Quantification of nematic cell polarity in three-dimensional tissues
Fig 4
Biaxial order of sinusoidal network correlates with nematic cell polarity.
(A) Central lines of the sinusoidal network in the liver lobule (same section of mouse liver tissue as in Fig 3; central vein: cyan, portal vein: orange). (B) The local anisotropy of the sinusoidal network is visualized by cuboids with equivalent moments-of-inertia tensor (using spherical regions of interest centered at each hepatocyte position of 20 μm radius). (C) Co-orientational order between apical nematic cell polarity and local anisotropy of the sinusoidal network, quantified in terms of the co-orientational order parameters introduced in Eq (10), where the principal axes are given by the axes of hepatic cell polarity for individual hepatocytes (n = a2, m = a1, l = a3), and the reference axes are given by the axes of the local anisotropy of the sinusoidal network (w = s2, v = s1, u = s3); (mean±s.d., n = 12 tissue samples). We find co−S > 0, showing that the ring axis a2 of hepatic cell polarity is preferentially aligned parallel to the plane axis s2 of the sinusoidal network, i.e., the ring axis is normal to the local layered organization of the sinusoidal network. Fluctuations of the ring axis are biased away from the preferred sinusoid axis s1, corresponding to co−P > 0. Note that s1 is approximately aligned with the direction of blood flow [15], while s2 is approximately parallel to the large veins. The COOP co−D and co−C characterize any additional alignment of the bipolar axis a1 of hepatic cell polarity; we find that co−D and co−C are not significantly different from zero. The inset shows a typical hepatocyte with apical membrane (green), basal membrane (magenta), and the sinusoidal network in a spherical region of interest of radius 20 μm centered at the position of the hepatocyte (magenta). While the apical membrane defines the cell polarity axes, the local sinusoidal network defines local reference axes, used in the definition of the COOP. (D) Spherical distribution of the apical ring axis of hepatocyte polarity a2 (represented as antipodal pairs of cyan dots on the unit sphere) and apical bipolar axis a1 (golden dots), relative to the reference frame of local sinusoidal network anisotropy, represented by the local preferred sinusoid axis s1 (red) and the plane axis s2 of the sinusoidal network that characterized its layered organization (blue).