The origin of allometric scaling laws in biology

The empirical rules relating metabolic rate and body size are described in terms of (i) a scaling exponent, which refers to the ratio of the fractional change in metabolic rate to a change in body size, (ii) a proportionality constant, which describes the rate of energy expenditure in an organism of unit mass. This article integrates the chemiosmotic theory of energy transduction with the methods of quantum statistics to propose a molecular mechanism which, in sharp contrast to competing models, explains both the variation in scaling exponents and the taxon-specific differences in proportionality constants. The new model is universal in the sense that it applies to unicellular organisms, plants and animals.     

The size–grain hypothesis and interspecific scaling in ants

1. The size–grain hypothesis maintains that as terrestrial walking organisms decrease in size, their environment becomes less planar and more rugose. The benefits of long legs (efficient, speedy movement over a planar environment) may thus decrease with smaller body size, while the costs (larger cross-sectional area limiting access to the interstitial environment) are enhanced.
2. A prediction from this hypothesis – that leg size should increase proportionately with body mass – is examined. Ants are among the smallest walking animals and extend the size gradient five orders of magnitude beyond the traditional 'mouse to elephant' curve. The mass of 135 species of worker ants spans 3·7 orders of magnitude (0·008–53 mg). Larger ants tended to be slimmer and longer legged. Ant subfamilies varied in their scaling relationships, but four out of five showed a positive allometry for hind leg length ( b > 0·33). Mammals, in contrast, show isometry for leg length over six orders of magnitude.
3. It is suggested that ants make a transition from living in an interstitial environment when small to a planar environment when large, a habit continued by most terrestrial mammals. Head length and pronotum width are robust estimators of mass in ants.     

Metabolic scaling of antibiotics in reptiles: Basis and limitations

The allometric equation y = a · x^b has been used to scale many morphological and physiological attributes relative to body mass. For instance, in eutherian mammals, the equation P_met = 70M_b^0.75 has been used to describe the relationship between metabolic rate (P_met) and body mass (M_b). Similar equations have been derived for squamate reptiles. Recently, this relationship between metabolic rate and body mass has been used in determining appropriate dosages and dosing intervals of antibiotics both intraspecifically for different sized reptiles and interspecifically for those reptiles in which antibiotic pharmacokinetic studies have not been performed. Although this is a simple mathematical process, a number of problems surface when this approach is examined closely. First, the mass constant (a) in reptiles varies from 1–5 for snakes and 6–10 for lizards. No such information is available for chelonians or crocodilians. Unless the mass constant for the unknown species approximates that of the known species, inappropriate dosages and intervals of administration will be calculated. Second, pharmacokinetic differences may exist between widely divergent species, independent of metabolic rate. Third, all available pharmacokinetic studies and metabolic allometric equations are derived from clinically healthy reptiles. Differences more than likely exist between healthy and ill reptiles in regard to uptake, distribution, and elimination of drugs and overall metabolism. While metabolic scaling of antibiotics is a potentially useful and practical tool in drug dosing, these limitations must be considered when dosing an ill reptile. Until more scientifically derived information is available for demonstrating the accuracy of metabolic scaling of antibiotics in reptiles, the clinician will need to understand the limitations of this approach. © 1996 Wiley-Liss, Inc.     

Uncertainty in allometric exponent estimation: a case study in scaling metabolic rate with body mass

Many factors could influence the allometric scaling exponent β estimation, but have not been explored systematically. We investigated the influences of three factors on the estimate of β based on a data set of 626 species of basal metabolic rate and mass in mammals. The influence of sampling error was tested by re-sampling with different sample sizes using a Monte Carlo method. Small random errors were introduced to measured data to examine their influence on parameter estimations. The influence of analysis method was also evaluated by applying nonlinear and linear regressions to the original data. Results showed that a relative large sample size was required to lower statistical inference errors. When sample size n was 10% of the base population size (n=63), 35% of the samples supported β=2/3, 39% supported β=3/4, and 15% rejected β=0.711, even though the base population had a β=0.711. The controversy surrounding the estimation of β in the literature could be partially attributable to such small sample sizes in many studies. Measurement errors in body mass and base metabolic rate, especially in body mass, could largely increase alpha and beta errors. Analysis methods also affected parameter estimations. Nonlinear regressions provided better estimates of the scaling exponent that were significantly higher than these commonly estimated by linear regressions. This study demonstrated the importance of the quantity and quality of data as well as analysis method in power law analysis, raising caution in interpreting power law results. Meta-data synthesis using data from independent studies seems to be a proper approach in the future, but caution should be taken to make sure that such measurements are made using similar protocols.     

Scaling of offspring number and mass to plant and animal size: model and meta-analysis

The scaling of reproductive parameters to body size is important for understanding ecological and evolutionary patterns. Here, we derived allometric relationships for the number and mass of seeds, eggs and neonates from an existing model on population production. In a separate meta-analysis, we collected 79 empirical regressions on offspring mass and number covering different taxa and various habitats. The literature review served as a validation of the model, whereas, vice versa, consistency of isolated regressions with each other and related ecological quantities was checked with the model. The total offspring mass delivered in a reproductive event scaled to adult size with slopes in the range of about 3/4 to 1. Exponents for individual seed, egg and neonate mass varied around 1/2 for most heterotherms and between 3/4 and 1 for most homeotherms. The scaling of the progeny number released in a sowing, clutch or litter was opposite to that of their size. The linear regressions fitted into a triangular envelope where maximum offspring mass is limited by the size of the adult. Minimum seed and egg size scaled with weight exponents of approximately 0 up to 1/4. These patterns can be explained by the influence of parents on the fate of their offspring, covering the continuum of r-strategists (pelagic–aquatic, arial, most invertebrates, heterotherms) and K-strategists (littoral–terrestrial, some invertebrates, homeotherms).     

Scaling of gas exchange cycle frequency in insects

Previously, it has been suggested that insect gas exchange cycle frequency (fC) is mass independent, making insects different from most other animals where periods typically scale as mass-0.25. However, the claim for insects is based on studies of only a few closely related taxa encompassing a relatively small size range. Moreover, it is not known whether the type of gas exchange pattern (discontinuous versus cyclic) influences the fC-mass scaling relationship. Here, we analyse a large database to examine interspecific fC-mass scaling. In addition, we investigate the effect of mode of gas exchange on the fC-scaling relationship using both conventional and phylogenetically independent approaches. Cycle frequency is scaled as mass(-0.280) (when accounting for phylogenetic non-independence and gas exchange pattern), which did not differ significantly from mass(-0.25). The slope of the fC-mass relationship was shallower with a significantly lower intercept for the species showing discontinuous gas exchange than for those showing the cyclic pattern, probably due to lower metabolic rates in the former. Insects therefore appear no different from other animals insofar as the scaling of gas exchange fC is concerned, although gas exchange fC may scale in distinct ways for different patterns.     

Validation of an individualised model of human thermoregulation for predicting responses to cold air

Most computer models of human thermoregulation are population based. Here, we individualised the Fiala model [Fiala et al. (2001) Int J Biometeorol 45:143–159] with respect to anthropometrics, body fat, and metabolic rate. The predictions of the adapted multisegmental thermoregulatory model were compared with measured skin temperatures of individuals. Data from two experiments, in which reclining subjects were suddenly exposed to mild to moderate cold environmental conditions, were used to study the effect on dynamic skin temperature responses. Body fat was measured by the three-compartment method combining underwater weighing and deuterium dilution. Metabolic rate was determined by indirect calorimetry. In experiment 1, the bias (mean difference) between predicted and measured mean skin temperature decreased from 1.8°C to −0.15°C during cold exposure. The standard deviation of the mean difference remained of the same magnitude (from 0.7°C to 0.9°C). In experiment 2 the bias of the skin temperature changed from 2.0±1.09°C using the standard model to 1.3±0.93°C using individual characteristics in the model. The inclusion of individual characteristics thus improved the predictions for an individual and led to a significantly smaller systematic error. However, a large part of the discrepancies in individual response to cold remained unexplained. Possible further improvements to the model accomplished by inclusion of more subject characteristics (i.e. body fat distribution, body shape) and model refinements on the level of (skin) blood perfusion, and control functions, are discussed.     

Power-law scaling in dimension-to-biomass relationship of fish schools

Motivated by the finding that there is some biological universality in the relationship between school geometry and school biomass of various pelagic fishes in various conditions, I here establish a scaling law for school dimensions: the school diameter increases as a power-law function of school biomass. The power-law exponent is extracted through the data collapse, and is close to 35. This value of the exponent implies that the mean packing density decreases as the school biomass increases, and the packing structure displays a mass-fractal dimension of 53. By exploiting an analogy between school geometry and polymer chain statistics, I examine the behavioral algorithm governing the swollen conformation of large-sized schools of pelagics, and I explain the value of the exponent.     

Metabolic thermogenesis and evaporative water loss in the Hwamei Garrulax canorus

Basal metabolic rate (BMR) is thought to be a major hub in the network of physiological mechanisms connecting life history traits. Evaporative water loss (EWL) is a physiological indicator that is widely used to measure water relations in inter- or intraspecific studies of birds in different environments. In this study, we examined the physiological responses of summer-acclimatized Hwamei Garrulax canorus to temperature by measuring their body temperature (T_b), metabolic rate (MR) and EWL at ambient temperatures (T_a) between 5 and 40 °C. Overall, we found that mean body temperature was 42.4 °C and average minimum thermal conductance (C) was 0.15 ml O₂ g⁻¹ h⁻¹ °C⁻¹ measured between 5 and 20 °C. The thermal neutral zone (TNZ) was 31.8–35.3 °C and BMR was 181.83 ml O₂ h⁻¹. Below the lower critical temperature, MR increased linearly with decreasing T_a according to the relationship: MR (ml O₂ h⁻¹)=266.59–2.66 T_a. At T_as above the upper critical temperature, MR increased with T_a according to the relationship: MR (ml O₂ h⁻¹)=−271.26+12.85 T_a. EWL increased with T_a according to the relationship: EWL (mg H₂O h⁻¹)=−19.16+12.64 T_a and exceeded metabolic water production at T_a>14.0 °C. The high T_b and thermal conductance, low BMR, narrow TNZ, and high evaporative water production/metabolic water production (EWP/MWP) ratio in the Hwamei are consistent with the idea that this species is adapted to warm, mesic climates, where metabolic thermogenesis and water conservation are not strong selective pressures.     

Metabolic rate during foraging in the honeybee

Summary The metabolic rate of free-flying honeybees, Apis mellifera ligustica, was determined by means of a novel respirometric device that allowed measurement of CO₂ produced by bees foraging under controlled reward at an artificial food source. Metabolic rate increased with reward (sugar flow rate) at the food source. In addition, there was no clear-cut dependence of metabolic rate on load carried during the visit, neither as crop load nor as supplementary weights attached to the thorax. The hypothesis that metabolic rate, as well as foraging and recruiting activities, depend on the motivational state of the foraging bee determined by the reward at the food source is discussed.Abbreviations CL crop load (fuel load at the FSS) - FC (=CL-W_c), fuel consumed during the visit - FSS food source simulator - FSS +dome, respirometric chamber - NVT non-visit time - TT titration time - VT visit time - W_c (=W_f-W_i) load carried at the end of the visit - W_f final weight of the forager - W_i initial weight of the forager     

Temporal variability and scaling behaviour in the fine-root system of kiwifruit (Actinidia deliciosa)

We show that temporal variability in root populations can depend upon the scale of measurement (particularly the sampled soil volume). The presence of roots in a range of volumes of soil was studied using perspex tubes installed horizontally into the soil around three mature kiwifruit vines. Roots intercepting lines scored on the tubes were counted using a periscope. For small volumes of soil (c. 2–4 cm³) the root counts varied with time in a very irregular manner, and as the interval between measurements increased the autocorrelation between the measurements decayed rapidly. At about half of the locations monitored there was no significant autocorrelation between measurements 27 d apart. Linear interpolation in these time series was unreliable, and where the correlation dimension could be resolved it was usually non-integer (suggesting chaotic behaviour). The time series measured at different locations were poorly correlated, indicating weak coordination. As the observed soil volume increased, the coordination between locations improved, the autocorrelation function increased, and linear interpolation errors decreased (although these remained substantial). Clearly there are considerable fundamental constraints on our ability to predict the root behaviour of kiwifruit vines at scales that are appropriate for mechanistic models of nutrient and water uptake. We discuss the need for a new conceptual model of the fine-root systems of kiwifruit and similar species.     

Determinants of rate variation in mammalian DNA sequence evolution

Attempts to analyze variation in the rates of molecular evolution among mammalian lineages have been hampered by paucity of data and by nonindependent comparisons. Using phylogenetically independent comparisons, we test three explanations for rate variation which predict correlations between rate variation and generation time, metabolic rate, and body size. Mitochondrial and nuclear genes, protein coding, rRNA, and nontranslated sequences from 61 mammal species representing 14 orders are used to compare the relative rates of sequence evolution. Correlation analyses performed on differences in genetic distance since common origin of each pair against differences in body mass, generation time, and metabolic rate reveal that substitution rate at fourfold degenerate sites in two out of three protein sequences is negatively correlated with generation time. In addition, there is a relationship between the rate of molecular evolution and body size for two nuclear-encoded sequences. No evidence is found for an effect of metabolic rate on rate of sequence evolution. Possible causes of variation in substitution rate between species are discussed.     

Tooth scaling and evolutionary dwarfism: an investigation of allometry in human pygmies

Gould has predicted that in rapidly dwarfed lineages the postcanine teeth exhibit a different scaling pattern than is the normal interspecific trend. His prediction of strong negative allometry has not been frequently tested in quantitative detail. Here we present results of scaling analyses of the molar teeth in African pygmies compared with other Africans of larger size and in Philippine pygmies compared with Filipinos of larger size. We find a pattern of strong negative allometry of tooth size to skull and body size in both these comparisons. This scaling pattern is explained by recourse to the developmental bases (known or inferred) of dwarfing in these populations. Body size decrease is related to low levels of the growth control substance insulin-like growth factor I (IGF-I), which does not appear to affect the size of the dentition. The implications of such developmental information for our understanding of allometric patterns in general, and dwarfing events in particular, are discussed.     

Plant physiology in theory and practice: An analysis of the WBE model for vascular plants

The theoretical model of West, Brown and Enquist (hereafter WBE) proposed the fractal geometry of the transport system as the origin of the allometric scaling laws observed in nature. The WBE model has either been criticized for some restrictive and biologically unrealistic constraints or its reliability debated on the evidence of empirical tests. In this work, we revised the structure of the WBE model for vascular plants, highlighting some critical assumptions and simplifications and discuss them with regard to empirical evidence from plant anatomy and physiology. We conclude that the WBE model had the distinct merit of shedding light on some important features such as conduit tapering. Nonetheless, it is over-simplistic and a revised model would be desirable with an ontogenetic perspective that takes some important phenomena into account, such as the transformation of the inner sapwood into heartwood and the effect of hydraulic constraints in limiting the growth in height.     

Geometrical constraints in the scaling relationships between genome size, cell size and cell cycle length in herbaceous plants

Plant nuclear genome size (GS) varies over three orders of magnitude and is correlated with cell size and growth rate. We explore whether these relationships can be owing to geometrical scaling constraints. These would produce an isometric GS-cell volume relationship, with the GS-cell diameter relationship with the exponent of 1/3. In the GS-cell division relationship, duration of processes limited by membrane transport would scale at the 1/3 exponent, whereas those limited by metabolism would show no relationship. We tested these predictions by estimating scaling exponents from 11 published datasets on differentiated and meristematic cells in diploid herbaceous plants. We found scaling of GS-cell size to almost perfectly match the prediction. The scaling exponent of the relationship between GS and cell cycle duration did not match the prediction. However, this relationship consists of two components: (i) S phase duration, which depends on GS, and has the predicted 1/3 exponent, and (ii) a GS-independent threshold reflecting the duration of the G1 and G2 phases. The matches we found for the relationships between GS and both cell size and S phase duration are signatures of geometrical scaling. We propose that a similar approach can be used to examine GS effects at tissue and whole plant levels.     

Size-dependent survivorship in the web-building spiderAgelena limbata

Summary Stage-specific mortality rates and mortality factors for the web-building spiderAgelena limbata, which is suggested to be food-limited, were studied, and the relationship between body size of spiders and survivorship for instar 3 to adults was examined. The mortality rate of the egg sac stage including eggs, deutova (prenymphal stage), and overwintering instar 1 nymphs was low. The low mortality of this stage was partly due to maternal care that reduced the mortality caused by predation and/or abiotic factors. From emergence of instar 1 nymphs from egg sacs to reproduction, the stagespecific mortality rates were almost constant, 32–47%, and the time-specific mortality rates were also constant. These results suggest a Deevey (1947) type II survivorship curve inA. limbata, in contrast to other reports on the wandering or burrowing spiders which suggested type III curves. Important mortality factors for nymphs and adults were parasitism by an ichneumonid wasp and predation by spiders. There were great variations in body size (carapace width) ofA. limbata in the field. Smaller individuals survived at a lower rate to the next stage than larger individuals. This tendency was clearer for the population living under poorer prey availability.A. limbata was unlikely to starve to death in the field because every stage ofA. limbata could survive starvation for a long time in the laboratory, 22–65 days on average. I suggest that the size-dependent survivorship of this spider is associated with vulnerability of smaller individuals to parasitism and predation.     

Ecological significance of hypometabolism in nonhuman primates: allometry, adaptation, and deviant diets

The "Kleiber relationship" describes the interspecific allometry between body size and metabolism. Like other allometric relationships, the Kleiber relationship not only summarizes scaling effects across species but also provides a standard by which species can be compared. One well-noted deviation from the Kleiber relationship is "hypometabolism": metabolic rates below that expected for a given size. It has been suggested in the literature that hypometabolism may be a primitive mammalian trait, a thermoregulatory adaptation, an adaptation to arboreal folivory, or an adaptation to a diet that is deviant for body size. Data on primate physiology and behavior are used to evaluate these hypotheses. Only the deviant-diet hypothesis is supported by the data on nonhuman primates. Indeed, the Jarman-Bell relationship, which is the basis for this hypothesis, provides a more coherent explanation of correlated features of animal physiology and behavior than do the alternative models. Hypometabolism may be an energy-conserving adaptation to a variety of nutritional stresses. The present analysis underscores the point that metabolic rate, like foraging behavior, should be thought of as evolutionarily labile.