### Introduction

- Top of page
- Summary
- Introduction
- WBE network model
- Network parameters from optimization
- Critical assumptions
- Discussion and conclusions
- Acknowledgements
- References
- Supporting Information

Kleiber's empirical law, stating that the basal or resting metabolic rate in organisms is approximately a power law of body mass *M*, i.e*. B* = *B*_{0}*M*^{a} (with *a* ≈ 0·75 and *B*_{0} a normalization constant whose units depend on the value of *a*), is one of the most intriguing and important allometric relations in ecology. It predicts how larger organisms have an increasingly lower rate of energy use per unit of body mass. Kleiber's law has been found to apply across a wide variety of taxa (Peters 1983; Calder 1984; Schmidt-Nielsen 1984; Savage *et al.* 2004 but see Dodds, Rothman & Weitz 2001; White & Seymour 2003; Bokma 2004 for exceptions), and it has far-reaching consequences at various levels of biological organization, ranging from organismal physiology, life span and evolution, to whole-ecosystem functioning (Brown *et al.* 2004; Whitfield 2006).

In 1997, West, Brown & Enquist (1997) (WBE) presented an appealing mechanistic explanation for the observed value *a* = 0·75. This led to wide-ranging application of Kleiber's empirical law in may fields of biology, as it was now ‘upgraded’ to a law based on first principles (Brown *et al*. 2004; Martínez del Rio 2008). WBE argued that the whole-organism metabolic rate of animals is constrained by the architecture of their internal transport network, which dictates the rate at which nutrients and oxygen are delivered to the cells. They hypothesized that, for species of different size, evolution through natural selection has resulted in an optimized fractal-like network that has minimal transport costs (energy dissipation) for a given flow and amount of transport fluid (blood). Assuming further that the network is ‘space filling’ to optimally service all cells, they arrived at a predicted value of *a =* 3/4 for the pulsatile blood flow that occurs in mammals and birds.

Unfortunately, the WBE article is written in a highly condensed way where not all steps are fully explained. In a previous article (Etienne, Apol & Olff 2006), we have reformulated and explained the basic structure of the original WBE model and demonstrated that 3/4-power scaling is indeed obtained if the transport network possesses the following four main properties: (1) its capillary size is invariant with body mass, (2) it is space filling, (3) its volume scales linearly with body mass and (4) it shows preservation of cross-sectional area upon branching. In other words, the network must have specific values for some internal network scaling exponents. This moves the problem of explaining metabolic 3/4-power scaling to understanding these key network properties. West and co-workers (West *et al*. 1997, 2000; West 1999; West & Brown 2004, 2005) claimed that the last two properties originate from (evolutionary) optimization, that is, minimizing transport costs (dissipation), and as a consequence minimizing wave reflections. That Kleiber's law is the result of evolution through natural selection operating within the context of universal constraints imposed by basic physical laws is probably the most appealing aspect of the WBE theory. However, this evolutionary basis of the theory has never been thoroughly scrutinized (Dodds *et al.* 2001; Kozlowski & Konarzewski 2004, 2005; Etienne *et al.* 2006), probably because of the difficulty of the mathematics involved and the highly condensed way it was presented.

Here we do so by fully reconstructing the WBE model (West *et al.* 1997), including the optimization part. First we briefly develop the relation between metabolic rate and network scaling properties along the lines presented in Etienne *et al.* (2006). Second, we show how the network properties are obtained via optimization. We present the essential logical steps in the model and fully assess all derivations of West *et al.*, for the first time by explicitly providing the pulsatile dissipation function and performing the corresponding optimization of the network. We find that following the logic and assumptions of the WBE model results in a biologically irrelevant network, i.e. a single vessel. We moreover find that at least five points in this derivation are highly questionable from a physical, physiological and/or logical point of view, but which are nonetheless critical to obtaining 3/4-power scaling of basal metabolism. These points concern the incorrect definition and incomplete optimization of the dissipation function by WBE, the assumed fixed vessel wall-thickness to radius ratio, the space-filling network and WBE's derivation of isometric mass scaling of blood volume. We discuss potential modifications of the model (in the spirit of the robustness analysis advocated by Martínez del Rio (2008)), which however opens a Pandora's box of possible metabolic exponents. Hence, the current framework of the WBE model cannot account for the observed universal metabolic scaling relation, and at least a more integrated approach including the modelling of oxygen transport from the capillaries to the cells and from the external environment to the network is needed. Or, explanations for Kleiber's law have to come from entirely different lines of argument.

### WBE network model

- Top of page
- Summary
- Introduction
- WBE network model
- Network parameters from optimization
- Critical assumptions
- Discussion and conclusions
- Acknowledgements
- References
- Supporting Information

In a previous article (Etienne *et al.* 2006), we reformulated the WBE model – except for the optimization part – and rederived its predicted 3/4-power metabolic scaling in terms of network scaling properties in a more straightforward way than in the original article (West *et al.* 1997). To also explore the optimization part, we will now summarize and generalize this scheme. It should be noted that we do not develop a new or different model in this and the previous article (as suggested by Martínez del Rio (2008)), but follow exactly the line of argument, and optimization procedure that WBE suggested would lead to 3/4-power scaling. Table 1 provides the conversion between our notation and that of West and co-workers.

Table 1. Conversion of notation used in this article to that in West, Brown & Enquist (1997) (WBE), West & Brown (2005) (WB) and Womersley (1955, 1958). Symbols that are not listed either have the same meaning in all the articles, or appear only in this article Property | This article | WBE | WB | Womersley |
---|

Level | *k* + 1 | *k* | *k* | – |

Level range | *k* = 1 ... *C* | *k* = 0 ... *N* | *k* = 0 ... *N* | – |

Number of levels | *C* | *N* + 1 | *N* + 1 | – |

Branching number | *v*_{k}_{+1} | *n*_{k} | *n* | – |

Radius ratio | ρ_{k }_{+ 1} | β_{k} | – | – |

Length ratio | λ_{k+1} | γ_{k} | – | – |

Metabolic exponent | *a* | *a* | *b* | – |

Volume flow | *Q*_{k}_{+1} | *Q̇*_{k} | *Q*_{k} | *Q* |

Average fluid velocity | *u*_{k}_{+1} | *ū*_{k} | – | *u* |

Service volume | *V*_{s,k+1} | – | *v*_{k} | – |

Capillary number, radius, etc. | *N*_{C}, *r*_{C}, ... | *N*_{C}, *r*_{C}, ... | *N*_{N}, *r*_{N}, ... | – |

Lagrange multipliers | Λ_{b}, Λ_{k}, Λ_{M} | λ, λ_{k}, λ_{M} | – | – |

‘Viscous function’ | | | | |

(Complex) reflection factor | | – | – | Λ = (1 − *λ*)/(1 + *λ*) |

Complex wave velocity | *c̃*_{k} | *c* | *c* | *c* |

Moens-Korteweg *a*_{inter}=*a*_{intra}−*δ* velocity | *c*_{0,k} | *c*_{0} | *c*_{0} | *c*_{0} |

Real wave velocity | *c*_{1,k} | – | – | *c*_{1} |

Mass density fluid (~wall) | ρ_{Μ} | ρ | ρ | ρ |

Consider an organism with a closed branching transport network consisting of *C* branching levels (e.g. in vertebrates, level 1 is the aorta, level *C* is the capillaries, but the principle also applies to other taxa with closed vascular networks). At level *k*, there are *N*_{k} vessels of radius *r*_{k}, length *l*_{k}, wall thickness *h*_{k}, and cross-sectional area , in which the transport fluid (blood in vertebrates) flows with average velocity *u*_{k} due to a pressure gradient ∇*p*_{k} (see Fig. 1). We define several scaling ratios that relate these vessel properties across adjacent branching levels: *v*_{k} ≡ *N*_{k}_{+1}/*N*_{k} is the branching ratio, ρ_{k} ≡ *r*_{k}_{+1}/*r*_{k} the radius or diameter ratio, λ_{k} ≡ *l*_{k}_{+1}/*l*_{k} the length ratio and η_{k} ≡ *h*_{k}_{+1}/*h*_{k} the wall-thickness ratio. We now define the internal network scaling exponents *c*_{r}, *c*_{l} and *c*_{h} by expressing the scaling ratios ρ_{k}, λ_{k} and η_{k} in terms of the branching ratio *v*_{k} as

- (1)

These scaling exponents form the critical connection between the topological dimensions of the network and its spatial dimensions (Mandelbrot 1982), in this case, to what extent does branching reduce vessel diameters, lengths and wall radii in the network. The minus signs are included for convenience since in general 0 < ρ_{k}, λ_{k}, η_{k} < 1 and *v*_{k} > 1.

The metabolic rate of the whole organism *B* is proportional to the total volume flow *Q*_{tot} through this network,

which is the so-called Fick equation (Milnor 1990), where *f*_{0} is the concentration difference of metabolites (oxygen) in the blood between the arteries and the veins (arteriovenous difference). In fact, *f*_{0}*Q*_{tot} is precisely the amount of oxygen per unit of time that is disappearing from the arterial system, and hence the respiration rate of the organism, which is proportional to its energy use. WBE assume that *f*_{0}, as well as the capillary properties *r*_{C}, *l*_{C}, *h*_{C} and *u*_{C} (their radius, length, wall thickness and average fluid velocity) are independent of body mass. Hence, the body-mass dependence of the total flow,

- (3)

and therefore also the body-mass dependence of the metabolic rate, is determined by the body-mass dependence of the number of capillaries *N*_{C}. Unfortunately, no straightforward theoretical predictions or good observational data are available for this dependency, as this would directly yield the ‘true’ flow-limited scaling of metabolism with body size. However, because the number of capillaries is subject to topological and geometrical constraints from the network (e.g. every group of capillaries must be supported by a larger vessel, that is again supported by a vessel, etc, up to the aorta), the body-mass dependence of *N*_{C} can be evaluated from some other global property of the network, of which we do know its body-mass dependence experimentally and/or theoretically. In the WBE model, the total blood volume contained in the network is taken for this purpose. With eqn 1 it follows after some straightforward algebra (Appendix S1 in Supporting Information) that the number of capillaries *N*_{C} can be expressed in terms of body mass (via the blood volume *V*_{b}), the internal scaling exponents (*c*_{r} and *c*_{l}) and the capillary dimensions as

- (4)

where is a generalization of the topological *S*_{3}-property (Etienne *et al.* 2006, see Appendix S1), which is just a number in the order of unity that depends on the scaling behaviour of the network (see also Fig. S1). We furthermore assume that blood volume *V*_{b} scales allometrically with body mass,

and that the topological property *S*_{3} is approximately independent of body mass (Etienne *et al.* 2006). The latter requires that 2*c*_{r} + *c*_{l} – 1 > 0 so that *S*_{3} quickly saturates with increasing number of levels *k* (Appendix S1). Therefore, the mass dependence of the total volume flow within the network and thus of the metabolic rate is defined through the mass dependence of the blood volume. Hence combining eqns 2, 3 and 4 with the above assumptions, the metabolic rate of the organism is given by

where the metabolic scaling exponent *a* is determined just by two network scaling exponents and by *b* as

- (7)

The normalization constant (the intercept of Kleiber's law, which was not explored by WBE) depends both on the network scaling exponents and on the properties of the capillaries,

- (8)

West *et al.* (1997) assumed *c*_{l} = 1/3 (space-filling network, based on the argument of spatial efficiency of the network), and argued that optimization of the network leads to *b* = 1 (isometric scaling of blood volume with body mass) and *c*_{r}=1/2 (preservation of cross-sectional area). If these values are assumed, *a* = 3/4 is obtained as in Kleiber's law (Etienne *et al.* 2006). However, as we will show next, there are severe problems with deriving these values by optimization of the network.

### Network parameters from optimization

- Top of page
- Summary
- Introduction
- WBE network model
- Network parameters from optimization
- Critical assumptions
- Discussion and conclusions
- Acknowledgements
- References
- Supporting Information

WBE assume that the process of evolution via natural selection has resulted in organisms with minimum maintenance costs required for their size thus maximizing energy available for growth and reproduction. To overcome the very slow process of diffusion over distances larger than ∼1 mm, a convective transport network is an effective solution (Calder 1984). The design of the transport network is likely to be evolutionarily optimal with respect to both costs and efficiency. With regard to the costs, West and co-workers therefore posed the plausible hypothesis that evolution for organisms of different size has resulted in a network that requires minimal transport costs per unit of flow and per unit of body mass to deliver oxygen and metabolites to the cells (West *et al*. 1997, 2000; West 1999; West & Brown 2005). This means that the dimensions of the vessels (all *N*_{k}, *r*_{k}, *l*_{k} and *h*_{k}) – or rather, their scaling behavior within the network (the exponents *c*_{r}, *c*_{l} and *c*_{h}) – are assumed to follow from minimizing the energy dissipation during transport. Yet, a network could be optimally designed with a certain internal scaling behavior to minimize transport costs but with an inefficient spatial arrangement of the (terminal) units (capillaries) that must deliver the metabolites to the cells (see Fig. 2a). Therefore, besides transport costs also the spatial efficiency has to be considered: the network has to be optimally embedded within the body volume with the capillaries spanning the whole three-dimensional volume to reduce diffusion distances (see Fig. 2b). To ensure this, WBE hypothesize that an optimal network is ‘space filling’. A group of cells that receives oxygen and nutrients from a single capillary is called a ‘service volume’*V*_{s,C} (West *et al.* 1997). If the capillaries are spatially arranged in a specific way to maximize supply efficiency, the vessels at branching level *C* – 1 (one step before the capillaries) must be spatially arranged approximately in the same way, because both levels are physically connected at the branching points. The same reasoning applies to the arrangement of vessels of levels *C* – 2, *C* – 3, etc. This spatial relation means that the *N*_{k}_{+1} service volumes of level *k* + 1 occupy approximately the same volume as the *N*_{k} service volumes of level *k* or any other level. Hence, via the argument of spatial efficiency of resource delivery, the properties of the service volume of the capillaries yield a specific relation between the network scaling parameters: the total service volume *X*_{k} ≡ *N*_{k}V_{s,k} is constant.

WBE argue that because *r*_{k} << *l*_{k}, the vessel length is the only length scale relevant for spatial efficiency, so that the volume occupied per vessel should scale as a sphere with vessel length as its diameter, so hence optimal space filling of the network is equivalent to (approximate) preservation of the total ‘service volume’ per level across all levels. From eqn 1 one can easily see that this implies that *c*_{l} = 1/3. WBE use this relation as an additional constraint during the cost minimization procedure (West *et al*. 1997, 2000; West 1999).

Apart from the preserved service volume *X*_{k} in the network, also some other quantities must be constrained during the optimization. First, the total net volume flow *Q*_{tot} through the network is assumed fixed so that the goal function (i.e. transport costs or energy dissipation) per unit of flow is optimized. Second, the body volume and mass of the organism are assumed fixed. Third, because we want to find the optimal arrangement of the network within this volume, the total size of the network (which is related to body mass) must be fixed. A suitable proxy of network size is the blood volume *V*_{b}. Optimization of the dissipation function with all these constraints is accomplished via the method of Lagrange multipliers (see Appendix S3).

The mathematical solution of pulsatile flow through a blood vessel with elastic walls caused by an applied sinusoidal pressure wave was first solved completely by Womersley (Milnor 1989, Nichols & O’Rourke 2005) by linearizing the Navier-Stokes equations, coupled to the dynamics of the vessel walls (see Appendix S2). The character of the flow is captured by the Womersley number where ω is the angular frequency of the heart beat and ρ_{M} and µ the density and viscosity of the blood (Caro *et al.* 1978; Milnor 1989; Nichols & O’Rourke 2005; see also Appendix S2): for α_{k} >> 1 (e.g. in the aorta), the flow is strongly pulsatile, and the fluid oscillates more or less like a solid core; for α_{k} < 1 (e.g. in the capillaries), the flow is quasi-steady with a parabolic velocity profile. Because of the wave character of pressure and flow, (partial) reflection of the forward waves may occur at branching points in the network, because in general the characteristic impedances *Z̃*_{C} at both sides of the junction are different (Milnor 1989; Nichols & O’Rourke 2005). A network with an architecture causing strong reflections will be highly inefficient (large dissipation). The degree of reflection is expressed by the reflection factor , the ratio of the backward and the forward pressure wave at the junction (Appendix S2).

Using the linearized Navier-Stokes equations, we find that the total energy dissipation within the network is the sum of the steady and pulsatile components: (Appendix S2). In general, the dissipation per unit length is given by the product of driving force (pressure difference −∇*p*_{k}dz) and flux (volume flow ) per vessel times the number of vessels *N*_{k} (Milnor 1989). This must be averaged over one period of oscillation 2π/ω (accomplished by multiplying the complex pressure gradient by the complex conjugate of the volume flow and taking one half of the real part), and integrated over the length of each vessel (from *z*_{k–}_{1} to *z*_{k} = z_{k–}_{1}* + l*_{k}, see Fig. 1):

- (9)

Complex quantities (Appendix S2) are indicated by a tilde (~). For steady flow, where −∇*p*_{k}=Δp_{k}/*l*_{k} and *Œ*_{k} are both real and independent of time and position in the vessel, eqn 9 correctly reduces to the familiar expression for Poiseuille flow (Fung 1984; Milnor 1989; West *et al.* 1997; Nichols & O’Rourke 2005)

- (10)

The pulsatile component of the dissipation is much more complicated. Evaluating the integral in eqn 9 (Appendix S2), we finally arrive at

- (11)

- (12)

In this expression, | | and *θ*_{Γ,k} are the absolute value and phase of the reflection factor and *a*_{k} is the damping factor of the attenuated pressure wave; the amplitude of the wave is damped by a factor when traveling a distance *l*_{k}. Furthermore, |*P̃*_{C} | is the oscillatory pressure amplitude at the beginning of the capillaries (which can also be expressed in terms of the initial pressure amplitude at the beginning of the aorta |*P̃*_{1} | see Appendix S2), and σ and *E* are the Poisson ratio and Young's modulus. On the one hand, the energy dissipation depends directly on the vessel dimensions *N*_{k}, *r*_{k}, *l*_{k} and *h*_{k} at each level *k*; also the damping factors *a*_{k }and ≡−*iJ*_{0}(*i*^{3/2}α_{k})/*J*_{2}(*i*^{3/2}α_{k}) are functions of the Womersley number α_{k} and hence of *r*_{k} (Appendix S2). On the other hand, the reflection factors of level *k* are not only functions of the properties *N*_{k}, *r*_{k} and *h*_{k} of the same level, but also of those of the next level, *N*_{k+}_{1}, *r*_{k+}_{1} and *h*_{k+}_{1}, and so they are functions of the scaling ratios *v*_{k}, ρ_{k} and η_{k} (Appendix S2).

The relative contribution of the steady and pulsatile parts of the energy dissipation per level changes from the aorta (*k* = 1) via the larger arteries to the microcirculation and finally the capillaries (*k* = *C*). At the aortic side of the network, the pulsatile contribution dominates, whereas at the capillary side the steady contribution is most important (Lighthill 1975; Fung 1984; West *et al.* 1997). This is partly due to damping of the pressure waves (attenuation) in the vessels, and also due to the change of pulsatility of the oscillating flow (the magnitude of α_{k}) because the vessel radius decreases in the direction of the capillaries. The part of the network that contributes most to the total blood volume is the aortic side (West *et al.* 1997; Etienne *et al.* 2006); this is linked to the fact that the cumulative *S*_{3}-function quickly saturates with increasing *k* (eqn 4 and Appendix S1, Fig. S1), meaning that the capillaries contribute relatively little. Hence, the internal network scaling exponents *c*_{r} and *c*_{l}, that are important for the metabolic scaling *a* (eqn 7), are virtually only determined by the aortic side of the network. During the optimization procedure, we will therefore consider only the pulsatile part of the dissipation in the limit of large Womersley number, corresponding to the conditions at the aortic side.

We use the analogy with electrical transmission-line theory (Taylor 1957; Milnor 1989; West *et al.* 1997; West 1999; West & Brown 2005) that suggests that elimination of wave reflections is a necessary (but not sufficient) condition to minimize energy loss. We therefore perform the minimization in two stages. In the first stage, we minimize the dissipation with respect to the reflection factor (that is a function of *v*_{k}, ρ_{k} and η_{k}), both its amplitude and phase. In the second stage, we further optimize the remaining dissipation function with respect to the vessel dimensions *N*_{k}, *r*_{k}, *l*_{k} and *h*_{k}, taking into account the constraints that follow from the first stage.

- (13)

By analogy with transmission line theory, the same result can also be obtained by impedance matching (Milnor 1989; Appendix S2), i.e. by equating the total characteristic impedances *Z̃*_{C} at levels *k* and *k* + 1: *Z̃*_{C,k}=*Z̃*_{C,k+1}/*v*_{k} (Milnor 1989, Appendix S5). For large Womersley number α_{k} these impedances are real, where *c*_{0,k}is the Moens-Korteweg velocity (Milnor 1989; Papageorgiou *et al.* 1990; West *et al*. 1997; Nichols & O’Rourke 2005; Appendix S2). Combining these three relations directly gives eqn 13.

In stage 2 of the optimization process, we minimize the pulsatile energy dissipation function under the condition of zero wave reflections (from the first stage) with the constraints of fixed body mass, fixed blood volume and fixed service volume . To account for the extra relation between *N*_{k}, *r*_{k} and *h*_{k} because of the absence of wave reflections as obtained in the first stage, we have to include preservation of (eqn 13) as an additional constraint in the second optimization process (see Appendix S5). WBE (1997, 2000) do not take this second step. Instead, they state that the wall thickness is a fixed proportion of the vessel radius, so *h*_{k}/*r*_{k} is (at least in the aortic side of the network) assumed to be a constant. They provide no theoretical or empirical evidence for this assumption. With their additional assumption, the three internal scaling exponents *c*_{r}, *c*_{l} and *c*_{h} are fully defined by the three network constraints: space filling (so *c*_{l} = 1/3), zero wave reflections (so 5*c*_{r} – *c*_{h} = 2), and a constant ratio *h*_{k}/*r*_{k} (so *c*_{r} = c_{h}). This yields *c*_{r} = 1/2, which implies that *a =* 3/4 in agreement with Kleiber's law; see case A in Fig. 3.

However, there are several essential steps in this derivation that need further scrutiny. In the following section, we discuss five critical assumptions of the model.

### Discussion and conclusions

- Top of page
- Summary
- Introduction
- WBE network model
- Network parameters from optimization
- Critical assumptions
- Discussion and conclusions
- Acknowledgements
- References
- Supporting Information

We have presented theoretical evidence that full optimization along the logic of the WBE model leads to an irrelevant network with *a* = 0. By slightly relaxing assumptions on spatial efficiency and wall thickness, isometric scaling (*a* = 1) of organismal metabolic rate with body mass is predicted (case E), with network scaling parameters that are in agreement with measurements. However, observations on the allometric scaling exponent of basal metabolic rate are centered in the range *a* = 0·65 – 0·75 (Peters 1983; Calder 1984; Schmidt-Nielsen 1984; White & Seymour 2003; Savage *et al.* 2004). This large discrepancy between theoretically predicted and observed values creates a serious paradox. First we assess the possibilities to reconcile our prediction that *a* = 1 with its observed values, while remaining within the WBE framework of network models, i.e. assuming that metabolic rate is limited by resource supply through the internal transport network (supply limitation). Second, we discuss possibilities to generalize the framework. Third, we briefly discuss alternative explanations for allometric scaling of metabolism when this framework would be rejected.

Is it possible to reconcile our prediction that evolutionary optimization leads to *a* = 1 with the apparently conflicting observational evidence that *a* = 0·65 – 0·75, while remaining within the framework of supply-limited network models? This is indeed possible if intra- and inter-specific allometric scaling are different, where the solution may lie in predictable differences between species in the quantity *B*_{0}. Let *a*_{intra} be the intra-specific scaling exponent, and *a*_{inter} be the scaling exponent across species. If the evolution optimizes the network only among individuals within species (as this is where natural selection operates), then *a*_{intra}* =* 1 is predicted for every species. In that case, average metabolic rates across species can still scale allometrically with body mass (*a*_{inter} < 1) if the intra-specific prefactor *B*_{0intra} ∝ *M*^{−δ} declines allometrically with average body mass across different species with exponent δ. This means that the metabolic rate across species is with *B*_{00} the (mass-independent) prefactor of the inter-specific relation, and the inter-specific exponent thus becomes *a*_{inter}=*a*_{intra}−δ. This requires that the morphological and physiological parameters that occur in *B*_{0intra} (see eqn 8) such as the capillary properties *r*_{C}, *l*_{C}, *h*_{C} and *u*_{C} and/or the arteriovenous concentration difference *f*_{0} (a capillary efficiency) depend on the average body mass of a species. In this case, the network model predicts that intra-specific scaling is isometric (*a*_{intra}* =* 1) but is silent on inter-specific scaling which can be allometric (*a*_{inter} < 1). Is there evidence that these properties depend on between-species body mass differences and thus could yield δ ≈ 0·25 resulting in *a* = *a*_{inter} ≈ 0·75?

There is indeed some experimental evidence that *r*_{C} and *l*_{C} vary with body size across species. Dawson (2001, 2003) claims that for mammals *r*_{C} and *l*_{C} scale with *M*^{0·08} and *M*^{0·21}, respectively, which gives for case E in Fig. 3 the value δ ≈ 0·21, which leads to *a* ≈ 0·79, so much closer to the observed values. Interestingly, since for case E the number of capillaries scales as *N*_{C} ∝ *M*, the total generalized service volume scales as which – for the choice *d*_{h} = 0 (i.e. setting ) – yields almost isometric scaling: *X*_{C} ∝ *M*^{1·03}. The size of red blood cells is fairly constant per vertebrate group (mammals, birds, reptiles, amphibians), suggesting that at least the capillary radii do not change very much with mass (Schmidt-Nielsen 1984). Schmidt-Nielsen also inferred that *u*_{C} might even scale as *M*^{0·2}, also affecting the value of *a*. Hence, dependence on body mass of capillary properties could potentially solve the paradox. Furthermore, if the capillary efficiency *f*_{0} that occurs in the Fick relation (eqn 2) were mass dependent, then the total cardiac output* Q*_{tot} and metabolic rate *B* would display a different body-mass scaling (as suggested by Patterson *et al.* 1965; Milnor 1989, 1990), which could also solve the paradox. Spatz (1991) e.g. gives for mammals a mass scaling ∝ *M*^{−}^{0·11} because of allometric differences in oxygen unloading (Schmidt-Nielsen 1984). The experimental evidence on intra-specific metabolic scaling is less conclusive. In a recent review, Glazier (2005) showed that mammals display a large variation in exponent: 0·38 < *a*_{intra} < 1·11 with average *a*_{intra} ≈ 2/3, similar to squamate reptiles. On the other hand, varanid lizards as well as juvenile mammals and pelargic invertebrates have a scaling exponent close to one. It should be noted however that this argument leads to a very different mechanistic explanation for the observed 3/4 slope than given by the WBE model. In this case, larger species have a lower per-mass metabolic rate not because of a different optimal network architecture that unavoidably limits supply to the cells, but because of selection during evolution for e.g. different capillary dimensions and/or capillary efficiency (that could even be caused by active down-regulation of metabolism, see Suarez & Darveau 2005). Explaining these body mass dependencies would require a different class of (diffusion and demand) models, and also a different underlying evolutionary framework (e.g. based on life-history trade-offs).

Another possibility to reconcile the theoretical prediction and experimental value of *a* is by generalizing the network model, to include the possibility of optimizing a goal function other than dissipation (e.g. the total drag force on the vessel walls), or to use a network property other than blood volume to link body-mass to vessel dimensions (e.g. the total mass of vessel walls, see Fig. 3). This seems a route worthwhile exploring.

A third option, while still remaining within the framework of supply-limiting networks, is to use a more integrated model. The WBE model is, as the authors admit (West *et al.* 2000), a simplified null-model that captures the essential details. Yet, the model seems to be too simple to undisputedly arrive at the seminal result: *a* = 3/4. Therefore we suggest the use of a more integrated model that includes a description of mass transport from the capillaries to the cells. In this way, the spatial efficiency of the network (service volume) can be included mathematically in a more realistic way. The ultimate evolutionary goal of the network is to transport resources in an efficient way to the target cells, not only to the capillaries. Also (part of) the temperature dependence of the metabolic rate (Brown *et al.* 2004) or even stoichiometry might be included in a natural way into such a network framework. A second part of such an integrated model would be the transport of oxygen from the lungs to the blood (lung capillaries, see e.g. Santillan 2003). A third part could consist of constraints that arise from the specific body shape of organisms, which may especially affect the internal network scaling at the aortic side and therefore the metabolic scaling exponent.

If we however fully reject the assumption that metabolism is constrained by the supply through the network, we also have to reject the WBE model as a mechanistic explanation of the between-species metabolic 3/4-power scaling, in favour of alternatives (see e.g. Agutter & Wheatley 2004; Glazier 2005, van der Meer 2006). Even if natural selection optimizes the network architecture to minimize its transport or maintenance costs, the mass-dependence of metabolic rate may be explained by different processes, such as basal transport processes across membranes (Demetrius 2006).

In our view, more anatomical and physiological data are required to solve the paradox between our new predictions and the available observations of allometric scaling. As outlined earlier, predicted and observed differences between intra- and inter-specific scaling of metabolic rate can provide further insight in this discussion. The intercept in allometric scaling relations deserves much more attention from a physiological perspective, besides the role of temperature and stoichiometry (see Brown *et al.* 2004). At the same time, there are interesting new theoretical possibilities left to discover.