Flow similarity, stochastic branching, and quarter power scaling in plants

Abstract

The origin of allometric scaling patterns that are multiples of 1/4 has long fascinated biologists. While not universal, scaling relationships with exponents that are close to multiples of 1/4 are common and have been described in all major clades. Foremost among these relationships is the 3/4 scaling of metabolism with mass which underpins the 1/4 power dependence of biological rates and times. Several models have been advanced to explain the underlying mechanistic drivers of such patterns, but questions regarding a disconnect between model structures and empirical data have limited their widespread acceptance. Notable among these is a fractal branching model which predicts power law scaling of both metabolism and physical dimensions. While a power law is a useful first approximation to many datasets, non-linearity in some large data compilations suggest the possibility of more complex or alternative mechanisms. Here, we first show that quarter power scaling can be derived using only the preservation of volume flow rate and velocity as model constraints. Applying our model to the specific case of land plants, we show that incorporating biomechanical principles and allowing different parts of plant branching networks to be optimized to serve different functions predicts non-linearity in allometric relationships, and helps explain why interspecific scaling exponents covary along a fractal continuum. We also demonstrate that while branching may be a stochastic process, due to the conservation of volume, data may still be consistent with the expectations for a fractal network when one examines subtrees within a tree. Data from numerous sources at the level of plant shoots, stems, petioles, and leaves show strong agreement with our model predictions. This novel theoretical framework provides an easily testable alternative to current general models of plant metabolic allometry.