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.