The time bend of curved allometries

The simulated body mass (w) distribution for placental mammals over time (curves left plot), with dots being the global maximum estimates from Smith et al. (2010). The dashed colour lines mark the time of the simulated allometries (coloured curves) in the right plot, that shows also the observed (dots) relationship between the basal metabolic rate (BMR) and body mass (w). Red: 50 million years ago (MA); Green: 30MA; Blue: 0MA. From Witting (2016b); data from McNab (2008).

Another challenge to our understanding of body mass evolution is the finding that the most studied allometry, i.e., the relationship between basal metabolism and mass in mammals, is not a straight line on the double logarithmic plot, but a convexly bend curve (Kolokotrones et al., 2010). Does this curvature reflect a bug in our theory, or is it simply a logical consequence of the natural selection of mass?

This is examined in a new release on bioRxiv (Witting, 2016b), where I use the “allometries are selected by the selection of metabolism and mass”-hypothesis (Witting, 2016a) to simulate the evolution of the body mass distribution for placental and marsupial mammals over 65 million years.

Following the extinction of the dinosaurs at the Cretaceous-Palaeogene (K-Pg) boundary 65 million years ago, I let both clades diversity across ecological niches with a body mass distribution that is constrained by the maximum mass of the clade over time and the current minimum mass. This allows me to calculate the inter-specific allometry as it evolves over time.

The predicted allometries are linear shortly after the diversification where the majority of the body mass variation is selected by the evolution of diverse resource handling across ecological niches. But this evolution stops when the species become fully adapted to their niches, with a body mass distribution that evolves over time to be more and more affected by the background selection of mass specific metabolism.

While this background selection is expected to be invariant of mass on the per generation time-scale of natural selection, the relative increase in physical time is largest in the smaller species because they evolve over a larger number of generations. And as the metabolic increase is selected into mass, it follows that the left-hand side of the allometry is bend upward over time with respect to both metabolism and mass.

The bend, from an exponential rate of increase in mass specific metabolism of 9.3x10^-9 on the per generation time-scale, explains the curvature of the metabolic allometry in placental mammals. A rate of increase that is about an order of magnitude smaller explains the smaller curvature in marsupials; predicting placentals with a higher metabolism than marsupials. This agrees 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. 2016a. The natural selection of metabolism and mass selects allometric transitions from prokaryotes to mammals. Preprint at bioRxiv https://dx.doi.org/10.1101/084624.
• Witting, L. 2016b. The natural selection of metabolism explains curvature in allometric scaling. Preprint at bioRxiv https://dx.doi.org/10.1101/090191.