Demystifying the West, Brown &amp; Enquist model of the allometry of metabolism

Etienne, Rampal S.; Apol, M. E. F.; Olff, Han

doi:10.1111/j.1365-2435.2006.01136.x

Cited by 47 publications

(51 citation statements)

References 17 publications

(38 reference statements)

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“…So our branching model, a significant modification of WBE's, predicts the quarterpower scaling exponents including metabolic rate scaling as M Our hierarchical branching model retains the fractal-like design of WBE but shows how the self-similar branching through the hierarchy of arteries arises naturally, with the number of capillaries (proportional to the metabolic rate) scaling as the cube (in 3D animals) of the ratio of the aorta length (∼M 1/3 ) to the capillary length (∼M 1/12 ). Thus, the aorta length scales as the Euclidean length of the organism while ensuring that blood volume scales linearly with M. The changes from WBE-having blood velocity scaling as M 1/12 rather than as M 0 , aorta length scaling as M 1/3 rather than M 1/4 , and average cross-sectional area of the network scaling as M 2/3 rather than M 3/4 -solve the problem of fitting the fractal-like network into an animal with a fundamentally Euclidean geometry (17).…”

Section: The Hierarchical Branching Networkmentioning

confidence: 99%

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A general basis for quarter-power scaling in animals

Banavar

Moses²,

Brown³

et al. 2010

Proc. Natl. Acad. Sci. U.S.A.

181

261

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It has been known for decades that the metabolic rate of animals scales with body mass with an exponent that is almost always <1, >2/3, and often very close to 3/4. The 3/4 exponent emerges naturally from two models of resource distribution networks, radial explosion and hierarchically branched, which incorporate a minimum of specific details. Both models show that the exponent is 2/3 if velocity of flow remains constant, but can attain a maximum value of 3/4 if velocity scales with its maximum exponent, 1/12. Quarterpower scaling can arise even when there is no underlying fractality. The canonical "fourth dimension" in biological scaling relations can result from matching the velocity of flow through the network to the linear dimension of the terminal "service volume" where resources are consumed. These models have broad applicability for the optimal design of biological and engineered systems where energy, materials, or information are distributed from a single source.A llometric scaling laws reflect changes in biological structure and function as animals diversified to span >12 orders of magnitude in body size. Since the seminal work of Kleiber in 1932, it has been known that the metabolic rate, B, or rate of energy use in most animals and plants scales as approximately the 3/4 power of body mass, M (1-8), and this B ∼ M 3/4 scaling has come to be known as Kleiber's rule. Most other biological rates and times, such as heart rates, reproductive rates, blood circulation times, and life times, scale with characteristic quarter powers, as M −1/4 and M 1/4 , respectively (4-9). This unusual fourth dimension came as a surprise to biologists, who expected that heat dissipation would cause metabolic rate to scale the same as body surface area and hence geometrically as the 2/3 power of volume or mass. Quarter-power scaling lacked a unified theoretical explanation until 1997, when West et al. (WBE) (10) produced a model based on optimizing resource supply and minimizing hydrodynamic resistance in the supply network.The WBE model stimulated a resurgence of interest in allometric scaling in biology. It initiated a lively debate about the empirical generality of 3/4-power metabolic scaling and its theoretical explanation (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26). A recent study revived the case for geometric scaling by showing that simple models of distribution networks generate metabolic scaling exponents of 2/3 (25). Nevertheless, the increasing number of empirical studies repeatedly finds exponents >2/3, <1, and often very close to 3/4 (14, 24, 26, 28-30). Here we show that an exponent of 3/4 emerges naturally as an upper bound for the scaling of metabolic rate in two simple models of vascular networks. Furthermore, quarter power scaling is shown to hold even when there is no underlying fractal network. A unique prediction of our models is that blood velocity scales as the 1/12 power of animal mass. These models have broad application to biological and human-engineered systems where resources ar...

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Section: The Hierarchical Branching Networkmentioning

confidence: 99%

“…It initiated a lively debate about the empirical generality of 3/4-power metabolic scaling and its theoretical explanation (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26). A recent study revived the case for geometric scaling by showing that simple models of distribution networks generate metabolic scaling exponents of 2/3 (25).…”

mentioning

confidence: 99%

A general basis for quarter-power scaling in animals

Banavar

Moses²,

Brown³

et al. 2010

Proc. Natl. Acad. Sci. U.S.A.

181

261

View full text Add to dashboard Cite

show abstract

“…There have been only a few attempts to give a mechanistic explanation for the statistical variability although both the scaling exponent and the normalization constant may have an interpretation based on biological processes (Kozłowski et al 2003;Etienne et al 2006;Mäkelä and Valentine 2006;Chown et al 2007;Enquist et al 2007;Price et al 2007). As an alternative, some process-based models use conditional values of a and b, or additional variables to modify the values of a and b, instead of attempting to give a direct interpretation of the values of a and b by themselves (Duursma et al 2007;Holdo 2007).…”

Section: Introductionmentioning

confidence: 99%

Precision of allometric scaling equations for trees can be improved by including the effect of ecological interactions

Kaitaniemi

Lintunen

2008

Trees

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Allometric scaling relationships of the form Y = aX b are widely utilized in many types of models and analyses of tree structure. They are often viewed as static relationships where both the scaling exponent (b) and the normalization constant (a) obtain empirical values that are fixed within a single set of data. Among different sets of data, their values can show environmental variability. However, there have been only few attempts to give a mechanistic interpretation for this variability. We used field data to demonstrate how the scaling relationships in trees can be modified by ecological interactions. Moreover, we show how such processes can be incorporated into the scaling models to improve the fit and the information content of the scaling equations. When fixed theoretical scaling exponents were used instead of empirical exponents and when the effect of competitive interactions between trees was described by separate submodels that predicted the value of the normalisation constant in the scaling equations, it was possible to obtain 4-10% improvement in the model fit of three different structural scaling relationships. Our results suggest that unexplained variation in the values of the scaling parameters can be substituted by an identified effect of ecological factors on the value of the normalisation constant. This agrees with recent theoretical suggestions stating that ecological factors can directly influence the value of normalisation constants.

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“…First, several rigorous empirical analyses, involving body sizes spanning several orders of magnitude, have shown that b often deviates substantially from 3/4, varying significantly among different taxonomic groups of animals and plants (Glazier 2005;Reich et al 2006;White et al 2006White et al , 2007, and among different physiological states (Glazier 2005;Niven & Scharlemann 2005;White & Seymour 2005;Makarieva et al 2005aMakarieva et al , 2006b). Second, the models supporting the so-called 3/4-power law appear to have flawed assumptions and serious mathematical inconsistencies that have not yet been resolved, despite much debate (Dodds et al 2001;Kozlowski & Konarzewski 2004, 2005Brown et al 2005;Makarieva et al 2005bMakarieva et al , 2006aPainter 2005b,c;Banavar et al 2006;Chaui-Berlinck 2006, 2007Etienne et al 2006;Savage et al 2007).…”

Section: Introductionmentioning

confidence: 99%

Effects of metabolic level on the body size scaling of metabolic rate in birds and mammals

2008

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Metabolic rate is traditionally assumed to scale with body mass to the 3/4-power, but significant deviations from the '3/4-power law' have been observed for several different taxa of animals and plants, and for different physiological states. The recently proposed 'metabolic-level boundaries hypothesis' represents one of the attempts to explain this variation. It predicts that the power (log-log slope) of metabolic scaling relationships should vary between 2/3 and 1, in a systematic way with metabolic level. Here, this hypothesis is tested using data from birds and mammals. As predicted, in both of these independently evolved endothermic taxa, the scaling slope approaches 1 at the lowest and highest metabolic levels (as observed during torpor and strenuous exercise, respectively), whereas it is near 2/3 at intermediate resting and coldinduced metabolic levels. Remarkably, both taxa show similar, approximately U-shaped relationships between the scaling slope and the metabolic (activity) level. These predictable patterns strongly support the view that variation of the scaling slope is not merely noise obscuring the signal of a universal scaling law, but rather is the result of multiple physical constraints whose relative influence depends on the metabolic state of the organisms being analysed.

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Demystifying the West, Brown & Enquist model of the allometry of metabolism

Cited by 47 publications

References 17 publications

A general basis for quarter-power scaling in animals

A general basis for quarter-power scaling in animals

Precision of allometric scaling equations for trees can be improved by including the effect of ecological interactions

Effects of metabolic level on the body size scaling of metabolic rate in birds and mammals

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