Statistics of eigenvalue dispersion indices: Quantifying the magnitude of phenotypic integration

The paradox of predictability provides a bridge between micro- and macroevolution

The relationship between the evolutionary dynamics observed in contemporary populations (microevolution) and evolution on timescales of millions of years (macroevolution) has been a topic of considerable debate. Historically, this debate centers on inconsistencies between microevolutionary processes and macroevolutionary patterns. Here, we characterize a striking exception: emerging evidence indicates that standing variation in contemporary populations and macroevolutionary rates of phenotypic divergence is often positively correlated. This apparent consistency between micro- and macroevolution is paradoxical because it contradicts our previous understanding of phenotypic evolution and is so far unexplained. Here, we explore the prospects for bridging evolutionary timescales through an examination of this "paradox of predictability." We begin by explaining why the divergence-variance correlation is a paradox, followed by data analysis to show that the correlation is a general phenomenon across a broad range of temporal scales, from a few generations to tens of millions of years. Then we review complementary approaches from quantitative genetics, comparative morphology, evo-devo, and paleontology to argue that they can help to address the paradox from the shared vantage point of recent work on evolvability. In conclusion, we recommend a methodological orientation that combines different kinds of short-term and long-term data using multiple analytical frameworks in an interdisciplinary research program. Such a program will increase our general understanding of how evolution works within and across timescales.

Distribution theories for genetic line of least resistance and evolvability measures

Quantitative genetic theory on multivariate character evolution predicts that a population's response to directional selection is biased towards the major axis of the genetic covariance matrix G-the so-called genetic line of least resistance. Inferences on the genetic constraints in this sense have traditionally been made by measuring the angle of deviation of evolutionary trajectories from the major axis or, more recently, by calculating the amount of genetic variance-the Hansen-Houle evolvability-available along the trajectories. However, there have not been clear practical guidelines on how these quantities can be interpreted, especially in a high-dimensional space. This study summarizes pertinent distribution theories for relevant quantities, pointing out that they can be written as ratios of quadratic forms in evolutionary trajectory vectors by taking G as a parameter. For example, a beta distribution with appropriate parameters can be used as a null distribution for the squared cosine of the angle of deviation from a major axis or subspace. More general cases can be handled with the probability distribution of ratios of quadratic forms in normal variables. Apart from its use in hypothesis testing, this latter approach could potentially be used as a heuristic tool for looking into various selection scenarios, like directional and/or correlated selection, as parameterized with the mean and covariance of selection gradients.

Phenotypic covariation predicts diversification in an adaptive radiation of pupfishes

Phenotypic covariation among suites of traits may constrain or promote diversification both within and between species, yet few studies have empirically tested this relationship. In this study, we investigate whether phenotypic covariation of craniofacial traits is associated with diversification in an adaptive radiation of pupfishes found only on San Salvador Island, Bahamas (SSI). The radiation includes generalist, durophagous, and lepidophagous species. We compared phenotypic variation and covariation (i.e., the P matrix) between (1) allopatric populations of generalist pupfish from neighboring islands and estuaries in the Caribbean, (2) SSI pupfish allopatric lake populations with only generalist pupfish, and (3) SSI lake populations containing the full radiation in sympatry. Additionally, we examine patterns observed in the P matrices of two independent lab-reared F2 hybrid crosses of the two most morphologically distinct members of the radiation to make inferences about the underlying mechanisms contributing to the variation in craniofacial traits in SSI pupfishes. We found that the P matrix of SSI allopatric generalist populations exhibited higher levels of mean trait correlation, constraints, and integration with simultaneously lower levels of flexibility compared to allopatric generalist populations on other Caribbean islands and sympatric populations of all three species on SSI. We also document that while many craniofacial traits appear to result from additive genetic effects, variation in key traits such as head depth, maxilla length, and lower jaw length may be produced via non-additive genetic mechanisms. Ultimately, this study suggests that differences in phenotypic covariation significantly contribute to producing and maintaining organismal diversity.

Innovation and elaboration on the avian tree of life

Widely documented, megaevolutionary jumps in phenotypic diversity continue to perplex researchers because it remains unclear whether these marked changes can emerge from microevolutionary processes. Here, we tackle this question using new approaches for modeling multivariate traits to evaluate the magnitude and distribution of elaboration and innovation in the evolution of bird beaks. We find that elaboration, evolution along the major axis of phenotypic change, is common at both macro- and megaevolutionary scales, whereas innovation, evolution away from the major axis of phenotypic change, is more prominent at megaevolutionary scales. The major axis of phenotypic change among species beak shapes at megaevolutionary scales is an emergent property of innovation across clades. Our analyses suggest that the reorientation of phenotypes via innovation is a ubiquitous route for divergence that can arise through gradual change alone, opening up further avenues for evolution to explore.

Exact expressions and numerical evaluation of average evolvability measures for characterizing and comparing G matrices

Theory predicts that the additive genetic covariance (G ) matrix determines a population's short-term (in)ability to respond to directional selection-evolvability in the Hansen-Houle sense-which is typically quantified and compared via certain scalar indices called evolvability measures. Often, interest is in obtaining the averages of these measures across all possible selection gradients, but explicit formulae for most of these average measures have not been known. Previous authors relied either on approximations by the delta method, whose accuracy is generally unknown, or Monte Carlo evaluations (including the random skewers analysis), which necessarily involve random fluctuations. This study presents new, exact expressions for the average conditional evolvability, average autonomy, average respondability, average flexibility, average response difference, and average response correlation, utilizing their mathematical structures as ratios of quadratic forms. The new expressions are infinite series involving top-order zonal and invariant polynomials of matrix arguments, and can be numerically evaluated as their partial sums with, for some measures, known error bounds. Whenever these partial sums numerically converge within reasonable computational time and memory, they will replace the previous approximate methods. In addition, new expressions are derived for the average measures under a general normal distribution for the selection gradient, extending the applicability of these measures into a substantially broader class of selection regimes.

Diversification of the ruminant skull along an evolutionary line of least resistance

Clarifying how microevolutionary processes scale to macroevolutionary patterns is a fundamental goal in evolutionary biology, but these analyses, requiring comparative datasets of population-level variation, are limited. By analyzing a previously published dataset of 2859 ruminant crania, we find that variation within and between ruminant species is biased by a highly conserved mammalian-wide allometric pattern, CREA (craniofacial evolutionary allometry), where larger species have proportionally longer faces. Species with higher morphological integration and species more biased toward CREA have diverged farther from their ancestors, and Ruminantia as a clade diversified farther than expected in the direction of CREA. Our analyses indicate that CREA acts as an evolutionary "line of least resistance" and facilitates morphological diversification due to its alignment with the browser-grazer continuum. Together, our results demonstrate that constraints at the population level can produce highly directional patterns of phenotypic evolution at the macroevolutionary scale. Further research is needed to explore how CREA has been exploited in other mammalian clades.

An effect size for comparing the strength of morphological integration across studies

Understanding how and why phenotypic traits covary is a major interest in evolutionary biology. Biologists have long sought to characterize the extent of morphological integration in organisms, but comparing levels of integration for a set of traits across taxa has been hampered by the lack of a reliable summary measure and testing procedure. Here, we propose a standardized effect size for this purpose, calculated from the relative eigenvalue variance,V r e l $V_{rel}$ . First, we evaluate several eigenvalue dispersion indices under various conditions, and show that onlyV r e l $V_{rel}$ remains stable across samples size and the number of variables. We then demonstrate thatV r e l $V_{rel}$ accurately characterizes input patterns of covariation, so long as redundant dimensions are excluded from the calculations. However, we also show that the variance of the sampling distribution ofV r e l $V_{rel}$ depends on input levels of trait covariation, makingV r e l $V_{rel}$ unsuitable for direct comparisons. As a solution, we propose transformingV r e l $V_{rel}$ to a standardized effect size (Z-score) for representing the magnitude of integration for a set of traits. We also propose a two-sample test for comparing the strength of integration between taxa, and show that this test displays appropriate statistical properties. We provide software for implementing the procedure, and an empirical example illustrates its use.

Solving the grand challenge of phenotypic integration: allometry across scales

Phenotypic integration is a concept related to the cascade of trait relationships from the lowest organizational levels, i.e. genes, to the highest, i.e. whole-organism traits. However, the cause-and-effect linkages between traits are notoriously difficult to determine. In particular, we still lack a mathematical framework to model the relationships involved in the integration of phenotypic traits. Here, we argue that allometric models developed in ecology offer testable mathematical equations of trait relationships across scales. We first show that allometric relationships are pervasive in biology at different organizational scales and in different taxa. We then present mechanistic models that explain the origin of allometric relationships. In addition, we emphasized that recent studies showed that natural variation does exist for allometric parameters, suggesting a role for genetic variability, selection and evolution. Consequently, we advocate that it is time to examine the genetic determinism of allometries, as well as to question in more detail the role of genome size in subsequent scaling relationships. More broadly, a possible-but so far neglected-solution to understand phenotypic integration is to examine allometric relationships at different organizational levels (cell, tissue, organ, organism) and in contrasted species.

Evolvability and Macroevolution: Overview and Synthesis

Evolvability is best addressed from a multi-level, macroevolutionary perspective through a comparative approach that tests for among-clade differences in phenotypic diversification in response to an opportunity, such as encountered after a mass extinction, entering a new adaptive zone, or entering a new geographic area. Analyzing the dynamics of clades under similar environmental conditions can (partially) factor out shared external drivers to recognize intrinsic differences in evolvability, aiming for a macroevolutionary analog of a common-garden experiment. Analyses will be most powerful when integrating neontological and paleontological data: determining differences among extant populations that can be hypothesized to generate large-scale, long-term contrasts in evolvability among clades; or observing large-scale differences among clade histories that can by hypothesized to reflect contrasts in genetics and development observed directly in extant populations. However, many comparative analyses can be informative on their own, as explored in this overview. Differences in clade-level evolvability can be visualized in diversity-disparity plots, which can quantify positive and negative departures of phenotypic productivity from stochastic expectations scaled to taxonomic diversification. Factors that evidently can promote evolvability include modularity—when selection aligns with modular structure or with morphological integration patterns; pronounced ontogenetic changes in morphology, as in allometry or multiphase life cycles; genome size; and a variety of evolutionary novelties, which can also be evaluated using macroevolutionary lags between the acquisition of a trait and phenotypic diversification, and dead-clade-walking patterns that may signal a loss of evolvability when extrinsic factors can be excluded. High speciation rates may indirectly foster phenotypic evolvability, and vice versa. Mechanisms are controversial, but clade evolvability may be higher in the Cambrian, and possibly early in the history of clades at other times; in the tropics; and, for marine organisms, in shallow-water disturbed habitats.

Detecting (non)parallel evolution in multidimensional spaces: angles, correlations and eigenanalysis

Parallelism between evolutionary trajectories in a trait space is often seen as evidence for repeatability of phenotypic evolution, and angles between trajectories play a pivotal role in the analysis of parallelism. However, properties of angles in multidimensional spaces have not been widely appreciated by biologists. To remedy this situation, this study provides a brief overview on geometric and statistical aspects of angles in multidimensional spaces. Under the null hypothesis that trajectory vectors have no preferred directions (i.e. uniform distribution on hypersphere), the angle between two independent vectors is concentrated around the right angle, with a more pronounced peak in a higher-dimensional space. This probability distribution is closely related to t- and beta distributions, which can be used for testing the null hypothesis concerning a pair of trajectories. A recently proposed method with eigenanalysis of a vector correlation matrix can be connected to the test of no correlation or concentration of multiple vectors, for which simple test procedures are available in the statistical literature. Concentration of vectors can also be examined by tools of directional statistics such as the Rayleigh test. These frameworks provide biologists with baselines to make statistically justified inferences for (non)parallel evolution.