A Scaling Law Governing Branching Morphogenesis in Neuronal Dendrites

Abstract

The systematic variation of diameters in branched networks has tantalized biologists since the discovery of da Vincis rule for trees. Da Vincis rule can be formulated as a power law with exponent two: the square of the mother branchs 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 and in neurons. The laws have been derived theoretically, based on optimality arguments, but, for the most part, have not been tested rigorously. In the case of neuronal dendrites, diameter changes across branch points have functional implications for the spread of electrical signals: for example, Ralls law with an exponent of 3/2 maximizes propagation speeds of action potentials across branch points. Using a super-resolution method to measure the diameters of all dendrites in highly branched Drosophila Class IV sensory neurons, we have tested Ralls law and shown it to be false. In its place, we have discovered a new 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 dendrite diameter. The law accords with microtubules providing force and transport for dendrite tip growth. That the observed scaling differs from Ralls law suggests that constraints imposed by cell biological mechanisms may impact electrical signaling in neurons. Our new scaling law generalizes to other branched processes such as the vasculature of plants and the circulatory system of animals. Significance StatementTo study the systematic variation of dendrite diameters, we have established a super-resolution method that allows us to resolve dendrite diameters in Drosophila Class IV dendritic arborization neurons, a model cell for studying branching morphogenesis. Interestingly, they do not follow any of the known scaling laws. We propose a new scaling law that follows from two concepts: there is an incremental cross-sectional area needed to support each terminal branch, and there is a minimum branch diameter. The law is consistent with dendrite growing by tip extension and being supported by microtubule-based transport. If the law generalizes to other neurons, it may facilitate segmentation in connectomic studies.