Body mass allometries

Given the rescaling of the life history with mass and metabolism, we can calculate the exponents of the final body mass allometries for a variety of situations. These are given as the sum of the exponents for the metabolic-rescaling and mass-rescaling allometries, with the former representing the importance of metabolism for the evolution of mass, and the latter the rescaling of the life history in response to the evolutionary changes in mass.

These exponents are listed in Table 1, where they are denoted by a hat over the trait symbol. Apart from the home range exponent that is always one, the exponents depend on the β_β^• exponent for the pre-mass component of mass specific metabolism (β_β). They are shown for the two extreme cases where all the variation in net energy (ε), and thus also body mass, is caused either by variation in metabolic pace (β_β^•=1) or by variation in resource handling/availability (β_β^•=0). The intermediate case with similar importance of handling and pace (β_β^•=1/2) is also shown, together with cases where post-mass metabolism is independent of mass (b=0). The latter depends on the spatial dimensionality of the interactive behaviour, with b=0 for β_β^•=1/2d.

The majority of post-mass exponents are given as fractions, where 2d is the common denominator. The most well-known set of allometric exponents for multicellular animals, i.e., the set where b=-1/4, is evolving for two dimensional interactions when all of the variation in body mass is caused by variation in the handling and availability of resources across ecological niches.

Table 1 The exponents of the inter-specific allometries as determined by the spatial dimensionality and the pre-mass exponent of mass specific metabolism: β:mass specific metabolism; β_β:pre-mass component of β; α:resource handling; τ:time periods; p:survival; R:lifetime reproduction; r:population dynamic growth rate; h:home range; n:abundance. From Witting (2017).

Evidence

The often observed Kleiber scaling in taxa of multicellular animals (with 1/4 and 1/6 exponents) is a special case where the final allometries resemble mass-rescaling. This is because the majority of the body mass variation evolves from the handling of different resources across a variety of niches, with only minor variation in the pre-mass component of metabolism.

Allometries on other scales include metabolic-rescaling. With an apparent 5/6 exponent for mass specific metabolism (DeLong et al., 2010), prokaryotes follow the prediction for the smallest single celled self-replicators. They are predicted to have a metabolic-rescaling exponent of unity (β_β^•=1) and body mass variation that follows from evolutionary differences in mass specific metabolism.

Having a b-exponent that declines from about 0.61 to -0.20 with an increasing mass (Witting, 2017), protozoa follow the prediction for larger single celled self-replicators with interactive competition and a metabolic-rescaling exponent that declines from one to zero.

And mass specific metabolism is invariant of mass on the macro evolutionary scale from prokaryotes to mammals (Makarieva et al., 2005, 2008; Kiorboe and Hirst, 2014). This follows from evolution around an upper metabolic bound, where the metabolic decline from mass-rescaling is balanced by a metabolic increase from primary selection on the net energy of the organism.

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References

• DeLong, J.P., J.G. Okie, M.E. Moses, R.M. Sibly and J.H. Brown 2010. Shifts in metabolic scaling, production, and efficiency across major evolutionary transitions of life. Proceedings of the National Academy of Sciences 107:12941--12945.
• rboe and Hirst, 2014Kiorboe:Hirst:2014Kiørboe, T., and A.G. Hirst 2014. Shifts in mass scaling of respiration, feeding, and growth rates across life-form transitions in marine pelagic organisms. The American Naturalist 183:E118--E130.
• Makarieva, A.M., V.G. Gorshkov and L.Bai-Lian 2005. Energetics of the smallest: do bacteria breathe at the same rate as whales. Proceedings of the Royal Society B 272:2219--2224.
• Makarieva, A.M., V.G. Gorshkov, B.Li, S.L. Chown, P.B. Reich and V.M. Gavrilov 2008. Mean mass-specific metabolic rates are strikingly similar across life's major domains: Evidence for life's metabolic optimum. εm Proceedings of the National Academy of Sciences 105:16994--16999.
• Witting, L. 2017. The natural selection of metabolism and mass selects allometric transitions from prokaryotes to mammals. Theoretical Population Biology 117:23--42, https://dx.doi.org/10.1016/j.tpb.2017.08.005.