ALLOMETRIC CONSTRAINTS AND THE EVOLUTION OF ALLOMETRY

Morphological traits often covary within and among species according to simple power laws referred to as allometry. Such allometric relationships may result from common growth regulation, and this has given rise to the hypothesis that allometric exponents may have low evolvability and constrain trait evolution. We formalize hypotheses for how allometry may constrain morphological trait evolution across taxa, and test these using more than 300 empirical estimates of static (within‐species) allometric relations of animal morphological traits. Although we find evidence for evolutionary changes in allometric parameters on million‐year, cross‐species time scales, there is limited evidence for microevolutionary changes in allometric slopes. Accordingly, we find that static allometries often predict evolutionary allometries on the subspecies level, but less so across species. Although there is a large body of work on allometry in a broad sense that includes all kinds of morphological trait–size relationships, we found relatively little information about the evolution of allometry in the narrow sense of a power relationship. Despite the many claims of microevolutionary changes of static allometries in the literature, hardly any of these apply to narrow‐sense allometry, and we argue that the hypothesis of strongly constrained static allometric slopes remains viable.     

Is non‐loglinear allometry a statistical artifact?

The allometric equation, y = ax^b, is commonly fitted to data indirectly by transforming predictor (x) and response (y) variables to logarithms, fitting a straight line to the transformations, and then back‐transforming (exponentiating) the resulting equation to the original arithmetic scale. Sometimes, however, transformation fails to linearize the observations, thereby giving rise to what has come to be known as non‐loglinear allometry. A smooth curve for observations displayed on a log–log plot is usually interpreted to mean that the scaling exponent in the allometric equation is a continuously changing function of body size, whereas a breakpoint between two (or more) linear segments on a log–log plot is typically taken to mean that the exponent changes abruptly, coincident with some important milestone in development. I applied simple graphical and statistical procedures in re‐analyses of three well‐known examples of non‐loglinear allometry, and showed in every instance that the relationship between predictor and response can be described in the original scale by simple functions with constant values for the exponent b. In no instance does the allometric exponent change during the course of development. Transformation of data to logarithms created new distributions that actually obscured the relationships between predictor and response variables in these investigations, and led to erroneous perceptions of growth. Such confounding effects of transformation are not limited to non‐loglinear allometry but are common to all applications of the allometric method. © 2012 The Linnean Society of London, Biological Journal of the Linnean Society, 2012, ?? , ??–??.     

The consequences of size dependent foraging for food web topology

A general question in biology is how processes at one scale, for example that of individual organisms, influence patterns at larger scales, for example communities of interacting individuals. Here we ask how changing the size‐dependence of the foraging behaviour of individuals can influence the structure of food webs. We assembled communities using a model in which species interactions are determined by allometric foraging rules of (1) handling time and (2) attack rates, and also (3) the distribution of body sizes. We systematically varied these three factors and examined their effects on three community level, food web allometries: the generality ‐ mass correlation, the vulnerability ‐ mass correlation and the trophic height ‐ mass correlation. The results demonstrate how allometries of individual foraging behaviour (handling time and attack rates) are linked across scales of organisation: different community level allometries are influenced by different individual level allometries. For example, generality allometries in the community are most affected by the individual allometric relationships of the attack rate, whereas trophic level allometries in the community are more strongly influenced by variation in individual handling time allometries. Importantly, we also find that the shape of the body size distribution from which species are drawn has a substantial influence on how these links between scales operate. This study suggests that understanding the variation of size structure among ecological networks requires knowledge about the causes of variation in individual foraging behaviour and determinants of the regional body size distribution.     

Scaling of Frond Form in Hawaiian Tree Fern Cibotium glaucum: Compliance with Global Trends and Application for Field Estimation

Large‐fronded tree ferns are critical components of many tropical forests. We investigated frond and whole‐plant allometries for Hawaiian keystone species Cibotium glaucum, for prediction and to compare with global scaling relationships. We found that C. glaucum fronds maintain geometric proportionality across a wide range of plant and frond sizes. These relationships result in strong allometries that permit rapid field estimation of frond size from simple linear dimensions. C. glaucum frond allometries complied with intra‐ and interspecific global trends for leaf area versus mass established for much smaller‐leafed species, indicating ‘diminishing returns’ in photosynthetic area per investment in mass for larger fronds. The intraspecific trend was related to declining water content in larger fronds, but not to a significantly larger investment in stipe or rachis relative to lamina. However, C. glaucum complied with the global interspecific trends for greater allocation to support structures in larger leaves. Allometries for frond number and size versus plant height showed that as plants increase in height, frond production and/or retention progressively declines, and the increases of leaf size tend to level off. These frond and whole plant‐level relationships indicate the potential for estimating frond area and mass at landscape scale to enrich studies of forest dynamics.     

The evolution of static allometry in sexually selected traits

Although it has been the subject of verbal theory since Darwin, the evolution of morphological trait allometries remains poorly understood, especially in the context of sexual selection. Here we present an allocation trade-off model that predicts the optimal pattern of allometry under different selective regimes. We derive a general solution that has a simple and intuitive interpretation and use it to investigate several examples of fitness functions. Verbal arguments have suggested cost or benefit scenarios under which sexual selection on signal or weapon traits may favor larger individuals with disproportionately larger traits (i.e., positive allometry). However, our results suggest that this is necessarily true only under a precisely specified set of conditions: positive allometry will evolve when the marginal fitness gains from an increase in relative trait size are greater for large individuals than for small ones. Thus, the optimal allometric pattern depends on the precise nature of net selection, and simple examples readily yield isometry, positive or negative allometry, or polymorphisms corresponding to sigmoidal scaling. The variety of allometric patterns predicted by our model is consistent with the diversity of patterns observed in empirical studies on the allometries of sexually selected traits. More generally, our findings highlight the difficulty of inferring complex underlying processes from simple emergent patterns.     

Multivariate allometry of the robust australopithecine zygomatic foramen: Bootstrap approach to confidence limits

The oddly rounded and disproportionate temporal fossa (or zygomatic foramen) in hyper-robust australopithecine fossils has elicited comment. Is this feature a scaled, positively-allometric version of hominoid anatomy in general or a unique adaptation? Multivariate allometry employing log principal component analysis is directed at the question, but confidence limits of the sampling error associated with these estimates are either dependent upon questionable multivariate theory or else unattainable by theory. The bootstrap method is therefore employed to estimate these aspects of dispersion. The size and shape of the hyper-robust australopithecine zygomatic foramen cannot be explained by simple extrapolation of primate interspecific allometries; although the positively and negatively allometric aspects of scaling are similar, the actual coefficients of scaling are significantly larger in the crucial variable in the australopithecines.     

Crown allometries are less responsive than stem allometry to tree size and habitat variations in an Indian monsoon forest

While theoretical allometric models postulate universal scaling exponents, empirical relationships between tree dimensions show marked variability that reflects changes in the biomass allocation pattern. As growth of the various tree compartments may be controlled by different functions, it is hypothesized that they may respond differently to factors of variation, resulting in variable tree morphologies and potentially in trade-offs between allometric relationships. We explore the variability of tree stem and crown allometries using a dataset of 1,729 trees located in an undisturbed wet evergreen forest of the Western Ghats, India. We specifically test whether species adult stature, terrain slope, tree size and crown light exposure affect the relationships between stem diameter and stem height (stem allometry), and between stem diameter and crown width, crown area and crown volume (crown allometries). Results show that both stem and crown allometries are subject to variations in relation to both endogenous (tree size, species adult stature) and exogenous (terrain slope, crown light exposure) factors. Stem allometry appears to be more affected by these factors than are crown allometries, including the stem diameter–crown volume relationship, which proved to be particularly stable. Our results support the idea that height is a prevailing adjustment factor for a tree facing variable growth (notably light) conditions, while stem diameter–crown volume allometry responds more to internal metabolic constraints. We ultimately discuss the various sources of variability in the stem and crown allometries of tropical trees that likely play an important role in forest community dynamics.     

Sexual selection and allometry: a critical reappraisal of the evidence and ideas

One of the most pervasive ideas in the sexual selection literature is the belief that sexually selected traits almost universally exhibit positive static allometries (i.e., within a sample of conspecific adults, larger individuals have disproportionally larger traits). In this review, I show that this idea is contradicted by empirical evidence and theory. Although positive allometry is a typical attribute of some sexual traits in certain groups, the preponderance of positively allometric sexual traits in the empirical literature apparently results from a sampling bias reflecting a fascination with unusually exaggerated (bizarre) traits. I review empirical examples from a broad range of taxa illustrating the diversity of allometric patterns exhibited by signal, weapon, clasping and genital traits, as well as nonsexual traits. This evidence suggests that positive allometry may be the exception rather than the rule in sexual traits, that directional sexual selection does not necessarily lead to the evolution of positive allometry and, conversely, that positive allometry is not necessarily a consequence of sexual selection, and that many sexual traits exhibit sex differences in allometric intercept rather than slope. Such diversity in the allometries of secondary sexual traits is to be expected, given that optimal allometry should reflect resource allocation trade-offs, and patterns of sexual and viability selection on both trait size and body size. An unbiased empirical assessment of the relation between sexual selection and allometry is an essential step towards an understanding of this diversity.     

Ontogenetic scaling of fish metabolism in the mouse-to-elephant mass magnitude range

Intraspecific or ontogenetic analyses of mass-metabolism relationships do not often conform to the same allometric correlations as those seen in interspecific analyses. A commonly cited reason for this discrepancy is that ontogenetic studies examine smaller mass ranges than interspecific studies, and are therefore not statistically comparable. In this study the metabolic rate of yellowtail kingfish was measured from 0.6 mg-2.2 kg, a mass range comparable to that between a mouse and an elephant. Linear regression of the log transformed data resulted in a scaling exponent of 0.90 and high correlation coefficient. Statistical and information theory comparisons of three other models showed that a segmented linear regression and curvilinear quadratic function were an improvement over a simple linear regression. This confirmed previous observations that the metabolic scaling exponent of fish changes during ontogeny. Ammonia excretion rates were also measured and scaled linearly with an exponent of 0.87. The data showed that the metabolism of yellowtail kingfish during ontogeny did not scale with the commonly cited 2/3 or 3/4 mass exponent. This demonstrates that differences between interspecific and ontogenetic allometries are not necessarily statistical artefacts.     

Variation in the ontogenetic allometry of horn length in bovids along a body mass continuum

Allometric relationships describe the proportional covariation between morphological, physiological, or life‐history traits and the size of the organisms. Evolutionary allometries estimated among species are expected to result from species differences in ontogenetic allometry, but it remains uncertain whether ontogenetic allometric parameters and particularly the ontogenetic slope can evolve. In bovids, the nonlinear evolutionary allometry between horn length and body mass in males suggests systematic changes in ontogenetic allometry with increasing species body mass. To test this hypothesis, we estimated ontogenetic allometry between horn length and body mass in males and females of 19 bovid species ranging from ca. 5 to 700 kg. Ontogenetic allometry changed systematically with species body mass from steep ontogenetic allometries over a short period of horn growth in small species to shallow allometry with the growth period of horns matching the period of body mass increase in the largest species. Intermediate species displayed steep allometry over long period of horn growth. Females tended to display shallower ontogenetic allometry with longer horn growth compared to males, but these differences were weak and highly variable. These findings show that ontogenetic allometric slope evolved across species possibly as a response to size‐related changes in the selection pressures acting on horn length and body mass.     

Relationships among ontogenetic,static, and evolutionary allometry

The relationship between ontogenetic, static, and evolutionary levels of allometry is investigated. Extrapolation from relative size relationships in adults to relative growth in ontogeny depends on the variability of slopes and intercepts of ontogenetic vectors relative to variability in length of the vector. If variability in slopes and intercepts is low relative to variability in length, ontogenetic and static allometries will be similar. The similarity of ontogenetic and static allometries was tested by comparing the first principal component, or size vector, for correlations among 48 cranial traits in a cross-sectional ontogenetic sample of rhesus macaques from Cayo Santiago with a static sample from which all age- and sex-related variation had been removed. The vector correlation between the components is high but significantly less than one while two of three allometric patterns apparent in the ontogenetic component are not discernable in the static component. This indicates that there are important differences in size and shape relationships among adults and within ontogenies. Extrapolation from intra- or interspecific phenotypic allometry to evolutionary allometry is shown to depend on the similarity of genetic and phenotypic allometry patterns. Similarity of patterns was tested by comparing the first principal components of the phenotypic, genetic, and environmental correlation matrices calculated using standard quantitative genetic methods. The patterns of phenotypic, genetic, and environmental allometry are dissimilar; only the environmental allometries show ontogenetic allometric patterns. This indicates that phenotypic allometry may not be an accurate guide to patterns of evolutionary change in size and shape.     

Across ecosystem comparisons of size structure: methods,approaches and prospects

Understanding how ecological communities are structured and how this may vary between different types of ecosystems is a fundamental question in ecology. We develop a general framework for quantifying size‐structure within and among different ecosystem types (e.g. terrestrial, freshwater or marine), via the use of a suite of bivariate relationships between organismal size and properties of individuals, populations, assemblages, pair‐wise interactions, and network topology. Each of these relationships can be considered a dimension of size‐structure, along which real communities lie on a continuous scale. For example, the strength, slope, or elevation of the body mass‐versus‐abundance or predator size‐versus‐prey size relationships may vary systematically among ecosystem types. We draw on examples from the literature and suggest new ways to use allometries for comparing among ecosystem types, which we illustrate by applying them to published data. Finally, we discuss how dimensions of size‐structure are interconnected and how we could approach this complex hierarchy systematically. We conclude: (1) there are multiple dimensions of size‐structure; (2) communities may be size‐structured in some of these dimensions, but not necessarily in others; (3) across‐system comparisons via rigorous quantitative statistical methods are possible, and (4) insufficient data are currently available to illuminate thoroughly the full extent and nature of differences in size‐structure among ecosystem types.     

Artificial selection on allometry: change in elevation but not slope

To what extent within-species (static) allometries constitute a constraint on evolution is the subject of a long-standing debate in evolutionary biology. A prerequisite for the constraint hypothesis is that static allometries are hard to change. Several studies have attempted to test this hypothesis with artificial-selection experiments, but their results remain inconclusive due to various methodological issues. Here, we present results from an experiment in which we selected independently on the slope and the elevation of the allometric relationship between caudal-fin size and body size in male guppies (Poecilia reticulata). After three episodes of selection, the allometric elevation (i.e. intercept at constant slope) had diverged markedly between the lines selected to increase or decrease it, and showed a realized heritability of 50%. In contrast, the allometric slope remained unaffected by selection. These results suggest that the allometric elevation is more evolvable than the allometric slope, this latter representing a potential constraint on adaptive trait evolution. To our knowledge, this study is the first artificial-selection experiment that directly tests the evolvability of static allometric slopes.     

EVOLUTION OF STATIC ALLOMETRIES: ADAPTIVE CHANGE IN ALLOMETRIC SLOPES OF EYE SPAN IN STALK‐EYED FLIES

Julian Huxley showed that within‐species (static) allometric (power‐law) relations can arise from proportional growth regulation with the exponent in the power law equaling the factor of proportionality. Allometric exponents may therefore be hard to change and act as constraints on the independent evolution of traits. In apparent contradiction to this, many empirical studies have concluded that static allometries are evolvable. Many of these studies have been based, however, on a broad definition of allometry that includes any monotonic shape change with size, and do not falsify the hypothesis of constrained narrow‐sense allometry. Here, we present the first phylogenetic comparative study of narrow‐sense allometric exponents based on a reanalysis of data on eye span and body size in stalk‐eyed flies (Diopsidae). Consistent with a role in sexual selection, we found strong evidence that male slopes were tracking “optima” based on sexual dimorphism and relative male trait size. This tracking was slow, however, with estimated times of 2–3 million years for adaptation to exceed ancestral influence on the trait. Our results are therefore consistent with adaptive evolution on million‐year time scales, but cannot rule out that static allometry may act as a constraint on eye‐span adaptation at shorter time scales.     

Nutrient‐dependent allometric plasticity in a male‐diphenic mite

Male secondary sexual traits often scale allometrically with body size. These allometries can be variable within species and may shift depending on environmental conditions, such as food quality. Such allometric plasticity has been hypothesized to initiate local adaptation and evolutionary diversification of scaling relationships, but is under‐recorded, and its eco‐evolutionary effects are not well understood. Here, we tested for allometric plasticity in the bulb mite (Rhizoglyphus robini), in which large males tend to develop as armed adult fighters with thickened third legs, while small males become adult scramblers without thickened legs. We first examined the ontogenetic timing for size‐ and growth‐dependent male morph determination, using experimentally amplified fluctuations in growth rate throughout juvenile development. Having established that somatic growth and body size determine male morph expression immediately before metamorphosis, we examined whether the relationship between adult male morph and size at metamorphosis shifts with food quality. We found that the threshold body size for male morph expression shifts toward lower values with deteriorating food quality, confirming food‐dependent allometric plasticity. Such allometric plasticity may allow populations to track prevailing nutritional conditions, potentially facilitating rapid evolution of allometric scaling relationships.     

Developmental causes of allometry: new models and implications for phenotypic plasticity and evolution

Shapes change during development because tissues, organs, and various anatomical features differ in onset, rate, and duration of growth. Allometry is the study of the consequences of differences in the growth of body parts on morphology, although the field of allometry has been surprisingly little concerned with understanding the causes of differential growth. The power-law equation y?=?ax(b), commonly used to describe allometries, is fundamentally an empirical equation whose biological foundation has been little studied. Huxley showed that the power-law equation can be derived if one assumes that body parts grow with exponential kinetics, for exactly the same amount of time. In life, however, the growth of body parts is almost always sigmoidal, and few, if any, grow for exactly the same amount of time during ontogeny. Here, we explore the shapes of allometries that result from real growth patterns and analyze them with new allometric equations derived from sigmoidal growth kinetics. We use an extensive ontogenetic dataset of the growth of internal organs in the rat from birth to adulthood, and show that they grow with Gompertz sigmoid kinetics. Gompertz growth parameters of body and internal organs accurately predict the shapes of their allometries, and that nonlinear regression on allometric data can accurately estimate the underlying kinetics of growth. We also use these data to discuss the developmental relationship between static and ontogenetic allometries. We show that small changes in growth kinetics can produce large and apparently qualitatively different allometries. Large evolutionary changes in allometry can be produced by small and simple changes in growth kinetics, and we show how understanding the development of traits can greatly simplify the interpretation of how they evolved.     

Grass allometry and estimation of above‐ground biomass in tropical alpine tussock grasslands

The puna/páramo grasslands span across the highest altitudes of the tropical Andes, and their ecosystem dynamics are still poorly understood. In this study we examined the above‐ground biomass and developed species specific and multispecies power‐law allometric equations for four tussock grass species in Peruvian high altitude grasslands, considering maximum height (h_max), elliptical crown area and elliptical basal area. Although these predictors are commonly used among allometric literature, they have not previously been used for estimating puna grassland biomass. Total above‐ground biomass was estimated to be of 6.7 ± 0.2 Mg ha^?1 (3.35 ± 0.1 Mg C ha^?1). All allometric relationships fitted to similar power‐law models, with basal area and crown area as the most influential predictors, although the fit improved when tussock maximum height was included in the model. Multispecies allometries gave better fits than the other species‐specific equations, but the best equation should be used depending on the species composition of the target grassland. These allometric equations provide an useful approach for measuring above‐ground biomass and productivity in high‐altitude Andean grasslands, where destructive sampling can be challenging and difficult because of the remoteness of the area. These equations can be also applicable for establishing above‐ground reference levels before the adoption of carbon compensation mechanisms or grassland management policies, as well as for measuring the impact of land use changes in Andean ecosystems.     

Allocation tradeoffs and life histories: a conceptual and graphical framework

Tradeoffs – negative reciprocal causal relationships in net benefits between trait magnitudes – have not always been studied in depth appropriate to their central role in life‐history analysis. Here we focus on allocation tradeoffs, in which acquisition of a limiting resource requires allocation of resource to alternative traits. We identify the components of this allocation process and emphasize the importance of quantifying them. We then propose categorizing allocation tradeoffs into linear, concave and convex relationships based on the way that resource allocation yields trait magnitudes under the tradeoff. Linear relationships are over‐represented in the literature because of typically small data sets over restricted ranges of trait magnitudes, an emphasis on simple correlation analysis, and a failure to remove variation associated with acquisition of the limiting resource in characterizing the tradeoff. (We provide methods for controlling these acquisition effects.) Non‐linear relationships have been documented and are expected under plausible conditions that we summarize. We note ways that shifting environments and biological features yield plasticity of tradeoff graphs. Finally, we illustrate these points using case studies and close with priorities for future work.     

A matter of measurements: challenges and approaches in the comparative analysis of static allometries

Comparisons of static allometries are frequently used to gain insights into patterns and processes underlying morphological and developmental evolution. A study by J. L. Tomkins and coworkers, recently published in the American Naturalist, examined complex nonlinear allometries in three insect species in which males are dimorphic in the expression of secondary sexual traits. Employing a novel approach to analyzing male allometries in these organisms, the authors were able to show that developmental reprogramming of trait primordia is not necessary to explain allometric scaling in two of the species examined, contrary to several previous studies on the same species. Instead, male dimorphisms could be explained by simple exponential growth, an important result that carries with it major evolutionary and developmental implications. Using this study as an example, I highlight some of the methodological challenges involved in analyzing and comparing static allometries and in inferring the developmental processes that underlie them. I end by discussing how correct application of hypothesis testing, on one side, and basic anatomy and developmental biology, on the other, should guide how morphology is measured.