Research

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

Page R Painter

Office of Environmental Health Hazard Assessment, California Environmental Protection, Agency, P.O. Box 4010, Sacramento CA 95812, USA

author email corresponding author email

Theoretical Biology and Medical Modelling 2005, 2:45doi:10.1186/1742-4682-2-45

Published:	10 November 2005

Abstract

Background

Recent theories have attempted to derive the value of the exponent α 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).

Methods

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.

Results

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.

Conclusion

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