Analytic theories of allometric scaling

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.

How Metabolic Rate Relates to Cell Size

Simple Summary

The metabolic conversion of resources into living structures and processes is fundamental to all living systems. The rate of metabolism (‘fire of life’) is critical for supporting the rates of various biological processes (‘pace of life’), but why it varies considerably within and among species is little understood. Much of this variation is related to body size, but such ‘metabolic scaling’ relationships also vary extensively. Numerous explanations have been offered, but no consensus has yet been reached. Here, I critically review explanations concerning how cell size and number and their establishment by cell expansion and multiplication may affect metabolic rate and its scaling with body mass. Numerous lines of evidence suggest that cell size and growth can affect metabolic rate at any given body mass, as well as how it changes with increasing body mass during growth or evolution. Mechanisms causing negative associations between cell size and metabolic rate may involve reduced resource supply and/or demand in larger cells, but more research is needed. A cell-size perspective not only helps to explain some (but not all) variation in metabolic rate and its body-mass scaling, but may also foster the conceptual integration of studies of ontogenetic development and body-mass scaling.

Abstract

Metabolic rate and its covariation with body mass vary substantially within and among species in little understood ways. Here, I critically review explanations (and supporting data) concerning how cell size and number and their establishment by cell expansion and multiplication may affect metabolic rate and its scaling with body mass. Cell size and growth may affect size-specific metabolic rate, as well as the vertical elevation (metabolic level) and slope (exponent) of metabolic scaling relationships. Mechanistic causes of negative correlations between cell size and metabolic rate may involve reduced resource supply and/or demand in larger cells, related to decreased surface area per volume, larger intracellular resource-transport distances, lower metabolic costs of ionic regulation, slower cell multiplication and somatic growth, and larger intracellular deposits of metabolically inert materials in some tissues. A cell-size perspective helps to explain some (but not all) variation in metabolic rate and its body-mass scaling and thus should be included in any multi-mechanistic theory attempting to explain the full diversity of metabolic scaling. A cell-size approach may also help conceptually integrate studies of the biological regulation of cellular growth and metabolism with those concerning major transitions in ontogenetic development and associated shifts in metabolic scaling.

A quantitative genetics perspective on the body-mass scaling of metabolic rate

Widely observed allometric scaling (log-log slope<1) of metabolic rate (MR) with body mass (BM) in animals has been frequently explained using functional mechanisms, but rarely studied from the perspective of multivariate quantitative genetics. This is unfortunate, given that the additive genetic slope (bA) of the MR-BM relationship represents the orientation of the 'line of least genetic resistance' along which MR and BM may most likely evolve. Here, we calculated bA in eight species. Although most bA values were within the range of metabolic scaling exponents reported in the literature, uncertainty of each bA estimate was large (only one bA was significantly lower than 3/4 and none were significantly different from 2/3). Overall, the weighted average for bA (0.667±0.098 95% CI) is consistent with the frequent observation that metabolic scaling exponents are negatively allometric in animals (b<1). Although bA was significantly positively correlated with the phenotypic scaling exponent (bP) across the sampled species, bP was usually lower than bA, as reflected in a (non-significantly) lower weighted average for bP (0.596±0.100). This apparent discrepancy between bA and bP resulted from relatively shallow MR-BM scaling of the residuals [weighted average residual scaling exponent (be)=0.503±0.128], suggesting regression dilution (owing to measurement error and within-individual variance) causing a downward bias in bP. Our study shows how the quantification of the genetic scaling exponent informs us about potential constraints on the correlated evolution of MR and BM, and by doing so has the potential to bridge the gap between micro- and macro-evolutionary studies of scaling allometry.

Metabolic scaling: individual versus intraspecific scaling of Nile tilapia (Oreochromis niloticus)

We examined intraspecific scaling of the resting metabolic rate (RMR) of Nile tilapia (Oreochromis niloticus) under different culture conditions and further explored the allometric relationships between organ mass (heart, liver, brain, gills, viscera, and red muscles) and blood parameters (erythrocyte size and red blood cell counts) and body mass. Oreochromis niloticus were bred in individual and group cultures. The scaling exponent of the RMR in the individual cultures was b = 0.620-0.821 (n = 30) and that in the group culture was b = 0.770 [natural logarithm (ln) RMR = 0.770 ln M - 1.107 (n = 76)]. The results of the two experimental methods were similar and were not significantly different from 0.75 (3/4), as predicted by the metabolic theory of ecology. The active and inactive organs were scaled with body mass by an exponent of 0.940 and 1.012, respectively. There was no significant relationship between the blood parameters and body mass. These results suggest that the differences in the culture methods may not have affected the allometric scaling of O. niloticus metabolism. The proportion of active and inactive organs contributed to allometric changes in the metabolic rate with body mass. Red blood cells in fish are not generally representative, and cell size can only partially explain the allometric scaling of metabolism.

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.

Plant respiration: Controlled by photosynthesis or biomass?

Two simplifying hypotheses have been proposed for whole-plant respiration. One links respiration to photosynthesis; the other to biomass. Using a first-principles carbon balance model with a prescribed live woody biomass turnover, applied at a forest research site where multidecadal measurements are available for comparison, we show that if turnover is fast the accumulation of respiring biomass is low and respiration depends primarily on photosynthesis; while if turnover is slow the accumulation of respiring biomass is high and respiration depends primarily on biomass. But the first scenario is inconsistent with evidence for substantial carry-over of fixed carbon between years, while the second implies far too great an increase in respiration during stand development-leading to depleted carbohydrate reserves and an unrealistically high mortality risk. These two mutually incompatible hypotheses are thus both incorrect. Respiration is not linearly related either to photosynthesis or to biomass, but it is more strongly controlled by recent photosynthates (and reserve availability) than by total biomass.

Theoretical considerations in determining allometric growth within instars of crustaceans, with special reference to Americamysis bahia

Allometry in crustaceans is typically considered growth over several instars primarily because crustaceans are presumed to grow only during ecdysis (discontinuous growth). Using theoretical distributions of the sizes of two morphometric variables over several instars, four theoretical instar allometry models are postulated: continuous allometry (indiscrete and discrete); discontinuous allometry (indiscrete and discrete); mixed allometry (simple or complex); and two-rate continuous allometry. The estimates of proportions of allometry within the instars are determined using Y = f(X) and X = f(Y) for variables X and Y. The amount of allometry in each variable is estimated using the mean ± standard deviation on the independent variable. Application of these theoretical instar allometry models using carapace and abdomen sizes in six instars indicates Americamysis bahia experiences two-rate continuous allometry, rather than "traditional" discontinuous allometry, with 85% or more of total growth occurring in the intermolt phase, and with the abdomen accounting for about 60% of the expansion.

Examination of the Effects of Curvature in Geometrical Space on Accuracy of Scaling Derived Projections of Plant Biomass Units: Applications to the Assessment of Average Leaf Biomass in Eelgrass Shoots

Conservation of eelgrass relies on transplants and evaluation of success depends on nondestructive measurements of average leaf biomass in shoots among other variables. Allometric proxies offer a convenient way to assessments. Identifying surrogates via log transformation and linear regression can set biased results. Views conceive this approach to be meaningful, asserting that curvature in geometrical space explains bias. Inappropriateness of correction factor of retransformation bias could also explain inconsistencies. Accounting for nonlinearity of the log transformed response relied on a generalized allometric model. Scaling parameters depend continuously on the descriptor. Joining correction factor is conceived as the partial sum of series expansion of mean retransformed residuals leading to highest reproducibility strength. Fits of particular characterizations of the generalized curvature model conveyed outstanding reproducibility of average eelgrass leaf biomass in shoots. Although nonlinear heteroscedastic regression resulted also to be suitable, only log transformation approaches can unmask a size related differentiation in growth form of the leaf. Generally, whenever structure of regression error is undetermined, choosing a suitable form of retransformation correction factor becomes elusive. Compared to customary nonparametric characterizations of this correction factor, present form proved more efficient. We expect that offered generalized allometric model along with proposed correction factor form provides a suitable analytical arrangement for the general settings of allometric examination.

Differentiating causality and correlation in allometric scaling: ant colony size drives metabolic hypometry

Metabolic rates of individual animals and social insect colonies generally scale hypometrically, with mass-specific metabolic rates decreasing with increasing size. Although this allometry has wide ranging effects on social behaviour, ecology and evolution, its causes remain controversial. Because it is difficult to experimentally manipulate body size of organisms, most studies of metabolic scaling depend on correlative data, limiting their ability to determine causation. To overcome this limitation, we experimentally reduced the size of harvester ant colonies (Pogonomyrmex californicus) and quantified the consequent increase in mass-specific metabolic rates. Our results clearly demonstrate a causal relationship between colony size and hypometric changes in metabolic rate that could not be explained by changes in physical density. These findings provide evidence against prominent models arguing that the hypometric scaling of metabolic rate is primarily driven by constraints on resource delivery or surface area/volume ratios, because colonies were provided with excess food and colony size does not affect individual oxygen or nutrient transport. We found that larger colonies had lower median walking speeds and relatively more stationary ants and including walking speed as a variable in the mass-scaling allometry greatly reduced the amount of residual variation in the model, reinforcing the role of behaviour in metabolic allometry. Following the experimental size reduction, however, the proportion of stationary ants increased, demonstrating that variation in locomotory activity cannot solely explain hypometric scaling of metabolic rates in these colonies. Based on prior studies of this species, the increase in metabolic rate in size-reduced colonies could be due to increased anabolic processes associated with brood care and colony growth.

Modeling of the metabolic energy dissipation for restricted tumor growth

Energy dissipation mostly represents unwanted outcome but in the biochemical processes it may alter the biochemical pathways. However, it is rarely considered in the literature although energy dissipation and its alteration due to the changes in cell microenvironment may improve methods for guiding chemical and biochemical processes in the desired directions. Deeper insight into the changes of metabolic activity of tumor cells exposed to osmotic stress or irradiation may offer the possibility of tumor growth reduction. In this work effects of the osmotic stress and irradiation on the thermodynamical affinity of tumor cells and their damping effects on metabolic energy dissipation were investigated and modeled. Although many various models were applied to consider the tumor restrictive growth they have not considered the metabolic energy dissipation. In this work a pseudo rheological model in the form of "the metabolic spring-pot element" is formulated to describe theoretically the metabolic susceptibility of tumor spheroid. This analog model relates the thermodynamical affinity of cell growth with the volume expansion of tumor spheroid under isotropic loading conditions. Spheroid relaxation induces anomalous nature of the metabolic energy dissipation which causes the damping effects on cell growth. The proposed model can be used for determining the metabolic energy "structure" in the context of restrictive cell growth as well as for predicting optimal doses for cancer curing in order to tailor the clinical treatment for each person and each type of cancer.

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.

Biomarkers of chondriome topology and function: implications for the extension of healthy aging

Multiple theories of aging (e.g., free radical, error catastrophe, mitochondrial) are complementary but fail to provide adequate models that comprehensively predict lifelong aging processes and that are valid across species. Hayflick (PLoS Genet 3(12):2351-2354, 2007) described six universal characteristics of aging that focus upon post-reproductive molecular entropy. Here we present a thermodynamic potential model of aging in which the energetic and topological properties of the mitochondrion drive functional and structural stabilities within living systems. Using multivariate regressions of physiological assessments from the National Health and Nutrition Examination Survey, VO₂ max consistently declined with age regardless of gender or race, although it had a significantly greater decline for African American females. Percent fat (negative), hematocrit (negative), and urine creatinine (negative) were strongly and significantly associated with VO₂ max and male aging, although cholesterol (positive) was an additional factor for African American males. Bioenergetic measures such as VO₂ max can be useful for physical assessments to promote healthy aging.

Interspecific Differences in Metabolic Rate and Metabolic Temperature Sensitivity Create Distinct Thermal Ecological Niches in Lizards (Plestiodon)

Three congeneric lizards from the southeastern United States (Plestiodon fasciatus, P. inexpectatus, and P. laticeps) exhibit a unique nested distribution. All three skink species inhabit the US Southeast, but two extend northward to central Ohio (P. fasciatus and P. laticeps) and P. fasciatus extends well into Canada. Distinct interspecific differences in microhabitat selection and behavior are associated with the cooler temperatures of the more Northern ranges. We hypothesized that interspecific differences in metabolic temperature sensitivity locally segregates them across their total range. Resting oxygen consumption was measured at 20°, 25° and 30°C. Plestiodon fasciatus, from the coolest habitats, exhibited greatly elevated oxygen consumption compared to the other species at high ecologically-relevant temperatures (0.10, 0.17 and 0.83 ml O₂^. g^-1. h^-1 at 20°, 25° and 30°C, respectively). Yet, P. inexpectatus, from the warmest habitats, exhibited sharply decreased oxygen consumption compared to the other species at lower ecologically-relevant temperatures (0.09, 0.27 and 0.42 ml O₂^. g^-1. h^-1 at 20°, 25° and 30°C, respectively). Plestiodon laticeps, from both open and closed microhabitats and intermediate latitudinal range, exhibited oxygen consumptions significantly lower than the other two species (0.057, 0.104 and 0.172 ml O₂^. g^-1. h^-1 at 20°, 25° and 30°C, respectively). Overall, Plestiodon showed metabolic temperature sensitivities (Q₁₀s) in the range of 2–3 over the middle of each species’ normal temperature range. However, especially P. fasciatus and P. inexpectatus showed highly elevated Q₁₀s (9 to 25) at the extreme ends of their temperature range. While morphologically similar, these skinks are metabolically distinct across the genus’ habitat, likely having contributed to their current distribution.

Olive and grape seed extract prevents post-traumatic osteoarthritis damages and exhibits in vitro anti IL-1β activities before and after oral consumption

Polyphenols exert a large range of beneficial effects in the prevention of age-related diseases. We sought to determine whether an extract of olive and grape seed standardized according to hydroxytyrosol (HT) and procyanidins (PCy) content, exerts preventive anti-osteoathritic effects. To this aim, we evaluated whether the HT/PCy mix could (i) have in vitro anti-inflammatory and chondroprotective actions, (ii) exert anti-osteoarthritis effects in two post-traumatic animal models and (iii) retain its bioactivity after oral administration. Anti-inflammatory and chondroprotective actions of HT/PCy were tested on primary cultured rabbit chondrocytes stimulated by interleukin-1 beta (IL-1β). The results showed that HT/PCy exerts anti-inflammatory and chondroprotective actions in vitro. The preventive effect of HT/PCy association was assessed in two animal models of post-traumatic OA in mice and rabbits. Diet supplementation with HT/PCy significantly decreased the severity of post-traumatic osteoarthritis in two complementary mice and rabbit models. The bioavailability and bioactivity was evaluated following gavage with HT/PCy in rabbits. Regular metabolites from HT/PCy extract were found in sera from rabbits following oral intake. Finally, sera from rabbits force-fed with HT/PCy conserved anti-IL-1β effect, suggesting the bioactivity of this extract. To conclude, HT/PCy extract may be of clinical significance for the preventive treatment of osteoarthritis.

A model for allometric scaling of mammalian metabolism with ambient heat loss

Background

Allometric scaling, which represents the dependence of biological traits or processes on body size, is a long-standing subject in biological science. However, there has been no study to consider heat loss to the ambient and an insulation layer representing mammalian skin and fur for the derivation of the scaling law of metabolism.

Methods

A simple heat transfer model is proposed to analyze the allometry of mammalian metabolism. The present model extends existing studies by incorporating various external heat transfer parameters and additional insulation layers. The model equations were solved numerically and by an analytic heat balance approach.

Results

A general observation is that the present heat transfer model predicted the 2/3 surface scaling law, which is primarily attributed to the dependence of the surface area on the body mass. External heat transfer effects introduced deviations in the scaling law, mainly due to natural convection heat transfer, which becomes more prominent at smaller mass. These deviations resulted in a slight modification of the scaling exponent to a value < 2/3.

Conclusion

The finding that additional radiative heat loss and the consideration of an outer insulation fur layer attenuate these deviation effects and render the scaling law closer to 2/3 provides in silico evidence for a functional impact of heat transfer mode on the allometric scaling law in mammalian metabolism.

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

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

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

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

Modeling for critically ill patients: An introduction for beginners

Models are mathematical tools used to describe real-world features. Therapeutic interventions in the field of critical care medicine may easily be translated into such models. Indeed, numerous variables influencing drug pharmacokinetics and pharmacodynamics are systematically documented in the intensive care unit over time. Organ failure, fluid shifts, other drug administration, and renal replacement therapy may cause changes in physiological values, such as body weight and composition, temperature, serum protein levels, arterial pH, and renal or hepatic function. Trials assessing the efficacy and safety of novel drugs usually exclude critically ill patients, and guidelines regarding drug dosage rarely apply to such patients. Modeling in the critically ill may allow physicians to inform decisions related to therapeutic interventions, particularly relating to infectious diseases. However, few clinicians are familiar with these methods. Here, we present a current overview of population pharmacokinetic and pharmacodynamic models applicable in critically ill patients aimed at nonspecialists and then emphazize recent potential of modeling for optimizing treatments and care in the intensive care unit.

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.

A new simplified allometric approach for predicting the biological half-life of radionuclides in reptiles

A major source of uncertainty in the estimation of radiation dose to wildlife is the prediction of internal radionuclide activity concentrations. Allometric (mass-dependent) relationships describing biological half-life (T1/2b) of radionuclides in organisms can be used to predict organism activity concentrations. The establishment of allometric expressions requires experimental data which are often lacking. An approach to predict the T1/2b in homeothermic vertebrates has recently been proposed. In this paper we have adapted this to be applicable to reptiles. For Cs, Ra and Sr, over a mass range of 0.02-1.5 kg, resultant predictions were generally within a factor of 6 of reported values demonstrating that the approach can be used when measured T1/2b data are lacking. However, the effect of mass on reptilian radionuclide T1/2b is minimal. If sufficient measured data are available for a given radionuclide then it is likely that these would give a reasonable estimate of T1/2b in any reptile species.

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.

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.

A conceptual framework for organismal biology: linking theories, models, and data

Implicit or subconscious theory is especially common in the biological sciences. Yet, theory plays a variety of roles in scientific inquiry. First and foremost, it determines what does and does not count as a valid or interesting question or line of inquiry. Second, theory determines the background assumptions within which inquiries are pursued. Third, theory provides linkages among disciplines. For these reasons, it is important and useful to develop explicit theories for biology. A general theory of organisms is developed, which includes 10 fundamental principles that apply to all organisms, and 6 that apply to multicellular organisms only. The value of a general theory comes from its utility to help guide the development of more specific theories and models. That process is demonstrated by examining two domains: ecoimmunology and development. For the former, a constitutive theory of ecoimmunology is presented, and used to develop a specific model that explains energetic trade-offs that may result from an immunological response of a host to a pathogen. For the latter, some of the issues involved in trying to devise a constitutive theory that covers all of development are explored, and a more narrow theory of phenotypic novelty is presented. By its very nature, little of a theory of organisms will be new. Rather, the theory presented here is a formal expression of nearly two centuries of conceptual advances and practice in research. Any theory is dynamic and subject to debate and change. Such debate will occur as part of the present, initial formulation, as the ideas presented here are refined. The very process of debating the form of the theory acts to clarify thinking. The overarching goal is to stimulate debate about the role of theory in the study of organisms, and thereby advance our understanding of them.

Intraspecific mass scaling of metabolic rates in grass carp (Ctenopharyngodon idellus)

We assessed the intraspecific mass scaling of standard metabolic rate (SMR), maximum metabolic rate (MMR), excess post-exercise oxygen consumption (EPOC), and erythrocyte size in grass carp (Ctenopharyngodon idellus), with body masses ranging from 4.0 to 459 g. SMR and MMR scaled with body mass with similar exponents, but neither exponent matched the expected value of 0.75 or 1, respectively. Erythrocyte size scaled with body mass with a very low exponent (0.090), suggests that while both cell number and cell size contribute to the increase in body mass, cell size plays a smaller role. The similar slopes of MMR and SMR in grass carp suggest a constant factorial aerobic scope (FAS) as the body grows. SMR was negatively correlated with FAS, indicating a tradeoff between SMR and FAS. Smaller fish recovered faster from the exhaustive exercises, and the scaling exponent of EPOC was 1.075, suggesting a nearly isometric increase in anaerobic capacity. Our results provide support for the cell size model and suggest that variations of erythrocyte size may partly contribute to the intraspecific scaling of SMR. The scaling exponent of MMR was 0.863, suggesting that the metabolism of non-athletic fish species is less reliant on muscular energy expenditure, even during strenuous exercise.

Intraspecific scaling of the resting and maximum metabolic rates of the crucian carp (Carassius auratus)

The question of how the scaling of metabolic rate with body mass (M) is achieved in animals is unresolved. Here, we tested the cell metabolism hypothesis and the organ size hypothesis by assessing the mass scaling of the resting metabolic rate (RMR), maximum metabolic rate (MMR), erythrocyte size, and the masses of metabolically active organs in the crucian carp (Carassius auratus). The M of the crucian carp ranged from 4.5 to 323.9 g, representing an approximately 72-fold difference. The RMR and MMR increased with M according to the allometric equations RMR = 0.212M (0.776) and MMR = 0.753M (0.785). The scaling exponents for RMR (b r) and MMR (b m) obtained in crucian carp were close to each other. Thus, the factorial aerobic scope remained almost constant with increasing M. Although erythrocyte size was negatively correlated with both mass-specific RMR and absolute RMR adjusted to M, it and all other hematological parameters showed no significant relationship with M. These data demonstrate that the cell metabolism hypothesis does not describe metabolic scaling in the crucian carp, suggesting that erythrocyte size may not represent the general size of other cell types in this fish and the metabolic activity of cells may decrease as fish grows. The mass scaling exponents of active organs was lower than 1 while that of inactive organs was greater than 1, which suggests that the mass scaling of the RMR can be partly due to variance in the proportion of active/inactive organs in crucian carp. Furthermore, our results provide additional evidence supporting the correlation between locomotor capacity and metabolic scaling.

Exploring network scaling through variations on optimal channel networks

Metabolic allometry, a common pattern in nature, is a close to 3/4-power scaling law between metabolic rate and body mass in organisms, across and within species. An analogous relationship between metabolic rate and water volume in river networks has also been observed. Optimal channel networks (OCNs), at local optima, accurately model many scaling properties of river systems, including metabolic allometry. OCNs are embedded in 2D space; this work extends the model to three dimensions. In this paper we compare characteristics of 3D OCNs with 2D OCNs and with organic metabolic networks, studying the scaling behaviors of area, length, volume, and energy. We find that the 3D OCN has predictable characteristics analogous to those of the 2D version, as well as scaling properties similar to metabolic networks in biological organisms.

Cellular differentiation and individuality in the 'minor' multicellular taxa

Biology needs a concept of individuality in order to distinguish organisms from parts of organisms and from groups of organisms, to count individuals and compare traits across taxa, and to distinguish growth from reproduction. Most of the proposed criteria for individuality were designed for 'unitary' or 'paradigm' organisms: contiguous, functionally and physiologically integrated, obligately sexually reproducing multicellular organisms with a germ line sequestered early in development. However, the vast majority of the diversity of life on Earth does not conform to all of these criteria. We consider the issue of individuality in the 'minor' multicellular taxa, which collectively span a large portion of the eukaryotic tree of life, reviewing their general features and focusing on a model species for each group. When the criteria designed for unitary organisms are applied to other groups, they often give conflicting answers or no answer at all to the question of whether or not a given unit is an individual. Complex life cycles, intimate bacterial symbioses, aggregative development, and strange genetic features complicate the picture. The great age of some of the groups considered shows that 'intermediate' forms, those with some but not all of the traits traditionally associated with individuality, cannot reasonably be considered ephemeral or assumed transitional. We discuss a handful of recent attempts to reconcile the many proposed criteria for individuality and to provide criteria that can be applied across all the domains of life. Finally, we argue that individuality should be defined without reference to any particular taxon and that understanding the emergence of new kinds of individuals requires recognizing individuality as a matter of degree.

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

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

Relationship between fish size and metabolic rate in the oxyconforming inanga Galaxias maculatus reveals size-dependent strategies to withstand hypoxia

The relationship between metabolic rate and body size in animals is unlikely to be a constant but is instead shaped by a variety of intrinsic (i.e., physiological) and extrinsic (i.e., environmental) factors. This study examined the effect of environmental oxygen tension on oxygen consumption as a function of body mass in the galaxiid fish, inanga (Galaxias maculatus). As an oxyconformer, this fish lacks overt intrinsic regulation of oxygen consumption, eliminating this as a factor affecting the scaling relationship at different oxygen tensions. The relationship between oxygen consumption rate and body size was best described by a power function, with an exponent of 0.82, higher than the theoretical values of 0.66 or 0.75. The value of this exponent was significantly altered by environmental P(O2), first increasing as P(O2) decreased and then declining at the lowest P(O2) tested. These data suggest that the scaling exponent is species specific and regulated by extrinsic factors. Furthermore, the external P(O2) at which fish lost equilibrium was related to fish size, an effect explained by the scaling of anaerobic capacity with fish mass. Therefore, although bigger fish were forced to depress aerobic metabolism more rapidly than small fish when exposed to progressive hypoxia, they were better able to enact anaerobic metabolism, potentially extending their survival in hypoxia.

The relationship between body mass and field metabolic rate among individual birds and mammals

Summary

The authors provide the first comprehensive empirical analysis of the scaling relationship between field metabolic rate and body mass in individual birds and mammals. The analysis reveals the importance of heterogeneity in the scaling exponent, with consequences for biomass and nutrient flow through communities, and the structure and functioning of whole ecosystems.

Intraspecific variation in the metabolic scaling exponent in ectotherms: testing the effect of latitudinal cline, ontogeny and transgenerational change in the land snail Cornu aspersum

The strong dependence of metabolic rates on body mass has attracted the interest of ecological physiologists, as it has important implications to many aspects of biology including species variations in body size, the evolution of life history, and the structure and function of biological communities. The great diversity of observed scaling exponents has led some authors to conclude that there is no single universal scaling exponent, but instead it ranges from 2/3 to 1. Most of the telling evidence against the universality of power scaling exponents comes from ontogenetic changes. Nevertheless, there could be other sources of phenotypic variation that influence this allometric relationship at least at the intraspecific level. In order to explore the general concept of the metabolic scaling in terrestrial molluscs we tested the role of several biological and methodological sources of variation on the empirically estimated scaling exponent. Specifically, we measured a proxy of metabolic rate (CO(2) production) in 421 individuals, during three generations, in three different populations. Additionally, we measured this scaling relationship in 208 individuals at five developmental stages. Our results suggest that the metabolic scaling exponent at the intraspecific level does not have a single stationary value, but instead it shows some degree of variation across geographic distribution, transgenerational change and ontogenetic stages. The major differences in the metabolic scaling exponent that we found were at different developmental stages of snails, because ontogeny involves increases in size at different rates, which in turn, generate differential energy demands.

Form and metabolic scaling in colonial animals

Benthic colonial organisms exhibit a wide variation in size and shape and provide excellent model systems for testing the predictions of models that describe the scaling of metabolic rate with organism size. We test the hypothesis that colony form will influence metabolic scaling and its derivatives by characterising metabolic and propagule production rates in three species of freshwater bryozoans that vary in morphology and module organisation and which demonstrate two- and three-dimensional growth forms. The results were evaluated with respect to predictions from two models for metabolic scaling. Isometric metabolic scaling in two-dimensional colonies supported predictions of a model based on dynamic energy budget theory (DEB) and not those of a model based on fractally branching supply networks. This metabolic isometry appears to be achieved by equivalent energy budgets of edge and central modules, in one species (Cristatella mucedo) via linear growth and in a second species (Lophopus crystallinus) by colony fission. Allometric scaling characterised colonies of a three-dimensional species (Fredericella sultana), also providing support for the DEB model. Isometric scaling of propagule production rates for C. mucedo and F. sultana suggests that the number of propagules produced in colonies increases directly with the number of modules within colonies. Feeding currents generated by bryozoans function in both food capture and respiration, thus linking metabolic scaling with dynamics of self-shading and resource capture. Metabolic rates fundamentally dictate organismal performance (e.g. growth, reproduction) and, as we show here, are linked with colony form. Metabolic profiles and associated variation in colony form should therefore influence the outcome of biotic interactions in habitats dominated by colonial animals and may drive patterns of macroevolution.

Influence of cell size and DNA content on growth rate and photosystem II function in cryptic species of Ditylum brightwellii

DNA content and cell volume have both been hypothesized as controls on metabolic rate and other physiological traits. We use cultures of two cryptic species of Ditylum brightwellii (West) Grunow with an approximately two-fold difference in genome size and a small and large culture of each clone obtained by isolating small and large cells to compare the physiological consequences of size changes due to differences in DNA content and reduction in cell size following many generations of asexual reproduction. We quantified the growth rate, the functional absorption cross-section of photosystem II (PSII), susceptibility of PSII to photoinactivation, PSII repair capacity, and PSII reaction center proteins D1 (PsbA) and D2 (PsbD) for each culture at a range of irradiances. The species with the smaller genome has a higher growth rate and, when acclimated to growth-limiting irradiance, has higher PSII repair rate capacity, PSII functional optical absorption cross-section, and PsbA per unit protein, relative to the species with the larger genome. By contrast, cell division rates vary little within clonal cultures of the same species despite significant differences in average cell volume. Given the similarity in cell division rates within species, larger cells within species have a higher demand for biosynthetic reductant. As a consequence, larger cells within species have higher numbers of PSII per unit protein (PsbA), since PSII photochemically generates the reductant to support biosynthesis. These results suggest that DNA content, as opposed to cell volume, has a key role in setting the differences in maximum growth rate across diatom species of different size while PSII content and related photophysiological traits are influenced by both growth rate and cell size.

Design standards for engineered tissues

Traditional technologies are required to meet specific, quantitative standards of safety and performance. In tissue engineering, similar standards will have to be developed to enable routine clinical use and customized tissue fabrication. In this essay, we discuss a framework of concepts leading towards general design standards for tissue-engineering, focusing in particular on systematic design strategies, control of cell behavior, physiological scaling, fabrication modes and functional evaluation.

Human brain mass: similar body composition associations as observed across mammals

OBJECTIVES

A classic association is the link between brain mass and body mass across mammals that has now been shown to derive from fat-free mass (FFM) and not fat mass (FM). This study aimed to establish for the first time the associations between human brain mass and body composition and to compare these relations with those established for liver as a reference organ.

METHODS

Subjects were 112 men and 148 women who had brain and liver mass measured by magnetic resonance imaging with FM and FFM measured by dual-energy X-ray absorptiometry.

RESULTS

Brain mass scaled to height (H) with powers of ≤0.6 in men and women; liver mass and FFM both scaled similarly as H(~2) . The fraction of FFM as brain thus scaled inversely to height (P < 0.001) while liver mass/FFM was independent of height. After controlling for age, brain, and liver mass were associated with FFM while liver was additionally associated with FM (all models P ≤ 0.01). After controlling for age and sex, FFM accounted for ~5% of the variance in brain mass while levels were substantially higher for liver mass (~60%). Brain mass was significantly larger (P < 0.001) in men than in women, even after controlling for age and FFM.

CONCLUSIONS

As across mammals, human brain mass associates significantly, although weakly, with FFM and not FM; the fraction of FFM as brain relates inversely to height; brain differs in these relations from liver, another small high metabolic rate organ; and the sexual dimorphism in brain mass persists even after adjusting for age and FFM.