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.     

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。     

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 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.     

Growth-size scaling relationships of woody plant species differ from predictions of the Metabolic Ecology Model

The Metabolic Ecology Model predicts that tree diameter ( D ) growth ( dD/dt ) scales with D ^1/3. Using data on diameter growth and height–diameter relationships for 56 and 40 woody species, respectively, from forests throughout New Zealand, we tested one prediction and two assumptions of this model: (i) the exponent of the growth–diameter scaling relationship equals 1/3 and is invariant among species and growth forms, (ii) small and large individuals are invariant in their exponents and (iii) tree height scales with D ^2/3. We found virtually no support for any prediction or assumption: growth–diameter scaling exponents varied substantially among species and growth forms, correlated positively with species' maximum height, and shifted significantly with increasing individual size. Tree height did not scale invariantly with diameter. Based on a quantitative test, violation of these assumptions alone could not explain the model's poor fit to our data, possibly reflecting multiple, unsound assumptions, as well as unaccounted-for variation that should be incorporated.     

代谢异速生长理论及其在微生物生态学领域的应用

新陈代谢是生物的基本生理过程,影响生物在不同环境中参与物质循环和能量转化的过程.代谢速率作为生物体重要的生命过程指标,几乎影响所有的生物活性速率,且在很多研究中均表现出异速生长现象.所谓代谢异速是指生物体代谢速率与其个体大小(或质量)之间存在的幂函数关系.代谢异速生长理论的提出,从机制模型角度解释了代谢异速关系这一普遍存在的生命现象.该理论利用分形几何学及流体动力学等原理,从生物能量学角度阐释了异速生长规律的机理,证实了3/4权度指数的存在;但同时有研究表明,权度指数因环境因素等影响处于2/3-1范围之间而非定值.随着研究工作的深入,代谢异速生长理论研究从起初的宏观动植物领域拓展到了微生物领域,在研究微生物的代谢异速生长理论时,可将微生物的可操作分类单元(Operational taxonomic unit,OTU)或具有特定功能的功能群视为一个微生物个体,基于其遗传多样性和功能多样性特征进行表征,以便于将微生物群落多样性与其生态功能性联系起来,使该理论在微生物生态学领域得到有效的补充和完善.尽管细菌具有独特的生物学特性,但与宏观生物系统中观测到的现象表现出明显的一致性.有研究表明,3个农田土壤细菌基于遗传多样性的OTU数的平均周转率分别为0.71、0.80和0.84,介于2/3与1之间,可能与生物代谢异速指数有一定关联,为微生物代谢异速指数的研究提出了一个参考解决方案.鉴于微生物个体特征和生物学特性,在分析代谢速率与个体大小关系中,从微生物单位个体的定义、个体大小表征到计量单位的统一,仍需更多的理论支持.分析了代谢异速生长理论在微生物与生态系统功能关系研究中的可能应用,延伸了该理论的应用范围,并对尚待加强的研究问题进行了评述和展望.     

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.     

Phylogeny affects estimation of metabolic scaling in mammals

Abstract.— The relationship between body size and metabolic rate is a crucial issue in organismal biology and evolution. There has been considerable debate over whether the scaling exponent of the relationship is 0.75 (Kleiber's Law) or 0.67. Here we show that determination of this exponent for mammals depends on both the evolutionary tree and the regression model used in the comparative analysis. For example, more recent molecular-based phylogenies tend to support a 0.67 exponent, whereas older phylogenies, mostly based on morphological data, suggest a 0.75 exponent. However, molecular phylogenies yield more variable results than morphological phylogenies and thus are not currently helping to resolve the issue.     

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).     

Metabolic constraints on the body size scaling of extreme population densities

Pest outbreaks, harmful algal blooms and population collapses are extreme events with critical consequences for ecosystems. Therefore, understanding the ecological mechanisms underlying these extreme events is crucial. We evaluated theoretical predictions on the size scaling and variance of extreme population abundance by combining (i) the generalized extreme value (GEV) theory and (ii) the resource-limited metabolic restriction hypothesis for population abundance. Using the phytoplankton data from the L4 station in the English Channel, we showed a negative size scaling of the expected value of maximal density, whose confidence interval included the predicted metabolic scaling (α = −1) supporting theoretical predictions. The role of resources and temperature in the distribution of the size–abundance pattern and residuals was well characterized by the GEV distribution. This comprehensive modelling framework will allow to elucidate community structure and fluctuations and provide unbiased return times estimates, thereby improving the prediction accuracy of the timing of the population outbreaks.     

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.     

Cryptic individual scaling relationships and the evolution of morphological scaling

Morphological scaling relationships between organ and body size—also known as allometries—describe the shape of a species, and the evolution of such scaling relationships is central to the generation of morphological diversity. Despite extensive modeling and empirical tests, however, the modes of selection that generate changes in scaling remain largely unknown. Here, we mathematically model the evolution of the group‐level scaling as an emergent property of individual‐level variation in the developmental mechanisms that regulate trait and body size. We show that these mechanisms generate a “cryptic individual scaling relationship” unique to each genotype in a population, which determines body and trait size expressed by each individual, depending on developmental nutrition. We find that populations may have identical population‐level allometries but very different underlying patterns of cryptic individual scaling relationships. Consequently, two populations with apparently the same morphological scaling relationship may respond very differently to the same form of selection. By focusing on the developmental mechanisms that regulate trait size and the patterns of cryptic individual scaling relationships they produce, our approach reveals the forms of selection that should be most effective in altering morphological scaling, and directs researcher attention on the actual, hitherto overlooked, targets of selection.     

某部机关中老年干部代谢综合征现况调查

目的:了解某部机关中老年干部代谢综合征(Metabolic syndrome,MS)的患病情况,为该类人群疾病的防治提供依据。方法:收集2013年4~5月年在解放军第309医院体检的452例某部机关中老年干部查体资料,按2007年《中国成人血脂异常防治指南》提出的代谢综合征诊断标准进行诊断,分析代谢综合征及代谢指标异常患病情况。结果:受检人群MS患病率11.73%,男性高于女性(P0.05)。年龄组患病率以60~69岁组最高(23.08%)。单项代谢异常检出率从高至低依次为血脂异常(41.15%)、超重和/肥胖(38.5%)、血压升高(25.0%)和血糖升高(9.73%)。不同年龄组代谢指标异常分布情况不同。代谢异常类型以血压升高+血脂升高+超重/肥胖模式人数最多。除MS外,仍有34.52%的人群存在1-2种代谢指标异常。结论:MS可防可控,应注重以胰岛素抵抗(Insulin Resistance,IR)为靶点的多危险因素综合治疗。     

Unimodal size scaling of phytoplankton growth and the size dependence of nutrient uptake and use

Phytoplankton size structure is key for the ecology and biogeochemistry of pelagic ecosystems, but the relationship between cell size and maximum growth rate (μ_max) is not yet well understood. We used cultures of 22 species of marine phytoplankton from five phyla, ranging from 0.1 to 10⁶ μm³ in cell volume (V_cell), to determine experimentally the size dependence of growth, metabolic rate, elemental stoichiometry and nutrient uptake. We show that both μ_max and carbon‐specific photosynthesis peak at intermediate cell sizes. Maximum nitrogen uptake rate (V_maxN) scales isometrically with V_cell, whereas nitrogen minimum quota scales as V_cell^0.84. Large cells thus possess high ability to take up nitrogen, relative to their requirements, and large storage capacity, but their growth is limited by the conversion of nutrients into biomass. Small species show similar volume‐specific V_maxN compared to their larger counterparts, but have higher nitrogen requirements. We suggest that the unimodal size scaling of phytoplankton growth arises from taxon‐independent, size‐related constraints in nutrient uptake, requirement and assimilation.     

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.     

Nitrogen/phosphorus leaf stoichiometry and the scaling of plant growth

We adopted previous N : P stoichiometric models for zooplankton relative growth to predict the relative growth rates of the leaves μ _L of vascular plants assuming that annual leaf growth in dry mass is dictated by how leaf nitrogen N _L is allocated to leaf proteins and how leaf phosphorus P _L is allocated to rRNA. This model is simplified provided that N _L scales as some power function of P _L across the leaves of different species. This approach successfully predicted the μ _L of 131 species of vascular plants based on the observation that, across these species, N _L scaled, on average, as the 3/4 power of P _L, i.e. N _L ∝  P     . When juxtaposed with prior allometric theory and observations, our findings suggest that a transformation in N : P stoichiometry occurs when the plant body undergoes a transition from primary to secondary growth.