Evolution and stability of the G-matrix on a landscape with a moving optimum

In quantitative genetics, the genetic architecture of traits, described in terms of variances and covariances, plays a major role in determining the trajectory of evolutionary change. Hence, the genetic variance-covariance matrix (G-matrix) is a critical component of modern quantitative genetics theory. Considerable debate has surrounded the issue of G-matrix constancy because unstable G-matrices provide major difficulties for evolutionary inference. Empirical studies and analytical theory have not resolved the debate. Here we present the results of stochastic models of G-matrix evolution in a population responding to an adaptive landscape with an optimum that moves at a constant rate. This study builds on the previous results of stochastic simulations of G-matrix stability under stabilizing selection arising from a stationary optimum. The addition of a moving optimum leads to several important new insights. First, evolution along genetic lines of least resistance increases stability of the orientation of the G-matrix relative to stabilizing selection alone. Evolution across genetic lines of least resistance decreases G-matrix stability. Second, evolution in response to a continuously changing optimum can produce persistent maladaptation for a correlated trait, even if its optimum does not change. Third, the retrospective analysis of selection performs very well when the mean G-matrix (G) is known with certainty, indicating that covariance between G and the directional selection gradient beta is usually small enough in magnitude that it introduces only a small bias in estimates of the net selection gradient. Our results also show, however, that the contemporary G-matrix only serves as a rough guide to G. The most promising approach for the estimation of G is probably through comparative phylogenetic analysis. Overall, our results show that directional selection actually can increase stability of the G-matrix and that retrospective analysis of selection is inherently feasible. One major remaining challenge is to gain a sufficient understanding of the G-matrix to allow the confident estimation of G.     

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.     

The mutation matrix and the evolution of evolvability

Evolvability is a key characteristic of any evolving system, and the concept of evolvability serves as a unifying theme in a wide range of disciplines related to evolutionary theory. The field of quantitative genetics provides a framework for the exploration of evolvability with the promise to produce insights of global importance. With respect to the quantitative genetics of biological systems, the parameters most relevant to evolvability are the G-matrix, which describes the standing additive genetic variances and covariances for a suite of traits, and the M-matrix, which describes the effects of new mutations on genetic variances and covariances. A population's immediate response to selection is governed by the G-matrix. However, evolvability is also concerned with the ability of mutational processes to produce adaptive variants, and consequently the M-matrix is a crucial quantitative genetic parameter. Here, we explore the evolution of evolvability by using analytical theory and simulation-based models to examine the evolution of the mutational correlation, r(mu), the key parameter determining the nature of genetic constraints imposed by M. The model uses a diploid, sexually reproducing population of finite size experiencing stabilizing selection on a two-trait phenotype. We assume that the mutational correlation is a third quantitative trait determined by multiple additive loci. An individual's value of the mutational correlation trait determines the correlation between pleiotropic effects of new alleles when they arise in that individual. Our results show that the mutational correlation, despite the fact that it is not involved directly in the specification of an individual's fitness, does evolve in response to selection on the bivariate phenotype. The mutational variance exhibits a weak tendency to evolve to produce alignment of the M-matrix with the adaptive landscape, but is prone to erratic fluctuations as a consequence of genetic drift. The interpretation of this result is that the evolvability of the population is capable of a response to selection, and whether this response results in an increase or decrease in evolvability depends on the way in which the bivariate phenotypic optimum is expected to move. Interestingly, both analytical and simulation results show that the mutational correlation experiences disruptive selection, with local fitness maxima at -1 and +1. Genetic drift counteracts the tendency for the mutational correlation to persist at these extreme values, however. Our results also show that an evolving M-matrix tends to increase stability of the G-matrix under most circumstances. Previous studies of G-matrix stability, which assume nonevolving M-matrices, consequently may overestimate the level of instability of G relative to what might be expected in natural systems. Overall, our results indicate that evolvability can evolve in natural systems in a way that tends to result in alignment of the G-matrix, the M-matrix, and the adaptive landscape, and that such evolution tends to stabilize the G-matrix over evolutionary time.     

A quantitative genetic approach for predicting ecological change in biological communities

Ecological studies of communities have become increasingly focused on the role of genetics. These studies often conclude that genetics and evolution play an important role in community structure and function. For instance, studies have shown that the structure of insect communities associated with a host plant is heritable and therefore can potentially evolve. However, when studying communities of interacting species two problems are faced: (1) the traits that determine the outcomes of these interactions are often unknown, and (2) communities are normally highly multidimensional (n-dimensional for n species). In order to surmount these problems, we adapt a commonly used approach for studying the evolution of multivariate quantitative traits to the study of biological communities. Specifically, we propose utilizing a community-based genetic covariance matrix (G-matrix) and an associated vector of community selection gradients for predicting changes in community composition, where the “traits” under study are the abundances, or other properties, of various interacting species. This approach capitalizes on the relative ease with which data on the abundance of individuals interacting with individuals of a focal species (e.g., abundances of various herbivorous insects on a plant) can be collected and on the utility of the quantitative genetic approach for predicting multidimensional evolution. In order to evaluate the utility and accuracy of the G-matrix approach for predicting the evolution of communities, we develop and analyze numerical simulations of evolving communities. Results of these simulations show that an approach based on community G-matrices and selection gradients provides a rich understanding of how underlying genetics shape community structure and, in many cases, accurately predicts how community structure changes over time.     

Rapid divergence of genetic variance-covariance matrix within a natural population

The matrix of genetic variances and covariances (G matrix) represents the genetic architecture of multiple traits sharing developmental and genetic processes and is central for predicting phenotypic evolution. These predictions require that the G matrix be stable. Yet the timescale and conditions promoting G matrix stability in natural populations remain unclear. We studied stability of the G matrix in a 20-year evolution field experiment, where a population of the cosmopolitan parthenogenetic soil nematode Acrobeloides nanus was subjected to drift and divergent selection (benign and stress environments). Selection regime did not influence the level of absolute genetic constraints: under both regimes, two genetic dimensions for three life-history traits were identified. A substantial response to selection in principal components structure and in general matrix pattern was indicated by three statistical methods. G structure was also influenced by drift, with higher divergence under benign conditions. These results show that the G matrix might evolve rapidly in natural populations. The observed high dynamics of G structure probably represents the general feature of asexual species and limits the predictive power of G in phenotypic evolution analyses.     

Covariance structure in the skull of Catarrhini: a case of pattern stasis and magnitude evolution

The study of the genetic variance/covariance matrix (G-matrix) is a recent and fruitful approach in evolutionary biology, providing a window of investigating for the evolution of complex characters. Although G-matrix studies were originally conducted for microevolutionary timescales, they could be extrapolated to macroevolution as long as the G-matrix remains relatively constant, or proportional, along the period of interest. A promising approach to investigating the constancy of G-matrices is to compare their phenotypic counterparts (P-matrices) in a large group of related species; if significant similarity is found among several taxa, it is very likely that the underlying G-matrices are also equivalent. Here we study the similarity of covariance and correlation structure in a broad sample of Old World monkeys and apes (Catarrhini). We made phylogenetically structured comparisons of correlation and covariance matrices derived from 39 skull traits, ranging from between species to the superfamily level. We also compared the overall magnitude of integration between skull traits (r²) for all Catarrhini genera. Our results show that P-matrices were not strictly constant among catarrhines, but the amount of divergence observed among taxa was generally low. There was significant and positive correlation between the amount of divergence in correlation and covariance patterns among the 30 genera and their phylogenetic distances derived from a recently proposed phylogenetic hypothesis. Our data demonstrate that the P-matrices remained relatively similar along the evolutionary history of catarrhines, and comparisons with the G-matrix available for a New World monkey genus (Saguinus) suggests that the same holds for all anthropoids. The magnitude of integration, in contrast, varied considerably among genera, indicating that evolution of the magnitude, rather than the pattern of inter-trait correlations, might have played an important role in the diversification of the catarrhine skull.     

Evolution in stressful environments. I. Phenotypic variability, phenotypic selection, and response to selection in five distinct environmental stresses

Considerable debate has accompanied efforts to integrate the selective impacts of environmental stresses into models of life-history evolution. This study was designed to determine if different environmental stresses have consistent phenotypic effects on life-history characters and whether selection under different stresses leads to consistent evolutionary responses. We created lineages of a wild mustard (Sinapis arvensis) that were selected for three generations under five stress regimes (high boron, high salt, low light, low water, or low nutrients) or under near-optimal conditions (control). Full-sibling families from the six selection histories were divided among the same six experimental treatments. In that test generation, lifetime plant fecundity and six phenotypic traits were measured for each plant. Throughout this greenhouse study, plants were grown individually and stresses were applied from the early seedling stage through senescence. Although all stresses consistently reduced lifetime fecundity and most size- and growth-related traits, different stresses had contrasting effects on flowering time. On average, stress delayed flowering compared to favorable conditions, although plants experiencing low nutrient stress flowered earliest and those experiencing low light flowered latest. Contrary to expectations of Grime's triangle model of life-history evolution, this ruderal species does not respond phenotypically to poor environments by flowering earlier. Most stresses enhanced the evolutionary potential of the study population. Compared with near-optimal conditions, stresses tended to increase the opportunity for selection as well as phenotypic variance, although both of these quantities were reduced in some stresses. Rather than favoring traits characteristic of stress tolerance, such as slow growth and delayed reproduction, phenotypic selection favored stress-avoidance traits: earlier flowering in all five stress regimes and faster seedling height growth in three stresses. Phenotypic correlations reinforced direct selection on these traits under stress, leading to predicted phenotypic change under stress, but no significant selection in the control environment. As a result of these factors, selection under stress resulted in an evolutionary shift toward earlier flowering. Environmental stresses may drive populations of ruderal plant species like S. arvensis toward a stress-avoidance strategy, rather than toward stress tolerance. Further studies will be needed to determine when selection in stressful environments leads to these alternative life-history strategies.     

Convergent evolution of phenotypic integration and its alignment with morphological diversification in Caribbean Anolis ecomorphs

The adaptive landscape and the G-matrix are keys concepts for understanding how quantitative characters evolve during adaptive radiation. In particular, whether the adaptive landscape can drive convergence of phenotypic integration (i.e., the pattern of phenotypic variation and covariation summarized in the P-matrix) is not well studied. We estimated and compared P for 19 morphological traits in eight species of Caribbean Anolis lizards, finding that similarity in P among species was not correlated with phylogenetic distance. However, greater similarity in P among ecologically similar Anolis species (i.e., the trunk-ground ecomorph) suggests the role of convergent natural selection. Despite this convergence and relatively deep phylogenetic divergence, a large portion of eigenstructure of P is retained among our eight focal species. We also analyzed P as an approximation of G to test for correspondence with the pattern of phenotypic divergence in 21 Caribbean Anolis species. These patterns of covariation were coincident, suggesting that either genetic constraint has influenced the pattern of among-species divergence or, alternatively, that the adaptive landscape has influenced both G and the pattern of phenotypic divergence among species. We provide evidence for convergent evolution of phenotypic integration for one class of Anolis ecomorph, revealing yet another important dimension of evolutionary convergence in this group.     

SELECTION ON MOTHERS AND OFFSPRING: WHOSE PHENOTYPE IS IT AND DOES IT MATTER?

Reproductive and early life-history traits can be considered aspects of either offspring or maternal phenotype, and their evolution will therefore depend on selection operating through offspring and maternal components of fitness. Furthermore, selection at these levels may be antagonistic, with optimal offspring and maternal fitness occurring at different phenotypic values. We examined selection regimes on the correlated traits of birth weight, birth date, and litter size in Soay sheep (Ovis aries) using data from a long-term study of a free-living population on the archipelago of St. Kilda, Scotland. We tested the hypothesis that selective constraints on the evolution of the multivariate phenotype arise through antagonistic selection, either acting at offspring and maternal levels, or on correlated aspects of phenotype. All three traits were found to be under selection through variance in short-term and lifetime measures of fitness. Analysis of lifetime fitness revealed strong positive directional selection on birth weight and weaker selection for increased birth date at both levels. However, there was also evidence for stabilizing selection on these traits at the maternal level, with reduced fitness at high phenotypic values indicating lower phenotypic optima for mothers than for offspring. Additionally, antagonistic selection was found on litter size. From the offspring's point of view it is better to be born a singleton, whereas maternal fitness increases with average litter size. The decreased fitness of twins is caused by their reduced birth weight; therefore, this antagonistic selection likely results from trade-offs between litter size and birth weight that have different optimal resolutions with respect to offspring and maternal fitness. Our results highlight how selection regimes may vary depending on the assignment of reproductive and early life-history traits to either offspring or maternal phenotype.     

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.     

HOMOGENEITY OF THE GENETIC VARIANCE-COVARIANCE MATRIX FOR ANTIPREDATOR TRAITS IN TWO NATURAL POPULATIONS OF THE GARTER SNAKE THAMNOPHIS ORDINOIDES

Quantitative genetic models of evolution rely on the genetic variance-covariance matrix to predict the phenotypic response to selection. Both prospective and retrospective studies of phenotypic evolution across generations rely on assumptions about the constancy of patterns of genetic covariance through time. In the absence of robust theoretical predictions about the stability of genetic covariances, this assumption must be tested with empirical comparisons of genetic parameters among populations and species. Genetic variance-covariance matrices were estimated for a suite of antipredator traits in two populations of the northwestern garter snake, Thamnophis ordinoides. The characters studied include color pattern and antipredator behaviors that interact to facilitate escape from predators. Significant heritabilities for all traits were detected in both populations. Genetic correlations and covariances were found among behaviors in both populations and between color pattern and behavior in one of the populations. Phenotypic means differed among populations, but pairwise comparisons revealed no heterogeneity of genetic parameters between the populations. The structure of the genetic variance-covariance matrix has apparently not changed significantly during the divergence of these two populations.     

Dissecting High-Dimensional Phenotypes with Bayesian Sparse Factor Analysis of Genetic Covariance Matrices

Quantitative genetic studies that model complex, multivariate phenotypes are important for both evolutionary prediction and artificial selection. For example, changes in gene expression can provide insight into developmental and physiological mechanisms that link genotype and phenotype. However, classical analytical techniques are poorly suited to quantitative genetic studies of gene expression where the number of traits assayed per individual can reach many thousand. Here, we derive a Bayesian genetic sparse factor model for estimating the genetic covariance matrix (G-matrix) of high-dimensional traits, such as gene expression, in a mixed-effects model. The key idea of our model is that we need consider only G-matrices that are biologically plausible. An organism’s entire phenotype is the result of processes that are modular and have limited complexity. This implies that the G-matrix will be highly structured. In particular, we assume that a limited number of intermediate traits (or factors, e.g., variations in development or physiology) control the variation in the high-dimensional phenotype, and that each of these intermediate traits is sparse – affecting only a few observed traits. The advantages of this approach are twofold. First, sparse factors are interpretable and provide biological insight into mechanisms underlying the genetic architecture. Second, enforcing sparsity helps prevent sampling errors from swamping out the true signal in high-dimensional data. We demonstrate the advantages of our model on simulated data and in an analysis of a published Drosophila melanogaster gene expression data set.     

Selection on fruit traits is mediated by the interplay between frugivorous birds,fruit flies,parasitoid wasps and seed‐dispersing ants

Every organism on Earth must cope with a multitude of species interactions both directly and indirectly throughout its life cycle. However, how selection from multiple species occupying different trophic levels affects diffuse mutualisms has received little attention. As a result, how a given species amalgamates the combined effects of selection from multiple mutualists and antagonists to enhance its own fitness remains little understood. We investigated how multispecies interactions (frugivorous birds, ants, fruit flies and parasitoid wasps) generate selection on fruit traits in a seed dispersal mutualism. We used structural equation models to assess whether seed dispersers (frugivorous birds and ants) exerted phenotypic selection on fruit and seed traits in the spiny hackberry (Celtis ehrenbergiana), a fleshy‐fruited tree, and how these selection regimes were influenced by fruit fly infestation and wasp parasitoidism levels. Birds exerted negative correlational selection on the combination of fruit crop size and mean seed weight, favouring either large crops with small seeds or small crops with large seeds. Parasitoids selected plants with higher fruit fly infestation levels, and fruit flies exerted positive directional selection on fruit size, which was positively correlated with seed weight. Therefore, higher parasitoidism indirectly correlated with higher plant fitness through increased bird fruit removal. In addition, ants exerted negative directional selection on mean seed weight. Our results show that strong selection on phenotypic traits may still arise in perceived diffuse species interactions. Overall, we emphasize the need to consider diverse direct and indirect partners to achieve a better understanding of the mechanisms driving phenotypic trait evolution in multispecies interactions.     

Testing the prediction from sexual selection of a positive genetic correlation between human mate preferences and corresponding traits

Sexual selection can cause evolution in traits that affect mating success, and it has thus been implicated in the evolution of human physical and behavioural traits that influence attractiveness. We use a large sample of identical and nonidentical female twins to test the prediction from mate choice models that a trait under sexual selection will be positively genetically correlated with preference for that trait. Six of the eight preferences we investigated, i.e. height, hair colour, intelligence, creativity, exciting personality, and religiosity, exhibited significant positive genetic correlations with the corresponding traits, while the personality measures ‘easy going’ and ‘kind and understanding’ exhibited no phenotypic or genetic correlation between preference and trait. The positive results provide important evidence consistent with the involvement of sexual selection in the evolution of these human traits.     

Differences in the temporal dynamics of phenotypic selection among fitness components in the wild

The balance of selection acting through different fitness components (e.g. fecundity, mating success, survival) determines the potential tempo and trajectory of adaptive evolution. Yet the extent to which the temporal dynamics of phenotypic selection may vary among fitness components is poorly understood. Here, we compiled a database of 3978 linear selection coefficients from temporally replicated studies of selection in wild populations to address this question. Across studies, we find that multi-year selection through mating success and fecundity is stronger than selection through survival, but varies less in direction. We also report that selection through mating success varies more in long-term average strength than selection through either survival or fecundity. The consistency in direction and stronger long-term average strength of selection through mating success and fecundity suggests that selection through these fitness components should cause more persistent directional evolution relative to selection through survival. Similar patterns were apparent for the subset of studies that evaluated the temporal dynamics of selection on traits simultaneously using several different fitness components, but few such studies exist. Taken together, these results reveal key differences in the temporal dynamics of selection acting through different fitness components, but they also reveal important limitations in our understanding of how selection drives adaptive evolution.     

EVOLUTION OF VARIATION AND VARIABILITY UNDER FLUCTUATING,STABILIZING, AND DISRUPTIVE SELECTION

How variation and variability (the capacity to vary) may respond to selection remain open questions. Indeed, effects of different selection regimes on variational properties, such as canalization and developmental stability are under debate. We analyzed the patterns of among‐ and within‐individual variation in two wing‐shape characters in populations of Drosophila melanogaster maintained under fluctuating, disruptive, and stabilizing selection for more than 20 generations. Patterns of variation in wing size, which was not a direct target of selection, were also analyzed. Disruptive selection dramatically increased phenotypic variation in the two shape characters, but left phenotypic variation in wing size unaltered. Fluctuating and stabilizing selection consistently decreased phenotypic variation in all traits. In contrast, within‐individual variation, measured by the level of fluctuating asymmetry, increased for all traits under all selection regimes. These results suggest that canalization and developmental stability are evolvable and presumably controlled by different underlying genetic mechanisms, but the evolutionary responses are not consistent with an adaptive response to selection on variation. Selection also affected patterns of directional asymmetry, although inconsistently across traits and treatments.     

The evolution of population stability as a by-product of life-history evolution

Proposed mechanisms for the evolution of population stability include group selection through longterm persistence, individual selection acting directly on stability determining the demographic parameters, and the evolution of stability as a by-product of life-history evolution. None of these hypotheses currently has clear empirical support. Using two sets of Drosophila melanogaster populations, we provide experimental evidence of stability evolving as a correlated response to selection on traits not directly related to demography. Four populations (FEJs) were selected for faster development and early reproduction for 125 generations, and the other four (JBs) were ancestral controls. All FEJ and JB populations have been maintained on discrete generations at moderate density, thus eliminating differential selection on stability determining demographic parameters. We derived eight small populations from each FEJ and JB population, and subjected four small populations each to either stabilizing or destabilizing food regimes. Census data on these 64 small populations over 20 generations clearly showed that the FEJ populations have significantly less temporal fluctuations in their numbers in both food regimes compared to their controls. This greater stability of the FEJ populations is probably a by-product of the evolution of reduced fecundity and pre-adult survivorship, as a correlated response to selection for rapid development.     

Evolution of morphological integration in the skull of Carnivora (Mammalia): Changes in Canidae lead to increased evolutionary potential of facial traits

Morphological integration refers to the fact that different phenotypic traits of organisms are not fully independent from each other, and tend to covary to different degrees. The covariation among traits is thought to reflect properties of the species' genetic architecture and thus can have an impact on evolutionary responses. Furthermore, if morphological integration changes along the history of a group, inferences of past selection regimes might be problematic. Here, we evaluated the stability and evolution of the morphological integration of skull traits in Carnivora by using evolutionary simulations and phylogenetic comparative methods. Our results show that carnivoran species are able to respond to natural selection in a very similar way. Our comparative analyses show that the phylogenetic signal for pattern of integration is lower than that observed for morphology (trait averages), and that integration was stable throughout the evolution of the group. That notwithstanding, Canidae differed from other families by having higher integration, evolvability, flexibility, and allometric coefficients on the facial region. These changes might have allowed canids to rapidly adapt to different food sources, helping to explain not only the phenotypic diversification of the family, but also why humans were able to generate such a great diversity of dog breeds through artificial selection.     

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.