Assessment of genotypic differences in the response to season and food limitation
All families were reared under four different environmental conditions mimicking the wet or dry season (temperature), and with or without food limitation. In the analyses, the traits measured were viewed as specific to each of these four environments (character state; Lynch and Walsh 1998). Because we used a full-sib design, we could only estimate genetic (co)variances, which include – apart from additive genetic effects – dominance and maternal effects. Thus, the heritability we estimated must be considered as the upper estimate under the assumptions that dominance and maternal variance components are zero. Family members were reared in different common environments (sleeves), thereby minimizing resemblance across relatives due to rearing effects. Each trait has an environment-specific genetic variance with genetic covariances between the environment-specific trait expressions. Thus, there are in total four genetic variance (VG) and six genetic covariances (CG) between all combinations of food limitation (n = no stress; s = stress) and season (wet and dry defined by thermal conditions). The (co)variances can be summarized as the matrix,
The genetic covariances can be scaled to a genetic correlation following the standard definition, for example, rG(n,wet − s, wet) = CG(n,wet − s, wet)/√(VG(n,wet) × VG(s,wet)). Because we estimate compounded genetic (co)variances (see above), this matrix is an approximation of the G matrix with additive genetic (co)variances. Whereas the VG estimate can be considered as an upper estimate for the additive genetic variance, the CG estimate lacks mathematical relationship to the additive genetic covariances because the covariances on the numerous levels (additive genetic, dominance, maternal effect) compounded in CG need to align. Hence, our estimates of CG and thus of rG must be interpret with caution and are not therefore the main focus of this study.
Interactions between genotype and environment (GEI, defined here by the unique combination of season and food limitation) can take two nonmutually exclusive forms (Lynch and Walsh 1998). First, the relative ranking of the breeding values may change between environments. In general, a negative or low genetic covariance between environments indicates that the ranking of genotypes is changed (crossing reaction norms). Second, additive genetic variances may be specific to the environment. However, changes in additive genetic variances across environment are likely to be subjected to scaling, where not only the additive genetic variance changes but also the residual variance (and hence the phenotypic variance). Approaches to study changes in genetic architecture independently from scaling include standardization of variance components with the traits mean (coefficient of variation) or standardization of trait values with their variance (variance standardization; Lynch & Walsh 1998). Here, we follow the latter approach and thereby explore a form of GEI which creates changes in heritability across the four environments. The matrix (eq. (1)) was estimated by defining the linear mixed model,
where y is a vector of observations on all individuals, β is a vector of fixed effects, X represents a design matrix (of 0s and 1s) relating to the appropriate fixed effects to each individual, u is a vector of random effects, Z is a design matrix relating the appropriate random effects to each individual, and e is a vector of residual errors. is defined as the matrix for vector u, and its elements (the genetic (co)variances) can be estimated by using information on the coefficient of coancestry Θij between individuals i and j, which is directly obtained from the pedigree. All individuals measured were the descendants of butterflies mated in a full-sib cross. There were 56 base parents with a total of 1206 descendants. The genetic effects in environment E (i.e., the combination of food limitation or no food limitation, and dry or wet season conditions experienced) were assumed to be normally distributed with mean of zero (i.e., defined relative to the environment-specific fixed-effect mean) and with an genetic variance of σ2A,E. This variance (and the additive genetic covariance between all E) was estimated by REML from the variance–covariance matrix of additive genetic effects which is equal to Aσ2, where A has elements,
The fixed-effect structure of equation (2) accounted for variation in age of the butterflies when they entered the final instar and for variation between the sexes. We thus considered all data on both sexes in order to maximize our power to detect changes in genetic variances across environments. By fitting “sex” as a fixed effect, we corrected only for the difference in the mean trait expression between the sexes and thereby assumed that the between-sex genetic correlations for traits did not differ from +1 (no gene-by-sex interaction). In addition, for RMR we included the total fat-free dry weight of the individual as a fixed effect. Residuals were assumed to be heterogeneous (environment specific) and not correlated across environments.
Variances in a linear mixed model are conditional upon the fixed-effect structure. Mixed model phenotypic variance is, in this case, the sum of the REML genetic (including dominance and maternal variances) and residual variances. We incorporated variance scaling by standardizing the raw data prior to analysis to have a REML variance of unity (1) in each environment. This was done by first running a model that only included the fixed effects, where the four residual variances (assumed to be uncorrelated across environments) estimated the environment-specific REML variances. In further analyses, the data were divided by the environment-specific REML standard deviation. By doing so, all trait values become dimensionless (expressed in unit REML phenotypic SD) and the diagonal in equation (1) thereby consisted of the upper estimates of the trait-specific heritabilities.
Given the four environments considered, there are, for each trait, 15 models to consider and, in addition, the null model with residuals only (no heritability). Models were implemented in ASReml (VSN International, Hemel Hempstead, UK), which provides the log likelihood of the mixed model. Because many of the models are not nested, model comparison relied on an information theoretical approach based on the Akaike information criterion (AIC; Akaike 1974; Wagner et al. 1997; Burnham and Anderson 2002). AIC was calculated as −2log(L) + 2K, where log(L) was the model's log likelihood and K the number of parameters estimated. All models were ranked in ascending order based on their AIC, where a difference in AIC of more than two compared to the model with the lowest AIC was considered as evidence of deterioration in model fit (Burnham and Anderson 2002). In ASReml, constraining the diagonal of the matrix (eq. (1)) does not constrain the genetic covariances, which were left unconstrained in all models. We therefore did not consider the covariances or the fixed effects in calculating K because these parameters were estimated in all models and hence are factored out when doing model comparisons based on AIC. Thus, we calculated K as the number of genetic variances estimates, and K ranged from 0 (model with residuals only) to 4 (environmental-specific variances). Akaike weights w for model i was calculated as wi = exp(ΔAICi)/Σexp(ΔAIC), where ΔAICi is the difference in AIC between model i and the top model (i.e., the model with the lowest AIC). Models ranked within two AIC units of the top model were considered as reasonable candidate models (Burnham and Anderson 2002). Because one typically finds some level of support for multiple candidate models, model averaging is advocated to provide more precise estimates (Burnham and Anderson 2002). We model averaged the estimates of the genetic and residual variances across all 15 models, where the model-averaged variance V* was calculated by weighing Vi, the variance estimate of model i, such that V* = Σ(Vi × wi) with its model-averaged standard error SE* calculated as (Burnham and Anderson 2002).