Functional Ecology 2004 18 , 257–282 © 2004 British Ecological Society 257 Blackwell Publishing, Ltd.Oxford, UKFECFunctional Ecology0269-8463British Ecological Society, 20044 2004182Original ArticleQuarter-power scaling in biologyV. M. Savage et al. FORUM The predominance of quarter-power scaling in biology V. M. SAVAGE,*‡† J. F. GILLOOLY,§ W. H. WOODRUFF,*‡ G. B. WEST,*‡ A. P. ALLEN,§ B. J. ENQUIST¶ and J. H. BROWN*§ * The Santa Fe Institute, 1399 Hyde Park Road., Santa Fe, NM 87501 USA, ‡ Los Alamos National Laboratory, Los Alamos, NM 87545 USA, § Department of Biology, The University of New Mexico, Albuquerque, NM 87131 USA, and ¶ Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, AZ 85721, USA Summary1. Recent studies have resurrected the debate over the value for the allometric scalingexponent that relates whole-organism metabolic rate to body size. Is it 3 / 4 or 2 / 3 ? Thisquestion has been raised before and resolved in favour of 3 / 4 . Like previous ones, recentclaims for a value of 2 / 3 are based almost entirely on basal metabolic rate (BMR) inmammals. 2. Here we compile and analyse a new, larger data set for mammalian BMR. We showthat interspecific variation in BMR, as well as field metabolic rates of mammals, andbasal or standard metabolic rates for many other organisms, including vertebrates,invertebrates, protists and plants, all scale with exponents whose confidence intervalsinclude 3 / 4 and exclude 2 / 3 . Our analysis of maximal metabolic rate gives a slope that isgreater than and confidence intervals that exclude both 3 / 4 and 2 / 3 . 3. Additionally, numerous other physiological rates that are closely tied to metabolismin a wide variety of organisms, including heart and respiratory rates in mammals, scaleas M − 1/4 . 4. The fact that quarter-power allometric scaling is so pervasive in biology suggeststhat different allometric relations have a common, mechanistic origin and provides anempirical basis for theoretical models that derive these scaling exponents. Key-words : Body size, metabolic rates, physiological times, quarter-power scaling Functional Ecology (2004) 18 , 257–282 Introduction Many fundamental characteristics of organisms scalewith body size as power laws of the form: Y = Y 0 M b , eqn 1where Y is some characteristic such as metabolic rate,stride length or life span, Y 0 is a normalization constant, M is body mass and b is the allometric scaling exponent.A longstanding puzzle in biology is why the exponent b is usually some simple multiple of 1 / 4 rather than amultiple of 1 / 3 , as would be expected from Euclideanscaling.Renewed interest in allometry is due at least in part torecent theories that purport to explain the quarter-powerscaling (West, Brown & Enquist 1997, 1999a; Banavar,Maritan & Rinaldo 1999; Banavar et al . 2002). Thesetheories derive the scaling for metabolic rate based onthe designs of resource distribution networks, such asanimal and plant vascular systems. In particular, themodel of West et al . assumes that these networks havethree properties: (1) they branch hierarchically to sup-ply all parts of three dimensional organisms; (2) theyhave terminal units, such as capillaries or petioles, thatdo not vary with body size; and (3) natural selection hasoptimized hydrodynamic flow through the network sothat the work required to distribute resources has beenminimized. This model predicts many other character-istics of plant and animal circulatory systems, includ-ing dimensions of vessels, total volume of fluid, rates offlow and delivery times. This model has been extendedto explain the quarter-power scaling of many biologicaltraits, including mitochondrial densities (West, Woodruff& Brown 2002), ontogenetic growth rates (West, Brown& Enquist 2001), the partitioning and allocation ofproduction between plant organs such as roots, stems,leaves, and reproductive structures (Enquist & Niklas2002; Niklas & Enquist 2002), times of life-history events(Gillooly et al . 2002; Savage et al . 2004), and populationgrowth rates (Savage et al . 2004).Ever since the seminal studies of Kleiber (1932) andBrody et al . (1934, 1945), some biologists have questioned †Author to whom correspondence should be sent. E-mail:[email protected] V. M. Savage et al. © 2004 British Ecological Society, Functional Ecology , 18 , 257–282 whether the exponent for whole-organism metabolicrate really is 3 / 4 or whether it might be 2 / 3 as expectedfrom Euclidean geometric scaling (Heusner 1982a,b,1987, 1991; Kooijman 2000; Dodds, Rothman & Weitz2001; White & Seymour 2003). These questions havefocused on metabolic rate because it is such a funda-mental characteristic for all organisms. It is the rate atwhich energy and materials are transformed withinorganisms and exchanged with the environment.In the present study, we evaluate the evidence for thescaling exponents for basal metabolic rate (BMR) andother traits. We analyse three kinds of data. First, wecompile and analyse a new comprehensive data setfor the basal metabolic rate of mammals. Second, wepresent analyses of field and maximal metabolic ratesfor mammals, because these rates are more relevant tothe normal function of free-living mammals thanBMR, and we present reanalyses of data for mamma-lian heart and respiratory rates. Third, we performmeta-analyses (i.e. we calculate the mean and standarderror) of scaling exponents reported in the literaturefor other biological rates and times, some of which canbe measured more accurately than BMR. Finally, weidentify problems with recent studies that have claimedthat BMR of mammals scales as M 2/3 (at least overa limited range of M ) (Dodds et al . 2001; White &Seymour 2003). We conclude that the evidence sup-ports the pervasiveness of quarter-power allometricscaling in biology and, by extension, the models ofWest et al . (1997, 1999a). Historical perspective The idea that power laws characterize size-relatedvariation is old and well established in biology. Rubner(1883) originally observed that metabolic rate dependedon organismal body size and proposed that the rela-tionship followed from a surface-area rule (see alsoBergman 1847). In the 1920s Julian Huxley investigatedthe body size dependence of ontogenetic growth andother biological attributes and coined the term ‘allo-metric equation’ for equation 1 (Huxley 1932; see alsoThompson 1942). In the