Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant?

Abstract

Geoffrey West and colleagues published a series of papers, mostly in Nature, Science and Proceeding of the National Academy of Sciences of the USA, on scaling of different aspects of physiology and life history in organisms as distinct from each other as plants and animals (West, Brown & Enquist 1997, 1999a,b; Gillooly et al. 2001). Recently they even extended their considerations down to cells, mitochondria and respiratory complexes (West, Woodruff & Brown 2002). All these papers invoke the 3/4 scaling of metabolic rate claimed to be derived in their first and basic paper (West et al. 1997; WBE hereafter). It is therefore extremely important to check the legitimacy of their model. First we show that to make WBE's model mathematically consistent either metabolic rate must be directly proportional to body mass (recall that the model is aimed to explain the 3/4 exponent for metabolic rate) or one of the basic model assumptions, that is, the size-invariance of terminal supplying vessels, must be violated. Then we show that animals built according to WBE's model cannot represent a broad range of sizes, because for large animals the volume of blood vessels would exceed body volume. Later we demonstrate that many features of the plant vascular system, insect tracheal system, vertebrate lung or vertebrate cardiovascular system do not conform to WBE's model assumptions. Finally, we argue that 3/4 scaling for metabolic rate is by no means universal, and therefore WBE's model was built to explain a non-existent pattern. WBE write: We develop a quantitative model that explains the origin and ubiquity of quarter-power scaling; it predicts the essential features of transport systems, such as mammalian blood vessels and bronchial trees, plant vascular systems, and insect tracheal tubes. It is based on three unifying principles or assumptions: First, in order for the network to supply the entire volume of the organism, a space-filling fractal-like branching pattern is required. Second, the final branch of the network (such as the capillary in the circulatory system) is a size-invariant unit and third, the energy required to distribute resources is minimized. Size-invariance here means that capillary number scales isometrically (with exponent 1) with size. Although WBE acknowledge that 'Biological networks vary in the properties of the tube (elastic to rigid), the fluid transported (liquid to gas), and the nature of the pump …' they mainly use the terms of the cardiovascular system to describe both the structure and function of the network. Their network consists of a hierarchy of branching vessels starting, in the case of the cardiovascular system, from the aorta and ending with capillaries. The assumption that the final branch is size-invariant causes the number of levels to be a function of body size (or vice versa): more levels are required to fill a larger volume with the same density of final vessels. Further on, terms characteristic of the cardiovascular system will be given in italics if the system is more general or if it cannot represent the cardiovascular system because of some additional assumptions. The consecutive levels of branches of the network are indexed with the symbol k, where k = 0 for aorta and k = c for capillaries. A branch k has length lk and radius rk. Flow velocity through a vessel of the level k equals ūk, and the number of such vessels is Nk. The volume rate of flow in a single vessel at each level is Q̇k, and because the fluid (or gas) must be conserved throughout the system, the following relation must hold (eqn 2 in WBE): Because equation 1 holds for all levels, Now, denoting metabolic rate by B and body mass by M, a new assumption is formulated: 'Because the fluid transports oxygen and nutrients for metabolism, Q̇0 ∝ B; thus, if B ∝ Ma (where a will later be determined to be 3/4), then Q̇0 ∝ Ma.Equation 2 therefore predicts that the total number of capillaries must scale as B, that is, Nc ∝ Ma.' The implicit part of this assumption is that the entire group of organisms compared shares the same exponent for the relationship between metabolic rate B and body mass M (although at this point of the considerations they may differ in their intercepts). It is not clear at this point whether the amount of oxygen or nutrients is proportional to the fluid volume, although it seems from further considerations that such an implicit assumption has also been made. It is likely that in animals the amount of oxygen per blood volume will be larger for small animals often having smaller erythrocytes and/or higher haemoglobin concentration (Kostelecka-Myrcha & Cholostiakow-Gromek 2001; Gregory 2002). At this point the origin of exponent a is not specified, and a < 1 may reflect transport ability per fluid volume decreasing with size. Now WBE introduce scale factors βk for vessel radius and γk for vessel length: βk ≡ rk+1/rk; γk ≡ lk+1/lk. The next assumption they make is that the scaling factor for length doe