Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant?

Size matters in metabolic scaling: Critical role of the thermodynamic efficiency of ATP synthesis and its dependence on mitochondrial H+ leak across mammalian species

In this last article of the trilogy, the unified biothermokinetic theory of ATP synthesis developed in the previous two papers is applied to a major problem in comparative physiology, biochemistry, and ecology-that of metabolic scaling as a function of body mass across species. A clear distinction is made between intraspecific and interspecific relationships in energy metabolism, clearing up confusion that had existed from the very beginning since Kleiber first proposed his mouse-to-elephant rule almost a century ago. It is shown that the overall mass exponent of basal/standard metabolic rate in the allometric relationship [Formula: see text] is composed of two parts, one emerging from the relative intraspecific constancy of the slope (b), and the other (b') arising from the interspecific variation of the mass coefficient, a(M) with body size. Quantitative analysis is shown to reveal the hidden underlying relationship followed by the interspecific mass coefficient, a(M)=P0M0.10, and a universal value of P0=3.23 watts, W is derived from empirical data on mammals from mouse to cattle. The above relationship is shown to be understood only within an evolutionary biological context, and provides a physiological explanation for Cope's rule. The analysis also helps in fundamentally understanding how variability and a diversity of scaling exponents arises in allometric relations in biology and ecology. Next, a molecular-level understanding of the scaling of metabolism across mammalian species is shown to be obtained by consideration of the thermodynamic efficiency of ATP synthesis η, taking mitochondrial proton leak as a major determinant of basal metabolic rate in biosystems. An iterative solution is obtained by solving the mathematical equations of the biothermokinetic ATP theory, and the key thermodynamic parameters, e.g. the degree of coupling q, the operative P/O ratio, and the metabolic efficiency of ATP synthesis η are quantitatively evaluated for mammals from rat to cattle. Increases in η (by ∼15%) over a 2000-fold body size range from rat to cattle, primarily arising from an ∼3-fold decrease in the mitochondrial H+ leak rate are quantified by the unified ATP theory. Biochemical and mechanistic consequences for the interpretation of basal metabolism, and the various molecular implications arising are discussed in detail. The results are extended to maximum metabolic rate, and interpreted mathematically as a limiting case of the general ATP theory. The limitations of the analysis are pointed out. In sum, a comprehensive quantitative analysis based on the unified biothermokinetic theory of ATP synthesis is shown to solve a central problem in biology, physiology, and ecology on the scaling of energy metabolism with body size.

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.

Evolutionary scaling of maximum growth rate with organism size

Data from nearly 1000 species reveal the upper bound to rates of biomass production achievable by natural selection across the Tree of Life. For heterotrophs, maximum growth rates scale positively with organism size in bacteria but negatively in eukaryotes, whereas for phototrophs, the scaling is negligible for cyanobacteria and weakly negative for eukaryotes. These results have significant implications for understanding the bioenergetic consequences of the transition from prokaryotes to eukaryotes, and of the expansion of some groups of the latter into multicellularity. The magnitudes of the scaling coefficients for eukaryotes are significantly lower than expected under any proposed physical-constraint model. Supported by genomic, bioenergetic, and population-genetic data and theory, an alternative hypothesis for the observed negative scaling in eukaryotes postulates that growth-diminishing mutations with small effects passively accumulate with increasing organism size as a consequence of associated increases in the power of random genetic drift. In contrast, conditional on the structural and functional features of ribosomes, natural selection has been able to promote bacteria with the fastest possible growth rates, implying minimal conflicts with both bioenergetic constraints and random genetic drift. If this extension of the drift-barrier hypothesis is correct, the interpretations of comparative studies of biological traits that have traditionally ignored differences in population-genetic environments will require revisiting.

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.

Solving the grand challenge of phenotypic integration: allometry across scales

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

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.

Towards 3D basic theories of plant forms

Allometric, metabolic, and biomechanical theories are the critical foundations for scientifically deciphering plant forms. Their concrete laws, however, are found to deviate for plenty of plant specimens. This phenomenon has not been extensively studied, due to technical restrictions. This bottleneck now can be overcome by the state-of-the-art three-dimensional (3D) mapping technologies, such as fine-scale terrestrial laser scanning. On these grounds, we proposed to reexamine the basic theories regarding plant forms, and then, we case validated the feasibility of upgrading them into 3D modes. As an in-time enlightening of 3D revolutionizing the related basic subject, our theoretical prospect further sorted out the potential challenges as the cutting points for advancing its future exploration, which may enable 3D reconstruction of the basic theories of plant forms and even boost life science.

In this Perspective, the authors discuss how state-of-the-art three-dimensional mapping technologies such as fine-scale terrestrial laser scanning can help us understand the theories of plant forms.

On the rules of life and Kleiber's law: the macroscopic relationship between materials and energy

Efforts to accommodate the growth in global energy consumption within a fragile biosphere are primarily focused on managing the transition towards a low-carbon energy mix. We show evidence that a more fundamental problem exists through a scaling relation, akin to Kleiber's Law, between society's energy consumption and material stocks. Humanity's energy consumption scales at 0.78 of its material stocks, which implies predictable environmental pressure regardless of the energy mix. If true, future global energy scenarios imply vast amounts of materials and corresponding environmental degradation, which have not been adequately acknowledged. Thus, limits to energy consumption are needed regardless of the energy mix to stabilize human intervention in the biosphere.

Recent advances in blood rheology: a review

Due to the potential impact on the diagnosis and treatment of various cardiovascular diseases, work on the rheology of blood has significantly expanded in the last decade, both experimentally and theoretically. Experimentally, blood has been confirmed to demonstrate a variety of non-Newtonian rheological characteristics, including pseudoplasticity, viscoelasticity, and thixotropy. New rheological experiments and the development of more controlled experimental protocols on more extensive, broadly physiologically characterized, human blood samples demonstrate the sensitivity of aspects of hemorheology to several physiological factors. For example, at high shear rates the red blood cells elastically deform, imparting viscoelasticity, while at low shear rates, they form "rouleaux" structures that impart additional, thixotropic behavior. In addition to the advances in experimental methods and validated data sets, significant advances have also been made in both microscopic simulations and macroscopic, continuum, modeling, as well as novel, multiscale approaches. We outline and evaluate the most promising of these recent developments. Although we primarily focus on human blood rheology, we also discuss recent observations on variations observed across some animal species that provide some indication on evolutionary effects.

Sportive lemurs elevate their metabolic rate during challenging seasons and do not enter regular heterothermy

Animals experience seasonal changes of environmental and ecological conditions in most habitats. Fluctuations in ambient temperature have a strong influence on thermoregulation, particularly on small endothermic mammals. However, different mammalian species cope differently with these changes. Understanding the physiological responses of organisms to different seasons and analysing the mechanisms that account for intra- and inter-specific differences and the ecological consequences of these variations is important to predict species responses to climatic changes. Consequences of climatic changes will be most pronounced in climatically already challenging habitats, such as the dry regions of western Madagascar. We aimed to identify the seasonal responses and adaptive possibilities in energy budgeting of Lepilemur edwardsi, a small primate of this habitat, by measuring metabolic rate (MR; open-flow respiratory) and skin temperature in the field during different seasons. Resting metabolism was generally low, but our study did not detect any signs of regular heterothermic episodes, despite the fact that these are known in other sympatrically living lemurs with a similar lifestyle. Surprisingly, L. edwardsi responded by elevating its resting MR in the poor-resourced dry season, compared to the better-resourced wet season, presumably to master detoxification of their increasingly toxic diet. As body mass decreased over this time, this strategy is obviously not energetically balanced on the long term. This is cause for concern, as it suggests that L. edwardsi has a very small leeway to adjust to changing conditions as experienced due to climate change, as dry season are expected to become longer and hotter, straining water budgets and food quality even more. Moreover, our findings highlight the importance of studying physiological parameters directly in the field and under differing climatic conditions.

Lizards as models to explore the ecological and neuroanatomical correlates of miniaturization

Extreme body size reductions bring about unorthodox anatomical arrangements and novel ways in which animals interact with the environment. Drawing from studies of vertebrates and invertebrates, we provide a theoretical framework for miniaturization to inform hypotheses using lizards as a study system. Through this approach, we demonstrate the repeated evolution of miniaturization across 11 families and a tendency for miniaturized species to occupy terrestrial microhabitats, possibly driven by physiological constraints. Differences in gross brain morphology between two gecko species demonstrate a proportionally larger telencephalon and smaller olfactory bulbs in the miniaturized species, though more data are needed to generalize this trend. Our study brings into light the potential contributions of miniaturized lizards to explain patterns of body size evolution and its impact on ecology and neuroanatomy. In addition, our findings reveal the need to study the natural history of miniaturized species, particularly in relation to their sensory and physiological ecology.

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.

Dynamic allometric scaling of tree biomass and size

Allometric scaling laws critically examine structure-function relationships. In estimating the forest biomass carbon and its response under climate change, the issue of scaling has resulted in difficulties when modelling the biomass for different-sized trees, especially large ones, and has not yet been solved in either theory or practice. Here, we propose the concept of a dynamic allometric scaling relationship between stem biomass and above-ground biomass The allometric curve approaches an asymptote with an increase in tree size. An asymptotic allometric equation is presented that has a better fit to the data than the simple power-law allometric equation. The non-constant exponent is determined by the change in the biomass ratio for different organs and is governed by the dynamic allometric coefficient. This study presents a methodological framework to theoretically characterize allometric relationships and provides new insights in understanding the general scaling pattern and carbon sequestration capacity of large trees across global forests.

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.

Scaling in Colloidal and Biological Networks

Scaling and dimensional analysis is applied to networks that describe various physical systems. Some of these networks possess fractal, scale-free, and small-world properties. The amount of information contained in a network is found by calculating its Shannon entropy. First, we consider networks arising from granular and colloidal systems (small colloidal and droplet clusters) due to pairwise interaction between the particles. Many networks found in colloidal science possess self-organizing properties due to the effect of percolation and/or self-organized criticality. Then, we discuss the allometric laws in branching vascular networks, artificial neural networks, cortical neural networks, as well as immune networks, which serve as a source of inspiration for both surface engineering and information technology. Scaling relationships in complex networks of neurons, which are organized in the neocortex in a hierarchical manner, suggest that the characteristic time constant is independent of brain size when interspecies comparison is conducted. The information content, scaling, dimensional, and topological properties of these networks are discussed.

Axial anatomy of the leaf midrib provides new insights into the hydraulic architecture and cavitation patterns of Acer pseudoplatanus leaves

The structure of leaf veins is typically described by a hierarchical scheme (e.g. midrib, 1st order, 2nd order), which is used to predict variation in conduit diameter from one order to another whilst overlooking possible variation within the same order. We examined whether xylem conduit diameter changes within the same vein order, with resulting consequences for resistance to embolism. We measured the hydraulic diameter (Dh), and number of vessels (VN) along the midrib and petioles of leaves of Acer pseudoplatanus, and estimated the leaf area supplied (Aleaf-sup) at different points of the midrib and how variation in anatomical traits affected embolism resistance. We found that Dh scales with distance from the midrib tip (path length, L) with a power of 0.42, and that VN scales with Aleaf-sup with a power of 0.66. Total conductive area scales isometrically with Aleaf-sup. Embolism events along the midrib occurred first in the basipetal part and then at the leaf tip where vessels are narrower. The distance from the midrib tip is a good predictor of the variation in vessel diameter along the 1st order veins in A. pseudoplatanus leaves and this anatomical pattern seems to have an effect on hydraulic integrity since wider vessels at the leaf base embolize first.

Inferring vascular architecture of the wheat spikelet based on resource allocation in the branched headt (bht-A1) near isogenic lines

Substantial genetic and physiological efforts were made to understand the causal factors of floral abortion and grain filling problem in wheat. However, the vascular architecture during wheat spikelet development is surprisingly under-researched. We used the branched headt near-isogenic lines, FL-bht-A1-NILs, to visualise the dynamics of spikelet fertility and dry matter accumulation in spikelets sharing the same rachis node (henceforth Primary Spikelet, PSt, and Secondary Spikelet, SSt). The experiment was conducted after grouping FL-bht-A1-NILs into two groups, where tillers were consistently removed from one group. Our results show differential spikelet fertility and dry matter accumulation between the PSt and SSt, but also showed a concomitant improvement after de-tillering. This suggests a tight regulation of assimilate supply and dry matter accumulation in wheat spikelets. Since PSt and SSt share the same rachis node, the main vascular bundle in the rachis/rachilla is expected to bifurcate to connect each spikelet/floret to the vascular system. We postulate that the vascular structure in the wheat spikelet might even follow Murray's law, where the wide conduits assigned at the base of the spikelet feed the narrower conduits of the distal florets. We discuss our results based on the two modalities of the vascular network systems in plants.

Local adaptation reduces the metabolic cost of environmental warming

Metabolism shapes the ecosystem role of organisms by dictating their energy demand and nutrient recycling potential. Metabolic theory (MTE) predicts consumer metabolic and recycling rates will rise with warming, especially if body size declines, but it ignores potential for adaptation. We measured metabolic and nutrient excretion rates of individuals from populations of a globally invasive fish that colonized sites spanning a wide temperature range (19-37°C) on two continents within the last 100 yr. Fish body size declined across our temperature gradient and MTE predicted large rises in population energy demand and nutrient recycling. However, we found that the allometry and temperature dependency of metabolism varied in a countergradient pattern with local temperature in a way that offset predictions of MTE. Scaling of nutrient excretion was more variable and did not track temperature. Our results suggest that adaptation can reduce the metabolic cost of warming, increasing the prospects for population persistence under extreme warming scenarios.

Misconceptions and issues regarding allometric scaling during the drug development process

INTRODUCTION

Allometry is the study of size and its consequences. The simple hypothesis of allometric scaling is that all physiological parameters are proportional to body size or body mass. This review examines the development of theory-based allometry or fixed exponents (0.75 and 1.0 for basal metabolic rate and volume, respectively) and the evidence for or against the theory. The main focus of this report is to show the readers that there is enough evidence from experimental data that negate the concept of theory-based allometry in biology, physiology, and pharmacokinetics. Areas covered: In this review, the history of the development of theoretical allometry and the strong evidence against theory-based allometry demonstrated by experimental data is provided. During drug development, allometry is applied to both inter-species (from animals to humans) and intra-species (adults to children) scaling. These two forms of allometric scaling in the context of theory-based allometry are discussed and provide insight on scientific progress that refute theory-based allometry. Expert opinion: Theory-based allometry is a mere theory and experimental data and real-life observations strongly negate the existence of such a theory. Pharmacostatistical and physiological models based on theory-based allometry can be misleading and incorrect because the theory-based allometric exponent 0.75 is not universal. The exponents of allometry are data dependent and are not fixed in the universe.

Prediction of Clearance, Volume of distribution, and Half-life of Drugs in Extremely Low to Low Birth Weight Neonates: An Allometric Approach

BACKGROUND AND OBJECTIVES

More than 20 million infants worldwide (15.5 % of all births) are born with low birth weight. Low birth weight is associated with poor growth in childhood and a higher incidence of adult diseases, such as type 2 diabetes, hypertension and cardiovascular disease. The objective of this study was to evaluate the predictive performance of allometric models to predict clearance, volume of distribution, and half-life in extremely low to low birth weight neonates (<1 to 2.5 kg body weight).

METHODS

Several allometric models were used to predict clearance (4 models), volume of distribution (2 models), and half-life (2 models) in extremely low to low birth weight neonates. From the literature, clearance, volume of distribution, and half-life values for 16 drugs for these neonates were obtained. The predictive performance of these allometric models was evaluated by comparing the predicted values of the aforementioned pharmacokinetic parameters with the observed pharmacokinetic parameters in an individual neonate. For the evaluation of the predictive performance of these allometric models, there were 16 drugs with 36 (n = 279), 34 (n = 261), and 31 (n = 197) weight groups for clearance, volume of distribution, and half-life, respectively.

RESULTS

The prediction error of ≤50 % for mean clearance, volume of distribution, and half-life values were 69, 79, and 58 %, respectively, by proposed allometric models. The prediction error of ≤50 % for mean clearance, volume of distribution, and half-life values by theoretical allometric exponents were 0 % (exponent = 0.75), 71 % (exponent = 1.0), and 0 % (exponent = 0.25), respectively. In this analysis, out of 16 drugs, there were three drugs (ibuprofen, zidovudine, and buprenorphine) which are metabolized by glucuronidation and one drug (furosemide) is renally secreted. The predicted clearances of these four drugs were substantially higher by the proposed allometric methods. It seems that drugs of these physiological characteristics may require different method(s) to improve the prediction of clearance in the neonates.

CONCLUSIONS

Overall, the proposed allometric methods can predict mean pharmacokinetic parameters of drugs in extremely low to low birth weight neonates with reasonable accuracy and are of practical value during neonatal drug development.

Effects of fat and exoskeletal mass on the mass scaling of metabolism in Carabidae beetles

The rate at which organisms metabolize resources and consume oxygen is tightly linked to body mass. Typically, there is a sub-linear allometric relationship between metabolic rates and body mass (mass-scaling exponent b < 1). The origin of this pattern remains one of the most intriguing and hotly debated topics in evolutionary physiology. A decrease in mass-specific metabolic rates in larger organisms might reflect disproportionate increases in body components with low metabolic activity, such as storage and skeletal tissues. Addressing this hypothesis, we studied standard metabolic rates, body mass, and fat and exoskeletal mass in males and females from 15 species of Carabidae beetles. There was a sub-linear allometric relationship of metabolic rate with body mass: b = 0.72 (phylogeny not considered), b = 0.54 (phylogeny considered). The latter exponent was significantly lower than 0.75, which is sometimes regarded as a universal exponent value in the mass scaling of metabolic rates. Contrary to our hypothesis, the relative contribution of fat and the exoskeleton to body mass decreased, rather than increased with body mass, as indicated by the sub-linear allometric mass scaling of both components (b < 1). Supporting the role of metabolically inert body components in shaping metabolic scaling, the exponents (b) for metabolism became slightly smaller (b = 0.70, phylogeny not considered; 0.52, phylogeny considered) when we removed lipids and the exoskeleton from body mass calculations and considered only the lean mass of soft tissue in the mass scaling. Overall, our results indicate that, in beetles, the relative content of metabolically inert components changes across species according to species-specific body mass. Nevertheless, we did not find evidence that this changing contribution plays a central role in the origin of interspecific metabolic scaling in carabids. Our findings stress the need for finding alternative explanations, at least in carabids, for the origin of the mass scaling of metabolic rates.

Ecology of ontogenetic body-mass scaling of gill surface area in a freshwater crustacean

Several studies have documented ecological effects on intraspecific and interspecific body-size scaling of metabolic rate. However, little is known about how various ecological factors may affect the scaling of respiratory structures supporting oxygen uptake for metabolism. To our knowledge, our study is the first to provide evidence for ecological effects on the scaling of a respiratory structure among conspecific populations of any animal. We compared the body-mass scaling of gill surface area (SA) among eight spring-dwelling populations of the amphipod crustacean Gammarus minus Although gill SA scaling was not related to water temperature, conductivity or G. minus population density, it was significantly related to predation regime (and secondarily to pH). Body-mass scaling slopes for gill SA were significantly lower in four populations inhabiting springs with fish predators than for four populations in springs without fish (based on comparing means of the population slopes, or slopes calculated from pooled raw data for each comparison group). As a result, gill SA was proportionately smaller in adult amphipods from springs with versus without fish. This scaling difference paralleled similar differences in the scaling exponents for the rates of growth and resting metabolic rate. We hypothesized that gill SA scaling is shallower in fish-containing versus fishless spring populations of G. minus because of effects of size-selective predation on size-specific growth and activity that in turn affect the scaling of oxygen demand and concomitantly the gill capacity (SA) for oxygen uptake. Although influential theory claims that metabolic scaling is constrained by internal body design, our study builds on previous work to show that the scaling of both metabolism and the respiratory structures supporting it may be ecologically sensitive and evolutionarily malleable.

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.

Basal Metabolic Rate of the Black-Faced Sheathbill (Chionis minor): Intraspecific Variation in a Phylogenetically Distinct Island Endemic

Metabolic rate is a fundamental characteristic of all organisms. It covaries most significantly with activity, body mass, seasonality, and temperature. Nonetheless, substantial additional variation in metabolic rate, especially either resting rate or basal rate, is associated with a range of factors including phylogenetic position, ecological distinctiveness, range position, and diet. Understanding this variation is a key goal of physiological ecology. The black-faced sheathbill is a phylogenetically distinct, high-latitude, island-endemic bird occurring exclusively on several archipelagos in the southern Indian Ocean. Here we examined the idea that the unique phylogenetic position and ecology of the black-faced sheathbill may lead to a basal metabolic rate (BMR) different from that predicted by its body mass. When compared with BMR data available for all birds and a subset of island species, it was clear that the BMR of the black-faced sheathbill on subantarctic Marion Island, estimated at 15°C using indirect calorimetry (2.370 ± 0.464 W, mean ± SD; n = 22), for a group of birds with a mean mass of 459 ± 64 g, is no different from that expected based on body mass. However, variation in BMR, associated with habitat use and diet, even when correcting for variation in mass, was found. Sheathbills foraging year-round in comparatively resource-rich king penguin colonies have a higher BMR (2.758 ± 0.291 W, n = 12) than sheathbills that split their foraging between rockhopper penguin colonies and the intertidal zone (2.047 ± 0.303 W, n = 10), which are poorer in resources. Because these populations coexist at relatively small spatial extents (the entire island is 290 km(2)), other factors seem unlikely as causes of this variation.

Body size and meta-community structure: the allometric scaling of parasitic worm communities in their mammalian hosts

In this paper we derive from first principles the expected body sizes of the parasite communities that can coexist in a mammal of given body size. We use a mixture of mathematical models and known allometric relationships to examine whether host and parasite life histories constrain the diversity of parasite species that can coexist in the population of any host species. The model consists of one differential equation for each parasite species and a single density-dependent nonlinear equation for the affected host under the assumption of exploitation competition. We derive threshold conditions for the coexistence and competitive exclusion of parasite species using invasion criteria and stability analysis of the resulting equilibria. These results are then used to evaluate the range of parasites species that can invade and establish in a target host and identify the 'optimal' size of a parasite species for a host of a given body size; 'optimal' is defined as the body size of a parasite species that cannot be outcompeted by any other parasite species. The expected distributions of parasites body sizes in hosts of different sizes are then compared with those observed in empirical studies. Our analysis predicts the relative abundance of parasites of different size that establish in the host and suggests that increasing the ratio of parasite body size to host body size above a minimum threshold increases the persistence of the parasite population.

Metabolic allometric scaling model. Combining cellular transportation and heat dissipation constraints

Living organisms need energy to be "alive". Energy is produced by biochemical processing of nutrients. The rate of energy production is called metabolic rate. Metabolism is very important from evolutionary, ecological perspectives, and for organisms' development and functioning. It depends on different parameters, of which organisms' mass is considered as one of the most important. Simple relationships between the mass of organisms and their metabolic rates were empirically discovered a while ago. Such dependence is described by a power function, whose exponent is called allometric scaling coefficient. With the increase of mass the metabolic rate usually increases slower; if mass increases by two times, the metabolic rate increases less than two times. This fact has far reaching implications for organization of life. The fundamental biological and biophysical mechanisms underlying this phenomenon are still not well understood. Here, we show that one of such primary mechanisms relates to transportation of substances, like nutrients and waste, at a cellular level. We show how variations in cell size and associated cellular transportation costs explain the known variance of allometric exponent. The introduced model also includes heat dissipation constraints. The model agrees with experimental observations and reconciles experimental results across different taxa. It ties metabolic scaling to organismal and environmental characteristics; helps defining perspective directions of future researches; allows predicting allometric exponents based on characteristics of organisms and environments they live in.

Growth, ageing and scaling laws of coronary arterial trees

Despite the well-known design principles of vascular systems, it is unclear whether the vascular arterial tree obeys some scaling constraints during normal growth and ageing in a given species. Based on the micro-computed tomography measurements of coronary arterial trees in mice at different ages (one week to more than eight months), we show a constant exponent of 3/4, but age-dependent scaling coefficients in a length-volume scaling law (Lc=K(length-volume) · Vc³/⁴; Lc is the crown length, Vc is the crown volume, K(length-volume) is the age-dependent scaling coefficient) during normal growth and ageing. The constant 3/4 exponent represents the self-similar fractal-like branching pattern (i.e. basic mechanism to regulate the development of vascular trees within a species), whereas the age-dependent scaling coefficients characterize the structural growth or resorption of vascular trees during normal growth or ageing, respectively. This study enhances the understanding of age-associated changes in vascular structure and function.

Biogeographical patterns of biomass allocation in leaves, stems, and roots in China's forests

To test whether there are general patterns in biomass partitioning in relation to environmental variation when stand biomass is considered, we investigated biomass allocation in leaves, stems, and roots in China's forests using both the national forest inventory data (2004-2008) and our field measurements (2011-2012). Distribution patterns of leaf, stem, and root biomass showed significantly different trends according to latitude, longitude, and altitude, and were positively and significantly correlated with stand age and mean annual precipitation. Trade-offs among leaves, stems, and roots varied with forest type and origin and were mainly explained by stand biomass. Based on the constraints of stand biomass, biomass allocation was also influenced by forest type, origin, stand age, stand density, mean annual temperature, precipitation, and maximum temperature in the growing season. Therefore, after stand biomass was accounted for, the residual variation in biomass allocation could be partially explained by stand characteristics and environmental factors, which may aid in quantifying carbon cycling in forest ecosystems and assessing the impacts of climate change on forest carbon dynamics in China.

Home is where the shell is: predicting turtle home range sizes

Home range is the area traversed by an animal in its normal activities. The size of home ranges is thought to be tightly linked to body size, through size effect on metabolic requirements. Due to the structure of Eltonian food pyramids, home range sizes of carnivores are expected to exceed those of herbivorous species. The habitat may also affect home range size, with reduced costs of locomotion or lower food abundance in, for example, aquatic habitats selecting for larger home ranges. Furthermore, home range of males in polygamous species may be large due to sexual selection for increased reproductive output. Comparative studies on home range sizes have rarely been conducted on ectotherms. Because ectotherm metabolic rates are much lower than those of endotherms, energetic considerations of metabolic requirements may be less important in determining the home range sizes of the former, and other factors such as differing habitats and sexual selection may have an increased effect. We collected literature data on turtle home range sizes. We used phylogenetic generalized least squares analyses to determine whether body mass, sex, diet, habitat and social structure affect home range size. Turtle home range size increases with body mass. However, body mass explains relatively little of the variation in home range size. Aquatic turtles have larger home ranges than semiaquatic species. Omnivorous turtles have larger home ranges than herbivores and carnivores, but diet is not a strong predictor. Sex and social structure are unrelated to home range size. We conclude that energetic constraints are not the primary factor that determines home range size in turtles, and energetic costs of locomotion in different habitats probably play a major role.

Testing Foundations of Biological Scaling Theory Using Automated Measurements of Vascular Networks

Scientists have long sought to understand how vascular networks supply blood and oxygen to cells throughout the body. Recent work focuses on principles that constrain how vessel size changes through branching generations from the aorta to capillaries and uses scaling exponents to quantify these changes. Prominent scaling theories predict that combinations of these exponents explain how metabolic, growth, and other biological rates vary with body size. Nevertheless, direct measurements of individual vessel segments have been limited because existing techniques for measuring vasculature are invasive, time consuming, and technically difficult. We developed software that extracts the length, radius, and connectivity of in vivo vessels from contrast-enhanced 3D Magnetic Resonance Angiography. Using data from 20 human subjects, we calculated scaling exponents by four methods—two derived from local properties of branching junctions and two from whole-network properties. Although these methods are often used interchangeably in the literature, we do not find general agreement between these methods, particularly for vessel lengths. Measurements for length of vessels also diverge from theoretical values, but those for radius show stronger agreement. Our results demonstrate that vascular network models cannot ignore certain complexities of real vascular systems and indicate the need to discover new principles regarding vessel lengths.

Vascular networks distribute resources and constrain metabolic rate. Founded on a few key principles, biological scaling theories predict characteristic patterns for vascular networks as they branch from large to small vessels. These theories also predict seemingly unrelated phenomena, such as size limits on mammals. However, vascular networks are difficult to measure because there are billions of vessels that range in size from meters to micrometers. To test the foundations of biological scaling theories, we developed software that quickly measures thousands of in vivo vessels based on MRI. Data for vessel radii match predicted patterns but lengths do not. Our work suggests the need for new theoretical principles and should facilitate comparisons across organisms, spatial scales, and healthy and diseased tissue.

Heterogeneity of cells may explain allometric scaling of metabolic rate

The origin of allometric scaling of metabolic rate is a long-standing question in biology. Several models have been proposed for explaining the origin; however, they have advantages and disadvantages. In particular, previous models only demonstrate either two important observations for the allometric scaling: the variability of scaling exponents and predominance of 3/4-power law. Thus, these models have a dispute over their validity. In this study, we propose a simple geometry model, and show that a hypothesis that total surface area of cells determines metabolic rate can reproduce these two observations by combining two concepts: the impact of cell sizes on metabolic rate and fractal-like (hierarchical) organization. The proposed model both theoretically and numerically demonstrates the approximately 3/4-power law although several different biological strategies are considered. The model validity is confirmed using empirical data. Furthermore, the model suggests the importance of heterogeneity of cell size for the emergence of the allometric scaling. The proposed model provides intuitive and unique insights into the origin of allometric scaling laws in biology, despite several limitations of the model.

Is the WBE model appropriate for semi-arid shrubs subjected to clear cutting?

It is crucial to understand the adaptive mechanisms of woody plants facing periodic drought to assess their vulnerability to the increasing climate variability predicted in the Sahel. Guiera senegalensis J.F.Gmel is a semi-evergreen Combretaceae commonly found in Sahelian rangelands, fallows and crop fields because of its value as an agroforestry species. We compared canopy leafing, and allometric measurements of leaf area, stem area and stem length and their relationships with leaf water potential, stomatal conductance (gs) and soil-to-leaf hydraulic conductance (KS-L), in mature and current-year resprouts of G. senegalensis in Sahelian Niger. In mature shrubs, seasonal drought reduced the ratio of leaf area to cross-sectional stem area (AL : AS), mainly due to leaf shedding. The canopy of the current-year resprouts remained permanently leafed as the shrubs produced leaves and stems continuously, and their AL : AS ratio increased throughout the dry season. Their KS-L increased, whereas gs decreased. West, Brown and Enquist's (WBE) model can thus describe allometric trends in the seasonal life cycle of undisturbed mature shrubs, but not that of resprouts. Annual clear cutting drives allometric scaling relationships away from theoretical WBE predictions in the current-year resprouts, with scaling exponents 2.5 times greater than those of mature shrubs. High KS-L (twice that of mature shrubs) supports this intensive regeneration process. The adaptive strategy described here is probably common to many woody species that have to cope with both severe seasonal drought and regular disturbance over the long term.

Kleiber's Law: How the Fire of Life ignited debate, fueled theory, and neglected plants as model organisms

Size is a key feature of any organism since it influences the rate at which resources are consumed and thus affects metabolic rates. In the 1930s, size-dependent relationships were codified as "allometry" and it was shown that most of these could be quantified using the slopes of log-log plots of any 2 variables of interest. During the decades that followed, physiologists explored how animal respiration rates varied as a function of body size across taxa. The expectation was that rates would scale as the 2/3 power of body size as a reflection of the Euclidean relationship between surface area and volume. However, the work of Max Kleiber (1893-1976) and others revealed that animal respiration rates apparently scale more closely as the 3/4 power of body size. This phenomenology, which is called "Kleiber's Law," has been described for a broad range of organisms, including some algae and plants. It has also been severely criticized on theoretical and empirical grounds. Here, we review the history of the analysis of metabolism, which originated with the works of Antoine L. Lavoisier (1743-1794) and Julius Sachs (1832-1897), and culminated in Kleiber's book The Fire of Life (1961; 2. ed. 1975). We then evaluate some of the criticisms that have been leveled against Kleiber's Law and some examples of the theories that have tried to explain it. We revive the speculation that intracellular exo- and endocytotic processes are resource delivery-systems, analogous to the supercellular systems in multicellular organisms. Finally, we present data that cast doubt on the existence of a single scaling relationship between growth and body size in plants.

Metabolic scaling in animals: methods, empirical results, and theoretical explanations

Life on earth spans a size range of around 21 orders of magnitude across species and can span a range of more than 6 orders of magnitude within species of animal. The effect of size on physiology is, therefore, enormous and is typically expressed by how physiological phenomena scale with mass(b). When b ≠ 1 a trait does not vary in direct proportion to mass and is said to scale allometrically. The study of allometric scaling goes back to at least the time of Galileo Galilei, and published scaling relationships are now available for hundreds of traits. Here, the methods of scaling analysis are reviewed, using examples for a range of traits with an emphasis on those related to metabolism in animals. Where necessary, new relationships have been generated from published data using modern phylogenetically informed techniques. During recent decades one of the most controversial scaling relationships has been that between metabolic rate and body mass and a number of explanations have been proposed for the scaling of this trait. Examples of these mechanistic explanations for metabolic scaling are reviewed, and suggestions made for comparing between them. Finally, the conceptual links between metabolic scaling and ecological patterns are examined, emphasizing the distinction between (1) the hypothesis that size- and temperature-dependent variation among species and individuals in metabolic rate influences ecological processes at levels of organization from individuals to the biosphere and (2) mechanistic explanations for metabolic rate that may explain the size- and temperature-dependence of this trait.

Prediction of drug clearance in children: an evaluation of the predictive performance of several models

The objective of this study is to evaluate the predictive performance of several models to predict drug clearance in children ≤5 years of age. Six models (allometric model (data-dependent exponent), fixed exponent of 0.75 model, maturation model, body weight-dependent model, segmented allometric model, and age-dependent exponent model) were evaluated in this study. From the literature, the clearance values for six drugs from neonates to adults were obtained. External data were used to evaluate the predictive performance of these models in children ≤5 years of age. With the exception of a fixed exponent of 0.75, the mean predicted clearance in most of the age groups was within ≤50% prediction error. Individual clearance prediction was erratic by all models and cannot be used reliably to predict individual clearance. Maturation, body weight-dependent, and segmented allometric models to predict clearances of drugs in children ≤5 years of age are of limited practical value during drug development due to the lack of availability of data. Age-dependent exponent model can be used for the selection of first-in-children dose during drug development.

Allometric scaling relationship between above- and below-ground biomass within and across five woody seedlings

Allometric biomass allocation theory predicts that leaf biomass (M L ) scaled isometrically with stem (M S ) and root (M R ) biomass, and thus above-ground biomass (leaf and stem) (M A ) and root (M R ) scaled nearly isometrically with below-ground biomass (root) for tree seedlings across a wide diversity of taxa. Furthermore, prior studies also imply that scaling constant should vary with species. However, litter is known about whether such invariant isometric scaling exponents hold for intraspecific biomass allocation, and how variation in scaling constants influences the interspecific scaling relationship between above- and below-ground biomass. Biomass data of seedlings from five evergreen species were examined to test scaling relationships among biomass components across and within species. Model Type II regression was used to compare the numerical values of scaling exponents and constants among leaf, stem, root, and above- to below-ground biomass. The results indicated that M L and M S scaled in an isometric or a nearly isometric manner with M R , as well as M A to M R for five woody species. Significant variation was observed in the Y-intercepts of the biomass scaling curves, resulting in the divergence for intraspecific scaling and interspecific scaling relationships for M L versus M S and M L versus M R , but not for M S versus M R and M A versus M R . We conclude, therefore, that a nearly isometric scaling relationship of M A versus M R holds true within each of the studied woody species and across them irrespective the negative scaling relationship between leaf and stem.

Critical PO2 is size-independent in insects: implications for the metabolic theory of ecology

Insects, and all animals, exhibit hypometric scaling of metabolic rate, with larger species having lower mass-specific metabolic rates. The metabolic theory of ecology (MTE) is based on models ascribing hypometric scaling of metabolic rate to constrained O₂ supply systems in larger animals. We compiled critical PO₂ of metabolic and growth rates for more than 40 insect species with a size range spanning four orders of magnitude. Critical PO₂ values vary from far below 21kPa for resting animals to near 21kPa for growing or flying animals and are size-independent, demonstrating that supply capacity matches oxygen demand. These data suggest that hypometric scaling of resting metabolic rate in insects is not driven by constraints on oxygen availability.

Scaling relationships between leaf mass and total plant mass across Chinese forests

Biomass partitioning is important for illustrating terrestrial ecosystem carbon flux. West, Brown and Enquist (WBE) model predicts that an optimal 3/4 allometric scaling of leaf mass and total biomass of individual plants will be applied in diverse communities. However, amount of scientific evidence suggests an involvement of some biological and environmental factors in interpreting the variation of scaling exponent observed in empirical studies. In this paper, biomass information of 1175 forested communities in China was collected and categorized into groups in terms of leaf form and function, as well as their locations to test whether the allocation pattern was conserved or variable with internal and/or environmental variations. Model Type II regression protocol was adopted to perform all the regressions. The results empirically showed that the slopes varied significantly across diverse forested biomes, between conifer and broadleaved forests, and between evergreen and deciduous forests. Based on the results, leaf form and function and their relations to environments play a significant role in the modification of the WBE model to explore more accurate laws in nature.