Allometric scaling laws of metabolism

Selectively advantageous instability in biotic and pre-biotic systems and implications for evolution and aging

Rules of biology typically involve conservation of resources. For example, common patterns such as hexagons and logarithmic spirals require minimal materials, and scaling laws involve conservation of energy. Here a relationship with the opposite theme is discussed, which is the selectively advantageous instability (SAI) of one or more components of a replicating system, such as the cell. By increasing the complexity of the system, SAI can have benefits in addition to the generation of energy or the mobilization of building blocks. SAI involves a potential cost to the replicating system for the materials and/or energy required to create the unstable component, and in some cases, the energy required for its active degradation. SAI is well-studied in cells. Short-lived transcription and signaling factors enable a rapid response to a changing environment, and turnover is critical for replacement of damaged macromolecules. The minimal gene set for a viable cell includes proteases and a nuclease, suggesting SAI is essential for life. SAI promotes genetic diversity in several ways. Toxin/antitoxin systems promote maintenance of genes, and SAI of mitochondria facilitates uniparental transmission. By creating two distinct states, subject to different selective pressures, SAI can maintain genetic diversity. SAI of components of synthetic replicators favors replicator cycling, promoting emergence of replicators with increased complexity. Both classical and recent computer modeling of replicators reveals SAI. SAI may be involved at additional levels of biological organization. In summary, SAI promotes replicator genetic diversity and reproductive fitness, and may promote aging through loss of resources and maintenance of deleterious alleles.

The Relevance of Time in Biological Scaling

Various phenotypic traits relate to the size of a living system in regular but often disproportionate (allometric) ways. These "biological scaling" relationships have been studied by biologists for over a century, but their causes remain hotly debated. Here, I focus on the patterns and possible causes of the body-mass scaling of the rates/durations of various biological processes and life-history events, i.e., the "pace of life". Many biologists have regarded the rate of metabolism or energy use as the master driver of the "pace of life" and its scaling with body size. Although this "energy perspective" has provided valuable insight, here I argue that a "time perspective" may be equally or even more important. I evaluate various major ways that time may be relevant in biological scaling, including as (1) an independent "fourth dimension" in biological dimensional analyses, (2) a universal "biological clock" that synchronizes various biological rates/durations, (3) a scaling method that uses various biological time periods (allochrony) as scaling metrics, rather than various measures of physical size (allometry), as traditionally performed, (4) an ultimate body-size-related constraint on the rates/timing of biological processes/events that is set by the inevitability of death, and (5) a geological "deep time" approach for viewing the evolution of biological scaling patterns. Although previously proposed universal four-dimensional space-time and "biological clock" views of biological scaling are problematic, novel approaches using allochronic analyses and time perspectives based on size-related rates of individual mortality and species origination/extinction may provide new valuable insights.

Accounting for body mass effects in the estimation of field metabolic rates from body acceleration

Dynamic body acceleration (DBA), measured through animal-attached tags, has emerged as a powerful method for estimating field metabolic rates of free-ranging individuals. Following respirometry to calibrate oxygen consumption rate (ṀO2) with DBA under controlled conditions, predictive models can be applied to DBA data collected from free-ranging individuals. However, laboratory calibrations are generally performed on a relatively narrow size range of animals, which may introduce biases if predictive models are applied to differently sized individuals in the field. Here, we tested the mass dependence of the ṀO2-DBA relationship to develop an experimental framework for the estimation of field metabolic rates when organisms differ in size. We performed respirometry experiments with individuals spanning one order of magnitude in body mass (1.74-17.15 kg) and used a two-stage modelling process to assess the intraspecific scale dependence of the ṀO2-DBA relationship and incorporate such dependencies into the coefficients of ṀO2 predictive models. The final predictive model showed scale dependence; the slope of the ṀO2-DBA relationship was strongly allometric (M1.55), whereas the intercept term scaled closer to isometry (M1.08). Using bootstrapping and simulations, we evaluated the performance of this coefficient-corrected model against commonly used methods of accounting for mass effects on the ṀO2-DBA relationship and found the lowest error and bias in the coefficient-corrected approach. The strong scale dependence of the ṀO2-DBA relationship indicates that caution must be exercised when models developed using one size class are applied to individuals of different sizes.

Repeatability and heritability of resting metabolic rate in a long-lived amphibian

Resting metabolic rate (RMR), i.e. spent energy necessary to maintain basic life functions, is a basic component of energy budget in ectotherms. The evolution of RMR through natural selection rests on the premise of its non-zero repeatability and heritability, i.e. consistent variation within individual lifetimes and resemblance between parents and their offspring, respectively. Joint estimates of RMR repeatability and heritability are missing in ectotherms, however, which precludes estimations of the evolutionary potential of this trait. We examined RMR repeatability and heritability in a long-lived ectotherm, the alpine newt (Ichthyosaura alpestris). Individual RMR was repeatable over both six-month (0.28 ± 0.09 [SE]) and five-year (0.16 ± 0.07) periods. While there was no resemblance between parent and offspring RMR (0.21 ± 0.34), the trait showed similarity among offspring within families (broad-sense heritability; 0.25 ± 0.09). Similar repeatability and broad-sense heritability values in parental and offspring generations, respectively, and non-conclusive narrow-sense heritability suggest the contribution of non-additive genetic factors to total phenotypic variance in this trait. We conclude that RMR evolutionary trajectories are shaped by other processes than natural selection in this long-lived ectotherm.

Spatial variance-mass allometry of population density in felids from camera-trapping studies worldwide

Power laws are cornerstone relationships in ecology and evolutionary biology. The density-mass allometry (DMA), which predicts an allometric scaling of population abundance, and Taylor’s law (TL), which predicts a decrease in the population abundance variation along with a decrease in population density, have enhanced our knowledge of inter- and intra-specific variation in population abundance. When combined, these two power laws led to the variance-mass allometry (VMA), which states that larger species have lower spatial variation in population density than smaller species. The VMA has been predicted through theoretical models, however few studies have investigated if this law is also supported by empirical data. Here, to formally test the VMA, we have used the population density estimates obtained through worldwide camera trapping studies for an emblematic and ecologically important carnivorous taxa, the Felidae family. Our results showed that the VMA law hold in felids, as well as the TL and the DMA laws; bigger cat species showed less variation for the population density than smaller species. These results have important implications for the conservation of wildlife population and confirm the validity of important ecological concepts, like the allometric scaling of population growth rate and the slow-fast continuum of life history strategies.

Mitochondrial performance of a continually growing marine bivalve, Mytilus edulis, depends on body size

Allometric decline of mass-specific metabolic rate with increasing body size in organisms is a well-documented phenomenon. Despite a long history of research, the mechanistic causes of metabolic scaling with body size remain under debate. Some hypotheses suggest that intrinsic factors such as allometry of cellular and mitochondrial metabolism may contribute to the organismal-level metabolic scaling. The aim of our present study was to determine the metabolic allometry at the mitochondrial level using a continually growing marine ectotherm, the mussel Mytilus edulis, as a model. Mussels from a single cohort that considerably differed in body size were selected, implying faster growth in the larger specimens. We determined the body mass-dependent scaling of the mitochondrial proton leak respiration, respiration in the presence of ADP indicative of the oxidative phosphorylation (OXPHOS), and maximum activity of the mitochondrial electron transport system (ETS) and cytochrome c oxidase (COX). Respiration was measured at normal (15°C), and elevated (27°C) temperatures. The results demonstrated a pronounced allometric increase in both proton leak respiration and OXPHOS activity of mussel mitochondria. Mussels with faster growth (larger body size) showed an increase in OXPHOS rate, proton leak respiration rate, and ETS and COX activity (indicating an overall improved mitochondrial performance) and higher respiratory control ratio (indicating better mitochondrial coupling and potentially lower costs of mitochondrial maintenance at the same OXPHOS capacity) compared with slower growing (smaller) individuals. Our data show that the metabolic allometry at the organismal level cannot be directly explained by mitochondrial functioning.

The relationship of mammal survivorship and body mass modeled by metabolic and vitality theories

A model describes the relationship between mammal body mass and survivorship by combining replicative senescence theory postulating a cellular basis of aging, metabolic theory relating metabolism to body mass, and vitality theory relating survival to vitality loss and extrinsic mortality. In the combined framework, intrinsic mortality results from replicative senescence of the hematopoietic stem cells and extrinsic mortality results from environmental challenges. Because the model expresses the intrinsic and extrinsic rates with different powers of body mass, across the spectrum of mammals, survivorship changes from Type I to Type II curve shapes with decreasing body mass. Fitting the model to body mass and maximum lifespan data of 494 nonvolant mammals yields allometric relationships of body mass to the vitality parameters, from which full survivorship profiles were generated from body mass alone. Because maximum lifespan data is predominantly derived from captive populations, the generated survivorship curves were dominated by intrinsic mortality. Comparison of the mass-derived and observed survivorship curves provides insights into how specific populations deviate from the aggregate of populations observed under captivity.

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.

Weighing trees with lasers: advances, challenges and opportunities

Terrestrial laser scanning (TLS) is providing exciting new ways to quantify tree and forest structure, particularly above-ground biomass (AGB). We show how TLS can address some of the key uncertainties and limitations of current approaches to estimating AGB based on empirical allometric scaling equations (ASEs) that underpin all large-scale estimates of AGB. TLS provides extremely detailed non-destructive measurements of tree form independent of tree size and shape. We show examples of three-dimensional (3D) TLS measurements from various tropical and temperate forests and describe how the resulting TLS point clouds can be used to produce quantitative 3D models of branch and trunk size, shape and distribution. These models can drastically improve estimates of AGB, provide new, improved large-scale ASEs, and deliver insights into a range of fundamental tree properties related to structure. Large quantities of detailed measurements of individual 3D tree structure also have the potential to open new and exciting avenues of research in areas where difficulties of measurement have until now prevented statistical approaches to detecting and understanding underlying patterns of scaling, form and function. We discuss these opportunities and some of the challenges that remain to be overcome to enable wider adoption of TLS methods.

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.

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.

High-Throughput Tissue Bioenergetics Analysis Reveals Identical Metabolic Allometric Scaling for Teleost Hearts and Whole Organisms

Organismal metabolic rate, a fundamental metric in biology, demonstrates an allometric scaling relationship with body size. Fractal-like vascular distribution networks of biological systems are proposed to underlie metabolic rate allometric scaling laws from individual organisms to cells, mitochondria, and enzymes. Tissue-specific metabolic scaling is notably absent from this paradigm. In the current study, metabolic scaling relationships of hearts and brains with body size were examined by improving on a high-throughput whole-organ oxygen consumption rate (OCR) analysis method in five biomedically and environmentally relevant teleost model species. Tissue-specific metabolic scaling was compared with organismal routine metabolism (RMO₂), which was measured using whole organismal respirometry. Basal heart OCR and organismal RMO₂ scaled identically with body mass in a species-specific fashion across all five species tested. However, organismal maximum metabolic rates (MMO₂) and pharmacologically-induced maximum cardiac metabolic rates in zebrafish Danio rerio did not show a similar relationship with body mass. Brain metabolic rates did not scale with body size. The identical allometric scaling of heart and organismal metabolic rates with body size suggests that hearts, the power generator of an organism’s vascular distribution network, might be crucial in determining teleost metabolic rate scaling under routine conditions. Furthermore, these findings indicate the possibility of measuring heart OCR utilizing the high-throughput approach presented here as a proxy for organismal metabolic rate—a useful metric in characterizing organismal fitness. In addition to heart and brain OCR, the current approach was also used to measure whole liver OCR, partition cardiac mitochondrial bioenergetic parameters using pharmacological agents, and estimate heart and brain glycolytic rates. This high-throughput whole-organ bioenergetic analysis method has important applications in toxicology, evolutionary physiology, and biomedical sciences, particularly in the context of investigating pathogenesis of mitochondrial diseases.

Impacts of genetic correlation on the independent evolution of body mass and skeletal size in mammals

Background

Mammals show a predictable scaling relationship between limb bone size and body mass. This relationship has a genetic basis which likely evolved via natural selection, but it is unclear how much the genetic correlation between these traits in turn impacts their capacity to evolve independently. We selectively bred laboratory mice for increases in tibia length independent of body mass, to test the hypothesis that a genetic correlation with body mass constrains evolutionary change in tibia length.

Results

Over 14 generations, we produced mean tibia length increases of 9-13%, while mean body mass was unchanged, in selectively bred mice and random-bred controls. Using evolutionary scenarios with different selection and quantitative genetic parameters, we also found that this genetic correlation impedes the rate of evolutionary change in both traits, slowing increases in tibia length while preventing decreases in body mass, despite the latter’s negative effect on fitness.

Conclusions

Overall, results from this ongoing selection experiment suggest that parallel evolution of relatively longer hind limbs among rodents, for example in the context of strong competition for resources and niche partitioning in heterogeneous environments, may have occurred very rapidly on geological timescales, in spite of a moderately strong genetic correlation between tibia length and body mass.

Electronic supplementary material

The online version of this article (doi:10.1186/s12862-014-0258-0) contains supplementary material, which is available to authorized users.

Dimensional analysis yields the general second-order differential equation underlying many natural phenomena: the mathematical properties of a phenomenon's data plot then specify a unique differential equation for it

Background

This study uses dimensional analysis to derive the general second-order differential equation that underlies numerous physical and natural phenomena described by common mathematical functions. It eschews assumptions about empirical constants and mechanisms. It relies only on the data plot’s mathematical properties to provide the conditions and constraints needed to specify a second-order differential equation that is free of empirical constants for each phenomenon.

Results

A practical example of each function is analyzed using the general form of the underlying differential equation and the observable unique mathematical properties of each data plot, including boundary conditions. This yields a differential equation that describes the relationship among the physical variables governing the phenomenon’s behavior. Complex phenomena such as the Standard Normal Distribution, the Logistic Growth Function, and Hill Ligand binding, which are characterized by data plots of distinctly different sigmoidal character, are readily analyzed by this approach.

Conclusions

It provides an alternative, simple, unifying basis for analyzing each of these varied phenomena from a common perspective that ties them together and offers new insights into the appropriate empirical constants for describing each phenomenon.

Mass and volume growth of larval insect tracheal system within a single instar

Organisms must accommodate oxygen delivery to developing tissues as body mass increases during growth. In insects, the growth of the respiratory system has been assumed to occur only when it molts, whereas body mass and volume increase during the larval stages between molts. This decouples whole body growth from the growth of the oxygen supply system. This assumption is derived from the observation that the insect respiratory system is an invagination of the exoskeleton, which must be shed during molts for continued growth to occur. Here, we provide evidence that this assumption is incorrect. We found that the respiratory system increases substantially in both mass and volume within the last larval instar of Manduca sexta larvae, and that the growth of the respiratory system changes with diet quality, potentially as a consequence of shifting metabolic demands.

Analytic theories of allometric scaling

During the 13 years since it was first advanced, the fractal network theory (FNT), an analytic theory of allometric scaling, has been subjected to a wide range of methodological, mathematical and empirical criticisms, not all of which have been answered satisfactorily. FNT presumes a two-variable power-law relationship between metabolic rate and body mass. This assumption has been widely accepted in the past, but a growing body of evidence during the past quarter century has raised questions about its general validity. There is now a need for alternative theories of metabolic scaling that are consistent with empirical observations over a broad range of biological applications. In this article, we briefly review the limitations of FNT, examine the evidence that the two-variable power-law assumption is invalid, and outline alternative perspectives. In particular, we discuss quantum metabolism (QM), an analytic theory based on molecular-cellular processes. QM predicts the large variations in scaling exponent that are found empirically and also predicts the temperature dependence of the proportionality constant, issues that have eluded models such as FNT that are based on macroscopic and network properties of organisms.

An exploration on greenhouse gas and ammonia production by insect species suitable for animal or human consumption

Background

Greenhouse gas (GHG) production, as a cause of climate change, is considered as one of the biggest problems society is currently facing. The livestock sector is one of the large contributors of anthropogenic GHG emissions. Also, large amounts of ammonia (NH₃), leading to soil nitrification and acidification, are produced by livestock. Therefore other sources of animal protein, like edible insects, are currently being considered.

Methodology/Principal Findings

An experiment was conducted to quantify production of carbon dioxide (CO₂) and average daily gain (ADG) as a measure of feed conversion efficiency, and to quantify the production of the greenhouse gases methane (CH₄) and nitrous oxide (N₂O) as well as NH₃ by five insect species of which the first three are considered edible: Tenebrio molitor, Acheta domesticus, Locusta migratoria, Pachnoda marginata, and Blaptica dubia. Large differences were found among the species regarding their production of CO₂ and GHGs. The insects in this study had a higher relative growth rate and emitted comparable or lower amounts of GHG than described in literature for pigs and much lower amounts of GHG than cattle. The same was true for CO₂ production per kg of metabolic weight and per kg of mass gain. Furthermore, also the production of NH₃ by insects was lower than for conventional livestock.

Conclusions/Significance

This study therefore indicates that insects could serve as a more environmentally friendly alternative for the production of animal protein with respect to GHG and NH₃ emissions. The results of this study can be used as basic information to compare the production of insects with conventional livestock by means of a life cycle analysis.

The intraspecific scaling of metabolic rate with body mass in fishes depends on lifestyle and temperature

Metabolic energy fuels all biological processes, and therefore theories that explain the scaling of metabolic rate with body mass potentially have great predictive power in ecology. A new model, that could improve this predictive power, postulates that the metabolic scaling exponent (b) varies between 2/3 and 1, and is inversely related to the elevation of the intraspecific scaling relationship (metabolic level, L), which in turn varies systematically among species in response to various ecological factors. We test these predictions by examining the effects of lifestyle, swimming mode and temperature on intraspecific scaling of resting metabolic rate among 89 species of teleost fish. As predicted, b decreased as L increased with temperature, and with shifts in lifestyle from bathyal and benthic to benthopelagic to pelagic. This effect of lifestyle on b may be related to varying amounts of energetically expensive tissues associated with different capacities for swimming during predator-prey interactions.

The terrestrial evolution of metabolism and life - by the numbers

Background

Allometric scaling relating body mass to metabolic rate by an exponent of the former (Kleiber's Law), commonly known as quarter-power scaling (QPS), is controversial for claims made on its behalf, especially that of its universality for all life. As originally formulated, Kleiber was based upon the study of heat; metabolic rate is quantified in watts (or calories per unit time). Techniques and technology for metabolic energy measurement have been refined but the math has not. QPS is susceptible to increasing deviations from theoretical predictions to data, suggesting that there is no single, universal exponent relevant to all of life. QPS's major proponents continue to fail to make good on hints of the power of the equation for understanding aging.

Essentialist-deductivist view

If the equation includes a term for efficiency in the exponent, thereby ruling out thermogenesis as part of metabolism, its heuristic power is greatly amplified, and testable deductive inferences are generated. If metabolic rate is measured in watts and metabolic efficiency is a redox-coupling ratio, then the equation is essentially about the energy storage capacity of organic molecules. The equation is entirely about the essentials of all life: water, salt, organic molecules, and energy. The water and salt provide an electrochemical salt bridge for the transmission of energy into and through the organic components. The equation, when graphed, treats the organic structure as battery-like, and relates its recharge rate and electrical properties to its longevity.

Conclusion

The equation models the longevity-extending effects of caloric restriction, and shows where those effects wane. It models the immortality of some types of cells, and supports the argument for the origin of life being at submarine volcanic vents and black smokers. It clarifies how early life had to change to survive drifting to the surface, and what drove mutations in its ascent. It does not deal with cause and effect; it deals with variables in the essentials of all life, and treats life as an epiphenomenon of those variables. The equation describes how battery discharge into the body can increase muscle mass, promote fitness, and extend life span, among other issues.

Fasting heat production and energy cost of standing activity in veal calves

Metabolic body size of veal calves is still calculated by using the 0·75 exponent and no data were available to determine energy cost of physical activity during the whole fattening period. Data from two trials focusing on protein and/or energy requirements were used to determine the coefficient of metabolic body size and the energy cost of standing activity in male Prim'Holstein calves. Total heat production was measured by indirect calorimetry in ninety-five calves weighing 60–265 kg and was divided using a modelling approach between components related to the BMR, physical activity and feed intake. The calculation of the energy cost of standing activity was based on quantifying the physical activity by using force sensors on which the metabolism cage was placed and on the interruption of an IR beam allowing the determination of standing or lying position of the calf. The best exponent relating zero activity fasting heat production (FHP0) to metabolic body size was 0·85, which differed significantly from the traditionally used 0·75. Per additional kJ metabolizable energy (ME) intake, FHP0 increased by 0·28 kJ; at a conventional daily 650 kJ/kg body weight (BW)0·85 ME intake, daily FHP0 averaged 310 kJ/kg BW0·85. Calves stood up sixteen times per day; total duration of standing increased from 5·1 to 6·4 h per day as animals became older. The hourly energy cost of standing activity was proportional to BW0·65 and was estimated as 12·4 kJ/kg BW0·65. These estimates allow for a better estimation of the maintenance energy requirements in veal calves.

Traditional allometric analysis fails to provide a valid predictive model for mammalian metabolic rates

The field of biological allometry was energized by the publication in 1997 of a theoretical model purporting to explain 3/4-power scaling of metabolic rate with body mass in mammals. This 3/4-power scaling exponent, which was first reported by Max Kleiber in 1932, has been derived repeatedly in empirical research by independent investigators and has come to be known as`Kleiber's Law'. The exponent was estimated in virtually every instance,however, by fitting a straight line to logarithmic transformations of data and by then re-expressing the resulting equation in the arithmetic scale. Because this traditional method may yield inaccurate and misleading estimates for parameters in the allometric equation, we re-examined the comprehensive data set that led Savage and colleagues to reaffirm the view that the metabolic rate of mammals scales to the 3/4-power of body mass. We found that a straight line fitted to logged data for the basal metabolic rate (BMR) of mammals ranging in size from a 2.4 g shrew to a 3672 kg elephant does not satisfy assumptions underlying the analysis and that the allometric equation obtained by back-transformation underestimates BMR for the largest species in the sample. Thus, the concept of 3/4-power scaling of metabolic rate to body mass is not well supported because the underlying statistical model does not apply to mammalian species spanning the full range in body size. Our findings have important implications with respect to methods and results of other studies that used the traditional approach to allometric analysis.