General quantitative genetic methods for comparative biology: phylogenies,taxonomies and multi‐trait models for continuous and categorical characters

Although many of the statistical techniques used in comparative biology were originally developed in quantitative genetics, subsequent development of comparative techniques has progressed in relative isolation. Consequently, many of the new and planned developments in comparative analysis already have well‐tested solutions in quantitative genetics. In this paper, we take three recent publications that develop phylogenetic meta‐analysis, either implicitly or explicitly, and show how they can be considered as quantitative genetic models. We highlight some of the difficulties with the proposed solutions, and demonstrate that standard quantitative genetic theory and software offer solutions. We also show how results from Bayesian quantitative genetics can be used to create efficient Markov chain Monte Carlo algorithms for phylogenetic mixed models, thereby extending their generality to non‐Gaussian data. Of particular utility is the development of multinomial models for analysing the evolution of discrete traits, and the development of multi‐trait models in which traits can follow different distributions. Meta‐analyses often include a nonrandom collection of species for which the full phylogenetic tree has only been partly resolved. Using missing data theory, we show how the presented models can be used to correct for nonrandom sampling and show how taxonomies and phylogenies can be combined to give a flexible framework with which to model dependence.     

QUANTITATIVE GENETICS AND POPULATION DYNAMICS

This study examines the dynamics of a competition and a host-parasite model in which the interactions are determined by quantitative characters. Both models are extensions of one-dimensional difference equations that can exhibit complicated dynamics. Compared to these basic models, the phenotypic variability given by the quantitative characters reduces the size of the density fluctuations in asexual populations. With sexual reproduction, which is described by modeling the genetics of the quantitative character explicitly with many haploid loci that determine the character additively, this reduction in fitness variance is magnified. Moreover, quantitative genetics can induce simple dynamics. For example, the sexual population can have a two-cycle when the asexual system is chaotic. This paper discusses the consequences for the evolution of sex. The higher mean growth rate implied by the lower fitness variance in sexual populations is an advantage that can overcome a twofold intrinsic growth rate of asexuals. The advantage is bigger when the asexual population contains only a subset of the phenotypes present in the sexual population, which conforms with the tangled bank theory for the evolution of sex and shows that tangled bank effects also occur in host-parasite systems. The results suggest that explicitly describing the genetics of a quantitative character leads to more flexible models than the usual assumption of normal character distributions, and therefore to a better understanding of the character's impact on population dynamics.     

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.     

Gene flow and genetic drift in urban environments

Evidence is growing that human modification of landscapes has dramatically altered evolutionary processes. In urban population genetic studies, urbanization is typically predicted to act as a barrier that isolates populations of species, leading to increased genetic drift within populations and reduced gene flow between populations. However, urbanization may also facilitate dispersal among populations, leading to higher genetic diversity within, and lower differentiation between, urban populations. We reviewed the literature on nonadaptive urban evolution to evaluate the support for each of these urban fragmentation and facilitation models. In a review of the literature with supporting quantitative analyses of 167 published urban population genetics studies, we found a weak signature of reduced within‐population genetic diversity and no evidence of consistently increased between‐population genetic differentiation associated with urbanization. In addition, we found that urban landscape features act as barriers or conduits to gene flow, depending on the species and city in question. Thus, we speculate that dispersal ability of species and environmental heterogeneity between cities contributes to the variation exhibited in our results. However, >90% of published studies reviewed here showed an association of urbanization with genetic drift or gene flow, highlighting the strong impact of urbanization on nonadaptive evolution. It is clear that species biology and city heterogeneity obscure patterns of genetic drift and gene flow in a quantitative analysis. Thus, we suggest that future research makes comparisons of multiple cities and nonurban habitats, and takes into consideration species' natural history, environmental variation, spatial modelling and marker selection.     

The past,present and future of reproductive skew theory and experiments

A major evolutionary question is how reproductive sharing arises in cooperatively breeding species despite the inherent reproductive conflicts in social groups. Reproductive skew theory offers one potential solution: each group member gains or is allotted inclusive fitness equal to or exceeding their expectation from reproducing on their own. Unfortunately, a multitude of skew models with conflicting predictions has led to confusion in both testing and evaluating skew theory. The confusion arises partly because one set of models (the ‘transactional’ type) answer the ultimate evolutionary question of what ranges of reproductive skew can yield fitness‐enhancing solutions for all group members. The second set of models (‘compromise’) give an evolutionarily proximate, game‐theoretic evolutionarily stable state (ESS) solution that determines reproductive shares based on relative competitive abilities. However, several predictions arising from compromise models require a linear payoff to increased competition and do not hold with non‐linear payoffs. Given that for most species it may be very difficult or impossible to determine the true relationship between effort devoted to competition and reproductive share gained, compromise models are much less predictive than previously appreciated. Almost all skew models make one quantitative prediction (e.g. realized skew must fall within ranges predicted by transactional models), and two qualitative predictions (e.g. variation in relatedness or competitive ability across groups affects skew). A thorough review of the data finds that these three predictions are relatively rarely supported. As a general rule, therefore, the evolution of cooperative breeding appears not to be dependent on the ability of group members to monitor relatedness or competitive ability in order to adjust their behaviour dynamically to gain reproductive share. Although reproductive skew theory fails to predict within‐group dynamics consistently, it does better at predicting quantitative differences in skew across populations or species. This suggests that kin selection can play a significant role in the evolution of sociality. To advance our understanding of reproductive skew will require focusing on a broader array of factors, such as the frequency of mistaken identity, delayed fitness payoffs, and selection pressures arising from across‐group competition. We furthermore suggest a novel approach to investigate the sharing of reproduction that focuses on the underlying genetics of skew. A quantitative genetics approach allows the partitioning of variance in reproductive share itself or that of traits closely associated with skew into genetic and non‐genetic sources. Thus, we can determine the heritability of reproductive share and infer whether it actually is the focus of natural selection. We view the ‘animal model’ as the most promising empirical method where the genetics of reproductive share can be directly analyzed in wild populations. In the quest to assess whether skew theory can provide a framework for understanding the evolution of sociality, quantitative genetics will be a central tool in future research.     

A synthetic theory of molecular evolution

According to the neo-Darwinian view of evolution evolution rate nu depends solely on the environment variation rate gamma, whereas in the non-Darwinian view evolution rate is determined mainly by the mutation rate mu. We have studied two kinds of population genetics models which exhibit both types of evolution in different parametric regions: one is a dynamical model representing infinite population, and the other is a Markov process model representing a nearly monomorphic finite population. In the infinite population model, after proving general time-derivative and mu-derivative formulas for the population average of quantitative traits, we show that if the mutation rate is adaptively determined, mu must be larger than nu in the stationary state. Loads of evolution are obtained in both regions. A high evolution rate such as nu = 1 per genome per generation is consistent with Haldane's value of tolerable load if and only if the functional constraint is not large and selection is weak, independent of whether the evolution is neo-Darwinian or non-Darwinian. As the selection intensity increases, nu is shown to change discontinuously from nearly mu to gamma at the transition point. In the finite population model, the transition of v is not discontinuous, but is very steep. On the other hand, no steep change of polymorphism takes place at the transition point. The steepness of the transition in our model suggests that real molecular evolution can be divided into either neo-Darwinian or non-Darwinian,and that the intermediate type of evolution is rather rare.     

Characterizing morphological (co)variation using structural equation models: Body size,allometric relationships and evolvability in a house sparrow metapopulation

Body size plays a key role in the ecology and evolution of all organisms. Therefore, quantifying the sources of morphological (co)variation, dependent and independent of body size, is of key importance when trying to understand and predict responses to selection. We combine structural equation modeling with quantitative genetics analyses to study morphological (co)variation in a meta‐population of house sparrows (Passer domesticus). As expected, we found evidence of a latent variable “body size,” causing genetic and environmental covariation between morphological traits. Estimates of conditional evolvability show that allometric relationships constrain the independent evolution of house sparrow morphology. We also found spatial differences in general body size and its allometric relationships. On islands where birds are more dispersive and mobile, individuals were smaller and had proportionally longer wings for their body size. Although on islands where sparrows are more sedentary and nest in dense colonies, individuals were larger and had proportionally longer tarsi for their body size. We corroborated these results using simulations and show that our analyses produce unbiased allometric slope estimates. This study highlights that in the short term allometric relationships may constrain phenotypic evolution, but that in the long term selection pressures can also shape allometric relationships.     

A general theory for the evolutionary dynamics of virulence

Most theory on the evolution of virulence is based on a game-theoretic approach. One potential shortcoming of this approach is that it does not allow the prediction of the evolutionary dynamics of virulence. Such dynamics are of interest for several reasons: for experimental tests of theory, for the development of useful virulence management protocols, and for understanding virulence evolution in situations where the epidemiological dynamics never reach equilibrium and/or when evolutionary change occurs on a timescale comparable to that of the epidemiological dynamics. Here we present a general theory similar to that of quantitative genetics in evolutionary biology that allows for the easy construction of models that include both within-host mutation as well as superinfection and that is capable of predicting both the short- and long-term evolution of virulence. We illustrate the generality and intuitive appeal of the theory through a series of examples showing how it can lead to transparent interpretations of the selective forces governing virulence evolution. It also leads to novel predictions that are not possible using the game-theoretic approach. The general theory can be used to model the evolution of other pathogen traits as well.     

Geonomics: Forward-Time,Spatially Explicit,and Arbitrarily Complex Landscape Genomic Simulations

Understanding the drivers of spatial patterns of genomic diversity has emerged as a major goal of evolutionary genetics. The flexibility of forward-time simulation makes it especially valuable for these efforts, allowing for the simulation of arbitrarily complex scenarios in a way that mimics how real populations evolve. Here, we present Geonomics, a Python package for performing complex, spatially explicit, landscape genomic simulations with full spatial pedigrees that dramatically reduces user workload yet remains customizable and extensible because it is embedded within a popular, general-purpose language. We show that Geonomics results are consistent with expectations for a variety of validation tests based on classic models in population genetics and then demonstrate its utility and flexibility with a trio of more complex simulation scenarios that feature polygenic selection, selection on multiple traits, simulation on complex landscapes, and nonstationary environmental change. We then discuss runtime, which is primarily sensitive to landscape raster size, memory usage, which is primarily sensitive to maximum population size and recombination rate, and other caveats related to the model’s methods for approximating recombination and movement. Taken together, our tests and demonstrations show that Geonomics provides an efficient and robust platform for population genomic simulations that capture complex spatial and evolutionary dynamics.     

Parent-offspring conflict and co-adaptation: behavioural ecology meets quantitative genetics

The evolution of the complex and dynamic behavioural interactions between caring parents and their dependent offspring is a major area of research in behavioural ecology and quantitative genetics. While behavioural ecologists examine the evolution of interactions between parents and offspring in the light of parent-offspring conflict and its resolution, quantitative geneticists explore the evolution of such interactions in the light of parent-offspring co-adaptation due to combined effects of parental and offspring behaviours on fitness. To date, there is little interaction or integration between these two fields. Here, we first review the merits and limitations of each of these two approaches and show that they provide important complementary insights into the evolution of strategies for offspring begging and parental resource provisioning. We then outline how central ideas from behavioural ecology and quantitative genetics can be combined within a framework based on the concept of behavioural reaction norms, which provides a common basis for behavioural ecologists and quantitative geneticists to study the evolution of parent-offspring interactions. Finally, we discuss how the behavioural reaction norm approach can be used to advance our understanding of parent-offspring conflict by combining information about the genetic basis of traits from quantitative genetics with key insights regarding the adaptive function and dynamic nature of parental and offspring behaviours from behavioural ecology.     

A unifying framework for metapopulation dynamics

Many biologically important processes, such as genetic differentiation, the spread of disease, and population stability, are affected by the (natural or enforced) subdivision of populations into networks of smaller, partly isolated, subunits. Such "metapopulations" can have extremely complex dynamics. We present a new general model that uses only two functions to capture, at the metapopulation scale, the main behavior of metapopulations. We show how complex, structured metapopulation models can be translated into our generalized framework. The metapopulation dynamics arising from some important biological processes are illustrated: the rescue effect, the Allee effect, and what we term the "antirescue effect." The antirescue effect captures instances where high migration rates are deleterious to population persistence, a phenomenon that has been largely ignored in metapopulation conservation theory. Management regimes that ignore a significant antirescue effect will be inadequate and may actually increase extinction risk. Further, consequences of territoriality and conspecific attraction on metapopulation-level dynamics are investigated. The new, simplified framework can incorporate knowledge from epidemiology, genetics, and population biology in a phenomenological way. It opens up new possibilities to identify and analyze the factors that are important for the evolution and persistence of the many spatially subdivided species.     

The genetic signature of rapid range expansions: How dispersal,growth and invasion speed impact heterozygosity and allele surfing

As researchers collect spatiotemporal population and genetic data in tandem, models that connect demography and dispersal to genetics are increasingly relevant. The dominant spatiotemporal model of invasion genetics is the stepping-stone model which represents a gradual range expansion in which individuals jump to uncolonized locations one step at a time. However, many range expansions occur quickly as individuals disperse far from currently colonized regions. For these types of expansion, stepping-stone models are inappropriate. To more accurately reflect wider dispersal in many organisms, we created kernel-based models of invasion genetics based on integrodifference equations. Classic theory relating to integrodifference equations suggests that the speed of range expansions is a function of population growth and dispersal. In our simulations, populations that expanded at the same speed but with spread rates driven by dispersal retained more heterozygosity along axes of expansion than range expansions with rates of spread that were driven primarily by population growth. To investigate surfing we introduced mutant alleles in wave fronts of simulated range expansions. In our models based on random mating, surfing alleles remained at relatively low frequencies and surfed less often compared to previous results based on stepping-stone simulations with asexual reproduction.     

Inferring number of populations and changes in connectivity under the n-island model

Inferring the demographic history of species is one of the greatest challenges in populations genetics. This history is often represented as a history of size changes, ignoring population structure. Alternatively, when structure is assumed, it is defined a priori as a population tree and not inferred. Here we propose a framework based on the IICR (Inverse Instantaneous Coalescence Rate). The IICR can be estimated for a single diploid individual using the PSMC method of Li and Durbin (2011). For an isolated panmictic population, the IICR matches the population size history, and this is how the PSMC outputs are generally interpreted. However, it is increasingly acknowledged that the IICR is a function of the demographic model and sampling scheme with limited connection to population size changes. Our method fits observed IICR curves of diploid individuals with IICR curves obtained under piecewise stationary symmetrical island models. In our models we assume a fixed number of time periods during which gene flow is constant, but gene flow is allowed to change between time periods. We infer the number of islands, their sizes, the periods at which connectivity changes and the corresponding rates of connectivity. Validation with simulated data showed that the method can accurately recover most of the scenario parameters. Our application to a set of five human PSMCs yielded demographic histories that are in agreement with previous studies using similar methods and with recent research suggesting ancient human structure. They are in contrast with the view of human evolution consisting of one ancestral population branching into three large continental and panmictic populations with varying degrees of connectivity and no population structure within each continent.Subject terms: Population genetics, Biological models, Population genetics     

Genome size, quantitative genetics and the genomic basis for flower size evolution in Silene latifolia

BACKGROUND AND AIMS: The overall goal of this paper is to construct an overview of the genetic basis for flower size evolution in Silene latifolia. It aims to examine the relationship between the molecular bases for flower size and the underlying assumption of quantitative genetics theory that quantitative variation is ultimately due to the impact of a number of structural genes. SCOPE: Previous work is reviewed on the quantitative genetics and potential for response to selection on flower size, and the relationship between flower size and nuclear DNA content in S. latifolia. These earlier findings provide a framework within which to consider more recent analyses of a joint quantitative trait loci (QTL) analysis of flower size and DNA content in this species. KEY RESULTS: Flower size is a character that fits the classical quantitative genetics model of inheritance very nicely. However, an earlier finding that flower size is correlated with nuclear DNA content suggested that quantitative aspects of genome composition rather than allelic substitution at structural loci might play a major role in the evolution of flower size. The present results reported here show that QTL for flower size are correlated with QTL for DNA content, further corroborating an earlier result and providing additional support for the conclusion that localized variations in DNA content underlie evolutionary changes in flower size. CONCLUSIONS: The search image for QTL should be broadened to include overall aspects of genome regulation. As we prepare to enter the much-heralded post-genomic era, we also need to revisit our overall models of the relationship between genotype and phenotype to encompass aspects of genome structure and composition beyond structural genes.     

How accurate are population models? Lessons from landscape‐scale tests in a fragmented system

There is a growing debate about the ability of Population Viability Analysis (PVA) to predict the risk of extinction. Previously, the debate has focused largely on models where spatial variation and species movement are ignored. We present a synthesis of the key results for an array of different species for which detailed tests of the accuracy of PVA models were completed. These models included spatial variation in habitat quality and the movement of individuals across a landscape. The models were good approximations for some species, but poor for others. Predictive ability was limited by complex processes typically overlooked in spatial population models, these being interactions between landscape structure and life history attributes. Accuracy of models could not be determined a priori, although model tests indicated how they might be improved. Importantly, model predictions were poor for some species that are among the best‐studied vertebrates in Australia. This indicated that although the availability of good life history data is a key part of PVA other factors also influence model accuracy. We were also able to draw broad conclusions about the sorts of populations and life history characteristics where model predictions are likely to be less accurate. Predictions of extinction risk are often essential for real‐world population management. Therefore, we believe that although PVA has been shown to be less than perfect, it remains a useful tool particularly in the absence of alternative approaches. Hence, tests of PVA models should be motivated by the cycle of testing and improvement.     

The evolution of isochores: evidence from SNP frequency distributions

The large-scale systematic variation in nucleotide composition along mammalian and avian genomes has been a focus of the debate between neutralist and selectionist views of molecular evolution. Here we test whether the compositional variation is due to mutation bias using two new tests, which do not assume compositional equilibrium. In the first test we assume a standard population genetics model, but in the second we make no assumptions about the underlying population genetics. We apply the tests to single-nucleotide polymorphism data from noncoding regions of the human genome. Both models of neutral mutation bias fit the frequency distributions of SNPs segregating in low- and medium-GC-content regions of the genome adequately, although both suggest compositional nonequilibrium. However, neither model fits the frequency distribution of SNPs from the high-GC-content regions. In contrast, a simple population genetics model that incorporates selection or biased gene conversion cannot be rejected. The results suggest that mutation biases are not solely responsible for the compositional biases found in noncoding regions.     

Fixation probabilities of additive alleles in diploid populations

The calculation of the survival probability of a selectively advantageous allele is a central part of the quantitative theory of genetic evolution. However, several areas of investigation in population genetics theory, including the generalized neutrality theory, the concept of Muller's ratchet, and the risk of extinction of sexually reproducing populations due to the accumulation of deleterious mutations, rely on the calculation of the survival probability of selectively disadvantageous mutant genes. The calculation of these probabilities in the standard Wright-Fisher model of genetic evolution appears to be intractable, and yet is a key element in the above investigations. In this paper we find bounds for the fixation probability of deleterious and advantageous additive mutants, as well as finding close approximations for these probabilities. In addition, we derive analytical estimates for the relative error of our approximations and compare our results with those from numerical computation. Our results justify the diffusion approximation for the fixation probability of a single mutant.     

Point Processes for Fine‐Scale Spatial Genetics and Molecular Ecology

This paper introduces point processes into fine‐scale spatial genetics and molecular ecology. Datasets given in the form of a complete map of individuals and their genotypes can be analyzed by means of the theory of marked or multivariate point processes. Beginning with reformulation of conventional spatial autocorrelation statistics in genetics by the language of point processes, this paper first shows an example of point process models that describe spatial patterns of both tree locations and their genotypes, on the assumption of limited seed dispersal and long pollen movement. The results show that isolation‐by‐distance slightly occurs from the assumption above, and more importantly, an increment of the degree of clustering of trees reduces the degree of genetic clustering. Next, the point process model is applied to field data of secondary forest regenerated after seed tree harvesting, and tests the hypothesis that the current population was formed only from a small number of seed trees. The hypothesis was not acceptable, instead, the alternative assuming advance reproduction conducted prior to the harvesting is supported. The results of this first trial of point process models suggest that point processes can provide a useful mathematical methodology in fine‐scale spatial genetics and molecular ecology.     

The factors that favor adaptive habitat construction versus non‐adaptive environmental conditioning

Adaptive habitat construction is a process by which individuals alter their environment so as to increase their (inclusive) fitness. Such alterations are a subset of the myriad ways that individuals condition their environment. We present an individual‐based model of habitat construction to explore what factors might favor selection when the benefits of environmental alterations are shared by individuals of the same species. Our results confirm the predictions of inclusive fitness and group selection theory and expectations based on previous models that construction will be more favored when its benefits are more likely to be directed to self or near kin. We found that temporal variation had no effect on the evolution of construction. For spatial heterogeneity, construction was disfavored when the spatial pattern of movement did not match the spatial pattern of environmental heterogeneity, especially when there was spatial heterogeneity in the optimal amount of construction. Under those conditions, very strong selection was necessary to favor genetic differentiation of construction propensity among demes. We put forth a constitutive theory for the evolution of adaptive habitat construction that unifies our model with previous verbal and quantitative models into a formal conceptual framework.