Monte Carlo simulation of OLS and linear mixed model inference of phenotypic effects on gene expression.

BACKGROUND: Self-contained tests estimate and test the association between a phenotype and mean expression level in a gene set defined a priori. Many self-contained gene set analysis methods have been developed but the performance of these methods for phenotypes that are continuous rather than discrete and with multiple nuisance covariates has not been well studied. Here, I use Monte Carlo simulation to evaluate the performance of both novel and previously published (and readily available via R) methods for inferring effects of a continuous predictor on mean expression in the presence of nuisance covariates. The motivating data are a high-profile dataset which was used to show opposing effects of hedonic and eudaimonic well-being (or happiness) on the mean expression level of a set of genes that has been correlated with social adversity (the CTRA gene set). The original analysis of these data used a linear model (GLS) of fixed effects with correlated error to infer effects of Hedonia and Eudaimonia on mean CTRA expression. METHODS: The standardized effects of Hedonia and Eudaimonia on CTRA gene set expression estimated by GLS were compared to estimates using multivariate (OLS) linear models and generalized estimating equation (GEE) models. The OLS estimates were tested using O'Brien's OLS test, Anderson's permutation [Formula: see text]-test, two permutation F-tests (including GlobalAncova), and a rotation z-test (Roast). The GEE estimates were tested using a Wald test with robust standard errors. The performance (Type I, II, S, and M errors) of all tests was investigated using a Monte Carlo simulation of data explicitly modeled on the re-analyzed dataset. RESULTS: GLS estimates are inconsistent between data sets, and, in each dataset, at least one coefficient is large and highly statistically significant. By contrast, effects estimated by OLS or GEE are very small, especially relative to the standard errors. Bootstrap and permutation GLS distributions suggest that the GLS results in downward biased standard errors and inflated coefficients. The Monte Carlo simulation of error rates shows highly inflated Type I error from the GLS test and slightly inflated Type I error from the GEE test. By contrast, Type I error for all OLS tests are at the nominal level. The permutation F-tests have ∼1.9X the power of the other OLS tests. This increased power comes at a cost of high sign error (∼10%) if tested on small effects. DISCUSSION: The apparently replicated pattern of well-being effects on gene expression is most parsimoniously explained as "correlated noise" due to the geometry of multiple regression. The GLS for fixed effects with correlated error, or any linear mixed model for estimating fixed effects in designs with many repeated measures or outcomes, should be used cautiously because of the inflated Type I and M error. By contrast, all OLS tests perform well, and the permutation F-tests have superior performance, including moderate power for very small effects.

Repeatability of locomotor performance and morphology-locomotor performance relationships.

There is good evidence that natural selection drives the evolution of locomotor performance, but the processes that generate the among-individual variation for selection to act on are relatively poorly understood. We measured prolonged swimming performance, Ucrit, and morphology in a large cohort (n=461) of wild-type zebrafish (Danio rerio) at ∼6 months and again at ∼9 months. Using mixed-model analyses to estimate repeatability as the intraclass correlation coefficient, we determined that Ucrit was significantly repeatable (r=0.55; 95% CI: 0.45-0.64). Performance differences between the sexes (males 12% faster than females) and changes with age (decreasing 0.07% per day) both contributed to variation in Ucrit and, therefore, the repeatability estimate. Accounting for mean differences between sexes within the model decreased the estimate of Ucrit repeatability to 21% below the naïve estimate, while fitting age in the models increased the estimate to 14% above the naïve estimate. Greater consideration of factors such as age and sex is therefore necessary for the interpretation of performance repeatability in wild populations. Body shape significantly predicted Ucrit in both sexes in both assays, with the morphology-performance relationship significantly repeatable at the population level. However, morphology was more strongly predicative of performance in older fish, suggesting a change in the contribution of morphology relative to other factors such as physiology and behaviour. The morphology-performance relationship changed with age to a greater extent in males than females.

Flowing water affects fish fast-starts: escape performance of the Hawaiian stream goby, Sicyopterus stimpsoni.

Experimental measurements of escape performance in fishes have typically been conducted in still water; however, many fishes inhabit environments with flow that could impact escape behavior. We examined the influences of flow and predator attack direction on the escape behavior of fish, using juveniles of the amphidromous Hawaiian goby Sicyopterus stimpsoni In nature, these fish must escape ambush predation while moving through streams with high-velocity flow. We measured the escape performance of juvenile gobies while exposing them to a range of water velocities encountered in natural streams and stimulating fish from three different directions. Frequency of response across treatments indicated strong effects of flow conditions and attack direction. Juvenile S. stimpsoni had uniformly high response rates for attacks from a caudal direction (opposite flow); however, response rates for attacks from a cranial direction (matching flow) decreased dramatically as flow speed increased. Mechanical stimuli produced by predators attacking in the same direction as flow might be masked by the flow environment, impairing the ability of prey to detect attacks. Thus, the likelihood of successful escape performance in fishes can depend critically on environmental context.

Performance trade-offs and individual quality in decathletes.

Many constraints of organismal design at the cell and organ level, including muscle fiber types, musculoskeletal gearing and control-surface geometry, are believed to cause performance trade-offs at the whole-organism level. Contrary to this expectation, positive correlations between diverse athletic performances are frequently found in vertebrates. Recently, it has been proposed that trade-offs between athletic performances in humans are masked by variation in individual quality and that underlying trade-offs are revealed by adjusting the correlations to 'control' quality. We argue that quality is made up of both intrinsic components, due to the causal mapping between morpho-physiological traits and performance, and extrinsic components, due to variation in training intensity, diet and pathogens. Only the extrinsic component should be controlled. We also show that previous methods to estimate 'quality-free' correlations perform poorly. We show that Wright's factor analysis recovers the correct quality-free correlation matrix and use this method to estimate quality-free correlations among the 10 events of the decathlon using a dataset of male college athletes. We found positive correlations between all decathlon events, which supports an axis that segregates 'good athletes' from 'bad athletes'. Estimates of quality-free correlations are mostly very small (<0.1), suggesting large, quality-free independence between events. Because quality must include both intrinsic and extrinsic components, the physiological significance of these adjusted correlations remains obscure. Regardless, the underlying architecture of the functional systems and the physiological explanation of both the un-adjusted and adjusted correlations remain to be discovered.

The effect of unmeasured confounders on the ability to estimate a true performance or selection gradient (and other partial regression coefficients).

Multiple regression of observational data is frequently used to infer causal effects. Partial regression coefficients are biased estimates of causal effects if unmeasured confounders are not in the regression model. The sensitivity of partial regression coefficients to omitted confounders is investigated with a Monte-Carlo simulation. A subset of causal traits is "measured" and their effects are estimated using ordinary least squares regression and compared to their expected values. Three major results are: (1) the error due to confounding is much larger than that due to sampling, especially with large samples, (2) confounding error shrinks trivially with sample size, and (3) small true effects are frequently estimated as large effects. Consequently, confidence intervals from regression are poor guides to the true intervals, especially with large sample sizes. The addition of a confounder to the model improves estimates only 55% of the time. Results are improved with complete knowledge of the rank order of causal effects but even with this omniscience, measured intervals are poor proxies for true intervals if there are many unmeasured confounders. The results suggest that only under very limited conditions can we have much confidence in the magnitude of partial regression coefficients as estimates of causal effects.

Body fineness ratio as a predictor of maximum prolonged-swimming speed in coral reef fishes.

The ability to sustain high swimming speeds is believed to be an important factor affecting resource acquisition in fishes. While we have gained insights into how fin morphology and motion influences swimming performance in coral reef fishes, the role of other traits, such as body shape, remains poorly understood. We explore the ability of two mechanistic models of the causal relationship between body fineness ratio and endurance swimming-performance to predict maximum prolonged-swimming speed (Umax ) among 84 fish species from the Great Barrier Reef, Australia. A drag model, based on semi-empirical data on the drag of rigid, submerged bodies of revolution, was applied to species that employ pectoral-fin propulsion with a rigid body at U max. An alternative model, based on the results of computer simulations of optimal shape in self-propelled undulating bodies, was applied to the species that swim by body-caudal-fin propulsion at Umax . For pectoral-fin swimmers, Umax increased with fineness, and the rate of increase decreased with fineness, as predicted by the drag model. While the mechanistic and statistical models of the relationship between fineness and Umax were very similar, the mechanistic (and statistical) model explained only a small fraction of the variance in Umax . For body-caudal-fin swimmers, we found a non-linear relationship between fineness and Umax , which was largely negative over most of the range of fineness. This pattern fails to support either predictions from the computational models or standard functional interpretations of body shape variation in fishes. Our results suggest that the widespread hypothesis that a more optimal fineness increases endurance-swimming performance via reduced drag should be limited to fishes that swim with rigid bodies.

An integrative model of evolutionary covariance: a symposium on body shape in fishes.

A major direction of current and future biological research is to understand how multiple, interacting functional systems coordinate in producing a body that works. This understanding is complicated by the fact that organisms need to work well in multiple environments, with both predictable and unpredictable environmental perturbations. Furthermore, organismal design reflects a history of past environments and not a plan for future environments. How complex, interacting functional systems evolve, then, is a truly grand challenge. In accepting the challenge, an integrative model of evolutionary covariance is developed. The model combines quantitative genetics, functional morphology/physiology, and functional ecology. The model is used to convene scientists ranging from geneticists, to physiologists, to ecologists, to engineers to facilitate the emergence of body shape in fishes as a model system for understanding how complex, interacting functional systems develop and evolve. Body shape of fish is a complex morphology that (1) results from many developmental paths and (2) functions in many different behaviors. Understanding the coordination and evolution of the many paths from genes to body shape, body shape to function, and function to a working fish body in a dynamic environment is now possible given new technologies from genetics to engineering and new theoretical models that integrate the different levels of biological organization (from genes to ecology).

A general model of functional constraints on phenotypic evolution.

A general model of the functional constraints on the rate and direction of phenotypic evolution is developed using a decomposition of the Lande-Arnold model of multivariate phenotypic evolution. The important feature of the model is the F matrix of performance coefficients reflecting the causal relationship between morphophysiological (m-p) and functional performance traits. The structure of F, which reflects the functional architecture of the organism, constrains the shape of the adaptive landscape and thus the rate and direction of m-p trait evolution. The rate of m-p trait evolution is a function of the pattern of coefficients in a row of F. The sums and variances of these rows are related to current concepts of evolvability. The direction of m-p trait evolution through m-p trait space is a function of the functional covariances among m-p traits. The functional covariance between a pair of m-p traits is a measure of how much the traits function together and is computed as the covariance between rows of F. Finally, it is shown that genetic covariances between m-p traits and performance traits are a function of the F matrix, but a G matrix that includes these covariances cannot be used to model functional constraints effectively.

Constraints on adaptive evolution: the functional trade-off between reproduction and fast-start swimming performance in the Trinidadian guppy (Poecilia reticulata).

The empirical study of natural selection reveals that adaptations often involve trade-offs between competing functions. Because natural selection acts on whole organisms rather than isolated traits, adaptive evolution may be constrained by the interaction between traits that are functionally integrated. Yet, few attempts have been made to characterize how and when such constraints are manifested or whether they limit the adaptive divergence of populations. Here we examine the consequences of adaptive life-history evolution on locomotor performance in the live-bearing guppy. In response to increased predation from piscivorous fish, Trinidadian guppies evolve an increased allocation of resources toward reproduction. These populations are also under strong selection for rapid fast-start swimming performance to evade predators. Because embryo development increases a female's wet mass as she approaches parturition, an increased investment in reproductive allocation should impede fast-start performance. We find evidence for adaptive but constrained evolution of fast-start swimming performance in laboratory trials conducted on second-generation lab-reared fish. Female guppies from high-predation localities attain a faster acceleration and velocity and travel a greater distance during fast-start swimming trials. However, velocity and distance traveled decline more rapidly over the course of pregnancy in these same females, thus reducing the magnitude of divergence in swimming performance between high- and low-predation populations. This functional trade-off between reproduction and swimming performance reveals how different aspects of the phenotype are integrated and highlights the complexity of adaptation at the whole-organism level.

Dynamics of pectoral fin rowing in a fish with an extreme rowing stroke: the threespine stickleback (Gasterosteus aculeatus).

The dynamics of pectoral fin rowing in the threespine stickleback are investigated by measuring the instantaneous force balance on freely swimming fish throughout the stroke cycle and comparing the measured forces with fin motions and an unsteady, blade-element model of pectoral fin propulsion. Both measured and modeled forces suggest that attached vortex and circulatory forces and not inertial (added mass) forces dominate the force balance. Peak forces occur at midstrokes. There is no evidence for large force peaks at the stroke transitions due to either rapid fin rotation (supination) or rapid fin closure against the body. The energetics of pectoral fin rowing are estimated using the unsteady blade-element model and an indirect method based on the center of mass dynamics. The results indicate that the mechanical efficiency of pectoral fin rowing is low (0.1-0.3) relative to a flapping mechanism and possibly relative to axial undulation at comparable speeds.

Multi-trait Selection, Adaptation, and Constraints on the Evolution of Burst Swimming Performance.

Whole organism performance represents the integration of numerous physiological, morphological, and behavioral traits. How adaptive changes in performance evolve therefore requires an understanding of how selection acts on multiple integrated traits. Two approaches that lend themselves to studying the evolution of performance in natural populations are the use of quantitative genetics models for estimating the strength of selection acting on multiple quantitative traits and ecological genetic comparisons of populations exhibiting phenotypic differences correlated with environmental variation. In both cases, the ultimate goal is to understand how suites of traits and trade-offs between competing functions respond to natural selection. Here we consider how these two complimentary approaches can be applied to study the adaptive evolution of escape performance in fish. We first present an extension of Arnold's (1983) quantitative genetic approach that explicitly considers how trade-offs between different components of performance interact with the underlying genetics. We propose that such a model can reveal the conditions under which multiple selection pressures will cause adaptive change in traits that influence more than one component of fitness. We then review work on the Atlantic silversides and Trinidadian guppies as two case studies where an ecological genetics approach has been successfully applied to evaluate how the evolution of escape performance trades-off with other components of fitness. We conclude with the general lesson that whole organism performance is embedded in a complex phenotype, and that the net outcome of selection acting on different aspects of the organism will often result in a compromise among competing influences.

Rotational lift: something different or more of the same?

This paper addresses the question, do the rotational forces in the hovering fruit fly Drosophila melanogaster reflect something different (the Magnus effect) or more of the same (circulatory-and-attached-vortex force)? The results of an unsteady blade-element model using empirically derived force coefficients from translating (root-oscillating) wings are compared with recent results derived from both the measured forces on a dynamically scaled Drosophila wing and the computational fluid dynamic (CFD)-modeled forces on a virtual Drosophila wing. The behavior of the forces in all three models during wing rotation supports the hypothesis that rotational lift is not a novel aerodynamic mechanism but is caused by the same fluid-dynamic mechanism that occurs during wing translation. A comparison of the unsteady model with a quasi-steady model that employs empirically derived rotational coefficients further supports the hypothesis that rotational forces are more of the same. Finally, the overall similarity of the results between the unsteady model, the physical wing model and the CFD model suggests that the unsteady model can be used to explore the performance consequences of kinematic variation and to investigate locomotor control in freely moving animals.

Fluid dynamics of flapping aquatic flight in the bird wrasse: three-dimensional unsteady computations with fin deformation.

Many fishes that swim with the paired pectoral fins use fin-stroke parameters that produce thrust force from lift in a mechanism of underwater flight. These locomotor mechanisms are of interest to behavioral biologists, biomechanics researchers and engineers. In the present study, we performed the first three-dimensional unsteady computations of fish swimming with oscillating and deforming fins. The objective of these computations was to investigate the fluid dynamics of force production associated with the flapping aquatic flight of the bird wrasse Gomphosus varius. For this computational work, we used the geometry of the wrasse and its pectoral fin, and previously measured fin kinematics, as the starting points for computational investigation of three-dimensional (3-D) unsteady fluid dynamics. We performed a 3-D steady computation and a complete set of 3-D quasisteady computations for a range of pectoral fin positions and surface velocities. An unstructured, grid-based, unsteady Navier-Stokes solver with automatic adaptive remeshing was then used to compute the unsteady flow about the wrasse through several complete cycles of pectoral fin oscillation. The shape deformation of the pectoral fin throughout the oscillation was taken from the experimental kinematics. The pressure distribution on the body of the bird wrasse and its pectoral fins was computed and integrated to give body and fin forces which were decomposed into lift and thrust. The velocity field variation on the surface of the wrasse body, on the pectoral fins and in the near-wake was computed throughout the swimming cycle. We compared our computational results for the steady, quasi-steady and unsteady cases with the experimental data on axial and vertical acceleration obtained from the pectoral fin kinematics experiments. These comparisons show that steady state computations are incapable of describing the fluid dynamics of flapping fins. Quasi-steady state computations, with correct incorporation of the experimental kinematics, are useful when determining trends in force production, but do not provide accurate estimates of the magnitudes of the forces produced. By contrast, unsteady computations about the deforming pectoral fins using experimentally measured fin kinematics were found to give excellent agreement, both in the time history of force production throughout the flapping strokes and in the magnitudes of the generated forces.

Performance limits of labriform propulsion and correlates with fin shape and motion.

Labriform locomotion, which is powered by oscillating the paired pectoral fins, varies along a continuum from rowing the fins back and forth to flapping the fins up and down. It has generally been assumed (i) that flapping is more mechanically efficient than rowing, a hypothesis confirmed by a recent simulation experiment, and (ii) that flapping should be associated with wing-shaped fins while rowing should be associated with paddle-shaped fins. To determine whether these hypotheses and the results of the simulation experiment are consistent with natural variation, we compared the steady swimming performance (critical swimming speed) of four species of labrid fish (Cirrhilabrus rubripinnis, Pseudocheilinus octotaenia, Gomphosus varius and Halichoeres bivittatus) selected to form two pairs of closely related species that vary in fin shape and in the direction of fin motion. The results were consistent with expectations. Within each pair, the species with the best swimming performance also had (i) a fin shape characterized by a higher aspect ratio, a longer leading edge relative to the trailing edge fin rays and the center of fin area located closer to the fin base, and (ii) a steeper (more dorsoventral) stroke plane.

Kinematics, dynamics, and energetics of rowing and flapping propulsion in fishes.

The shape and motion of the pectoral fins vary considerably among fishes that swim in the labriform mode. Pectoral fin motion in fishes is highly variable, but one conspicuous axis of this variation is the rowing-flapping axis. At one extreme of this axis, paddle-shaped fins row back and forth in a plane that is parallel to fish motion, while at the other extreme, wing-shaped fins flap up and down in a plane that is perpendicular to fish motion. We have used two fish, the threespine stickleback (Gasterosteus aculeatus) and the bird wrasse (Gomphosus varius), that fall near the extremes of the rowing-flapping axis to study the dynamic, energetic, and ecological and evolutionary consequences of this kinematic variation. Our work confirms some traditionally held assumptions about rowing and flapping dynamics and energetics but reject others. A computer simulation experiment of virtual rowing and flapping appendages makes several predictions about differences in maneuvering performance and swimming energetics between rowing and flapping, which, in turn, make predictions about the behavior and ecological distribution of fishes that vary along the rowing-flapping axis. Both laboratory and field studies of labrid swimming ability and distribution support these predictions.

Functional morphology and virtual models: physical constraints on the design of oscillating wings, fins, legs, and feet at intermediate reynolds numbers.

Why do some animals swim by rowing appendages back and forth while others fly by flapping them up and down? One hypothesis suggests the answer lies in the sharply divergent physical environments encountered by small, slow animals, and large, fast animals. Flapping appendages allow large animals to move through a fluid environment quickly and efficiently. As size and speed decrease, however, viscous drag increasingly dominates the force balance, with negative consequences for both rowing and flapping appendages. Nevertheless, comparative data suggest that flapping does not occur in animals at Reynolds numbers (Re) less than about 15. I used a computer simulation experiment to address the question, "Below what Re is rowing more effective than flapping?" The simulation, which employed a simple quasi-steady, blade-element model of virtual oscillating appendages, has several important results. First, the mechanical efficiency of both rowing and flapping decrease dramatically with scale. Second, the performance of rowing can increase substantially by taking advantage of several dynamic shape modifications, including area and span reduction during the recovery stroke. Finally, the relative performance of rowing and flapping is dependent on the advance ratio, which is a function of the travel speed relative to the oscillation frequency. The model predicts that rowing is more efficient than flapping at Re < 20 for animals moving throughout the range of typically observed advance ratios.

EVOLUTION OF PELVIC REDUCTION IN THREESPINE STICKLEBACK FISH: A TEST OF COMPETING HYPOTHESES.

Reimchen hypothesized that pelvic reduction in threespine stickleback is favored by an absence of piscivorous fishes and the resulting increase in predation by insects, but Giles hypothesized that the predation regime is unimportant and that a low dissolved calcium concentration favors evolution of pelvic reduction. Substantial pelvic reduction in threespine stickleback sampled from 179 lakes around Cook Inlet, Alaska is strongly associated both with an absence of predatory fishes and a low calcium concentration. However, the association of pelvic reduction with low calcium concentration appears to be contingent on the absence of predatory fishes. These results emphasize the importance of interactions between seemingly unrelated environmental variables for selection of a single trait. However, these results also conflict with some observations elsewhere and do not rule out the possibility that other environmental factors are important for selection for pelvic reduction in threespine stickleback.