single path increased the χ2 to at least 59 (Fig. 3), giving a highly significant difference (< 0.0001) between the proposed and reduced models. The very large effect of sex on gonad size could obscure differences between the sexes in path coefficients; therefore, we repeated the analysis for each sex separately (Fig. 4). To maintain one degree of freedom, the number of paths was reduced by the elimination of the path from wing morph to gonad mass. We assessed the importance of this path by repeating the analysis with this path inserted and the path from wing morph to DLM mass removed. For the models shown, the chi-square values were significant (females, inline image = 18.13, < 0.0001; males, inline image = 4.45, = 0.035), but the alternate indexes of fit indicated an excellent fit (females, NFI = 0.990, CFI = 0.990, RMSEA = 0.099; males, NFI = 0.998, CFI = 0.999, RMSEA = 0.040). We attribute the significant chi-square values to the large sample size that allows us to pick up very small effects (Tabachnick & Fidell, 2007). Changing the path from the wing to the DLM mass to a path from the wing to gonad mass considerably reduced the fit of the model (females, inline image = 643.7; males, inline image = 1732.8).

Figure 4.

 Path diagrams for females and males separately. Short-winged individuals coded as 0 and long-winged individuals as 1. Females coded as 0 and males as 1. The coefficients from the different analyses are separated by slashes: phenotypic variation/coefficients from the use of individual best linear unbiased predictors (BLUPs)/coefficients from the use of Dam mean BLUPs. All coefficients are highly significant (< 0.001).

The three path models indicate that (i) there is a significant positive effect of size on both DLM mass and gonad mass, (ii) wing morph has a large effect on DLM mass, (iii) DLM mass has a negative effect on gonad mass, the effect being much greater in females than in males and (iv) the positive correlation that we observed between DLM mass and testis mass is a consequence of the positive scaling effect of body size on these two traits and obscures the underlying trade-off between DLM mass and gonad mass. The partial correlation between DLM mass and testis mass is significantly negative in LW males (−0.13) but significantly positive in SW males (0.13). This indicates the overriding effect of the trade-off in LW males, which is replaced by a scaling effect in SW males.

Heritabilities

As neither dam nor cage effects were significant, we excluded them from the estimation of genetic parameters. Heritabilities were estimated both separately by wing morph and sex and as combined estimates using wing morph and sex as fixed effects (interactions were included when both wing morph and sex were included as fixed effects, Table 2). Males and females differ in proportion LW, and hence, differences on the initial scale of estimation, namely the 0, 1 scale, may exist by virtue of this difference whereas not existing on the underlying liability scale. Because of the differences in proportion between the sexes, the raw data cannot be combined and a single estimate made using the Dempster & Lerner (1950) approach. As the sample sizes were roughly equal (1834 females, 2206 males), an approximate combined estimate can be made as inline image with inline image, where the symbols ‘F’ and ‘M’ refer to females and males, respectively. It is evident from the two separate estimates that there is no significant difference between the sexes, and thus, a combined estimate is warranted (Table 2). Estimation using a trait as a fixed effect assumes that the heritability is the same for each component (e.g. male and female) and that the genetic correlation between the trait components is one. If this is not the case, the combined heritability estimate will be biased downward.

Table 2.   Heritability estimates for traits involved in the trade-off, by sex and wing morph. ‘Combined’ denotes combined estimates derived from including wing morph, sex or both as fixed effects. Standard errors are in parentheses. Nonsignificant heritabilities are shown in italics.
TraitWing morphFemalesMalesCombined
  1. SW, short-winged; LW, long-winged.

  2. *< 0.001.

Head widthLW0.38 (0.07)0.19 (0.06)0.22 (0.04)
SW0.37 (0.08)0.32 (0.06)0.32 (0.05)
Combined0.34 (0.07)0.24 (0.04)0.26 (0.03)
Gonad massLW0.40 (0.07)0.34 (0.08)0.25 (0.05)
SW0.50 (0.08)0.36 (0.06)0.23 (0.04)
Combined0.42 (0.05)0.36 (0.05)0.22 (0.03)
DLM massLW0.29 (0.07)0.27 (0.07)0.25 (0.04)
SW0.00 (0.00)*0.09 (0.04)0.00 (0.00)*
Combined0.18 (0.04)0.15 (0.03)0.14 (0.02)
WM liability 0.62 (0.08)0.69 (0.08)0.65 (0.08)

The heritabilities were statistically higher than zero for all of the traits examined with the exception of DLM mass in the SW morph for females and for both sexes combined (i.e. using sex as a fixed effect). In SW males, the heritability of DLM mass was significant but very small. In contrast, the heritability of DLM mass is highly significant and much larger in the LW morph for both males and females. This difference between wing morphs in heritability of DLM mass is not mirrored by similar differences in the other two traits. The LW and SW morphs have similar heritabilities for both head width and gonad mass, and the combined estimates with wing morph as a fixed effect are not significantly lower than the separate estimates for each wing morph. This suggests that the genetic correlations between wing morphs are very close or equal to one for both head width and gonad mass.

As noted earlier, males and females showed similar heritabilities for wing morph liability. The heritabilities of head width are slightly higher in females than in males, but the differences are relatively small, and the combined estimates, with sex as a fixed effect, are consistent with between-sex genetic correlations close to 1.0. The heritabilities of gonad mass are also slightly higher in females than in males, and here, the heritabilities obtained using sex as a fixed effect are lower than the estimates for either sex, suggesting that the genetic correlation between the sexes is likely to be less than unity. DLM mass shows the curious pattern noted above: The sexes show similar heritabilities within the LW morph but low or no heritabilities in the SW morph.

Genetic correlations between the sexes and wing morphs

The genetic correlations between wing morphs for head width and gonad mass were significantly positive in both sexes and did not differ significantly from 1.0 (Table 3). These results are consistent with the heritability analysis and validate using combined estimates, with wing morph as a fixed effect for these two traits. The two wing morphs appear to share a common genetic architecture for head width and gonad mass.

Table 3.   Genetic correlations, rg (SE), between the sexes and wing morphs. ‘Combined’ denotes estimates derived from including wing morph or sex as fixed effects.
TraitMorphCombinedTests†
Between wing morphsr> 0r< 1
FemaleMale
  1. SW, short-winged; LW, long-winged.

  2. *0.01 < < 0.05, otherwise < 0.0001.

  3. †In all cases, where rg can be estimated, the same qualitative answer applies across each cell of a given row.

Head width0.93 (0.23)0.76 (0.20)1.10 (0.16)YN
Gonad mass0.79 (0.13)1.16 (0.16)0.81 (0.11)YN
DLM massNE−0.04 (0.30)NENY
 Between sexes   
LWSW   
Head width0.68 (0.16)*0.79 (0.11)*0.81 (0.08)YY
Gonad mass0.33 (0.15)*0.34 (0.15)*0.37 (0.10)YY
DLM mass1.13 (0.14)NE1.02 (0.12)YN
WM liabilityNANA0.89 (0.05)*YY

In contrast, the genetic correlation between wing morphs for DLM mass was not significant in males and could not be estimated in females (Table 3). These results can be attributed to the very low variation in DLM mass in SW individuals, particularly females, and are consistent with the heritability analysis in suggesting that the genetic architecture for DLM mass differs between wing morphs.

The pattern of genetic correlations between the sexes suggests a converse pattern of constraint (Table 3). Only DLM mass in LW individuals has a between-sex genetic correlation that does not differ significantly from 1.0. Thus, whereas the genetic architecture for this trait differs between wing morphs, it does not appear to differ between sexes. In contrast, the between-sex correlations for head width and gonad mass differ significantly from both zero and unity. Correlations between sexes that are consistently < 1.0 combined with the sexual differences in heritabilities noted previously strongly suggest sex-specific genetic architecture for head width and gonad mass and the potential for the further evolution of sexual dimorphism in these traits.

Genetic correlations among traits within morphs

The pattern of genetic correlation of wing morph liability with DLM mass and head width is similar in the two sexes (Table 4). As would be expected, the strongest correlation is with DLM mass. These correlations are high but significantly < 1.0, indicating some potential for independent evolution of wing morph liability and muscle mass. Head width is also positively correlated with wing morph liability in both sexes (LW individuals average slightly larger than SW individuals), and these correlations are much lower than the correlations with DLM mass. Gonad mass shows a different pattern. It is negatively correlated with wing morph liability in females, but the correlation does not differ from zero in males. This reflects the large effect of wing morph on ovary mass but not on testes mass, which we noted in Table 1 and Fig. 2.

Table 4.   Genetic correlations, rg (SE), among traits within sexes, with wing morphs (WM) treated separately, ignored (B1) or entered as a fixed effect (B2). Correlations in bold are significantly different from zero.
Trait 1Trait 2WMrg
FemalesMales
  1. SW, short-winged; LW, long-winged.

  2. *0.002.

  3. †Not estimable because estimated heritability of DLM mass negative.

WM liabilityHW 0.32 (0.10)0.42 (0.09)
WM liabilityGonad mass−0.29 (0.10)−0.04 (0.10)
WM liabilityDLM mass0.91 (0.04)0.95 (0.02)
HWGonad massLW0.00* (0.15)0.52 (0.16)
HWGonad massSW0.50 (0.12)0.53 (0.10)
HWGonad massB10.17(0.10)0.52 (0.08)
HWGonad massB20.26 (0.10)0.56 (0.08)
HWDLM massLW0.33 (0.15)0.18 (0.21)
HWDLM massSWNE†0.46 (0.19)
HWDLM massB10.34 (0.10)0.39 (0.09)
HWDLM massB20.22 (0.13)0.19 (0.14)
Gonad massDLM massLW−0.66 (0.10)−0.14 (0.18)
Gonad massDLM massSWNE0.66 (0.18)
Gonad massDLM massB1−0.43 (0.08)−0.07 (0.10)
Gonad massDLM massB2−0.38 (0.11)−0.10 (0.13)

The genetic correlations between head width, gonad mass and DLM mass show both wing morph- and sex-specific variation (Table 4). The estimated genetic correlation between head width and gonad mass is zero in LW females but approximately 0.5 in the other three wing morph/sex combinations. The genetic correlation between head width and DLM mass is similar between males and females when wing morph is either ignored or entered as a fixed factor but could not be estimated within SW females because almost all had no measurable DLM. The genetic correlations between gonad mass and DLM mass support our hypothesis that both scale effects and a trade-off are important in determining these interrelationships. Negative correlations in the LW morph indicate that the effect of a trade-off between the two traits predominates, particularly in females. The significant positive correlation in SW males indicates that scale effects predominate, as expected from the lack of function in the flight muscles of this morph. As mentioned earlier, we could not estimate the genetic correlation between gonad mass and DLM mass in SW females because of the absence of variation in DLM.

The genetic correlations are highly correlated with the phenotypic correlations (= 0.927, F1,14 = 86, <0.0001: Mantel test, < 0.0001 based on 10 000 randomizations, Fig. 5), and the regression of rg on rp does not differ from the 1:1 line [rg = 0.03 (SE = 0.05) + 1.19 (0.13)rp].

Figure 5.

 Top panel: Comparison of phenotypic and genetic correlations. Bottom panel: Comparison of path coefficients from the separate-sex path models using phenotypic and best linear unbiased predictor (BLUP) values. The dashed line is the 1 : 1 line, and the solid line is the fitted regression line.

Path analysis using BLUPs

Given that the within-sex heritabilities of head width, gonad mass and DLM mass ranged from 0.15 to 0.43 (Table 2), the simulation results predict correlations of 0.6–0.8 between our estimates of individual BLUPs and the actual values, with dam means having correlations ranging from 0.8 to 0.9 (Fig. 1). The heritability of wing morph liability varied from 0.62 to 0.69 giving correlations for the individual and dam BLUPS of approximately 0.7 and 0.8. Correlations < 0.8 indicate that genetic and phenotypic effects may not be well separated, and hence, caution needs to be exercised in contrasting the path models produced from the phenotypic values and those from the individual BLUPs. On the other hand, the dam means are expected to be very highly correlated with the true values and hence should provide a reasonable path model of genetic effects. The path coefficients using the individual BLUPs give the same picture as that obtained using the phenotypic values (Figs 3 and 4). The fit was somewhat better than with the phenotypic data both with respect to the χ2 values (females, inline image = 6.70, = 0.01; males, inline image = 2.30, = 0.129) and the alternate indexes of fit (females, NFI = 0.997, CFI = 0.997, RMSEA = 0.057; males, NFI = 0.999, CFI = 1.000, RMSEA = 0.024). The regression of observed genetic correlations on predicted was highly significant (= 0.935, F1,10 = 143, <0.00001) and not significantly different from the line of equality [Predicted = 0.08 (SE = 0.04) + 0.99 (0.08) Observed]. Using dam means gives a very similar fit (females, inline image = 3.52, = 0.061, NFI = 0.992, CFI = 0.994, RMSEA = 0.086; males, inline image = 1.88, = 0.170; NFI = 0.997, CFI = 0.998, RMSEA = 0.051). We conclude that our proposed causal model reflects the interrelationships at both the phenotypic and genetic levels.

Discussion

Our results confirm the presence of a significant trade-off at both the phenotypic and genetic levels between flight capability and reproduction in females and males of the sand cricket. We found that ovary mass is negatively correlated with wing morph liability and with DLM mass and the latter correlation occurs both overall and within LW females. This pattern is significant for both phenotypic and genetic correlations and is consistent with previous estimates for this and other populations of G. firmus (Roff, 1984, 1990a; Stirling et al., 2001; Roff et al., 2002). Although we found the same trade-off in males, it is considerably weaker and only apparent in LW males after adjustment for size effects, whereas in SW males, the scaling effects overcome any trade-off between gonad mass and DLM mass.

Except for DLM mass in SW females, which shows virtually no phenotypic variation, all traits show significant heritabilities in both sexes and both wing morphs. While this would suggest considerable potential for the traits to evolve independently within each sex and wing morph, we found a web of strong genetic correlations between and within morphs that would be expected to cause strong correlated responses and constrain the evolutionary trajectories of both the target and correlated traits. Genetic correlations of 1.0 constrain evolution to a particular direction, namely along the line connecting the two traits. The genetic correlations between the wing morphs for head width and gonad mass are very high and not significantly different from one, indicating that evolutionary divergence of these traits between wing morphs, if not prevented, is at least greatly slowed. Note that the high genetic correlations for gonad mass occur in spite of very large differences in the mean values within morphs, especially for females. Clearly, past selection has been successful in causing divergence in gonad mass between LW and SW adults, but the high genetic correlations now present are likely to greatly restrict further divergent evolution. By comparison, divergence in DLM mass between wing morphs is less constrained. DLM mass has become essentially wing morph limited in its expression, with only remnant muscles present in SW adults. In SW females, DLM mass is so small and invariant that no correlation between wing morphs could be estimated. In males, the correlation between wing morphs is almost zero and not statistically significant, indicating that DLM mass can evolve independently in the two morphs. Given the moderate and significant heritability of DLM mass in LW adults, we would predict that selection favouring an increased or decreased investment in DLM in LW crickets could produce an evolutionary response without significantly affecting the SW morph.

In their meta-analysis, Poissant et al. (2009) found an average genetic correlation between the sexes of 0.8 among nonlife history traits and 0.4 for life history traits. Our data are consistent with these findings: The average correlation between the sexes for the morphological traits (head width, DLM mass, wing morph liability) is 0.91 (SE = 0.06, using combined wing morph estimates from Table 3) whereas for the life history trait (gonad mass) it is only 0.37. However, with one exception (DLM mass in the LW morph), all of the genetic correlations between the sexes are significantly less than one. Thus, separate evolutionary trajectories of the sexes for gonad size and head width are less impeded than those for the two wing morphs.

Each sex and wing morph can be characterized by a suite of traits that are functionally correlated, such as ovary size and DLM mass in LW females, and this correlation is presumed to be a consequence of selection on physiological and life history traits (Roff & Fairbairn, 1991, 2001, 2007a; Zera & Denno, 1997). The present analysis has shown that the patterns of genetic correlation among the traits are in accord with this functional pattern and that whereas strong genetic correlations may prevent divergent evolution between wing morphs, lower correlations between the sexes can permit sex-specific evolutionary pathways.

Evolutionary trajectories are contingent on both heritabilities and genetic correlations. Our results show that extreme integration can occur among some traits while others remain free to vary. Similarly, and most relevant to the goals of our study, the patterns of genetic integration do not always map simply onto patterns of discrete phenotypic variation: Morphs that are discrete phenotypically are not genetically independent. Although it is obvious that past selection has caused marked divergence of the four discrete morphs defined by sex and wing dimorphism in G. firmus, strong genetic correlations between morphs can be expected to constrain and in the case of wing morphs even prevent further divergence. For most of the traits involved in the trade-off between reproduction and flight capacity, selection on one morph (male, female, LW or SW) will produce correlated responses in the other morphs. Further, the correlated responses will occur not only in the target trait but also in the suite of correlated traits. Thus, for example, selection on wing morph liability would be expected to produce a correlated response in DLM mass. Similarly, selection on ovary mass in females would be expected to produce a response in wing morph liability. We have observed these responses in previous selection experiments (Fairbairn & Roff, 1990; Roff et al., 1999). A challenge for future research is to determine the extent to which the observed pattern of genetic integration within and among morphs is maintained by current patterns of selection and hence is adaptive vs. the alternative that these represent, at least in part, persistent functional constraints.

Above, we have considered how genetic and phenotypic correlations may constrain or direct evolutionary change. A complementary approach is to address the problem by generating a causal model and testing the fit of this model with the observed data. This approach was first formulated by Wright (1934a,b) as path analysis and has been subsequently developed into a more comprehensive statistical tool called structural equation modelling (Tabachnick & Fidell, 2007). Path analysis using phenotypic values is used extensively in animal and plant breeding to assess the relationship between economically important traits and correlated traits that allow more precise selection criteria. Such analyses frequently use data from inbred lines, varieties or hybrids (e.g. Bisht & Gahalain, 2009). BLUPs are also frequently used to increase the precision of selection criteria in these fields (Hofer, 1998; Piepho et al., 2008), but we have found no study that combined both approaches. Path analysis has also been used in the analysis of natural selection and fitness relationships in wild populations (Kingsolver & Schemske, 1991; Scheiner et al., 2000; van Tienderen, 2000; Coulson et al., 2003). More recently, BLUPs have been used in selection analysis of wild populations within the context of the animal model, although this approach has serious limitations (Postma, 2006; Hadfield et al., 2010). As in the case of the applied literature, we have not located any studies on nondomestic species that combined path analysis with the use of BLUPs. The use of path analysis to test interrelationships leading to trade-offs has been used with phenotypic data (e.g. Crnokrak & Roff, 2000; Langerhans, 2009), and Latta & McCain (2009) applied the approach using inbred lines of Avena barbata, which can be considered a path analysis of genetic effects.

In the present paper, we have used path analysis at both the phenotypic and genetic level to test a proposed model of causal relationships and trade-offs in the sand cricket. Our analyses show that the proposed model is well supported at both levels: In fact, the models based on the genetic data are better supported than those based on the phenotypic data. Significantly, the results from the phenotypic model are congruent with those from the genetic model, which is important because, while selection acts on the phenotype, the response to selection depends on the underlying genetic structure. Estimation of genetic and phenotypic correlations is a prerequisite for predicting evolutionary response using the multivariate breeder’s equation. The covariance matrices used in the latter approach are statistical descriptions of relationships among traits, whereas path analysis provides a causal explanation for these relationships. Thus, the two approaches are complementary and together can greatly improve our understanding of the evolution of trait combinations.

Acknowledgments

We are grateful for the technical assistance of M. Walsh, E. King and I. Saglam in the collection of data used in the present analysis. The manuscript was improved by the constructive comments of two anonymous reviewers, and we are grateful for their advice. This research was supported by NSF grant DEB044510 to DJF and DAR.

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