Lack of Evidence for 3/4 Scaling of Metabolism in Terrestrial Plants

Scaling, as the translation of information across spatial, temporal, and organizational scales, is essential to predictions and understanding in all sciences and has become a central issue in ecology. A large body of theoretical and empirical evidence concerning allometric scaling in terrestrial individual plants and plant communities has been constructed around the fractal volume-filling theory of West, Brown, and Enquist (the WBE model). One of the most thought-provoking findings has been that the metabolic rates of plants, like those of animals, scale with their size as a 3/4 power law. The earliest, single most-important study cited in support of the application of the WBE model to terrestrial plants claims that whole-plant resource use in terrestrial plants scales as the 3/4 power of total mass, as predicted by the WBE model.However, in the present study we show that empirical data actually do not support such a claim. More recent studies cited as evidence for 3/4 scaling also suffer from several statistical and data-related problems. Using a forest biomass dataset including 1 266 plots of 17 main forest types across China, we explored the scaling exponents between tree productivity and tree mass and found no universal value across forest stands. We conclude that there is not sufficient evidence to support the existence of a single constant scaling exponent for the metabolism-biomass relationship for terrestrial plants.     

Scaling Relationships between Leaf Mass and Total Plant Mass across Chinese Forests

Biomass partitioning is important for illustrating terrestrial ecosystem carbon flux. West, Brown and Enquist (WBE) model predicts that an optimal 3/4 allometric scaling of leaf mass and total biomass of individual plants will be applied in diverse communities. However, amount of scientific evidence suggests an involvement of some biological and environmental factors in interpreting the variation of scaling exponent observed in empirical studies. In this paper, biomass information of 1175 forested communities in China was collected and categorized into groups in terms of leaf form and function, as well as their locations to test whether the allocation pattern was conserved or variable with internal and/or environmental variations. Model Type II regression protocol was adopted to perform all the regressions. The results empirically showed that the slopes varied significantly across diverse forested biomes, between conifer and broadleaved forests, and between evergreen and deciduous forests. Based on the results, leaf form and function and their relations to environments play a significant role in the modification of the WBE model to explore more accurate laws in nature.     

Allometric scaling among tree components in Pinus massoniana stands with different sites

The WBE model was used to predict intraspecific scaling relationships among mean branch, needle, stem, root, and above-ground masses across eight stands of Pinus massoniana to test whether the scaling exponent was (1) dependent on site and (2) in accordance with WBE theory. The results showed that mean stem and root masses as well as mean above-ground and root masses scaled in a near-isometric manner across sites, except at two sites, which exhibited an exponent slightly less than unity. Mean needle mass scaled as 3/4 power of mean stem mass, except at one site, which exhibited an exponent slightly higher than 3/4. Mean branch mass scaled isometrically with mean stem mass at each site. These results supported the WBE theory. However, mean branch mass across sites scaled neither as 3/4 nor 1 power of mean stem mass, indicating that the scaling relationship predicted by WBE theory for these two components did not hold in P. massoniana stands.     

Sizing up allometric scaling theory

Metabolic rate, heart rate, lifespan, and many other physiological properties vary with body mass in systematic and interrelated ways. Present empirical data suggest that these scaling relationships take the form of power laws with exponents that are simple multiples of one quarter. A compelling explanation of this observation was put forward a decade ago by West, Brown, and Enquist (WBE). Their framework elucidates the link between metabolic rate and body mass by focusing on the dynamics and structure of resource distribution networks-the cardiovascular system in the case of mammals. Within this framework the WBE model is based on eight assumptions from which it derives the well-known observed scaling exponent of 3/4. In this paper we clarify that this result only holds in the limit of infinite network size (body mass) and that the actual exponent predicted by the model depends on the sizes of the organisms being studied. Failure to clarify and to explore the nature of this approximation has led to debates about the WBE model that were at cross purposes. We compute analytical expressions for the finite-size corrections to the 3/4 exponent, resulting in a spectrum of scaling exponents as a function of absolute network size. When accounting for these corrections over a size range spanning the eight orders of magnitude observed in mammals, the WBE model predicts a scaling exponent of 0.81, seemingly at odds with data. We then proceed to study the sensitivity of the scaling exponent with respect to variations in several assumptions that underlie the WBE model, always in the context of finite-size corrections. Here too, the trends we derive from the model seem at odds with trends detectable in empirical data. Our work illustrates the utility of the WBE framework in reasoning about allometric scaling, while at the same time suggesting that the current canonical model may need amendments to bring its predictions fully in line with available datasets.     

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。     

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.     

Computer simulations support a core prediction of a contentious plant model

? Premise of the study: An overarching but vigorously debated plant model proposed by the West, Brown, Enquist (WBE) theory predicts the scaling relationships for numerous botanical phenomena. However, few studies have evaluated this model's basic assumptions, one of which is that natural selection has resulted in hierarchal networks that minimize the energy required to distribute nutrients internally and have thus produced highly efficient organisms. ? Methods: If these core assumptions are correct, an "idealized" plant complying with all of the scaling relationships emerging from the WBE plant model should rapidly outcompete other plants, even those that differ slightly from it. To test this reasoning, a computer model was used to simulate competition between an idealized WBE plant, a generic "average" angiosperm (GA), and one of seven variants of the idealized WBE plant, each being similar to the GA in one of the GA's scaling parameters. ? Key results: Replicate simulations show that the idealized WBE plant rapidly outcompetes all other plants under light-shade and open-field conditions. However, changing only one of the WBE's scaling parameters results in death or in the coexistence of WBE and GA plants. ? Conclusions: These simulations support a core assumption of the WBE plant model and suggest why this idealized plant has not evolved.     

Scaling relationship between tree respiration rates and biomass

The WBE theory proposed by West, Brown and Enquist predicts that larger plant respiration rate, R, scales to the three-quarters power of body size, M. However, studies on the R versus M relationship for larger plants (i.e. trees larger than saplings) have not been reported. Published respiration rates of field-grown trees (saplings and larger trees) were examined to test this relationship. Our results showed that for larger trees, aboveground respiration rates R_A scaled as the 0.82-power of aboveground biomass M_A, and that total respiration rates R_T scaled as the 0.85-power of total biomass M_T, both of which significantly deviated from the three-quarters scaling law predicted by the WBE theory, and which agreed with 0.81–0.84-power scaling of biomass to respiration across the full range of measured tree sizes for an independent dataset reported by Reich et al. (Reich et al. 2006 Nature 439, 457–461). By contrast, R scaled nearly isometrically with M in saplings. We contend that the scaling exponent of plant metabolism is close to unity for saplings and decreases (but is significantly larger than three-quarters) as trees grow, implying that there is no universal metabolic scaling in plants.     

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.     

植物代谢速率与个体生物量关系研究进展

植物的各项生理生态功能（例如,呼吸、生长和繁殖）都与个体生物量成异速生长关系。West, Brown及Enquist基于分形网络结构理论所提出的WBE模型认为：植物的代谢（呼吸）速率正比于个体生物量的3/4次幂。然而,恒定的“3/4异速生长指数”与实测数据、植物生理生态学等研究之间存在矛盾,引发激烈的争论。论文分析了不同回归方法对代谢指数的影响,重点对植物代谢速率与个体生物量异速生长关系研究进展进行了综述,分析并得出了植物代谢指数在小个体时接近1.0,并随着生物量的增加而系统减小,且其密切依赖于氮含量的调控的结论。据此,提出了进一步深入研究植物代谢速率个体生物量关系需要解决的一些科学问题。     

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.     

Gap creation and regenerative processes driving diversity and structure of mangrove ecosystems

Turnover within both mangrove and terrestrial forests is driven by standdevelopment in conjunction with factors influencing tree death andreplacement at various temporal and spatial scales. Development interrestrial forests appears comparable with that in mangroves but turnoverseems to differ considerably between these broad forest types. The mostimportant difference is in the character of small forest gaps. Gaps arecommon in terrestrial forests but those in mangroves rarely involve falls oflarge older trees in the first instance. Instead, mangrove trees usually diestanding in small clusters of mixed age cohorts. Identifying a common causefor gap creation in mangroves might be important towards understandingwhat drives forest turnover but there is a greater need to quantify thisprocess. Small-scale disturbance in mangrove forests is poorly quantified butpreliminary evidence implies that its' importance may have been greatlyunder-estimated. Based on available observations, a conceptual model ofmangrove forest development and gap regeneration is proposed. The modelhelps explain the peculiar characteristics and structure of mangrove forestsand how these forests might respond to changing environmental conditionsand disturbance at various landscape scales.     

Scaling of tree height and trunk diameter as a function of ring number

 Phipps (1967) showed, in suppressed maples and oaks, that on average the cross-sectional area of successive tree rings is independent of ring number. From that empirical result I argued that mean ring width and mean ring diameter scale as the inverse square root and square root, respectively, of ring number (Prothero 1997). Here I give an illustrative argument as to how macroscopic trunk variables may be related quantitatively to microscopic ring variables. First, evidence is presented consistent with the theory of Greenhill (1881) and the empirical evidence of McMahon (1973) that tree height in a diversity of species scales as about the two-thirds power of trunk diameter. Here I report further evidence bearing on the same question. From these combined results I suggest that tree height, and by implication trunk height, and other parameters governing macroscopic tree trunk proportions, can all be expressed as powers of a microscopic variable, namely ring number, where the scaling exponents are all proper fractions. I suggest that at least one of these relationships is, in principle, non-adaptive. Received: 9 June 1998 / Accepted: 4 February 1999     

Ecological scaling laws link individual body size variation to population abundance fluctuation

Scaling research has seen remarkable progress in the past several decades. Many scaling relationships were discovered within and across individual and population levels, such as species–abundance relationship, Taylor's law, and density mass allometry. However none of these established patterns incorporate individual variation in the formulation. Individual body size variation is a key evolutionary phenomenon and closely related to ecological diversity and species adaptation. Using a macroecological approach, I test 57 Long‐Term Ecological Research data sets and show that a power‐law and a generalized power‐law function describe well the mean‐variance scaling of individual body mass. This relationship connects Taylor's law and density mass allometry, and leads to a new scaling pattern between the individual body size variation and population abundance fluctuation, which is confirmed using freshwater fish and forest tree data. Underlying mechanisms and implications of the proposed scaling relationships are discussed. This synthesis shows that integration and extension of existing ecological laws can lead to the discovery of new scaling patterns and complete our understanding of the relation between individual trait and population abundance. Synthesis Scaling relationships are useful for community ecology as they reveal ubiquitous patterns across different levels of biological organizations. This work extends and integrates two existing scaling laws: Taylor's law and density‐mass allometry, and derives a new variance allometry between individual body mass and population abundance. The result shows that diverse individual body size is associated with stable population fluctuation, reflecting the effect of individual traits on population characteristics. Confirmed by several empirical data sets, these scaling relationships suggest new ways to study the underlying mechanisms of Taylor's law and have profound implications for fisheries and other applied sciences.     

Variant Scaling Relationship for Mass-Density Across Tree-Dominated Communities

The past few decades have seen a resurgence of Interest in biological allometry. Specifically, a number of recent studies has suggested a -4/3 Invariant scaling relationship between mass and density that Is universally valid for tree-dominated communities, regardless of their phyietic affiliation or habitat. In the present study, we test this scaling relationship using a comprehensive forest biomass database, Including 1 266 plots of six blomes and 17 forest types across China. The present study shows that the scaling exponent of the massdensity relationship varies across different tree-dominated communities and habitats. This great variability In the scaling exponent makes any generalization unwarranted. Although Inappropriate regression methods can lead to flawed estimation of the scaling exponent, inconsistency of theoretical framework and empirical patterns may have undermined the validity of previous work.     

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。     

Exoskeletal chitin scales isometrically with body size in terrestrial insects

The skeletal system of animals provides the support for a variety of activities and functions. For animals such as mammals, which have endoskeletons, research has shown that skeletal investment (mass) scales with body mass to the 1.1 power. In this study, we ask how exoskeletal investment in insects scales with body mass. We measured the body mass and mass of exoskeletal chitin of 551 adult terrestrial insects of 245 species, with dry masses ranging from 0.0001 to 2.41 g (0.0002–6.13 g wet mass) to assess the allometry of exoskeletal investment. Our results showed that exoskeletal chitin mass scales isometrically with dry body mass across the Insecta as M_chitin = a M, where b = 1.03 ± 0.04, indicating that both large and small terrestrial insects allocate a similar fraction of their body mass to chitin. This isometric chitin‐scaling relationship was also evident at the taxonomic level of order, for all insect orders except Coleoptera. We additionally found that the relative exoskeletal chitin investment, indexed by the coefficient, a, varies with insect life history and phylogeny. Exoskeletal chitin mass tends to be proportionally less and to increase at a lower rate with mass in flying than in nonflying insects (M_{flying insect chitin} = ?0.56 × M; M_{nonflying insect chitin} = ?0.55 × M), and to vary with insect order. Isometric scaling (b = 1) of insect exoskeletal chitin suggests that the exoskeleton in insects scales differently than support structures of most other organisms, which have a positive allometry (b > 1) (e.g., vertebrate endoskeleton, tree secondary tissue). The isometric pattern that we document here additionally suggests that exoskeletal investment may not be the primary limit on insect body size. J. Morphol., 2010. © 2010 Wiley‐Liss, Inc.     

A New Model for Size-Dependent Tree Growth in Forests

Tree growth, especially diameter growth of tree stems, is an important issue for understanding the productivity and dynamics of forest stands. Metabolic scaling theory predicted that the 2/3 power of stem diameter at a certain time is a linear function of the 2/3 power of the initial diameter and that the diameter growth rate scales to the 1/3 power of the initial diameter. We tested these predictions of the metabolic scaling theory for 11 Japanese secondary forests at various growth stages. The predictions were not supported by the data, especially in younger stands. Alternatively, we proposed a new theoretical model for stem diameter growth on the basis of six assumptions. All these assumptions were supported by the data. The model produced a nearly linear to curvilinear relationship between the 2/3 power of stem diameters at two different times. It also fitted well to the curvilinear relationship between diameter growth rate and the initial diameter. Our model fitted better than the metabolic scaling theory, suggesting the importance of asymmetric competition among trees, which has not been incorporated in the metabolic scaling theory.     

Power laws and plant trait variation in spatio-temporally heterogeneous environments

A challenge

Variation is ubiquitous in nature across all spatial and temporal scales and underlies prominent ecological and evolutionary theories. Although understanding the causes and consequences of trait variation is a central goal of trait-based ecology, the scaling of trait variance across space and time (variance scaling) is unresolved.

A solution

We argue that characterizing trait variance across spatio-temporal scales using a combination of prominent power laws can elucidate the role of environmental variability in trait variation and potential mechanisms driving trait patterns. In particular, the species–time–area relationship and Taylor's power law help to establish a generalizable framework for developing and testing variance scaling theory. Finally, we outline priority research questions and tractable systems for answering them. Successional forests, long-term forest monitoring networks and censuses of short-lived taxa are ideal for coupling high-resolution environmental data with measurements of trait variance across scales to test the models proposed here.

Main conclusions

Characterizing the behaviour of variance across spatio-temporal scales is feasible and a prerequisite for developing a predictive theory of trait-based ecology.