# Quantification of Nematic Cell Polarity in Three-dimensional Tissues

## TL;DR

This paper introduces a new framework for quantifying complex nematic cell polarity patterns in 3D tissues, revealing local coordination and alignment with tissue structures, and models liver tissue as a biaxial liquid crystal.

## Contribution

It develops a novel algorithmic approach using multipole expansion and co-orientational order parameters to analyze nematic cell polarity in 3D tissues.

## Key findings

- Nematic cell polarity axes show local coordination in liver tissue.
- Polarity axes align with the sinusoidal blood transport network.
- Liver tissue can be modeled as a biaxial liquid crystal.

## Abstract

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.