  How epithelial cells coordinate their polarity to form functional tissues is
an open question in cell biology. Here, we characterize a unique type of
polarity found in liver tissue, nematic cell polarity, which is different from
vectorial cell polarity in simple, sheet-like epithelia. We propose a
conceptual and algorithmic framework to characterize complex patterns of
polarity proteins on the surface of a cell in terms of a multipole expansion.
To rigorously quantify previously observed tissue-level patterns of nematic
cell polarity (Morales-Navarette et al., eLife 8:e44860, 2019), we introduce
the concept of co-orientational order parameters, which generalize the known
biaxial order parameters of the theory of liquid crystals. Applying these
concepts to three-dimensional reconstructions of single cells from
high-resolution imaging data of mouse liver tissue, we show that the axes of
nematic cell polarity of hepatocytes exhibit local coordination and are aligned
with the biaxially anisotropic sinusoidal network for blood transport. Our
study characterizes liver tissue as a biological example of a biaxial liquid
crystal. The general methodology developed here could be applied to other
tissues or in-vitro organoids.
