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.