Fig 1.
Multipole decomposition of surface patterns.
(A, B) Schematic of vectorial and nematic cell polarity, respectively. (C) Multipole decomposition of a distribution on a sphere into spherical harmonics, see Eq (1). (D) Prototypical membrane distribution of vectorial polarity type with respective Mollweide projection and spherical power spectrum. The spherical power spectrum shows the power ||Fi||2 of the each mode Fi corresponding to the contribution from spherical harmonics of order i, i = 0, 1, 2, 3, 4, see Eq (2). (E) Same as panel D but for a ring-like surface distribution. Here, the second mode of the spherical power spectrum dominates. (E’) The second mode of the spherical power spectrum also dominates for the analogous case of a bipolar surface distribution. (F) Spherical projection, Mollweide projection and spherical power spectrum for an epithelial tubular cell from kidney tissue, as well as averaged power spectrum for a population of cells (n = 286, mean±2 ⋅ s.e.m., corresponding to 95% confidence interval). (G) Same as panel F, but for a hepatocyte from mouse liver tissue, as well as a population of hepatocytes (n = 9983).
Fig 2.
Spatial patterns of nematic cell polarity.
We visualize surface distributions by cuboids that have the same moments of inertia tensor. Opposite faces of these cuboids are colored red, green, and blue, respectively, corresponding to the principal axes of inertia (ordered in increasing order). These principal axes of inertia correspond exactly to the principal axes of the nematic tensor A introduced in Eq (3) (ordered in decreasing order), such that the bipolar axis a1 (golden) is normal to the red face, and the ring axis a2 (cyan) is normal to the blue face of the cuboids (see S1E Appendix for details). (A) Idealized bipolar distribution. The bipolar axis a1 (golden) marks the principal axis of inertia of this surface distribution with largest eigenvalue α1, hence the smallest moment of inertia. We represent this bipolar surface distribution by a cuboid with same moments of inertia tensor. Thus, the bipolar axis corresponds to the red face (with smallest area). (B) Idealized ring-like distribution. The ring axis a2 (cyan) marks the principal axis of inertia with the smallest eigenvalue α2, hence the largest moment of inertia. In the cuboid representation of this ring-like distribution, the ring axis corresponds to the blue face of the cuboid (with largest area). (C) Bipolar and ring axis of a typical hepatocyte. From left to right: Apical membrane distribution for a typical hepatocyte, spherical projection, Mollweide projection, and cuboid representation. Shown are two distinguished principal axes of inertia a1 and a2, corresponding to the bipolar and ring nematic cell polarity axes, respectively. In the cuboid representation of the hepatocyte, the bipolar axis axis a1 corresponds again to the red face, whereas the ring axis a2 corresponds to the blue face. (D) For each hepatocyte in a tissue sample, the corresponding cuboid is plotted, revealing ordered patterns at the liver lobule level. (E) Orientational order becomes even more apparent after spatial averaging, which was performed using a Gaussian kernel with standard deviation of 20 μm and omitting the cell in the center (kernel sketched to scale, blue), see S1F Appendix for details. In panels (D) and (E), a central vein (CV, cyan) and a portal vein (PV, orange) are shown, which serve as landmarks within a liver lobule.
Fig 3.
Four biaxial co-orientational order parameters applied to liver tissue.
(A) Ensemble of first principal axes n, represented as pairs of antipodal points (cyan) on the unit sphere; the ensemble displays prolate nematic order with respect to the first reference axis w (blue). This type of order is characterized by a positive value of the co-orientational order parameter co−S, co−S > 0. First reference axis w (blue line), second reference axis v (red), third reference axis u (green). (B) Example of a phase biaxial distribution of first principal axes n (cyan dots) with nematic alignment towards the first reference axis w (blue) and strong anisotropic fluctuations biased towards the third reference axis u (green), corresponding to a positive value of the phase-biaxiality order parameter co−P, co−P > 0. (C) Example of oblate nematic order of n with respect to the first reference axis w, corresponding to co−S < 0. (D) Ensemble of tripods of principal axes n, m, l that displays prolate nematic order of the first principal axis n (cyan dots) with respect to the first reference axis w (blue), but no additional order of the second principal axis m (golden dots); third principal axis not shown. Since there is no additional order of m, we have co−D = co−C = 0. (E) Example of molecular biaxial order quantified by the co-orientational order parameter co−C. Here, the first principal axis n (cyan dots) displays prolate nematic order as in panel D, while the second principal axis m (golden dots) is additionally biased towards the second reference axis v (red), corresponding to co−C > 0. (F) A second type of molecular biaxial order is measured by the co-orientational order parameter co−D. Here, the first principal axis n (cyan dots) exhibits nematic order with respect to the first reference axis w (blue). Fluctuations of the second principal axis m (golden dots) are also biased towards w, corresponding to co−D < 0. (G) Co-orientational order parameters quantify biaxial order of hepatocytes in liver tissue (mean±s.d., n = 12 tissue samples). The local reference system was chosen as a local average with punctured Gaussian kernel, see text for details. (H) Spherical distribution of apical ring axis a2 (antipodal pairs of cyan dots) and apical bipolar axis a1 (golden dots) of hepatocyte cell polarity relative to the reference axes w = 〈a2〉loc (blue), v = 〈a1〉loc (red), u = v × w (green), illustrating the quantitative analysis in panel G.
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).
Fig 5.
Minimal interaction model reproduces biaxial order parameters for hepatocyte/sinusoid co-alignment.
(A) Simulated co-orientational order parameters (COOP) between nematic cell polarity axes and local anisotropy of the sinusoidal network as function of the dimensionless interaction strength λ (solid lines) for the minimal interaction model given in Eq (12). The color code for co−S, co−P, co−D, co−C corresponds to that of Figs 3 and 4; the insets reshow the spherical distribution plots from Fig 3A and 3B for the cases co−S > 0 and co−P > 0. Shaded regions indicate mean±s.d. of experimental values from Fig 4C. The range of λ for which all four order parameters agree in simulation and experiment is highlighted in gray. Note that the model prediction for co−C (golden solid line) coincides with the prediction for co−D (brown solid line) and is thus not visible. (B) Graphical summary of co-orientational order between hepatocyte polarity and the local anisotropy of the sinusoidal network. The anisotropy of the sinusoidal network defines local reference axes: a preferred sinusoid axis s1 (red, approximately parallel to the mean direction of blood flow), a plane axis s2 (blue, characterizing a layered organization of the sinusoidal network), and a third axis s3 = s1 × s2 perpendicular to the other two axes (green). Nematic cell polarity of hepatocytes defines two nematic axes for each cell: a ring axis a2 and a bipolar axis a1, of which the ring axis co-aligns with the sinusoidal network. The ring axis a2 (cyan) aligns parallel to the plane axis s2 of the local sinusoidal network. This reflects a sandwich architecture of the two intertwined networks of bile canaliculi and sinusoids, with alternating layers enriched in apical membrane of hepatocytes, which represent contact surfaces to the bile canaliculi network, and endothelial cells forming the sinusoidal network, which is quantified by co−S > 0. Fluctuations of the ring axis a2 are not random, but are biased towards the third reference axis s3 of the sinusoidal network (green), and thus away from the preferred axis s1 (red), as quantified by co−P > 0. These anisotropic fluctuations supposedly reflect an effective mutual repulsion between the sinusoidal and the bile canaliculi network. For the bipolar axis a1 of hepatocyte polarity, we do not find (unexpected) co-orientational order relative to the sinusoidal network, corresponding to co−D ≈ 0 and co−C ≈ 0.