Why Extensile and Contractile Tissues Could be Hard to Tell Apart

TL;DR

The study tackles the mismatch between contractile single-cell behavior and extensile-looking tissue dynamics in epithelial monolayers. It introduces a two-tensor continuum model with a cell-shape tensor $\mathbf{R}$ and a nematic activity tensor $\mathbf{Q}$, coupled through the active stress $\boldsymbol{\Pi}^{active}=-\zeta \mathbf{Q}$ to analyze confined flows, defect dynamics, and wall alignment. Across unidirectional channel flow, the dancing state, and wall interactions, the results show that the flow- and shape-patterns for both extensile and contractile activity can resemble extensile signatures, making the sign of the active stress unrecoverable from single-field observations. Consequently, simultaneous measurements of stress fields (e.g., traction forces) are essential to unambiguously determine the nature of forces acting within epithelial layers.

Abstract

Active nematic models explain the topological defects and flow patterns observed in epithelial tissues, but the nature of active stress-whether it is extensile or contractile, a key parameter of the theory-is not well established experimentally. Individual cells are contractile, yet tissue-level behavior often resembles extensile nematics. To address this discrepancy, we use a continuum theory with two-tensor order parameters that distinguishes cell shape from active stress. We show that correlating cell shape and flow, whether in coherent flows in channels, near topological defects, or at rigid boundaries, cannot unambiguously determine the type of active stress. Our results demonstrate that simultaneous measurements of stress and cell shape are essential to fully interpret experiments investigating the nature of the physical forces acting within epithelial cell layers.

Figures (4)

• Figure 1: Top to bottom: velocity, active stress director, and cell shape profiles along a cross-section of the channel during unidirectional flow for (a) an extensile and (b) a contractile cellular continuum. Note the coordinate system is always oriented so that the $x$ axis corresponds to the direction of flow.
• Figure 2: Comparison of channel flows in an extensile and a contractile system: profiles of (a) $\theta_{\mathbf{Q}}$, the angle between the active stress director and the flow direction, (b) $\theta_{\mathbf{R}}$, the angle between the long axis of the shape and the flow direction, (c) velocity, and (d) shape aspect ratio. Insets of panels (a) and (b) illustrate the definitions of the angles $\theta_{\mathbf{Q}}$ and $\theta_{\mathbf{R}}$: they are the smallest angle between flow direction (channel axis) and the director/shape long axis, defined such that the value of the angle increases as the director/shape is rotated counterclockwise. That is, the inset of (a) shows a positive angle and the inset of (b) a negative angle. (e,f) Schematic comparing cell elongation in extensile and contractile systems: (e) extensile active flows (black curved arrows) extend an initially isotropic cell (dashed red circle to ellipse) along the active stress director (double headed blue arrow), whereas (f) contractile flows extend the cell perpendicular to the active stress director.
• Figure 3: (a) Schematic of the dancing state, illustrating vortices in the velocity profile (green) and the trajectories (orange and gray) of $+1/2$ defects (cyan). For an extensile system: (b) Zoom on a stress defect (cyan) and its corresponding shape defect (magenta); blue lines and red ellipses show the active stress director and shape field respectively. (c) Distribution of angles between the orientation of stress defects and their nearest shape defect, $\phi$. (d) Distribution of angles between stress defects and the local velocity field, $\beta$. (e) Distribution of angles between shape defects and the local velocity field, $\gamma$. (f)-(i) Corresponding plots for a contractile system. Insets on panels (c)-(e) illustrate the relevant angles.
• Figure 4: Distribution of angles relative to the wall tangent (shown in the insets) for (a) the active stress director field, $\alpha_{\mathbf{Q}}$ and (b) the long axis of the cell shape, $\alpha_{\mathbf{R}}$ in a turbulent field produced by extensile active stresses in a very wide channel. (c,d) Corresponding distributions in a contractile system. A snapshot of typical active stress and shape fields near the wall is shown in Fig. S4 SI.