| 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 |
Return to Power Laws.