Functional Ecology 2006 20 , 394–399 394 © 2006 The Authors.Journal compilation© 2006 British Ecological Society Blackwell Publishing LtdOxford, UKFECFunctional Ecology0269-84632006 The Authors. Journal compilation © 2006 British Ecological Society 4 2006202FORUM Allometry of metabolismR. S. Etienne et al. FORUM Demystifying the West, Brown & Enquist model of the allometry of metabolism RAMPAL S. ETIENNE*†, M. EMILE F. APOL* and HAN OLFF Community and Conservation Ecology Group, Centre for Ecological and Evolutionary Studies, University of Groningen, PO Box 14, 9750 AA Haren, The Netherlands Summary1. The allometry of metabolic rate has long been one of the key relationships in ecology.While its existence is generally agreed on, the exact value of the scaling exponent, andthe key mechanisms that determine its value, are still hotly debated. 2. The network model of West, Brown & Enquist ( Science 276 , 122–126, 1997) predictsa value of 3 / 4 but, although appealing, this model has not been generally accepted. 3. Here we reconstruct the model and derive the exponent in a clearer and much morestraightforward way that requires weaker assumptions than the original model. Specifi-cally, self-similarity of the network is not required. Our formulation can even be usedif one or several assumptions of West et al . (1997) are considered invalid. 4. Moreover, we provide a formula for the proportionality constant (i.e. the interceptof the allometric scaling relation) that shows explicitly where factors as temperatureand stoichiometry affect metabolism. Key-words : Allometric scaling, metabolic theory, organismal transport network, fractal topology Functional Ecology (2006) 20 , 394–399doi: 10.1111/j.1365-2435.2006.01136.x Introduction It has been long recognized that for many taxa (e.g.mammals) allometric relations of the form Y = Y 0 M x exist, where Y is an organismal property (e.g. growthrate, metabolic rate or life span), M is the body mass, Y 0 is a taxonomic-group specific constant, and x is acharacteristic exponent (Calder 1984; Peters 1983).Probably the most important of these allometricrelations is the relation for the basal metabolic rate,because many other allometric relations depend onit, and particularly on the value of x . Both empiricaland theoretical studies have been carried out to studythis exponent. Empirical studies report values rangingfrom 2 / 3 to 3 / 4 (e.g. Dodds, Rothman & Weitz 2001;Savage et al . 2004) and both extremes of this range havegained theoretical support. An old and simple argu-ment for the value 2 / 3 (Rubner 1883) is the following. Inan organism at steady state (i.e. constant temperature),the heat produced by metabolism must equal the heatdissipated to the environment via the organism’s bodysurface. Thus the metabolic rate is proportional to thebody surface area, which scales with the 2 / 3 power ofbody size. This very simple model ignores the fact thatmetabolic processes require resources (e.g. oxygen) anddoes not seem to do justice to the complex structuresthat have evolved to transport resources to the cells.Almost a decade ago, West, Brown & Enquist (1997)used the fact that many taxa have fractal-like networksfor resource transport to predict a value of 3 / 4 for theallometric exponent x . This model, although appealingand a stimulus for follow-up studies (see reviews byBrown et al . 2004; Marquet et al . 2005), has still notbeen generally accepted, as evidenced by a vigorous recentdebate in Functional Ecology between the originalauthors on the one hand (hereafter called WBE) andKozlowski & Konarzewski (hereafter called K&K)on the other (Kozlowski & Konarzewski 2004, 2005;Brown, West & Enquist 2005). Other authors havealso heavily criticized the assumptions and derivationof this model (e.g. Banavar, Maritan & Rinaldo 1999;Dodds et al . 2001). The main message emerging fromthis debate is that the model as formulated by WBEis not at all clear. It sorely needs a thorough recon-struction for a correct understanding and subsequentempirical testing of (elements of) the model andfurther theoretical development. In this paper we aimto provide such a thorough reconstruction. We describethe structure of the transport network, WBE’s assump-tions (regardless of whether they are plausible ornot) and their mathematical translations. For optimaltransparency, we sometimes deviate from the notation †Author to whom correspondence should be addressed.E-mail: [email protected]*R.S.E. and M.E.F.A. contributed equally to this paper.395 Allometry of metabolism © 2006 The Authors.Journal compilation© 2006 British Ecological Society, Functional Ecology , 20 , 394–399 of WBE. See Table 1 for conversion of our notation tothat of West, Brown & Enquist (1997). Furthermore,we generalize the model in three ways. First, we formu-late the model in such a way that the derivation ofthe allometric exponent of x = 3 / 4 does not require thebranching network distributing resources to the cellsto be self-similar. Second, where the proportionalityconstant Y 0 is usually ignored, we present a formulafor this constant. Third, our formulation can be usedas a basis for models that make different assumptionsfrom those of WBE, making the theory amenableto rigorous testing. We discuss how the disagreementbetween WBE and K&K can be understood in thelight of the reconstructed, generalized model. Model      Assumption 1: The transport system of oxygen has a fractal-like branching topology as shown in Fig. 1 West et al. (1997) make a somewhat stronger assump-tion than assumption 1, by requiring the network to bea self-similar fractal, but self-similarity is not necessary,as will become clear below. For illustrative purposes wetake the blood transport system as an example, but thetopology also applies to other hierarchical transportsystems found in organisms. The first level in the branch-ing system (aorta) has value k = 1 and the last level(capillaries) has value k = C . Each level consists of N k identical pipes (vessels) of radius r k , length l k , cross-sectionalarea A k = and volume V k = . Each vessel at level k splits at a node into ν k vessels at level k + 1. Hence, N k is a product of all ν i at levels up to (but not including)level k , that is N k = and generally N 1 = 1. Con-versely, the number of