Why are metabolic scaling exponents so controversial? Quantifying variance and testing hypotheses

A metabolic perspective on polyploid invasion and the emergence of life histories: Insights from a mechanistic model

PREMISE

Whole-genome duplication (WGD, polyploidization) has been identified as a driver of genetic and phenotypic novelty, having pervasive consequences for the evolution of lineages. While polyploids are widespread, especially among plants, the long-term establishment of polyploids is exceedingly rare. Genome doubling commonly results in increased cell sizes and metabolic expenses, which may be sufficient to modulate polyploid establishment in environments where their diploid ancestors thrive.

METHODS

We developed a mechanistic simulation model of photosynthetic individuals to test whether changes in size and metabolic efficiency allow autopolyploids to coexist with, or even invade, ancestral diploid populations. Central to the model is metabolic efficiency, which determines how energy obtained from size-dependent photosynthetic production is allocated to basal metabolism as opposed to somatic and reproductive growth. We expected neopolyploids to establish successfully if they have equal or higher metabolic efficiency as diploids or to adapt their life history to offset metabolic inefficiency.

RESULTS

Polyploid invasion was observed across a wide range of metabolic efficiency differences between polyploids and diploids. Polyploids became established in diploid populations even when they had a lower metabolic efficiency, which was facilitated by recurrent formation. Competition for nutrients is a major driver of population dynamics in this model. Perenniality did not qualitatively affect the relative metabolic efficiency from which tetraploids tended to establish.

CONCLUSIONS

Feedback between size-dependent metabolism and energy allocation generated size and age differences between plants with different ploidies. We demonstrated that even small changes in metabolic efficiency are sufficient for the establishment of polyploids.

Unraveling power-law scaling through exponential cell division dynamics

A primary objective of biology is the development of universal laws that define how organic form develops and how it evolves as a function of size, both ontogenetically and across evolutionary time. Scaling theory has been essential in reaching this goal by giving a complete perspective point, particularly in illuminating the fundamental biological features produced within scaling exponents defining families of equations. Nonetheless, the theoretical basis of the allometric equation within scaling theory are inadequately explained, particularly when it comes to establishing links between micro-level processes at the cellular level and macro-level phenomena. We proposed an unlimited cell bipartition, resulting in an exponential growth in cell numbers during an individual's lifespan, to bridge this conceptual gap between cellular processes and allometric scaling. The power-law scaling between body mass and organ weight was produced by the synchronous exponential increments and the allometric exponent is rate of logarithmic cell proliferation rate. Substituting organ weight for erythrocyte weight aided in the development of a power-law scaling relationship between body mass and metabolic rate. Furthermore, it is critical to understand how cell size affects the exponent in power-law scaling. We find that a bigger exponent will result from an increase in the average weight of organ cells or a decrease in the average weight of all cells. Furthermore, cell proliferation dynamics showed a complex exponential scaling between body mass and longevity, defying the previously reported power-law scaling. We discovered a quadratic link between longevity and logarithmic body mass. Notably, all of the parameters included in these relationships are explained by indices linked to cell division and embryonic development. This research adds to our understanding of the complex interaction between cellular processes and overarching scaling phenomena in biology.

Metabolic scaling as an emergent outcome of variation in metabolic rate

The allometric scaling of metabolic rate and what drives it are major questions in biology with a long history. Since the metabolic rate at any level of biological organization is an emergent property of its lower-level constituents, it is an outcome of the intrinsic heterogeneity among these units and the interactions among them. However, the influence of lower-level heterogeneity on system-level metabolic rate is difficult to investigate, given the tightly integrated body plan of unitary organisms. In this context, social insects such as honeybees can serve as important model systems because unlike unitary organisms, these superorganisms can be taken apart and reassembled in different configurations to study metabolic rate and its various drivers at different levels of organization. This commentary discusses the background of such an approach and how combining it with artificial selection to generate heterogeneity in metabolic rate with an analytical framework to parse out the different mechanisms that contribute to the effects of heterogeneity can contribute to the various models of metabolic scaling. Finally, the absence of the typical allometric scaling relationship among different species of honeybees is discussed as an important prospect for deciphering the role of top-down ecological factors on metabolic scaling. This article is part of the theme issue 'The evolutionary significance of variation in metabolic rates'.

How and Why Does Metabolism Scale with Body Mass?

Most explanations for the relationship between body size and metabolism invoke physical constraints; such explanations are evolutionarily inert, limiting their predictive capacity. Contemporary approaches to metabolic rate and life history lack the pluralism of foundational work. Here, we call for reforging of the lost links between optimization approaches and physiology.

Embracing the complexity of leaf shape: a commentary on 'Anatomical determinants of gas exchange and hydraulics vary with leaf shape in soybean'

This article comments on:

Bishal G. Tamang, Yanqun Zhang, Michelle A. Zambrano and Elizabeth A. Ainsworth Anatomical determinants of gas exchange and hydraulics vary with leaf shape in soybean, Annals of Botany, Volume 131, Issue 6, 9 May 2023, Pages 909–920, https://doi.org/10.1093/aob/mcac118

Variable metabolic scaling breaks the law: from 'Newtonian' to 'Darwinian' approaches

Life's size and tempo are intimately linked. The rate of metabolism varies with body mass in remarkably regular ways that can often be described by a simple power function, where the scaling exponent (b, slope in a log-linear plot) is typically less than 1. Traditional theory based on physical constraints has assumed that b is 2/3 or 3/4, following natural law, but hundreds of studies have documented extensive, systematic variation in b. This overwhelming, law-breaking, empirical evidence is causing a paradigm shift in metabolic scaling theory and methodology from ‘Newtonian’ to ‘Darwinian’ approaches. A new wave of studies focuses on the adaptable regulation and evolution of metabolic scaling, as influenced by diverse intrinsic and extrinsic factors, according to multiple context-dependent mechanisms, and within boundary limits set by physical constraints.

Parameter estimation of the Solow-Swan fundamental differential equation

Background

The Solow-Swan model describes the long-term growth of the capital to labor ratio by the fundamental differential equation of Solow-Swan theory. In conventional approaches, this equation was fitted to data using additional information, such as the rates of population growth, capital depreciation, or saving. However, this was not the best possible fit.

Objectives

Using the method of least squares, what is the best possible fit of the fundamental equation to the time-series of the capital to labor ratios? Are the best-fit parameters economically sound?

Method

For the data, we used the Penn-World Table in its 2021 version and compared six countries and three definitions of the capital to labor ratio. For optimization, we used a custom-made variant of the method of simulated annealing. We also compared different optimization methods and calibrations.

Results

When comparing different methods of optimization, our custom-made tool provided reliable parameter estimates. In terms of R-squared they improved upon the parameter estimates of the conventional approach. Except for the USA, the best-fit values of the exponent were unplausible, as they suggested a too large elasticity of output. However, using a different calibration resulted in more plausible values of the best-fit exponent also for France and Pakistan, but not for Argentina and Japan.

Conclusion

Our results have shown a discrepancy between the best-fit parameters obtained from optimization and the parameter values that are deemed plausible in economy. We propose a research program to resolve this issue by investigating if suitable calibrations may generate economically plausible best-fit parameter values.

Metabolic Scaling in Birds and Mammals: How Taxon Divergence Time, Phylogeny, and Metabolic Rate Affect the Relationship between Scaling Exponents and Intercepts

Simple Summary

This study is based on a large dataset and re-evaluates data on the metabolic rate, providing new insights into the similarities and differences across different groups of birds and mammals. We compared six taxonomic groups of mammals and birds according to their energetic characteristics and the geological time of evolutionary origin. The overall metabolic rate of a taxonomic group increases with the geological time of evolutionary origin. The terrestrial mammals and flightless birds have almost equal metabolic levels. The higher the metabolic rate in a group, the less it increases within increasing body size in this group.

Abstract

Analysis of metabolic scaling in currently living endothermic animal species allowed us to show how the relationship between body mass and the basal metabolic rate (BMR) has evolved in the history of endothermic vertebrates. We compared six taxonomic groups according to their energetic characteristics and the time of evolutionary divergence. We transformed the slope of the regression lines to the common value and analyzed three criteria for comparing BMR of different taxa regardless of body size. Correlation between average field metabolic rate (FMR) of the group and its average BMR was shown. We evaluated the efficiency of self-maintenance in ordinary life (defined BMR/FMR) in six main groups of endotherms. Our study has shown that metabolic scaling in the main groups of endothermic animals correlates with their evolutionary age: the younger the group, the higher the metabolic rate, but the rate increases more slowly with increasing body weight. We found negative linear relationship for scaling exponents and the allometric coefficient in five groups of endotherms: in units of mL O₂/h per g, in relative units of allometric coefficients, and also in level or scaling elevation. Mammals that diverged from the main vertebrate stem earlier have a higher “b” exponent than later divergent birds. A new approach using three criteria for comparing BMR of different taxa regardless of body mass will be useful for many biological size-scaling relationships that follow the power function.

Exceptional parallelisms characterize the evolutionary transition to live birth in phrynosomatid lizards

Viviparity, an innovation enhancing maternal control over developing embryos, has evolved >150 times in vertebrates, and has been proposed as an adaptation to inhabit cold habitats. Yet, the behavioral, physiological, morphological, and life history features associated with live-bearing remain unclear. Here, we capitalize on repeated origins of viviparity in phrynosomatid lizards to tease apart the phenotypic patterns associated with this innovation. Using data from 125 species and phylogenetic approaches, we find that viviparous phrynosomatids repeatedly evolved a more cool-adjusted thermal physiology than their oviparous relatives. Through precise thermoregulatory behavior viviparous phrynosomatids are cool-adjusted even in warm environments, and oviparous phrynosomatids warm-adjusted even in cool environments. Convergent behavioral shifts in viviparous species reduce energetic demand during activity, which may help offset the costs of protracted gestation. Whereas dam and offspring body size are similar among both parity modes, annual fecundity repeatedly decreases in viviparous lineages. Thus, viviparity is associated with a lower energetic allocation into production. Together, our results indicate that oviparity and viviparity are on opposing ends of the fast-slow life history continuum in both warm and cool environments. In this sense, the ‘cold climate hypothesis’ fits into a broader range of energetic/life history trade-offs that influence transitions to viviparity.

There have been five independent transitions from egg laying to live birth in the phrynosomatid lizards. Here, Domínguez-Guerrero et al. identify parallel changes in physiology, life history and behaviour that characterize these transitions to live birth.

First successful hybridization experiment between native European weatherfish (Misgurnus fossilis) and non-native Oriental weatherfish (M. anguillicaudatus) reveals no evidence for postzygotic barriers

The European weatherfish Misgurnus fossilis (Linnaeus, 1758) is a threatened freshwater species in large parts of Europe and might come under pressure from currently establishing exotic weatherfish species. Additional threats might arise if those species hybridize which has been questioned in previous research. Regarding the hybridization of M. fossilis × M. anguillicaudatus (Cantor, 1842), we demonstrate that despite the considerable genetic distance between parental species, the estimated long divergence time and different ploidy levels do not represent a postzygotic barrier for hybridization of the European and Oriental weatherfish. The paternal species can be easily differentiated based on external pigment patterns with hybrids showing intermediate patterns. No difference in standard metabolic rate, indicating a lack of hybrid vigour, renders predictions of potential threats to the European weatherfish from hybridization with the Oriental weatherfish difficult. Therefore, the genetic and physiological basis of invasiveness via hybridization remains elusive in Misgurnus species and requires further research. The existence of prezygotic reproductive isolation mechanisms and the fertility of F1 hybrids remains to be tested to predict the potential threats of globally invasive Oriental weatherfish species.

Concerted evolution of body mass, cell size and metabolic rate among carabid beetles

Alterations in cell number and size are apparently associated with the body mass differences between species and sexes, but we rarely know which of the two mechanisms underlies the observed variance in body mass. We used phylogenetically informed comparisons of males and females of 19 Carabidae beetle species to compare body mass, resting metabolic rate, and cell size in the ommatidia and Malpighian tubules. We found that the larger species or larger sex (males or females, depending on the species) consistently possessed larger cells in the two tissues, indicating organism-wide coordination of cell size changes in different tissues and the contribution of these changes to the origin of evolutionary and sex differences in body mass. The species or sex with larger cells also exhibited lower mass-specific metabolic rates, and the interspecific mass scaling of metabolism was negatively allometric, indicating that large beetles with larger cells spent relatively less energy on maintenance than small beetles. These outcomes also support existing hypotheses about the fitness consequences of cell size changes, postulating that the low surface-to-volume ratio of large cells helps decrease the energetic demand of maintaining ionic gradients across cell membranes. Analyses with and without phylogenetic information yielded similar results, indicating that the observed patterns were not biased by shared ancestry. Overall, we suggest that natural selection does not operate on each trait independently and that the linkages between concerted cell size changes in different tissues, body mass and metabolic rate should thus be viewed as outcomes of correlational selection.

Universal rules of life: metabolic rates, biological times and the equal fitness paradigm

Here we review and extend the equal fitness paradigm (EFP) as an important step in developing and testing a synthetic theory of ecology and evolution based on energy and metabolism. The EFP states that all organisms are equally fit at steady state, because they allocate the same quantity of energy, ~ 22.4 kJ/g/generation to the production of offspring. On the one hand, the EFP may seem tautological, because equal fitness is necessary for the origin and persistence of biodiversity. On the other hand, the EFP reflects universal laws of life: how biological metabolism - the uptake, transformation and allocation of energy - links ecological and evolutionary patterns and processes across levels of organisation from: (1) structure and function of individual organisms, (2) life history and dynamics of populations, and (3) interactions and coevolution of species in ecosystems. The physics and biology of metabolism have facilitated the evolution of millions of species with idiosyncratic anatomy, physiology, behaviour and ecology but also with many shared traits and tradeoffs that reflect the single origin and universal rules of life.

Water content quantitatively affects metabolic rates over the course of plant ontogeny

Plant metabolism determines the structure and dynamics of ecological systems across many different scales. The metabolic theory of ecology quantitatively predicts the scaling of metabolic rate as a function of body size and temperature. However, the role of tissue water content has been neglected even though hydration significantly affects metabolism, and thus ecosystem structure and functioning. Here, we use a general model based on biochemical kinetics to quantify the combined effects of water content, body size and temperature on plant metabolic rates. The model was tested using a comprehensive dataset from 205 species across 10 orders of magnitude in body size from seeds to mature large trees. We show that water content significantly influences mass-specific metabolic rates as predicted by the model. The scaling exponents of whole-plant metabolic rate vs body size numerically converge onto 1.0 after water content is corrected regardless of body size or ontogenetic stage. The model provides novel insights into how water content together with body size and temperature quantitatively influence plant growth and metabolism, community dynamics and ecosystem energetics.

Coevolution of body size and metabolic rate in vertebrates: a life-history perspective

Despite many decades of research, the allometric scaling of metabolic rates (MRs) remains poorly understood. Here, we argue that scaling exponents of these allometries do not themselves mirror one universal law of nature but instead statistically approximate the non-linearity of the relationship between MR and body mass. This 'statistical' view must be replaced with the life-history perspective that 'allows' organisms to evolve myriad different life strategies with distinct physiological features. We posit that the hypoallometric allometry of MRs (mass scaling with an exponent smaller than 1) is an indirect outcome of the selective pressure of ecological mortality on allocation 'decisions' that divide resources among growth, reproduction, and the basic metabolic costs of repair and maintenance reflected in the standard or basal metabolic rate (SMR or BMR), which are customarily subjected to allometric analyses. Those 'decisions' form a wealth of life-history variation that can be defined based on the axis dictated by ecological mortality and the axis governed by the efficiency of energy use. We link this variation as well as hypoallometric scaling to the mechanistic determinants of MR, such as metabolically inert component proportions, internal organ relative size and activity, cell size and cell membrane composition, and muscle contributions to dramatic metabolic shifts between the resting and active states. The multitude of mechanisms determining MR leads us to conclude that the quest for a single-cause explanation of the mass scaling of MRs is futile. We argue that an explanation based on the theory of life-history evolution is the best way forward.

The relationship between eDNA particle concentration and organism abundance in nature is strengthened by allometric scaling

Organism abundance is a critical parameter in ecology, but its estimation is often challenging. Approaches utilizing eDNA to indirectly estimate abundance have recently generated substantial interest. However, preliminary correlations observed between eDNA concentration and abundance in nature are typically moderate in strength with significant unexplained variation. Here, we apply a novel approach to integrate allometric scaling coefficients into models of eDNA concentration and organism abundance. We hypothesize that eDNA particle production scales nonlinearly with mass, with scaling coefficients < 1. Wild populations often exhibit substantial variation in individual body size distributions; we therefore predict that the distribution of mass across individuals within a population will influence population-level eDNA production rates. To test our hypothesis, we collected standardized body size distribution and mark-recapture abundance data using whole-lake experiments involving nine populations of brook trout. We correlated eDNA concentration with three metrics of abundance: density (individuals/ha), biomass (kg/ha) and allometrically scaled mass (ASM) (∑(individual mass^0.73 )/ha). Density and biomass were both significantly positively correlated with eDNA concentration (adj. r²  = 0.59 and 0.63, respectively), but ASM exhibited improved model fit (adj. r²  = 0.78). We also demonstrate how estimates of ASM derived from eDNA samples in "unknown" systems can be converted to biomass or density estimates with additional size-structure data. Future experiments should empirically validate allometric scaling coefficients for eDNA production, particularly where substantial intraspecific size distribution variation exists. Incorporating allometric scaling may improve predictive models to the extent that eDNA concentration may become a reliable indicator of abundance in nature.

Elastohydrodynamic Scaling Law for Heart Rates

Animal hearts are soft shells that actively pump blood to oxygenate tissues. Here, we propose an allometric scaling law for the heart rate based on the idea of elastohydrodynamic resonance of a fluid-loaded soft active elastic shell that buckles and contracts axially when twisted periodically. We show that this picture is consistent with numerical simulations of soft cylindrical shells that twist-buckle while pumping a viscous fluid, yielding optimum ejection fractions of 35%-40% when driven resonantly. Our scaling law is consistent with experimental measurements of heart rates over 2 orders of magnitude, and provides a mechanistic basis for how metabolism scales with organism size. In addition to providing a physical rationale for the heart rate and metabolism of an organism, our results suggest a simple design principle for soft fluidic pumps.

Parameterized maximum entropy models predict variability of metabolic scaling across tree communities and populations

The maximum entropy theory of ecology (METE) applies the concept of "entropy" from information theory to predict macroecological patterns. The energetic predictions of the METE rely on predetermined metabolic scaling from external theories, and this reliance diminishes the testability of the theory. In this work, I build parameterized METE models by treating the metabolic scaling exponent as a free parameter, and I use the maximum-likelihood method to obtain empirically plausible estimates of the exponent. I test the models using the individual tree data from an oak-dominated deciduous forest in the northeastern United States and from a tropical forest in central Panama. My analysis shows that the metabolic scaling exponents predicted from the parameterized METE models deviate from that of the metabolic theory of ecology and exhibit large variation, at both community and population levels. Assemblage and population abundance may act as ecological constraints that regulate the individual-level metabolic scaling behavior. This study provides a novel example of the use of the parameterized METE models to reveal the biological processes of individual organisms. The implication and possible extensions of the parameterized METE models are discussed.

Scaling of movements with body mass in a small opossum: evidence for an optimal body size in mammals

Movement by mammals generally increases with body size, described by a positive exponent scaling with either home range area or day range distances. Below ca. 100 g, however, interspecific comparisons suggest a negative scaling, increasing movement with decreasing body size. Such a pattern is expected from the rising costs of thermoregulation below ca. 100 g, implying that it should also be observed in intraspecific comparisons. We tested this hypothesis by investigating the scaling exponent of daily home range with body mass for a small (&lt; 100 g) marsupial, the gray slender mouse opossum, Marmosops incanus. We tracked 85 opossums (56 M, 29 F) with a spool-and-line device between August 1998 and October 2005 in the Serra dos Órgãos National Park, a region of Atlantic Forest in the State of Rio de Janeiro, Brazil. Individual paths were mapped and daily home ranges quantified by the minimum convex polygon encompassing each path. We formulated linear models and compared them using Akaike information criteria. The best-supported model for females had only climatic season as a main determinant of daily home range, whereas the best model for males had body mass and reproductive season as the main effects. As predicted, the scaling exponent of daily home range with body mass of males was negative, in contrast with positive intraspecific exponents for opossums &gt; 100 g estimated in a previous study. The inversion in scaling relationships around 100 g in opossums supports the rising costs of thermoregulation as the main cause of this general pattern in mammals. Effects of body mass are generally weak in intraspecific comparisons, but might still be detected after standardizing other effects, opening new possibilities for testing macroecological models at smaller scales.

Espera-se que a quantidade de movimento de mamíferos aumente com o tamanho corporal, descrita por um expoente de escala positivo, tanto para área de vida quanto para distâncias diárias de deslocamento. Abaixo de ca. 100 g, comparações interespecíficas sugerem um expoente negativo, áreas de movimento aumentando com menores tamanhos de corpo. Este padrão é apoiado pelo custo crescente de termorregulação abaixo de ca. 100 g, que implica que também ocorreria em comparações intraespecíficas. Testamos esta hipótese investigando o expoente de escala da área de vida diária com a massa corporal em um pequeno (&lt; 100 g) marsupial, Marmosops incanus. Indivíduos foram rastreados com carretel-de-rastreamento entre agosto de 1998 e outubro de 2005, no Parque Nacional da Serra dos Órgãos, uma região de Mata Atlântica no Estado do Rio de Janeiro, Brasil. Trajetórias individuais foram mapeadas e sua área de vida diária medida pelo polígono convexo mínimo envolvendo cada trajetória. Oitenta e cinco indivíduos foram rastreados, sendo 29 fêmeas e 56 machos. Modelos lineares foram formulados e comparados com o Critério de Informação de Akaike. O modelo com maior suporte para fêmeas teve apenas estação climática como determinante principal da área de vida diária, enquanto o melhor modelo para machos teve massa corporal e estação reprodutiva como efeitos principais. O expoente de escala de área de vida diária com massa corporal de machos foi negativo, contrastando como os expoentes positivos para marsupiais &gt; 100 g estimados em um estudo anterior. A inversão de relações de escala em torno de 100 g nestes marsupiais apoia que custos crescentes de termorregulação sejam a causa principal deste padrão geral em mamíferos. Os efeitos da massa corporal são geralmente fracos em comparações intraespecíficas, mas podem ser detectados após a exclusão de outros efeitos, abrindo novas possibilidades para testar modelos macroecológicos em escalas menores.

Best-fitting growth curves of the von Bertalanffy-Pütter type

Introduction: A large body of literature aims at identifying growth models that fit best to given mass-at-age data. The von Bertalanffy-Pütter differential equation is a unifying framework for the study of growth models.

Problem: The most common growth models used in poultry science literature fit into this framework, as these models correspond to different exponent-pairs (e.g., Brody, Gompertz, logistic, Richards, and von Bertalanffy models). Here, we search for the optimal exponent-pairs (a and b) amongst all possible exponent-pairs and expect a significantly better fit of the growth curve to concrete mass-at-age data.

Method: Data fitting becomes more difficult, as there is a large region of nearly optimal exponent-pairs. We therefore develop a fully automated optimization method, with computation time of about 1 to 2 wk per data-set. For the proof of principle, we applied it to literature data about 217 male meat-type chickens, Athens Canadian Random Bred, that were reared under controlled conditions and weighed 28 times during a time span of 170 D.

Results: We compared 2 methods of data fitting, least squares using the sum of squared errors (SSE), which is common in literature, and a variant using the sum of squared log-errors SSElog. For these data, the optimal exponent-pairs were (0.43, 4.06) for SSE = 2,208.6 (31% improvement over literature values for the residual standard deviation) and (0.89, 0.93) for SSElog = 0.04599. Both optimal exponents were clearly distinct from the exponent-pairs of the common models in literature. This finding was reinforced by considering the region of nearly optimal exponents.

Discussion: We explain, why we recommend using SSElog for data fitting and we discuss prognosis, where data from the first 8 wk of growth would not be enough.

Strong Evidence for an Intraspecific Metabolic Scaling Coefficient Near 0.89 in Fish

As an example of applying the evidential approach to statistical inference, we address one of the longest standing controversies in ecology, the evidence for, or against, a universal metabolic scaling relationship between metabolic rate and body mass. Using fish as our study taxa, we curated 25 studies with measurements of standard metabolic rate, temperature, and mass, with 55 independent trials and across 16 fish species and confronted this data with flexible random effects models. To quantify the body mass - metabolic rate relationship, we perform model selection using the Schwarz Information Criteria (ΔSIC), an established evidence function. Further, we formulate and justify the use of ΔSIC intervals to delineate the values of the metabolic scaling relationship that should be retained for further consideration. We found strong evidence for a metabolic scaling coefficient of 0.89 with a ΔSIC interval spanning 0.82 to 0.99, implying that mechanistically derived coefficients of 0.67, 0.75, and 1, are not supported by the data. Model selection supports the use of a random intercepts and random slopes by species, consistent with the idea that other factors, such as taxonomy or ecological or lifestyle characteristics, may be critical for discerning the underlying process giving rise to the data. The evidentialist framework applied here, allows for further refinement given additional data and more complex models.

Abyssal deposit-feeding rates consistent with the metabolic theory of ecology

The Metabolic Theory of Ecology (MTE) posits that metabolic rate controls ecological processes, such as the rate of resource uptake, from the individual‐ to the ecosystem‐scale. Metabolic rate has been found empirically to be an exponential function of whole organism body mass. We test a fundamental assumption of MTE, whether resource uptake scales to metabolism, by examining detritivores accessing a single common resource pool, an ideal study case. We used an existing empirical model of ingestion for aquatic deposit feeders adjusted for temperature to test whether ingestion by abyssal deposit feeders conforms to MTE‐predicted feeding rates. We estimated the sediment deposit‐feeding rates of large invertebrates from two abyssal study sites using time‐lapse photography, and related those rates to body mass, environmental temperature, and sediment organic matter content using this framework. Ingestion was significantly related to individual wet mass, with a mass‐scaling coefficient of 0.81, with 95% confidence intervals that encompass the MTE‐predicted value of 0.75, and the same pattern determined in other aquatic systems. Our results also provide insight into the potential mechanism through which this fundamental assumption operates. After temperature correction, both deep‐ and shallow‐water taxa might be summarized into a single mass‐scaled ingestion rate.

Consistent temperature dependence of functional response parameters and their use in predicting population abundance

Global warming is one of the greatest threats to the persistence of populations: increased metabolic demands should strengthen pairwise species interactions, which could destabilize food webs at the higher organizational levels. Quantifying the temperature dependence of consumer-resource interactions is thus essential for predicting ecological responses to warming. We explored feeding interactions between different predator-prey pairs in controlled-temperature chambers and in a system of naturally heated streams. We found consistent temperature dependence of attack rates across experimental settings, though the magnitude and activation energy of attack rate were specific to each predator, which varied in mobility and foraging mode. We used these parameters along with metabolic rate measurements to estimate energetic efficiency and population abundance with warming. Energetic efficiency accurately estimated field abundance of a mobile predator that struggled to meet its metabolic demands, but was a poor predictor for a sedentary predator that operated well below its energetic limits. Temperature effects on population abundance may thus be strongly dependent on whether organisms are regulated by their own energy intake or interspecific interactions. Given the widespread use of functional response parameters in ecological modelling, reconciling outcomes from laboratory and field studies increases the confidence and precision with which we can predict warming impacts on natural systems.

Physiological diversity, biodiversity patterns and global climate change: testing key hypotheses involving temperature and oxygen

Documenting and explaining global patterns of biodiversity in time and space have fascinated and occupied biologists for centuries. Investigation of the importance of these patterns, and their underpinning mechanisms, has gained renewed vigour and importance, perhaps becoming pre-eminent, as we attempt to predict the biological impacts of global climate change. Understanding the physiological features that determine, or constrain, a species' geographical range and how they respond to a rapidly changing environment is critical. While the ecological patterns are crystallizing, explaining the role of physiology has just begun. The papers in this volume are the primary output from a Satellite Meeting of the Society of Experimental Biology Annual Meeting, held in Florence in July 2018. The involvement of two key environmental factors, temperature and oxygen, was explored through the testing of key hypotheses. The aim of the meeting was to improve our knowledge of large-scale geographical differences in physiology, e.g. metabolism, growth, size and subsequently our understanding of the role and vulnerability of those physiologies to global climate warming. While such an aim is of heuristic interest, in the midst of our current biodiversity crisis, it has an urgency that is difficult to overstate. This article is part of the theme issue 'Physiological diversity, biodiversity patterns and global climate change: testing key hypotheses involving temperature and oxygen'.

Snake venom potency and yield are associated with prey-evolution, predator metabolism and habitat structure

Snake venom is well known for its ability to incapacitate and kill prey. Yet, potency and the amount of venom available varies greatly across species, ranging from the seemingly harmless to those capable of killing vast numbers of potential prey. This variation is poorly understood, with comparative approaches confounded by the use of atypical prey species as models to measure venom potency. Here, we account for such confounding issues by incorporating the phylogenetic similarity between a snake's diet and the species used to measure its potency. In a comparative analysis of 102 species we show that snake venom potency is generally prey-specific. We also show that venom yields are lower in species occupying three dimensional environments and increases with body size corresponding to metabolic rate, but faster than predicted from increases in prey size. These results underline the importance of physiological and environmental factors in the evolution of predator traits.

Ecological and evolutionary implications of allometric growth in stomach size of brachyuran crabs

Individual characteristics often scale allometrically with organismal body size and the form of this scaling can be influenced by ecological and evolutionary factors. Examining the specific form of this scaling can therefore yield important insights into organismal ecology and evolution and the ability of organisms to respond to future environmental changes. We examine the intraspecific allometric scaling of stomach volume with body mass for 17 species of brachyuran crabs. We also examine how this scaling is influenced by dietary strategy, maximum body size, and activity level, all while controlling for phylogenetic relationships between the species. We show that the slope and intercept of the allometric scaling relationships vary across species and are influenced by all three ecological factors examined here, as well as by evolutionary relationships. These results highlight potential divergent strategies in stomach growth taken by different groups of crabs and highlight potential limitations that may be imposed on the ability of this group of organisms to respond to warming trends expected with climate change.

Measuring metabolic rates of small terrestrial organisms by fluorescence-based closed-system respirometry

We explore a recent, innovative variation of closed-system respirometry for terrestrial organisms, whereby oxygen partial pressure (P_O2 ) is repeatedly measured fluorometrically in a constant-volume chamber over multiple time points. We outline a protocol that aligns this technology with the broader literature on aerial respirometry, including the calculations required to accurately convert O₂ depletion to metabolic rate (MR). We identify a series of assumptions, and sources of error associated with this technique, including thresholds where O₂ depletion becomes limiting, that impart errors to the calculation and interpretation of MR. Using these adjusted calculations, we found that the resting MR of five species of angiosperm seeds ranged from 0.011 to 0.640 ml g^-1 h^-1, consistent with published seed MR values. This innovative methodology greatly expands the lower size limit of terrestrial organisms that can be measured, and offers the potential for measuring MR changes over time as a result of physiological processes of the organism.

Response to formal comment on Myhrvold (2016) submitted by Griebeler and Werner (2017)

Griebeler and Werner offer a formal comment on Myhrvold, 2016 defending the conclusions of Werner and Griebeler, 2014. Although the comment criticizes several aspects of methodology in Myhrvold, 2016, all three papers concur on a key conclusion: the metabolism of extant endotherms and ectotherms cannot be reliably classified using growth-rate allometry, because the growth rates of extant endotherms and ectotherms overlap. A key point of disagreement is that the 2014 paper concluded that despite this general case, one can nevertheless classify dinosaurs as ectotherms from their growth rate allometry. The 2014 conclusion is based on two factors: the assertion (made without any supporting arguments) that the comparison with dinosaurs must be restricted only to extant sauropsids, ignoring other vertebrate groups, and that extant sauropsid endotherm and ectotherm growth rates in a data set studied in the 2014 work do not overlap. The Griebeler and Werner formal comment presents their first arguments in support of the restriction proposition. In this response I show that this restriction is unsupported by established principles of phylogenetic comparison. In addition, I show that the data set studied in their 2014 work does show overlap, and that this is visible in one of its figures. I explain how either point effectively invalidates the conclusion of their 2014 paper. I also address the other methodological criticisms of Myhrvold 2016, and find them unsupported.

On the exponent in the Von Bertalanffy growth model

Von Bertalanffy proposed the differential equation m'(t) = p × m(t) ^a  - q × m(t) for the description of the mass growth of animals as a function m(t) of time t. He suggested that the solution using the metabolic scaling exponent a = 2/3 (Von Bertalanffy growth function VBGF) would be universal for vertebrates. Several authors questioned universality, as for certain species other models would provide a better fit. This paper reconsiders this question. Based on 60 data sets from literature (37 about fish and 23 about non-fish species) it optimizes the model parameters, in particular the exponent 0 ≤ a < 1, so that the model curve achieves the best fit to the data. The main observation of the paper is the large variability in the exponent, which can vary over a very large range without affecting the fit to the data significantly, when the other parameters are also optimized. The paper explains this by differences in the data quality: variability is low for data from highly controlled experiments and high for natural data. Other deficiencies were biologically meaningless optimal parameter values or optimal parameter values attained on the boundary of the parameter region (indicating the possible need for a different model). Only 11 of the 60 data sets were free of such deficiencies and for them no universal exponent could be discerned.

The Evolution of Energetic Scaling across the Vertebrate Tree of Life

Metabolism is the link between ecology and physiology-it dictates the flow of energy through individuals and across trophic levels. Much of the predictive power of metabolic theories of ecology derives from the scaling relationship between organismal size and metabolic rate. There is growing evidence that this scaling relationship is not universal, but we have little knowledge of how it has evolved over macroevolutionary time. Here we develop a novel phylogenetic comparative method to investigate how often and in which clades the macroevolutionary dynamics of the metabolic scaling have changed. We find strong evidence that the metabolic scaling relationship has shifted multiple times across the vertebrate phylogeny. However, shifts are rare and otherwise strongly constrained. Importantly, both the estimated slope and intercept values vary widely across regimes, with slopes that spanned across theoretically predicted values such as 2/3 or 3/4. We further tested whether traits such as ecto-/endothermy, genome size, and quadratic curvature with body mass (i.e., energetic constraints at extreme body sizes) could explain the observed pattern of shifts. Though these factors help explain some of the variation in scaling parameters, much of the remaining variation remains elusive. Our results lay the groundwork for further exploration of the evolutionary and ecological drivers of major transitions in metabolic strategy and for harnessing this information to improve macroecological predictions.

A general model for metabolic scaling in self-similar asymmetric networks

How a particular attribute of an organism changes or scales with its body size is known as an allometry. Biological allometries, such as metabolic scaling, have been hypothesized to result from selection to maximize how vascular networks fill space yet minimize internal transport distances and resistances. The West, Brown, Enquist (WBE) model argues that these two principles (space-filling and energy minimization) are (i) general principles underlying the evolution of the diversity of biological networks across plants and animals and (ii) can be used to predict how the resulting geometry of biological networks then governs their allometric scaling. Perhaps the most central biological allometry is how metabolic rate scales with body size. A core assumption of the WBE model is that networks are symmetric with respect to their geometric properties. That is, any two given branches within the same generation in the network are assumed to have identical lengths and radii. However, biological networks are rarely if ever symmetric. An open question is: Does incorporating asymmetric branching change or influence the predictions of the WBE model? We derive a general network model that relaxes the symmetric assumption and define two classes of asymmetrically bifurcating networks. We show that asymmetric branching can be incorporated into the WBE model. This asymmetric version of the WBE model results in several theoretical predictions for the structure, physiology, and metabolism of organisms, specifically in the case for the cardiovascular system. We show how network asymmetry can now be incorporated in the many allometric scaling relationships via total network volume. Most importantly, we show that the 3/4 metabolic scaling exponent from Kleiber’s Law can still be attained within many asymmetric networks.

We present a model for incorporating geometrically asymmetric branching into biological resource distribution networks. Our work shows how space-filling and fluid flow principles constrain allowed branching morphologies within the context of our model. Simultaneously, we demonstrate that there is a wide range of asymmetrically branching network architectures that still give rise to 3/4 metabolic scaling exponents.

Mass scaling of metabolic rates in carabid beetles (Carabidae) – the importance of phylogeny, regression models and gas exchange patterns

The origin of the allometric relationship between standard metabolic rate (MR) and body mass (M), often described as MR=aMb, remains puzzling and interpretation of the mass-scaling exponent, b may depend on the methodological approach, shapes of residuals, coefficient of determination (r2) and sample size. We investigated the mass scaling of MRs within and between species of Carabidae beetles. We used ordinary least squares (OLS) regression, phylogenetically generalized least squares (PGLS) regression and standardized major axis (SMA) regression to explore the effects of different model-fitting methods and data clustering caused by phylogenetic clades (grade shift) and gas exchange patterns (discontinuous, cyclic and continuous). At the interspecific level, the relationship between MR and M was either negatively allometric (b&lt;1) or isometric (b=1), depending on the fitting method. At the intraspecific level, the relationship either did not exist or was isometric or positively allometric (b&gt;1), and the fit was significantly improved after the analysed dataset was split according to gas exchange patterns. The studied species originated from two distinct phylogenetic clades that had different intercepts but a common scaling exponent (OLS, 0.61) that was much shallower than the scaling exponent for the combined dataset for all species (OLS, 0.71). The best scaling exponent estimates were obtained by applying OLS while accounting for grade shifts or by applying PGLS. Overall, we show that allometry of MR in insects can depend heavily on the model fitting method, the structure of phylogenetic non-independence and ecological factors that elicit different modes of gas exchange.

Mass, phylogeny, and temperature are sufficient to explain differences in metabolic scaling across mammalian orders?

Whether basal metabolic rate-body mass scaling relationships have a single exponent is highly discussed, and also the correct statistical model to establish relationships. Here, we aimed (1) to identify statistically best scaling models for 17 mammalian orders, Marsupialia, Eutheria and all mammals, and (2) thereby to prove whether correcting for differences in species' body temperature and their shared evolutionary history improves models and their biological interpretability. We used the large dataset from Sieg et al. (The American Naturalist174, 2009, 720) providing species' body mass (BM), basal metabolic rate (BMR) and body temperature (T). We applied different statistical approaches to identify the best scaling model for each taxon: ordinary least squares regression analysis (OLS) and phylogenetically informed analysis (PGLS), both without and with controlling for T. Under each approach, we tested linear equations (log-log-transformed data) estimating scaling exponents and normalization constants, and such with a variable normalization constant and a fixed exponent of either ⅔ or ¾, and also a curvature. Only under temperature correction, an additional variable coefficient modeled the influence of T on BMR. Except for Pholidata and Carnivora, in all taxa studied linear models were clearly supported over a curvature by AICc. They indicated no single exponent at the level of orders or at higher taxonomic levels. The majority of all best models corrected for phylogeny, whereas only half of them included T. When correcting for T, the mathematically expected correlation between the exponent (b) and the normalization constant (a) in the standard scaling model y = a x^b was removed, but the normalization constant and temperature coefficient still correlated strongly. In six taxa, T and BM correlated positively or negatively. All this hampers a disentangling of the effect of BM, T and other factors on BMR, and an interpretation of linear BMR-BM scaling relationships in the mammalian taxa studied.

Common metabolic constraints on dive duration in endothermic and ectothermic vertebrates

Dive duration in air-breathing vertebrates is thought to be constrained by the volume of oxygen stored in the body and the rate at which it is consumed (i.e., "oxygen store/usage hypothesis"). The body mass-dependence of dive duration among endothermic vertebrates is largely supportive of this model, but previous analyses of ectothermic vertebrates show no such body mass-dependence. Here we show that dive duration in both endotherms and ectotherms largely support the oxygen store/usage hypothesis after accounting for the well-established effects of temperature on oxygen consumption rates. Analyses of the body mass and temperature dependence of dive duration in 181 species of endothermic vertebrates and 29 species of ectothermic vertebrates show that dive duration increases as a power law with body mass, and decreases exponentially with increasing temperature. Thus, in the case of ectothermic vertebrates, changes in environmental temperature will likely impact the foraging ecology of divers.

Physics of metabolic organization

We review the most comprehensive metabolic theory of life existing to date. A special focus is given to the thermodynamic roots of this theory and to implications that the laws of physics-such as the conservation of mass and energy-have on all life. Both the theoretical foundations and biological applications are covered. Hitherto, the foundations were more accessible to physicists or mathematicians, and the applications to biologists, causing a dichotomy in what always should have been a single body of work. To bridge the gap between the two aspects of the same theory, we (i) adhere to the theoretical formalism, (ii) try to minimize the amount of information that a reader needs to process, but also (iii) invoke examples from biology to motivate the introduction of new concepts and to justify the assumptions made, and (iv) show how the careful formalism of the general theory enables modular, self-consistent extensions that capture important features of the species and the problem in question. Perhaps the most difficult among the introduced concepts, the utilization (or mobilization) energy flow, is given particular attention in the form of an original and considerably simplified derivation. Specific examples illustrate a range of possible applications-from energy budgets of individual organisms, to population dynamics, to ecotoxicology.

Temperature mediates continental-scale diversity of microbes in forest soils

Climate warming is increasingly leading to marked changes in plant and animal biodiversity, but it remains unclear how temperatures affect microbial biodiversity, particularly in terrestrial soils. Here we show that, in accordance with metabolic theory of ecology, taxonomic and phylogenetic diversity of soil bacteria, fungi and nitrogen fixers are all better predicted by variation in environmental temperature than pH. However, the rates of diversity turnover across the global temperature gradients are substantially lower than those recorded for trees and animals, suggesting that the diversity of plant, animal and soil microbial communities show differential responses to climate change. To the best of our knowledge, this is the first study demonstrating that the diversity of different microbial groups has significantly lower rates of turnover across temperature gradients than other major taxa, which has important implications for assessing the effects of human-caused changes in climate, land use and other factors.

Climate warming has a wide range of effects on biodiversity. Here, Zhou et al. show that although variation in environmental temperature is a primary driver of soil microbial biodiversity, microbes show much lower rates of turnover across temperature gradients than other major taxa.

Making the most of what we have: application of extrapolation approaches in radioecological wildlife transfer models

We will never have data to populate all of the potential radioecological modelling parameters required for wildlife assessments. Therefore, we need robust extrapolation approaches which allow us to make best use of our available knowledge. This paper reviews and, in some cases, develops, tests and validates some of the suggested extrapolation approaches. The concentration ratio (CRproduct-diet or CRwo-diet) is shown to be a generic (trans-species) parameter which should enable the more abundant data for farm animals to be applied to wild species. An allometric model for predicting the biological half-life of radionuclides in vertebrates is further tested and generally shown to perform acceptably. However, to fully exploit allometry we need to understand why some elements do not scale to expected values. For aquatic ecosystems, the relationship between log10(a) (a parameter from the allometric relationship for the organism-water concentration ratio) and log(Kd) presents a potential opportunity to estimate concentration ratios using Kd values. An alternative approach to the CRwo-media model proposed for estimating the transfer of radionuclides to freshwater fish is used to satisfactorily predict activity concentrations in fish of different species from three lakes. We recommend that this approach (REML modelling) be further investigated and developed for other radionuclides and across a wider range of organisms and ecosystems. Ecological stoichiometry shows potential as an extrapolation method in radioecology, either from one element to another or from one species to another. Although some of the approaches considered require further development and testing, we demonstrate the potential to significantly improve predictions of radionuclide transfer to wildlife by making better use of available data.

An explanation of the relationship between mass, metabolic rate and characteristic length for placental mammals

The Mass, Metabolism and Length Explanation (MMLE) was advanced in 1984 to explain the relationship between metabolic rate and body mass for birds and mammals. This paper reports on a modernized version of MMLE. MMLE deterministically computes the absolute value of Basal Metabolic Rate (BMR) and body mass for individual animals. MMLE is thus distinct from other examinations of these topics that use species-averaged data to estimate the parameters in a statistically best fit power law relationship such as BMR = a(bodymass) (b) . Beginning with the proposition that BMR is proportional to the number of mitochondria in an animal, two primary equations are derived that compute BMR and body mass as functions of an individual animal's characteristic length and sturdiness factor. The characteristic length is a measureable skeletal length associated with an animal's means of propulsion. The sturdiness factor expresses how sturdy or gracile an animal is. Eight other parameters occur in the equations that vary little among animals in the same phylogenetic group. The present paper modernizes MMLE by explicitly treating Froude and Strouhal dynamic similarity of mammals' skeletal musculature, revising the treatment of BMR and using new data to estimate numerical values for the parameters that occur in the equations. A mass and length data set with 575 entries from the orders Rodentia, Chiroptera, Artiodactyla, Carnivora, Perissodactyla and Proboscidea is used. A BMR and mass data set with 436 entries from the orders Rodentia, Chiroptera, Artiodactyla and Carnivora is also used. With the estimated parameter values MMLE can calculate characteristic length and sturdiness factor values so that every BMR and mass datum from the BMR and mass data set can be computed exactly. Furthermore MMLE can calculate characteristic length and sturdiness factor values so that every body mass and length datum from the mass and length data set can be computed exactly. Whether or not MMLE can calculate a sturdiness factor value so that an individual animal's BMR and body mass can be simultaneously computed given its characteristic length awaits analysis of a data set that simultaneously reports all three of these items for individual animals. However for many of the addressed MMLE homogeneous groups, MMLE can predict the exponent obtained by regression analysis of the BMR and mass data using the exponent obtained by regression analysis of the mass and length data. This argues that MMLE may be able to accurately simultaneously compute BMR and mass for an individual animal.

Energy and the Scaling of Animal Space Use

Daily animal movements are usually limited to a discrete home range area that scales allometrically with body size, suggesting that home-range size is shaped by metabolic rates and energy availability across species. However, there is little understanding of the relative importance of the various mechanisms proposed to influence home-range scaling (e.g., differences in realm productivity, thermoregulation, locomotion strategy, dimensionality, trophic guild, and prey size) and whether these extend beyond the commonly studied birds and mammals. We derive new home-range scaling relationships for fishes and reptiles and use a model-selection approach to evaluate the generality of home-range scaling mechanisms across 569 vertebrate species. We find no evidence that home-range allometry varies consistently between aquatic and terrestrial realms or thermoregulation strategies, but we find that locomotion strategy, foraging dimension, trophic guild, and prey size together explain 80% of the variation in home-range size across vertebrates when controlling for phylogeny and tracking method. Within carnivores, smaller relative prey size among gape-limited fishes contributes to shallower scaling relative to other predators. Our study reveals how simple morphological traits and prey-handling ability can profoundly influence individual space use, which underpins broader-scale patterns in the spatial ecology of vertebrates.