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A central debate in biology has been the allometric scaling of metabolicrate. Kleiber's observation that animals' basal metabolic rate scales to the3/4-power of body mass (Kleiber's rule) has been the prevailing hypothesis inthe last eight decades. Increasingly, more evidences are supporting thealternative 2/3-power scaling rule, especially for smaller animals. The2/3-rule dates back to before Kleiber's time and was thought to originate fromthe surface to volume relationship in Euclidean geometry. In this study, weshow 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 systemsthat obey Murray's law. Murray's law describes the branching pattern of energyoptimized vascular network under laminar flow. It is generally regarded asbeing closely followed by blood vessels. Our analysis agrees with experimentalobservations and recent numerical analyses that showed a curvature in metabolicscaling. When applied to metabolic data, our model accurately produces theobserved 2/3-scaling rule for small animals of ~10 kg or less and the 3/4-rulefor all animals excluding the smallest ones (~15 g). The model has broadimplications to the ongoing debate. It proves that both the 3/4- and2/3-exponents are phenomenological approximations of the same scaling rulewithin their applicable mass ranges, and that the 2/3-rule does not originatefrom the classical surface law.

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