Blouch: Bayesian Linear Ornstein-Uhlenbeck Models for Comparative Hypotheses.

Relationships among species in the tree of life can complicate comparative methods and testing adaptive hypotheses. Models based on the Ornstein-Uhlenbeck process permit hypotheses about adaptation to be tested by allowing traits to either evolve towards fixed adaptive optima (e.g., regimes or niches) or track continuously changing optima that can be influenced by other traits. These models allow estimation of the effects of both adaptation and phylogenetic inertia - resistance to adaptation due to any source - on trait evolution, an approach known as the "adaptation-inertia" framework. However, previous applications of this framework, and most approaches suggested to deal with the issue of species non-independence, are based on a maximum likelihood approach and thus it is difficult to include information based on prior biological knowledge in the analysis, which can affect resulting inferences. Here I present Blouch, (Bayesian Linear Ornstein-Uhlenbeck Models for Comparative Hypotheses), which fits allometric and adaptive models of continuous trait evolution in a Bayesian framework based on fixed or continuous predictors and incorporates measurement error. I first briefly discuss the models implemented in Blouch, and then the new applications for these models provided by a Bayesian framework. This includes the advantages of assigning biologically meaningful priors when compared to non-Bayesian approaches, allowing for varying effects (intercepts and slopes), and multilevel modeling. Validations on simulated data show good performance in recovering the true evolutionary parameters for all models. To demonstrate the workflow of Blouch on an empirical dataset, I test the hypothesis that the relatively larger antlers of larger bodied deer are the result of more intense sexual selection that comes along with their tendency to live in larger breeding groups. While results show that larger bodied deer that live in larger breeding groups have relatively larger antlers, deer living in the smallest groups appear to have a different and steeper scaling pattern of antler size to body size than other groups. These results are contrary to previous findings and may argue that a different type of sexual selection or other selective pressures govern optimum antler size in the smallest breeding groups.

Three modes of evolution?Remarks on rates of evolution and time scaling.

Rates of evolution get smaller when they are measured over longer time intervals. As first shown by Gingerich, rates of morphological change measured from fossil time series show a robust minus-one scaling with time span, implying that evolutionary changes are just as large when measured over a hundred years as when measured over a hundred-thousand years. On even longer time scales, however, the scaling shifts toward a minus-half exponent consistent with evolution behaving as Brownian motion, as commonly observed in phylogenetic comparative studies. Here, I discuss how such scaling patterns arise, and I derive the patterns expected from standard stochastic models of evolution. I argue that observed shifts cannot be easily explained by simple univariate models, but require shifts in mode of evolution as time scale is changing. To illustrate this idea, I present a hypothesis about three distinct, but connected, modes of evolution. I analyze the scaling patterns predicted from this, and use the results to discuss how rates of evolution should be measured and interpreted. I argue that distinct modes of evolution at different time scales act to decouple micro- and macroevolution, and criticize various attempts at extrapolating from one to the other.

A stochastic model for topographically influenced cell migration.

Migrating cells traverse a range of topographic configurations presented by the native extracellular environment to conduct their physiologic functions. It is well documented cells can modulate their behaviour in response to different topographic features, finding promising applications in biomaterial and bioimplant design. It is useful, in these areas of research, to be able to predict which topographic arrangements could be used to promote certain patterns of migration prior to laboratory experimentation. Despite a profusion of study and interest shown in these fields by experimentalists, the related modelling literature is as yet relatively sparse and tend to focus more on either cell-matrix interaction or morphological responses of cells. We propose a mathematical model for individual cell migration based on an Ornstein-Uhlenbeck process, and set out to see if the model can be used to predict migration patterns on 2-d isotropic and anisotropic topographies, whose characteristics can be broadly described as either uniform flat, uniform linear with variable ridge density or non-uniform disordered with variable feature density. Results suggest the model is capable of producing realistic patterns of migration for flat and linear topographic patterns, with calibrated output closely approximating NIH3T3 fibroblast migration behaviour derived from an experimental dataset, in which migration linearity increased with ridge density and average speed was highest at intermediate ridge densities. Exploratory results for non-uniform disordered topographies suggest cell migration patterns may adopt disorderedness present in the topography and that 'distortion' introduced to linear topographic patterns may not impede linear guidance of migration, given its magnitude is bounded within certain limits. We conclude that an Ornstein-Uhlenbeck based model for topographically influenced migration may be useful to predict patterns of migration behaviour for certain isotropic (flat) and anisotropic (linear) topographies in the NIH3T3 fibroblast cell line, but additional investigation is required to predict with confidence migration patterns for non-uniform disordered topographic arrangements.

Uncovering the mosaic evolution of the carnivoran skeletal system.

The diversity of vertebrate skeletons is often attributed to adaptations to distinct ecological factors such as diet, locomotion, and sensory environment. Although the adaptive evolution of skull, appendicular skeleton, and vertebral column is well studied in vertebrates, comprehensive investigations of all skeletal components simultaneously are rarely performed. Consequently, we know little of how modes of evolution differ among skeletal components. Here, we tested if ecological and phylogenetic effects led to distinct modes of evolution among the cranial, appendicular and vertebral regions in extant carnivoran skeletons. Using multivariate evolutionary models, we found mosaic evolution in which only the mandible, hindlimb and posterior (i.e. last thoracic and lumbar) vertebrae showed evidence of adaptation towards ecological regimes whereas the remaining skeletal components reflect clade-specific evolutionary shifts. We hypothesize that the decoupled evolution of individual skeletal components may have led to the origination of distinct adaptive zones and morphologies among extant carnivoran families that reflect phylogenetic hierarchies. Overall, our work highlights the importance of examining multiple skeletal components simultaneously in ecomorphological analyses. Ongoing work integrating the fossil and palaeoenvironmental record will further clarify deep-time drivers that govern the carnivoran diversity we see today and reveal the complexity of evolutionary processes in multicomponent systems.

Dynamical analysis of a stochastic maize streak virus epidemic model with logarithmic Ornstein-Uhlenbeck process.

To describe the transmission dynamics of maize streak virus infection, in the paper, we first formulate a stochastic maize streak virus infection model, in which the stochastic fluctuations are depicted by a logarithmic Ornstein-Uhlenbeck process. This approach is reasonable to simulate the random impacts of main parameters both from the biological significance and the mathematical perspective. Then we investigate the detailed dynamics of the stochastic system, including the existence and uniqueness of the global solution, the existence of a stationary distribution, the exponential extinction of the infected maize and infected leafhopper vector. Especially, by solving the five-dimensional algebraic equations corresponding to the stochastic system, we obtain the specific expression of the probability density function near the quasi-endemic equilibrium of the stochastic system, which provides valuable insights into the stationary distribution. Finally, the model is discretized using the Milstein higher-order numerical method to illustrate our theoretical results numerically. Our findings provide a groundwork for better methods of preventing the spread of this type of virus.

phylosem: A fast and simple R package for phylogenetic inference and trait imputation using phylogenetic structural equation models.

Phylogenetic comparative methods (PCMs) can be used to study evolutionary relationships and trade-offs among species traits. Analysts using PCM may want to (1) include latent variables, (2) estimate complex trait interdependencies, (3) predict missing trait values, (4) condition predicted traits upon phylogenetic correlations and (5) estimate relationships as slope parameters that can be compared with alternative regression methods. The Comprehensive R Archive Network (CRAN) includes well-documented software for phylogenetic linear models (phylolm), phylogenetic path analysis (phylopath), phylogenetic trait imputation (Rphylopars) and structural equation models (sem), but none of these can simultaneously accomplish all five analytical goals. We therefore introduce a new package phylosem for phylogenetic structural equation models (PSEM) and summarize features and interface. We also describe new analytical options, where users can specify any combination of Ornstein-Uhlenbeck, Pagel's-δ and Pagel's-λ transformations for species covariance. For the first time, we show that PSEM exactly reproduces estimates (and standard errors) for simplified cases that are feasible in sem, phylopath, phylolm and Rphylopars and demonstrate the approach by replicating a well-known case study involving trade-offs in plant energy budgets.

When can we reconstruct the ancestral state? Beyond Brownian motion.

Reconstructing the ancestral state of a group of species helps answer many important questions in evolutionary biology. Therefore, it is crucial to understand when we can estimate the ancestral state accurately. Previous works provide a necessary and sufficient condition, called the big bang condition, for the existence of an accurate reconstruction method under discrete trait evolution models and the Brownian motion model. In this paper, we extend this result to a wide range of continuous trait evolution models. In particular, we consider a general setting where continuous traits evolve along the tree according to stochastic processes that satisfy some regularity conditions. We verify these conditions for popular continuous trait evolution models including Ornstein-Uhlenbeck, reflected Brownian Motion, bounded Brownian Motion, and Cox-Ingersoll-Ross.

Learning Retention Mechanisms and Evolutionary Parameters of Duplicate Genes from Their Expression Data.

Learning about the roles that duplicate genes play in the origins of novel phenotypes requires an understanding of how their functions evolve. A previous method for achieving this goal, CDROM, employs gene expression distances as proxies for functional divergence and then classifies the evolutionary mechanisms retaining duplicate genes from comparisons of these distances in a decision tree framework. However, CDROM does not account for stochastic shifts in gene expression or leverage advances in contemporary statistical learning for performing classification, nor is it capable of predicting the parameters driving duplicate gene evolution. Thus, here we develop CLOUD, a multi-layer neural network built on a model of gene expression evolution that can both classify duplicate gene retention mechanisms and predict their underlying evolutionary parameters. We show that not only is the CLOUD classifier substantially more powerful and accurate than CDROM, but that it also yields accurate parameter predictions, enabling a better understanding of the specific forces driving the evolution and long-term retention of duplicate genes. Further, application of the CLOUD classifier and predictor to empirical data from Drosophila recapitulates many previous findings about gene duplication in this lineage, showing that new functions often emerge rapidly and asymmetrically in younger duplicate gene copies, and that functional divergence is driven by strong natural selection. Hence, CLOUD represents a major advancement in classifying retention mechanisms and predicting evolutionary parameters of duplicate genes, thereby highlighting the utility of incorporating sophisticated statistical learning techniques to address long-standing questions about evolution after gene duplication.

Special Care Is Needed in Applying Phylogenetic Comparative Methods to Gene Trees with Speciation and Duplication Nodes.

How gene function evolves is a central question of evolutionary biology. It can be investigated by comparing functional genomics results between species and between genes. Most comparative studies of functional genomics have used pairwise comparisons. Yet it has been shown that this can provide biased results, as genes, like species, are phylogenetically related. Phylogenetic comparative methods should be used to correct for this, but they depend on strong assumptions, including unbiased tree estimates relative to the hypothesis being tested. Such methods have recently been used to test the "ortholog conjecture," the hypothesis that functional evolution is faster in paralogs than in orthologs. Although pairwise comparisons of tissue specificity (τ) provided support for the ortholog conjecture, phylogenetic independent contrasts did not. Our reanalysis on the same gene trees identified problems with the time calibration of duplication nodes. We find that the gene trees used suffer from important biases, due to the inclusion of trees with no duplication nodes, to the relative age of speciations and duplications, to systematic differences in branch lengths, and to non-Brownian motion of tissue specificity on many trees. We find that incorrect implementation of phylogenetic method in empirical gene trees with duplications can be problematic. Controlling for biases allows successful use of phylogenetic methods to study the evolution of gene function and provides some support for the ortholog conjecture using three different phylogenetic approaches.

Phylogenetic analysis of adaptation in comparative physiology and biomechanics: overview and a case study of thermal physiology in treefrogs.

Comparative phylogenetic studies of adaptation are uncommon in biomechanics and physiology. Such studies require data collection from many species, a challenge when this is experimentally intensive. Moreover, researchers struggle to employ the most biologically appropriate phylogenetic tools for identifying adaptive evolution. Here, we detail an established but greatly underutilized phylogenetic comparative framework - the Ornstein-Uhlenbeck process - that explicitly models long-term adaptation. We discuss challenges in implementing and interpreting the model, and we outline potential solutions. We demonstrate use of the model through studying the evolution of thermal physiology in treefrogs. Frogs of the family Hylidae have twice colonized the temperate zone from the tropics, and such colonization likely involved a fundamental change in physiology due to colder and more seasonal temperatures. However, which traits changed to allow colonization is unclear. We measured cold tolerance and characterized thermal performance curves in jumping for 12 species of treefrogs distributed from the Neotropics to temperate North America. We then conducted phylogenetic comparative analyses to examine how tolerances and performance curves evolved and to test whether that evolution was adaptive. We found that tolerance to low temperatures increased with the transition to the temperate zone. In contrast, jumping well at colder temperatures was unrelated to biogeography and thus did not adapt during dispersal. Overall, our study shows how comparative phylogenetic methods can be leveraged in biomechanics and physiology to test the evolutionary drivers of variation among species.

Modelling group movement with behaviour switching in continuous time.

This article presents a new method for modelling collective movement in continuous time with behavioural switching, motivated by simultaneous tracking of wild or semi-domesticated animals. Each individual in the group is at times attracted to a unobserved leading point. However, the behavioural state of each individual can switch between 'following' and 'independent'. The 'following' movement is modelled through a linear stochastic differential equation, while the 'independent' movement is modelled as Brownian motion. The movement of the leading point is modelled either as an Ornstein-Uhlenbeck (OU) process or as Brownian motion (BM), which makes the whole system a higher-dimensional Ornstein-Uhlenbeck process, possibly an intrinsic non-stationary version. An inhomogeneous Kalman filter Markov chain Monte Carlo algorithm is developed to estimate the diffusion and switching parameters and the behaviour states of each individual at a given time point. The method successfully recovers the true behavioural states in simulated data sets , and is also applied to model a group of simultaneously tracked reindeer (Rangifer tarandus).

Slow Resting State Fluctuations Enhance Neuronal and Behavioral Responses to Looming Sounds.

We investigate both experimentally and using a computational model how the power of the electroencephalogram (EEG) recorded in human subjects tracks the presentation of sounds with acoustic intensities that increase exponentially (looming) or remain constant (flat). We focus on the link between this EEG tracking response, behavioral reaction times and the time scale of fluctuations in the resting state, which show considerable inter-subject variability. Looming sounds are shown to generally elicit a sustained power increase in the alpha and beta frequency bands. In contrast, flat sounds only elicit a transient upsurge at frequencies ranging from 7 to 45 Hz. Likewise, reaction times (RTs) in an audio-tactile task at different latencies from sound onset also present significant differences between sound types. RTs decrease with increasing looming intensities, i.e. as the sense of urgency increases, but remain constant with stationary flat intensities. We define the reaction time variation or "gain" during looming sound presentation, and show that higher RT gains are associated with stronger correlations between EEG power responses and sound intensity. Higher RT gain further entails higher relative power differences between loom and flat in the alpha and beta bands. The full-width-at-half-maximum of the autocorrelation function of the eyes-closed resting state EEG also increases with RT gain. The effects are topographically located over the central and frontal electrodes. A computational model reveals that the increase in stimulus-response correlation in subjects with slower resting state fluctuations is expected when EEG power fluctuations at each electrode and in a given band are viewed as simple coupled low-pass filtered noise processes jointly driven by the sound intensity. The model assumes that the strength of stimulus-power coupling is proportional to RT gain in different coupling scenarios, suggesting a mechanism by which slower resting state fluctuations enhance EEG response and shorten reaction times.

Exact solution for the Anisotropic Ornstein-Uhlenbeck process.

Active-Matter models commonly consider particles with overdamped dynamics subject to a force (speed) with constant modulus and random direction. Some models also include random noise in particle displacement (a Wiener process), resulting in diffusive motion at short time scales. On the other hand, Ornstein-Uhlenbeck processes apply Langevin dynamics to the particles' velocity and predict motion that is not diffusive at short time scales. Experiments show that migrating cells have gradually varying speeds at intermediate and long time scales, with short-time diffusive behavior. While Ornstein-Uhlenbeck processes can describe the moderate-and long-time speed variation, Active-Matter models for over-damped particles can explain the short-time diffusive behavior. Isotropic models cannot explain both regimes, because short-time diffusion renders instantaneous velocity ill-defined, and prevents the use of dynamical equations that require velocity time-derivatives. On the other hand, both models correctly describe some of the different temporal regimes seen in migrating biological cells and must, in the appropriate limit, yield the same observable predictions. Here we propose and solve analytically an Anisotropic Ornstein-Uhlenbeck process for polarized particles, with Langevin dynamics governing the particle's movement in the polarization direction and a Wiener process governing displacement in the orthogonal direction. Our characterization provides a theoretically robust way to compare movement in dimensionless simulations to movement in experiments in which measurements have meaningful space and time units. We also propose an approach to deal with inevitable finite-precision effects in experiments and simulations.

Latent Ornstein-Uhlenbeck models for Bayesian analysis of multivariate longitudinal categorical responses.

We propose a Bayesian latent Ornstein-Uhlenbeck (OU) model to analyze unbalanced longitudinal data of binary and ordinal variables, which are manifestations of fewer continuous latent variables. We focus on the evolution of such latent variables when they continuously change over time. Existing approaches are limited to data collected at regular time intervals. Our proposal makes use of an OU process for the latent variables to overcome this limitation. We show that assuming real eigenvalues for the drift matrix of the OU process, as is frequently done in practice, can lead to biased estimates and/or misleading inference when the true process is oscillating. In contrast, our proposal allows for both real and complex eigenvalues. We illustrate our proposed model with a motivating dataset, containing patients with amyotrophic lateral sclerosis disease. We were interested in how bulbar, cervical, and lumbar functions evolve over time.

Serial correlation structures in latent linear mixed models for analysis of multivariate longitudinal ordinal responses.

We propose a latent linear mixed model to analyze multivariate longitudinal data of multiple ordinal variables, which are manifestations of fewer continuous latent variables. We focus on the latent level where the effects of observed covariates on the latent variables are of interest. We incorporate serial correlation into the variance component rather than assuming independent residuals. We show that misleading inference may be drawn when misspecifying the variance component. Furthermore, we provide a graphical tool depicting latent empirical semi-variograms to detect serial correlation for latent stationary linear mixed models. We apply our proposed model to examine the treatment effect on patients having the amyotrophic lateral sclerosis disease. The result shows that the treatment can slow down progression of latent cervical and lumbar functions.

Judge it by its shape: a pollinator-blind approach reveals convergence in petal shape and infers pollination modes in the genus Erythrina.

PREMISE: Pollinators are thought to exert selective pressures on plants, mediating the evolution of convergent floral shape often recognized as pollination syndromes. However, little is known about the accuracy of using petal shape for inferring convergence in pollination mode without a priori pollination information. Here we studied the genus Erythrina L. as a test case to assess whether ornithophyllous pollination modes (hummingbirds, passerines, sunbirds, or mixed pollination) can be inferred based on the evolutionary analysis of petal shape. METHODS: We characterized the two-dimensional dissected shape of standard, keel, and wing petals from 106 Erythrina species using geometric morphometrics and reconstructed a phylogenetic tree of 83 Erythrina species based on plastid trnL-F and nuclear ribosomal ITS sequences. We then used two phylogenetic comparative methods based on Ornstein-Uhlenbeck models, SURFACE and l1OU, to infer distinct morphological groups using petal shape and identify instances of convergent evolution. The effectiveness of these methods was evaluated by comparing the groups inferred to known pollinators. RESULTS: We found significant petal shape differences between hummingbird- and passerine-pollinated Erythrina species. Our analyses also revealed that petal combinations generally provided better inferences of pollinator types than individual petals and that the method and optimization criterion can affect the results. CONCLUSIONS: We show that model-based approaches using petal shape can detect convergent evolution of floral shape and relatively accurately infer pollination modes in Erythrina. The inference power of the keel petals argues for a deeper investigation of their role in the pollination biology of Erythrina and other bird-pollinated legumes.

Energetic tradeoffs control the size distribution of aquatic mammals.

Four extant lineages of mammals have invaded and diversified in the water: Sirenia, Cetacea, Pinnipedia, and Lutrinae. Most of these aquatic clades are larger bodied, on average, than their closest land-dwelling relatives, but the extent to which potential ecological, biomechanical, and physiological controls contributed to this pattern remains untested quantitatively. Here, we use previously published data on the body masses of 3,859 living and 2,999 fossil mammal species to examine the evolutionary trajectories of body size in aquatic mammals through both comparative phylogenetic analysis and examination of the fossil record. Both methods indicate that the evolution of an aquatic lifestyle is driving three of the four extant aquatic mammal clades toward a size attractor at ∼500 kg. The existence of this body size attractor and the relatively rapid selection toward, and limited deviation from, this attractor rule out most hypothesized drivers of size increase. These three independent body size increases and a shared aquatic optimum size are consistent with control by differences in the scaling of energetic intake and cost functions with body size between the terrestrial and aquatic realms. Under this energetic model, thermoregulatory costs constrain minimum size, whereas limitations on feeding efficiency constrain maximum size. The optimum size occurs at an intermediate value where thermoregulatory costs are low but feeding efficiency remains high. Rather than being released from size pressures, water-dwelling mammals are driven and confined to larger body sizes by the strict energetic demands of the aquatic medium.

Relationship between Age and Value of Information for a Noisy Ornstein-Uhlenbeck Process.

The age of information (AoI) has been widely used to quantify the information freshness in real-time status update systems. As the AoI is independent of the inherent property of the source data and the context, we introduce a mutual information-based value of information (VoI) framework for hidden Markov models. In this paper, we investigate the VoI and its relationship to the AoI for a noisy Ornstein-Uhlenbeck (OU) process. We explore the effects of correlation and noise on their relationship, and find logarithmic, exponential and linear dependencies between the two in three different regimes. This gives the formal justification for the selection of non-linear AoI functions previously reported in other works. Moreover, we study the statistical properties of the VoI in the example of a queue model, deriving its distribution functions and moments. The lower and upper bounds of the average VoI are also analysed, which can be used for the design and optimisation of freshness-aware networks. Numerical results are presented and further show that, compared with the traditional linear age and some basic non-linear age functions, the proposed VoI framework is more general and suitable for various contexts.

Phylogenetic Curved Optimal Regression for Adaptive Trait Evolution.

Regression analysis using line equations has been broadly applied in studying the evolutionary relationship between the response trait and its covariates. However, the characteristics among closely related species in nature present abundant diversities where the nonlinear relationship between traits have been frequently observed. By treating the evolution of quantitative traits along a phylogenetic tree as a set of continuous stochastic variables, statistical models for describing the dynamics of the optimum of the response trait and its covariates are built herein. Analytical representations for the response trait variables, as well as their optima among a group of related species, are derived. Due to the models' lack of tractable likelihood, a procedure that implements the Approximate Bayesian Computation (ABC) technique is applied for statistical inference. Simulation results show that the new models perform well where the posterior means of the parameters are close to the true parameters. Empirical analysis supports the new models when analyzing the trait relationship among kangaroo species.

Application of OU processes to modelling temporal dynamics of the human microbiome, and calculating optimal sampling schemes.

BACKGROUND: The vast majority of microbiome research so far has focused on the structure of the microbiome at a single time-point. There have been several studies that measure the microbiome from a particular environment over time. A few models have been developed by extending time series models to accomodate specific features in microbiome data to address questions of stability and interactions of the microbime time series. Most research has observed the stability and mean reversion for some microbiomes. However, little has been done to study the mean reversion rates of these stable microbes and how sampling frequencies are related to such conclusions. In this paper, we begin to rectify this situation. We analyse two widely studied microbial time series data sets on four healthy individuals. We choose to study healthy individuals because we are interested in the baseline temporal dynamics of the microbiome. RESULTS: For this analysis, we focus on the temporal dynamics of individual genera, absorbing all interactions in a stochastic term. We use a simple stochastic differential equation model to assess the following three questions. (1) Does the microbiome exhibit temporal continuity? (2) Does the microbiome have a stable state? (3) To better understand the temporal dynamics, how frequently should data be sampled in future studies? We find that a simple Ornstein-Uhlenbeck model which incorporates both temporal continuity and reversion to a stable state fits the data for almost every genus better than a Brownian motion model that contains only temporal continuity. The Ornstein-Uhlenbeck model also fits the data better than modelling separate time points as independent. Under the Ornstein-Uhlenbeck model, we calculate the variance of the estimated mean reversion rate (the speed with which each genus returns to its stable state). Based on this calculation, we are able to determine the optimal sample schemes for studying temporal dynamics. CONCLUSIONS: There is evidence of temporal continuity for most genera; there is clear evidence of a stable state; and the optimal sampling frequency for studying temporal dynamics is in the range of one sample every 0.8-3.2 days.