Multi-scaling allometry in human development, mammalian morphology, and tree growth

Various animal and plant species exhibit allometric relationships among their respective traits, wherein one trait undergoes expansion as a power-law function of another due to constraints acting on growth processes. For instance, the acknowledged consensus posits that tree height scales with the two-thirds power of stem diameter. In the context of human development, it is posited that body weight scales with the second power of height. This prevalent allometric relationship derives its nomenclature from fitting two variables linearly within a logarithmic framework, thus giving rise to the term "power-law relationship." Here, we challenge the conventional assumption that a singular power-law equation adequately encapsulates the allometric relationship between any two traits. We strategically leverage quantile regression analysis to demonstrate that the scaling exponent characterizing this power-law relationship is contingent upon the centile within these traits' distributions. This observation fundamentally underscores the proposition that individuals occupying disparate segments of the distribution may employ distinct growth strategies, as indicated by distinct power-law exponents. We introduce the innovative concept of "multi-scale allometry" to encapsulate this newfound insight. Through a comprehensive reevaluation of (i) the height-weight relationship within a cohort comprising 7, 863, 520 Japanese children aged 5-17 years for which the age, sex, height, and weight were recorded as part of a national study, (ii) the stem-diameter-height and crown-radius-height relationships within an expansive sample of 498, 838 georeferenced and taxonomically standardized records of individual trees spanning diverse geographical locations, and (iii) the brain-size-body-size relationship within an extensive dataset encompassing 1, 552 mammalian species, we resolutely substantiate the viability of multi-scale allometric analysis. This empirical substantiation advocates a paradigm shift from uni-scaling to multi-scaling allometric modeling, thereby affording greater prominence to the inherent growth processes that underlie the morphological diversity evident throughout the living world.

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

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

Interactive effects of intrinsic and extrinsic factors on metabolic rate

Metabolism energizes all biological processes, and its tempo may importantly influence the ecological success and evolutionary fitness of organisms. Therefore, understanding the broad variation in metabolic rate that exists across the living world is a fundamental challenge in biology. To further the development of a more reliable and holistic picture of the causes of this variation, we review several examples of how various intrinsic (biological) and extrinsic (environmental) factors (including body size, cell size, activity level, temperature, predation and other diverse genetic, cellular, morphological, physiological, behavioural and ecological influences) can interactively affect metabolic rate in synergistic or antagonistic ways. Most of the interactive effects that have been documented involve body size, temperature or both, but future research may reveal additional 'hub factors'. Our review highlights the complex, intimate inter-relationships between physiology and ecology, knowledge of which can shed light on various problems in both disciplines, including variation in physiological adaptations, life histories, ecological niches and various organism-environment interactions in ecosystems. We also discuss theoretical and practical implications of interactive effects on metabolic rate and provide suggestions for future research, including holistic system analyses at various hierarchical levels of organization that focus on interactive proximate (functional) and ultimate (evolutionary) causal networks. This article is part of the theme issue 'The evolutionary significance of variation in metabolic rates'.

Respiration kinetics and allometric scaling in the demosponge Halichondria panicea

BACKGROUND

The aquiferous system in sponges represents one of the simplest circulatory systems used by animals for the internal uptake and distribution of oxygen and metabolic substrates. Its modular organization enables sponges to metabolically scale with size differently than animals with an internal circulatory system. In this case, metabolic rate is typically limited by surface to volume constraints to maintain an efficient supply of oxygen and food. Here, we consider the linkeage between oxygen concentration, the respiration rates of sponges and sponge size.

RESULTS

We explored respiration kinetics for individuals of the demosponge Halichondria panicea with varying numbers of aquiferous modules (nmodules = 1-102). From this work we establish relationships between the sponge size, module number, maximum respiration rate (Rmax) and the half-saturation constant, Km, which is the oxygen concentration producing half of the maximum respiration rate, Rmax. We found that the nmodules in H. panicea scales consistently with sponge volume (Vsp) and that Rmax increased with sponge size with a proportionality > 1. Conversly, we found a lack of correlation between Km and sponge body size suggesting that oxygen concentration does not control the size of sponges.

CONCLUSIONS

The present study reveals that the addition of aquiferous modules (with a mean volume of 1.59 ± 0.22 mL) enables H. panicea in particular, and likely demosponges in general, to grow far beyond constraints limiting the size of their component modules and independent of ambient oxygen levels.

Metabolic size scaling reflects growth performance effects on age-size relationships in mussels (Mytilus galloprovincialis)

Body-size scaling of metabolic rate in animals is typically allometric, with mass exponents that vary to reflect differences in the physiological status of organisms of both endogenous and environmental origin. Regarding the intraspecific analysis of this relationship in bivalve molluscs, one important source of metabolic variation comes from the large inter-individual differences in growth performance characteristic of this group. In the present study, we aimed to address the association of growth rate differences recorded among individual mussels (Mytilus galloprovincialis) with variable levels of the standard metabolic rate (SMR) resulting in growth-dependent shift in size scaling relationships. SMR was measured in mussels of different sizes and allometric functions fitting SMR vs. body-mass relationships were compared both inter- and intra-individually. The results revealed a metabolic component (the overhead of growth) attributable to the differential costs of maintenance of feeding and digestion structures between fast and slow growers; these costs were estimated to amount to a 3% increase in SMR per unit of increment in the weight specific growth rate. Scaling exponents computed for intraindividual SMR vs body-mass relationships had a common value b = 0.79 (~ ¾); however, when metabolic effects caused by differential growth were discounted, this value declined to 0.67 (= ⅔), characteristic of surface dependent processes. This last value of the scaling exponent was also recorded for the interindividual relationships of both standard and routine metabolic rates (SMR and RMR) after long-lasting maintenance of mussels under optimal uniform conditions in the laboratory. The above results were interpreted based on the metabolic level boundaries (MLB) hypothesis.

How Metabolic Rate Relates to Cell Size

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.

Growth and Mortality as Causes of Variation in Metabolic Scaling Among Taxa and Taxonomic Levels

Metabolic rate (MR) usually changes (scales) out of proportion to body mass (BM) as MR = aBMb, where a is a normalisation constant and b is the scaling exponent that reflects how steep this change is. This scaling relationship is fundamental to biology, but over a century of research has provided little consensus on the value of b, and why it appears to vary among taxa and taxonomic levels. By analysing published data on fish and taking an individual-based approach to metabolic scaling, I show that variation in growth of fish under naturally restricted food availability can explain variation in within-individual (ontogenetic) b for standard (maintenance) metabolic rate (SMR) of brown trout (Salmo trutta), with the fastest growers having the steepest metabolic scaling (b ≈ 1). Moreover, I show that within-individual b can vary much more widely than previously assumed from work on different individuals or different species, from -1 to 1 for SMR among individual brown trout. The negative scaling of SMR for some individuals was caused by reductions in metabolic rate in a food limited environment, likely to maintain positive growth. This resulted in a mean within-individual b for SMR that was significantly lower than the across-individual ("static") b, a difference that also existed for another species, cunner (Tautogolabrus adspersus). Interestingly, the wide variation in ontogenetic b for SMR among individual brown trout did not exist for maximum (active) metabolic rate (MMR) of the same fish, showing that these two key metabolic traits (SMR and MMR) can scale independently of one another. I also show that across-species ("evolutionary") b for SMR of 134 fishes is significantly steeper (b approaching 1) than the mean ontogenetic b for the brown trout and cunner. Based on these interesting findings, I hypothesise that evolutionary and static metabolic scaling can be systematically different from ontogenetic scaling, and that the steeper evolutionary than ontogenetic scaling for fishes arises as a by-product of natural selection for fast-growing individuals with steep metabolic scaling (b ≈ 1) early in life, where size-selective mortality is high for fishes. I support this by showing that b for SMR tends to increase with natural mortality rates of fish larvae within taxa.

Actual and Model-Predicted Growth of Sponges—With a Bioenergetic Comparison to Other Filter-Feeders

Sponges are one of the earliest-evolved and simplest groups of animals, but they share basic characteristics with more advanced and later-evolved filter-feeding invertebrates, such as mussels. Sponges are abundant in many coastal regions where they filter large amounts of water for food particles and thus play an important ecological role. Therefore, a better understanding of the bioenergetics and growth of sponges compared to other filter-feeders is important. While the filtration (pumping) rates of many sponge species have been measured as a function of their size, little is known about their rate of growth. Here, we use a bioenergetic growth model for demosponges, based on the energy budget and observations of filtration (F) and respiration rates (R). Because F versus dry weight (W) can be expressed as F = a1Wb1 and the maintenance respiratory rate can be expressed as Rm = a2Wb2, we show that if b1~ b2the growth rate can be expressed as: G = aWb1, and, consequently, the weight-specific growth rate is µ = G/W = aWb1−1 = aWb where the constant a depends on ambient sponge-available food particles (free-living bacteria and phytoplankton with diameter < ostia diameter). Because the exponent b1 is close to 1, then b ~ 0, which implies µ = a and thus exponential growth as confirmed in field growth studies. Exponential growth in sponges and in at least some bryozoans is probably unique among filter-feeding invertebrates. Finally, we show that the F/R-ratio and the derived oxygen extraction efficiency in these sponges are similar to other filter-feeding invertebrates, thus reflecting a comparable adaptation to feeding on a thin suspension of bacteria and phytoplankton.

Biological scaling analyses are more than statistical line fitting

The magnitude of many biological traits relates strongly and regularly to body size. Consequently, a major goal of comparative biology is to understand and apply these 'size-scaling' relationships, traditionally quantified by using linear regression analyses based on log-transformed data. However, recently some investigators have questioned this traditional method, arguing that linear or non-linear regression based on untransformed arithmetic data may provide better statistical fits than log-linear analyses. Furthermore, they advocate the replacement of the traditional method by alternative specific methods on a case-by-case basis, based simply on best-fit criteria. Here, I argue that the use of logarithms in scaling analyses presents multiple valuable advantages, both statistical and conceptual. Most importantly, log-transformation allows biologically meaningful, properly scaled (scale-independent) comparisons of organisms of different size, whereas non-scaled (scale-dependent) analyses based on untransformed arithmetic data do not. Additionally, log-based analyses can readily reveal biologically and theoretically relevant discontinuities in scale invariance during developmental or evolutionary increases in body size that are not shown by linear or non-linear arithmetic analyses. In this way, log-transformation advances our understanding of biological scaling conceptually, not just statistically. I hope that my Commentary helps students, non-specialists and other interested readers to understand the general benefits of using log-transformed data in size-scaling analyses, and stimulates advocates of arithmetic analyses to show how they may improve our understanding of scaling conceptually, not just statistically.

Growth and metabolic performance of black soldier fly larvae grown on low and high-quality substrates

We have measured growth and respiration in black soldier fly (BSF) larvae fed with mixtures of a low-quality substrate (degassed sludge, DS), and a high-quality substrate (chicken feed, CF) in order to elucidate how substrate quality affect larval metabolism and feed conversion into new biomass. The BSF larvae grew faster and became larger the higher the content of CF was in the substrate. Growth followed a sigmoidal curve from where the specific growth rate was determined and compared to measured specific respiration rates, in order to estimate costs of growth, maintenance metabolism, rates of feed assimilation, and net growth efficiency. Specific feed assimilation rates were similar on all substrate mixtures. Maximal specific growth rates were also affected only little unless the larvae were grown in pure DS. In contrast, the cost of growth and the maintenance metabolism were larger the higher the proportion of DS was. High specific growth rates were, in addition, sustained for shorter periods of the time the more DS was included in the substrate mixtures. In effect, higher proportions of the assimilated feed were spent on respiratory purposes instead of being converted into larval biomass and the net growth efficiency decreased the more DS was included in the substrate mixtures. We conclude that substrate quality may affect the conversion of feed into new biomass via alterations of the metabolic performance of BSF larvae and thereby the overall performance of BSF larval cultures.

Temperature effects on metabolic scaling of a keystone freshwater crustacean depend on fish-predation regime

According to the metabolic theory of ecology, metabolic rate, an important indicator of the pace of life, varies with body mass and temperature as a result of internal physical constraints. However, various ecological factors may also affect metabolic rate and its scaling with body mass. Although reports of such effects on metabolic scaling usually focus on single factors, the possibility of significant interactive effects between multiple factors requires further study. In this study, we show that the effect of temperature on the ontogenetic scaling of resting metabolic rate of the freshwater amphipod Gammarus minus depends critically on habitat differences in predation regime. Increasing temperature tends to cause decreases in the metabolic scaling exponent (slope) in population samples from springs with fish predators, but increases in population samples from springs without fish. Accordingly, the temperature sensitivity of metabolic rate is not only size-specific, but also its relationship to body size shifts dramatically in response to fish predators. We hypothesize that the dampened effect of temperature on the metabolic rate of large adults in springs with fish, and of small juveniles in springs without fish are adaptive evolutionary responses to differences in the relative mortality risk of adults and juveniles in springs with versus without fish predators. Our results demonstrate a complex interaction among metabolic rate, body mass, temperature and predation regime. The intraspecific scaling of metabolic rate with body mass and temperature is not merely the result of physical constraints related to internal body design and biochemical kinetics, but rather is ecologically sensitive and evolutionarily malleable.

Linking scaling laws across eukaryotes

Scaling laws relating body mass to species characteristics are among the most universal quantitative patterns in biology. Within major taxonomic groups, the 4 key ecological variables of metabolism, abundance, growth, and mortality are often well described by power laws with exponents near 3/4 or related to that value, a commonality often attributed to biophysical constraints on metabolism. However, metabolic scaling theories remain widely debated, and the links among the 4 variables have never been formally tested across the full domain of eukaryote life, to which prevailing theory applies. Here we present datasets of unprecedented scope to examine these 4 scaling laws across all eukaryotes and link them to test whether their combinations support theoretical expectations. We find that metabolism and abundance scale with body size in a remarkably reciprocal fashion, with exponents near ±3/4 within groups, as expected from metabolic theory, but with exponents near ±1 across all groups. This reciprocal scaling supports "energetic equivalence" across eukaryotes, which hypothesizes that the partitioning of energy in space across species does not vary significantly with body size. In contrast, growth and mortality rates scale similarly both within and across groups, with exponents of ±1/4. These findings are inconsistent with a metabolic basis for growth and mortality scaling across eukaryotes. We propose that rather than limiting growth, metabolism adjusts to the needs of growth within major groups, and that growth dynamics may offer a viable theoretical basis to biological scaling.

Body size and temperature effects on standard metabolic rate for determining metabolic scope for activity of the polychaete Hediste (Nereis) diversicolor

Considering the ecological importance and potential value of Hediste diversicolor, a better understanding of its metabolic rate and potential growth rates is required. The aims of this study are: (i) to describe key biometric relationships; (ii) to test the effects of temperature and body size on standard metabolic rate (as measure by oxygen consumption) to determine critical parameters, namely Arrhenius temperature (T_A ), allometric coefficient (b) and reaction rate; and (iii) to determine the metabolic scope for activity (MSA) of H. diversicolor for further comparison with published specific growth rates. Individuals were collected in a Mediterranean lagoon (France). After 10 days of acclimatization, 7 days at a fixed temperature and 24 h of fasting, resting oxygen consumption rates (VO₂ ) were individually measured in the dark at four different temperatures (11, 17, 22 and 27 °C) in worms weighing from 4 to 94 mgDW (n = 27 per temperature). Results showed that DW and L3 were the most accurate measurements of weight and length, respectively, among all the metrics tested. Conversion of WW (mg), DW (mg) and L3 (mm) were quantified with the following equations: DW = 0.15 × WW, L3 = 0.025 × TL(mm) + 1.44 and DW = 0.8 × L3^3.68. Using an equation based on temperature and allometric effects, the allometric coefficient (b) was estimated at 0.8 for DW and at 2.83 for L3. The reaction rate (VO₂ ) equaled to 12.33 µmol gDW^-1 h^-1 and 0.05 µmol mm L3^-1 h^-1 at the reference temperature (20 °C, 293.15 K). Arrhenius temperature (T_A ) was 5,707 and 5,664 K (for DW and L3, respectively). Metabolic scope for activity ranged from 120.1 to 627.6 J gDW^-1 d^-1. Predicted maximum growth rate increased with temperature, with expected values of 7-10% in the range of 15-20 °C. MSA was then used to evaluate specific growth rates (SGR) in several experiments. This paper may be used as a reference and could have interesting applications in the fields of aquaculture, ecology and biogeochemical processes.

Why does offspring size affect performance? Integrating metabolic scaling with life-history theory

Within species, larger offspring typically outperform smaller offspring. While the relationship between offspring size and performance is ubiquitous, the cause of this relationship remains elusive. By linking metabolic and life-history theory, we provide a general explanation for why larger offspring perform better than smaller offspring. Using high-throughput respirometry arrays, we link metabolic rate to offspring size in two species of marine bryozoan. We found that metabolism scales allometrically with offspring size in both species: while larger offspring use absolutely more energy than smaller offspring, larger offspring use proportionally less of their maternally derived energy throughout the dependent, non-feeding phase. The increased metabolic efficiency of larger offspring while dependent on maternal investment may explain offspring size effects-larger offspring reach nutritional independence (feed for themselves) with a higher proportion of energy relative to structure than smaller offspring. These findings offer a potentially universal explanation for why larger offspring tend to perform better than smaller offspring but studies on other taxa are needed.

Growth Retardation and Altered Isotope Composition As Delayed Effects of PCB Exposure in Daphnia magna

Trophic magnification factor (TMF) analysis employs stable isotope signatures to derive biomagnification potential for environmental contaminants. This approach relies on species δ(15)N values aligning with their trophic position (TP). This, however, may not always be true, because toxic exposure can alter growth and isotope allocation patterns. Here, effects of PCB exposure (mixture of PCB18, PCB40, PCB128, and PCB209) on δ(15)N and δ(13)C as well as processes driving these effects were explored using the cladoceran Daphnia magna. A two-part experiment assessed effects of toxic exposure during and after exposure; juvenile daphnids were exposed during 3 days (accumulation phase) and then allowed to depurate for 4 days (depuration phase). No effects on survival, growth, carbon and nitrogen content, and stable isotope composition were observed after the accumulation phase, whereas significant changes were detected in adults after the depuration phase. In particular, a significantly lower nitrogen content and a growth inhibition were observed, with a concomitant increase in δ(15)N (+0.1 ‰) and decrease in δ(13)C (-0.1 ‰). Although of low magnitude, these changes followed the predicted direction indicating that sublethal effects of contaminant exposure can lead to overestimation of TP and hence underestimated TMF.

Shape shifting predicts ontogenetic changes in metabolic scaling in diverse aquatic invertebrates

Metabolism fuels all biological activities, and thus understanding its variation is fundamentally important. Much of this variation is related to body size, which is commonly believed to follow a 3/4-power scaling law. However, during ontogeny, many kinds of animals and plants show marked shifts in metabolic scaling that deviate from 3/4-power scaling predicted by general models. Here, we show that in diverse aquatic invertebrates, ontogenetic shifts in the scaling of routine metabolic rate from near isometry (bR = scaling exponent approx. 1) to negative allometry (bR < 1), or the reverse, are associated with significant changes in body shape (indexed by bL = the scaling exponent of the relationship between body mass and body length). The observed inverse correlations between bR and bL are predicted by metabolic scaling theory that emphasizes resource/waste fluxes across external body surfaces, but contradict theory that emphasizes resource transport through internal networks. Geometric estimates of the scaling of surface area (SA) with body mass (bA) further show that ontogenetic shifts in bR and bA are positively correlated. These results support new metabolic scaling theory based on SA influences that may be applied to ontogenetic shifts in bR shown by many kinds of animals and plants.

The effect of parameter variability in the allometric projection of leaf growth rates for eelgrass (Zostera marina L.) II: the importance of data quality control procedures in bias reduction

Background

Eelgrass grants important ecological benefits including a nursery for waterfowl and fish species, shoreline stabilization, nutrient recycling and carbon sequestration. Upon the exacerbation of deleterious anthropogenic influences, re-establishment of eelgrass beds has mainly depended on transplantation. Productivity estimations provide valuable information for the appraisal of the restoration of ecological functions of natural populations. Assessments over early stages of transplants should preferably be nondestructive. Allometric scaling of eelgrass leaf biomass in terms of matching length provides a proxy that reduces leaf biomass and productivity estimations to simple measurements of leaf length and its elongation over a period. We examine how parameter variability impacts the accuracy of the considered proxy and the extent on what data quality and sample size influence the uncertainties of the involved allometric parameters.

Methods

We adapted a Median Absolute Deviation data quality control procedure to remove inconsistencies in the crude data. For evaluating the effect of parametric uncertainty we performed both a formal exploration and an analysis of the sensitivity of the allometric projection method to parameter changes. We used parameter estimates obtained by means of nonlinear regression from crude as well as processed data.

Results

We obtained reference leaf growth rates by allometric projection using parameter estimates produced by the crude data, and then considered changes in fitted parameters bounded by the modulus of the vector of the linked standard errors, we found absolute deviations up to 10 % of reference values. After data quality control, the equivalent maximum deviation was under 7 % of corresponding reference rates. Therefore, the addressed allometric method is robust. Even the smaller sized samples in the quality controlled dataset produced better accuracy levels than the whole set of crude data.

Conclusions

We propose quality control of data as a highly recommended step in the overall procedure that leads to reliable allometric surrogates of eelgrass leaf growth rates. The proliferation of inconsistent replicates in the crude data points towards the importance of discarding incomplete leaves. We also recommend avoiding errors in estimating the biomass of small leaves for which precision of the used analytical scale might be an issue.

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.

On the analytical solution for the Pütter-Bertalanffy growth equation

This study develops the basic idea of Pütter and Bertalanffy addressing the allometric scaling of anabolism and catabolism on somatic growth dynamics. We proposed a standardized form of the Pütter-Bertalanffy equation (PBE), which is given as the extended model of Richards function, and subsequently solved it. The analytical solution of the PBE was defined by an incomplete beta function and can take a wide range of shapes in its growth curve. The mathematical behavior of PBE due to the change in parameter values was briefly discussed. Most forms of solution consistently hold the implicit functional type with respect to the variable of body size.

Jellyfish body plans provide allometric advantages beyond low carbon content

Jellyfish form spectacular blooms throughout the world’s oceans. Jellyfish body plans are characterised by high water and low carbon contents which enables them to grow much larger than non-gelatinous animals of equivalent carbon content and to deviate from non-gelatinous pelagic animals when incorporated into allometric relationships. Jellyfish have, however, been argued to conform to allometric relationships when carbon content is used as the metric for comparison. Here we test the hypothesis that differences in allometric relationships for several key functional parameters remain for jellyfish even after their body sizes are scaled to their carbon content. Data on carbon and nitrogen contents, rates of respiration, excretion, growth, longevity and swimming velocity of jellyfish and other pelagic animals were assembled. Allometric relationships between each variable and the equivalent spherical diameters of jellyfish and other pelagic animals were compared before and after sizes of jellyfish were standardised for their carbon content. Before standardisation, the slopes of the allometric relationships for respiration, excretion and growth were the same for jellyfish and other pelagic taxa but the intercepts differed. After standardisation, slopes and intercepts for respiration were similar but excretion rates of jellyfish were 10× slower, and growth rates 2× faster than those of other pelagic animals. Longevity of jellyfish was independent of size. The slope of the allometric relationship of swimming velocity of jellyfish differed from that of other pelagic animals but because they are larger jellyfish operate at Reynolds numbers approximately 10× greater than those of other pelagic animals of comparable carbon content. We conclude that low carbon and high water contents alone do not explain the differences in the intercepts or slopes of the allometric relationships of jellyfish and other pelagic animals and that the evolutionary longevity of jellyfish and their propensity to form blooms is facilitated by their unique body plans.

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.

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.

No evidence of mass dependency of specific organ metabolic rate in healthy humans

BACKGROUND

In humans, resting energy expenditure (REE) can be calculated from organ and tissue masses using constant specific organ metabolic rates. However, interspecies data suggest allometric relations between body mass and organ metabolic rate with higher specific metabolic rates in mammals with a smaller body mass.

OBJECTIVE

The objective was to compare the accuracy of REE prediction with the use of either constant or body mass-dependent specific organ metabolic rates.

DESIGN

Healthy subjects (79 women, 75 men) within the normal range of fat mass (FM) expected for a healthy body mass index and aged 18-78 y were stratified into tertiles of body mass. Fifty subjects were grouped as tertile 1 (<66.3 kg), 52 as tertile 2 (> or =66.3 to < or =77.2 kg), and 52 as tertile 3 (>77.2 kg). Magnetic resonance imaging was used to assess the volume of 4 internal organs (brain, heart, liver, and kidneys). REE was measured by indirect calorimetry (REE(m)) and compared with REE calculated from previously published constant (REE(c1)) and body mass-dependent organ metabolic rates (REE(c2)).

RESULTS

REE(m) increased significantly with weight tertile (tertile 1: 5536 +/- 529 kJ/d; tertile 2: 6389 +/- 672 kJ/d; tertile 3: 7467 +/- 745 kJ/d; P < 0.01). The deviation REE(m)-REE(c1) did not differ between weight tertiles (tertile 1: 66 +/- 382 kJ/d; tertile 2: 167 +/- 507 kJ/d; tertile 3: 86 +/- 480 kJ/d; NS) and showed no relation with body mass (r = -0.05, NS). By contrast, REE(m)-REE(c2) increased with increasing weight tertile (tertile 1: -45 +/- 369 kJ/d; tertile 2: 150 +/- 503 kJ/d; tertile 3: 193 +/- 482 kJ/d; P < 0.05) and correlated significantly with body mass (r = 0.16, P < 0.05).

CONCLUSION

Our data do not support a lower specific organ metabolic rate in humans with a larger body mass than in those with a smaller body mass.

Scaling of metabolism in Helix aspersa snails: changes through ontogeny and response to selection for increased size

Though many are convinced otherwise, variability of the size-scaling of metabolism is widespread in nature, and the factors driving that remain unknown. Here we test a hypothesis that the increased expenditure associated with faster growth increases metabolic scaling. We compare metabolic scaling in the fast- and slow-growth phases of ontogeny of Helix aspersasnails artificially selected or not selected for increased adult size. The selected line evolved larger egg and adult sizes and a faster size-specific growth rate, without a change in the developmental rate. Both lines had comparable food consumption but the selected snails grew more efficiently and had lower metabolism early in ontogeny. Attainment of lower metabolism was accompanied by decreased shell production, indicating that the increased growth was fuelled partly at the expense of shell production. As predicted,the scaling of oxygen consumption with body mass was isometric or nearly isometric in the fast-growing (early) ontogenetic stage, and it became negatively allometric in the slow-growing (late) stage; metabolic scaling tended to be steeper in selected (fast-growing) than in control (slow-growing)snails; this difference disappeared later in ontogeny. Differences in metabolic scaling were not related to shifts in the scaling of metabolically inert shell. Our results support the view that changes in metabolic scaling through ontogeny and the variability of metabolic scaling between organisms can be affected by differential growth rates. We stress that future approaches to this phenomenon should consider the metabolic effects of cell size changes which underlie shifts in the growth pattern.

A conceptual framework for mapping quantitative trait Loci regulating ontogenetic allometry

Although ontogenetic changes in body shape and its associated allometry has been studied for over a century, essentially nothing is known about their underlying genetic and developmental mechanisms. One of the reasons for this ignorance is the unavailability of a conceptual framework to formulate the experimental design for data collection and statistical models for data analyses. We developed a framework model for unraveling the genetic machinery for ontogenetic changes of allometry. The model incorporates the mathematical aspects of ontogenetic growth and allometry into a maximum likelihood framework for quantitative trait locus (QTL) mapping. As a quantitative platform, the model allows for the testing of a number of biologically meaningful hypotheses to explore the pleiotropic basis of the QTL that regulate ontogeny and allometry. Simulation studies and real data analysis of a live example in soybean have been performed to investigate the statistical behavior of the model and validate its practical utilization. The statistical model proposed will help to study the genetic architecture of complex phenotypes and, therefore, gain better insights into the mechanistic regulation for developmental patterns and processes in organisms.

Allometric exponents do not support a universal metabolic allometry

The debate about the value of the allometric scaling exponent (b) relating metabolic rate to body mass (metabolic rate = a x mass(b)) is ongoing, with published evidence both for and against a 3/4-power scaling law continuing to accumulate. However, this debate often revolves around a dichotomous distinction between the 3/4-power exponent predicted by recent models of nutrient distribution networks and a 2/3 exponent predicted by Euclidean surface-area-to-volume considerations. Such an approach does not allow for the possibility that there is no single "true" exponent. In the present study, we conduct a meta-analysis of 127 interspecific allometric exponents to determine whether there is a universal metabolic allometry or if there are systematic differences between taxa or between metabolic states. This analysis shows that the effect size of mass on metabolic rate is significantly heterogeneous and that, on average, the effect of mass on metabolic rate is stronger for endotherms than for ectotherms. Significant differences between scaling exponents were also identified between ectotherms and endotherms, as well as between metabolic states (e.g., rest, field, and exercise), a result that applies to b values estimated by ordinary least squares, reduced major axis, and phylogenetically correct regression models. The lack of support for a single exponent model suggests that there is no universal metabolic allometry and represents a significant challenge to any model that predicts only a single value of b.

A critical understanding of the fractal model of metabolic scaling

The exponent of the scaling of metabolic rate with body mass has been the subject of debate for more than a century. The argument is at two levels, one concerning questions of empirical support for the exponent and the other, how to derive it theoretically. At this second level, the exponent is usually treated as the outcome of an underlying physical burden and approached as the search for a natural law emerging within energetic and geometric constraints. Recently, a model relying on fractal geometry was proposed as a general explanation for the phenomenon. In the present study, a reanalysis of the fractal model is performed to verify its validity. All the conditions that allow for the connection between the geometric proposition and the allometric exponent are evaluated, as well as the energy loss minimization procedure put forward in the model. It is demonstrated that the minimization procedure is mathematically incorrect and ill-posed. Also, it is shown that none of the connecting conditions are fulfilled. Therefore, it is concluded that the fractal model lacks self-consistency and correct statement: it relies on strong assumptions of homogeneity in morpho-physiological features among organisms instead of demonstrating them, as claimed by its authors. It is proposed that empiricists and theoreticians should rather evaluate the frameworks for addressing metabolic scaling phenomena.

Beyond the '3/4-power law': variation in the intra- and interspecific scaling of metabolic rate in animals

In this review I show that the '3/4-power scaling law' of metabolic rate is not universal, either within or among animal species. Significant variation in the scaling of metabolic rate with body mass is described mainly for animals, but also for unicells and plants. Much of this variation, which can be related to taxonomic, physiological, and/or environmental differences, is not adequately explained by existing theoretical models, which are also reviewed. As a result, synthetic explanatory schemes based on multiple boundary constraints and on the scaling of multiple energy-using processes are advocated. It is also stressed that a complete understanding of metabolic scaling will require the identification of both proximate (functional) and ultimate (evolutionary) causes. Four major types of intraspecific metabolic scaling with body mass are recognized [based on the power function R=aMb, where R is respiration (metabolic) rate, a is a constant, M is body mass, and b is the scaling exponent]: Type I: linear, negatively allometric (b<1); Type II: linear, isometric (b=1); Type III: nonlinear, ontogenetic shift from isometric (b=1), or nearly isometric, to negatively allometric (b<1); and Type IV: nonlinear, ontogenetic shift from positively allometric (b>1) to one or two later phases of negative allometry (b<1). Ontogenetic changes in the metabolic intensity of four component processes (i.e. growth, reproduction, locomotion, and heat production) appear to be important in these different patterns of metabolic scaling. These changes may, in turn, be shaped by age (size)-specific patterns of mortality. In addition, major differences in interspecific metabolic scaling are described, especially with respect to mode of temperature regulation, body-size range, and activity level. A 'metabolic-level boundaries hypothesis' focusing on two major constraints (surface-area limits on resource/waste exchange processes and mass/volume limits on power production) can explain much, but not all of this variation. My analysis indicates that further empirical and theoretical work is needed to understand fully the physiological and ecological bases for the considerable variation in metabolic scaling that is observed both within and among species. Recommended approaches for doing this are discussed. I conclude that the scaling of metabolism is not the simple result of a physical law, but rather appears to be the more complex result of diverse adaptations evolved in the context of both physico-chemical and ecological constraints.

Universal scaling of respiratory metabolism, size and nitrogen in plants

The scaling of respiratory metabolism to body size in animals is considered to be a fundamental law of nature, and there is substantial evidence for an approximate (3/4)-power relation. Studies suggest that plant respiratory metabolism also scales as the (3/4)-power of mass, and that higher plant and animal scaling follow similar rules owing to the predominance of fractal-like transport networks and associated allometric scaling. Here, however, using data obtained from about 500 laboratory and field-grown plants from 43 species and four experiments, we show that whole-plant respiration rate scales approximately isometrically (scaling exponent approximately 1) with total plant mass in individual experiments and has no common relation across all data. Moreover, consistent with theories about biochemically based physiological scaling, isometric scaling of whole-plant respiration rate to total nitrogen content is observed within and across all data sets, with a single relation common to all data. This isometric scaling is unaffected by growth conditions including variation in light, nitrogen availability, temperature and atmospheric CO2 concentration, and is similar within or among species or functional groups. These findings suggest that plants and animals follow different metabolic scaling relations, driven by distinct mechanisms.

Diet, phylogeny, and basal metabolic rate in phyllostomid bats

Aside from the pervasive effects of body mass, much controversy exists as to what factors account for interspecific variation in basal metabolic rates (BMR) of mammals; however, both diet and phylogeny have been strongly implicated. We examined variation in BMR within the New World bat family Phyllostomidae, which shows the largest diversity of food habits among mammalian families, including frugivorous, nectarivorous, insectivorous, carnivorous and blood-eating species. For 27 species, diet was taken from the literature and BMR was either measured on animals captured in Brazil or extracted from the literature. Conventional (nonphylogenetic) analysis of covariance (ANCOVA), with body mass as the covariate, was first used to test the effects of diet on BMR. In this analysis, which assumes that all species evolved simultaneously from a single ancestor (i.e., a "star" phylogeny), diet exerted a strong effect on mass-independent BMR: nectarivorous bats showed higher mass-independent BMR than other bats feeding on fruits, insects or blood. In phylogenetic ANCOVAs via Monte Carlo computer simulation, which assume that species are part of a branching hierarchical phylogeny, no statistically significant effect of diet on BMR was observed. Hence, results of the nonphylogenetic analysis were misleading because the critical values for testing the effect of diet were underestimated. However, in this sample of bats, diet is perfectly confounded with phylogeny, because the four dietary categories represent four separate subclades, which greatly reduces statistical power to detect a diet (= subclade) effect. But even if diet did appear to exert an influence on BMR in this sample of bats, it would not be logically possible to separate this effect from the possibility that the dietary categories differ for some other reason (i.e., another synapomorphy of one or more of the subclades). Examples such as this highlight the importance of considering phylogenetic relationships when designing new comparative studies, as well as when analyzing existing data sets. We also discuss some possible reasons why BMR may not coadapt with diet.

Allometric cascade: a model for resolving body mass effects on metabolism

Expanding upon a preliminary communication (Nature 417 (2002) 166), we here further develop a "multiple-causes model" of allometry, where the exponent b is the sum of the influences of multiple contributors to control. The relative strength of each contributor, with its own characteristic value of b(i), is determined by c(i), the control contribution or control coefficient. A more realistic equation for the scaling of metabolism with body size thus can be written as BMR=MR(0)Sigmac(i)(M/M(0))(bi), where MR(0) is the "characteristic metabolic rate" of an animal with a "characteristic body mass", M(0). With M(0) of 1 unit mass (usually kg), MR(0) takes the place of the value a, found in the standard scaling equation, b(i) is the scaling exponent of the process i, and c(i) is its control contribution to overall flux, or the control coefficient of the process i. One can think of this as an allometric cascade, with the b exponent for overall energy metabolism being determined by the b(i) and c(i) values for key steps in the complex pathways of energy demand and energy supply. Key intrinsic factors (such as neural and endocrine processes) or ecological extrinsic factors are considered to act through this system in affecting allometric scaling of energy turnover. Applying this model to maximum vs. BMR data for the first time explains the differing scaling behaviour of these two biological states in mammals, both in the absence and presence of intrinsic regulators such as thyroid hormones (for BMR) and catecholamines (for maximum metabolic rate).