A theoretical interpretation of length-biomass allometry of predominantly bidimensional seaweeds

Experimental studies on bidimensional seaweeds revealed a scaling exponent of 0.472 for their length-biomass allometry. This was significantly higher than the value 0.25, which was proposed earlier as universal for all primary producers, based on the data for unicellular microalgae and vascular plants. Later, an exponent of 0.5 was theoretically derived, which agreed, to some extent, with experimental findings. Here, it is shown that there exists a power-law relation between the two perpendicular length parameters along the directions of growth of a bidimensional organism. The length-biomass allometric parameters can be expressed in terms of this power index. A relation between the allometric scaling exponent and allometric constant, involving the mass per unit area, has been obtained analytically. A method is proposed to determine the power index experimentally. Some mathematical expressions, relating mass, length and other parameters, have been formulated and these would be useful for experimental purposes in allometric studies. Analyzing images from an experimental study, a lot of parameters, regarding flat seaweeds, have been determined by analytical and numerical techniques.     

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.     

异速生长模型研究概述

最近,关于异速生长模型的讨论再次成为焦点,讨论热点为异速生长指数的取值及其理论解释.本文综述了WBE 97、BMR(99)模型的相关研究,重点介绍了MGL模型及由此模型得到的结果:个体整体的新陈代谢率与个体的质量没有明显依赖关系,其标度指数不是一个固定的值,而是一个区间[2/3,1].考虑的视角从个体整体的新陈代谢率转到单位质量的新陈代谢率,通过对不同物种、不同环境的单位质量新陈代谢率的研究,发现对大多数物种,其值落在一个具有普适性的上、下界的区间内;认为存在单位质量的新陈代谢率最小值确定了个体的大小,并建立基于该最小值的描述个体大小与温度关系的数学模型,该模型得到实验数据验证.     

The observed range for temporal mean-variance scaling exponents can be explained by reproductive correlation

The mean-variance scaling relationship known as Taylor's power law has been well documented empirically over the past four decades but a general theoretical explanation for the phenomenon does not exist. Here we provide an explanation that relates empirical patterns of temporal mean-variance scaling to individual level reproductive behavior. Initially, we review the scaling behavior of population growth models to establish theoretical limits for the scaling exponent b that is in agreement with the empirically observed range (1≤b≤2). We go on to show that the degree of reproductive covariance among individuals determines the scaling exponent b. Independent reproduction results in an exponent of one, while completely correlated reproduction results in the upper limit of two. Intermediate exponents, which are common empirically, can be generated through the decay of reproductive covariance with increasing population size. Finally, we describe how the link between reproductive correlation and the scaling exponent provides a way to infer properties of individual-level reproductive behavior, such as the relative influence of demographic stochasticity, from a macroecological pattern.     

Modeling fractal structure of city-size distributions using correlation functions

Zipf's law is one the most conspicuous empirical facts for cities, however, there is no convincing explanation for the scaling relation between rank and size and its scaling exponent. Using the idea from general fractals and scaling, I propose a dual competition hypothesis of city development to explain the value intervals and the special value, 1, of the power exponent. Zipf's law and Pareto's law can be mathematically transformed into one another, but represent different processes of urban evolution, respectively. Based on the Pareto distribution, a frequency correlation function can be constructed. By scaling analysis and multifractals spectrum, the parameter interval of Pareto exponent is derived as (0.5, 1]; Based on the Zipf distribution, a size correlation function can be built, and it is opposite to the first one. By the second correlation function and multifractals notion, the Pareto exponent interval is derived as [1, 2). Thus the process of urban evolution falls into two effects: one is the Pareto effect indicating city number increase (external complexity), and the other the Zipf effect indicating city size growth (internal complexity). Because of struggle of the two effects, the scaling exponent varies from 0.5 to 2; but if the two effects reach equilibrium with each other, the scaling exponent approaches 1. A series of mathematical experiments on hierarchical correlation are employed to verify the models and a conclusion can be drawn that if cities in a given region follow Zipf's law, the frequency and size correlations will follow the scaling law. This theory can be generalized to interpret the inverse power-law distributions in various fields of physical and social sciences.     

Ontogenic growth of the haemopoietic stem cell pool in humans

Recently, the size of the active stem cell pool has been predicted to scale allometrically with the adult mass of mammalian species with a 3/4 power exponent, similar to what has been found to occur for the resting metabolic rate across species. Here we investigate the allometric scaling of human haemopoietic stem cells (HSCs) during ontogenic growth and predict a linear scaling with body mass. We also investigate the allometric scaling of resting metabolic rate during growth in humans and find a linear scaling with mass similar to that of the haemopoietic stem cell pool. Our findings suggest a common underlying organizational principle determining the linear scaling of both the stem cell pool and resting metabolic rate with mass during ontogenic growth within the human species, combined with a 3/4 scaling with adult mass across mammalian species. It is possible that such common principles remain valid for haemopoiesis in other mammalian species.     

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.     

FURTHER EVIDENCE FOR ANOMALOUS SIZE SCALING OF RESPIRATION IN PHYTOPLANKTON1

Respiration per unit mass decreases as organism size increases among metazoans and heterotrophic unicells. The rate of decrease is described by a power function of organism mass; the exponent of the power function is 0.75 (Three-fourths Rule). Previously unanalyzed respiration rates for 11 species of phytoplankton ranging in size over four orders of magnitude show a size-scaling exponent of 1.13 (SE, ±0.15), which is statistically different from 0.75. This result confirms the result of an earlier study of eight phytoplankton species indicating that size scaling of respiration is absent or minimal in phytoplankton, in contrast to the pattern of heterotrophic unicells. The size-related range of respiration rates per unit mass across the full size spectrum of phytoplankton would be approximately 18–fold if respiration were scaled according to the Three-fourths Rule. If respiration does not scale with size or scales minimally with size, as suggested by present evidence, the size-related range of rates will be much smaller or negligible. The apparent anomaly of size scaling for phytoplankton respiration is potentially of great ecological and adaptive significance in unicellular algae.     

Scaling properties of cell and organelle size

How size is controlled is a fundamental question in biology. In this review, we discuss the use of scaling relationships—for example, power laws of the form y ∝ xα—to provide a framework for comparison and interpretation of size measurements. Such analysis can illustrate the biological and physical principles underlying observed trends, as has been proposed for the allometric dependence of metabolic rate or limb structure on organism mass. Techniques for measuring size at smaller length-scales continue to improve, leading to more data on the control of size in cells and organelles. Size scaling of these structures is expected to influence growth patterns, functional capacity, and intracellular transport. Furthermore, organelles such as the nucleus, mitochondria, and endoplasmic reticulum show widely varying morphologies which affect their scaling properties. We provide brief summaries of these issues for individual organelles, and conclude with a discussion on how to apply this concept to better understand the mechanisms of size control in the cellular environment.     

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.     

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.     

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.     

Determinate growth and the scaling of photosynthetic energy intake in the solitary coral Fungia concinna (Verrill)

For many marine invertebrates, the maximum size of an individual is influenced heavily by environmental factors and may be limited by energetic constraints. In this study, an energetic model developed originally for anemones was applied to the free-living scleractinian Fungia concinna (Verrill) from Moorea (French Polynesia) to test the hypothesis that energetic constraints limit the size of this solitary coral. The modified model assumed that photosynthesis was the primary source of metabolic energy, and that metabolic costs were represented by aerobic respiration; these sources and sinks of energy were compared using daily energy budgets that were analyzed using double logarithmic regressions of energy against coral size. With this approach, energy limitation is characterized by a scaling exponent for energetic cost (b_cost) that is larger than the scaling exponent for energy intake (b_intake). For the size range of F. concinna studied, b_intake = 0.73 ± 0.09 and b_cost = 0.46 ± 0.10, thereby demonstrating that large individuals accumulated an energetic surplus, even when the expenditure associated with host tissue and symbiont growth was included in the model. The surplus of energy that this coral acquires as it grows appears to be driven by the scaling of traits associated functionally with the scaling of respiration and photosynthesis. Specifically, tissue biomass displayed a strong positive allometry with respect to surface area (i.e., b > 1), and this constraint on surface area may be the mechanistic basis of the low scaling exponent for metabolic cost. In contrast, the capacity for autotrophy - defined indirectly as Symbiodinium population density and chlorophyll content - increased isometrically with surface area, and likely contributed to the higher scaling exponent for intake relative to cost. Our results suggest that growth in F. concinna is not limited strictly by energy, but instead maximum size must be determined by alternative physiological or ecological constraints.     

The terrestrial evolution of metabolism and life – by the numbers

Background  

Allometric scaling relating body mass to metabolic rate by an exponent of the former (Kleiber's Law), commonly known as quarter-power scaling (QPS), is controversial for claims made on its behalf, especially that of its universality for all life. As originally formulated, Kleiber was based upon the study of heat; metabolic rate is quantified in watts (or calories per unit time). Techniques and technology for metabolic energy measurement have been refined but the math has not. QPS is susceptible to increasing deviations from theoretical predictions to data, suggesting that there is no single, universal exponent relevant to all of life. QPS's major proponents continue to fail to make good on hints of the power of the equation for understanding aging.     

Pitfalls and challenges of estimating population growth rate from empirical data: consequences for allometric scaling relations

The intrinsic rate of increase is a fundamental concept in population ecology, and a variety of problems require that estimates of population growth rate be obtained from empirical data. However, depending on the extent and type of data available (e.g. time series, life tables, life history traits), several alternative empirical estimators of population growth rate are possible. Because these estimators make different assumptions about the nature of age‐dependent mortality and density‐dependence of population dynamics, among other factors, these quantities capture fundamentally different aspects of population growth and are not interchangeable. Nevertheless, they have been routinely commingled in recent ecoinformatic analyses relating to allometry and conservation biology. Here we clarify some of the confusion regarding the empirical estimation of population growth rate and present separate analyses of the frequency distributions and allometric scaling of three alternative, non‐interchangeable measures of population growth. Studies of allometric scaling of population growth rate with body size are additionally sensitive to the statistical line fitting approach used, and we find that different approaches yield different allometric scaling slopes. Across the mix of population growth estimators and line fitting techniques, we find scattered and limited support for the key allometric prediction from the metabolic theory of ecology, namely that log₁₀(population growth rate) should scale as ?0.25 power of log₁₀(body mass). More importantly, we conclude that the question of allometric scaling of population growth rate with body size is highly sensitive to previously unexamined assumptions regarding both the appropriate population growth parameter to be compared and the line fitting approach used to examine the data. Finally, we suggest that the ultimate test of allometric scaling of maximum population growth rates with body size has not been done and, moreover, may require data that are not currently available.     

NEW LIGHT ON THE SCALING OF METABOLIC RATE WITH THE SIZE OF ALGAE

The scaling of metabolic rate with the size of algae has been discussed and researched at length. The observation that algae usually have exponents b in the equation R = a· W ^{− b} (where R is the specific growth rate, W is the organism [cell] biomass, and a and b are constants) equal to or higher than the value of −0.25 for many other organisms is generally related to resource-saturated (maximal) values of R. Recent work has shown that the exponent b for light-limited growth is more negative than −0.25. This was predicted from considerations of the package effect in photon absorption, as modulated by the volume-specific pigment content of the cells, and the photosynthetic unit size. Further work is needed to extrapolate these findings to fluctuating light environments. This minireview puts the recent work into a broader context and suggests how further work could quantify the roles of optical thickness and of spatial and temporal variations in the radiation field in determining metabolic rates.     

Scaling relations for a functionally two-dimensional plant: Chamaesyce setiloba (Euphorbiaceae)

Many characteristics of plants and animals scale with body size as described by allometric equations of the form Y = βM(α), where Y is an attribute of the organism, β is a coefficient that varies with attribute, M is a measure of organism size, and α is another constant, the scaling exponent. In current models, the frequently observed quarter-power scaling exponents are hypothesized to be due to fractal-like structures. However, not all plants or animals conform to the assumptions of these models. Therefore, they might be expected to have different scaling relations. We studied one such plant, Chamaesyce setiloba, a prostrate annual herb that grows to functionally fill a two-dimensional space. Number of leaves scaled slightly less than isometrically with total aboveground plant mass (α ≈ 0.9) and substantially less than isometrically with dry total stem mass (α = 0.82), showing reduced allocation to leaf as opposed to stem tissue with increasing plant size. Additionally, scalings of the lengths and radii of parent and daughter branches differed from those predicted for three-dimensional trees and shrubs. Unlike plants with typical three-dimensional architectures, C. setiloba has distinctive scaling relations associated with its particular prostrate herbaceous growth form.     

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.     

Ontogenetic and interspecific scaling of consumption in insects

The uptake of resources from the environment is a basic feature of all life. Consumption rate has been found to scale with body size with an exponent close to unity across diverse organisms. However, past analyses have ignored the important distinction between ontogenetic and interspecific size comparisons. Using principles of dynamic energy budget theory, we present a mechanistic model for the body mass scaling of consumption, which separates interspecific size effects from ontogenetic size effects. Our model predicts uptake to scale with surface‐area (mass^2/3) during ontogenetic growth but more quickly (between mass^3/4 and mass¹) for interspecific comparisons. Available data for 41 insect species on consumption and assimilation during ontogeny provides strong empirical support for our theoretical predictions. Specifically, consumption rate scaled interspecifically with an exponent close to unity (0.89) but during ontogenetic growth scaled more slowly with an exponent of 0.70. Assimilation rate (consumption minus defecation) through ontogeny scaled more slowly than consumption due to a decrease in assimilation efficiency as insects grow. Our results highlight how body size imposes different constraints on metabolism depending on whether the size comparison is ontogenetic or inter‐specific. Synthesis One of the most robust patterns in biology is the effect of body size on metabolism – a relationship that underlies the rapidly emerging field of metabolic ecology. However, the precise energetic constraints imposed by body size have been notoriously difficult to entangle. Here we show that the constraints imposed on metabolism by body size are different depending on whether the size comparison is ontogenetic or interspecific. Using a single unifying theory of animal metabolism and a newly compiled data set on insect consumption and assimilation rates, we show that interspecific comparisons generally lead to the estimation of higher scaling exponents compared with ontogenetic comparisons. Our results help to explain large variation in estimated metabolic scaling exponents and will encourage future studies in metabolic ecology to make the important distinction between ontogenetic and evolutionary size changes.     

Allometric scaling in animals and plants

In this paper we give a derivation for the allometric scaling relation between the metabolic rate and the mass of animals and plants. We show that the characteristic scaling exponent of 3/4 occurring in this relation is a result of the distribution of sources and sinks within the living organism. We further introduce a principle of least mass and discuss the kind of flows that arise from it.