Bayesian approaches in evolutionary quantitative genetics

The study of evolutionary quantitative genetics has been advanced by the use of methods developed in animal and plant breeding. These methods have proved to be very useful, but they have some shortcomings when used in the study of wild populations and evolutionary questions. Problems arise from the small size of data sets typical of evolutionary studies, and the additional complexity of the questions asked by evolutionary biologists. Here, we advocate the use of Bayesian methods to overcome these and related problems. Bayesian methods naturally allow errors in parameter estimates to propagate through a model and can also be written as a graphical model, giving them an inherent flexibility. As packages for fitting Bayesian animal models are developed, we expect the application of Bayesian methods to evolutionary quantitative genetics to grow, particularly as genomic information becomes more and more associated with environmental data.     

Developmental quantitative genetic models of evolutionary change

Discussions about evolutionary change in developmental processes or morphological structures are predicated on specific quantitative genetic models whose parameters predict whether evolutionary change can occur, its relative rate and direction, and if correlated change will occur in other related and unrelated structures. The appropriate genetic model should reflect the relevant genetical and developmental biology of the organisms, yet be simple enough in its parameters so that deductions can be made and hypotheses tested. As a consequence, the choice of the most appropriate genetic model for polygenically controlled traits is a complex tissue and the eventual choice of model is often a compromise between completeness of the model and computational expediency. Herein, we discuss several developmental quantitative genetic models for the evolution of development and morphology. The models range from the classical direct effects model to complex epigenetic models. Further, we demonstrate the algebraic equivalency of the Cowley and Atchley epigenetic model and Wagner's developmental mapping model. Finally, we propose a new multivariate model for continuous growth trajectories. The relative efficacy of these various models for understanding evolutionary change in developmental and morphological traits is discussed. © 1994 Wiley-Liss, Inc.     

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

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

Fluctuating selection and its (elusive) evolutionary consequences in a wild rodent population

Temporal fluctuations in the strength and direction of selection are often proposed as a mechanism that slows down evolution, both over geological and contemporary timescales. Both the prevalence of fluctuating selection and its relevance for evolutionary dynamics remain poorly understood however, especially on contemporary timescales: unbiased empirical estimates of variation in selection are scarce, and the question of how much of the variation in selection translates into variation in genetic change has largely been ignored. Using long‐term individual‐based data for a wild rodent population, we quantify the magnitude of fluctuating selection on body size. Subsequently, we estimate the evolutionary dynamics of size and test for a link between fluctuating selection and evolution. We show that, over the past 11 years, phenotypic selection on body size has fluctuated significantly. However, the strength and direction of genetic change have remained largely constant over the study period; that is, the rate of genetic change was similar in years where selection favoured heavier vs. lighter individuals. This result suggests that over shorter timescales, fluctuating selection does not necessarily translate into fluctuating evolution. Importantly however, individual‐based simulations show that the correlation between fluctuating selection and fluctuating evolution can be obscured by the effect of drift, and that substantially more data are required for a precise and accurate estimate of this correlation. We identify new challenges in measuring the coupling between selection and evolution, and provide methods and guidelines to overcome them.     

Optimal selection on two quantitative trait loci with linkage

A mathematical approach to optimize selection on multiple quantitative trait loci (QTL) and an estimate of residual polygenic effects was applied to selection on two linked or unlinked additive QTL. Strategies to maximize total or cumulative discounted response over ten generations were compared to standard QTL selection on the sum of breeding values for the QTL and an estimated breeding value for polygenes, and to phenotypic selection. Optimal selection resulted in greater response to selection than standard QTL or phenotypic selection. Tight linkage between the QTL (recombination rate 0.05) resulted in a slightly lower response for standard QTL and phenotypic selection but in a greater response for optimal selection. Optimal selection capitalized on linkage by emphasizing selection on favorable haplotypes. When the objective was to maximize total response after ten generations and QTL were unlinked, optimal selection increased QTL frequencies to fixation in a near linear manner. When starting frequencies were equal for the two QTL, equal emphasis was given to each QTL, regardless of the difference in effects of the QTL and regardless of the linkage, but the emphasis given to each of the two QTL was not additive. These results demonstrate the ability of optimal selection to capitalize on information on the complex genetic basis of quantitative traits that is forthcoming.     

Expression quantitative trait loci: present and future

The last few years have seen the development of large efforts for the analysis of genome function, especially in the context of genome variation. One of the most prominent directions has been the extensive set of studies on expression quantitative trait loci (eQTLs), namely, the discovery of genetic variants that explain variation in gene expression levels. Such studies have offered promise not just for the characterization of functional sequence variation but also for the understanding of basic processes of gene regulation and interpretation of genome-wide association studies. In this review, we discuss some of the key directions of eQTL research and its implications.     

A method to optimize selection on multiple identified quantitative trait loci

A mathematical approach was developed to model and optimize selection on multiple known quantitative trait loci (QTL) and polygenic estimated breeding values in order to maximize a weighted sum of responses to selection over multiple generations. The model allows for linkage between QTL with multiple alleles and arbitrary genetic effects, including dominance, epistasis, and gametic imprinting. Gametic phase disequilibrium between the QTL and between the QTL and polygenes is modeled but polygenic variance is assumed constant. Breeding programs with discrete generations, differential selection of males and females and random mating of selected parents are modeled. Polygenic EBV obtained from best linear unbiased prediction models can be accommodated. The problem was formulated as a multiple-stage optimal control problem and an iterative approach was developed for its solution. The method can be used to develop and evaluate optimal strategies for selection on multiple QTL for a wide range of situations and genetic models.     

Sex ratio of leopards taken in trophy hunting: genetic data from Tanzania

The sex ratio of leopards, Panthera pardus, taken by trophyhunters in Tanzania is examined. We used sex specific molecularmarkers to analyze 77 samples collected from animals shot betweenthe years 1995–1998 and found that 28.6% were females, despitethe fact that only males are allowed on licenses and all skinswere tagged as males. The model used for quota setting assumesthat only males are shot, but the effect of this violation ofquotas is unknown. Off-take in Tanzania does not currently fillquotas, but when off-take approach maximum levels, compliancewith set quotas and regulations will be critical for sustainableharvest.     

Molecular ecological approaches to studying the evolutionary impact of selective harvesting in wildlife

Harvesting of wildlife populations by humans is usually targeted by sex, age or phenotypic criteria, and is therefore selective. Selective harvesting has the potential to elicit a genetic response from the target populations in several ways. First, selective harvesting may affect population demographic structure (age structure, sex ratio), which in turn may have consequences for effective population size and hence genetic diversity. Second, wildlife-harvesting regimes that use selective criteria based on phenotypic characteristics (e.g. minimum body size, horn length or antler size) have the potential to impose artificial selection on harvested populations. If there is heritable genetic variation for the target characteristic and harvesting occurs before the age of maturity, then an evolutionary response over time may ensue. Molecular ecological techniques offer ways to predict and detect genetic change in harvested populations, and therefore have great utility for effective wildlife management. Molecular markers can be used to assess the genetic structure of wildlife populations, and thereby assist in the prediction of genetic impacts by delineating evolutionarily meaningful management units. Genetic markers can be used for monitoring genetic diversity and changes in effective population size and breeding systems. Tracking evolutionary change at the phenotypic level in the wild through quantitative genetic analysis can be made possible by genetically determined pedigrees. Finally, advances in genome sequencing and bioinformatics offer the opportunity to study the molecular basis of phenotypic variation through trait mapping and candidate gene approaches. With this understanding, it could be possible to monitor the selective impacts of harvesting at a molecular level in the future. Effective wildlife management practice needs to consider more than the direct impact of harvesting on population dynamics. Programs that utilize molecular genetic tools will be better positioned to assess the long-term evolutionary impact of artificial selection on the evolutionary trajectory and viability of harvested populations.     

Model selection for quantitative trait loci mapping in a full-sib family

Statistical methods for mapping quantitative trait loci (QTLs) in full-sib forest trees, in which the number of alleles and linkage phase can vary from locus to locus, are still not well established. Previous studies assumed that the QTL segregation pattern was fixed throughout the genome in a full-sib family, despite the fact that this pattern can vary among regions of the genome. In this paper, we propose a method for selecting the appropriate model for QTL mapping based on the segregation of different types of markers and QTLs in a full-sib family. The QTL segregation patterns were classified into three types: test cross (1:1 segregation), F₂ cross (1:2:1 segregation) and full cross (1:1:1:1 segregation). Akaike’s information criterion (AIC), the Bayesian information criterion (BIC) and the Laplace-empirical criterion (LEC) were used to select the most likely QTL segregation pattern. Simulations were used to evaluate the power of these criteria and the precision of parameter estimates. A Windows-based software was developed to run the selected QTL mapping method. A real example is presented to illustrate QTL mapping in forest trees based on an integrated linkage map with various segregation markers. The implications of this method for accurate QTL mapping in outbred species are discussed.     

Dissecting the pleiotropic consequences of a quantitative trait nucleotide

The downstream consequences of a single quantitative trait polymorphism can provide important insight into the molecular basis of a trait. However, the molecular consequences of a polymorphism may be complex and only a subset of these may influence the trait of interest. In natural isolates of Saccharomyces cerevisiae , a nonsynonymous polymorphism in cystathione β-synthase ( CYS4 ) causes a deficiency in both cysteine and glutathione that results in rust-colored colonies and drug-dependent growth defects. Using a single-nucleotide allele replacement, we characterized the effects of this polymorphism on gene expression levels across the genome. To determine whether any of the differentially expressed genes are necessary for the production of rust-colored colonies, we screened the yeast deletion collection for genes that enhance or suppress rust coloration. We found that genes in the sulfur assimilation pathway are required for the production of rust color but not the drug-sensitivity phenotype. Our results show that a single quantitative trait polymorphism can generate a complex set of downstream changes, providing a molecular basis for pleiotropy.     

Estimating evolutionary parameters when viability selection is operating

Some individuals die before a trait is measured or expressed (the invisible fraction), and some relevant traits are not measured in any individual (missing traits). This paper discusses how these concepts can be cast in terms of missing data problems from statistics. Using missing data theory, I show formally the conditions under which a valid evolutionary inference is possible when the invisible fraction and/or missing traits are ignored. These conditions are restrictive and unlikely to be met in even the most comprehensive long-term studies. When these conditions are not met, many selection and quantitative genetic parameters cannot be estimated accurately unless the missing data process is explicitly modelled. Surprisingly, this does not seem to have been attempted in evolutionary biology. In the case of the invisible fraction, viability selection and the missing data process are often intimately linked. In such cases, models used in survival analysis can be extended to provide a flexible and justified model of the missing data mechanism. Although missing traits pose a more difficult problem, important biological parameters can still be estimated without bias when appropriate techniques are used. This is in contrast to current methods which have large biases and poor precision. Generally, the quantitative genetic approach is shown to be superior to phenotypic studies of selection when invisible fractions or missing traits exist because part of the missing information can be recovered from relatives.     

Social structure modulates the evolutionary consequences of social plasticity: A social network perspective on interacting phenotypes

Organisms express phenotypic plasticity during social interactions. Interacting phenotype theory has explored the consequences of social plasticity for evolution, but it is unclear how this theory applies to complex social structures. We adapt interacting phenotype models to general social structures to explore how the number of social connections between individuals and preference for phenotypically similar social partners affect phenotypic variation and evolution. We derive an analytical model that ignores phenotypic feedback and use simulations to test the predictions of this model. We find that adapting previous models to more general social structures does not alter their general conclusions but generates insights into the effect of social plasticity and social structure on the maintenance of phenotypic variation and evolution. Contribution of indirect genetic effects to phenotypic variance is highest when interactions occur at intermediate densities and decrease at higher densities, when individuals approach interacting with all group members, homogenizing the social environment across individuals. However, evolutionary response to selection tends to increase at greater network densities as the effects of an individual's genes are amplified through increasing effects on other group members. Preferential associations among similar individuals (homophily) increase both phenotypic variance within groups and evolutionary response to selection. Our results represent a first step in relating social network structure to the expression of social plasticity and evolutionary responses to selection.     

Predicting the evolution of sexual size dimorphism

11 , Evolution 34 : 292–305) equations for predicting the evolution of sexual size dimorphism (SSD) through frequency‐dependent sexual selection, and frequency‐independent natural selection, were tested against results obtained from a stochastic genetic simulation model. The SSD evolved faster than predicted, due to temporary increases in the genetic variance brought about by directional selection. Predictions for the magnitude of SSD at equilibrium were very accurate for weak sexual selection. With stronger sexual selection the total response was greater than predicted. Large changes in SSD can occur without significant long‐term change in the genetic correlation between the sexes. Our results suggest that genetic correlations constrain both the short‐term and long‐term evolution of SSD less than predicted by the Lande model.     

Predicting trait values and measuring selection in complex life histories: reproductive allocation decisions in Soay sheep

Accurate prediction of life history phenomena and characterisation of selection in free-living animal populations are fundamental goals in evolutionary ecology. In density regulated, structured populations, where individual state influences fate, simple and widely used approaches based on individual lifetime measures of fitness are difficult to justify. We combine recently developed structured population modelling tools with ideas from modern evolutionary game theory (adaptive dynamics) to understand selection on allocation of female reproductive effort to singletons or twins in a size-structured population of feral sheep. In marked contrast to the classical selection analyses, our model-based approach predicts that the female allocation strategy is under negligible directional selection. These differences arise because classical selection analysis ignores components of offspring fitness and fails to consider selection over the complete life cycle.     

Predicting evolutionary responses to climate change in the sea

An increasing number of short‐term experimental studies show significant effects of projected ocean warming and ocean acidification on the performance on marine organisms. Yet, it remains unclear if we can reliably predict the impact of climate change on marine populations and ecosystems, because we lack sufficient understanding of the capacity for marine organisms to adapt to rapid climate change. In this review, we emphasise why an evolutionary perspective is crucial to understanding climate change impacts in the sea and examine the approaches that may be useful for addressing this challenge. We first consider what the geological record and present‐day analogues of future climate conditions can tell us about the potential for adaptation to climate change. We also examine evidence that phenotypic plasticity may assist marine species to persist in a rapidly changing climate. We then outline the various experimental approaches that can be used to estimate evolutionary potential, focusing on molecular tools, quantitative genetics, and experimental evolution, and we describe the benefits of combining different approaches to gain a deeper understanding of evolutionary potential. Our goal is to provide a platform for future research addressing the evolutionary potential for marine organisms to cope with climate change.