A spectrum of p-atic symmetries and defects in confluent epithelia

TL;DR

It is demonstrated that epithelial tissues should not be described by nematic order alone, but instead host a spectrum of p-atic symmetries, and the first direct experimental evidence for this multivalency of order is provided.

Abstract

Topological defects provide a unifying language to describe how orientational order breaks down in active and living matter. Considering cells as elongated particles confluent, epithelial tissues can be interpreted as nematic fields and its defects have been linked to extrusion, migration, and morphogenetic transformations. Yet, epithelial cells are not restricted to nematic order: their irregular shapes can express higher rotational symmetries, giving rise to p-atic order. Here we introduce a framework to extract p-atic fields and their defects directly from experimental images. Applying this method to MDCK cells, we find that all symmetries generate defects.No strong positional or orientational correlations are found between nematic and hexatic defects, suggesting that different symmetries coexist largely independently. These results demonstrate that epithelial tissues should not be described by nematic order alone, but instead host a spectrum of p-atic symmetries. Our work provides the first direct experimental evidence for this multivalency of order and offers a route to test and refine emerging p-atic liquid crystal theories of living matter.

Figures (18)

• Figure 1: Shape classification of cells in wild-type MDCK cell monolayer. a) Raw experimental data - Full data of Frame 0. b) Raw experimental data - used snippet of Frame 0 for visualization c) - g) Minkowski tensor for cells in b), visualized using $\vartheta_p$ for $p = 2,3,4,5,6$, respectively. The rotation of the $p$-atic director indicates the orientation.
• Figure 2: Illustration of how we go from the orientation of cells (left column) to a fine orientation field (middle columns), which we then visualize as a LIC image (the two right most columns) and use to detect defects (right column). In this illustration a coarse-graining radius $r_{avg}$ of $1.5r_{max}(t)$ is used. Positively charged defects are indicated by an open circle and negatively charged defects by a closed circle. Different colors are used for different $p$, corresponding to the color scheme used in Figure \ref{['fig:2_Minkowski']}. The pictures on the left correspond to the pictures on Figure \ref{['fig:2_Minkowski']}.
• Figure 3: $+\frac{1}{p}$ and $-\frac{1}{p}$ defects shown as described in Palacios_IEEE_2011. For the LIC filter the vtkSurfaceLICMapper from vtkvtkBook was used, which is implemented based on the ideas from Laramee_IEEE_2003. Positively charged defects are indicated by an open circle and negatively charged defects by a closed circle. Different colors are used for different $p$, corresponding to the color scheme used in Figure \ref{['fig:2_Minkowski']}.
• Figure 4: Defect position for different coarse-graining radii $r_{avg}$ at different times. Different colors are used for different $p$, corresponding to the color scheme used in Figure \ref{['fig:2_Minkowski']}. Positively charged defects are indicated by an open circle and negatively charged defects by a closed circle. The raw experimental image for Frame 0 can be seen in Figure \ref{['fig:2_Minkowski']}$a)$, the one for Frame 25 in Figure \ref{['fig:frame_25']} and the one for Frame 50 in Figure \ref{['fig:frame_50']}.
• Figure 5: Number of $-\frac{1}{p}$ and $+\frac{1}{p}$ defects over time for different coarse-graining radii $r_{avg}$. Different colors are used for different $p$, corresponding to the color scheme used in Figure \ref{['fig:2_Minkowski']}.