A modified Murray's law for insect respiration

Insects breathe through a complex, ramifying tracheal network that carries oxygen directly to the cells, without using a closed cardiovascular system. The transport of respiratory gases occurs primarily by diffusion in insects, but can be enhanced in some species through rhythmic tracheal compression (RTC), where abdominal contractions collapse large tracheae and drive advective airflow. Murray’s law, derived for vascular systems, predicts that the cube of the parent vessel radius equals the sum of the cubes of the daughter radii, with flow scaling as Q ~ R³. Despite being an important classical result in mathematical biophysics, Murray’s law neglects key features of insect respiration: diffusive transport, slip flow, and unsteady advection during RTC. Here, we introduce modifications to Murray’s law that incorporate these features, as outlined in Khan & Staples (Nature Reviews Physics, 2025). Minimizing the total power dissipated during transport, we derive a version of Murray’s law that incorporates the Peclet number (Pe), Knudsen number (Kn), and Womersley number (Wo), accounting for diffusive, rarefied, and unsteady effects. The modified law predicts scaling exponents that range from 1 to 5, consistent with preliminary bifurcation scaling measurements in the entomology literature. In the no-slip (Kn = 0) limit, the unsteady formulation may also inform pulsatile vascular flow modeling, offering a unified framework for branched biological networks under diverse transport conditions.