Curvature in metabolic scaling


For more than three-quarters of a century it has been assumed1 that basal metabolic rate increases as body mass raised to some power p. However, there is no broad consensus regarding the value of p: whereas many studies have asserted that p is 3/4 (refs 1–4; ‘Kleiber’s law’), some have argued that it is 2/3 (refs 5–7), and others have found that it varies depending on factors like environment and taxonomy6,8,9,10,11,12,13,14,15,16. Here we show that the relationship between mass and metabolic rate has convex curvature on a logarithmic scale, and is therefore not a pure power law, even after accounting for body temperature. This finding has several consequences. First, it provides an explanation for the puzzling variability in estimates of p, settling a long-standing debate. Second, it constitutes a stringent test for theories of metabolic scaling. A widely debated model17 based on vascular system architecture fails this test, and we suggest modifications that could bring it into compliance with the observed curvature. Third, it raises the intriguing question of whether the scaling relation limits body size.

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Figure 1: Curvature in metabolic scaling.
Figure 2: Scaling exponent depends on mass range.
Figure 3: Modified WBE models.


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We thank the staff of the library of the Museum of Comparative Zoology at Harvard University for their assistance and access to their resources. A. Duncan assisted in indexing and copying original printed materials.

Author Contributions T.K. carried out the statistical analysis, performed the analytical calculations, and annotated the data of ref. 8; T.K., V.S. and E.J.D. extended and analysed vascular scaling models; T.K., V.S., E.J.D. and W.F. interpreted the results and wrote the paper.

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Correspondence to Walter Fontana.

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Supplementary information

Supplementary Information

This Supplementary Information file comprises: 1 Regression Coefficients for Various Data Sets; 2 Regression Diagnostics; 3 Linear Regression and the Slope of a Quadratic Function; 4 Extending the West-Brown-Enquist model. It also includes Supplementary Figures 1-5 with legends and Supplementary References. (PDF 4758 kb)

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This file contains curvature data. (XLS 295 kb)

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Kolokotrones, T., Van Savage, Deeds, E. et al. Curvature in metabolic scaling. Nature 464, 753–756 (2010).

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