Estimating the biological half-life for radionuclides in homoeothermic vertebrates: a simplified allometric approach

The application of allometric, or mass-dependent, relationships within radioecology has increased with the evolution of models to predict the exposure of organisms other than man. Allometry presents a method of addressing the lack of empirical data on radionuclide transfer and metabolism for the many radionuclide–species combinations which may need to be considered. However, sufficient data across a range of species with different masses are required to establish allometric relationships and this is not always available. Here, an alternative allometric approach to predict the biological half-life of radionuclides in homoeothermic vertebrates which does not require such data is derived. Biological half-life values are predicted for four radionuclides and compared to available data for a range of species. All predictions were within a factor of five of the observed values when the model was parameterised appropriate to the feeding strategy of each species. This is an encouraging level of agreement given that the allometric models are intended to provide broad approximations rather than exact values. However, reasons why some radionuclides deviate from what would be anticipated from Kleiber’s law need to be determined to allow a more complete exploitation of the potential of allometric extrapolation within radioecological models.     

Abstracts of papers to be presented at the Fiftieth Annual Meeting of the American Association of physical anthropologists Detroit,Michigan April 22–25, 1981

The critical problem confronting all allometric studies is the choice of an appropriate size variable, especially when body mass or some other measure of total size is unavailable. A method proposed by Jolicoeur (1963a,b) claims to generate an internal size variable by a principal components analysis of the covariance matrix of logarithmically transformed data, from which allometric coefficients can be computed. Despite the current popularity of this method, the precise relationship and degree of compatibility between such multivariate coefficients and the exponent of the bivariate power function (Y = βX^α) is unknown. This study evaluates the comparability and interpretability of allometric values computed by Jolicoeur's procedure and by standard bivariate regressions (least squares and major axis). Two primate data sets with known measures of size were utilized for these purposes: (1) longitudinal growth data from radiographs of two species of capuchin monkeys, Cebus apella and Cebus albifrons; and (2) interspecific osteometric data from a series of adult lemurs, species of prosimians from Madagascar. Consistent differences exist between multivariate and bivariate allometric coefficients for both ontogenetic and static data sets. Multivariate analysis underestimated the coefficients in the former and overestimated them in the latter. The internal size variable generated by principal components analysis is clearly not equivalent to, and hence not a suitable substitute for, known measures of size. Moreover, multivariate coefficients are very sensitive to the composition of variables in a given data set; α values of a variable changed appreciably depending on the other variables included in the analysis. The multivariate coefficients are also sample-specific, and provide misleading biological information when compared between samples (e.g., between species of capuchin monkey). For allometric investigations designed to evaluate scaling parameters relative to total size, alternative analytical solutions to the Jolicoeur method should be considered.     

Pitfalls and challenges of estimating population growth rate from empirical data: consequences for allometric scaling relations

The intrinsic rate of increase is a fundamental concept in population ecology, and a variety of problems require that estimates of population growth rate be obtained from empirical data. However, depending on the extent and type of data available (e.g. time series, life tables, life history traits), several alternative empirical estimators of population growth rate are possible. Because these estimators make different assumptions about the nature of age‐dependent mortality and density‐dependence of population dynamics, among other factors, these quantities capture fundamentally different aspects of population growth and are not interchangeable. Nevertheless, they have been routinely commingled in recent ecoinformatic analyses relating to allometry and conservation biology. Here we clarify some of the confusion regarding the empirical estimation of population growth rate and present separate analyses of the frequency distributions and allometric scaling of three alternative, non‐interchangeable measures of population growth. Studies of allometric scaling of population growth rate with body size are additionally sensitive to the statistical line fitting approach used, and we find that different approaches yield different allometric scaling slopes. Across the mix of population growth estimators and line fitting techniques, we find scattered and limited support for the key allometric prediction from the metabolic theory of ecology, namely that log₁₀(population growth rate) should scale as ?0.25 power of log₁₀(body mass). More importantly, we conclude that the question of allometric scaling of population growth rate with body size is highly sensitive to previously unexamined assumptions regarding both the appropriate population growth parameter to be compared and the line fitting approach used to examine the data. Finally, we suggest that the ultimate test of allometric scaling of maximum population growth rates with body size has not been done and, moreover, may require data that are not currently available.     

An allometric method for the projection of eelgrass leaf biomass production rates

We demonstrate that an allometric model for eelgrass leaf-growth rates can be derived from data on leaf architecture and growth form. Using this construct, we produced indirect assessments of growth rates of leaves that we call projections, which can be easily obtained in terms of allometric parameters and proxy values for leaf area, expressed as the product of leaf length and width. These projections of leaf-growth rates displayed a high level of correspondence with values observed in our data, as well as with other sets of reference data. A comparison with growth rates obtained by using the plastochrone index method showed that our model provides more accurate estimations while using a simpler methodology. Our results also show that whenever allometric parameters for the scaling of eelgrass leaf dry weight in terms of leaf area are available, the proposed model provides an accurate, cost-effective and non-destructive alternative to assessments based on traditional or plastochrone methods.     

Allometric growth in the Praying mantis Stagmatoptera biocellata

The growth of ten exoskeletal structures has been studied throughout the post-embryonic development in Stagmatoptera biocellata. The selection of an appropriate statistical procedure to analyse the data was discussed. In spite of theoretical prediction the least-squares regression technique turned out to be as efficient as the Bartlett (1949) procedure for studying growth. Longitudinal treatment of longitudinal data proved to be the best method for this kind of analysis. All the dimensions investigated showed some statistically significant degree of allometric growth and could be ascribed to certain class of allometry. The adaptative value of some allometric growth patterns was discussed in terms of physiological efficiency. A hypothesis put forth by Balderrama & Maldonado (1973) about the anatomic relationships between the parts involved in the mantis strike, received clear support from this analysis. A quantitative assessment of Dyar's (1890) and Przibram's (1912) rules and their modifications led us to conclude that they are untenable at least for the mantid Stagmatoptera biocellata. However, some periods of constant growth ratio could be identified during the post-embryonic development and the concept of maximum growth period was put forward as a complementary element for typifying structures.     

Aspects of line-fitting in bivariate allometric analyses

One of the fundamental problems involved in analyses of the scaling effects of body size (allometric analysis is the choice of an appropriate best-fit line in bivariate logarithmic plots. Following a discussion of some basic aspects of allometric analysis, the tow mai procedures for the determination of a best-fit line - the least-squares regression and the major axis - are examined with respect to their different properties and underlying models. It is important to distinguish intraspecific from interspecific scaling and to recognize the distinction between use of a best-fit line to define a relationship and use of the line for prediction. An alternative model to the bivariate normal distribution, referred to as the 'extruded normal distribution', is presented and its implications are examined with respect to two test cases (scaling of basal metabolic rate in human males; scaling of population density in mammals).     

On the use of log‐transformation versus nonlinear regression for analyzing biological power laws

Xiao and colleagues re‐examined 471 datasets from the literature in a major study comparing two common procedures for fitting the allometric equation y = ax^b to bivariate data (Xiao et al., 2011). One of the procedures was the traditional allometric method, whereby the model for a straight line fitted to logarithmic transformations of the original data is back‐transformed to form a two‐parameter power function with multiplicative, lognormal, heteroscedastic error on the arithmetic scale. The other procedure was standard nonlinear regression, whereby a two‐parameter power function with additive, normal, homoscedastic error is fitted directly to untransformed data by nonlinear least squares. Xiao and colleagues articulated a simple (but explicit) protocol for fitting and comparing the alternative models, and then used the protocol to examine each of the datasets in their compilation. The traditional method was said to provide a better fit in 69% of the cases and an equivalent fit in another 15%, so the investigation appeared to validate findings from a large majority of prior studies on allometric variation. However, focus for the investigation by Xiao and colleagues was overly narrow, and statistical models apparently were not validated graphically in the scale of measurement. The present study re‐examined a subset of the cases using a larger pool of candidate models and graphical validation, and discovered complexities that were overlooked in their investigation. Some datasets that appeared to be described better by the traditional method actually were unsuited for use in an allometric analysis, whereas other datasets were not described adequately by a two‐parameter power function, regardless of how the model was fitted. Thus, conclusions reached by Xiao and colleagues are not well supported and their paradigm for fitting allometric equations is unreliable. Future investigations of allometric variation should adopt a more holistic approach and incorporate graphical validation on the original arithmetic scale. © 2014 The Linnean Society of London, Biological Journal of the Linnean Society, 2014, 113 , 1167–1178.     

Rejection of non-adaptive hypotheses for intraspecific variation in trophic morphology in gape-limited predators

Recent studies have interpreted intraspecific divergence in relative head sizes in snakes as evidence for adaptation of the trophic apparatus in gape-limited predators to local prey size. However, such variation might also arise from non-adaptive processes (such as allometry, correlated response, genetic drift, or non-adaptive phenotypic plasticity). We test predictions from these alternative hypotheses using data on the allometric relationship between head size and body size in two wide-ranging snake species: eight populations of adders ( Vipera berus ) and 30 populations of common gartersnakes ( Thamnophis sirtalis ). Our data enable strong rejection of the alternative (non-adaptive) hypotheses, because the relationship between head and body size differed significantly among populations, the geographic distance separating pairs of populations explained less than 1.5% of their divergence in allometric coefficients, and the within-population allometric coefficients were higher than the among-population coefficients in each species. In addition, the geographical variability of allometric coefficients in females did not parallel that in males, suggesting that allometric coefficients have evolved independently in the two sexes. Phenotypic plasticity also cannot explain the data, because laboratory studies show that the allometric relationship between head size and body size is relatively insensitive to differing growth rates. We conclude that the intraspecific head size divergence in these snakes is better explained by spatially heterogeneous selection to optimize prey handling ability, than by non-adaptive processes.     

Allometry and adaptation of body proportions and stature in African pygmies

We have analyzed the growth allometry of external body proportions in Efe pygmies from Zaire and combined these data with values from the literature for comparable dimensions in adult pygmies and nonpygmies. We sequentially tested the hypotheses that adult proportion differences between 1) male vs. female Efe, and 2) pygmies vs. nonpygmies result from ontogenetic scaling, or the differential extension of common patterns of growth allometry. Results indicate an almost complete concordance of allometric trajectories for male and female Efe. These preliminary analyses also strongly suggest that adult nonpygmy Africans generally differ from pygmies in their terminal size and correlated allometric consequences, rather than in more fundamental alterations of underlying patterns of growth. Biacromial diameter emerges as the measurement most likely to depart from this general pattern. These results provide further evidence that shifts in systemic growth hormones yielding differences in terminal overall body size may be accompanied by global and coordinated allometric transformations. Certain proportion differences previously interpreted by some as specific evidence of primitive retention in pygmies in fact reflect simple growth allometric correlates of the derived rapid size decrease in these groups. Selected divergent body proportions characterizing adult pygmies, previously interpreted by some as independent evidence of climatic adaptation, also reflect such allometric correlates of ontogenetic scaling. We critically assess arguments that the small overall body size of pygmies was specifically selected for reasons of thermoregulatory efficiency, and consider an alternative or complementary scenario, based on selection for small size in order to reduce caloric requirements. © 1996 Wiley-Liss, Inc.     

Allometry, antilog transformations, and the perils of prediction on the original scale

Biologists often use allometric equations that take the form of power functions (e.g., Y = aM(b), where M stands for mass and a and b are empirically fitted constants). Typically, these allometric equations are fitted by taking the antilog of log-log regressions. Predictions from these allometric equations are biased, and the bias my be appreciable. Methods for making predictions that correct for the bias are available, but they have rarely, if ever, been used by ecological and evolutionary physiologists. Just as physiologists would not use an instrument that was not properly calibrated, they should not use allometric equations to make predictions unless they account for the bias of those predictions. We analyzed 20 interspecific and 10 intraspecific data sets. We compared predictions from standard allometric equations with those from several alternative methods. Our analyses suggest that the bias of predictions from interspecific data sets may be substantial. For the intraspecific data sets we analyzed, the bias was likely to be small. Biologists, including ecological and evolutionary physiologists, should exercise care when using allometric equations to make predictions, particularly given that methods to adjust for bias are easily implemented.     

Allometry and prediction in hominoids: a solution to the problem of intervening variables.

To avoid misinterpretation of allometric exponents determined from interspecific allometric comparisons, specific conditions must be met with respect to the common reference variable. Body weight is considered to be the best general indication of overall size and is hence widely acknowledged to be the most suitable reference variable. However, because of the paucity of recorded body weights for museum specimens, various comparative studies have used other size indicators as intervening variables, although the allometric relationships to body size/weight were often unknown and possibly differed between species. Because of differences in the scaling properties of alternative intervening variables across the species investigated, conflicting conclusions may be drawn if different variables are chosen as substitutes for overall size. This is illustrated with two examples. In this study, series of skeletons with associated body weights of Gorilla, Pan, Pongo, and Homo were investigated. Both ontogenetic and static adult allometric relationships between several widely used reference variables and body weight were determined. Neither these variables nor additional estimators investigated in this study displayed allometric exponents and coefficients similar enough across species to justify direct interspecific comparison. To generate an alternative size estimator for both ontogenetic and static interspecific investigations, equations for combined sexes were derived to predict body weight from various long bone dimensions for individual hominoid species. From a total of 25 predictors, 12 prediction equations per species (six for nonadults and six for adults) were selected according to their relative suitability for reliable prediction of body weight. It is shown that the derived reference variable "predicted body weight" avoids problems of intervening variables, is valid for any interspecific ontogenetic and static allometric comparison, and displays less fluctuation in comparison to actual body weight.     

On the analysis of non‐linear allometries

Abstract 1. Non‐linear allometries are those where a log–log scatterplot of trait size against body size deviates from simple linearity. These are found in many insects, including the horns of beetles, the forceps of earwigs, and the heads of certain castes of ant. 2. Non‐linear allometries are often associated with polyphenism that is itself related to behaviour: for example, the alternative mating tactics displayed by many species of beetle are widely associated with dimorphisms in horn size. 3. This paper critically reviews the current techniques used to analyse these datasets. 4. Recommendations include the use of scatterplots and assessment of the goodness of fit of simple linear models as an initial screen for non‐linear allometry. The use of recently developed algorithms for ‘segmented' regression to analyse continuous allometric relationships, and a pragmatic approach to the analysis of discontinuous relationships that recognises that there is no simple way to distinguish between morphs in some cases, and that all of the proposed methods for doing so have some drawbacks. 5. Worked examples of the analysis of two sets of data from animals that have been the subject of controversy regarding the nature of their allometric relationships are given: further worked examples are provided as online Supporting Information.     

Evaluating scaling models in biology using hierarchical Bayesian approaches

Theoretical models for allometric relationships between organismal form and function are typically tested by comparing a single predicted relationship with empirical data. Several prominent models, however, predict more than one allometric relationship, and comparisons among alternative models have not taken this into account. Here we evaluate several different scaling models of plant morphology within a hierarchical Bayesian framework that simultaneously fits multiple scaling relationships to three large allometric datasets. The scaling models include: inflexible universal models derived from biophysical assumptions (e.g. elastic similarity or fractal networks), a flexible variation of a fractal network model, and a highly flexible model constrained only by basic algebraic relationships. We demonstrate that variation in intraspecific allometric scaling exponents is inconsistent with the universal models, and that more flexible approaches that allow for biological variability at the species level outperform universal models, even when accounting for relative increases in model complexity.     

Allometric disparity in rodent evolution

In this study, allometric trajectories for 51 rodent species, comprising equal representatives from each of the major clades (Ctenohystrica, Muroidea, Sciuridae), are compared in a multivariate morphospace (=allometric space) to quantify magnitudes of disparity in cranial growth. Variability in allometric trajectory patterns was compared to measures of adult disparity in each clade, and dietary habit among the examined species, which together encapsulated an ecomorphological breadth. Results indicate that the evolution of allometric trajectories in rodents is characterized by different features in sciurids compared with muroids and Ctenohystrica. Sciuridae was found to have a reduced magnitude of inter‐trajectory change and growth patterns with less variation in allometric coefficient values among members. In contrast, a greater magnitude of difference between trajectories and an increased variation in allometric coefficient values was evident for both Ctenohystrica and muroids. Ctenohystrica and muroids achieved considerably higher adult disparities than sciurids, suggesting that conservatism in allometric trajectory modification may constrain morphological diversity in rodents. The results provide support for a role of ecology (dietary habit) in the evolution of allometric trajectories in rodents.     

Evolution of extreme ontogenetic allometric diversity and heterochrony in pythons,a clade of giant and dwarf snakes

Ontogenetic allometry, how species change with size through their lives, and heterochony, a decoupling between shape, size, and age, are major contributors to biological diversity. However, macroevolutionary allometric and heterochronic trends remain poorly understood because previous studies have focused on small groups of closely related species. Here, we focus on testing hypotheses about the evolution of allometry and how allometry and heterochrony drive morphological diversification at the level of an entire species‐rich and diverse clade. Pythons are a useful system due to their remarkably diverse and well‐adapted phenotypes and extreme size disparity. We collected detailed phenotype data on 40 of the 44 species of python from 1191 specimens. We used a suite of analyses to test for shifts in allometric trajectories that modify morphological diversity. Heterochrony is the main driver of initial divergence within python clades, and shifts in the slopes of allometric trajectories make exploration of novel phenotypes possible later in divergence history. We found that allometric coefficients are highly evolvable and there is an association between ontogenetic allometry and ecology, suggesting that allometry is both labile and adaptive rather than a constraint on possible phenotypes.     

An analysis of structure: biomass structure relationships for characteristic species of the western Kalahari,Botswana

Savannah ecosystems are important carbon stocks on the Earth, and their quantification is crucial for understanding the global impact of climate and land‐use changes in savannahs. The estimation of aboveground/belowground plant biomass requires tested allometric relationships that can be used to determine total plant biomass as a function of easy‐to‐measure morphological indicators. Despite recent advances in savannah ecology, research on allometric relations in savannahs remains confined to a few site‐specific studies where basal area is typically used as the main morphometric parameter with plant biomass. We investigate allometric relations at four sites along a 950‐km transect in the Kalahari across mean rainfall gradient 170 mm yr^?1–550 mm yr^?1. Using data from 342 harvested trees/shrubs, we relate basal area, height and crown diameter to aboveground biomass. These relationships are strongest in trees and weakest in small shrubs. Strong allometric relationships are also determined for morphologically similar groups of woody vegetation. We show that crown diameter can be used as an alternative to basal area in allometric relationships with plant biomass. This finding may enhance the ability to determine aboveground biomass over large areas using high‐resolution aerial or satellite imagery without requiring ground‐based measurements of basal area.     

Interpreting the evolutionary regression: the interplay between observational and biological errors in phylogenetic comparative studies

Regressions of biological variables across species are rarely perfect. Usually, there are residual deviations from the estimated model relationship, and such deviations commonly show a pattern of phylogenetic correlations indicating that they have biological causes. We discuss the origins and effects of phylogenetically correlated biological variation in regression studies. In particular, we discuss the interplay of biological deviations with deviations due to observational or measurement errors, which are also important in comparative studies based on estimated species means. We show how bias in estimated evolutionary regressions can arise from several sources, including phylogenetic inertia and either observational or biological error in the predictor variables. We show how all these biases can be estimated and corrected for in the presence of phylogenetic correlations. We present general formulas for incorporating measurement error in linear models with correlated data. We also show how alternative regression models, such as major axis and reduced major axis regression, which are often recommended when there is error in predictor variables, are strongly biased when there is biological variation in any part of the model. We argue that such methods should never be used to estimate evolutionary or allometric regression slopes.     

A theoretical interpretation of length-biomass allometry of predominantly bidimensional seaweeds

Experimental studies on bidimensional seaweeds revealed a scaling exponent of 0.472 for their length-biomass allometry. This was significantly higher than the value 0.25, which was proposed earlier as universal for all primary producers, based on the data for unicellular microalgae and vascular plants. Later, an exponent of 0.5 was theoretically derived, which agreed, to some extent, with experimental findings. Here, it is shown that there exists a power-law relation between the two perpendicular length parameters along the directions of growth of a bidimensional organism. The length-biomass allometric parameters can be expressed in terms of this power index. A relation between the allometric scaling exponent and allometric constant, involving the mass per unit area, has been obtained analytically. A method is proposed to determine the power index experimentally. Some mathematical expressions, relating mass, length and other parameters, have been formulated and these would be useful for experimental purposes in allometric studies. Analyzing images from an experimental study, a lot of parameters, regarding flat seaweeds, have been determined by analytical and numerical techniques.     

A standardized reanalysis of molecular phylogenetic hypotheses of Gobioidei

Gobioidei is a suborder of perciform fishes with about 2000 species distributed worldwide. Despite the evolutionary and ecological importance of gobioids, their phylogenetic inter- and intrarelationships are still poorly understood. Only a few studies (either morphological or molecular) have tackled the phylogeny of Gobioidei as a whole. Of these, only six studies thus far have addressed gobioid intrarelationships based on molecular data (each using different taxon sampling, genes, outgroups and method of phylogenetic inference), yielding contrasting results regarding the phylogenetic relationships among major lineages. In this study, we have reanalysed data from four of these molecular phylogenetic studies of Gobioidei under standardization criteria (same outgroup and methods of phylogenetic inference) in order to assess the robustness of their results, as well as to identify which parts of the gobioid tree are least resolved. Results from all datasets reanalysed in this study are generally similar to those of the respective original studies, and suggest broad patterns of phylogenetic relationships among gobioid lineages. However, there are numerous topological discrepancies among the four studies, support is low for many phylogenetic relationships and topology tests are unable to reject the vast majority of alternative topologies tested. The concatenation of datasets yields a relatively robust phylogeny of major lineages of Gobioidei, but there are some issues of overlap and missing data, which are ameliorated with the inclusion of additional homologous sequences from GenBank that increase dataset completeness. Because both monophyly of major gobioid groups and phylogenetic relationships among them cannot be fully resolved, it is clear that further phylogenetic research is needed, and this should be accompanied by a major taxonomic revision of the Gobioidei. Nevertheless, even with the relatively unstable nature of the available molecular phylogenies, there are some monophyletic units that can be identified, and a basic structure of the gobioid tree appears evident.