^{**}=7/3ψ

# The evolution of curved allometries

The primary selection of metabolism explains curvature in allometric scaling

Curvature in allometric scaling was documented by Kolokotrones et al. (2010) for the maybe best studied allometry, i.e., the relationship between the basal metabolic rate and body mass in mammals.

The curved allometry is explained by a simulation of body mass selection in placental mammals over the past 65 million years (Witting, 2016). The best fit of the current allometry is given by the blue curve in the left plot in Fig. 1. The upward bend in the metabolism of the smaller species implies an overall exponent that is smaller than 0.75 should a linear allometry be fitted to the data. The overall linear exponent is 0.72 across the entire range of simulated body masses, and it increases to 0.74 for the upper half of the body mass distribution, and declines to 0.67 for the lower half.

The estimated rate of exponential increase in the pre-mass component of mass specific metabolism is 9.3x10^{-9} (95% CI: 7.3x10^{-9} - 1.1x10^{-8}) on the per generation time-scale. The bend is more apparent in placental than marsupial mammals (MacKay, 2011), and this reflects a per generation rate of increase that is about an order of magnitude larger in placentals (Witting, 2016). From the differences in the curvature of the metabolic allometry we conclude that placentals have evolved a higher metabolism than marsupials; in agreement with an average metabolism that is 30% larger in placentals relative to marsupials of similar size (McNab, 2008).

### References

- Kolokotrones, T., V.Savage, E.J. Deeds and W.Fontana 2010. Curvature in metabolic scaling. Nature 464:753--756.
- McNab, B.K. 2008. An analysis of the factors that influence the level and scaling of mammalian BMR. Comparative Biochemical Physiology A 151:5--28.
- Smith, F.A., A.G. Boyer, J.H. Brown, D.P. Costa, T.Dayan, S.K.M. Ernest, A.R. Evans, M.Fortelius, J.L. Gittleman, M.J. Hamilton, L.E. Harding, K.Lintulaakso, S.K. Lyons, C.McCain, J.G. Okie, J.J. Saarinen, R.M. Sibly, P.R. Stephens, J.Theodor and M.D. Uhen 2010. The evolution of maximum body size of terrestrial mammals. Science 330:1216--1219.
- Witting, L. 2016. The natural selection of metabolism explains curvature in allometric scaling. Preprint at bioRxiv http://dx.doi.org/10.1101/090191.