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.