Kleiber's Law - Value of The Exponent

Value of The Exponent

The exponent for Kleiber's law, which is called a power law, was a matter of dispute for many decades. It is still contested by a diminishing number as being ⅔ rather than the more widely accepted ¾. Because the law concerned the capture, use, and loss of energy by a biological system, the system's metabolic rate was, at first, taken to be ⅔, because energy was thought of mostly in terms of heat energy. Metabolic rate was expressed in energy per unit time, specifically calories per second. Two thirds expressed the relation of the square of the radius to the cube of the radius of a sphere, with the volume of the sphere increasing faster than the surface area, with increases in radius. This was purportedly the reason large creatures lived longer than small ones - that is, as they got bigger they lost less energy per unit volume through the surface, as radiated heat.

The problem with ⅔ as an exponent was that it did not agree with a lot of the data. There were many exceptions, and the concept of metabolic rate itself was poorly defined and difficult to measure. It seemed to concern more than rate of heat generation and loss. Since what was being considered was not necessarily Euclidean geometry, the appropriateness of ⅔ as an exponent was questioned. Kleiber himself came to favor ¾, and that is the number favored today by the foremost proponents of the law, despite that ¾ also does not agree with much of the data, and is also troubled with exceptions. Theoretical models presented by Geoffrey West, Brian Enquist, and James Brown, purport to show how the ¾ observation can emerge from the constraint of how resources are distributed through hierarchical branching networks. Their understanding of an organism's metabolic/respiratory chain is based entirely on blood-flow considerations. Their claims have been repeatedly criticized as mistaken, given that the role of fractal capillary branching is not demonstrated as fundamental to the exponent ¾; and that blood-flow claims severely limit the relevance of the equation to organisms greater than e - 6 (≈ .0025) grams when the simultaneous claim is made that the equation is relevant over 27 orders of magnitude, extending from bacteria, which do not have hearts, to whales or forests.