Curvature in metabolic scaling

Abstract

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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

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

References

  1. 1

    Kleiber, M. Body size and metabolism. Hilgardia 6, 315–353 (1932)

    CAS  Article  Google Scholar 

  2. 2

    Bartels, H. Metabolic rate of mammals equals the 0.75 power of their body weight. Exp. Biol. Med. 7, 1–11 (1982)

    Google Scholar 

  3. 3

    Feldman, H. A. & McMahon, T. A. The 3/4 mass exponent for energy metabolism is not a statistical artifact. Respir. Physiol. 52, 149–163 (1983)

    CAS  Article  Google Scholar 

  4. 4

    Savage, V. M. et al. The predominance of quarter-power scaling in biology. Funct. Ecol. 18, 257–282 (2004)

    Article  Google Scholar 

  5. 5

    Heusner, A. A. Energy metabolism and body size. I. Is the 0.75 mass exponent of Kleiber's equation a statistical artifact? Respir. Physiol. 48, 1–12 (1982)

    CAS  Article  Google Scholar 

  6. 6

    Dodds, P. S., Rothman, D. H. & Weitz, J. S. Re-examination of the “3/4-law'' of metabolism. J. Theor. Biol. 209, 9–27 (2001)

    CAS  Article  Google Scholar 

  7. 7

    White, C. R. & Seymour, R. S. Mammalian basal metabolic rate is proportional to body mass2/3 . Proc. Natl Acad. Sci. USA 100, 4046–4049 (2003)

    ADS  CAS  Article  Google Scholar 

  8. 8

    McNab, B. K. An analysis of the factors that influence the level and scaling of mammalian BMR. Comp. Biochem. Physiol. A 151, 5–28 (2008)

    Article  Google Scholar 

  9. 9

    Glazier, D. S. A unifying explanation for diverse metabolic scaling in animals and plants. Biol. Rev. Camb. Phil. Soc. 85, 111–138 (2009)

    Article  Google Scholar 

  10. 10

    White, C. R., Phillips, N. F. & Seymour, R. S. The scaling and temperature dependence of vertebrate metabolism. Biol. Lett. 2, 125–127 (2006)

    Article  Google Scholar 

  11. 11

    McNab, B. K. Complications inherent in scaling the basal rate of metabolism in mammals. Q. Rev. Biol. 63, 25–54 (1988)

    CAS  Article  Google Scholar 

  12. 12

    Bokma, F. Evidence against universal metabolic allometry. Funct. Ecol. 18, 184–187 (2004)

    Article  Google Scholar 

  13. 13

    Lovegrove, B. G. The zoogeography of mammalian basal metabolic rate. Am. Nat. 156, 201–219 (2000)

    Article  Google Scholar 

  14. 14

    Lovegrove, B. G. The influence of climate on the metabolic rate of small mammals: a slow-fast metabolic continuum. J. Comp. Physiol. B 173, 87–112 (2003)

    CAS  PubMed  Google Scholar 

  15. 15

    Heusner, A. A. Size and power in mammals. J. Exp. Biol. 160, 25–54 (1991)

    CAS  PubMed  Google Scholar 

  16. 16

    Sieg, A. E. et al. Mammalian metabolic allometry: do intraspecific variation, phylogeny, and regression models matter? Am. Nat. 174, 720–733 (2009)

    Article  Google Scholar 

  17. 17

    West, G. B., Brown, J. H. & Enquist, B. J. A general model for the origin of allometric scaling laws in biology. Science 276, 122–126 (1997)

    CAS  Article  Google Scholar 

  18. 18

    Lindstedt, S. L. & Calder, W. A. Body size, physiological time, and the longevity of homeothermic mammals. Q. Rev. Biol. 56, 1–16 (1981)

    Article  Google Scholar 

  19. 19

    Schmidt-Nielsen, K. Scaling: Why Is Animal Size So Important? (Cambridge Univ. Press, 1983)

    Google Scholar 

  20. 20

    Brown, J. H., Gillooly, J. F., Allen, A. P., Savage, V. M. & West, G. B. Toward a metabolic theory of ecology. Ecology 85, 1771–1789 (2004)

    Article  Google Scholar 

  21. 21

    McMahon, T. A. & Bonner, J. T. On Size and Life (Scientific American Library, 1983)

    Google Scholar 

  22. 22

    Peters, R. H. The Ecological Implications of Body Size (Cambridge Univ. Press, 1983)

    Google Scholar 

  23. 23

    Calder, W. A. Size, Function, and Life History (Dover, 1996)

    Google Scholar 

  24. 24

    Heusner, A. A. Energy metabolism and body size. II. Dimensional analysis and energetic non-similarity. Respir. Physiol. 48, 13–25 (1982)

    ADS  CAS  Article  Google Scholar 

  25. 25

    Robinson, W. R., Peters, R. H. & Zimmermann, J. The effects of body size and temperature on metabolic rate of organisms. Can. J. Zool. 61, 281–288 (1983)

    Article  Google Scholar 

  26. 26

    Gillooly, J. F. & Allen, A. P. Changes in body temperature influence the scaling of VO2,max and aerobic scope in mammals. Biol. Lett. 3, 99–102 (2007)

    PubMed  Google Scholar 

  27. 27

    Alberts, B. et al. Molecular Biology of the Cell 4th edn (Garland Science, 2002)

    Google Scholar 

  28. 28

    White, C. R., Blackburn, T. M. & Seymour, R. S. Phylogenetically informed analysis of the allometry of mammalian basal metabolic rate supports neither geometric nor quarter-power scaling. Evolution 63, 2658–2667 (2009)

    Article  Google Scholar 

  29. 29

    Grafen, A. The phylogenetic regression. Phil. Trans. R. Soc. Lond. B 326, 119–157 (1989)

    ADS  CAS  Article  Google Scholar 

  30. 30

    Savage, V. M., Deeds, E. J. & Fontana, W. Sizing up allometric scaling theory. PLoS Comp. Biol. 4, e1000171 (2008)

    ADS  MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Walter Fontana.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

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)

Supplementary Data

This file contains curvature data. (XLS 295 kb)

PowerPoint slides

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kolokotrones, T., Van Savage, Deeds, E. et al. Curvature in metabolic scaling. Nature 464, 753–756 (2010). https://doi.org/10.1038/nature08920

Download citation

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing