Stability of the G-matrix in a population experiencing pleiotropic mutation, stabilizing selection, and genetic drift

Abstract. Quantitative genetics theory provides a framework that predicts the effects of selection on a phenotype consisting of a suite of complex traits. However, the ability of existing theory to reconstruct the history of selection or to predict the future trajectory of evolution depends upon the evolutionary dynamics of the genetic variance-covariance matrix (G-matrix). Thus, the central focus of the emerging field of comparative quantitative genetics is the evolution of the G-matrix. Existing analytical theory reveals little about the dynamics of G, because the problem is too complex to be mathematically tractable. As a first step toward a predictive theory of G-matrix evolution, our goal was to use stochastic computer models to investigate factors that might contribute to the stability of G over evolutionary time. We were concerned with the relatively simple case of two quantitative traits in a population experiencing stabilizing selection, pleiotropic mutation, and random genetic drift. Our results show that G-matrix stability is enhanced by strong correlational selection and large effective population size. In addition, the nature of mutations at pleiotropic loci can dramatically influence stability of G. In particular, when a mutation at a single locus simultaneously changes the value of the two traits (due to pleiotropy) and these effects are correlated, mutation can generate extreme stability of G. Thus, the central message of our study is that the empirical question regarding G-matrix stability is not necessarily a general question of whether G is stable across various taxonomic levels. Rather, we should expect the G-matrix to be extremely stable for some suites of characters and unstable for others over similar spans of evolutionary time.     

Selective and genetic constraints on the evolution of body size in a stream-dwelling salmonid fish

To examine constraints on evolution of larger body size in two stunted populations of brook charr (Salvelinus fontinalis) from a single river in Cape Race, Newfoundland, Canada, we measured viability selection acting on length-at-age traits, and estimated quantitative genetic parameters in situ (following reconstruction of pedigree information from microsatellite data). Furthermore we tested for phenotypic differentiation between the populations, and for association of high juvenile growth with early maturity that is predicted by life history theory. Within each population, selection differentials and estimates of heritabilities for length-at-age traits suggested that evolution of larger size is prevented by both selective and genetic constraints. Between the populations, phenotypic differentiation was found in length-at-age and age of maturation traits, whereas early maturation was associated with increased juvenile growth (relative to adult growth) both within and between populations. The results suggest an adaptive plastic response in age of maturation to juvenile growth rates that have a largely environmental basis of determination.     

A combination of walk-back and optimum contribution selection in fish: a simulation study

The aim of this paper was to study the performance of a novel fish breeding scheme, which is a combination of walk-back and optimum contribution selection using stochastic simulation. In this walk-back selection scheme, batches of different sizes (50, 100, 1000, 5000 and 10 000) with the phenotypically superior fish from one tank with mixed families were genotyped to set up the pedigree. BLUP estimated breeding values were calculated. The optimum contribution selection method was used with the rate of inbreeding (ΔF) constrained to 0.005 or 0.01 per generation. If the constraint on ΔF could not be held, a second batch of fish was genotyped etc. Compared with the genotyping of all selection candidates (1000, 5000 or 10 000), the use of batches saves genotyping costs. The results show that two batches of 50 fish were often necessary. With a batch size of 100, genetic level was 76–92% of the genetic level achieved for schemes with all fish being genotyped and thus candidates for the optimum contribution selection step. More parents were selected for schemes with larger batches, resulting in a higher genetic gain, especially when all selection candidates were genotyped. There was little extra genetic gain in genotyping of 1000 fish instead of 100 for the larger schemes of 5000 and 10 000 candidates. The accuracy of breeding values was similar for all batch sizes (~0.30), but higher (~0.5) when all candidates were included. Since only the phenotypically most superior fish were genotyped, BLUP-EBV were biased. Compared with genotyping of all selection candidates, the use of batches saves genotyping costs, while simultaneously maintaining high genetic gains.     

The dilemma of Fisherian sexual selection: Mate choice for indirect benefits despite rarity and overall weakness of trait‐preference genetic correlation

Fisher's mechanism of sexual selection is a fundamental element of evolutionary theory. In it nonrandom mate choice causes a genetic covariance between a male trait and female preference for that trait and thereby generates a positive feedback process sustaining accelerated coevolution of the trait and preference. Numerous theoretical models of Fisher's mechanism have confirmed its mathematical underpinnings, yet biologists have often failed to find evidence for trait‐preference genetic correlation in populations in which the mechanism was expected to function. We undertook a survey of the literature to conduct a formal meta‐analysis probing the incidence and strength of trait‐preference correlation among animal species. Our meta‐analysis found significant positive genetic correlations in fewer than 20% of the species studied and an overall weighted correlation that is slightly positive. Importantly, a significant positive correlation was not found in any thorough study that included multiple subgroups. We discuss several ways in which the dynamic, multivariate nature of mate choice may reduce the trait‐preference genetic correlation predicted by Fisher's mechanism. We then entertain the possibilities that Fisherian‐like processes sometimes function without genetic correlation, and that mate choice may persist in a population as long as genetic correlation, and therefore Fisher's mechanism, occurs intermittently.     

Elimination of a genetic correlation between the sexes via artificial correlational selection

Genetic correlations between the sexes can constrain the evolution of sexual dimorphism and be difficult to alter, because traits common to both sexes share the same genetic underpinnings. We tested whether artificial correlational selection favoring specific combinations of male and female traits within families could change the strength of a very high between-sex genetic correlation for flower size in the dioecious plant Silene latifolia. This novel selection dramatically reduced the correlation in two of three selection lines in fewer than five generations. Subsequent selection only on females in a line characterized by a lower between-sex genetic correlation led to a significantly lower correlated response in males, confirming the potential evolutionary impact of the reduced correlation. Although between-sex genetic correlations can potentially constrain the evolution of sexual dimorphism, our findings reveal that these constraints come not from a simple conflict between an inflexible genetic architecture and a pattern of selection working in opposition to it, but rather a complex relationship between a changeable correlation and a form of selection that promotes it. In other words, the form of selection on males and females that leads to sexual dimorphism may also promote the genetic phenomenon that limits sexual dimorphism.     

An exact form of the breeder's equation for the evolution of a quantitative trait under natural selection

Starting with the Price equation, I show that the total evolutionary change in mean phenotype that occurs in the presence of fitness variation can be partitioned exactly into five components representing logically distinct processes. One component is the linear response to selection, as represented by the breeder's equation of quantitative genetics, but with heritability defined as the linear regression coefficient of mean offspring phenotype on parent phenotype. The other components are identified as constitutive transmission bias, two types of induced transmission bias, and a spurious response to selection caused by a covariance between parental fitness and offspring phenotype that cannot be predicted from parental phenotypes. The partitioning can be accomplished in two ways, one with heritability measured before (in the absence of) selection, and the other with heritability measured after (in the presence of) selection. Measuring heritability after selection, though unconventional, yields a representation for the linear response to selection that is most consistent with Darwinian evolution by natural selection because the response to selection is determined by the reproductive features of the selected group, not of the parent population as a whole. The analysis of an explicitly Mendelian model shows that the relative contributions of the five terms to the total evolutionary change depends on the level of organization (gene, individual, or mated pair) at which the parent population is divided into phenotypes, with each frame of reference providing unique insight. It is shown that all five components of phenotypic evolution will generally have nonzero values as a result of various combinations of the normal features of Mendelian populations, including biparental sex, allelic dominance, inbreeding, epistasis, linkage disequilibrium, and environmental covariances between traits. Additive genetic variance can be a poor predictor of the adaptive response to selection in these models. The narrow-sense heritability sigma2A/sigma2P should be viewed as an approximation to the offspring-parent linear regression rather than the other way around.     

A new individual-based spatial approach for identifying genetic discontinuities in natural populations

The population concept is central in evolutionary and conservation biology, but identifying the boundaries of natural populations is often challenging. Here, we present a new approach for assessing spatial genetic structure without the a priori assumptions on the locations of populations made by adopting an individual-centred approach. Our method is based on assignment tests applied in a moving window over an extensively sampled study area. For each individual, a spatially explicit probability surface is constructed, showing the estimated probability of finding its multilocus genotype across the landscape, and identifying putative migrants. Population boundaries are localized by estimating the mean slope of these probability surfaces over all individuals to identify areas with genetic discontinuities. The significance of the genetic discontinuities is assessed by permutation tests. This new approach has the potential to reveal cryptic population structure and to improve our ability to understand gene flow dynamics across landscapes. We illustrate our approach by simulations and by analysing two empirical datasets: microsatellite data of Ursus arctos in Scandinavia, and amplified fragment length polymorphism (AFLP) data of Rhododendron ferrugineum in the Alps.     

A test of the hypothesis that correlational selection generates genetic correlations

Theory predicts that correlational selection on two traits will cause the major axis of the bivariate G matrix to orient itself in the same direction as the correlational selection gradient. Two testable predictions follow from this: for a given pair of traits, (1) the sign of correlational selection gradient should be the same as that of the genetic correlation, and (2) the correlational selection gradient should be positively correlated with the value of the genetic correlation. We test this hypothesis with a meta-analysis utilizing empirical estimates of correlational selection gradients and measures of the correlation between the two focal traits. Our results are consistent with both predictions and hence support the underlying hypothesis that correlational selection generates a genetic correlation between the two traits and hence orients the bivariate G matrix.     

Use of the score test as a goodness-of-fit measure of the covariance structure in genetic analysis of longitudinal data

Model selection is an essential issue in longitudinal data analysis since many different models have been proposed to fit the covariance structure. The likelihood criterion is commonly used and allows to compare the fit of alternative models. Its value does not reflect, however, the potential improvement that can still be reached in fitting the data unless a reference model with the actual covariance structure is available. The score test approach does not require the knowledge of a reference model, and the score statistic has a meaningful interpretation in itself as a goodness-of-fit measure. The aim of this paper was to show how the score statistic may be separated into the genetic and environmental parts, which is difficult with the likelihood criterion, and how it can be used to check parametric assumptions made on variance and correlation parameters. Selection of models for genetic analysis was applied to a dairy cattle example for milk production.     

The genetic consequences of selection in natural populations

The selection coefficient, s, quantifies the strength of selection acting on a genetic variant. Despite this parameter's central importance to population genetic models, until recently we have known relatively little about the value of s in natural populations. With the development of molecular genetic techniques in the late 20th century and the sequencing technologies that followed, biologists are now able to identify genetic variants and directly relate them to organismal fitness. We reviewed the literature for published estimates of natural selection acting at the genetic level and found over 3000 estimates of selection coefficients from 79 studies. Selection coefficients were roughly exponentially distributed, suggesting that the impact of selection at the genetic level is generally weak but can occasionally be quite strong. We used both nonparametric statistics and formal random‐effects meta‐analysis to determine how selection varies across biological and methodological categories. Selection was stronger when measured over shorter timescales, with the mean magnitude of s greatest for studies that measured selection within a single generation. Our analyses found conflicting trends when considering how selection varies with the genetic scale (e.g., SNPs or haplotypes) at which it is measured, suggesting a need for further research. Besides these quantitative conclusions, we highlight key issues in the calculation, interpretation, and reporting of selection coefficients and provide recommendations for future research.     

Evidence of genetic selection for growth in new recruits of a marine fish

Abstract A cohort of Diplodus sargus, a coastal marine fish abundant in the Mediterranean Sea, has been surveyed from its settlement following the pelagic larval stage up to 4 months of age, when the juveniles are moving to adult habitats in order to assess selective processes. We followed the mortality by looking at the decrease in population abundance and, simultaneously, the genetic structure using allozymes and the growth associated with each genotype to test for a relationship between genotype and phenotype. The recruitment survey demonstrated that 80% of individuals arrived within a single night and that they show very similar age providing a discrete pulse of new recruits that we followed for changes in survival and allele frequencies. After 4 months, there was a total mortality of 80.8%, with the disappearance of 181 of 224 fish that initially colonized the rocky barrier. The decrease in number followed a logarithmic model with a maximum decrease in the early period (first 30 days). The model derived from the 4 months of data demonstrates that most of the mortality in the cohort occurs over the first 120 days following settlement and the model predicted a final abundance of 10 individuals after 1 year. Within the same period of 4 months, we observed significant decrease in multilocus heterozygosity. Such a decrease in heterozygosity partly resulted from a purge of the Pgm-80* allele. Together with this major change in a natural population, an aquarium experiment demonstrated that individuals with Pgm-80* alleles show significantly lower growth than other new recruits. We propose that the decrease in frequency of Pgm-80* in the natural environment is the result of targeted predation that eliminates smaller individuals and therefore individuals bearing Pgm-80*. The potential metabolic effect as well as a scenario that could lead to the maintenance of polymorphism is discussed.     

Population genetic simulation: Benchmarking frameworks for non-standard models of natural selection

Population genetic simulation has emerged as a common tool for investigating increasingly complex evolutionary and demographic models. Software capable of handling high-level model complexity has recently been developed, and the advancement of tree sequence recording now allows simulations to merge the efficiency and genealogical insight of coalescent simulations with the flexibility of forward simulations. However, frameworks utilizing these features have not yet been compared and benchmarked. Here, we evaluate various simulation workflows using the coalescent simulator msprime and the forward simulator SLiM, to assess resource efficiency and determine an optimal simulation framework. Three aspects were evaluated: (1) the burn-in, to establish an equilibrium level of neutral diversity in the population; (2) the forward simulation, in which temporally fluctuating selection is acting; and (3) the final computation of summary statistics. We provide typical memory and computation time requirements for each step. We find that the fastest framework, a combination of coalescent and forward simulation with tree sequence recording, increases simulation speed by over twenty times compared to classical forward simulations without tree sequence recording, although it does require six times more memory. Overall, using efficient simulation workflows can lead to a substantial improvement when modelling complex evolutionary scenarios—although the optimal framework ultimately depends on the available computational resources.     

Natural selection and quantitative genetics of life-history traits in Western women: a twin study

Whether contemporary human populations are still evolving as a result of natural selection has been hotly debated. For natural selection to cause evolutionary change in a trait, variation in the trait must be correlated with fitness and be genetically heritable and there must be no genetic constraints to evolution. These conditions have rarely been tested in human populations. In this study, data from a large twin cohort were used to assess whether selection will cause a change among women in a contemporary Western population for three life-history traits: age at menarche, age at first reproduction, and age at menopause. We control for temporal variation in fecundity (the "baby boom" phenomenon) and differences between women in educational background and religious affiliation. University-educated women have 35% lower fitness than those with less than seven years education, and Roman Catholic women have about 20% higher fitness than those of other religions. Although these differences were significant, education and religion only accounted for 2% and 1% of variance in fitness, respectively. Using structural equation modeling, we reveal significant genetic influences for all three life-history traits, with heritability estimates of 0.50, 0.23, and 0.45, respectively. However, strong genetic covariation with reproductive fitness could only be demonstrated for age at first reproduction, with much weaker covariation for age at menopause and no significant covariation for age at menarche. Selection may, therefore, lead to the evolution of earlier age at first reproduction in this population. We also estimate substantial heritable variation in fitness itself, with approximately 39% of the variance attributable to additive genetic effects, the remainder consisting of unique environmental effects and small effects from education and religion. We discuss mechanisms that could be maintaining such a high heritability for fitness. Most likely is that selection is now acting on different traits from which it did in pre-industrial human populations.     

A tale of two matrices: multivariate approaches in evolutionary biology

Two symmetric matrices underlie our understanding of microevolutionary change. The first is the matrix of nonlinear selection gradients (gamma) which describes the individual fitness surface. The second is the genetic variance-covariance matrix (G) that influences the multivariate response to selection. A common approach to the empirical analysis of these matrices is the element-by-element testing of significance, and subsequent biological interpretation of pattern based on these univariate and bivariate parameters. Here, I show why this approach is likely to misrepresent the genetic basis of quantitative traits, and the selection acting on them in many cases. Diagonalization of square matrices is a fundamental aspect of many of the multivariate statistical techniques used by biologists. Applying this, and other related approaches, to the analysis of the structure of gamma and G matrices, gives greater insight into the form and strength of nonlinear selection, and the availability of genetic variance for multiple traits.     

Climate change shifts natural selection and the adaptive potential of the perennial forb Boechera stricta in the Rocky Mountains

Heritable genetic variation is necessary for populations to evolve in response to anthropogenic climate change. However, antagonistic genetic correlations among traits may constrain the rate of adaptation, even if substantial genetic variation exists. We examine potential genetic responses to selection by comparing multivariate genetic variance–covariances of traits and fitness (multivariate Robertson–Price identities) across different environments in a reciprocal transplant experiment of the forb Boechera stricta in the Rocky Mountains. By transplanting populations into four common gardens arrayed along an elevational gradient, and exposing populations to control and snow removal treatments, we simulated future and current climates and snowmelt regimes. Genetic variation in flowering and germination phenology declined in plants moved downslope to warmer, drier sites, suggesting that these traits may have a limited ability to evolve under future climates. Simulated climate change via snow removal altered the strength of selection on flowering traits, but we found little evidence that genetic correlations among traits are likely to affect the rate of adaptation to climate change. Overall, our results suggest that climate change may alter the evolutionary potential of B. stricta, but reduced expression of genetic variation may be a larger impediment to adaptation than constraints imposed by antagonistic genetic correlations.     

The prediction of adaptive evolution: empirical application of the secondary theorem of selection and comparison to the breeder's equation

Adaptive evolution occurs when fitness covaries with genetic merit for a trait (or traits). The breeder's equation (BE), in both its univariate and multivariate forms, allows us to predict this process by combining estimates of selection on phenotype with estimates of genetic (co)variation. However, predictions are only valid if all factors causal for trait-fitness covariance are measured. Although this requirement will rarely (if ever) be met in practice, it can be avoided by applying Robertson's secondary theorem of selection (STS). The STS predicts evolution by directly estimating the genetic basis of trait-fitness covariation without any explicit model of selection. Here we apply the BE and STS to four morphological traits measured in Soay sheep (Ovis aries) from St. Kilda. Despite apparently positive selection on heritable size traits, sheep are not getting larger. However, although the BE predicts increasing size, the STS does not, which is a discrepancy that suggests unmeasured factors are upwardly biasing our estimates of selection on phenotype. We suggest this is likely to be a general issue, and that wider application of the STS could offer at least a partial resolution to the common discrepancy between naive expectations and observed trait dynamics in natural populations.     

Very low levels of direct additive genetic variance in fitness and fitness components in a red squirrel population

A trait must genetically correlate with fitness in order to evolve in response to natural selection, but theory suggests that strong directional selection should erode additive genetic variance in fitness and limit future evolutionary potential. Balancing selection has been proposed as a mechanism that could maintain genetic variance if fitness components trade off with one another and has been invoked to account for empirical observations of higher levels of additive genetic variance in fitness components than would be expected from mutation–selection balance. Here, we used a long‐term study of an individually marked population of North American red squirrels (Tamiasciurus hudsonicus) to look for evidence of (1) additive genetic variance in lifetime reproductive success and (2) fitness trade‐offs between fitness components, such as male and female fitness or fitness in high‐ and low‐resource environments. “Animal model” analyses of a multigenerational pedigree revealed modest maternal effects on fitness, but very low levels of additive genetic variance in lifetime reproductive success overall as well as fitness measures within each sex and environment. It therefore appears that there are very low levels of direct genetic variance in fitness and fitness components in red squirrels to facilitate contemporary adaptation in this population.     

driftsel: an R package for detecting signals of natural selection in quantitative traits

Approaches and tools to differentiate between natural selection and genetic drift as causes of population differentiation are of frequent demand in evolutionary biology. Based on the approach of Ovaskainen et al. (2011), we have developed an R package (driftsel ) that can be used to differentiate between stabilizing selection, diversifying selection and random genetic drift as causes of population differentiation in quantitative traits when neutral marker and quantitative genetic data are available. Apart from illustrating the use of this method and the interpretation of results using simulated data, we apply the package on data from three‐spined sticklebacks (Gasterosteus aculeatus) to highlight its virtues. driftsel can also be used to perform usual quantitative genetic analyses in common‐garden study designs.     

Non-linear selection response in Drosophila: a strategy for testing the rare-alleles model of quantitative genetic variability

Quantitative genetic theory predicts that variation due to rare alleles at many loci will generate a transient acceleration in the response to directional selection. We have tested this prediction by constructing experimental lines ofDrosophila melanogaster that carry positively selected ethanol resistance alleles at low frequencies, and then subjecting the lines to directional selection for ethanol resistance. Approximately 468,000 files were subjected to artificial selection over 30 generations. The predicted non-linear selection responses were observed in all experimental lines and replicates, on three genetic backgrounds. In contrast, un-selected controls and lines carrying random alleles at low frequencies on the same genetic backgrounds exhibited linear selection responses. These results demonstrate that non-linearities due to rare alleles are detectable and repeatable, provided that experiments are done on a sufficiently large scale. The results suggest that it may be possible to test for rare-alleles as a component of naturally occurring genetic variation by careful examination of selection response curves.     

Phenological shifts in North American red squirrels: disentangling the roles of phenotypic plasticity and microevolution

Phenological shifts are the most widely reported ecological responses to climate change, but the requirements to distinguish their causes (i.e. phenotypic plasticity vs. microevolution) are rarely met. To do so, we analysed almost two decades of parturition data from a wild population of North American red squirrels (Tamiasciurus hudsonicus). Although an observed advance in parturition date during the first decade provided putative support for climate change‐driven microevolution, a closer look revealed a more complex pattern. Parturition date was heritable [h² = 0.14 (0.07–0.21 (HPD interval)] and under phenotypic selection [β = ?0.14 ± 0.06 (SE)] across the full study duration. However, the early advance reversed in the second decade. Further, selection did not act on the genetic contribution to variation in parturition date, and observed changes in predicted breeding values did not exceed those expected due to genetic drift. Instead, individuals responded plastically to environmental variation, and high food [white spruce (Picea glauca) seed] production in the first decade appears to have produced a plastic advance. In addition, there was little evidence of climate change affecting the advance, as there was neither a significant influence of spring temperature on parturition date or evidence of a change in spring temperatures across the study duration. Heritable traits not responding to selection in accordance with quantitative genetic predictions have long presented a puzzle to evolutionary ecologists. Our results on red squirrels provide empirical support for one potential solution: phenotypic selection arising from an environmental, as opposed to genetic, covariance between the phenotypic trait and annual fitness.