Morphology of Fly Larval Class IV Dendrites Accords with a Random Branching and Contact Based Branch Deletion Model

Dendrites are branched neuronal processes that receive input signals from other neurons or the outside world [1]. To maintain connectivity as the organism grows, dendrites must also continue to grow. For example, the dendrites in the peripheral nervous system continue to grow and branch to maintain proper coverage of their receptor fields [2, 3, 4, 5]. One such neuron is the Drosophila melanogaster class IV dendritic arborization neuron [6]. The dendritic arbors of these neurons tile the larval surface [7], where they detect localized noxious stimuli, such as jabs from parasitic wasps [8]. In the present study, we used a novel measure, the hitting probability, to show that the class IV neuron forms a tight mesh that covers the larval surface. Furthermore, we found that the mesh size remains largely unchanged during the larval stages, despite a dramatic increase in overall size of the neuron and the larva. We also found that the class IV dendrites are dense (assayed with the fractal dimension) and uniform (assayed with the lacunarity) throughout the larval stages. To understand how the class IV neuron maintains its morphology during larval development, we constructed a mathematical model based on random branching and self-avoidance. We found that if the branching rate is uniform in space and time and that if all contacting branches are deleted, we can reproduce the branch length distribution, mesh size and density of the class IV dendrites throughout the larval stages. Thus, a simple set of statistical rules can generate and maintain a complex branching morphology during growth. In our brains, billions of neurons interact with each other to build a nervous system of unparalleled complexity and computational power. Neurons have dendrites, which are branched structures that receive synaptic or sensory inputs, and an axon, which send outputs to other neurons. The shape or morphology of individual neurons sets the number and types of interactions that a neuron can have and provides the structural basis of neuronal computation [9, 10, 11, 12, 13, 14]. Since many organisms continue to enlarge after the establishment of the body plan, it is critical for axons and dendrites to maintain their morphology as they grow. For example, interneurons of the grasshoper [2], motor neurons in moths [3] and mice [4] grow drastically in size yet maintain connections to their target cells. Futhermore, dendrites in the perpherial nervous system, like those of gold fish retinal ganglion cells [5], and dendritic arborization (da) sensory neurons of the fly larva [6], which are the subject of this work, grow to continually maintain coverage of their receptor fields. In this paper, we investigate the growth rules that are required to maintain the correct branching morphology as a dendrite grows. The da sensory neurons of the fly larva are a model system for studying dendritic arborization [15, 16, 17]. These dendrites innervate the extracellular matrix, which lies between the outer cuticle and the inner epidermal cell layer [7]. They tile the surface on the fly larva in a highly stereotyped manner and have four distinct morphological classes [18] (Fig. 1 A). Since it is easy to identify and image individual da neurons, these neurons have proven to be a powerful model system for studying dendrite morphology [15, 16]. In this paper, we address the question of how the morphology of the class IV da neurons (Fig. 1 B) is maintained during the larval stages. The class IV da neuron has highly branched dendrites [18], which detect potentially harmful stimuli, such as the ovipositor barb of parasitic wasps [8, 19]. The dendrites of the class IV neuron begin 1 ar X iv :1 61 1. 05 91 8v 1 [ qbi o. N C ] 1 7 N ov 2 01 6 morphogenesis during late embryogenesis ∼ 16 hrs After Egg Lay (AEL). By the time the larva hatches (∼ 22 hrs AEL at 25 ◦C), the class IV dendrites nearly cover its surface. The dendrites then continue to expand and branch as the larva grows (22− 126 hrs AEL), so that the neuron maintains its coverage of the larval surface [6]. In this work, we are seeking the growth rules that allow class IV dendrites to maintain their dense coverage of the larval surface. To this end, we have used a novel measure, the hitting probability, that quantifies the mesh size and two well-known measures of branching morphology: the fractal dimension [20] and lacunarity [21, 22] (see Definition of Morphometrics). We show that these measures remain largely invariant over larval stages, despite a several fold increase in larval length. Furthermore, we demonstrate that a model with simple rules for branching and self-avoidance can capture essential features of the establishment and maintenance of the dendrite’s morphology. 1 Experimental Results To characterize the morphology of fully-developed class IV dendrites, we imaged larvae expressing Cd4tdGFP under the ppk promoter (ppk-cd4-tdGFP) during the third instar stage (Fig. 1) using a laserscanning confocal microscope (See Material and Methods for details). Using NeuronStudio [23] and Fiji we traced the branches of the dendrites to produce skeletons. These skeletons were then analyzed to obtain the the mesh size, density and uniformity of class IV dendrites using parameters defined in the next section. Definition of Morphometrics Here we include simple definitions of the relevant morphometrics to aid comprehension. Hitting Probabiltiy H(B): The probability that a box of size B hits the dendrite. Mesh Size BH: The length at which 50% of all boxes hit the neuron. Fractal Dimension df: A measure of the space-fillingness of a shape. For a completely filled box df = 2, for a straight line df = 1, for branched shapes 1 ≤ df ≤ 2. Lacunarity Λ(B): A measure of density fluctuations as a function of length scale B. Lacunarity Length BΛ: The length at which Λ(B = BΛ) = 0.25, i.e. the length at which the neuron is uniform. The larger BΛ, the more variable the density of the neuron. Radius of Gyration Rg: A length scale that measures how spread out a shape is from its center. The larger Rg the more spread out the neuron. Persistence Length β: The characteristic length at which a branch bends. For mathematical definitions see Appendix. 1.1 Class IV dendrites have a small mesh size To characterize the mesh size of the dendrites, we developed a novel measure called the hitting probability H(B). H(B) measures the probability H that a randomly placed box of size B hits the dendrite (see Appendix for details). The hitting probability generalizes an earlier metric called the coverage index [6] by allowing for any box location and any box size. A typical hitting probability curve of a neuron (Fig. 1 D) H(B) increases monotonically with B, eventually reaching H = 1 as B approaches the size of the neuron. We define the characteristic mesh size BH as the box size at which half of all boxes hit the dendrite. In other words, BH is the maximum size of a stimulus that would go untouched, or undetected, on average, by the neuron. BH is similar to the mesh size in a cross-linked polymer network[24]. We found that BH = 8.4± 0.5μm (mean± SD, n = 14 neurons) for the mature dendrites of the class IV neuron. Thus, the mesh size is approximately equal to the diameter of the ovipositor barb of wasps that lay eggs in Drosophila larva (∼ 10μm, [8]). This indicates that the class IV dendrite has a high chance of detecting a wasp attack. Furthermore, the mesh size is small compared to the overall size of the neuron (∼ 500μm) and is similar to the mean branch length (see below).