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Re-examination of the “3/4-law” of Metabolism
Abstract
We examine the scaling law B∝Mαwhich connects organismal resting metabolic rate B with organismal mass M, where α is commonly held to be 3/4. Since simple dimensional analysis suggests α=2/3, we consider this to be a null hypothesis testable by empirical studies. We re-analyse data sets for mammals and birds compiled by Heusner, Bennett and Harvey, Bartels, Hemmingsen, Brody, and Kleiber, and find little evidence for rejecting α=2/3 in favor ofα =3/4. For mammals, we find a possible breakdown in scaling for larger masses reflected in a systematic increase in α. We also review theoretical justifications of α=3/4 based on dimensional analysis, nutrient-supply networks, and four-dimensional biology. We find that present theories for α=3/4 require assumptions that render them unconvincing for rejecting the null hypothesis that α=2/3.
... The next step in the study of co-varying ecological patterns is the identification of the mechanisms that determine the values of the independent exponents. For example, theoretical evidence [36] suggests that the value of z is affected by topological constraints posed by the ecological substrate. ...
... where m is the mass of an individual,m is the characteristic mass of the species and F (x) → 0 suitably fast for x → 0 and x → ∞, as detailed in Giometto et al. [9]. When intra-specific size distributions are described by (36), the results of our scaling frameworks hold exactly. The moment (j,k) is computed as: ...
... Furthermore, we show that the linking relationship in Eq. 1 is also unchanged. In fact, assuming that intra-specific size distributions are given by (36), the size spectrum is given by: ...
Scaling laws in ecology, intended both as functional relationships among ecologically-relevant quantities and the probability distributions that characterize their occurrence, have long attracted the interest of empiricists and theoreticians. Empirical evidence exists of power laws associated with the number of species inhabiting an ecosystem, their abundances and traits. Although their functional form appears to be ubiquitous, empirical scaling exponents vary with ecosystem type and resource supply rate. The idea that ecological scaling laws are linked had been entertained before, but the full extent of macroecological pattern covariations, the role of the constraints imposed by finite resource supply and a comprehensive empirical verification are still unexplored. Here, we propose a theoretical scaling framework that predicts the linkages of several macroecological patterns related to species' abundances and body sizes. We show that such framework is consistent with the stationary state statistics of a broad class of resource-limited community dynamics models, regardless of parametrization and model assumptions. We verify predicted theoretical covariations by contrasting empirical data and provide testable hypotheses for yet unexplored patterns. We thus place the observed variability of ecological scaling exponents into a coherent statistical framework where patterns in ecology embed constrained fluctuations.
... K&K continue by showing that, aside from this theoretical inconsistency, A variable scaling exponent is the result of evolutionary optimisation and an expected finding following the complexity of biology [53,54,56] There is little statistical support to reject b = 2/3 over b = 3/4 [9]. In mammals lighter than 10 kg and in birds, b = 2/3 cannot be rejected [35] The scaling exponent varies across ranges of bodyweight [10,35,[44][45][46][47] Kleiber finding a b-value higher than 2/3 could be explained by overrepresentation of Artiodactyla in his dataset [8] Empiric observations do not support the existence of one universal value for b, as a wide range of exponents has been reported [54] The network is of a fractal nature ...
... K&K continue by showing that, aside from this theoretical inconsistency, A variable scaling exponent is the result of evolutionary optimisation and an expected finding following the complexity of biology [53,54,56] There is little statistical support to reject b = 2/3 over b = 3/4 [9]. In mammals lighter than 10 kg and in birds, b = 2/3 cannot be rejected [35] The scaling exponent varies across ranges of bodyweight [10,35,[44][45][46][47] Kleiber finding a b-value higher than 2/3 could be explained by overrepresentation of Artiodactyla in his dataset [8] Empiric observations do not support the existence of one universal value for b, as a wide range of exponents has been reported [54] The network is of a fractal nature ...
... have raised doubts about the mathematical validity of the WBE framework. Dodds et al. have analysed theoretical derivations of the 3/4 power scaling law [35] and assessed the assumption made by WBE that the networks are fractal in nature. In WBE 1997, the network impedance (or resistance, Z) of pulsatile systems is minimised via a so-called Lagrange multiplier method. ...
Scaling approaches are used to describe or predict clearance for paediatric or obese populations from normal-weight adult values. Theoretical allometry assumes the existence of a universal bodyweight-based scaling relationship. Although theoretical allometry is highly disputed, it is commonly applied in pharmacological data analyses and clinical practice. The aim of the current review is to (1) increase pharmacologists’ understanding of theoretical allometry to better understand the (implicit) assumptions and (dis)advantages and (2) highlight important methodological considerations with the application of this methodology. Theoretical allometry originated in an empirical, and later debated, observation by Kleiber of a scaling exponent of 0.75 between basal metabolic rate and body mass of mammals. The mathematical framework of West, Brown, and Enquist provides one possible explanation for this value. To date, multiple key assumptions of this framework have been disputed or disproven, and an increasing body of evidence is emerging against the existence of one universal allometric exponent. The promise of ease and universality of use that comes with theoretical approaches may be the reason they are so strongly sought after and defended. However, ecologists have suggested that the theory should move from a ‘Newtonian approach’, in which physical explanations are sought for a universal law and variability is of minor importance, to a ‘Darwinian approach’, in which variability is considered of primary importance for which evolutionary explanations can be found. No scientific support was found for the application of allometry for within-species scaling, so the application of basal metabolic rate-based scaling principles to clearance scaling remains unsubstantiated. Recent insights from physiologically based modelling approaches emphasise the interplay between drugs with different properties and physiological variables that underlie drug clearance, which drives the variability in the allometric scaling exponent in the field of pharmacology. To deal with this variability, drug-specific or patient-specific adaptations to theoretical allometric scaling are proposed, that introduce empiric elements and reduce the universality of the theory. The use of allometric scaling with an exponent of 0.75 may hold empirical merit for paediatric populations, except for the youngest individuals (aged ≤ 5 years). Nevertheless, biological interpretations and extrapolation potential attributed to models based on 0.75 allometric scaling are theoretically unfounded, and merits of the empirical application of this function should, as for all models, always be supported by appropriate model validation procedures. In this respect, it is not the value of the allometric exponent but the description and prediction of individual clearance values and drug concentrations that are of primary interest.
... Originally, Kleiber's law proposed a universal exponent of 3/4 among all organisms (Kleiber, 1932). Although this rule has gained evidence from further studies (e.g., Hemmingsen, 1960;Kleiber, 1961;Savage et al., 2004;Schmidt-Nielsen, 1984), others have argued that the scaling exponent is 2/3 (e.g., Dodds et al., 2001;Heusner, 1982aHeusner, , 1991. To complicate this matter further, metabolic allometries are affected by phylogeny (Capellini et al., 2010;Clarke et al., 2010;DeLong et al., 2010;Farrell-Gray & Gotelli, 2005;Lovegrove, 2000;Uyeda et al., 2017), temperature (Addo-Bediako et al., 2002;Bueno & López-Urrutia, 2014;Chown & Nicolson, 2004;Clarke et al., 2010;Gillooly et al., 2001;Gillooly & Allen, 2007;Glazier, 2005;Killen et al., 2010;Lovegrove, 2003;Metz, 2023;Schmidt-Nielsen, 1984a; C. R. White et al., 2007;C. ...
... First, the Metabolic Theory predicts that the variation arises as a consequence of physical and chemical constraints. According to the surface-area-to-volume relationship originally proposed by Rubner (1883), a 2/3 exponent is expected (e.g., Dodds et al., 2001;Heusner, 1991;White & Seymour, 2003). The main idea behind this geometric scaling model is that heat production is coupled to heat loss via the body's surface area (Rubner, 1883). ...
... One of the prevalent power laws for metabolic allometry is based on the geometric surface-area-to-volume relationship and predicts an exponent of 2/3 (Dodds et al., 2001;Heusner, 1991;Rubner, 1883;C. R. White & Seymour, 2003). ...
As the "engine of life", metabolic rate is one of the most well-studied physiological traits across the animal kingdom. Metabolic rates have been linked to key physiological and life-history traits as well as to large-scale ecological processes. For decades, scientists have tried to understand and explain metabolic allometry-the relationship between metabolic rate MR and body mass M (MR = bM a). Two major theories are based on the body's surface-area-to-volume ratio or its nutrient supply network and propose a metabolic scaling exponent a of 2/3 or 3/4 across animals, respectively. However, recent evidence has challenged these theories and metabolic allometries are likely less constrained by physical laws than previously thought. Here, I uncover the inter-and intraspecific metabolic allometries in one of the oldest insect orders, the dragonflies and damselflies (Odonata). I measured the standard metabolic rate (SMR) of 359 field-collected individuals from 36 species in South Sweden. These data were combined with SMR measurements of previously published studies (Sweden and the US), leading to a compiled set of 654 individuals of 56 species. Taking phylogenetic relatedness, within-species variation and sample sizes into account, I show that the interspecific scaling exponent does not fit the traditional 2/3 or 3/4 power laws in Odonata. Instead, the interspecific scaling relationship is significantly steeper, with an allometric slope close to one. The metabolic scaling relationship does not differ statistically between suborders (dragonflies and damselflies), flight behaviors (fliers and perchers), sexes, or mated and unmated individuals. By contrast, the mean intraspecific slope for the metabolic allometry after phylogenetic correction was consistent with the expected 2/3 or 3/4 slopes. However, at the intraspecific level, the slopes and intercepts of the SMR-body mass relationship vary substantially across species. Several species deviated significantly in slope and/or intercept from the common interspecific allometric relationship. Using phylogenetic comparative analyses, I further show that the intraspecific slopes and intercepts are relatively stable across the Odonata phylogeny and only rare shifts of single species or small clades deviating from this common allometric trend have happened. Next, I reveal evidence for constraining forces operating over long-term evolutionary periods and several optima shifts in mass-specific SMR over 237 million years of evolution. One such supported optimum shift is located at the branch leading towards the hawker family (Aeshnidae), which exhibits relatively high SMR. This family comprises agile and fast fliers and, therefore, their high activity level might have caused the evolution of high SMR. However, no support for separate SMR optima for fliers and perchers was found. Similarly, there was no statistical evidence for different evolutionary optima between the two suborders (dragonflies and damselflies). Furthermore, I found evidence for rapid evolution of both mass-specific nucleus size and cell size across the Odonata phylogeny, with closely-related species often having dramatically different nucleus and cell sizes. Both these cellular traits seem to evolve in close concordance with each other but in discordance with SMR. Lastly, I used phylogenetic path analysis to investigate the causal relationships between morphological and cellular traits that are supposed to influence metabolic allometry. My results confirm the well-established direct effect of body mass on SMR and an indirect effect of body volume on SMR via body mass. By contrast, I neither found statistical support for the often-proposed relationship between genome size or cell size and SMR nor for a causal link between surface-area to SMR. Overall, these results question several well-established theories and mechanistic models and thus motivate a rethinking of metabolic allometries and their evolutionary drivers.
... Since that time, various allometric approaches have been p posed and used to determine the most efficacious human equivalent dose (HED) [4 450]. However, the identification of a definitive unified principle that effectively sc and optimizes different energy metabolism systems across animal species remains elu and highly controversial [441,[451][452][453][454]. ...
... Since that time, various allometric approaches have posed and used to determine the most efficacious human equivalent dose (H 450]. However, the identification of a definitive unified principle that effect and optimizes different energy metabolism systems across animal species rema and highly controversial [441,[451][452][453][454]. ...
... Since that time, various allometric approaches have been proposed and used to determine the most efficacious human equivalent dose (HED) [436][437][438][439][440][441][442][443][444][445][446][447][448][449][450]. However, the identification of a definitive unified principle that effectively scales and optimizes different energy metabolism systems across animal species remains elusive and highly controversial [441,[451][452][453][454]. ...
The swift rise in acceptance of molecular principles defining phase separation by a broad array of scientific disciplines is shadowed by increasing discoveries linking phase separation to pathological aggregations associated with numerous neurodegenerative disorders, including Alzheimer’s disease, that contribute to dementia. Phase separation is powered by multivalent macromolecular interactions. Importantly, the release of water molecules from protein hydration shells into bulk creates entropic gains that promote phase separation and the subsequent generation of insoluble cytotoxic aggregates that drive healthy brain cells into diseased states. Higher viscosity in interfacial waters and limited hydration in interiors of biomolecular condensates facilitate phase separation. Light, water, and melatonin constitute an ancient synergy that ensures adequate protein hydration to prevent aberrant phase separation. The 670 nm visible red wavelength found in sunlight and employed in photobiomodulation reduces interfacial and mitochondrial matrix viscosity to enhance ATP production via increasing ATP synthase motor efficiency. Melatonin is a potent antioxidant that lowers viscosity to increase ATP by scavenging excess reactive oxygen species and free radicals. Reduced viscosity by light and melatonin elevates the availability of free water molecules that allow melatonin to adopt favorable conformations that enhance intrinsic features, including binding interactions with adenosine that reinforces the adenosine moiety effect of ATP responsible for preventing water removal that causes hydrophobic collapse and aggregation in phase separation. Precise recalibration of interspecies melatonin dosages that account for differences in metabolic rates and bioavailability will ensure the efficacious reinstatement of the once-powerful ancient synergy between light, water, and melatonin in a modern world.
... Kleiber's law states that metabolism scales to the three-quarters power of mass (West et al., 1997;Dodds et al., 2001;Banavar et al., 2010). Its persistence across diverse species and ecosystems has fascinated successive generations of biologists and motivated many attempts to provide a general explanation, which often require new ways of viewing biological systems (West et al., 1997;Dodds et al., 2001;López-Urrutia et al., 2006;Banavar et al., 2010). ...
... Kleiber's law states that metabolism scales to the three-quarters power of mass (West et al., 1997;Dodds et al., 2001;Banavar et al., 2010). Its persistence across diverse species and ecosystems has fascinated successive generations of biologists and motivated many attempts to provide a general explanation, which often require new ways of viewing biological systems (West et al., 1997;Dodds et al., 2001;López-Urrutia et al., 2006;Banavar et al., 2010). Observations in recent years have shown that this metabolic principle for individual organisms also applies to the whole-ecosystem properties, such as predator-prey ratios (Hatton et al., 2015) or estuarine productivity (Nidzieko, 2018), implying a far greater generality and that in certain respects, ecosystems self-organize to behave like superorganisms. ...
... Observations in recent years have shown that this metabolic principle for individual organisms also applies to the whole-ecosystem properties, such as predator-prey ratios (Hatton et al., 2015) or estuarine productivity (Nidzieko, 2018), implying a far greater generality and that in certain respects, ecosystems self-organize to behave like superorganisms. However, the specificity of a 3/4-exponent is a challenging criterion for the admissibility of theoretical models, and the mathematical and biological validity of many candidate models is contested (Dodds et al., 2001). The mechanisms underlying this strikingly universal scaling law are, therefore, still unclear and, furthermore, may be different in different instances. ...
Kleiber’s 3/4-scaling law for metabolism with mass is one of the most striking regularities in biological sciences. Kleiber’s law has been shown to apply not only to individual organisms but also to communities and even the whole-ecosystem properties such as the productivity of estuaries. Might Kleiber’s law also then apply to lake ecosystems? Here, we show that for a collection of whole-lake primary production measurements, production scales to the 3/4 power of lake volume, consistent with Kleiber’s law. However, this relationship is not explicable by analogy to theories developed for individual organisms. Instead, we argue that dimensional analysis offers a simple explanation. After accounting for latitudinal gradients in temperature and insolation, whole-lake primary production scales isometrically with lake area. Because Earth’s topography is self-affine, meaning there are global-scale differences between vertical and horizontal scaling of topography, lake volume scales super-linearly with lake surface area. 3/4 scaling for primary production by volume then results from these other two scaling relationships. The identified relationship between the primary production and temperature- and insolation-adjusted area may be useful for constraining lakes’ global annual productivity and photosynthetic efficiency. More generally, this suggests that there are multiple paths to realizing the 3/4 scaling of metabolism rather than a single unifying law, at least when comparing across levels of biological organization.
... When constructing a model of the tumor-immune system dynamics, it is essential to first delineate the pattern of tumor growth. Various dynamic models of tumor growth have been previously proposed, such as the power law growth model [4][5][6], the Gompertz growth model [7][8][9], and the Logistic growth model [10][11][12]. ...
... and Growth rate of NK cells r 3 Constant source of dendritic cells a 1 Tumor death rate caused by NK cells a 2 NK cell death rate due to tumor interaction a 3 Activation rate of CD8 + T cells by NK-lysed tumor cells b 1 Tumor death rate induced by CD8 + T cells b 2 CD8 + T cell death rate caused by tumor interaction b 3 CD8 + T cell self-regulation and suppression rate c 1 CD8 + T cell activation rate induced by dendritic cells c 2 Dendritic cell inactivation rate related to the production of more CD8 + T cells d 1 The tumor cells' carrying capacity d 2 The NK cells' carrying capacity d 3 Natural death rate of CD8 + T cells d 4 Natural death rate of dendritic cells ...
To investigate the effects of environmental factors on tumor growth and the immune response, we have developed a stochastic model of the tumor–immune system, which encompasses tumor cells, NK cells, CD T cells, and dendritic cells. Initially, we analyzed the deterministic version of the system, deriving the threshold conditions for the local asymptotic stability of the equilibrium point in accordance with the stability theory of differential equations. For the stochastic version, we utilized Itô’s formula and Lyapunov analysis techniques to confirm the existence of a unique global positive solution and to identify sufficient conditions for the mean persistence of the system. Furthermore, we applied the stochastic maximum principle to devise optimal control strategies for the prevention and control of tumor cell proliferation. Our numerical simulations reveal that varying levels of noise intensity result in different outcomes for tumor cells, NK cells, CD T cells, and dendritic cells, including persistence and extinction. These findings offer critical insights that can guide the development of strategies for preventing and managing tumor progression.
... where MR is metabolic rate, M mass, a is the intercept, and b is the scaling factor. The nature of the exponent b is subject to an intensive and ongoing debate in ecology (Kozłowski et al. 2003;Glazier 2005Glazier , 2022Hoppeler and Weibel 2005;Dodds et al. 2001;Isaac and Carbone 2010;Niklas andKutschera 2015, White et al. 2011). Deviations from the 3 /4 scaling factor have been found for many groups of organisms and are particularly pronounced in modular colonial organisms such as bryozoans, which display high variability in their scaling factors with estimates as low as 0.5 . ...
... Deviations from the 3 /4 scaling factor have been found for many groups of organisms and are particularly pronounced in modular colonial organisms such as bryozoans, which display high variability in their scaling factors with estimates as low as 0.5 . Studies on metabolic rate in mammals and birds suggest that explicit use of neither Kleiber's 3 /4 law nor the earlier 2 /3 prediction of scaling factor is appropriate (Dodds et al. 2001). Colonial animals often display allometric scaling of respiration rate with colony size (Burgess et al. 2017). ...
Coloniality may grant colony members an energetic advantage in the form of lower individual respiration rates as colony size increases. If this occurs it should be apparent as negative allometric scaling of respiration with colony size, and colonial organisms should have scaling factors < 1. However, colonial members from phylum Rotifera have yet to be examined. To test if colonial rotifers possess allometric scaling relationships between respiration rate and colony size, we measured respiration rates for four solitary and three colonial rotifer species; from these respiration rates we estimated scaling factors. We found mixed evidence for allometric scaling of respiration rate in colonial rotifers. Both rotifers with allometric scaling of respiration rate, Conochilus hippocrepis and Lacinularia flosculosa, have extensive mucilaginous coverings. These coverings may represent an investment of colony members into a shared structure, lowering individual metabolic costs and thus respiratory needs. Additionally, we determined which traits are associated with allometric scaling of respiration. We compiled known scaling factors for animal phyla from a wide phylogenetic spectrum with colonial representatives and conducted a hierarchical mixed regression that included attributes of colonies. Traits associated with allometric scaling in colonial animals included colony shape, the presence of shared extrazooidal structures, and planktonic lifestyle. There are many other colonial rotifers and animal taxa for which allometric scaling factors have yet to be estimated, knowing these may enhance our understanding of the benefits of coloniality in animals.
... Glazier (2006) argues that Kleiber's Law, which the WBE theory explains, is not truly universal. Dodds et al. (2001) and Seymour (2003, 2005) argue that statistical evidence is consistent with λ = ⅔, rather than ¾. WBE have defended their research against these criticisms West and Brown 2005). ...
... The model assumed Q liver to be 20 mL/ kg/min (36), Q gut to be 18L/h and intestine surface area to be 6600 cm 2 in human (24), the human specific metabolic stability data (22) and as well as the species dependent GI transit. Other uptake and distribution flows were scaled with 0.75 as the scaling factor (41), and volume of distribution scaled with 1.0. Typical body-weights of 70 and 0.3 kg were assumed for humans and rats, respectively. ...
Intestinal lymphatic absorption is a crucial alternative to portal uptake for highly lipophilic drugs (log P > 5), bypassing first-pass metabolism. Unlike the portal-hepatic pathway, lymphatic uptake is rarely considered in physiologically based pharmacokinetic (PBPK) models for oral delivery. Our study developed an innovative Gastro-Intestinal (GI)-lymph-PBPK model that includes GI absorption, chylomicron extraction (CE) to rescue drugs from gut extraction (GE), and bypass hepatic extraction (HE). This model introduces CE clearance (CL CE ), competing with GE clearance, to estimate the drug proportion subjected to CE versus GE. PBPK analysis for Buprenorphine revealed extensive GE (0.87) and HE (0.58), explaining the low bioavailability (F%) of 5.28% in rats. Buprenorphine prodrugs activated CL CE , leading to CE ranging from 0.37 to 0.79, boosting oral F% to 39.9%-79.9% in rats. To translate from rat to human, our model considered species differences in GI transit time, formulation, food-dependent drug dissolution, allometric scaling in CL CE , and between species variability in gut metabolism. Using Halofantrine, we established an allometric scaling factor for CL CE at 1.1. Accounting for six times faster human gut metabolism, our model predicted an extremely low oral F% of 0.382% for Buprenorphine in humans. Incorporating the allometric scaled CL CE competing with the extensive gut metabolism, our model predicted Buprenorphine prodrugs remains effective in enabling substantial absorption boosts, with oral F% estimates ranging from 15.8% to 56.7% in humans. This study highlights the significant potential of GI-lymph-PBPK modeling in predicting intestinal lymphatic absorption and facilitating cross-species translation.
Graphical Abstract
... On the other hand, athletes with the same BMI as non-athletes generally have lower body fat. This is because muscle mass, which weighs more than fat, occupies less space in the body [4]. ...
The aim of this study was to see the effect of diet mainly non-vegetarian on Muslim population on basal metabolic rate, that is, BMR. It has been found that BMR increases with the consumption of non-vegetarian diet. The study found that the average BMR level in the group exceeded 30, placing them in the obese category. This suggests a significant link between diet, particularly a non-vegetarian one, and increased BMR. Given this correlation, it becomes clear that adopting a vegetarian diet could be beneficial for weight management. A vegetarian diet generally contains less saturated fat and more essential nutrients, which can support healthier weight management. By managing weight more effectively, one can also achieve a better BMR, which is crucial for overall metabolic health. The growing trend of people shifting towards vegetarianism is likely driven by the desire to improve health, reduce the risk of chronic diseases, and ultimately enhance longevity. A balanced vegetarian diet, rich in plant-based nutrients, supports not only weight management but also overall well-being. This shift in dietary patterns reflects a broader awareness of the long-term health benefits associated with plant-based eating. As more people recognize the connection between diet and health outcomes, the move toward vegetarianism is gaining momentum as a proactive choice for living a healthier and longer life. INTRODUCTION Basal metabolic rate (BMR) refers to the amount of energy expended by the body while at complete rest, which includes being in a comfortable environment with stable temperature and humidity, remaining awake, and sitting in a relaxed position for 10-12 hours after eating [1]. This rate represents the minimal energy needed to sustain essential functions such as heart rate, respiration, and kidney activity. On the other hand, Body Mass Index (BMI) is calculated by dividing an individual's weight in kilograms by the square of their height in meters.
... The explanation of Kleiber's law provided by WBE is not without critics (see Dodds et al. 2001, for example, or Kozłowski and Konarzewski 2004, 2005. As far as this paper is concerned, however, the key point is not whether WBE have successfully explained why Kleiber's law takes the form it does but rather that the field takes it seriously as a candidate for explanatory success (Lange 2023, p. 272) makes a parallel point about explanatory theories in historical linguistics). ...
Paradigmatic examples of laws drawn from physics are accompanied by a rich explanatory structure that supports unificatory efforts. Part of the reason these efforts have been successful has been their identification of conservation laws, whose greater necessity allows them to act as constraints. Within biology, much of the debate over lawhood has focused on whether restricted regularities should be treated as laws, rather than finding parallels to how physical laws are structured. This paper argues that extending that structure to biology is both possible and well-motivated by building on the functionalist approach to lawhood. Through two specific examples – the West Brown Enquist model of metabolism and the neutral theory of biodiversity – I show there are principles in biological theorising that can be interpreted as analogues of conservation laws. This reveals a similarity of nomological structure between physics and biology which is often obscured by methodological differences. Commitment to these laws in biology supports the view that theoretical progress can be made by focusing research efforts on unificatory principles.
... ‡ Considering the fractal nature of multicellular bodies, the scaling slightly alters to be sublinear with size, T ∼ M 1/4 ∼ L 3/4 (44). The origins and validity of such power laws remain open questions (45,46). But for us, it is only essential to note that the scaling of the macro-cascade is roughly linear and, at any rate, much less steep than that of the micro-cascade (T ∼ L 4 ). ...
Life is invasive, occupying all physically accessible scales, stretching between almost nothing (protons, electrons, and photons) and almost everything (the whole biosphere). Motivated by seventeenth-century insights into this infinity, this paper proposes a language to discuss life as an infinite double cascade of machines making machines. Using this simplified language, we first discuss the micro-cascade proposed by Leibniz, which describes how the self-reproducing machine of the cell is built of smaller submachines down to the atomic scale. In the other direction, we propose that a macro-cascade builds from cells larger, organizational machines, up to the scale of the biosphere. The two cascades meet at the critical point of 10 ³ s in time and 1 micron in length, the scales of a microbial cell. We speculate on how this double cascade evolved once a self-replicating machine emerged in the salty water of prebiotic earth.
... And why does the mass-specific meta-bolic rate decrease? These questions are much debated and still unresolved (West et al., 1997;Dodds et al., 2001;Kozlowski et al., 2005;Mori et al., 2010;Glazier, 2022). Recently, most investigators no longer regard metabolic scaling as following any single exponent (e.g., Glazier, 2005Glazier, , 2006Glazier, , 2022White, 2010;Gavrilov et al., 2022). ...
The deep sea, which encompasses the largest habitat on Earth, presents a set of extreme and unique environmental conditions, including high hydrostatic pressure, near-freezing temperatures, and perpetual darkness. These conditions pose significant challenges to the survival and energy management of its inhabitants. Deep-sea organisms have evolved a range of bioenergetic adaptations to negotiate these harsh conditions, ensuring efficient energy acquisition and utilization. This review examines the multifaceted strategies employed by deep-sea animals, focusing on three key areas: energy input, digestive and absorptive efficiency, and energy consumption. We examine the physical environment of the deep sea, highlighting vertical profiles of temperature, salinity, and dissolved oxygen, which contrast sharply with surface conditions. Physiological adaptations of deep-sea species, such as specialized digestive systems and enzyme modifications that function optimally under high pressure, are explored in detail. Furthermore, we discuss behavioral adaptations, including diurnal vertical migration, which optimize energy intake and reduce metabolic costs. Comparative analyses with shallow-water species provide insights into the evolutionary pressures that have shaped these adaptations. This review also addresses the concept of "power budgeting", in which energy expenditures for specific dynamic actions (SDAs) must be balanced with other metabolic demands. This comprehensive examination of bioenergetic adaptation in deep-sea organisms enhances our understanding of their resilience and adaptability, offering glimpses into the complex interplay between environmental constraints and biological processes in one of the most challenging habitats on the planet.
... However, the "correct" value of the p exponent remains a controversial issue since other values have also been proposed in the literature. For instance, a recent paper (Dodds et al., 2001) shows that p=3/4 , which was the classical value considered in the pioneer paper by Kleiber (1932), ...
We have previously reported that a universal growth law, as proposed by West and collaborators for all living organisms, appears to be able to describe also the growth of tumors in vivo. In contrast to the assumption of a fixed power exponent p (assumed by West et al. to be equal to 3/4), we show in this paper the dynamic evolution of p from 2/3 to 1, using experimental data from the cancer literature and in analogy with results obtained by applying scaling laws to the study of fragmentation of solids. The dynamic behaviour of p is related to the evolution of the fractal topology of neoplastic vascular systems and might be applied for diagnostic purposes to mark the emergence of a functionally sufficient (or effective) neo-angiogenetic structure.
... As organism size increases, the maturation time increases as well. If we let N denote the number of cells in the adult organism, then it has been suggested that the maturation time is proportional to either N 1/4 [47,48] or N 1/3 [49,50]. To account for both possibilities, we take the maturation time to scale as N α , where α = 1/4, 1/3. ...
This paper studies the mutation-selection balance in three simplified replication models. The first model considers a population of organisms replicating via the production of asexual spores. The second model considers a sexually replicating population that produces identical gametes. The third model considers a sexually replicating population that produces distinct sperm and egg gametes. All models assume diploid organisms whose genomes consist of two chromosomes, each of which is taken to be functional if equal to some master sequence, and defective otherwise. In the asexual population, the asexual diploid spores develop directly into adult organisms. In the sexual populations, the haploid gametes enter a haploid pool, where they may fuse with other haploids. The resulting immature diploid organisms then proceed to develop into mature organisms. Based on an analysis of all three models, we find that, as organism size increases, a sexually replicating population can only outcompete an asexually replicating population if the adult organisms produce distinct sperm and egg gametes. A sexual replication strategy that is based on the production of large numbers of sperm cells to fertilize a small number of eggs is found to be necessary in order to maintain a sufficiently low cost for sex for the strategy to be selected for over a purely asexual strategy. We discuss the usefulness of this model in understanding the evolution and maintenance of sexual replication as the preferred replication strategy in complex, multicellular organisms.
... Although they assumed B ∝ m p with p=3/4, their value of p is not universally accepted. Indeed there is strong evidence that growth may be consistent with a 2/3 law in the case of birds and small mammals [14]. Also, in a recent study, Makarieva, Gorshkov, and Li suggest that it is the process of energy consumption and not the transport of nutrients on fractal-like networks that determines the dependence of plant energetics on size [15]. ...
Most organisms grow according to simple laws, which in principle can be derived from energy conservation and scaling arguments, critically dependent on the relation between the metabolic rate B of energy flow and the organism mass m. Although this relation is generally recognized to be of the form B(m) = mp, the specific value of the exponent p is the object of an ongoing debate, with many mechanisms being postulated to support different predictions. We propose that multicellular tumor spheroids provide an ideal experimental model system for testing these allometric growth theories, especially under controlled conditions of malnourishment and applied mechanical stress.
... It is well established empirically that whole body metabolism of resting mammals scales with body volume (or mass) with an exponent close to 3/4, which is known as Kleiber's law [1,2,3,4]. The same exponent or its simple derivatives govern the scalings of respiratory and cardiovascular systems in mammals and some other physiological parameters in animals and plants [2,3,5]. ...
Brain is one of the most energy demanding organs in mammals, and its total metabolic rate scales with brain volume raised to a power of around 5/6. This value is significantly higher than the more common exponent 3/4 relating whole body resting metabolism with body mass and several other physiological variables in animals and plants. This article investigates the reasons for brain allometric distinction on a level of its microvessels. Based on collected empirical data it is found that regional cerebral blood flow CBF across gray matter scales with cortical volume V as , brain capillary diameter increases as , and density of capillary length decreases as . It is predicted that velocity of capillary blood is almost invariant (), capillary transit time scales as , capillary length increases as , and capillary number as , where is typically a small correction for medium and large brains, due to blood viscosity dependence on capillary radius. It is shown that the amount of capillary length and blood flow per cortical neuron are essentially conserved across mammals. These results indicate that geometry and dynamics of global neuro-vascular coupling have a proportionate character. Moreover, cerebral metabolic, hemodynamic, and microvascular variables scale with allometric exponents that are simple multiples of 1/6, rather than 1/4, which suggests that brain metabolism is more similar to the metabolism of aerobic than resting body. Relation of these findings to brain functional imaging studies involving the link between cerebral metabolism and blood flow is also discussed.
... where the leading term is for normalization ( = 1, following the framework derived in ref. 13), f (x) is a general scaling function constrained by the limits f (x) → 0 for x → ∞ and f (x) tends to a constant for x → 0, and c is the scaling exponent of the Kleiber's law (1) for mean mass A . Note that a similar formulation for Eq. 5 was already proposed by ref. 6. Distributions are evaluated for binned data along the A-axis (SI Appendix). ...
Allometric scaling relations are widely used to link biological processes to body size in nature. Several studies have shown that such scaling laws hold also for natural ecosystems, including individual trees and forests, riverine metabolism, and river network organization. However, the derivation of scaling laws for catchment-scale water and carbon fluxes has not been achieved so far. Here, we focus on scaling relations of catchment green metabolism, defined as the set of ecohydrological and biogeochemical processes through which vegetation assemblages in catchments maintain their structure and react to the surrounding environment. By revising existing plant size–density relationships and integrating them across large-scale domains, we show that the ecohydrological fluxes occurring at the catchment scale are invariant with respect to the above-ground vegetation biomass per unit area of the basin, while they scale linearly with catchment size. We thus demonstrate that the sublinear scaling of plant metabolism results in an isometric scaling at catchment and regional scales. Deviations from such predictions are further shown to collapse onto a common distribution, thus incorporating natural fluctuations due to resource limitations into a generalized scaling theory. Results from scaling arguments are supported by hyperresolution ecohydrological simulations and remote sensing observations.
... This intriguing observation suggests that allometric relationships emerge from fundamental properties in the way organisms are built (Elgin, 2006;Sousa et al., 2008;Glazier, 2014;Kempes et al., 2019), which fostered the development of mechanistic theories to explain the seemingly apparent nearuniversality of many scaling relationships. For example, the surface area theory predicts a ⅔ scaling exponent for how many traits and attributes, including metabolic rate and growth rate (⅔), population density (À⅔), and plant height (⅔), scale with changes in plant size, as a result of the disproportionate increase in volume with surface area (Rubner, 1883;White, 1981;Niklas, 1995;Dodds et al., 2001). By contrast, the metabolic scaling theory (MST) predicts how numerous physiological and anatomical traits scale with allometric exponents that are multiples of a quarter (e.g. ...
Understanding trait–trait coordination is essential for successful plant breeding and crop modeling. Notably, plant size drives variation in morphological, physiological, and performance‐related traits, as described by allometric laws in ecology. Yet, as allometric relationships have been limitedly studied in crops, how they influence and possibly limit crop performance remains unknown. Here, we review how an allometry perspective on crops gains insights into the phenotypic evolution during crop domestication, the breeding of varieties adapted to novel conditions, and the prediction of crop yields. As allometry is an active field of research, modeling and manipulating crop allometric relationships can help to develop more resilient and productive agricultural systems to face future challenges.
... However, there are many other ecological relationships that exhibit well documented power-laws, such as the relationship between metabolic rate and mass (Dodds et al 2001), community production and community biomass (Hatton et al 2015), population abundance and spatial variance (Taylor 1961), and species richness and area (Šizling & Storch 2004). ...
Group foraging is a commonly observed phenomenon across various taxa that has been shown to improve an individual's fitness. There are tradeoffs to foraging in groups, with benefits such as reducing predation risk and costs such as increased competition within and between groups. The combination of these factors determines the optimal group size. While these factors are known to be density dependent, group size itself has not been explicitly tested for density dependence. Here I use an ODE model to show that the optimal group size in a population can exhibit density dependence. My model predicts that the optimal group size displays a power-law relationship with abundance, with a coefficient that depends on how predators respond to changes in their prey abundance. I then tested this model using existing empirical data and found that results were consistent across both the empirical and theoretical results. Previous work has shown that power-law group formation can stabilize predator-prey dynamics. Here I identify the key driving factors behind this pattern of group formation. My work contributes to a growing body of work that groups, rather than individuals, play a primary role in the population dynamics of group-forming fauna. This also illustrates how an individual's behavior, namely the decision to join a group, can scale up to be a significant driver of population dynamics in birds and ungulates.
... The scaling of heat dissipation with size, both through ontogeny and between species, is empirically observed to be a power function of body mass where the exponent varies but is frequently near 3 / 4 . [25][26][27] Much thought has been invested in the underlying causes of this taxonomically general empirical pattern, 28-32 but it is regarded as unsolved. The dual-pool thermodynamic processes of DEB theory lead to metabolic rate being a weighted sum of the heat lost through assimilation, growth, and dissipation. ...
What makes living things special is how they manage matter, energy, and entropy. A general theory of organismal metabolism should therefore be quantified in these three currencies while capturing the unique way they flow between individuals and their environments. We argue that such a theory has quietly arrived -- 'Dynamic Energy Budget' (DEB) theory -- which conceptualises organisms as a series of macrochemical reactions that use energy to transform food into structured biomass and bioproducts while producing entropy. We show that such conceptualisation is deeply rooted in thermodynamic principles and that, with the help of a small set of biological assumptions, it underpins the emergence of fundamental ecophysiological phenomena, most notably the three-quarter power scaling of metabolism. Building on the subcellular nature of the theory, we unveil the eco-evolutionary relevance of coarse-graining biomass into qualitatively distinct, stoichiometricially fixed pools with implicitly regulated dynamics based on surface area-volume relations. We also show how generalised enzymes called 'synthesising units' and an information-based state variable called 'maturity' capture transitions between ecological and physiological metabolic interactions, and thereby transitions between unicellular and multicellular metabolic organisation. Formal theoretical frameworks make the constraints imposed by the laws of nature explicit, which in turn leads to better research hypotheses and avoids errors in reasoning. DEB theory uniquely applies thermodynamic formalism to organismal metabolism, linking biological processes across different scales through the transformation of matter and energy, the production of entropy, and the exchange of information. We propose ways in which the theory can inform trans-disciplinary efforts at the frontiers of the life sciences.
... To this day, the extrapolation of point measurements to a survey, or larger geographic areas, is challenging potentially leading to inaccurate regional assessments (Brown et al. 2004;Kearney and Porter 2009;Allen and Polimene 2011;Barneche et al. 2014). Hence, the applicability of the metabolic theory to the ecology of aquatic organisms is questioned by some researchers (Dodds et al. 2001;Tilman et al. 2004;Alcaraz 2016). Generally, ecosystem studies underexplore the spatial and source-based variability of aquatic organisms' MRs. ...
The mass-specific metabolic rates of the Antarctic pelagic tunicate Salpa thompsoni Foxton, 1961 were studied during March-April of 1998 and March 2002. The study revealed a large variation in metabolic rates and attempted explaining their variability. The main factors driving variability included density of tunicates in incubation containers (incubation density, e.g. the salp mass per unit volume of the respirometer), diel/circadian rhythms in salps, and spatial variability of their metabolic performance related to the feeding conditions. The mass-specific respiration rates of both salp life forms (solitaries and aggregates) appeared to be independent of their body mass. The salp-specific respiration rates at 3°C were strongly negatively influenced by their incubation density ranging between 2.0 and 90.4 gWW.L-1. Salp respiration rates adjusted to an incubation density of 3 gWW.L-1 in both solitary and aggregate forms followed similar circadian rhythms with the mean respiration rates of 79.5 and 41.5 μg O 2 gWW-1 h-1 , respectively. These specific respiration rates of S. thompsoni were assumed to be a statistical norm and were compared to actual field point respiration rates corrected for the salp density and diel variability during 1998 and 2002 near the Elephant Island, the South Orkney Islands, and in the Bransfield Strait. Calculated deviations from a statistical norm identified effects of food concentrations, i.e. proxy of the plankton community development and composition, on the salp population performance. Findings of this study highlighted the applicability of the metabolic theory in describing the salp ecological dynamics in the Southern Ocean.
... As early as the 1880s, Rubner proposed that an organism's metabolic rate is proportional to the 2/3 power of its body size [5]. This concept of a 2/3 proportional index has been widely accepted and proven [6][7][8][9][10]. Later, Huxley ...
Allometry reflects the quantitative relationship between the allocation of resources among different organs. Understanding patterns of forest biomass allocation is critical to comprehending global climate change and the response of terrestrial vegetation to climate change. By collecting and reorganizing the existing allometric models of tree species in China, we established a database containing over 3000 empirical allometric models. Based on this database, we analyzed the model parameters and the effect of climate on forest biomass allocation under the context of ‘optimal allocation theory’. We showed that (1) the average and median exponent of power functions for above-ground biomass were 2.344 and 2.385, respectively, which significantly deviated from the theoretical prediction of 2.667 by metabolic theory (p < 0.01). (2) The parameters of the allometric model were not constant, and not significantly correlated with temperature, precipitation, latitude, and elevation (p > 0.05), but were more closely related to individual size (p < 0.01). (3) Among different types of forests, the proportion of above-ground biomass in tropical rainforests and subtropical evergreen rainforests was significantly higher than that in temperate forests and boreal forests (p < 0.05). The proportion of trunk and branch biomass allocated to tropical rainforest was significantly higher than that of boreal forest (p < 0.05), while the proportion of root and leaf biomass allocated to tropical rainforest was significantly lower than that of boreal forest (p < 0.05). (4) The abiotic environment plays a crucial role in determining the allocation of plant biomass. The ratio of below-ground/above-ground biomass is significantly and negatively correlated with both temperature and rainfall (p < 0.01), and significantly and positively correlated with altitude and latitude (p < 0.01). This means that as temperature and rainfall increase, there is a decrease in the amount of biomass allocated to below-ground structures such as roots. On the other hand, as altitude and latitude increase, there is an increase in below-ground biomass allocation. These findings highlight the importance of considering the influence of abiotic factors on plant growth and development.
... The scaling coefficient for metabolic rates is not always À0.25 and depends on the range of body masses examined (111)(112)(113) and the groups examined (60,114). When the observed pattern for the scaling of metabolic rate is not À0.25, the logic of the rate of metabolism hypothesis leads to the prediction that the scaling of immune defenses should match the observed pattern of scaling for metabolic rates. ...
Most studies in comparative immunology involve investigations into the detailed mechanisms of the immune system of a non-model organism. Although this approach has been insightful, it has promoted a deep understanding of only a handful of species, thus inhibiting the recognition of broad taxonomic patterns. Here, we call for investigating the immune defenses of numerous species within a pointillist framework, that is, the meticulous, targeted collection of data from dozens species and investigation of broad patterns of organismal, ecological, and evolutionary forces shaping those patterns. Without understanding basic immunological patterns across species, we are limited in our ability to extrapolate and/or translate our findings to other organisms, including humans. We illustrate this point by focusing predominantly on the biological scaling literature with some integrations of the pace of life literature, as these perspectives have been the most developed within this framework. We also highlight how the more traditional approach in comparative immunology works synergistically with a pointillist approach, with each approach feeding back into the other. We conclude that the pointillist approach promises to illuminate comprehensive theories about the immune system and enhance predictions in a wide variety of domains, including host-parasite dynamics and disease ecology.
... 31 The estimated allometric exponent for the effect of body weight on CL/F (0.31) is lower than the typically anticipated 0.75 33 or 0.67. 34,35 The biological reason for this lower allometric exponent for tofacitinib in patients with JIA is not clear. Of note, the estimated allometric exponent is still within the range that has been reported previously (0.29-1.2). 36 The current dataset contains PK information from more than 200 pediatric patients with body weights ranging from 11.1 to 121.8 kg, with 11.4% of patients considered "obese" per CDC and Prevention criteria. ...
These analyses characterized tofacitinib pharmacokinetics (PKs) in children and adolescents with juvenile idiopathic arthritis (JIA). Data were pooled from phase I (NCT01513902), phase III (NCT02592434), and open‐label, long‐term extension (NCT01500551) studies of tofacitinib tablet/solution (weight‐based doses administered twice daily [b.i.d.]) in patients with JIA aged 2 to less than 18 years. Population PK modeling used a nonlinear mixed‐effects approach, with covariates identified using stepwise forward‐inclusion backward‐deletion procedures. Simulations were performed to derive dosing recommendations for children and adolescents with JIA. Two hundred forty‐six pediatric patients were included in the population PK model. A one‐compartment model with first‐order elimination and absorption with body weight as a covariate for oral clearance and apparent volume of distribution sufficiently described the data. Oral solution was associated with comparable average concentration ( C avg ) and slightly higher (113.9%) maximum concentration ( C max ) versus tablet, which was confirmed by a subsequent randomized, open‐label, bioavailability study conducted in healthy adult participants ( n = 12) by demonstrating adjusted geometric mean ratios (90% confidence interval) between oral solution and tablet of 1.04 (1.00–1.09) and 1.10 (1.00–1.21) for area under the curve extrapolated to infinity and C max , respectively (NCT04111614). A dosing regimen of 3.2 mg b.i.d. solution in patients 10 to less than 20 kg, 4 mg b.i.d. solution in patients 20 to less than 40 kg, and 5 mg b.i.d. tablet/solution in patients greater than or equal to 40 kg, irrespective of age, was proposed to achieve constant C avg across weight groups. In summary, population PK characterization informed a simplified tofacitinib dosing regimen that has been implemented in pediatric patients with JIA.
... This relation is usually expressed in terms of two parameters: a scaling exponent and the normalizing coefficient. For large mammals, the scaling exponent has been placed around 0.75, and for small mammals and birds, the exponent has been placed around 0.66 [44,45]. A normalizing coefficient has been demonstrated that has a large range of values depending on biological features, from plants to animals, including unicellular organisms. ...
This paper presents so-called thermoelectric generators (TEGs), which are considered thermal engines that transform heat into electricity using the Seebeck effect for this purpose. By using linear irreversible thermodynamics (LIT), it is possible to study the thermodynamic properties of TEGs for three different operating regimes: maximum power output (MPO), maximum ecological function (MEF) and maximum power efficiency (MPE). Then, by considering thermoelectricty, using the correspondence between the heat capacity of a solid and the metabolic rate, and taking the generation of energy by means of the metabolism of an organism as a process out of equilibrium, it is plausible to use linear irreversible thermodynamics (LIT) to obtain some interesting results in order to understand how metabolism is generated by a particle’s released energy, which explains the empirically studied allometric laws.
... Изучение ответных реакции популяций гидробионтов предполагает охват протяженных акваторий, но пространственная изменчивость измеряемых величин ЭО гидробионтов оставалась не изученной ввиду высокой вариабельности результатов измерений, их «зашумленности», причины которой были мало понятны. В результате применимость метаболической теории в экологии гидробионтов вызывает сомнения у некоторых исследователей (Dodds et al., 2001;Tilman et al., 2004;Alcaraz, 2016). Но необходимость оценить взаимосвязи функциональных характеристик планктона и особенностей функционирования экосистем по-прежнему являются очень актуальной (Calow, 1987;Brown et al., 2004;Kearney, Porter, 2009;De Bello et al., 2010;Allen, Polimene, 2011;Barneche et al., 2014;Hébert et al., 2017;Kiørboe T., Hirst, 2016). ...
In the monography the complex of the most structural-functional characteristics of the most mass representative of Antarctic ecosystem – krill Euphausia superba Dana – and of other main components of plankton community in its areal have been considered for the first time on the basis of many-year field and experimental observations in the Indian Ocean and the Atlantic parts of Antarctica. The organisms that make up the Antarctic plankton community with an emphasis on their taxonomic composition, abundance, general and special features of development have been studied. The scale of interannual fluctuations in their biomass and abundance is estimated, and the underlying mechanisms of this variability are discussed. The trophic structure of Antarctic zooplankton, which provides a better understanding of the production and destruction processes occurring in the pelagial of the Southern Ocean is described. Balance estimates of the flows of matter and energy in the ecosystem based on measurements carried out during the years of Antarctic research are given. Thus, a theoretical basis for further study and understanding of the Antarctic ecosystem functioning has been created. For hydrobiologists, ecologists, marine biologists.
... A recent effort that has garnered considerable attention is West, Brown, and Enquist's fractal similarity model, which assumes elastic similarity but goes on to make a suite of additional predictions including the scaling of leaf number and plant mass (Enquist & Niklas, 2001West et al., 1997West et al., , 1999. This body of work has catalyzed great interest in the origin of allometric scaling relationships, but many have questioned the model's scope and mechanistic underpinnings for plants (Dodds et al., 2001;Muller-Landau et al., 2006;Niklas, 2004;Petit & Anfodillo, 2009;Price et al., 2010Price et al., , 2012 Price et al. (2022) showed that several parts of plant branching networks including juvenile tree stems, interspecific terminal branches, and interspecific petioles have allometric relationships that are generally consistent with the predictions of the flow similarity model (detailed derivations and explanation can be found in Price et al., 2022). The model is based on the Hagen-Poiseuille equation for laminar flow in a cylindrical tube and relies on two principal model assumptions/constraints, velocity preservation and the conservation of volumetric flow rate (hence "flow similarity") which together predict a constant pressure drop (note that any two of these constraints predicts the third). ...
Allometric relationships for plants, plant organs and plant parts, have long generated interest among biologists. Several prominent theoretical models based on biomechanical and/or hydraulic arguments have been introduced with mixed support. Here, I test a more recent offering, flow similarity, which is based on the conservation of volumetric flow rate and velocity. Using dimensional data for 935 petioles from 43 angiosperm species, I show that both the intraspecific and interspecific petiole allometries are more closely aligned with the predictions of the flow similarity model than that of elastic or geometric similarity. Further, allometric covariation among empirical scaling exponents falls along predicted functions with clustering around the flow similarity predictions. This work adds to the body of literature highlighting the importance of hydraulics in understanding the physiological basis of plant allometries, identifies previously unknown central tendencies in petiole allometry, and helps to delineate the scope within which the flow similarity model may be applicable.
... Most research has focused on the exponent b, which dictates the pace at which organisms allocate resources to perform life-essential tasks, such as growth and reproduction (West & Brown, 2005). Although debates have been about whether b is equal to 0.67 or 0.75 (Dodds et al., 2001;Heusner, 1991;Kleiber, 1932;Savage et al., 2004), numerous empirical studies have shown evidence against the universality of b and revealed a large variation of b across taxa, lifestyles and environmental conditions (Bokma, 2004;Clarke & Johnston, 1999;Glazier, 2009;Isaac & Carbone, 2010;Killen et al., 2010;Post & Lee, 1996;White & Seymour, 2005). ...
Metabolic scaling provides valuable information about the physiological and ecological functions of organisms, although few studies have quantified the metabolic scaling exponent (b) of communities under natural conditions. Maximum entropy theory of ecology (METE) is a constraint‐based unified theory with the potential to empirically assess the spatial variation of the metabolic scaling.
Our main goal is to develop a novel method of estimating b within a community by integrating metabolic scaling and METE. We also aim to study the relationships between the estimated b and environmental variables across communities.
We developed a new METE framework to estimate b in 118 stream fish communities in the north‐eastern Iberian Peninsula. We first extended the original maximum entropy model by parameterizing b in the model prediction of the community‐level individual size distributions and compared our results with empirical and theoretical predictions. We then tested the effects of abiotic conditions, species composition and human disturbance on the spatial variation of community‐level b.
We found that community‐level b of the best maximum entropy models showed great spatial variability, ranging from 0.25 to 2.38. The mean exponent (b = 0.93) resembled the community‐aggregated mean values from three previous metabolic scaling meta‐analyses, all of which were greater than the theoretical predictions of 0.67 and 0.75. Furthermore, the generalized additive model showed that b reached maximum at the intermediate mean annual precipitation level and declined significantly as human disturbance intensified.
The parameterized METE is proposed here as a novel framework for estimating the metabolic pace of life of stream fish communities. The large spatial variation of b may reflect the combined effects of environmental constraints and species interactions, which likely have important feedback on the structure and function of natural communities. Our newly developed framework can also be applied to study the impact of global environmental pressures on metabolic scaling and energy use in other ecosystems.
... The team took off by accepting that "the vast majority of organisms exhibit scaling exponents very close to 3/4 for metabolic rate and to 1/4 for internal times and distances." This starting point was an important one because, as seen above, quite a few researchers did not agree with it, for different reasons (53,79,97,103,108,128,176,195,197,245,247,248). The authors proposed a novel model of distribution of resources organized according to a fractal-type pattern of branching tubes that reached all cells by filling all available spaces. ...
Although it is intuitive that large mammals need more food than smaller ones, it is not so obvious that, relative to their body mass, larger mammals consume less than smaller ones. In fact, on a per kg basis, the resting metabolic rate of a mouse is some 50 times higher than that of an elephant. The fact that metabolism could not be proportional to the mass of the animal was suggested by Sarrus and Rameaux in 1838. The first indication that oxygen consumption (or other indices of metabolic rate, Y) related to the animal body mass (M) according to an exponential of the type Y = a · Mb , where b was about 0.75, was presented by Max Kleiber in 1932. Two years later Samuel Brody had collected sufficient data to construct the first "mouse-to-elephant" metabolic curve. The physiological basis of the relationship has been the object of many hypotheses, often accompanied by a great deal of controversy. This historical essay traces the origin of the mouse-to-elephant metabolic function, recalling the earliest concepts of metabolism and its measurements to understand the body size dependency, which is still one of the most elusive phenomena in comparative physiology. A brief look at the metabolic scaling of nonmammalian organisms will be included to frame the mouse-to-elephant curve into a broader context and to introduce some interesting interpretations of the mammalian function. © 2023 American Physiological Society. Compr Physiol 13:4513-4558, 2023.
... The MTE has been criticized for numerous reasons (Dodds et al. 2001;Banavar et al. 2002;Hochachka et al. 2003;Konarzewski 2004, 2005;Glazier 2005Glazier , 2014Glazier , 2015Glazier , 2018aGlazier , 2018bDuncan et al. 2007;O'Connor et al. 2007;White et al. 2007;Clarke 2017;Kozłowski et al. 2020)-some conceptual, some analytic, and some because large groups of organisms do not seem to fit the paradigm (e.g., plants; Reich et al. 2006). We do not wish to outline or debate any of these published criticisms. ...
Many attempts at providing a single-scale exponent and mechanism to explain metabolic rate assert a monolithic selective mechanism for allometries, characterized by a universal allometric scale power (usually chosen to be 0.75). To test for the deviations from universal allometric scaling, we gathered data from previously published metabolic measurements on 903 bird species and performed regressions of log(basal metabolic rate) and log(body mass) for (1) all birds and (2) 20 monophyletic clades within birds. We constructed two Bayesian linear mixed models-one included ecological variables and the other included data for mammals from Sieg et al. (2009). Overall allometric patterns differed significantly among clades of birds, and some clades were not consistent with the 0.75 scale power. We were unable to find apparent physiological, morphological, phylogenetic, or ecological characteristics among clades, predicting a difference in allometry or consistency with any previously proposed universal allometry. The Bayesian analysis illuminated novel bivariate, clade-specific differences in scaling slope-intercept space, separating large groups of birds and mammals. While significantly related to basal metabolic rate, feeding guild and migratory tendency had small effects compared to clade and body mass. We propose that allometric hypotheses, in general, must extend beyond simple overarching mechanisms to allow for conflicting and interacting influences that produce allometric patterns at narrower taxonomic scales-perhaps including other processes whose optimization may interfere with that of the system proposed by the metabolic theory of ecology.
... The premise of the argument based on tooth positive allometry was that an animal's metabolic rate and thus its energy requirements increase as its volume to the power of 0.75. This scaling exponent has been predicted by several theories (Kleiber 1932;Feldman and McMahon 1983;West et al. 1997;Savage et al. 2004), but an exponent of 0.67 has been equally reported in mammals and derived from theoretical considerations (Heusner 1982;Dodds et al. 2001;White and Seymour 2003), the same exponent developed in assumption 3 in the "Introduction." This disagreement has important implications for the reasoning in the studies by Purnell (1993Purnell ( , 1994: if the metabolic rate in conodonts scaled as body mass to the power of 0.67 and not 0.75, then an isometric relationship between the food acquisition surface and the animal's dimensions might well be expected. ...
Conodont elements have high rates of morphological evolution, but the drivers of this disparity are debated. Positive allometric relationships between dimensions of food-processing surfaces and entire P 1 elements have been used to argue that these elements performed mechanical digestion. If involved in food processing, the surface of the element should grow at a rate proportional to the increase in energy requirements of the animal. This inference of function relies on the assumption that the energy requirements of the animal grew faster (≅ mass 0.75 ) than the tooth area (≅ mass 0.67 ). We reevaluate this assumption based on metabolic rates across animals and calculate the allometry in platform-bearing P 1 elements of Late Triassic co-occurring taxa, Metapolygnathus communisti and Epigondolella rigoi , using 3D models of ontogenetic series. Positive allometry is found in platform and element dimensions in both species, supporting a grasping-tooth hypothesis, based on the assumption that metabolic rate in conodonts scaled with body mass similarly to that in fish and ectotherms. We also calculate the curvature of the P 1 platform surface using the Dirichlet normal energy (DNE) as a proxy for diet. DNE values increase with body mass, supporting the assumption that conodont metabolic rates increased faster than mass 0.67 . We finally find that adults in both taxa differ in their food bases, which supports trophic diversification as an important driver of the remarkable disparity of conodont elements.
... Yet, for the same rate of increase on the timescale of natural selection, metabolic evolution in physical time accelerates more in the smaller species because they evolve over a larger number of generations. By simulating the evolutionary radiation of placental and marsupial mammals from the extinction of dinosaurs, Witting (2018) found that this body mass dependent evolutionary acceleration bends the inter-specific metabolic allometry over time, with the primary selection of metabolism predicting a curvature in the metabolic allometry of placentals, as documented by Hayssen and Lacy (1985), Dodds et al. (2001), Packard and Birchard (2008), Kolokotrones et al. (2010), andMacKay (2011). The metabolic scaling for marsupials, on the other hand, showed almost no curvature suggesting that placentals have more primary selected variation in mass-specific metabolism than marsupials. ...
I use data based life history models to illustrate natural selection causes for the evolution of inter-specific body mass allometries in birds and mammals. This illustrates i) how the primary selection of resource handling and mass-specific metabolism generates net energy for individuals, ii) how the selected net energy generates a population dynamic feedback selection where intra-specific interactive competition selects body masses that scale in proportion with net energy on the timescale of natural selection, iii) how the feedback selection of body mass buffers ecological variation in survival, iv) how the exponents of body mass allometries are selected from the dominant spatial dimensionality of the foraging ecology, v) how the population density allometry is affected by inter-specific competition, and vi) how primary selected metabolism bends the metabolic allometry and explains a metabolic invariance across major taxa of vertebrates.
... The use of scaling laws in biological physics has a long story that begins with the last of Galileo's book Dialogues Concerning Two New Sciences, published in 1638 [1]. In the past century many allometric scaling relations [2] have been advanced in the biological and physiological domains, some of them spanning several decades of variability in the relevant variables [3,4]. In recent years, ecologists have also observed that many ecosystems exhibit frequently power-law scaling relations between the number of species (S) and the available area (A) controlled by a characteristic non-universal exponent z in the form S ∼ A z , known as SAR curves [5,6]. ...
Here a statistical fractal model is introduced to explain the observed scaling along six orders of magnitude for the average linguistic diversity within geographic areas. It is conjectured that the same model could be relevant to explain some aspects of the species-area curves in ecological studies.
... Nevertheless, contrasting theories have recently arisen because of several reported exceptions in which the resting metabolic rate of the scaling exponent was significantly different 2 of 18 from the universal 3/4-power law [20][21][22][23]26,[37][38][39][40][41][42][43][44][45]. Deviations from the 3/4-power law are known to occur interspecifically (ranging from approximately 0 to >1, but mostly found between 2/3 [two-thirds] and 1) [20,21,23,26,[46][47][48][49], as well as intraspecifically [15,23,26]. ...
Metabolism is the fundamental process dictating material and energy fluxes through organisms. Several studies have suggested that resting metabolic scaling in various aquatic invertebrates is positively correlated with changes in body shape and the scaling of body surface area, which agrees with the surface area theory, but contradicts the negative correlations predicted by the resource–transport network theory. However, the relationship between resting metabolic scaling and respiration area, particularly in asymmetric fish that have undergone dramatically rapid metamorphosis, remains unclear. In this morphometric study in an asymmetric fish species (Paralichthys olivaceus), I compared my results with previous reports on resting metabolic scaling. I measured the respiratory area of P. olivaceus specimens aged 11–94 days (body weight, 0.00095–1.30000 g, respectively) to determine whether and how the resting metabolic scaling is associated with changes in body shape and respiratory area. Resting metabolic scaling might be more closely related to body surface area, because their slopes exactly corresponded with each other, than to respiratory area. Furthermore, confirming the surface area theory, it was linked to changes in body shape, but not from the resource–transport network theory. These findings provide new insights into the scaling mechanisms of area in relation to metabolism in asymmetric fish.
Although it is intuitive that large mammals need more food than smaller ones, it is not so obvious that, relative to their body mass, larger mammals consume less than smaller ones. In fact, on a per kg basis, the resting metabolic rate of a mouse is some 50 times higher than that of an elephant. The fact that metabolism could not be proportional to the mass of the animal was suggested by Sarrus and Rameaux in 1838. The first indication that oxygen consumption (or other indices of metabolic rate, Y ) related to the animal body mass ( M ) according to an exponential of the type Y = a · M b , where b was about 0.75, was presented by Max Kleiber in 1932. Two years later Samuel Brody had collected sufficient data to construct the first “mouse‐to‐elephant” metabolic curve. The physiological basis of the relationship has been the object of many hypotheses, often accompanied by a great deal of controversy. This historical essay traces the origin of the mouse‐to‐elephant metabolic function, recalling the earliest concepts of metabolism and its measurements to understand the body size dependency, which is still one of the most elusive phenomena in comparative physiology. A brief look at the metabolic scaling of nonmammalian organisms will be included to frame the mouse‐to‐elephant curve into a broader context and to introduce some interesting interpretations of the mammalian function. © 2023 American Physiological Society. Compr Physiol 13:4513‐4558, 2023.
Life on earth spans a size range of around 21 orders of magnitude across species and can span a range of more than 6 orders of magnitude within species of animal. The effect of size on physiology is, therefore, enormous and is typically expressed by how physiological phenomena scale with mass b . When b ≠ 1 a trait does not vary in direct proportion to mass and is said to scale allometrically. The study of allometric scaling goes back to at least the time of Galileo Galilei, and published scaling relationships are now available for hundreds of traits. Here, the methods of scaling analysis are reviewed, using examples for a range of traits with an emphasis on those related to metabolism in animals. Where necessary, new relationships have been generated from published data using modern phylogenetically informed techniques. During recent decades one of the most controversial scaling relationships has been that between metabolic rate and body mass and a number of explanations have been proposed for the scaling of this trait. Examples of these mechanistic explanations for metabolic scaling are reviewed, and suggestions made for comparing between them. Finally, the conceptual links between metabolic scaling and ecological patterns are examined, emphasizing the distinction between (1) the hypothesis that size‐ and temperature‐dependent variation among species and individuals in metabolic rate influences ecological processes at levels of organization from individuals to the biosphere and (2) mechanistic explanations for metabolic rate that may explain the size‐ and temperature‐dependence of this trait. © 2014 American Physiological Society. Compr Physiol 4:231‐256, 2014.
In the previous chapters we described how statistical laws have been used in complex-systems research: Chap. 1 provided a brief historical overview and a working definition of statistical laws; Chap. 2 listed examples from different disciplines, focusing on the similar role played by statistical laws in connecting data and models; and Chap. 3 introduced increasingly sophisticated quantitative methods that have been employed to test, fit, and explore statistical laws. In this last chapter, we will move away from the description of how statistical laws have been used and, instead, focus on the role they can and should play in the study of complex systems. We will propose different ways in which methods and interpretations can be coherently employed, highlight possible pitfalls, suggest good practices, and speculate about the future of statistical laws in data-driven research.
This chapter contains a case by case description of paradigmatic statistical laws. While acknowledging their distinctiveness, our focus is on the common aspects across different statistical laws, in particular the similar role they have played in different research areas. The aim is to facilitate a comparative analysis that highlights the significance of this concept in complex-systems studies, supporting the unified treatment proposed in this monograph. In each case, we briefly describe how these laws were proposed, the most prominent explanations for their origin, and some of their uses. While we attempt to refer to original work, and to give credit to the original proponents of the laws, the description should be interpreted as a historical narrative that justified (and still justifies) the use of statistical laws and not as an attempt to reproduce the historical steps involved in this process. For readers interested in specific statistical laws, we hope the content of this chapter will provide a contextual introduction and point to relevant work where more specific aspects are discussed.
This chapter introduces and critically discusses the quantitative (statistical) methods used to study statistical laws.
As discussed in Chaps. 3, 4, and 5, growth and remodeling of coronary vasculature are complex processes that occur during both normal physiology (exercise, pregnancy, aging, etc.) and in pathophysiology (hypertension, hypertrophy, heart failure, etc.). Mathematical models can be very useful in describing growth and remodeling of the coronary vasculature for understanding the underlying principles of the biological response to physical or chemical stimuli. The analytical models described in this chapter can also be useful for the diagnosis and treatment of coronary artery disease using percutaneous coronary intervention or coronary artery bypass surgery as described in Chaps. 7 and 9, respectively.
Complex patterns of genome evolution associated with the end-Cretaceous [Cretaceous-Paleogene (K–Pg)] mass extinction limit our understanding of the early evolutionary history of modern birds. Here, we analyzed patterns of avian molecular evolution and identified distinct macroevolutionary regimes across exons, introns, untranslated regions, and mitochondrial genomes. Bird clades originating near the K–Pg boundary exhibited numerous shifts in the mode of molecular evolution, suggesting a burst of genomic heterogeneity at this point in Earth’s history. These inferred shifts in substitution patterns were closely related to evolutionary shifts in developmental mode, adult body mass, and patterns of metabolic scaling. Our results suggest that the end-Cretaceous mass extinction triggered integrated patterns of evolution across avian genomes, physiology, and life history near the dawn of the modern bird radiation.
O estudo foi elaborado com a finalidade de estimar as exigências de proteína bruta para mantença, ganho e produção de ovos de codornas japonesas no pico de postura (80 a 110 dias de idade). Para a exigência de mantença foi usado delineamento inteiramente ao acaso com quatro tratamentos e seis repetições com seis aves. Os tratamentos foram: T1 = dieta basal fornecida à vontade; T2 = 75; T3 = 50 e T4 = 30% do consumo do T1. A metodologia usada para estimar a exigência de mantença foi a do abate comparativo. Para a estimativa da exigência de ganho de peso, dez grupos de seis codornas foram criados paralelamente, alimentados à vontade. Destas aves, um grupo, por vez, foi abatido aos 01, 03, 06, 09, 12, 15, 18, 21, 24, 27 e 30 dias do ensaio. Concomitantemente, para a estimativa da exigência de produção de ovos dez grupos de seis codornas foram criados paralelamente, alimentados à vontade, e sendo coletados os ovos a cada três dias. Com os dados de mantença, ganho e produção de ovos foi elaborado modelo de predição da exigência de proteína bruta: PB (g/ave/dia) = 6,99×P^0,67 + 0,662×GP + 0,228×MO. Onde: PB requerimento em proteína bruta; P é o peso da codorna em kg; GP ganho de peso em g/ave/dia e MO massa de ovos em g/dia.
O estudo foi elaborado com a finalidade de estimar as exigências em energia para mantença, ganho e produção de ovos de codornas japonesas no período de 80 a 110 dias de idade. Para a exigência de mantença foi usado delineamento inteiramente ao acaso com quatro tratamentos e seis repetições com seis aves. Os tratamentos foram: T1 = dieta basal fornecida à vontade; T2 = 75; T3 = 50 e T4 = 30% do consumo do T1. A metodologia usada para estimar a exigência de mantença foi a do abate comparativo. Para a estimativa da exigência de ganho de peso, dez grupos de seis codornas foram criados paralelamente alimentados à vontade. Destas aves, um grupo por vez foi abatido aos 01, 03, 06, 09, 12, 15, 18, 21, 24, 27 e 30 dias do ensaio. Concomitantemente, para a estimativa da exigência para produção de ovos, dez grupos com seis codornas foram criados paralelamente alimentados à vontade e os ovos coletados os ovos a cada três dias. Portanto, a determinação das exigências de mantença, ganho e produção de ovos foi elaborado modelo para predição da exigência de energia metabolizável (EM - kcal/ave/dia = 150,37×P^0,67 + 6,01×GP + 2,349×MO).
The laws governing life should be as simple as possible; however, theoretical investigations into allometric laws have become increasingly complex, with the long-standing debate over the scaling exponent in allometric laws persisting. This paper re-examines the same biological phenomenon using two different scales. On a macroscopic scale, a cell surface appears smooth, but on a smaller scale, it exhibits a fractal-like porous structure. To elaborate, a few examples are given. Employing the two-scale fractal theory, we theoretically predict and experimentally verify the scaling exponent values for basal, active, and maximal metabolic rates. This paper concludes that Rubner’s 2/3 law and Kleiber’s 3/4 law are two facets of the same truth, manifested across different scale approximations.
The metabolic scaling theory (MST) predicts quasi‐universal trait–size relationships in plants, characterised by a unique allometric exponent within and across large taxonomic scales. However, recent studies have identified variability in allometric relationships, without a clear understanding of the modulating role played by genetic variation and environment.
Here, we investigated (1) the allometric relationships for two central traits of MST, namely total leaf area and plant growth rate, in the model species Arabidopsis thaliana, (2) the variability of plant allometries between genotypes and (3) the plastic responses of plant allometries under water deficit, high temperature and their interaction.
Using a population of 120 genotypes, we found that intraspecific allometries adhered on average with MST predictions. However, a broad variability but a moderate plasticity in the allometric exponents was observed across genotypes and environments. Allometric exponents were impacted significantly, yet weakly, by water deficit, but not by high temperature. Moreover, genotypes that deviated from MST predictions exhibited more plasticity in trait–size relationships than genotypes that followed MST predictions.
Our study suggests that plant allometry is genetically variable and might be related to different adaptive strategies to cope with stressing conditions. Thus, our results highlights the need of assessing trait–size relationships within species to understand the mechanisms of plant adaptation to contrasted environments.
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Species in the mammalian order Carnivora exhibit a huge diversity of life histories with body sizes spanning more than three orders of magnitude. Despite this diversity, most terrestrial carnivores can be classified as either feeding on invertebrates and small vertebrates or on large vertebrates. Small carnivores feed predominantly on invertebrates probably because they are a superabundant: resource (sometimes 90% of animal biomass(1-3)); however, intake rates of invertebrate feeders are low, about one tenth of those of vertebrate feeders(4,5). Although small carnivores can subsist on this diet because of low absolute energy requirements, invertebrate feeding appears to be unsustainable for larger carnivores. Here we show, by reviewing the most common live prey in carnivore diets, that there is a striking transition from feeding on small prey (less than half of predator mass) to large prey (near predator mass), occurring at predator masses of 21.5-25kg. We test the hypothesis that this dichotomy is the
Resting or basal metabolic rates, compared across a wide range of organisms, scale with respect to body mass as approximately the 0.75 power. This relationship has recently been linked to the fractal geometry of the appropriate transport system or, in the case of birds and mammals, the blood vascular system. However, the structural features of the blood vascular system should more closely reflect maximal aerobic metabolic rates rather than submaximal function. Thus, the maximal aerobic metabolic rates of birds and mammals should also scale as approximately the 0.75 power. A review of the literature on maximal oxygen consumption and factorial aerobic scope (maximum oxygen consumption divided by basal metabolic rate) suggests that body mass influences the capacity of the cardiovascular system to raise metabolic rates above those at rest. The results show that the maximum sustainable metabolic rates of both birds and mammals are similar and scale as approximately the 0.88 +/- 0.02 power of body mass (and
Fluid transport systems mediate the transfer of materials both within an organism and between an organism and its environment.
The architecture of fluid transport systems is determined by the small distances over which transfer processes are effective
and by hydrodynamic and energetic constraints. All fluid transport systems within organisms exhibit one of two geometries,
a simple tube interrupted by a planar transfer region or a branched network of vessels linking widely distributed transfer
regions; each is determined by different morphogenetic processes. By exploiting the signal inherent in local shear stress
on the vessel walls, animals have repeatedly evolved a complex branching hierarchy of vessels approximating a globally optimal
system that minimizes the costs of the construction and maintenance of the fluid transport system.
This paper is the first in a series of three papers investigating the detailed geometry of river networks. Branching networks are a universal structure employed in the distribution and collection of material. Large-scale river networks mark an important class of two-dimensional branching networks, being not only of intrinsic interest but also a pervasive natural phenomenon. In the description of river network structure, scaling laws are uniformly observed. Reported values of scaling exponents vary, suggesting that no unique set of scaling exponents exists. To improve this current understanding of scaling in river networks and to provide a fuller description of branching network structure, here we report a theoretical and empirical study of fluctuations about and deviations from scaling. We examine data for continent-scale river networks such as the Mississippi and the Amazon and draw inspiration from a simple model of directed, random networks. We center our investigations on the scaling of the length of a subbasin's dominant stream with its area, a characterization of basin shape known as Hack's law. We generalize this relationship to a joint probability density, and provide observations and explanations of deviations from scaling. We show that fluctuations about scaling are substantial, and grow with system size. We find strong deviations from scaling at small scales which can be explained by the existence of a linear network structure. At intermediate scales, we find slow drifts in exponent values, indicating that scaling is only approximately obeyed and that universality remains indeterminate. At large scales, we observe a breakdown in scaling due to decreasing sample space and correlations with overall basin shape. The extent of approximate scaling is significantly restricted by these deviations, and will not be improved by increases in network resolution.
Arterial bifurcations in the cardiovascular system of a rat were studied, using a resin cast of the entire arterial tree. At each bifurcation, measurements were made of the diameters of the three vessels involved, the two branching angles, and the angle delta, which the parent artery makes with the plane containing the two branches. The results were found to be consistent with those reported previously in man and monkey. In addition, measurements of delta in the present study indicate that arterial bifurcations are mostly two dimensional.
Over 100 allometric regressions on rare [kg kg(-1) d(-1) = d(-1)], age [d] and density [kg km(-2)] parameters often used in ecological models were collected in a literature review. For each parameter, typical coefficients and exponents that covered most of the correlations found were selected. The parameters were checked for consistency by comparing them to each other, e.g. mortality versus age and to independent data, e.g. production and respiration rate constants versus production efficiency. Exponents for consumption, production and respiration rates on the one hand and slopes for maturation. average and maximum age on the other, scale to size with an exponent of the same magnitude but opposite sign. Most values are between +/- 0.25 and +/- 0.33. No consistent differences between parameters and species were found. Coefficients for production and respiration were largely consistent with those for ingestion and with independent data on assimilation and production efficiencies. The same was concluded for production versus mortality. for mortality versus age as well as maximum versus average production and consumption. Warm-blooded animals consume, produce and respite at rates about 10 times higher than equally sized cold-blooded species. The reverse uas noted for age and possibly density. Population density of individual species and biota density of geometric size classes are nearly independent of size for regressions that span a wide size range. The coefficients for density vary up to two orders of magnitude. depending on local conditions. Most allometric regressions reported apply to aquatic invertebrates or terrestrial birds and mammals of the biophageous food chain. Yet, the rate and age regressions that include other species do not indicate substantial deviations for bacteria, micro-and macrophytes and for terrestrial (saprophageous) invertebrates. The set of typical values derived may facilitate parameter estimation for generic models. For specific models, the regressions collected in the literature review may provide initial values for parameters.
The relationship between home range area and body size of terrestrial mammals is reconsidered in light of the concept of biological time. Biological time is an internal, body-mass-dependent, time scale to which the durations (of rates) of biological events are entrained. These events range from purely physiological (e.g., muscle contraction time) to purely ecological (e.g., time to traverse home range). Evidence is presented that home range size scales linearly to body mass for carnivores as it does for herbivores. This scaling supports the hypothesis that animals select their home range areas to meet metabolic demands integrated over biologically critical periods. Confounding variables in the home range-body mass regression include habitat productivity and methods of location. Data on home ranges derived from telemetry studies of terrestrial carnivores are presented and used to derive allometric equations for home range area. The exponents of these equations are shown to approximate 1.0, although intercept values vary with latitude and, presumably, habitat productivity. Social organization and behavior may also influence the relationship of home range area to metabolic needs for different sex and age categories within a species.
The principles of biological similarity have not been adequately defined. Previous studies have not been fully successful, mainly because the search for such principles centered around the idea that a few of them would apply to all animals, just as Newton's principles of mechanical similarity apply to all inanimate objects. However, this is not possible, so that the search has led up a number of blind alleys, ending with a failure to provide a fundamental and unified explanation—as contrasted with empirical justification—for the scaling of basal metabolic rate of many kinds of animals (e.g. mammals, birds, fish, certain small metazoa) according to a of body mass, M, rather than as (“surface law”) or M1·0; moreover, none of the previous theories can account for the fact that other kinds of animals (e.g. insects, snakes, hibernating mammals) do not obey the . Two basic characteristics of all animals are that in the water they are on “the verge of floating” and that movement is interwoven with the nature of animal life itself. These observations lead to the principle of constancy of body density (ρ ⋍ 1) and to the principle of similarity in some defined sense of the muscular apparatus—the universal generator of movement—across species, but only within classes of animals that have evolved similar methods of locomotion. Because muscle tissues are subject to elastic (“spring-type”) contraction forces, a general elastic similarity principle holds for muscle diameter, d, vs. a linear body dimension, L, i.e., d2 ∝ L3 (Galileo-Rashevsky principle; this principle is valid also for the dimensions of the trunk of animals without exoskeleton subject to a gravitational load, e.g. for land mammals but not for sea mammals). A final principle which, combined with the above, leads to the for muscle-power generation, is that time is scaled as the linear dimension, T ∝ L (in contrast to Newton's second principle of mechanical similarity for physical objects, ). This principle was introduced in the past either as an arbitrary assumption (Lambert & Teissier) or based on the empirical finding of constancy of muscle shortening velocity ( across mammalian species (Hill-McMahon). However, this principle cannot be valid for the classes of animals which do not obey the . This principle is derived here from the more fundamental principle of constancy, across similar species, of mechanical stress (force over cross-section) endured by contracting muscles. In species such as snakes, however, in which locomotion is generated in a different way, friction forces assume an important role and scaling of time as with mechanical similarity, , is obtained by assuming a similar velocity of the animals along their axis as size increases; using this scaling, the deviation of snakes from the is explained. Interestingly, though, such scaling may have limited the maximal body size of snakes. A yet different principle seems to be operating in insects, leading to the scaling: and a faster than body mass increase of basal metabolic rate. Finally, neither heat loss nor surface-related transport appear to be limiting and setting factors for metabolic rate—indeed “surface arguments” are entirely unbased. Only in some situations where the is not obeyed because of heat loss considerations (e.g. dogs adapted to the tropics or to the arctic cold) is a surface argument relevant.
Generalized additive models have the form η(x) = α + σ fj(xj), where η might be the regression function in a multiple regression or the logistic transformation of the posterior probability Pr(y = 1 | x) in a logistic regression. In fact, these models generalize the whole family of generalized linear models η(x) = β′x, where η(x) = g(μ(x)) is some transformation of the regression function. We use the local scoring algorithm to estimate the functions fj(xj) nonparametrically, using a scatterplot smoother as a building block. We demonstrate the models in two different analyses: a nonparametric analysis of covariance and a logistic regression. The procedure can be used as a diagnostic tool for identifying parametric transformations of the covariates in a standard linear analysis. A variety of inferential tools have been developed to aid the analyst in assessing the relevance and significance of the estimated functions: these include confidence curves, degrees of freedom estimates, and approximate hypothesis tests.The local scoring algorithm is analogous to the iterative reweighted least squares algorithm for solving likelihood and nonlinear regression equations. At each iteration, an adjusted dependent variable is formed and an additive regression model is fit using the backfitting algorithm. The backfitting algorithm cycles through the variables and estimates each coordinated function by smoothing the partial residuals.
A refinement of a model of the vascular system (Cohn, 1954) is discussed. We now consider that the major vessels arise from
the aorta as secondary branching vessels rather than by successive dichotomies of the aorta as previously considered. It is
shown that this change does not affect calculated values of flow, the number of branchings, or the relative radii of vessels
at a branching. A comparison is made of theoretical predictions with experimental data and good greement is found.
The method of analysis of this paper is used to extend the treatments of the previous paper to a more general case. It is
shown that the results previously obtained are valid.
Motion.- Segmental Movement and Vibrations.- External Flow: Dynamic Forces Acting on Moving Bodies.- Flying and Swimming.- Blood Flow in Heart, Lung, Arteries, and Veins.- Micro- and Macrocirculation.- Respiratory Gas Flow.- Basic Transport Equations According to Thermodynamics, Molecular Diffusion, Mechanisms in Membranes, and Multiphasic Structure.- Mass Transport in Capillaries, Tissues, Interstitial Space, Lymphatics, Indicator Dilution Method, and Peristalsis.- Description of Internal Deformation and Forces.- Stress, Strain, and Stability of Organs.- Strength, Trauma, and Tolerance.- Biomechanical Aspects of Growth and Tissue Engineering.- Author Index.- Subject Index.
Variation in resting metabolic rate is strongly correlated with differences in body weight among birds. The lowest taxonomic level at which most of the variance in resting metabolic rate and body weight is evident for the sample is among families within orders. The allometric exponent across family points is 0.67. This exponent accords with the surface area interpretation of metabolic scaling based on considerations of heat loss. Deviations of family points from this allometric line are used to examine how resting metabolic rates differ among taxa, and whether variation in resting metabolic rate is correlated with broad differences in ecology and behaviour. Despite the strong correlation between resting metabolic rate and body weight, there is evidence for adaptive departures from the allometric line, and possible selective forces are discussed.
The allometric scaling of active metabolic rate is compared with that of resting metabolic rate. The allometric exponents for the two levels of energy expenditure differ, demonstrating that active small-bodied birds require proportionately more energy per unit time above resting levels than do active large-bodied birds. No consistent evidence was found to indicate that the different methods used to estimate active metabolic rate result in systematic bias. Birds require more energy relative to body size when undertaking breeding activities than at other stages of the annual cycle.
Land mammals range in size from the 3-g shrew to the 3000-kg elephant. Despite this 106 range in weight, most land mammals have similar anatomy, physiology, biochemistry, and cellular structure. This similarity has allowed interspecies scaling of physiologic properties such as heart rate, blood flow, blood volume, organ size, and longevity. The equation that is the basis for scaling physiologic properties among mammals is the power equation Y = aWb, where Y is the physiologic variable of interest, W is body weight, and log a is the y-intercept and b is the slope obtained from the plot of log Y versus log W. Animals commonly used in preclinical drug studies (i.e., mice, rats, rabbits, monkeys, and dogs) do not eliminate drugs at the same rate that humans eliminate drugs; small mammals usually eliminate drugs faster than large mammals. Since drug elimination is intimately associated with physiologic properties that are well described among species, it seems reasonable to surmise that drug elimination can be scaled among mammals. Analysis of drug pharmacokinetics in numerous species demonstrates that drug elimination among species is predictable and, in general, obeys the power equation Y = aWa. Early papers on interspecies pharmacokinetic scaling normalized the x- and y-axes to illustrate the superimposability of pharmacokinetic curves from different species. More recently, the x- and y-axes have been left in the common units of concentration and time, and individual pharmacokinetic variables have been adjusted to predict pharmacokinetic profiles in an untested species, usually humans.
The concept of correlating pharmacokinetic parameters with body weight from different animal species has become a useful tool in drug development. The allometric approach is based on the power function, where the body weight of the species is plotted against the pharmacokinetic parameter(s) of interest. Clearance, volume of distribution, and elimination half-life are the three most frequently extrapolated pharmacokinetic parameters. Over the years, many approaches have been suggested to improve the prediction of these pharmacokinetic parameters in humans from animal data. A literature review indicates that there are different degrees of success with different methods for different drugs. Overall, though interspecies scaling requires refinement and better understanding, the approach has lot of potential during the drug development process.
An approach to quantitative work on optimal systems is considered. Desired optimal principles are utilized in constructing
a hypothetical system similar to the organ system considered. A comparison of this constructed system with the anatomical
system then gives an indication of the importance of the optimal principles in the form or function of the organ system considered.
These ideas are applied to the mammalian vascular system, and limiting values are obtained for some of its important component
parts. The constructed system gives good agreement with anatomical data for vessel radii, lengths, and hydrodynamic resistance
to flow.
The allometric equation P = aMb (P: standard metabolism, M: body mass, a: mass coefficient, and b: mass exponent) can be theoretically derived from the following relations: l/L = t/T = λ, m/M = λ3 where l and L are homologous lengths, t and T homologous times and λ is the coefficient of similitude of two animals. Animals are homomorphic when , a = constant, and when their density is the same. These conditions appear to be realized in mature mammals of the same species, but mammals of different species are not homomorphic. Homomorphism means that the physiological time-scale is not the same in small and large animals, but that the energy spent per unit mass and unit of physiological time remain the same in homomorphic animals [mass-specific physiological power, Φ]. The mass coefficient ‘a’ is equal to Φm therefore ‘a’ is physiologically the most significant parameter in the allometric equation. The physiological implications of Φ are discussed.
It is generally recognized that larger animals eat more, live longer, have larger offspring, and so on; but it is unusual to see these commonplace observations as a basis for scientific biology. A large number of empirically based relationships describe biological rates as simple functions of body size; and other such relations predict the intrinsic rate of population growth, animal speed, animal density, territory size, prey size, physiology, and morphology. Such equations almost always exist for mammals and birds, often for other vertebrates and invertebrates, sometimes for protozoa, algae, and bacteria, and occasionally even for plants. There are too many organisms to measure all aspects of the biology of every species of population, so scientists must depend on generalizations. Body size relations represent our most extensive and powerful assemblage of generalizations, but they have never been organized for use in ecology. This book represents the largest single compilation of interspecific size relations, and instructs the reader on the use of these relationships; their comparison, combination, and criticism. Both strengths and weaknesses of our current knowledge are discussed in order to indicate the many possible directions for further research. This important volume will therefore provide a point of departure toward a new applied ecology, giving quantitative solutions to real questions. It will interest advanced students of ecology and comparative physiology as well as professional biologists.
Benedict has shown that this law is already over ninety years old, Robiquet and Tillaye having formulated it quite clearly in 1839. The history of the surface law is given in the paper of (Harris and Benedict (1919)). We may here only briefly mention the different ways in which it has been found. The early writers derived the law from theoretical considerations on a rather small experimental basis, as did Bergmann, who in 1847 had already written a book on the subject. Respiration trials were carried out by Regnault and Reiset, and Rameaux based the surface law on measurements of the amount of air respired per minute by two thousand human beings of different sizes. (Rubner (1883)) demonstrated the law in accurate respiration trials on dogs and Richet rediscovered it empirically on rabbits. The latter writes (p. 223): “C’est aprèe coup seulement que je me suis avisé que la donnée surface était plus intéressante que la donnée poids.”
A review of growth rates of diatoms and dinoflagellates in light-saturated, nutrient-replete cultures at 20/sup 0/C confirms weak dependence on cell volume or mass. These maximal (intrinsic) rates are not linearly related to surface area or surface-to-volume ratio of the cells. The growth of most diatoms is materially faster than that of dinoflagellates; other algae fall in between or below the dinoflagellates. Small ciliates have appreciably higher intrinsic growth rates than algae of the same cell volume. The average food consumption per ciliate in the marine pelagic realm is inferred to be very low, so that the realized specific growth rates are much smaller than the intrinsic potentials. Also, a previously postulated refuge from predation, afforded by small size, is extended down to about 10-..mu..m/sup 3/ cell volume.
Given that all organisms share a common ancestry, why is it that they differ so greatly in their capacities to sense, remember, and respond to the world about them? How did we gain our ability to think and to feel? How do we differ from other organisms in these capacities? Our brain endows us with the faculties and the drive to ask these fundamental questions. The answers depend crucially on understanding how brains have evolved. This inquiry into brain evolution is interdisciplinary and multifaceted, based on converging evidence obtained from the study of the genetic regulation of development, the geological history of the earth, and the behavioral ecology of animals, as well as from direct anatomical and physiological studies of brains of animals of different species. From this investigation three themes will emerge: that the essential role of brains is to serve as a buffer against environmental variation; that every evolutionary advance in the nervous system has a cost; and that the development of the brain to the level of complexity we enjoy -- and that makes our lives so rich -- depended on the establishment of the human family as a social and reproductive unit.
I will begin by considering one of the basic problems faced by all organisms: how to find food and avoid hazards in a constantly changing world. This leads to the question of how nervous systems detect and integrate the vast array of information available to them and derive from this flood of data adaptive behavioral responses. The evolution of nervous systems depended on a unique mechanism for communication, the action potential, a self-renewing electrical signal that moves along specialized neural fibers called axons that serve as the wires connecting nerve cells. By permitting the development of large nervous systems, this mechanism for neuronal communication made possible the emergence of complex and diverse forms of animal life.
From this review we conclude the following: 1) The body weight of an organism is an adequate reference index for the correlation of morphological and physiological characteristics. In comparative physiology, body weight can be recommended as a unifying frame of reference, particularly if the ponderal scale includes several decades, in order to apply logarithmic scales for the variables involved. (See article). 2) The statistical analysis of the experimental data can be represented conveniently by means of the logarithmic equivalent of Huxley's allometric equation (y = a-Wb), which is the most simple and at the same time the most versatile mathematical expression for intra- or interspecies comparisons. The exponents (b) for the allometric equations can be predicted for all biological variables definable in terms of the MLT system of physics (M = mass, L = length, T = time) or of a four-dimensional system MLTt where t = temperature. 3) By means of dimensional analysis and the theory of biological similarity a range of similarity criteria can be established: a) mechanical or dynamic similarity, b) kinematic or biological similarity; and c) hydrodynamic or transport similarity. Most functions obey the so-called biological (kinematic) similarity, particularly when the concept of operational time is introduced into Lambert-Teissier's original theory. 4) A satisfactory correlation (r = 0.99) for 80 empirical allometric exponents (b) describing morphological and physiological characteristics of living beings was found. These results are discussed in relation to Rosen's optimality principles in biology. 5) Organisms should be considered as mixed regimes. This means that no single similarity criterion can predict the allometric exponent (b) of all functions that dimensionally belong to MLT or MLTt systems, despite the fact that in the great majority of cases kinematic similarity will satisfactorily predict the reduced exponent (b). Nevertheless, in some instances mechanical (dynamic) similarity must be applied, and in other circumstances hydrodynamic (transport) similarity. 6) Cellular or molecular levels are not in the domain of the present theory, since neither cell dimensions nor molecular processes (viz., blood viscosity, diffusion capacity) can be predicted by biological similarity criteria.
Basal metabolism and body mass are related by the metabolic power function: P = aMb, where P = basal metabolism in Watts, a = mass coefficient, M = body mass in kg, and b = mass exponent. The mass exponent of 117 dogs from the literature b dog = 0.885 +/- 0.024 (r = 0.960; F = 1387; df = 1,115). This mass exponent is significantly greater than the commonly accepted value of 0.75 for mammals. The dog's 95% confidence ellipse is compared with that of mammals with body mass (M) less than 3.2 kg (the lower limit of the mass range in dogs) and greater than 3.2 kg. When M greater than 3.2 kg the interspecific metabolic mass exponent (bi) in mammals is also significantly greater than 0.75 and not different from b dog (bi = 0.869 +/- 0.034; r = 0.919; F = 648; df = 1,120). In mammals M less than 3.2 kg bi is significantly smaller than 0.75 (bi = 0.634 +/- 0.010; r = 0.941; F = 4319; df = 1,561). These data show that in mammals the relationship between the logarithms of basal metabolism and body mass is not accurately described by a single regression line. They also indicate that the commonly accepted 0.75 mass exponent is not applicable to the prediction of basal metabolism in dogs and mammals. The relationship between body mass and maintenance energy metabolism (MEM) in 332 dogs shows that the prediction interval is too wide to reasonably predict MEM in individual dogs. However, the minimum maintenance energy metabolism (MMEM in Watts) can be accurately predicted by a simple algorithm: MMEM = 10.3 + 1.41 x M. The theoretical meaning of the basal metabolic power function is discussed.
Dogs are unique among mammals in having a 100-fold range in body weight for nonobese adults. This variation makes the calculation of the power function for metabolic body size and hence the allometry of energy requirements a particularly challenging subject. Several functions have been proposed from W0.68 to W0.88 (W = body weight in kg). In the present study we measured the heat output of 22 dogs representing seven breeds, aged 1-10 y with W from 5.8 to 48.8 kg, using a whole-body calorimeter specifically designed for this purpose. Regression of log energy output against log W gave the equation 678 W0.64 (r = 0.96; P less than 0.001), which is considered to represent resting energy expenditure (REE) as kJ/d. If estimates of the energy cost of activity are added to REE, new equations of 655 W0.69 (low activity) and 643 W0.73 (higher activity) are obtained, depending on the amount of activity included in the calculation. From these results we suggest that the allometry of energy requirements of adult dogs is a function of different exponents for REE and the energy cost of activity. It does not appear to exceed W0.75 and may be nearer to W0.67.
The relationship between basal metabolism P and body mass M of 391 mammalian species has been analysed by least-squares regression, robust regression and covariance analyses. This relationship is a power function:
where the mass exponent b is 0.678±0.007 (mean±S.D.) and the mass coefficient a takes different values. Theory of measurement revealed that the 2/3 mass exponent is due to an underlying dimensional relationship between the primary quantity of mass and the secondary quantity of power.
This paper shows that the 2/3 mass exponent is not the physiological problem of interest. It is not the slope of the metabolic regression line, but its location in the mass/power plane, that must be explained. This location is given by the value of the mass coefficient, the explanation of which is, and remains, the central question in comparative physiology.
The database used by Hemmingsen (1960) to compute energy metabolism in unicellular organisms was reassembled and submitted to linear (log-log) analysis. As Hemmingsen noted, this data set includes marine zygotes, which are not unicellular organisms. If no temperature correction factors are applied to the data the best-fit regression line has a slope of 0.698 +/- 0.024. Application of the temperature correction factors assumed to have been used by Hemmingsen gave a slope of 0.756 +/- 0.021, identical to the value he reported. The correlation coefficient is 0.97. The mean scatter about the regression line exceeds 100%. A revised set of temperature correction factors gave a slope of 0.730 +/- 0.021, suggesting that the value of almost exactly three-quarters obtained by Hemmingsen was probably fortuitous. The slope of the best-fit regression line is very sensitive to the inclusion of bacteria and flagellates. When the data points for these organisms are omitted from the calculation the slope decreases to 0.645 +/- 0.045. When the data points for bacteria, flagellates and marine zygotes are omitted, the slope drops to 0.608 +/- 0.025. The correlation coefficient (0.97), compared to the best-fit line reported by Hemmingsen, is unaffected; the mean deviation about the regression line drops to 40% and the points are evenly distributed about the regression line. Because of the small number of species for which measurements have been made, the existing database relating energy metabolism to cell size is not representative of unicellular organisms generally. It is concluded that the case for a three-quarters power rule expressing energy metabolism as a function of size in unicellular organisms generally is not at all persuasive.