![]() ( B) Maximum projection image of a class IV neuron from the A3 segment constructed from 76 stacks (separated by 0.2 μm) from a SDC microscope (Nikon spinning disk, 50-μm pinholes, 40× water-immersion objective, NA = 1.25) of a 92-h after-egg-lay larva ( Shi ts1 ppkCD4-tdGFP). One cell in abdominal segment 3 (A3) is boxed. ( A) Third-instar larva expressing GFP in its class IV neurons. Measurements of dendrite branch diameters by SDC and SIM. Thus, it is an open question whether scaling laws such as Eq. Experimental support for scaling laws, however, is scarce because of the difficulties of imaging entire branched networks and because intrinsic anatomical variability may obscure precise laws ( 14). Da Vinci’s law has been reformulated as the “pipe model” for plants, in which a fixed cross-section of stems and branches is required to support each unit amount of leaves ( 13). Rall’s law minimizes the propagation time of action potentials ( 11) and the decrement of graded electrical signals across dendrite branch points ( 12) ( SI Appendix) it assumes that the density of ion channels is constant, and the cost of building or maintaining the surface area of the dendrite is minimized. For example, Murray’s law for the vasculature minimizes the frictional dissipation associated with moving fluid through pipes, given a fixed volume of blood ( 7). Scaling laws have been derived theoretically using optimality arguments. ![]() 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. 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. The area proportionality accords with a requirement for microtubules to transport materials and nutrients for dendrite tip growth. 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. 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. ![]() The laws have been derived theoretically, based on optimality arguments, but, for the most part, have not been tested rigorously. 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). 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. The systematic variation of diameters in branched networks has tantalized biologists since the discovery of da Vinci’s rule for trees. ![]()
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