Here M is the organism’s final mass at maturity, 0 is the organism’s mass at birth, and is a taxon-specific constant. Their model results in a sigmoid “universal growth curve” (Fig. 7) for fraction of adult body mass as a function of time since birth. It does a good job of predicting the shape of growth curves for guinea pigs, guppies, hens, and cattle, as well as general sizes for nine other examples of fish, mammals and birds (West et al. 2001). This model, or one similar to it, might eventually help creationists explain the extreme longevity of pre-Flood humans, as discussed in Section VC. IV. WBE THEORY: CRITICISMS AND ALTERNATIVES The WBE theory is not without criticism. Glazier (2006) argues that Kleiber’s Law, which the WBE theory explains, is not truly universal. Dodds et al. (2001) and White and Seymour (2003, 2005) argue that statistical evidence is consistent with λ ⅔, rather than ¾. WBE have defended their research against these criticisms (Brown et al. 2005; West and Brown 2005). West and Brown (2005) point out that Calder (1984), McMahon and Bonner (1983), Schmidt-Nielsen (1986), and Peters (1993) all independently concluded that quarter-power scaling in biological systems was both real and ubiquitous. Kozłowski and Konarzewski (2004, 2005) and Etienne et al. (2006) have criticized the underlying logic of the WBE model. Hulbert (2014) argues that Kleiber’s law is an “empirical approximation”, not a rule or law. Price et al. (2011), Bentley et al. (2013), Tredennick et al. (2013) and Price et al. (2022), have noted observations that contradict or potentially contradict the theory, as has Niklas (1995, 1997). Nevertheless, Niklas thinks WBE are “on the right track”, so to speak, and he goes so far as to say that, although much work remains to be done, a unifying allometric theory regarding plant biology is “near at hand” (2004, p. 871). Nor is the WBE theory the only metabolic scaling theory. For instance, Price et al. (2022), have produced an alternate theory for land plants, one that utilizes velocity constraints and conservation of volume flow rate throughout the organism. Escala (2019) used dimensional analysis (McMahon and Bonner 1983) to obtain an analogue to Kleiber’s Law in which the rate of oxygen consumption is proportional to the product of body mass, some characteristic body frequency (such as respiration or heart rate), and the mass of oxygen utilized per unit of body mass. Unlike the WBE theory, his result Figure 7. The sigmoid growth curve derived by WBE, showing fraction of adult body mass as a function of (normalized) time τ since birth. Variable Symbol Predicted Exp. Empirical Exp. Reference Lung Volume .00 .05 Weibel Respiratory Frequency − /4 −0.25 −0.26, −0.28 Stahl, Tenney & Bartlett Volume Flow to Lung 3/4 0.75 0.80 Stahl Interpleural Pressure 0 0.00 0.004 Günther Trachea Diameter 3/8 0.0.375 0.39 Tenney & Bartlett Air Velocity in Trachea 0 0.00 0.02–0.04 Calder Tidal Volume .00 1.04 Stahl Power Dissipated 3/4 0.75 0.78 Stahl Number of Alveoli NA 3/4 0.75 ND Volume of Alveolus VA /4 0.25 ND Radius of Alveolus A /2 0.083 Surface Area of Alveolus AA /6 0.083 0.095 Gehr et al. Surface Area of Lung AL /2 0.92 Oxygen Diffusing Capacity .00 0.96, 0.99, 1.18 Weibel, Gehr et al., Stahl Total Airway Resistance −3/4 −0.75 −0.70 Stahl O2 Consumption Rate 3/4 0.75 0.76, 0.72 Stahl, Weibel Table 2. WBE predicted values of λ in respiratory allometric relationships Y Y0 Mλ, as well as experimentally-determined values (with references). HEBERT Allometric and metabolic scaling 2023 ICC 213
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