A unifying explanation for diverse metabolic scaling in animals and plants

The scaling of metabolic rate with body mass has long been a controversial topic. Some workers have claimed that the slope of log-log metabolic scaling relationships typically obeys a universal 3/4-power law resulting from the geometry of resource-transport networks. Others have attempted to explain the broad diversity of metabolic scaling relationships. Although several potentially useful models have been proposed, at present none successfully predicts the entire range of scaling relationships seen among both physiological states and taxonomic groups of animals and plants. Here I argue that our understanding may be aided by three shifts in focus: from explaining average tendencies to explaining variation between extreme boundary limits, from explaining the slope and elevation (metabolic level) of scaling relationships separately to showing how and why they are interrelated, and from focusing primarily on internal factors (e.g. body design) to a more balanced consideration of both internal and external (ecological) factors. By incorporating all of these shifts in focus, the recently proposed metabolic-level boundaries hypothesis appears to provide a useful way of explaining both taxonomic and physiological variation in metabolic scaling relationships. This hypothesis correctly predicts that the scaling slope should vary mostly between 2/3 and 1 and that it should be related to metabolic (activity) level according to an approximately U-shaped function. It also implies that the scaling of other energy-dependent biological processes should be related to the metabolic level of the organisms being examined. Some data are presented that support this implication, but further research is needed.     

Metabolic level and size scaling of rates of respiration and growth in unicellular organisms

1.  Metabolic rate is conventionally assumed to scale with body mass to the 3/4-power, independently of the metabolic level of the organisms being considered. However, recent analyses in a variety of animals and plants indicate that the power (log–log slope) of this relationship varies significantly with metabolic level, ranging from c . 2/3 to 1.
2.  Here I show that the scaling slopes of rates of respiration and growth are related to the metabolic level of a variety of unicellular organisms, as similarly occurs for respiration rates in multicellular organisms.
3.  The recently proposed 'metabolic-level boundaries hypothesis' provides insight into these effects of metabolic level. As predicted, the scaling slopes for resting (endogenous) respiration rate in prokaryotes, algae and protozoans are negatively related to metabolic level; and in protozoans, the scaling slope increases with starvation. Also as predicted, the scaling slopes of growth rate in algae and protozoans are negatively related to growth level. Unexpectedly, opposite effects of starvation on the metabolic scaling slopes of unicellular prokaryotes (compared to that of eukaryotes) may be a spurious result of respiration measurements that did not adequately consider the effects of rapid cell multiplication in prokaryotes with extremely short generation times.
4.  Analyses of both unicellular and multicellular organisms show that there is no universal metabolic scaling relationship, and that variation in metabolic scaling relationships is systematically and possibly universally related to metabolic level.     

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 (b_R = scaling exponent approx. 1) to negative allometry (b_R < 1), or the reverse, are associated with significant changes in body shape (indexed by b_L = the scaling exponent of the relationship between body mass and body length). The observed inverse correlations between b_R and b_L 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 (b_A) further show that ontogenetic shifts in b_R and b_A are positively correlated. These results support new metabolic scaling theory based on SA influences that may be applied to ontogenetic shifts in b_R shown by many kinds of animals and plants.     

Ontogenetic body-mass scaling of resting metabolic rate covaries with species-specific metabolic level and body size in spiders and snakes

According to common belief, metabolic rate usually scales with body mass to the 3/4-power, which is considered by some to be a universal law of nature. However, substantial variation in the metabolic scaling exponent (b) exists, much of which can be related to the overall metabolic level (L) of various taxonomic groups of organisms, as predicted by the recently proposed metabolic-level boundaries (MLB) hypothesis. Here the MLB hypothesis was tested using data for intraspecific (ontogenetic) body-mass scaling of resting metabolic rate in spiders and boid snakes. As predicted, in both animal groups b varies mostly between 2/3 and 1, and is significantly negatively related to L. L is, in turn, negatively related to species-specific body mass (M_m: estimated as the mass at the midpoint of a scaling relationship), and as a result, larger species tend to have steeper metabolic scaling slopes (b) than smaller species. After adjusting for the effects of M_m, b and L are still negatively related, though significantly only in the spiders, which exhibit a much wider range of L than the snakes. Therefore, in spiders and snakes the intraspecific scaling of metabolic rate with body mass itself scales with interspecific variation in both metabolic level and body mass.     

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.     

7种木本植物的分支指数与代谢指数

West、Brown和Enquist提出的植物分形网络模型(简称WBE模型)认为: 植物的分支指数(1/a, 1/b)决定植物的代谢指数, 当分支指数1/a、1/b分别为理论值2.0、3.0时, 代谢速率与个体大小的3/4次幂成正比, 但是恒定的3/4代谢指数并不能全面地反映植物的代谢情况。基于分支指数的协同变化, Price、Enquist和Savage对WBE模型进行扩展, 提出植物分支参数协同变化模型(简称PES模型)。该文借助于PES模型分析了7种木本植物的分支指数和代谢指数。结果表明: 物种间叶面积与叶生物量呈异速生长关系, 基于叶面积得到的分支指数1/a和代谢指数θ在物种间无显著差异, 基于叶生物量得到的分支指数1/a、1/b和代谢指数θ在物种间均存在显著差异, 但基于叶面积和叶生物量分别拟合出的整体分支指数1/a、1/b和代谢指数θ与理论值均无显著差异, 且用叶面积作为代谢速率的替代指标比用叶生物量分析得出的代谢指数与理论值更接近。今后研究应当关注植物叶面积与叶生物量的异速生长关系对植物代谢速率及相关功能特性的影响。     

Branching and metabolic exponents in seven woody plants

West、Brown和Enquist提出的植物分形网络模型(简称WBE模型)认为: 植物的分支指数(1/a, 1/b)决定植物的代谢指数, 当分支指数1/a、1/b分别为理论值2.0、3.0时, 代谢速率与个体大小的3/4次幂成正比, 但是恒定的3/4代谢指数并不能全面地反映植物的代谢情况。基于分支指数的协同变化, Price、Enquist和Savage对WBE模型进行扩展, 提出植物分支参数协同变化模型(简称PES模型)。该文借助于PES模型分析了7种木本植物的分支指数和代谢指数。结果表明: 物种间叶面积与叶生物量呈异速生长关系, 基于叶面积得到的分支指数1/a和代谢指数θ在物种间无显著差异, 基于叶生物量得到的分支指数1/a、1/b和代谢指数θ在物种间均存在显著差异, 但基于叶面积和叶生物量分别拟合出的整体分支指数1/a、1/b和代谢指数θ与理论值均无显著差异, 且用叶面积作为代谢速率的替代指标比用叶生物量分析得出的代谢指数与理论值更接近。今后研究应当关注植物叶面积与叶生物量的异速生长关系对植物代谢速率及相关功能特性的影响。     

Metabolic scaling in insects supports the predictions of the WBE model

The functional association between body size and metabolic rate (BS-MR) is one of the most intriguing issues in ecological physiology. An average scaling exponent of 3/4 is broadly observed across animal and plant taxa. The numerical value of 3/4 is theoretically predicted under the optimized version of West, Brown, and Enquist's vascular resource supply network model. Insects, however, have recently been proposed to express a numerically different scaling exponent and thus application of the WBE network model to insects has been rejected. Here, we re-analyze whether such variation is indeed supported by a global deviation across all insect taxa at the order and family levels to assess if specific taxa influence insect metabolic scaling. We show that a previous reported deviation is largely due to the effect of a single insect family (Termitidae). We conclude that the BS-MR relationship in insects broadly supports the core predictions of the WBE model. We suggest that the deviation observed within the termites warrants further investigation and may be due to either difficulty in accurately measuring termite metabolism and/or particularities of their life history. Future work on allometric scaling should assess the nature of variation around the central tendencies in scaling exponents in order to test if this variation is consistent with core assumptions and predictions of the WBE model that stem by relaxing its secondary optimizing assumptions that lead to the 3/4 exponent.     

On structural theories of basal metabolic rate

Because cells and organisms interface with the environment through surfaces, their design should be governed by surface laws. Yet, basal metabolic rate is not proportional to the 0·67-power of body mass (surface law) but to the 0·75-power of body mass. From the many theories that have derived a surface law, Teissier's dimensional analysis theory was probably the neatest. However, the surface law has been empirically invalidated. Moreover, Teissier assumed that times in the prototype animal and a similar one with different size are in the same ratio as their linear sizes. This is incorrect, however, because heart rates, being inverses of times, should be proportional to the 1/3-power of body mass—but are proportional to the 1/4-power of body mass, which is consistent with a 0·75-power law of basal metabolic rate. McMahon's recent attempt to explain the deviation of the empirical law from a surface law based entirely on structural considerations, is critically examined. It does not appear that purely structural considerations could explain the deviation between the empirical 0·75-law of basal metabolic rate and the surface law.     

Testing metabolic scaling theory using intraspecific allometries in Antarctic microarthropods

Quantitative scaling relationships among body mass, temperature and metabolic rate of organisms are still controversial, while resolution may be further complicated through the use of different and possibly inappropriate approaches to statistical analysis. We propose the application of a modelling strategy based on the theoretical approach of Akaike's information criteria and non‐linear model fitting (nlm). Accordingly, we collated and modelled available data at intraspecific level on the individual standard metabolic rate of Antarctic microarthropods as a function of body mass (M), temperature (T), species identity (S) and high rank taxa to which species belong (G) and tested predictions from metabolic scaling theory (mass‐metabolism allometric exponent b = 0.75, activation energy range 0.2–1.2 eV). We also performed allometric analysis based on logarithmic transformations (lm). Conclusions from lm and nlm approaches were different. Best‐supported models from lm incorporated T, M and S. The estimates of the allometric scaling exponent linking body mass and metabolic rate resulted in a value of 0.696 ± 0.105 (mean ± 95% CI). In contrast, the four best‐supported nlm models suggested that both the scaling exponent and activation energy significantly vary across the high rank taxa (Collembola, Cryptostigmata, Mesostigmata and Prostigmata) to which species belong, with mean values of b ranging from about 0.6 to 0.8. We therefore reached two conclusions: 1, published analyses of arthropod metabolism based on logarithmic data may be biased by data transformation; 2, non‐linear models applied to Antarctic microarthropod metabolic rate suggest that intraspecific scaling of standard metabolic rate in Antarctic microarthropods is highly variable and can be characterised by scaling exponents that greatly vary within taxa, which may have biased previous interspecific comparisons that neglected intraspecific variability.     

Increase in light interception cost and metabolic mass component of leaves are coupled for efficient resource use in the high altitude vegetation

Global syntheses of leaf trait scaling relationships report an increase in light interception costs or ‘diminishing returns’ with increase in leaf area. However, variation in light interception costs across ecological gradients and plant strategies to cope up with these costs are not adequately understood. We analyzed leaf area (A) – leaf dry mass (M), leaf water mass (W) – M and W – A scaling relationships in plants occurring in a high altitude region of western Himalaya across environmental gradients to understand changes in light interception cost and metabolic mass component. M represents light interception cost, whereas, W is considered as a proxy of metabolic mass component for liquid phase being the ultimate source of metabolic activity. Trait values were measured from 9278 leaves belonging to 136 dominant species occurring at different sites, slope aspects, elevations and habitat types. Overall, light interception cost increased with increasing A (scaling exponent (α) < 1 in A–M relationship) and metabolic mass component increased disproportionately high with increasing M and A. We found significant differences in scaling exponents of leaf trait relationship between sites, elevations, slope aspects and habitat types, indicating that increase in light interception cost was more evident at higher elevations, southern slopes and open habitats. Further, with increase in light interception cost, metabolic mass component also increased (α > 1 in W–M and W–A relationships). The changes in scaling exponents of various leaf trait relationships across ecological gradients indicated that vegetation of different regions have differences in light interception cost and metabolic mass component. Moreover, increasing light interception cost (increase in mechanical and hydraulic tissues) with increasing A and increasing metabolic mass (leaf thickness) with increasing A and M are favored in high altitude vegetation. This could be a key strategy of high altitude plants for efficient resource capture and use in harsh environments.     

Reexamination of a canonical model for plant organ biomass partitioning

A previously proposed "canonical" model for the scaling relations among leaf, stem, and root biomass (M(L), M(S), and M(R), respectively) asserts that the proportional relations M(L) ∝ M(S)(3/4) ∝ M(R)(3/4) and M(S) ∝ M(R) hold across seed plant species. This model is scrutinized by determining whether the scaling relations between M(L), M(S), and M(R) vs. basal stem diameter D(S) and between M(L), M(S), and M(R) vs. plant height h are logically consistent with previously predicted scaling exponents. For example, if M(L) is observed to scale as the 2-power of D(S) and the model asserts that M(L) ∝ M(S)(3/4), then M(S) must scale as the 8/3-power of D(S) if the model is valid. Using a large data base for species with self-supporting stems, statistical support was found for most such comparisons between predicted and observed scaling relationships. However, this judgement is predicated on (1) the assertion that the scaling exponents for M(R) with respect to D(S) (or h) are numerically "deflated" due to a systematic underestimate of fine and small root biomass and (2) the stringent protocol used to calculate the 95% confidence intervals of scaling exponents, which favors rejection of the model. In light of these features, the "canonical" model is logically consistent with the new scaling relations reported here. Therefore, the model is judged valid within the context of this evaluation.     

Trade-offs between the metabolic rate and population density of plants

The energetic equivalence rule, which is based on a combination of metabolic theory and the self-thinning rule, is one of the fundamental laws of nature. However, there is a progressively increasing body of evidence that scaling relationships of metabolic rate vs. body mass and population density vs. body mass are variable and deviate from their respective theoretical values of 3/4 and -3/4 or -2/3. These findings questioned the previous hypotheses of energetic equivalence rule in plants. Here we examined the allometric relationships between photosynthetic mass (M(p)) or leaf mass (M(L)) vs. body mass (beta); population density vs. body mass (delta); and leaf mass vs. population density, for desert shrubs, trees, and herbaceous plants, respectively. As expected, the allometric relationships for both photosynthetic mass (i.e. metabolic rate) and population density varied with the environmental conditions. However, the ratio between the two exponents was -1 (i.e. beta/delta = -1) and followed the trade-off principle when local resources were limited. Our results demonstrate for the first time that the energetic equivalence rule of plants is based on trade-offs between the variable metabolic rate and population density rather than their constant allometric exponents.     

WBE 模型及其在生态学中的应用：研究概述

介绍了WBE模型,综述了该模型在生态学中的应用进展。WBE模型,以及以该模型为基础的MTE模型,假设生物体为自相似分形网络结构,提出代谢速率和个体大小之间存在3/4指数关系,分别预测了从个体到生物圈多个尺度上的生物属性之间的异速生长关系,而且部分得到了验证。WBE模型的应用涵盖了个体组织生物量、年生长率,种群密度和生态系统单位面积产量、能量流动率等多个方面;即使在生物圈大尺度上,WBE模型也可用来预测试验中无法直接测量的特征变量的属性,如全球碳储量的估算等。至今,关于WBE和MTE模型仍然存在各种褒贬争论,讨论焦点主要集中于模型建立的前提假设以及权度指数的预测。今后的研究工作应规范试验技术和方法,考虑物种多样性和环境等因素的影响,提出符合各类生物的模型结构体系,使其具有更广泛的应用性和预测性。     

Intraspecific temperature dependence of the scaling of metabolic rate with body mass in fishes and its ecological implications

Metabolism constitutes a fundamental property of all organisms. Metabolic rate is commonly described to scale as a power function of body size and exponentially with temperature, thereby treating the effects of body size and temperature independently. Mounting evidence shows that the scaling of metabolic rate with body mass itself depends on temperature. Across‐species analyses in fishes suggest that the mass‐scaling exponent decreases with increasing temperature. However, whether this relationship holds at the within‐species level has rarely been tested. Here, we re‐analyse data on the metabolic rates of four freshwater fish species, two coregonids and two cyprinids, that cover wide ranges of body masses and their naturally experienced temperatures. We show that the standard metabolic rate of the coregonids is best fit when accounting for a linear temperature dependence of the scaling of metabolic rate with body mass, whereas a constant mass‐scaling exponent is supported in case of the cyprinids. Our study shows that phenotypic responses to temperature can result in temperature‐dependent scaling relationships at the species level and that these responses differ between taxa. Together with previous findings, these results indicate that evolutionarily adaptive and phenotypically plastic responses to temperature affect the scaling of metabolic rate with body mass in fishes.     

Evidence of variant intra- and interspecific scaling of tree crown structure and relevance for allometric theory

General scaling rules or constants for metabolic and structural plant allometry as assumed by the theory of Euclidian geometric scaling (2/3-scaling) or metabolic scaling (3/4-scaling) may meet human's innate propensity for simplicity and generality of pattern and processes in nature. However, numerous empirical works show that variability of crown structure rather than constancy is essential for a tree's success in coping with crowding. In order to link theory and empiricism, we analyzed the intra- and inter-specific scaling of crown structure for 52 tree species. The basis is data from 84 long-term plots of temperate monospecific forests under survey since 1870 and a set of 126 yield tables of angiosperm and gymnosperm forest tree species across the world. The study draws attention to (1) the intra-specific variation and correlation of the three scaling relationships: tree height versus trunk diameter, crown cross-sectional area versus trunk diameter, and tree volume versus trunk diameter, and their dependence on competition, (2) the inter-specific variation and correlation of the same scaling exponents ([Formula: see text] and [Formula: see text]) across 52 tree species, and (3) the relevance of the revealed variable scaling of crown structure for leaf organs and metabolic scaling. Our results arrive at suggesting a more extended metabolic theory of ecology which includes variability and covariation between allometric relationships as prerequisite for the individual plant's competitiveness.     

Revising the distributive networks models of West, Brown and Enquist (1997) and Banavar, Maritan and Rinaldo (1999): metabolic inequity of living tissues provides clues for the observed allometric scaling rules

Basic assumptions of two distributive network models designed to explain the 3/4 power scaling between metabolic rate and body mass are re-analysed. It is shown that these models could have consistently accounted for the observed scaling patterns if and only if body mass M had scaled as L4, where L is body length, in the model of Banavar et al. (1999, Nature 399, 130-132), or if spatial volume VF occupied by the distributive network had scaled as M3/4 in the model of West et al. (1997, Science 276, 122-126). Lack of agreement between these predictions and observational evidence invalidates both models rendering them mathematically controversial. It is further shown that consideration of distributive networks can nevertheless yield realistic values of scaling exponents under the major assumption that living organisms are designed so as to keep the mass-specific metabolic rate of important functional tissues in the vicinity of a size-independent optimum value. Mass-specific metabolic rate of subsidiary mechanical tissues can be small and vary with body mass. Different patterns of spatial distribution of metabolically active biomass within the organism result in different patterns of allometric scaling. From the available evidence the presumable optimum value of mass-specific metabolic rate of living matter is estimated to be in the vicinity of 1-10 W kg-1.     

A common genetic basis to the origin of the leaf economics spectrum and metabolic scaling allometry

Many facets of plant form and function are reflected in general cross‐taxa scaling relationships. Metabolic scaling theory (MST) and the leaf economics spectrum (LES) have each proposed unifying frameworks and organisational principles to understand the origin of botanical diversity. Here, we test the evolutionary assumptions of MST and the LES using a cross of two genetic variants of Arabidopsis thaliana. We show that there is enough genetic variation to generate a large fraction of variation in the LES and MST scaling functions. The progeny sharing the parental, naturally occurring, allelic combinations at two pleiotropic genes exhibited the theorised optimum ¾ allometric scaling of growth rate and intermediate leaf economics. Our findings: (1) imply that a few pleiotropic genes underlie many plant functional traits and life histories; (2) unify MST and LES within a common genetic framework and (3) suggest that observed intermediate size and longevity in natural populations originate from stabilising selection to optimise physiological trade‐offs.     

Methods for Identifying a Default Cross-Species Scaling Factor

The need to identify “toxicologically equivalent” doses across different species is a major issue in toxicology and risk assessment. In this article, we describe an approach for establishing default cross-species extrapolation factors used to scale oral doses across species for non-carcinogenic endpoints. This work represents part of an on-going effort to harmonize the way animal data are evaluated for carcinogenic and non-carcinogenic endpoints. In addition to considering default scaling factors, we also discuss how chemical-specific data (e.g., metabolic or mechanistic data) can be incorporated into the dose extrapolation process. After first examining the required properties of a default scaling methodology, we consider scaling approaches based on empirical relationships observed for particular classes of compounds and also more theoretical approaches based on general physiological principles (i.e, allometry). The available data suggest that the empirical and allometric approaches each provide support for the idea that toxicological risks are approximately equal when daily oral doses are proportional to body weight raised to the 3/4-power. We also discuss specific challenges for dose scaling related to different routes of exposure, acute versus chronic toxicity, and extrapolations related to particular life stages (e.g., childhood).