A scheme for 3-dimensional morphological reconstruction and force inference in the early <i>C. elegans</i> embryo

<div><p>In this study, we present a scheme for the reconstruction of cellular morphology and the inference of mechanical forces in the early <i>C. elegans</i> embryo. We have developed and bench-marked a morphological reconstruction scheme that transforms flourescence-based <i>in vivo</i> images of membranes into a point cloud of smoothed surface patches, which facilitates an accurate estimation of membrane curvatures and the angles between membranes. Assuming an isotropic and homogeneous distribution of tensions along individual membranes, we infer a pattern of forces that are 7% deviated from force balance at edges, and 10% deviated from the Young-Laplace relation across membranes. We demonstrate the stability of our inference scheme via a sensitivity analysis, and the reproducibility of our image-analysis and force inference pipelines.</p></div

Error plots for inferred forces.

<p>Plot of the errors in the force balance relations on the membranes (A-C) and junctions (D-E). In A, we show errors in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0199151#pone.0199151.e002" target="_blank">Eq (1)</a> by plotting 2<i>T</i>_<i>k</i><i>H</i>_<i>k</i> against Δ<i>P</i> for each face in the scatter plot. The two clusters correspond to the inner and outer membranes, with their respective average percent errors in the box. Errors for the faces are portrayed as a heat map on the embryo, with outer membranes in B and inner membranes in C. In D, we plot the percent error equations, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0199151#pone.0199151.e009" target="_blank">(2)</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0199151#pone.0199151.e010" target="_blank">(3)</a>, as a histogram, with the average percent errors in the box. The percent errors of edges are portrayed as a heat map on the edge junctions (E), where the heat map corresponds to the heat map in the histogram (D).</p

Schematics of a two-cell system and <i>in-silico</i> assesment of force inferences.

<p>On the left, the system depicts the cross-sectional slice of two cells (with pressures <i>P</i>₁ and <i>P</i>₂), with constant mean curvatures, <i>H</i>₁ = 1/<i>R</i>₁ and <i>H</i>₂ = 1/<i>R</i>₂, on the major membrane faces 1 and 2 (with tensions <i>T</i>₁ and <i>T</i>₂) separated by the interfacial membrane face 3 (with tension <i>T</i>₃) with constant mean curvature <i>H</i>₃. Note, all three membranes are patches of spherical membranes as they have constant mean curvatures. <i>d</i> denotes the radius of the circular junction between the 3 membrane faces. <i>θ</i>’s correspond to the angles between the 3 membranes along the circular junction. On the right, we measure the total error between the true tension and the inferred tension: .</p

Schematic for tension balance at a junction.

<p>The three membrane faces, illustrated as curved planes, intersect at a junction. The isotropic surface tensions act perpendicular to the junction. The dihedral angles of intersection between the faces prescribe the relation in Eqs <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0199151#pone.0199151.e009" target="_blank">(2)</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0199151#pone.0199151.e010" target="_blank">(3)</a>.</p

Inferred forces.

<p>Inferred tensions and pressures at the 7-cell stage. (A-C) Relative tensions on the outer membranes. (D-F) Relative tensions on the inner membrane and the relative pressures in cells (represented as colored circular region in the middle of each cell). (G-I) Pressures in the cells. In these images, all pressure values are rescaled to match the same scale as the relative tension. The three rows depict the embryo from different viewpoints.</p

Depiction of the average mechanical state and cell-shape tensors.

<p>Plot of the shape and stress tensors at the 7-cell stage. The copper gradient lines represent cell connectivity, while the colors ranging from black to copper corresponds the depth along the z-axis. The tensors (all of which are symmetric) are represented by their 3 orthogonal eigenvectors plotted as blue line segments. The length of the segments correspond to the magnitude of the eigenvalue. Compressive forces in the stress tensors are plotted as red lines, and are circled in red for clarity.</p

Sensitivity analysis and reproducibility.

<p>Plot of the error in the reprocessing at the 7 cell stage (top row). Scatter plots of original value against the reprocessed values, for the inferred pressure, tension, mean curvature, and mean angles. The percent error for each value is boxed. The eigenvalues of are plotted on a log-scale histogram (bottom row) with the 7-cell stage on the left and 12-cell stage on the right.</p

Workflow of embryo reconstruction.

<p>(<b>A</b>) Shows a cross-sectional grayscale image of a 7-cell embryo, the ilastik outputted probability map, and a smoothed point cloud representation. (<b>B</b>) 3D visualization of the binarized image and probability map, with a cross section (left) and the whole embryo (right). (<b>C</b>) Plot of all the junctions in an embryo. The effect of smoothing can be seen clearly here. (<b>D</b>) Depiction of a junction, along with the tangent vectors of the adjacent membranes. The sparser watershed-based junction is replaced by a denser and more robust point cloud representation. (<b>E</b>) shows the cross-sectional grayscale image of a 12-cell embryo, the probability map, and a smoothed point cloud representation.</p

Correction: Influenza A Virus Assembly Intermediates Fuse in the Cytoplasm

<p>Correction: Influenza A Virus Assembly Intermediates Fuse in the Cytoplasm</p

Generation of WSN PA-GFP influenza virus.

<p>Schematic of the reverse genetics pHW2000 plasmid created by inserting the coding region of full length GFP into the C-terminal domain of the WSN PA open reading frame (A). This insertion included a duplication of 162 nucleotides of WSN PA prior to the 5′ non-coding region (NCR) depicted in the black rectangle. Western blot of purified recombinant WT WSN or WSN PA-GFP virus grown in embryonated eggs (B). The GFP signal corresponds to the appropriate size of a PA-GFP fusion protein. Viral NP protein was used as a control to ensure loading of equivalent amounts of virus. Immunofluorescence of MDCK cells infected with WSN PA-GFP (MOI = 3) for 16 hpi with anti-influenza NP antibody (C). Images to the right are enlarged regions identified by the dashed square. White arrows show areas of co-localization. All scale bars are 5 µm.</p