The narrowing of dendrite branches across nodes follows a well-defined scaling law

Abstract

The systematic variation of diameters in branched networks has tantalized biologists since the discovery of da Vinci’s rule for trees. Da Vinci’s rule can be formulated as a power law with exponent two: The square of the mother branch’s diameter is equal to the sum of the squares of those of the daughters. Power laws, with different exponents, have been proposed for branching in circulatory systems (Murray’s law with exponent 3) and in neurons (Rall’s law with exponent 3/2). The laws have been derived theoretically, based on optimality arguments, but, for the most part, have not been tested rigorously. Using superresolution methods to measure the diameters of dendrites in highly branched Drosophila class IV sensory neurons, we have found that these types of power laws do not hold. In their place, we have discovered a different diameter-scaling law: The cross-sectional area is proportional to the number of dendrite tips supported by the branch plus a constant, corresponding to a minimum diameter of the terminal dendrites. The area proportionality accords with a requirement for microtubules to transport materials and nutrients for dendrite tip growth. The minimum diameter may be set by the force, on the order of a few piconewtons, required to bend membrane into the highly curved surfaces of terminal dendrites. Because the observed scaling differs from Rall’s law, we propose that cell biological constraints, such as intracellular transport and protrusive forces generated by the cytoskeleton, are important in determining the branched morphology of these cells.