Supply-demand balance in outward-directed networks and Kleiber's law

Recent theories have attempted to derive the value of the exponent alpha in the allometric formula for scaling of basal metabolic rate from the properties of distribution network models for arteries and capillaries. It has recently been stated that a basic theorem relating the sum of nutrient currents to the specific nutrient uptake rate, together with a relationship claimed to be required in order to match nutrient supply to nutrient demand in 3-dimensional outward-directed networks, leads to Kleiber's law (b = 3/4).

The validity of the supply-demand matching principle and the assumptions required to prove the basic theorem are assessed. The supply-demand principle is evaluated by examining the supply term and the demand term in outward-directed lattice models of nutrient and water distribution systems and by applying the principle to fractal-like models of mammalian arterial systems.

Application of the supply-demand principle to bifurcating fractal-like networks that are outward-directed does not predict 3/4-power scaling, and evaluation of water distribution system models shows that the matching principle does not match supply to demand in such systems. Furthermore, proof of the basic theorem is shown to require that the covariance of nutrient uptake and current path length is 0, an assumption unlikely to be true in mammalian arterial systems.

The supply-demand matching principle does not lead to a satisfactory explanation for the approximately 3/4-power scaling of mammalian basal metabolic rate.