(2) where B0 is a normalization constant. Basal metabolic rate is the rate at which an organism expends energy, while in a resting state, in order to support basic life functions such as tissue maintenance. For ectothermic (cold-blooded) animals, basal metabolic rate is sometimes referred to as standard metabolic rate, or SMR (Auer et al. 2014). Kleiber and Brody’s result was counter to the long-held “conventional wisdom”, as many biologists were expecting the exponent in Eq. (2) to be ⅔. Their reasoning was that endothermic (warm-blooded) animals must radiate body heat sufficiently quickly to avoid overheating. Since heat is radiated away from a body through its surface, biologists assumed this rate of heat dissipation was proportional to surface area. Surface area is proportional to the square of a linear dimension. And, as mass is proportional to volume, or the cube of the linear dimension, they expected metabolic rate to be proportional to mass raised to the ⅔ power. Although Kleiber’s Law was originally obtained for just mammals and birds, many consider it more widely applicable, to cold-blooded organisms, trees, unicellular organisms, and even molecular processes within the cell. If this is true, Kleiber’s Law accounts for a range of masses that varies by 27 orders of magnitude! Over time, numerous empirical relationships were revealed in addition to Kleiber’s Law. But these lacked an overarching theoretical framework to explain the observations. Plant biologist Karl Niklas (2004, p. 872) explains why such a theoretical framework is so important: If certain trends are size-dependent and ‘invariant’ with regard to phyletic affinity or habitat, they draw sharp attention to the existence of properties that are deeply rooted in all, or at least most living things. Identifying these properties using a first principles approach, therefore, has become something of a Holy Grail in biological allometry because any successful theory would unify as many diverse phenomena in biology as Einstein’s general theory of relativity has for physics. It is understandable, therefore, that numerous attempts have been made to provide an all-inclusive, unifying theory for broad interspecific trends. However, most have not held up against well-reasoned criticism or withstood empirical tests. Hence, allometry is of great possible interest to biologists, both creationist and evolutionist. III. THE WEST, ENQUIST, AND BROWN (WBE) ALLOMETRIC THEORY A. Overview Physicist Geoffrey West and biologists Brian Enquist and James Brown (West et al. 1997) have published a theoretical justification for many of the observed allometric scaling laws, including Kleiber’s Law. They then explained their metabolic scaling theory (MST) (Tredennick et al. 2013), in greater detail in subsequent papers (Brown et al. 2000; West et al. 2000). They also extended it to include allometric patterns in angiosperm trees (Enquist et al. 2000), as well as the ontogenetic growth of an organism over the course of its lifespan (West et al. 2001). They think the model also has ecological applications, and they provide a heuristic explanation for the fact that tree population density is inversely proportional to individual body mass raised to the ¾ power (Enquist and Niklas 2001; West and Brown 2005). With the exception of a very short book review (Hebert 2022), these developments have gone virtually unnoticed in the creation literature. The following discussion is an overview of the theory, its main assumptions, and its predictions. Detailed derivations of some of the theory’s key features are provided in the appendices. WBE modelled an organism’s fluid distribution network as a hierarchical branching network (Fig. 2) of N + 1 levels of interconnected cylindrical pipes. For animals, particularly mammals and birds, the highest level of the network, denoted by = 0, is a single pipe, such as the aorta within the human cardiovascular system. Note that this model only includes arteries; it does not attempt to take into account the venous network. This single pipe branches into a number of smaller pipes in the = 1 level and subsequent levels. Let N denote the number of pipes within the th level of the network. Each pipe in the th level will diverge into two or more pipes which are part of the + 1 level. The number of new pipes at each branching point is the “branching ratio”, denoted by n. If we assume the branching ratio n is a constant, say n 2 or n 3, then it follows that the number of pipes in each level of the network is N (Fig. 2). Because the network is assumed to be in steady state, and because an incompressible fluid cannot “pile up” at network junctions, the rate of fluid flow in the single pipe corresponding to = 0 must equal the total fluid flow in each level of the network: (3) Eq. (3) must hold for all values of in the network, including = N. Within each level, each pipe has a length and radius (Fig. 3). Fluid within each pipe is driven by a pressure gradient Δ . This pressure gradient can in principle be provided by a pulsatile pump, as in the case of the mammalian cardiovascular system, or it could be an osmotic vapor pressure gradient, as in the case of a vascular plant. Figure 2. Schematic showing the relationships between levels and branches within the West, Enquist, and Brown (WBE) hierarchical nutrient supply network. HEBERT Allometric and metabolic scaling 2023 ICC 208
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