Structure and stability of genetic variance–covariance matrices: A Bayesian sparse factor analysis of transcriptional variation in the three‐spined stickleback

The genetic variance–covariance matrix ( G ) is a quantity of central importance in evolutionary biology due to its influence on the rate and direction of multivariate evolution. However, the predictive power of empirically estimated G ‐matrices is limited for two reasons. First, phenotypes are high‐dimensional, whereas traditional statistical methods are tuned to estimate and analyse low‐dimensional matrices. Second, the stability of G to environmental effects and over time remains poorly understood. Using Bayesian sparse factor analysis (BSFG) designed to estimate high‐dimensional G ‐matrices, we analysed levels variation and covariation in 10,527 expressed genes in a large (n = 563) half‐sib breeding design of three‐spined sticklebacks subject to two temperature treatments. We found significant differences in the structure of G between the treatments: heritabilities and evolvabilities were higher in the warm than in the low‐temperature treatment, suggesting more and faster opportunity to evolve in warm (stressful) conditions. Furthermore, comparison of G and its phenotypic equivalent P revealed the latter is a poor substitute of the former. Most strikingly, the results suggest that the expected impact of G on evolvability—as well as the similarity among G ‐matrices—may depend strongly on the number of traits included into analyses. In our results, the inclusion of only few traits in the analyses leads to underestimation in the differences between the G ‐matrices and their predicted impacts on evolution. While the results highlight the challenges involved in estimating G , they also illustrate that by enabling the estimation of large G ‐matrices, the BSFG method can improve predicted evolutionary responses to selection.     

Comparison of genetic (co)variance matrices within and between Scabiosa canescens and S. columbaria

In the current study, we used bootstrap analyses and the common principal component (CPC) method of Flury (1988) to estimate and compare the G ‐matrix of Scabiosa columbaria and S. canescens populations. We found three major patterns in the G ‐matrices: (i) the magnitude of the (co)variances was more variable among characters than among populations, (ii) different populations showed high (co)variance for different characters, and (iii) there was a tendency for S. canescens to have higher genetic (co)variances than S. columbaria. The hypothesis of equal G ‐matrices was rejected in all comparisons and there was no evidence that the matrices differed by a proportional constant in any of the analyses. The two ‘species matrices’ were found to be unrelated, both for raw data and data standardized over populations, and there was significant between‐population variation in the G ‐matrix in both species. Populations of S. canescens showed conservation of structure (principal components) in their G ‐matrices, contrasting with the lack of common structure among the S. columbaria matrices. Given these observations and the results from previous studies, we propose that selection may be responsible for some of the variation between the G ‐matrices, at least in S. columbaria and at the between‐species level.     

The stability of the G‐matrix: The role of spatial heterogeneity

The temporal stability of the genetic variance‐covariance matrix ( G ) has been discussed for a long time in the evolutionary literature. A common assumption in all studies, including empirical ones, is that spatial heterogeneity is minor such that the population can be represented by a single mean and variance. We use the well‐established allocation‐acquisition model to analyze the effect of relaxing of this assumption, simulating a case where the population is divided into patches with a variance in quality between patches. This variance can in turn differ between years. We found that changes in spatial variance in patch quality over years can make the G ‐matrix vary substantially over years and that the estimated genetic correlations, evolvability, and response to selection are different dependent on whether spatial heterogeneity is taken into account or not. This will have profound implications for our ability to predict evolutionary change and understanding of the evolutionary process.     

A TEST OF THE CONJECTURE THAT G‐MATRICES ARE MORE STABLE THAN B‐MATRICES

The G‐matrix occupies an important position in evolutionary biology both as a summary of the inheritance of quantitative traits and as an ingredient in predicting how those traits will respond to selection and drift. Consequently, the stability of G has an important bearing on the accuracy of predicted evolutionary trajectories. Furthermore, G should evolve in response to stable features of the adaptive landscape and their trajectories through time. Although the stability and evolution of G might be predicted from knowledge of selection in natural populations, most empirical comparisons of G‐matrices have been made in the absence of such a priori predictions. We present a theoretical argument that within‐sex G‐matrices should be more stable than between‐sex B‐matrices because they are more powerfully exposed to multivariate stabilizing selection. We tested this conjecture by comparing estimates of B‐ and within‐sex G‐matrices among three populations of the garter snake Thamnophis elegans. Matrix comparisons using Flury's hierarchical approach revealed that within‐sex G‐matrices had four principal components in common (full CPC), whereas B‐matrices had only a single principal component in common and eigenvalues that were more variable among populations. These results suggest that within‐sex G is more stable than B , as predicted by our theoretical argument.     

Similarity in G matrix structure among natural populations of Arabidopsis lyrata

Understanding the stability of the G matrix in natural populations is fundamental for predicting evolutionary trajectories; yet, the extent of its spatial variation and how this impacts responses to selection remain open questions. With a nested paternal half‐sib crossing design and plants grown in a field experiment, we examined differences in the genetic architecture of flowering time, floral display, and plant size among four Scandinavian populations of Arabidopsis lyrata. Using a multivariate Bayesian framework, we compared the size, shape, and orientation of G matrices and assessed their potential to facilitate or constrain trait evolution. Flowering time, floral display and rosette size varied among populations and significant additive genetic variation within populations indicated potential to evolve in response to selection. Yet, some characters, including flowering start and number of flowers, may not evolve independently because of genetic correlations. Using a multivariate framework, we found few differences in the genetic architecture of traits among populations. G matrices varied mostly in size rather than shape or orientation. Differences in multivariate responses to selection predicted from differences in G were small, suggesting overall matrix similarity and shared constraints to trait evolution among populations.     

Microevolution in time and space: SNP analysis of historical DNA reveals dynamic signatures of selection in Atlantic cod

Little is known about how quickly natural populations adapt to changes in their environment and how temporal and spatial variation in selection pressures interact to shape patterns of genetic diversity. We here address these issues with a series of genome scans in four overfished populations of Atlantic cod (Gadus morhua) studied over an 80‐year period. Screening of >1000 gene‐associated single‐nucleotide polymorphisms (SNPs) identified 77 loci that showed highly elevated levels of differentiation, likely as an effect of directional selection, in either time, space or both. Exploratory analysis suggested that temporal allele frequency shifts at certain loci may correlate with local temperature variation and with life history changes suggested to be fisheries induced. Interestingly, however, largely nonoverlapping sets of loci were temporal outliers in the different populations and outliers from the 1928 to 1960 period showed almost complete stability during later decades. The contrasting microevolutionary trajectories among populations resulted in sequential shifts in spatial outliers, with no locus maintaining elevated spatial differentiation throughout the study period. Simulations of migration coupled with observations of temporally stable spatial structure at neutral loci suggest that population replacement or gene flow alone could not explain all the observed allele frequency variation. Thus, the genetic changes are likely to at least partly be driven by highly dynamic temporally and spatially varying selection. These findings have important implications for our understanding of local adaptation and evolutionary potential in high gene flow organisms and underscore the need to carefully consider all dimensions of biocomplexity for evolutionarily sustainable management.     

THE EVOLUTIONARY STABILITY OF CROSS‐SEX,CROSS‐TRAIT GENETIC COVARIANCES

Although knowledge of the selective agents behind the evolution of sexual dimorphism has advanced considerably in recent years, we still lack a clear understanding of the evolutionary durability of cross‐sex genetic covariances that often constrain its evolution. We tested the relative stability of cross‐sex genetic covariances for a suite of homologous contact pheromones of the fruit fly Drosophila serrata, along a latitudinal gradient where these traits have diverged in mean. Using a Bayesian framework, which allowed us to account for uncertainty in all parameter estimates, we compared divergence in the total amount and orientation of genetic variance across populations, finding divergence in orientation but not total variance. We then statistically compared orientation divergence of within‐sex ( G ) to cross‐sex ( B ) covariance matrices. In line with a previous theoretical prediction, we find that the cross‐sex covariance matrix, B , is more variable than either within‐sex G matrix. Decomposition of B matrices into their symmetrical and nonsymmetrical components revealed that instability is linked to the degree of asymmetry. We also find that the degree of asymmetry correlates with latitude suggesting a role for spatially varying natural selection in shaping genetic constraints on the evolution of sexual dimorphism.     

Temporal change in genetic structure and effective population size in steelhead trout (Oncorhynchus mykiss)

There is a wealth of published molecular population genetic studies, however, most do not include historic samples and thus implicitly assume temporal genetic stability. We tested for changes in genetic diversity and structure in three populations of steelhead trout (Oncorhynchus mykiss) from a northern British Columbia watershed using seven microsatellite loci over 40 years. We found little change in genetic diversity (mean allele numbers and observed and expected heterozygosity), despite large variation in the estimated numbers of steelhead returning to the watershed over the same time period. However, the temporal stability in genetic diversity is not reflected in population structure, which appears to be high among populations, yet significantly variable over time. The neighbour-joining tree showed that, overall, two of the populations (Zymoetz and Kispiox) clustered separately from the third (Babine); a finding which was not consistent with their geographical separation. The clustering pattern was also not temporally consistent. We used the temporal method to estimate the effective number of breeders (Nb ) for the three populations; our values (Nb = 17-102) were low for the large and presumed vigorous populations of steelhead trout sampled. The low Nb values were also not consistent with the generally high genetic diversity estimates, suggesting the possibility of intermittent gene flow among the three populations. The use of temporal analyses in population genetic samples should be a priority; first, to verify observed patterns in contemporary data, and second, to build a dataset of temporal analyses to allow generalizations to be made concerning temporal genetic stability and effective population size in natural populations.     

Can invasions occur without change? A comparison of G-matrices and selection in the peach-potato aphid,Myzus persicae

Most evolutionary research on biological invasions has focused on changes seen between the native and invaded range for a particular species. However, it is likely that species that live in human‐modified habitats in their native range might have evolved specific adaptations to those environments, which increase the likelihood of establishment and spread in similar human‐altered environments. From a quantitative genetic perspective, this hypothesis suggests that both native and introduced populations should reside at or near the same adaptive peak. Therefore, we should observe no overall changes in the G (genetic variance–covariance) matrices between native and introduced ranges, and stabilizing selection on fitness‐related traits in all populations. We tested these predictions comparing three populations of the worldwide pest Myzus persicae from the Middle East (native range) and the UK and Chile (separately introduced ranges). In general, our results provide mixed support for this idea, but further comparisons of other species are needed. In particular, we found that there has been some limited evolution in the studied traits, with the Middle East population differing from the UK and Chilean populations. This was reflected in the structure of the G ‐matrices, in which Chile differed from both UK and Middle East populations. Furthermore, the amount of genetic variation was massively reduced in Chile in comparison with UK and Middle East populations. Finally, we found no detectable selection on any trait in the three populations, but clones from the introduced ranges started to reproduce later, were smaller, had smaller offspring, and had lower reproductive fitness than clones from the native range.     

Stability of population structure and genetic diversity across generations assessed by microsatellites among sympatric populations of landlocked Atlantic salmon (Salmo salar L.)

It may often be necessary to perform genetic analyses of temporal replicates to estimate the significance of spatial variation independently from that of temporal variation in order to ensure the reliability of estimates of a defined population structure. Nevertheless, temporal studies of genetic diversity remain scarce in the literature relative to the plethora of empirical studies of population structure. In vertebrates, a limited number of studies have specifically assessed the temporal stability of population structure for more than one generation. In this study, we performed a microsatellite analysis of DNA obtained from archived scales to compare the population structure among four sympatric landlocked populations of Atlantic salmon ( Salmo salar ) over a time frame of three to five generations. The same patterns of allele frequency distribution, θ, R _ST and genetic distance estimates were observed among populations for two time periods, confirming the temporal stability of the population structure. Despite population declines and stocking during this period, no statistically significant changes in intrapopulation genetic diversity were apparent. This study illustrates the feasibility and usefulness of microsatellite analysis of temporal samples, not only to infer changes of intrapopulation genetic diversity, but also to assess the stability of population structure over a time frame of several generations.     

Similarity in temporal variation in sex‐biased dispersal over short and long distances in the dark‐eyed junco,Junco hyemalis

Patterns of sex‐biased dispersal (SBD) are typically consistent within taxa, for example female‐biased in birds and male‐biased in mammals, leading to theories about the evolutionary pressures that lead to SBD. However, generalizations about the evolution of sex biases tend to overlook that dispersal is mediated by ecological factors that vary over time. We examined potential temporal variation in between‐ and within‐population dispersal over an 11‐year period in a bird, the dark‐eyed junco (Junco hyemalis). We measured between‐population dispersal patterns using genetic assignment indices and found yearly variation in which sex was more likely to have immigrated. When we measured within‐population spatial genetic structure and mark–recapture dispersal distances, we typically found yearly SBD patterns that mirrored between‐population dispersal, indicating common eco‐evolutionary causes despite expected differences due to the scale of dispersal. However, in years without detectable between‐population sex biases, we found genetic similarity between nearby males within our population. This suggests that, in certain circumstances, ecological pressures may act on within‐population dispersal without affecting dispersal between populations. Alternatively, current analytical tools may be better able to detect within‐population SBD. Future work will investigate potential causes of the observed temporal variation in dispersal patterns and whether they have greater effects on within‐population dispersal.     

Paleogenetic records of Daphnia pulicaria in two North American lakes reveal the impact of cultural eutrophication

Understanding the evolutionary consequences of the green revolution, particularly in wild populations, is an important frontier in contemporary biology. Because human impacts have occurred at varying magnitudes or time periods depending on the study ecosystem, evolutionary histories may vary considerably among populations. Paleogenetics in conjunction with paleolimnology enable us to associate microevolutionary dynamics with detailed information on environmental change. We used this approach to reconstruct changes in the temporal population genetic structure of the keystone zooplankton grazer, Daphnia pulicaria, using dormant eggs extracted from sediments in two Minnesota lakes (South Center, Hill). The extent of agriculture and human population density in the catchment of these lakes has differed markedly since European settlement in the late 19th century and is reflected in their environmental histories reconstructed here. The reconstructed environments of these two lakes differed strongly in terms of environmental stability and their associated patterns of Daphnia population structure. We detected long periods of stability in population structure and environmental conditions in South Center Lake that were followed by a dramatic temporal shift in population genetic structure after the onset of European settlement and industrialized agriculture in its watershed. In particular, we noted a 24.3‐fold increase in phosphorus (P) flux between pre‐European and modern sediment P accumulation rates (AR) in this lake. In contrast, no such shifts were detected in Hill Lake, where the watershed was not as impacted by European settlement and rates of change were less directional with a much smaller increase in sediment P AR (2.3‐fold). We identify direct and indirect effects of eutrophication proxies on genetic structure in these lake populations and demonstrate the power of using this approach in understanding the consequences of anthropogenic environmental change on natural populations throughout historic time periods.     

Short‐term temporal instability in fine‐scale genetic structure of masu salmon

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Temporal genetic stability in natural populations of the waterflea Daphnia magna in response to strong selection pressure

Studies monitoring changes in genetic diversity and composition through time allow a unique understanding of evolutionary dynamics and persistence of natural populations. However, such studies are often limited to species with short generation times that can be propagated in the laboratory or few exceptional cases in the wild. Species that produce dormant stages provide powerful models for the reconstruction of evolutionary dynamics in the natural environment. A remaining open question is to what extent dormant egg banks are an unbiased representation of populations and hence of the species’ evolutionary potential, especially in the presence of strong environmental selection. We address this key question using the water flea Daphnia magna, which produces dormant stages that accumulate in biological archives over time. We assess temporal genetic stability in three biological archives, previously used in resurrection ecology studies showing adaptive evolutionary responses to rapid environmental change. We show that neutral genetic diversity does not decline with the age of the population and it is maintained in the presence of strong selection. In addition, by comparing temporal genetic stability in hatched and unhatched populations from the same biological archive, we show that dormant egg banks can be consulted to obtain a reliable measure of genetic diversity over time, at least in the multidecadal time frame studied here. The stability of neutral genetic diversity through time is likely mediated by the buffering effect of the resting egg bank.     

TYPE I ERROR RATES FOR TESTING GENETIC DRIFT WITH PHENOTYPIC COVARIANCE MATRICES: A SIMULATION STUDY

Studies of evolutionary divergence using quantitative genetic methods are centered on the additive genetic variance–covariance matrix ( G ) of correlated traits. However, estimating G properly requires large samples and complicated experimental designs. Multivariate tests for neutral evolution commonly replace average G by the pooled phenotypic within‐group variance–covariance matrix ( W ) for evolutionary inferences, but this approach has been criticized due to the lack of exact proportionality between genetic and phenotypic matrices. In this study, we examined the consequence, in terms of type I error rates, of replacing average G by W in a test of neutral evolution that measures the regression slope between among‐population variances and within‐population eigenvalues (the Ackermann and Cheverud [AC] test) using a simulation approach to generate random observations under genetic drift. Our results indicate that the type I error rates for the genetic drift test are acceptable when using W instead of average G when the matrix correlation between the ancestral G and P is higher than 0.6, the average character heritability is above 0.7, and the matrices share principal components. For less‐similar G and P matrices, the type I error rates would still be acceptable if the ratio between the number of generations since divergence and the effective population size (t/N_e) is smaller than 0.01 (large populations that diverged recently). When G is not known in real data, a simulation approach to estimate expected slopes for the AC test under genetic drift is discussed.     

Effects of the demographic transition on the genetic variances and covariances of human life‐history traits

The recent demographic transitions to lower mortality and fertility rates in most human societies have led to changes and even quick reversals in phenotypic selection pressures. This can only result in evolutionary change if the affected traits are heritable, but changes in environmental conditions may also lead to subsequent changes in the genetic variance and covariance (the G matrix) of traits. It currently remains unclear if there have been concomitant changes in the G matrix of life‐history traits following the demographic transition. Using 300 years of genealogical data from Finland, we found that four key life‐history traits were heritable both before and after the demographic transition. The estimated heritabilities allow a quantifiable genetic response to selection during both time periods, thus facilitating continued evolutionary change. Further, the G matrices remained largely stable but revealed a trend for an increased additive genetic variance and thus evolutionary potential of the population after the transition. Our results demonstrate the validity of predictions of evolutionary change in human populations even after the recent dramatic environmental change, and facilitate predictions of how our biology interacts with changing environments, with implications for global public health and demography.     

Characterizing the evolution of genetic variance using genetic covariance tensors

Determining how genetic variance changes under selection in natural populations has proved to be a very resilient problem in evolutionary genetics. In the same way that understanding the availability of genetic variance within populations requires the simultaneous consideration of genetic variance in sets of functionally related traits, determining how genetic variance changes under selection in natural populations will require ascertaining how genetic variance–covariance (G) matrices evolve. Here, we develop a geometric framework using higher-order tensors, which enables the empirical characterization of how G matrices have diverged among populations. We then show how divergence among populations in genetic covariance structure can then be associated with divergence in selection acting on those traits using key equations from evolutionary theory. Using estimates of G matrices of eight male sexually selected traits from nine geographical populations of Drosophila serrata, we show that much of the divergence in genetic variance occurred in a single trait combination, a conclusion that could not have been reached by examining variation among the individual elements of the nine G matrices. Divergence in G was primarily in the direction of the major axes of genetic variance within populations, suggesting that genetic drift may be a major cause of divergence in genetic variance among these populations.     

The effects of stochastic and episodic movement of the optimum on the evolution of the G‐matrix and the response of the trait mean to selection

Theoretical and empirical results demonstrate that the G ‐matrix, which summarizes additive genetic variances and covariances of quantitative traits, changes over time. Such evolution and fluctuation of the G ‐matrix could potentially have wide‐ranging effects on phenotypic evolution. Nevertheless, no studies have yet addressed G ‐matrix stability and evolution when movement of an intermediate optimum includes large, episodic jumps or stochasticity. Here, we investigate such scenarios by using simulation‐based models of G ‐matrix evolution. These analyses yield four important insights regarding the evolution and stability of the G ‐matrix. (i) Regardless of the model of peak movement, a moving optimum causes the G ‐matrix to orient towards the direction of net peak movement, so that genetic variance is enhanced in that direction (the variance enhancement effect). (ii) Peak movement skews the distribution of breeding values in the direction of movement, which impedes the response to selection. (iii) The stability of the G ‐matrix is affected by the overall magnitude and direction of peak movement, but modes and rates of peak movement have surprisingly small effects (the invariance principle). (iv) Both episodic and stochastic peak movement increase the probability that a population will fall below its carrying capacity and go extinct. We also present novel equations for the response of the trait mean to multivariate selection, which take into account the higher moments of the distribution of breeding values.     

Relaxation of herbivore‐mediated selection drives the evolution of genetic covariances between plant competitive and defense traits

Insect herbivores are important mediators of selection on traits that impact plant defense against herbivory and competitive ability. Although recent experiments demonstrate a central role for herbivory in driving rapid evolution of defense and competition‐mediating traits, whether and how herbivory shapes heritable variation in these traits remains poorly understood. Here, we evaluate the structure and evolutionary stability of the G matrix for plant metabolites that are involved in defense and allelopathy in the tall goldenrod, Solidago altissima. We show that G has evolutionarily diverged between experimentally replicated populations that evolved in the presence versus the absence of ambient herbivory, providing direct evidence for the evolution of G by natural selection. Specifically, evolution in an herbivore‐free habitat altered the orientation of G , revealing a negative genetic covariation between defense‐ and competition‐related metabolites that is typically masked in herbivore‐exposed populations. Our results may be explained by predictions of classical quantitative genetic theory, as well as the theory of acquisition‐allocation trade‐offs. The study provides compelling evidence that herbivory drives the evolution of plant genetic architecture.     

Population differentiation in G matrix structure due to natural selection in Rana temporaria

The additive genetic variance-covariance matrix (G) is a concept central to discussions about evolutionary change over time in a suite of traits. However, at the moment we do not know how fast G itself changes as a consequence of selection or how sensitive it is to environmental influences. We investigated possible evolutionary divergence and environmental influences on G using data from a factorial common-garden experiment where common frog (Rana temporaria) tadpoles from two divergent populations were exposed to three different environmental treatments. G-matrices were estimated using an animal model approach applied to data from a NCII breeding design. Matrix comparisons using both Flury and multivariate analysis of variance methods revealed significant differences in G matrices both between populations and between treatments within populations, the former being generally larger than the latter. Comparison of levels of population differentiation in trait means using Q(ST) indices with that observed in microsatellite markers (F(ST)) revealed that the former values generally exceeded the neutral expectation set by F(ST). Hence, the results suggest that intraspecific divergence in G matrix structure has occurred mainly due to natural selection.