1The 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, and the key mechanisms that determine its value, are still hotly debated.
2The network model of West, Brown & Enquist (Science 276, 122–126, 1997) predicts a value of 3/4 but, although appealing, this model has not been generally accepted.
3Here we reconstruct the model and derive the exponent in a clearer and much more straightforward way that requires weaker assumptions than the original model. Specifically, self-similarity of the network is not required. Our formulation can even be used if one or several assumptions of West et al. (1997) are considered invalid.
4Moreover, we provide a formula for the proportionality constant (i.e. the intercept of the allometric scaling relation) that shows explicitly where factors as temperature and stoichiometry affect metabolism.
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It has been long recognized that for many taxa (e.g. mammals) allometric relations of the form Y = Y0Mx exist, where Y is an organismal property (e.g. growth rate, metabolic rate or life span), M is the body mass, Y0 is a taxonomic-group specific constant, and x is a characteristic exponent (Calder 1984; Peters 1983). Probably the most important of these allometric relations is the relation for the basal metabolic rate, because many other allometric relations depend on it, and particularly on the value of x. Both empirical and theoretical studies have been carried out to study this exponent. Empirical studies report values ranging from 2/3 to 3/4 (e.g. Dodds, Rothman & Weitz 2001; Savage et al. 2004) and both extremes of this range have gained theoretical support. An old and simple argument for the value 2/3 (Rubner 1883) is the following. In an organism at steady state (i.e. constant temperature), the heat produced by metabolism must equal the heat dissipated to the environment via the organism's body surface. Thus the metabolic rate is proportional to the body surface area, which scales with the 2/3 power of body size. This very simple model ignores the fact that metabolic processes require resources (e.g. oxygen) and does not seem to do justice to the complex structures that 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 networks for resource transport to predict a value of 3/4 for the allometric exponent x. This model, although appealing and a stimulus for follow-up studies (see reviews by Brown et al. 2004; Marquet et al. 2005), has still not been generally accepted, as evidenced by a vigorous recent debate in Functional Ecology between the original authors on the one hand (hereafter called WBE) and Kozlowski & Konarzewski (hereafter called K&K) on the other (Kozlowski & Konarzewski 2004, 2005; Brown, West & Enquist 2005). Other authors have also heavily criticized the assumptions and derivation of this model (e.g. Banavar, Maritan & Rinaldo 1999; Dodds et al. 2001). The main message emerging from this debate is that the model as formulated by WBE is not at all clear. It sorely needs a thorough reconstruction for a correct understanding and subsequent empirical testing of (elements of) the model and further theoretical development. In this paper we aim to provide such a thorough reconstruction. We describe the structure of the transport network, WBE's assumptions (regardless of whether they are plausible or not) and their mathematical translations. For optimal transparency, we sometimes deviate from the notation of WBE. See Table 1 for conversion of our notation to that of West, Brown & Enquist (1997). Furthermore, we generalize the model in three ways. First, we formulate the model in such a way that the derivation of the allometric exponent of x = 3/4 does not require the branching network distributing resources to the cells to be self-similar. Second, where the proportionality constant Y0 is usually ignored, we present a formula for this constant. Third, our formulation can be used as a basis for models that make different assumptions from those of WBE, making the theory amenable to rigorous testing. We discuss how the disagreement between WBE and K&K can be understood in the light of the reconstructed, generalized model.
Table 1. Conversion of notation used in this article to those in West et al. (1997). Symbols that are not listed either have the same meaning in both papers, or appear only in this article
Quotient of number of vessels at levels k + 1 and k
Quotient of vessel radius at levels k + 1 and k
Quotient of vessel length at levels k + 1 and k
Metabolic scaling parameter
Blood flow at level k + 1
Velocity at level k + 1 averaged over the cross-sectional area
topology of the transport system
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 assumption than assumption 1, by requiring the network to be a self-similar fractal, but self-similarity is not necessary, as will become clear below. For illustrative purposes we take the blood transport system as an example, but the topology also applies to other hierarchical transport systems found in organisms. The first level in the branching system (aorta) has value k = 1 and the last level (capillaries) has value k = C. Each level consists of Nk identical pipes (vessels) of radius rk, length lk, cross-sectional area Ak= and volume Vk=. Each vessel at level k splits at a node into νk vessels at level k + 1. Hence, Nk is a product of all νi at levels up to (but not including) level k, that is Nk = and generally N1 = 1. Conversely, the number of branches originating from each vessel at level k is the quotient of the number of vessels at levels k + 1 and k:
( (eqn 1))
For the vessel radius rk and the vessel length lk we can define the quantities ρk and λk analogously:
( (eqn 2))
( (eqn 3))
We use Greek symbols to indicate that these quantities are quotients. These quantities are convenient because they allow us to express the radius, length and number of vessels at level k in terms of the radius, length and number of vessels at the last level, the capillary level (which has a special status in the model):
( (eqn 4))
( (eqn 5))
( (eqn 6))
We also define the following quantities:
( (eqn 7))
( (eqn 8))
Thus, the quantity αk represents the ratio of the total cross-sectional areas of levels k + 1 and k, and ϕk represents the ratio of the total of volumes around the vessels at levels k + 1 and k, the interpretation of which will be discussed below. These quantities will be shown to be useful in translating assumptions made by West et al. (1997) into mathematical terms.
Finally, we define the three basic topological quantities S1, S2 and S3 which will be helpful in our derivation below:
So far we have only used definitions and the assumption that the network has the topology of Fig. 1. We are now ready to take the first step towards the allometry of metabolism.
The total volume of the transport fluid, blood, contained at branching level k of the network, is given by:
( (eqn 13))
and hence the total blood volume is the sum over all levels:
( (eqn 14))
This leads to the following expression for the total number of capillaries:
( (eqn 15))
This equation will be inserted in the expression for the metabolic rate below. Note the 3/4 exponent that appears in this expression, as it will finally determine the allometric exponent of x = 3/4 under the assumptions of the model.
flow through the transport system
Let us denote by Qk the blood flow at level k, i.e. the total volume flowing through the vessel cross-sectional area Ak per unit of time. Let us further denote by uk the velocity of the fluid at level k, averaged over the cross-sectional area. We have:
Qk = Akuk( (eqn 16))
and the total flow through all the vessels at level k is then NkQk= NkAkuk.
Assumption 2: The proportionality between organismal metabolic rate and blood volume flow is independent of body size
The basal metabolic rate B is usually calculated from the rate of oxygen consumption by the organism. This rate is proportional to the total volume flow in the aorta, N1Q1, and is thus given by:
B=f0N1Q1( (eqn 17))
where f0 is the change in oxygen concentration due to metabolic processes in the cells. Assumption 2, also made by West et al. (1997), states that f0 is independent of body size.
Assumption 3: The transport fluid (blood) is incompressible
Water and blood are virtually incompressible, and also gases (e.g. in insect tracheae) can be treated this way provided that the transport velocity u is smaller than the speed of sound (Tritton 1988). For an incompressible fluid, conservation of mass throughout the network implies that (Berne & Levy 1986):
The terminal units, i.e. the capillaries, have properties that do not depend on body mass M. Thus rC, lC and uC do not depend on M. This is a crucial assumption of West et al. (1997), because the only remaining way the metabolic rate in equation (20) can depend on body size, apart from the dependence of Vb on body size, expressed by equation (21), is by its dependence on S2. This is however, prevented by assumption 6.
Assumption 6: The quantity S2 defined in equation (10) does not depend on body size, M
West et al. (1997) made three separate assumptions to support this assumption. First they assumed that the network is area-preserving. This means that the total area of all vessels at level k equals the total cross-sectional area of all vessels at any other level. As this also applies to level k + 1, this entails αk = 1 for all k (see the definition of αk in equation (7)). Second, West et al. (1997) assumed that the network is ‘space-filling’. The interpretation of this assumption is the main cause of the disagreement between WBE and K&K, as we will discuss below. West et al. (1997) interpret it as preservation of the quantity 4/3 πNk(lk/2)3 across all levels k (which is mathematically equivalent to preservation of Nk). This then leads to ϕk = 1 for all k (see the definition of ϕk in equation (8)). Third, they made an assumption that can be translated in our new framework as the assumption that the quantity S3 (see equation (11)) does not depend on body size M. The only way that S3 can depend on body size is through the number of levels C. In Fig. 2 we have plotted S3 vs the total number of levels C and we observe that S3 quickly converges to a constant value. Moreover, this value is less than an order of magnitude larger than the value for C = 1. This strongly supports the assumption that S3 does not depend on M. The three assumptions together imply that S2 does not depend on body size M. Assumption 6 may, however, also be supported by other sets of assumptions.
with b = 1 under assumption 4, see equation (22). The proportionality factor B0 is given by:
( (eqn 24))
which does not depend on body size if the assumptions of the model hold. This formula is a new result, not given by West et al. (1997), which may have far-reaching consequences, as we discuss below.
We have given a new presentation of the WBE network model, and derived a value of x = 3/4 for the allometric scaling exponent of metabolism, with more explicit statement of the assumptions. Of all assumptions, assumption 6 requires the most scrutiny. In turn, of the three supporting assumptions that can be used to justify assumption 6, the preservation of the quantity Nk across the hierarchical levels of the network, is the most questionable, and has been the topic of vigorous debate. The misunderstanding about this supporting assumption is caused by the fact that West et al. (1997) state that the network is ‘space-filling’ and that Nk is proportional to the ‘service volume’ of all vessels at level k. This terminology suggests an erroneous biological interpretation of Nk. ‘Service volume’ seems to correspond to a part of the body to which the network delivers oxygen, and ‘space-filling’ gives the impression that the whole body of the organism should be filled with such service volumes. Such interpretations indeed lead to a contradiction. If the volume to which oxygen is delivered is assumed to be proportional to NC and this volume fills all space (i.e. the whole body), it should naturally scale with body size M1. However, assumption 5 implies that is independent of body size, and the total model shows that NC scales with M3/4 (if b = 1), leading to an overall scaling of NC with M3/4. The only way to solve this contradiction is to drop the biological interpretation of NC as the total body volume and to reconsider the assumption that the quantity Nk is preserved across the hierarchical levels of the network. Figure 3 shows how this supporting assumption that the quantity Nk is the same for all levels k can be envisaged. The sum of volumes of all circles (spheres in three dimensions) at level k is, apart from a proportionality constant, given by Nk. Preservation of Nk thus means that it is a geometrical property of the network that the sum of the volumes around vessels remains constant rather than a biological property. This seems a plausible assumption, but theoretical support is scarce. Preservation of Nk is a stronger assumption than actually needed. All we need, is that the quantity S2 is roughly independent of body size M. This can be true, even if Nk decreases with k and thus ϕk < 1, because this can be countered by αk > 1. Particularly note that ϕk is raised to the power 1/3 in S2 which implies that any deviations of ϕk from unity will be reduced. Hence, quite a strong deviation from 1 is needed to make S2 strongly dependent on C and thus on body size.
but this is not necessary to make S2 independent of body size.
Under different assumptions from those of assumption 6, equation (14) will still hold, but the body size independence of S2 is then no longer valid. To obtain the allometric scaling exponent under these different assumptions, S1 must be expressed as a product of an (p being some exponent) and an such that is largely body size independent. (In the case of WBE's assumptions, we had p = 1/3 and = S2). The consequences of these assumptions and their implications need to be explored further.
We have provided a new formula for the proportionality factor of metabolic scaling, B0. If measurements can be carried out to estimate the quantities that make up this proportionality factor, we have an alternative test of the network model. The results could even indicate a (slight) body size dependence of this proportionality factor. It has already been suggested that B0 may depend on body size, e.g. through f0 (K & K 2004), or through lC and rC (Dawson 2001, 2003), which, respectively, violate assumptions 2 and 5. Our formulation of the model allows for easy incorporation of such deviations by way of equation (24). The quantity B0 is a crucial factor in themetabolic theory of ecology (Brown et al. 2004), in which B0 shows explicitly where factors as temperature and stoichiometry affect metabolism. Our formula shows that these factors should affect metabolism via f0, the proportionality constant that relates metabolic rate to the total blood flow through the aorta.
In sum, our reformulation of the network model and our new derivation indeed entails the exponent x = 3/4, and the assumptions under which this result is obtained are weaker than stated by West et al. (1997). Furthermore, if any of these assumptions is violated (and there is some evidence for this), our formulation provides a basis for incorporating such assumptions and finding the corresponding allometric scaling exponent, which then might differ from 3/4. Our reformulation opens up the theory to rigorous experimental testing. Finally, we have provided a mechanistic formula for the proportionality factor, B0, which plays a central role in the metabolic theory of ecology, and we have argued that, under alternative model assumptions, B0 might depend on body size.
Funding for M.E.F.A. was provided by the Netherlands Organisation for Scientific Research (NWO).