Spatial processes and evolutionary models: a critical review

Evolution is a fundamentally population level process in which variation, drift and selection produce both temporal and spatial patterns of change. Statistical model fitting is now commonly used to estimate which kind of evolutionary process best explains patterns of change through time using models like Brownian motion, stabilizing selection (Ornstein–Uhlenbeck) and directional selection on traits measured from stratigraphic sequences or on phylogenetic trees. But these models assume that the traits possessed by a species are homogeneous. Spatial processes such as dispersal, gene flow and geographical range changes can produce patterns of trait evolution that do not fit the expectations of standard models, even when evolution at the local‐population level is governed by drift or a typical OU model of selection. The basic properties of population level processes (variation, drift, selection and population size) are reviewed and the relationship between their spatial and temporal dynamics is discussed. Typical evolutionary models used in palaeontology incorporate the temporal component of these dynamics, but not the spatial. Range expansions and contractions introduce rate variability into drift processes, range expansion under a drift model can drive directional change in trait evolution, and spatial selection gradients can create spatial variation in traits that can produce long‐term directional trends and punctuation events depending on the balance between selection strength, gene flow, extirpation probability and model of speciation. Using computational modelling that spatial processes can create evolutionary outcomes that depart from basic population‐level notions from these standard macroevolutionary models.     

Adaptive evolution of reproductive and vegetative traits driven by breeding systems

The evolution of inflorescence size, a key trait in reproductive success, was studied in the genus Acer under a perspective of adaptive evolution. Breeding systems, hypothesized to indicate different levels of mating competition, were considered as the selective scenarios defining different optima of inflorescence size. Larger inflorescences, which increase male fitness by generating larger floral displays, were hypothesized to be selected under scenarios with higher competition with unisexuals. An identical approach was used to test if the same selective regimes could be driving the evolution of leaf size, a vegetative trait that was found to be correlated with inflorescence size. A Brownian motion model of inflorescence/leaf-size evolution (which cannot distinguish between changes caused by pure drift processes and changes caused by natural selection in rapidly and randomly changing environments) was compared with several adaptive Ornstein-Uhlenbeck (OU) models, which can quantify the effects of both stochasticity and natural selection. The best-fitting model for inflorescence/leaf-size evolution was an OU model with three optima that increased with the level of mating competition. Both traits evolved under the same selective regimes and in the same direction, confirming a pattern of correlated evolution. These results show that a selective regime hypothetically related to the evolution of a reproductive trait can also explain the evolution of a vegetative trait.     

Tempo and mode of performance evolution across multiple independent origins of adhesive toe pads in lizards

Understanding macroevolutionary dynamics of trait evolution is an important endeavor in evolutionary biology. Ecological opportunity can liberate a trait as it diversifies through trait space, while genetic and selective constraints can limit diversification. While many studies have examined the dynamics of morphological traits, diverse morphological traits may yield the same or similar performance and as performance is often more proximately the target of selection, examining only morphology may give an incomplete understanding of evolutionary dynamics. Here, we ask whether convergent evolution of pad‐bearing lizards has followed similar evolutionary dynamics, or whether independent origins are accompanied by unique constraints and selective pressures over macroevolutionary time. We hypothesized that geckos and anoles each have unique evolutionary tempos and modes. Using performance data from 59 species, we modified Brownian motion (BM) and Ornstein–Uhlenbeck (OU) models to account for repeated origins estimated using Bayesian ancestral state reconstructions. We discovered that adhesive performance in geckos evolved in a fashion consistent with Brownian motion with a trend, whereas anoles evolved in bounded performance space consistent with more constrained evolution (an Ornstein–Uhlenbeck model). Our results suggest that convergent phenotypes can have quite distinctive evolutionary patterns, likely as a result of idiosyncratic constraints or ecological opportunities.     

Experimental evidence that the Ornstein‐Uhlenbeck model best describes the evolution of leaf litter decomposability

Leaf litter decomposability is an important effect trait for ecosystem functioning. However, it is unknown how this effect trait evolved through plant history as a leaf ‘afterlife’ integrator of the evolution of multiple underlying traits upon which adaptive selection must have acted. Did decomposability evolve in a Brownian fashion without any constraints? Was evolution rapid at first and then slowed? Or was there an underlying mean-reverting process that makes the evolution of extreme trait values unlikely? Here, we test the hypothesis that the evolution of decomposability has undergone certain mean-reverting forces due to strong constraints and trade-offs in the leaf traits that have afterlife effects on litter quality to decomposers. In order to test this, we examined the leaf litter decomposability and seven key leaf traits of 48 tree species in the temperate area of China and fitted them to three evolutionary models: Brownian motion model (BM), Early burst model (EB), and Ornstein-Uhlenbeck model (OU). The OU model, which does not allow unlimited trait divergence through time, was the best fit model for leaf litter decomposability and all seven leaf traits. These results support the hypothesis that neither decomposability nor the underlying traits has been able to diverge toward progressively extreme values through evolutionary time. These results have reinforced our understanding of the relationships between leaf litter decomposability and leaf traits in an evolutionary perspective and may be a helpful step toward reconstructing deep-time carbon cycling based on taxonomic composition with more confidence.     

Stochasticity in Sexual Selection Enables Divergence: Implications for Moth Pheromone Evolution

Sexual selection has long been hypothesized to lead to allopatric speciation, and one possible mechanism for this is that its interaction with stochasticity, which perturbs the trait and preference equilibria, can result in different traits being preferred in different populations. Here we specifically examine the role that stochastic changes in sexual selection strength plays in the shift of predominance between pairs of preferences and traits within a single population. We first create a single-locus null model of stochasticity during frequency dependent selection and then compare it to a two-locus population genetic model with stochastic strengths of female preferences for male traits. We find some interesting differences between the two models, primarily that in the two-locus sexual selection model shifts between preference and trait regimes occur more often with both weak and strong preferences, compared to intermediate preference strengths. We discuss the implications of our results for the evolution of pheromone blends and male responses during speciation in moths, a case that seems to match the assumptions of our model.     

Phenotypic traits evolution and morphological traits associated with echolocation calls in cryptic horseshoe bats (Rhinolophidae)

Bats provide an excellent casestudy for studying evolution due to their remarkable flight and echolocation capabilities. In this study, we sought to understand the phenotypic evolution of key traits in Rhinolophidae (horseshoe bats) using phylogenetic comparative methods. We aimed to test the phylogenetic signals of traits, and evaluated the best-fit evolutionary models given the data for each trait considering different traits may evolve under different models (i.e., Brownian Motion [BM], Ornstein-Uhlenbeck [OU], and Early Burst [EB]) and reconstruct ancestral character states. We examined how phenotypic characters are associated with echolocation calls and minimum detectable prey size. We measured 34 traits of 10 Asian rhinolophids species (187 individuals). We found that the majority of traits showed a high phylogenetic signal based on Blomberg′s K and Pagel′s λ, but each trait may evolve under different evolutionary models. Sella traits were shown to evolve under stabilizing selection based on OU models, indicating sella traits have the tendency to move forward along the branches toward some medial value in equilibrium. Our findings highlight the importance of sella characters in association with echolocation call emissions in Rhinolophidae, as calls are important for spatial cognition and also influence dietary preferences. Minimum detectable prey size in Rhinolophidae was associated with call frequency, bandwidth, call duration, wingspan, and wing surface area. Ultimately, understanding trait evolution requires sensitivity due to the differential selective pressures which may apply to different characteristics.     

The Morris–Lecar neuron model embeds a leaky integrate-and-fire model

We show that the stochastic Morris–Lecar neuron, in a neighborhood of its stable point, can be approximated by a two-dimensional Ornstein–Uhlenbeck (OU) modulation of a constant circular motion. The associated radial OU process is an example of a leaky integrate-and-fire (LIF) model prior to firing. A new model constructed from a radial OU process together with a simple firing mechanism based on detailed Morris–Lecar firing statistics reproduces the Morris–Lecar Interspike Interval (ISI) distribution, and has the computational advantages of a LIF. The result justifies the large amount of attention paid to the LIF models.     

Likelihood-based clustering (LiBaC) for codon models, a method for grouping sites according to similarities in the underlying process of evolution

Models of codon evolution are useful for investigating the strength and direction of natural selection via a parameter for the nonsynonymous/synonymous rate ratio (omega = d(N)/d(S)). Different codon models are available to account for diversity of the evolutionary patterns among sites. Codon models that specify data partitions as fixed effects allow the most evolutionary diversity among sites but require that site partitions are a priori identifiable. Models that use a parametric distribution to express the variability in the omega ratio across site do not require a priori partitioning of sites, but they permit less among-site diversity in the evolutionary process. Simulation studies presented in this paper indicate that differences among sites in estimates of omega under an overly simplistic analytical model can reflect more than just natural selection pressure. We also find that the classic likelihood ratio tests for positive selection have a high false-positive rate in some situations. In this paper, we developed a new method for assigning codon sites into groups where each group has a different model, and the likelihood over all sites is maximized. The method, called likelihood-based clustering (LiBaC), can be viewed as a generalization of the family of model-based clustering approaches to models of codon evolution. We report the performance of several LiBaC-based methods, and selected alternative methods, over a wide variety of scenarios. We find that LiBaC, under an appropriate model, can provide reliable parameter estimates when the process of evolution is very heterogeneous among groups of sites. Certain types of proteins, such as transmembrane proteins, are expected to exhibit such heterogeneity. A survey of genes encoding transmembrane proteins suggests that overly simplistic models could be leading to false signal for positive selection among such genes. In these cases, LiBaC-based methods offer an important addition to a "toolbox" of methods thereby helping to uncover robust evidence for the action of positive selection.     

The effect of parity on morphological evolution among phrynosomatid lizards

The shift from egg laying to live‐bearing is one of the most well‐studied transitions in evolutionary biology. Few studies, however, have assessed the effect of this transition on morphological evolution. Here, we evaluated the effect of reproductive mode on the morphological evolution of 10 traits, among 108 species of phrynosomatid lizards. We assess whether the requirement for passing shelled eggs through the pelvic girdle has led to morphological constraints in oviparous species and whether long gestation times in viviparous species have led to constraints in locomotor morphology. We fit models to the data that vary both in their tempo (strength and rate of selection) and mode of evolution (Brownian or Ornstein‐Uhlenbeck) and estimates of trait optima. We found that most traits are best fit by a generalized multipeak OU model, suggesting differing trait optima for viviparous vs. oviparous species. Additionally, rates (σ²) of both pelvic girdle and forelimb trait evolution varied with parity; viviparous species had higher rates. Hindlimb traits, however, exhibited no difference in σ² between parity modes. In a functional context, our results suggest that the passage of shelled eggs constrains the morphology of the pelvic girdle, but we found no evidence of morphological constraint of the locomotor apparatus in viviparous species. Our results are consistent with recent lineage diversification analyses, leading to the conclusion that transitions to viviparity increase both lineage and morphological diversification.     

Phylogenetic patterns of trait and trait plasticity evolution: Insights from amphibian embryos

Environmental variation favors the evolution of phenotypic plasticity. For many species, we understand the costs and benefits of different phenotypes, but we lack a broad understanding of how plastic traits evolve across large clades. Using identical experiments conducted across North America, we examined prey responses to predator cues. We quantified five life‐history traits and the magnitude of their plasticity for 23 amphibian species/populations (spanning three families and five genera) when exposed to no cues, crushed‐egg cues, and predatory crayfish cues. Embryonic responses varied considerably among species and phylogenetic signal was common among the traits, whereas phylogenetic signal was rare for trait plasticities. Among trait‐evolution models, the Ornstein–Uhlenbeck (OU) model provided the best fit or was essentially tied with Brownian motion. Using the best fitting model, evolutionary rates for plasticities were higher than traits for three life‐history traits and lower for two. These data suggest that the evolution of life‐history traits in amphibian embryos is more constrained by a species’ position in the phylogeny than is the evolution of life history plasticities. The fact that an OU model of trait evolution was often a good fit to patterns of trait variation may indicate adaptive optima for traits and their plasticities.     

Detecting non-Brownian trait evolution in adaptive radiations

Many phylogenetic comparative methods that are currently widely used in the scientific literature assume a Brownian motion model for trait evolution, but the suitability of that model is rarely tested, and a number of important factors might affect whether this model is appropriate or not. For instance, we might expect evolutionary change in adaptive radiations to be driven by the availability of ecological niches. Such evolution has been shown to produce patterns of change that are different from those modelled by the Brownian process. We applied two tests for the assumption of Brownian motion that generally have high power to reject data generated under non-Brownian niche-filling models for the evolution of traits in adaptive radiations. As a case study, we used these tests to explore the evolution of feeding adaptations in two radiations of warblers. In one case, the patterns revealed do not accord with Brownian motion but show characteristics expected under certain niche-filling models.     

Mixed evidence for early bursts of morphological evolution in extant clades

Macroevolutionary theory predicts high rates of evolution should occur early in a clade's history as species exploit ecological opportunity. Evidence from the fossil record has shown a high prevalence of early bursts in morphological evolution, but recent work has provided little evidence for early high rates in the evolution of extant clades. Here, I test the prevalence of early bursts in extant data using phylogenetic comparative methods. Existing models are extended to allow a shift from a background Brownian motion (BM) process to an early burst process within subclades of phylogenies, rather than an early burst being applied to an entire phylogenetic tree. This nested early burst model is compared to other modes of evolution that can occur within subclades, such as evolution with a constraint (Ornstein‐Uhlenbeck model) and nested BM rate shift models. These relaxed models are validated using simulations and then are applied to body size evolution of three major clades of amniotes (mammals, squamates and aves) at different levels of taxonomic organization (order, family). Applying these unconstrained models greatly increases the support for early bursts within nested subclades, and so early bursts are the most common model of evolution when only one shift is analysed. However, the relative fit of early burst models is worse than models that allow for multiple shifts of the BM or OU process. No single‐shift or homogenous model is superior to models of multiple shifts in BM or OU evolution, but the patterns shown by these multirate models are generally congruent with patterns expected from early bursts.     

Carnivory maintains cranial dimorphism between males and females: Evidence for niche divergence in extant Musteloidea

The evolution and maintenance of sexual dimorphism has long been attributed to sexual selection. Niche divergence, however, serves as an alternative but rarely tested selective pressure also hypothesized to drive phenotypic disparity between males and females. We reconstructed ancestral social systems and diet and used Ornstein–Uhlenbeck (OU) modeling approaches to test whether niche divergence is stronger than sexual selection in driving the evolution of sexual dimorphism in cranial size and bite force across extant Musteloidea. We found that multipeak OU models favored different dietary regimes over social behavior and that the greatest degree of cranial size and bite force dimorphism were found in terrestrial carnivores. Because competition for terrestrial vertebrate prey is greater than other dietary groups, increased cranial size and bite force dimorphism reduces dietary competition between the sexes. In contrast, neither dietary regime nor social system influenced the evolution of sexual dimorphism in cranial shape. Furthermore, we found that the evolution of sexual dimorphism in bite force is influenced by the evolution of sexual dimorphism in cranial size rather than cranial shape. Overall, our results highlight niche divergence as an important mechanism that maintains the evolution of sexual dimorphism in musteloids.     

The impact of rate heterogeneity on inference of phylogenetic models of trait evolution

Rates of trait evolution are known to vary across phylogenies; however, standard evolutionary models assume a homogeneous process of trait change. These simple methods are widely applied in small‐scale phylogenetic studies, whereas models of rate heterogeneity are not, so the prevalence and patterns of potential rate variation in groups up to hundreds of species remain unclear. The extent to which trait evolution is modelled accurately on a given phylogeny is also largely unknown because studies typically lack absolute model fit tests. We investigated these issues by applying both rate‐static and variable‐rates methods on (i) body mass data for 88 avian clades of 10–318 species, and (ii) data simulated under a range of rate‐heterogeneity scenarios. Our results show that rate heterogeneity is present across small‐scaled avian clades, and consequently applying only standard single‐process models prompts inaccurate inferences about the generating evolutionary process. Specifically, these approaches underestimate rate variation, and systematically mislabel temporal trends in trait evolution. Conversely, variable‐rates approaches have superior relative fit (they are the best model) and absolute fit (they describe the data well). We show that rate changes such as single internal branch variations, rate decreases and early bursts are hard to detect, even by variable‐rates models. We also use recently developed absolute adequacy tests to highlight misleading conclusions based on relative fit alone (e.g. a consistent preference for constrained evolution when isolated terminal branch rate increases are present). This work highlights the potential for robust inferences about trait evolution when fitting flexible models in conjunction with tests for absolute model fit.     

Divergence and evolution of assortative mating in a polygenic trait model of speciation with gene flow

Assortative mating is an important driver of speciation in populations with gene flow and is predicted to evolve under certain conditions in few‐locus models. However, the evolution of assortment is less understood for mating based on quantitative traits, which are often characterized by high genetic variability and extensive linkage disequilibrium between trait loci. We explore this scenario for a two‐deme model with migration, by considering a single polygenic trait subject to divergent viability selection across demes, as well as assortative mating and sexual selection within demes, and investigate how trait divergence is shaped by various evolutionary forces. Our analysis reveals the existence of sharp thresholds of assortment strength, at which divergence increases dramatically. We also study the evolution of assortment via invasion of modifiers of mate discrimination and show that the ES assortment strength has an intermediate value under a range of migration‐selection parameters, even in diverged populations, due to subtle effects which depend sensitively on the extent of phenotypic variation within these populations. The evolutionary dynamics of the polygenic trait is studied using the hypergeometric and infinitesimal models. We further investigate the sensitivity of our results to the assumptions of the hypergeometric model, using individual‐based simulations.     

Evolutionary tracking is determined by differential selection on demographic rates and density dependence

Recent ecological forecasts predict that ~25% of species worldwide will go extinct by 2050. However, these estimates are primarily based on environmental changes alone and fail to incorporate important biological mechanisms such as genetic adaptation via evolution. Thus, environmental change can affect population dynamics in ways that classical frameworks can neither describe nor predict. Furthermore, often due to a lack of data, forecasting models commonly describe changes in population demography by summarizing changes in fecundity and survival concurrently with the intrinsic growth rate (r). This has been shown to be an oversimplification as the environment may impose selective pressure on specific demographic rates (birth and death) rather than directly on r (the difference between the birth and death rates). This differential pressure may alter population response to density, in each demographic rate, further diluting the information combined to produce r. Thus, when we consider the potential for persistence via adaptive evolution, populations with the same r can have different abilities to persist amidst environmental change. Therefore, we cannot adequately forecast population response to climate change without accounting for demography and selection on density dependence. Using a continuous‐time Markov chain model to describe the stochastic dynamics of the logistic model of population growth and allow for trait evolution via mutations arising during birth events, we find persistence via evolutionary tracking more likely when environmental change alters birth rather than the death rate. Furthermore, species that evolve responses to changes in the strength of density dependence due to environmental change are less vulnerable to extinction than species that undergo selection independent of population density. By incorporating these key demographic considerations into our predictive models, we can better understand how species will respond to climate change.     

Multi-locus Analysis of Genomic Time Series Data from Experimental Evolution

Genomic time series data generated by evolve-and-resequence (E&R) experiments offer a powerful window into the mechanisms that drive evolution. However, standard population genetic inference procedures do not account for sampling serially over time, and new methods are needed to make full use of modern experimental evolution data. To address this problem, we develop a Gaussian process approximation to the multi-locus Wright-Fisher process with selection over a time course of tens of generations. The mean and covariance structure of the Gaussian process are obtained by computing the corresponding moments in discrete-time Wright-Fisher models conditioned on the presence of a linked selected site. This enables our method to account for the effects of linkage and selection, both along the genome and across sampled time points, in an approximate but principled manner. We first use simulated data to demonstrate the power of our method to correctly detect, locate and estimate the fitness of a selected allele from among several linked sites. We study how this power changes for different values of selection strength, initial haplotypic diversity, population size, sampling frequency, experimental duration, number of replicates, and sequencing coverage depth. In addition to providing quantitative estimates of selection parameters from experimental evolution data, our model can be used by practitioners to design E&R experiments with requisite power. We also explore how our likelihood-based approach can be used to infer other model parameters, including effective population size and recombination rate. Then, we apply our method to analyze genome-wide data from a real E&R experiment designed to study the adaptation of D. melanogaster to a new laboratory environment with alternating cold and hot temperatures.     

Mathematical modeling of evolution of horizontally transferred genes

We describe a stochastic birth-and-death model of evolution of horizontally transferred genes in microbial populations. The model is a generalization of the stochastic model described by Berg and Kurland and includes five parameters: the rate of mutational inactivation, selection coefficient, invasion rate (i.e., rate of arrival of a novel sequence from outside of the recipient population), within-population horizontal transmission ("infection") rate, and population size. The model of Berg and Kurland included four parameters, namely, mutational inactivation, selection coefficient, population size, and "infection." However, the effect of "infection" was disregarded in the interpretation of the results, and the overall conclusion was that horizontally acquired sequences can be fixed in a population only when they confer a substantial selective advantage onto the recipient and therefore are subject to strong positive selection. Analysis of the present model in different domains of parameter values shows that, as long as the rate of within-population horizontal transmission is comparable to the mutational inactivation rate and there is even a low rate of invasion, horizontally acquired sequences can be fixed in the population or at least persist for a long time in a substantial fraction of individuals in the population even when they are neutral or slightly deleterious. The available biological data strongly suggest that intense within-population and even between-populations gene flows are realistic for at least some prokaryotic species and environments. Therefore, our modeling results are compatible with the notion of a pivotal role of horizontal gene transfer in the evolution of prokaryotes.