Why does metabolic rate scale with body size?/Allometric cascades

Darveau et al. reply

West et al. and Banavar et al. criticize our results on mathematical grounds, but they overlook the consistency of our multiple-cause model (concept) of metabolic scaling¹ with what is known from biochemical² and physiological³ analysis of metabolic control. Their single-cause explanations^4,5 are based on the assumption that whole-body metabolism in animals is exclusively supply-limited, whereas there are many factors that together explain the observed patterns of metabolic scaling^6,7. Our concept can accommodate these multiple causes, the range of metabolic scaling exponents observed in various taxa⁸, and variation in exponents due to physiological state⁷.

Allometric equations are mathematical descriptions of empirical relationships, rather than derived physical laws⁶. Our equation¹ is a first approximation that attempts to express our concept in mathematical terms. It does not distinguish between energy-demand processes that occur in parallel and supply processes that operate in series. It suffers from a semantic flaw that imposes units on the control coefficient, c_i.

In a modified equation (J. Endelman) to determine the basal metabolic rate, BMR = MR₀Σc_i${(\mathrm{MML:M}/{\mathrm{MML:M}}_{\mathrm{o}})}^{b_i}$, where MR₀ is the characteristic metabolic rate of an animal with a characteristic body mass M₀, c_i is rendered dimensionless while the exact meaning of the original equation¹ is retained. With M₀ of 1 unit mass, MR₀ now takes the place of the value a, as found in the standard scaling equation⁶ and in our original. For mammalian maximum metabolic rate, MMR, the same equation applies with a roughly tenfold higher MR₀. We were able to find the relevant b_i values and estimate c_i for various processes in mammals to demonstrate the utility of our model.

Although using mammalian data precludes extrapolation to non-mammalian species, our concept can be used to understand metabolic scaling in other taxa. Our equation is not a power-law function, but yields meaningful results when a biologically realistic range of b_i values is used in simulations. The examples we used yield results that are indistinguishable from power functions, reflected in r² values that are greater than 0.999. Lower r² values result when b_i values outside the biological range are used.

The inherent limitations of the data, and estimates based on them, offer new directions for experiments, and the shortcomings of our equation highlight the need for better ways to express our multiple-cause model. Branching distributive structures and supply limitations^4,5 may contribute to metabolic scaling, although supply limitations contribute minimally to BMR, which scales with an exponent that is close to 0.75. Supply limitations have a greater influence on MMR^3,9, but the allometric exponent for this is paradoxically higher⁷. These and the many factors that contribute to the allometric scaling of metabolic rates^6,7,10, as well as the observation that cellular metabolic rates in vitro decline with increasing body mass^10,11, should give pause to advocates of single-cause explanations for metabolic scaling.