Genetic constraints predict evolutionary divergence in Dalechampia blossoms

If genetic constraints are important, then rates and direction of evolution should be related to trait evolvability. Here we use recently developed measures of evolvability to test the genetic constraint hypothesis with quantitative genetic data on floral morphology from the Neotropical vine Dalechampia scandens (Euphorbiaceae). These measures were compared against rates of evolution and patterns of divergence among 24 populations in two species in the D. scandens species complex. We found clear evidence for genetic constraints, particularly among traits that were tightly phenotypically integrated. This relationship between evolvability and evolutionary divergence is puzzling, because the estimated evolvabilities seem too large to constitute real constraints. We suggest that this paradox can be explained by a combination of weak stabilizing selection around moving adaptive optima and small realized evolvabilities relative to the observed additive genetic variance.     

COMPARING STRENGTHS OF DIRECTIONAL SELECTION: HOW STRONG IS STRONG?

The fundamental equation in evolutionary quantitative genetics, the Lande equation, describes the response to directional selection as a product of the additive genetic variance and the selection gradient of trait value on relative fitness. Comparisons of both genetic variances and selection gradients across traits or populations require standardization, as both are scale dependent. The Lande equation can be standardized in two ways. Standardizing by the variance of the selected trait yields the response in units of standard deviation as the product of the heritability and the variance-standardized selection gradient. This standardization conflates selection and variation because the phenotypic variance is a function of the genetic variance. Alternatively, one can standardize the Lande equation using the trait mean, yielding the proportional response to selection as the product of the squared coefficient of additive genetic variance and the mean-standardized selection gradient. Mean-standardized selection gradients are particularly useful for summarizing the strength of selection because the mean-standardized gradient for fitness itself is one, a convenient benchmark for strong selection. We review published estimates of directional selection in natural populations using mean-standardized selection gradients. Only 38 published studies provided all the necessary information for calculation of mean-standardized gradients. The median absolute value of multivariate mean-standardized gradients shows that selection is on average 54% as strong as selection on fitness. Correcting for the upward bias introduced by taking absolute values lowers the median to 31%, still very strong selection. Such large estimates clearly cannot be representative of selection on all traits. Some possible sources of overestimation of the strength of selection include confounding environmental and genotypic effects on fitness, the use of fitness components as proxies for fitness, and biases in publication or choice of traits to study.     

Predicting evolutionary potential: A numerical test of evolvability measures

Despite sophisticated mathematical models, the theory of microevolution is mostly treated as a qualitative rather than a quantitative tool. Numerical measures of selection, constraints, and evolutionary potential are often too loosely connected to theory to provide operational predictions of the response to selection. In this paper, we study the ability of a set of operational measures of evolvability and constraint to predict short‐term selection responses generated by individual‐based simulations. We focus on the effects of selective constraints under which the response in one trait is impeded by stabilizing selection on other traits. The conditional evolvability is a measure of evolutionary potential explicitly developed for this situation. We show that the conditional evolvability successfully predicts rates of evolution in an equilibrium situation, and further that these equilibria are reached with characteristic times that are inversely proportional to the fitness load generated by the constraining characters. Overall, we find that evolvabilities and conditional evolvabilities bracket responses to selection, and that they together can be used to quantify evolutionary potential on time scales where the G‐matrix remains relatively constant.     

Comparing Evolvability and Variability of Quantitative Traits

There are two distinct reasons for making comparisons of genetic variation for quantitative characters. The first is to compare evolvabilities, or ability to respond to selection, and the second is to make inferences about the forces that maintain genetic variability. Measures of variation that are standardized by the trait mean, such as the additive genetic coefficient of variation, are appropriate for both purposes. Variation has usually been compared as narrow sense heritabilities, but this is almost always an inappropriate comparative measure of evolvability and variability. Coefficients of variation were calculated from 842 estimates of trait means, variances and heritabilities in the literature. Traits closely related to fitness have higher additive genetic and nongenetic variability by the coefficient of variation criterion than characters under weak selection. This is the reverse of the accepted conclusion based on comparisons of heritability. The low heritability of fitness components is best explained by their high residual variation. The high additive genetic and residual variability of fitness traits might be explained by the great number of genetic and environmental events they are affected by, or by a lack of stabilizing selection to reduce their phenotypic variance. Over one-third of the quantitative genetics papers reviewed did not report trait means or variances. Researchers should always report these statistics, so that measures of variation appropriate to a variety of situations may be calculated.     

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.     

Selection on skewed characters and the paradox of stasis

Observed phenotypic responses to selection in the wild often differ from predictions based on measurements of selection and genetic variance. An overlooked hypothesis to explain this paradox of stasis is that a skewed phenotypic distribution affects natural selection and evolution. We show through mathematical modeling that, when a trait selected for an optimum phenotype has a skewed distribution, directional selection is detected even at evolutionary equilibrium, where it causes no change in the mean phenotype. When environmental effects are skewed, Lande and Arnold's (1983) directional gradient is in the direction opposite to the skew. In contrast, skewed breeding values can displace the mean phenotype from the optimum, causing directional selection in the direction of the skew. These effects can be partitioned out using alternative selection estimates based on average derivatives of individual relative fitness, or additive genetic covariances between relative fitness and trait (Robertson–Price identity). We assess the validity of these predictions using simulations of selection estimation under moderate sample sizes. Ecologically relevant traits may commonly have skewed distributions, as we here exemplify with avian laying date — repeatedly described as more evolutionarily stable than expected — so this skewness should be accounted for when investigating evolutionary dynamics in the wild.     

Conserved phenotypic variation patterns, evolution along lines of least resistance, and departure due to selection in fossil rodents

Within a group of organisms, some morphologies are more readily generated than others due to internal developmental constraints. Such constraints can channel evolutionary changes into directions corresponding to the greatest intraspecific variation. Long-term evolutionary outputs, however, depend on the stability of these intraspecific patterns of variation over time and from the interplay between internal constraints and selective regimes. To address these questions, the relationship between the structure of phenotypic variance covariance matrices and direction of morphological evolution was investigated using teeth of fossil rodents. One lineage considered here leads to Stephanomys, a highly specialized genus characterized by a dental pattern supposedly favoring grass eating. Stephanomys evolved in the context of directional selection related to the climatic trend of global cooling causing an increasing proportion of grasslands in southwestern Europe. The initial divergence (up to approximately 6.5 mya) was channeled along the direction of greatest intraspecific variation, whereas after 6.5 mya, morphological evolution departed from the direction favored by internal constraints. This departure from the "lines of least resistance" was likely the consequence of an environmental degradation causing a selective gradient strong enough to overwhelm the constraints to phenotypic evolution. However, in a context of stabilizing selection, these constraints actually channel evolution, as exemplified by the lineage of Apodemus. This lineage retained a primitive diet and dental pattern over the last 10 myr. Limited morphological changes occurred nevertheless in accordance with the main patterns of intraspecific variation. The importance of these lines of least resistance directing long-term morphological evolution may explain parallel evolution of some dental patterns in murine evolution.     

Heritable genetic variation via mutation-selection balance: Lerch's zeta meets the abdominal bristle

Most quantitative traits in most populations exhibit heritable genetic variation. Lande proposed that high levels of heritable variation may be maintained by mutation in the face of stabilizing selection. Several analyses have appeared of two distinct models with n additive polygenic loci subject to mutation and stabilizing selection. Each is reviewed and a new analysis and model are presented. Lande and Fleming analyzed extensions of a model originally treated by Kimura which assumes a continuum of possible allelic effects at each locus. Latter and Bulmer analyzed a model with diallelic loci. The published analyses of these models lead to qualitatively different predictions concerning the dependence of the equilibrium genetic variance on the underlying biological parameters. A new asymptotic analysis of the Kimura model shows that the different predictions are not consequences of the number of alleles assumed but rather are attributable to assumptions concerning the relative magnitudes of per locus mutation rates, the phenotypic effects of mutation, and the intensity of selection. This conclusion is reinforced by analysis of a model with triallelic loci. None of the approximate analyses presented are mathematically rigorous. To quantify their accuracy and display the domains of validity for alternative approximations, numerically determined equilibria are presented. In addition, empirical estimates of mutation rates and selection intensity are reviewed, revealing weaknesses in both the data and its connection to the models. Although the mathematical results and underlying biological requirements of my analyses are quite different from those of Lande, the results do not refute his hypothesis that considerable additive genetic variance may be maintained by mutation-selection balance. However, I argue that the validity of this hypothesis can only be determined with additional data and mathematics.     

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.     

Quantitative genetic analysis of brain size variation in sticklebacks: support for the mosaic model of brain evolution

The mosaic model of brain evolution postulates that different brain regions are relatively free to evolve independently from each other. Such independent evolution is possible only if genetic correlations among the different brain regions are less than unity. We estimated heritabilities, evolvabilities and genetic correlations of relative size of the brain, and its different regions in the three-spined stickleback (Gasterosteus aculeatus). We found that heritabilities were low (average h² = 0.24), suggesting a large plastic component to brain architecture. However, evolvabilities of different brain parts were moderate, suggesting the presence of additive genetic variance to sustain a response to selection in the long term. Genetic correlations among different brain regions were low (average r_G = 0.40) and significantly less than unity. These results, along with those from analyses of phenotypic and genetic integration, indicate a high degree of independence between different brain regions, suggesting that responses to selection are unlikely to be severely constrained by genetic and phenotypic correlations. Hence, the results give strong support for the mosaic model of brain evolution. However, the genetic correlation between brain and body size was high (r_G = 0.89), suggesting a constraint for independent evolution of brain and body size in sticklebacks.     

Comparing G: multivariate analysis of genetic variation in multiple populations

The additive genetic variance–covariance matrix (G) summarizes the multivariate genetic relationships among a set of traits. The geometry of G describes the distribution of multivariate genetic variance, and generates genetic constraints that bias the direction of evolution. Determining if and how the multivariate genetic variance evolves has been limited by a number of analytical challenges in comparing G-matrices. Current methods for the comparison of G typically share several drawbacks: metrics that lack a direct relationship to evolutionary theory, the inability to be applied in conjunction with complex experimental designs, difficulties with determining statistical confidence in inferred differences and an inherently pair-wise focus. Here, we present a cohesive and general analytical framework for the comparative analysis of G that addresses these issues, and that incorporates and extends current methods with a strong geometrical basis. We describe the application of random skewers, common subspace analysis, the 4th-order genetic covariance tensor and the decomposition of the multivariate breeders equation, all within a Bayesian framework. We illustrate these methods using data from an artificial selection experiment on eight traits in Drosophila serrata, where a multi-generational pedigree was available to estimate G in each of six populations. One method, the tensor, elegantly captures all of the variation in genetic variance among populations, and allows the identification of the trait combinations that differ most in genetic variance. The tensor approach is likely to be the most generally applicable method to the comparison of G-matrices from any sampling or experimental design.     

Pleiotropic Models of Polygenic Variation, Stabilizing Selection, and Epistasis

We show that in polymorphic populations many polygenic traits pleiotropically related to fitness are expected to be under apparent ``stabilizing selection' independently of the real selection acting on the population. This occurs, for example, if the genetic system is at a stable polymorphic equilibrium determined by selection and the nonadditive contributions of the loci to the trait value either are absent, or are random and independent of those to fitness. Stabilizing selection is also observed if the polygenic system is at an equilibrium determined by a balance between selection and mutation (or migration) when both additive and nonadditive contributions of the loci to the trait value are random and independent of those to fitness. We also compare different viability models that can maintain genetic variability at many loci with respect to their ability to account for the strong stabilizing selection on an additive trait. Let V(m) be the genetic variance supplied by mutation (or migration) each generation, V(g) be the genotypic variance maintained in the population, and n be the number of the loci influencing fitness. We demonstrate that in mutation (migration)-selection balance models the strength of apparent stabilizing selection is order V(m)/V(g). In the overdominant model and in the symmetric viability model the strength of apparent stabilizing selection is approximately 1/(2n) that of total selection on the whole phenotype. We show that a selection system that involves pairwise additive by additive epistasis in maintaining variability can lead to a lower genetic load and genetic variance in fitness (approximately 1/(2n) times) than an equivalent selection system that involves overdominance. We show that, in the epistatic model, the apparent stabilizing selection on an additive trait can be as strong as the total selection on the whole phenotype.     

GENETIC ARCHITECTURE AND POSTZYGOTIC REPRODUCTIVE ISOLATION: EVOLUTION OF BATESON–DOBZHANSKY–MULLER INCOMPATIBILITIES IN A POLYGENIC MODEL

The Bateson–Dobzhansky–Muller model predicts that postzygotic isolation evolves due to the accumulation of incompatible epistatic interactions, but few studies have quantified the relationship between genetic architecture and patterns of reproductive divergence. We examined how the direction and magnitude of epistatic interactions in a polygenic trait under stabilizing selection influenced the evolution of hybrid incompatibilities. We found that populations evolving independently under stabilizing selection experienced suites of compensatory allelic changes that resulted in genetic divergence between populations despite the maintenance of a stable, high‐fitness phenotype. A small number of loci were then incompatible with multiple alleles in the genetic background of the hybrid and the identity of these incompatibility loci changed over the evolution of the populations. For F₁ hybrids, reduced fitness evolved in a window of intermediate strengths of epistatic interactions, but F₂ and backcross hybrids evolved reduced fitness across weak and moderate strengths of epistasis due to segregation variance. Strong epistatic interactions constrained the allelic divergence of parental populations and prevented the development of reproductive isolation. Because many traits with varying genetic architectures must be under stabilizing selection, our results indicate that polygenetic drift is a plausible hypothesis for the evolution of postzygotic reproductive isolation.     

Genetic constraints on the evolution of mate recognition under natural selection

Field populations of Drosophila serrata display reproductive character displacement in cuticular hydrocarbons (CHCs) when sympatric with Drosophila birchii. We have previously shown that the naturally occurring pattern of reproductive character displacement can be experimentally replicated by exposing field allopatric populations of D. serrata to experimental sympatry with D. birchii. Here, we tested whether the repeated evolution of reproductive character displacement in natural and experimental populations was a consequence of genetic constraints on the evolution of CHCs. The genetic variance-covariance (G) matrices for CHCs were determined for populations of D. serrata that had evolved in either the presence or absence of D. birchii under field and experimental conditions. Natural selection on mate recognition under both field and experimental sympatric conditions increased the genetic variance in CHCs consistent with a response to selection based on rare alleles. A close association between G eigenstructure and the eigenstructure of the phenotypic divergence (D) matrix in natural and experimental populations suggested that G matrix eigenstructure may have determined the direction in which reproductive character displacement evolved during the reinforcement of mate recognition.     

EMPIRICAL COMPARISON OF G MATRIX TEST STATISTICS: FINDING BIOLOGICALLY RELEVANT CHANGE

A central assumption of quantitative genetic theory is that the breeder's equation ( R = GP ⁻¹ S ) accurately predicts the evolutionary response to selection. Recent studies highlight the fact that the additive genetic variance–covariance matrix ( G ) may change over time, rendering the breeder's equation incapable of predicting evolutionary change over more than a few generations. Although some consensus on whether G changes over time has been reached, multiple, often-incompatible methods for comparing G matrices are currently used. A major challenge of G matrix comparison is determining the biological relevance of observed change. Here, we develop a "selection skewers" G matrix comparison statistic that uses the breeder's equation to compare the response to selection given two G matrices while holding selection intensity constant. We present a bootstrap algorithm that determines the significance of G matrix differences using the selection skewers method, random skewers, Mantel's and Bartlett's tests, and eigenanalysis. We then compare these methods by applying the bootstrap to a dataset of laboratory populations of Tribolium castaneum . We find that the results of matrix comparison statistics are inconsistent based on differing a priori goals of each test, and that the selection skewers method is useful for identifying biologically relevant G matrix differences.     

A G matrix analogue to capture the cumulative effects of nongenetic inheritance

The genetic variance‐covariance ( G ) matrix describes the variances and covariances of genetic traits under strict genetic inheritance. Genetically expressed traits often influence trait expression in another via nongenetic forms of transmission and inheritance, however. The importance of non‐genetic influences on phenotypic evolution is increasingly clear, but how genetic and nongenetic inheritance interact to determine the response to selection is not well understood. Here, we use the ‘reachability matrix’ – a key analytical tool of geometric control theory – to integrate both forms of inheritance, capturing how the consequences of generation‐lagged maternal effects accumulate. Building on the classic Lande and Kirkpatrick model that showed how nongenetic (maternal) inheritance fundamentally alters the expected path of phenotypic evolution, we make novel inferences through decomposition of the reachability matrix. In particular, we quantify how nongenetic inheritance affects the distribution (orientation and shape) of ellipses of phenotypic change and how these distributions influence subsequent evolution. This interweaving of phenotypic means and variances accumulates generation by generation and is described analytically by the reachability matrix, which acts as an analogue of G when genetic and nongenetic inheritance both act.     

Epistasis in polygenic traits and the evolution of genetic architecture under stabilizing selection

We consider the effects of epistasis in a polygenic trait in the balance of mutation and stabilizing selection. The main issues are the genetic variation maintained in equilibrium and the evolution of the mutational effect distribution. The model assumes symmetric mutation and a continuum of alleles at all loci. Epistasis is modeled proportional to pairwise products of the single-locus effects. A general analytical formalism is developed. Assuming linkage equilibrium, we derive results for the equilibrium mutation load and the genetic and mutational variance in the house of cards and the Gaussian approximation. The additive genetic variation maintained in mutation-selection balance is reduced by any pattern of the epistatic interactions. The mutational variance, in contrast, is often increased. Large differences in mutational effects among loci emerge, and a negative correlation among (standard mean) locus mutation effects and mutation rates is predicted. Contrary to the common view since Waddington, we find that stabilizing selection in general does not lead to canalization of the trait. We propose that canalization as a target of selection instead occurs at the genic level. Here, primarily genes with a high mutation rate are buffered, often at the cost of decanalization of other genes. An intuitive interpretation of this view is given in the discussion.     

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.     

Quantitative genetic variability maintained by mutation-stabilizing selection balance: sampling variation and response to subsequent directional selection

A model of genetic variation of a quantitative character subject to the simultaneous effects of mutation, selection and drift is investigated. Predictions are obtained for the variance of the genetic variance among independent lines at equilibrium with stabilizing selection. These indicate that the coefficient of variation of the genetic variance among lines is relatively insensitive to the strength of stabilizing selection on the character. The effects on the genetic variance of a change of mode of selection from stabilizing to directional selection are investigated. This is intended to model directional selection of a character in a sample of individuals from a natural or long-established cage population. The pattern of change of variance from directional selection is strongly influenced by the strengths of selection at individual loci in relation to effective population size before and after the change of regime. Patterns of change of variance and selection responses from Monte Carlo simulation are compared to selection responses observed in experiments. These indicate that changes in variance with directional selection are not very different from those due to drift alone in the experiments, and do not necessarily give information on the presence of stabilizing selection or its strength.     

Transforming the dilemma

How does natural selection lead to cooperation between competing individuals? The Prisoner's Dilemma captures the essence of this problem. Two players can either cooperate or defect. The payoff for mutual cooperation, R, is greater than the payoff for mutual defection, P. But a defector versus a cooperator receives the highest payoff, T, where as the cooperator obtains the lowest payoff, S. Hence, the Prisoner's Dilemma is defined by the payoff ranking T > R > P > S . In a well‐mixed population, defectors always have a higher expected payoff than cooperators, and therefore natural selection favors defectors. The evolution of cooperation requires specific mechanisms. Here we discuss five mechanisms for the evolution of cooperation: direct reciprocity, indirect reciprocity, kin selection, group selection, and network reciprocity (or graph selection). Each mechanism leads to a transformation of the Prisoner's Dilemma payoff matrix. From the transformed matrices, we derive the fundamental conditions for the evolution of cooperation. The transformed matrices can be used in standard frameworks of evolutionary dynamics such as the replicator equation or stochastic processes of game dynamics in finite populations.