Why does metabolic rate scale with body size?

Abstract

A long-standing problem has been the origin of quarter-power allometric scaling laws that relate many characteristics of organisms to their body mass^1,2 — specifically, whole-organism metabolic rate, B = aM^b, where M is body mass, a is a taxon-dependent normalization, and b ≈ 3/4 for animals and plants. Darveau et al.³ propose a multiple-cause model for mammalian metabolic rate as the “sum of multiple contributors”, B_i, which they assume to scale as B_i = $a_i{\mathrm{MML:M}}^{b_i}$, and obtain b ≈ 0.78 for the basal and 0.86 for the maximally active rate, ${\mathop V\limits^{\bullet}}{}_{{\rm O}_2}^{\max}$. We argue, however, that this scaling equation is based on technical, theoretical and conceptual errors, including misrepresentations of our published results^4,5.

Main

All of the results of Darveau et al.³ follow from their equation (2), B = $a\sum c_i{\mathrm{MML:M}}^{b_i}$, which they neither derive nor prove. As control coefficients⁶, c_i, and exponents, b_i, are dimensionless, this must be incorrect because it violates dimensional homogeneity, leading to different results for b that depend on the units of mass: for the basal rate, b = 0.76 when M is in kilograms, and b = 1.08 when M is in picograms.

Now, by definition, c_i ≡ ∂lnB/∂lnB_i, which leads to the standard sum rules⁶ Σc_i = 1 and Σc_iɛ_i = 0, where ɛ_i = b − b_i with b(M) ≡ dlnB/dlnM, the slope of lnB versus lnM, and b_i(M) ≡ dlnB_i/dlnM. This gives b = Σc_ib_i, the equations that Darveau et al. should have used to determine b from the empirical c_i and b_i. These formulae are very general, requiring no assumptions about how B and B_i scale, or whether the B_i are connected in parallel or in series. Darveau et al., however, use B = ΣB_i, implying that the B_i are added in parallel and, as such, their model is simply a consistency check on the conservation of energy, which requires all “ATP-utilizing processes”³ (in parallel) to sum to B and so must be trivially correct. This gives c_i = (a_i/a)${\mathrm{MML:M}}^{-{\epsilon }_i}$ and B = aΣ($c_i{\mathrm{MML:M}}^{{\epsilon }_i}$)${\mathrm{MML:M}}^{b_i}$, which is the (dimensionally) correct version of equation (2).

As Darveau et al. take a and a_i as constant, their c_i must scale as ${\mathrm{MML:M}}^{-{\epsilon }_i}$. However, they assume that c_i (and b_i) ∝ M⁰, which requires b (which equals Σc_ib_i)∝M⁰, thereby contradicting their equation (2), in which b depends on M. This inconsistency in the M-dependence of b is concealed in their plots, which cover only three orders of magnitude in M, over which b is almost constant (about 0.78 for basal). However, when their analysis is extended to the realistic eight orders of magnitude spanned by mammals, their b increases with M to an average value of about 0.85 and, for ${\mathop V\limits^{\bullet}}{}_{{\rm O}_2}^{\max}$, to about 0.98, which are both inconsistent with other data^1,2.

Darveau et al. have taken their value for b_i from empirical data, without explaining why B, or b_i, scales nonlinearly with M, or why most b_i ≈ 3/4. Understanding these features is the real challenge — the formulation of Darveau et al. is therefore hardly fundamental. By contrast, our theory^4,5 is grounded in basic principles of geometry, physics and biology, and offers a general unifying explanation for these and the other quarter-power scalings that are so pervasive in biology.