Chen et al. 10.1073/pnas.0506806103. |
Supporting Text
Supporting Figure 5
Supporting Figure 6
Supporting Figure 7
Supporting Figure 5
Fig. 5. Predictability of positions of neurons belonging to the same ganglion from wiring minimization. Each dot represents the mean positional deviation from actual of neurons within the given ganglion. Error bar shows the first standard deviation from the mean.
Fig. 6. (A) Mean of the absolute value of predicted–actual position for different values of the normalization coefficient (a ) and the wire length power (z ). Interestingly, the minimum average deviation is achieved by the quadratic cost function with a = 27, close to biologically justifiable 29.3. (B) Neuron positions for different values of normalization coefficient, a , at z = 2. Colors indicate which ganglia neurons belong to, according to the code of Fig. 1. For the sake of clarity only unpaired or left neurons are shown.
Fig. 7. (A) Optimal synaptic and neuronal layout in the shared-wire model. Neurons are vertically offset for clarity. Each neuron is shown by a blue line with cell body position indicated by a black triangle. Synapses are blue dots and fixed points are red crosses. A circle indicates actual cell body position. (B) Predicted distribution of synaptic density in the shared wire model. Unexpectedly, a peak in synaptic density emerges in the anterior region, which corresponds to the nerve ring. This result demonstrates predictive power of the wiring optimization approach.
Supporting Text
Connectivity Data.
For the current work, we compiled an updated version of Caenorhabditis elegans wiring diagram. Pivotal works published by White et al. (1) and Hall (2) had provided neuronal circuitry in the head and tail but lacked connection details for 58 motor neurons in the ventral cord of the worm. We compiled most of the missing data using original electron micrographs (EM) and handwritten notes from White and coworkers. The dorsal side of the worm around the midbody, however, was not previously documented. Using original thin worm sections prepared by White et al.(1), we generated new EM images and reconstructed neurons with processes in this region (1). The new version of the wiring diagram incorporates original data and new reconstructions, as well as updates based upon later work (O. Hobert and D.H.H.., unpublished work), and R. M. Durbin (http://elegans.swmed.edu/parts/neurodata.txt ).The wiring diagram of 279 nonpharyngeal neurons in C. elegans is now 97% complete, covering 6,393 chemical synapses, 890 electrical junctions, and 1,410 neuromuscular junctions. Over 3,000 connections, including chemical synapses, electrical junctions, and neuromuscular junctions, were added and/or updated from the previous version. Due to rather sparse sampling of data along lengths of the sublateral, canal-associated lateral, and midbody dorsal cords, connectivity ambiguities for a select few neurons remain.
The external connection cost consists of 200 sensory and motor neurons wired to 20 sensory organs, 95 body wall muscles, and one representative muscle for the vulva and anus, respectively. The identity of sensory neurons and the locations of their corresponding sensory organs are based on diagrams of amphids, phasmids, and putative touch sensors from the Wormatlas web site (www.wormatlas.org). A neuron is assumed to make a single connection to a given sensory organ.
The positions of muscles are used in two different ways in the paper (i) Location of external structures in optimization calculation (Eq. 3). (ii) Mapping of neuron-to-muscle connections for ventral cord motor neurons. For optimization, the position of each muscle is defined as the midpoint between anterior and posterior extremities of the sarcomere region (1, 3). Neuron-to-muscle connections for the first 32 muscles in the head are detailed by White (1). For the remaining muscles, direct neuron-to-muscle mapping is not available. In this case, we assume that motor neurons connect to muscles where positions of neuromuscular junctions overlap the sacromere region of a given muscle (3). Because more than one muscle can overlap the neuromuscular junction (NMJ) region of a single neuron, we approximate the number of connections to each muscle by taking the total number of NMJs made by a given neuron, divided by the number of muscles overlapping the NMJ region. For neurons lacking complete reconstruction, especially ones on the dorsal side of the worm, the number of neuron-to-muscle connections is assumed to be the average NMJ per muscle from fully reconstructed neurons of the same class. The last three body muscles in tail of the worm do not overlap with NMJ regions. The connections to these muscles are assigned to the posterior-most motor neurons in the ventral and dorsal cords.
Neuron Position.
We define neuron location by the center of the cell body projected onto the anterior–posterior (AP) axis of the worm. These positions are determined from various diagrams of neuronal cell bodies in the adult worm (www.wormatlas.org). We also divided neurons into three regions in the worm body: head, midbody, and tail. Head neurons have cell bodies located <25% along the AP axis from the head of the worm. Midbody neurons are located between 25% and 75% down the worm body from the head. Tail neurons are >75% down the worm body from the head.Synapse Position.
Despite the near completion of the wiring diagram, which maps connections between neurons, there is a lack of data specifying the location of individual synapses in the worm. Using neuron diagrams in ref. 1, information from handwritten notes designating the source animal of the reconstruction (N2U vs. N2Y and JSE), and crossreferences with ref. 2, we approximated synapse positions into three gross categories: head (<25% of body length from the nose), midbody (between 25% and 75%), and tail (>75%). This data set was created by looking at individual neurons by themselves. Positions of synapses across pairs of neurons are not reconciled into these gross categories. A synapse is considered to be in proximity of the cell body if both fall within the same defined areas of head, midbody, or tail.Ganglia Distribution.
The positions of neurons within each ganglion are more dispersed in the wiring cost minimization placement than the actual layout, especially for the three posterior-most ganglia. Fig. 5 shows the mean deviation and corresponding standard deviations between predicted and actual positions of neurons in each ganglion. The largest dispersion of deviations is attributed to neurons belonging to the preanal, dorsorectal and lumbar ganglia.General Power-Law Cost Function
. To test the robustness of the cost minimization results, we explored several alternative forms of the cost function where cost is proportional to different powers of wire length. Mathematically, we replaced the exponent, z , in Eqs. 2 and 3 with values between 1 and 4. Unlike the quadratic formulation, power-law cost functions with exponents ¹ 2 are not exactly solvable. However, if the exponent is greater than one, these cost functions are convex (4), meaning that any local minimum must be global as well, thus simplifying the minimization task.We minimized these cost functions by using several effective numerical techniques. First, we used the conjugate gradient method, which converges particularly well for exponents greater than about 1.5. Second, we used an iteration procedure based on the quadratic cost function (5). This technique converges particularly well when the exponent is below 3. Third, we used linear programming to solve the case of exponent equal to one by introducing additional variables (6). For those exponents where cost functions can be solved by different methods, we verified that the corresponding solutions coincide.
Fig. 6A shows the mean deviation between predicted and actual neuron position as a function of a and z . To understand the dependence of layout on these parameters, we show the "trajectories" of neuron locations for different a with z = 2 (Fig. 6B). In the limiting case of large a , neurons with sensory endings are located close to the fixed points. In the limiting case of small a , neurons bunch up unrealistically close to each other. Best agreement between predicted and actual is achieved at intermediate values of a . The fit gets worse for smaller a , because neurons from lumbar ganglion move too far forward and for larger a , because neurons in the nerve ring ganglia move too far backward
Shared-Wire Model
. In this formulation, each neuron is represented as a single straight wire with multiple synapses on it (Fig. 1E Inset). Wires belonging to synaptically coupled neurons must overlap. Then the cost function is the sum of each neuron’s wire length. By introducing variables corresponding to the front and back tips of each wire, this cost function can be solved by linear programming (6). In addition to the tips of wires, the solution yields locations of synapses (Fig. 7). When two synaptically coupled neurons had an extensive overlap region, we placed the synapse between them at the midpoint of that region. Because the actual locations of most synapses in the worm are not currently known, comparison with the data requires predicting cell body positions (see above, Neuron Position). The center of mass of synapses for each neuron was used as the position of its cell body. Multiplicity of connections may be included in this model by weighting synapses correspondingly.Lineage Analysis.
We constructed a matrix of "relatedness" between neurons expressed in terms of distances using published embryonic and postembryonic lineage trees (7, 8)."Relatedness" between two neurons is found by first identifying the lowest common progenitor cell. Then, for each cell, we count the number of cell divisions from the common ancestor where each division is defined as a single unit in length. The lineage distance is the total number of cell divisions from the two neurons with the initial division from the common progenitor counted only once. Two hundred seventy-nine nonpharyngeal neurons are included in the analysis. Postembryonic blast cells are mapped back to the embryonic lineage such that all cells can be traced to the anterior daughter of the fertilized egg P0. In cases where variability has been noted in the left-right pair of postembryonic cells, we assigned the precursors to what is most often observed in experiment (7). Specifically, blast cell P1 is assigned to the right and P2 is assigned to the left. Cells that appear to be random in left/right division have been arbitrarily assigned such that P3, P5, P7, and P9 are right, and P4, P6, P8, and P10 are left. AVFR has been assumed to come from P1.aaa and AVFL from W.aaa. Ambiguities of postembryonic P cells affect mostly ventral cord motor neurons.
Because cell divisions split lineage trees into left/right or AP branches, optimization results, not surprisingly, show distinct left and right neurons clusters (data not shown). This effect gives rise to large deviation between predicted and actual placement of neurons, because bilateral neurons are usually located near each other along the AP axis in the animal. We test the sensitivity of left/right lineage to layout prediction by calculating the solution for unpaired and left neurons only (186 neurons). This methodology improves the mean deviation slightly (24.5% compared with 26.1% for all 279 neurons) but is still a worse prediction than the wire-minimized solution based on neuron connectivity.
Data Files.
Data used in the paper are available from the authors at http://www.wormatlas.org/handbook/nshandbook.htm/nswiring.htm.1. White, J. G., Southgate, E., Thomson, J. N. & Brenner, S. (1986) Philos. Trans. R. Soc. London Ser. B 314, 1-340.
2. Hall, D. H. & Russell, R. L. (1991) J. Neurosci. 11, 1-22.
3. Dixon, S. J. & Roy, P. J. (2005) Development (Cambridge, U.K.) 132, 3079-3092.
4. Boyd, S. & Vandenberghe, L. (2004) Convex Optimization (Cambridge University Press, Cambridge, U.K.)..
5. Sigl, G., K, D. & Johannes, F. (1991) in Proc. ACM/IEEE Design Automation Conf., pp. 57-62.
6. Weis, B. & Mlynski, D. (1987) in Proc. IEEE Int. Symp. on Circuits and Systems, pp. 564-567.
7. Sulston, J. E. & Horvitz, H. R. (1977) Dev. Biol. 56, 110-156.
8. Sulston, J. E., Schierenberg, E., White, J. G. & Thomson, J. N. (1983) Dev. Biol. 100, 64-119.
9. Achacoso, T. B. & Yamamoto, W. S. (1992) AY’s Neuroanatomy of C. elegans for Computation (CRC, Boca Raton, FL).