Seed at cell centres, not nuclei: the cost of a standard shortcut in tessellation models of embryos

Space-filling tessellations are a standard way to turn a list of cell positions into a list of cell contacts. In Caenorhabditis elegans, where automated lineage tracing delivers nuclei and not cell outlines, the generators of those tessellations are routinely placed at the nuclei. We measure what that costs. In 60,701 cell-frames from two segmented embryos, a nucleus sits a median of 0.59 μm from its own cell's centre of volume, which is 26% of the cell's equivalent-sphere radius, and the offset is not random: its component along the outward direction from the embryo's centre is negative in 65% of cells and averages -0.148 μm, so nuclei sit systematically further out than the cells that contain them. The systematic part follows the same law in both embryos, about 1.5 n^-0.44 μm for an embryo of n cells. Running one space-filling model over 38 matched frames of one embryo from 4 to 328 cells, with only the seed positions changed, mean relative contact-area error is 0.134 to 0.526 seeded at measured centroids against 0.169 to 0.634 seeded at nuclei; centroid seeding is better in 37 of 38 frames (sign test p = 1.4\\times10^-10), and contact recall at 328 cells is 0.965 against 0.924. We reproduce the comparison in a second embryo (53 frames, 3 to 415 cells; centroids better in 48 of 53). We then ask what can be done when nuclei are all there is, and report that the two obvious corrections both fail: Lloyd relaxation of the seeds makes the fit worse, not better, because it drives the model toward a centroidal tessellation that real tissue is not, and a one-parameter inward radial shift, with constants fitted on the other embryo, changes nothing, because four fifths of the offset is cell-specific scatter that no function of nuclear positions can recover. We also report two things a modeller needs before reading any such number: the embryo-to-embryo variability of contact areas in the same deposit, which is the ceiling any model is measured against, and two faults in that deposit - a unit disagreement between the atlas and a later paper quantifying the same embryos, and an off-by-one row indexing that shifts contacts 1.39 minutes against volumes.