Very roughly, the surface area, A, of an animal scales as its linear size, L, squared: |
A ∼ L^{2} |
by which we mean there is a constant k_{1} with |
A = k_{1}⋅L^{2} |
Similarly, the mass, M, of an animal scales as |
M = k_{2}⋅L^{3} |
Heat dissipation occurs across the surface, so the total metabolic rate of an animal is proportional to L^{2}, hence to M^{2/3}. |
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 M^{1/4}, age of first reproduction as M^{3/4}, the time of embryonic development as M^{-1/4}, and the diameters of tree trunks and of aortas as M^{3/8}, 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 |
Return to Power Laws.