A common origin for 3/4- and 2/3-power rules in metabolic scaling

A central debate in biology has been the allometric scaling of metabolic rate. Kleiber's observation that animals' basal metabolic rate scales to the 3/4-power of body mass (Kleiber's rule) has been the prevailing hypothesis in the last eight decades. Increasingly, more evidences are supporting the alternative 2/3-power scaling rule, especially for smaller animals. The 2/3-rule dates back to before Kleiber's time and was thought to originate from the surface to volume relationship in Euclidean geometry. In this study, we show that both the 3/4- and 2/3-scaling rules have in fact one common origin. They are governed by animals' nutrient supply networks-their vascular systems that obey Murray's law. Murray's law describes the branching pattern of energy optimized vascular network under laminar flow. It is generally regarded as being closely followed by blood vessels. Our analysis agrees with experimental observations and recent numerical analyses that showed a curvature in metabolic scaling. When applied to metabolic data, our model accurately produces the observed 2/3-scaling rule for small animals of ~10 kg or less and the 3/4-rule for all animals excluding the smallest ones (~15 g). The model has broad implications to the ongoing debate. It proves that both the 3/4- and 2/3-exponents are phenomenological approximations of the same scaling rule within their applicable mass ranges, and that the 2/3-rule does not originate from the classical surface law.