Topology recapitulates morphogenesis of neuronal dendrites.

Branching allows neurons to make synaptic contacts with large numbers of other neurons, facilitating the high connectivity of nervous systems. Neuronal arbors have geometric properties such as branch lengths and diameters that are optimal in that they maximize signaling speeds while minimizing construction costs. In this work, we asked whether neuronal arbors have topological properties that may also optimize their growth or function. We discovered that for a wide range of invertebrate and vertebrate neurons the distributions of their subtree sizes follow power laws, implying that they are scale invariant. The power-law exponent distinguishes different neuronal cell types. Postsynaptic spines and branchlets perturb scale invariance. Through simulations, we show that the subtree-size distribution depends on the symmetry of the branching rules governing arbor growth and that optimal morphologies are scale invariant. Thus, the subtree-size distribution is a topological property that recapitulates the functional morphology of dendrites.

Counting fluorescently labeled proteins in tissues in the spinning-disk microscope using single-molecule calibrations.

Quantification of molecular numbers and concentrations in living cells is critical for testing models of complex biological phenomena. Counting molecules in cells requires estimation of the fluorescence intensity of single molecules, which is generally limited to imaging near cell surfaces, in isolated cells, or where motions are diffusive. To circumvent this difficulty, we have devised a calibration technique for spinning-disk confocal microscopy, commonly used for imaging in tissues, that uses single-step bleaching kinetics to estimate the single-fluorophore intensity. To cross-check our calibrations, we compared the brightness of fluorophores in the SDC microscope to those in the total internal reflection and epifluorescence microscopes. We applied this calibration method to quantify the number of end-binding protein 1 (EB1)-eGFP in the comets of growing microtubule ends and to measure the cytoplasmic concentration of EB1-eGFP in sensory neurons in fly larvae. These measurements allowed us to estimate the dissociation constant of EB1-eGFP from the microtubules as well as the GTP-tubulin cap size. Our results show the unexplored potential of single-molecule imaging using spinning-disk confocal microscopy and provide a straightforward method to count the absolute number of fluorophores in tissues that can be applied to a wide range of biological systems and imaging techniques.

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

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.