Allometric rates

Very roughly, the surface area, A, of an animal scales as its linear size, L, squared:
A ∼ L2
by which we mean there is a constant k1 with
A = k1⋅L2
Similarly, the mass, M, of an animal scales as
M = k2⋅L3
Heat dissipation occurs across the surface, so the total metabolic rate of an animal is proportional to L2, hence to M2/3.
The metabolic rate per unit mass then is proportional to M-1/3, or so argued Rubner in 1883.
Pulse rate is related to metabolic rate per unit mass, so smaller animals should have faster pulse rates and larger animals slower. Indeed, this is observed, familiar even. A mouse's heart beats very rapidly, a whale's heart very slowly. Add in the observation that most mammal hearts beat 1 to 2 billion times during the animal's life and we understand that in the absence of external perturbation (early death due to predation or disease, for example), a mouse has a shorter life than a person, who in turn has a shorter life than a whale.
This makes perfect sense, but careful measurements by Kleiber in 1932 revealed something different: for most animals, over a range of sizes spanning 21 orders of magnitude, the metabolic rate per unit mass varies as M-1/4 rather than as M-1/3.
Adding to the interest of this question, plants exhibit this M-1/4 metabolic scaling law. Also, other biological variables exhibit power law scalings with mass. Life-span scales as M1/4, age of first reproduction as M3/4, the time of embryonic development as M-1/4, and the diameters of tree trunks and of aortas as M3/8, for example.
The reason for the observed M-1/4 scaling is not understood, but several explanations have been proposed. One is based on the observation that the diffusion does not occur across a smooth 2-dimensional surface, but across the fractal boundary of the lungs. Because the dimension of the lungs is d > 2, the metabolic rate scales as Ld, so as Md/3. Then the metabolic rate per unit mass scales as Md/3 - 1. Because d > 2, d/3 - 1 > -1/3. Measurements of the dimension of the lungs suggest a metabolic rate/unit mass of about -1/4.

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