🔬 condensed matter

Hydrodynamic stability and pattern formation in hexatic epithelial layers

This paper investigates the hydrodynamic stability of hexatic epithelial layers, revealing that while high adhesion relative to active stress maintains a stable quiescent state, increased activity triggers a hierarchy of instabilities leading to counter-flowing lanes and eventually chaotic dynamics distinct from typical active turbulence.

Original authors: Josep-Maria Armengol-Collado, Leonardo Puggioni, Livio N. Carenza, Luca Giomi

Published 2026-08-03

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CC BY 4.0

Original authors: Josep-Maria Armengol-Collado, Leonardo Puggioni, Livio N. Carenza, Luca Giomi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a bustling city where the buildings aren't made of brick and mortar, but of living, breathing cells. In this city, the streets are crowded, and the buildings are constantly pushing and pulling on one another. This is the world of epithelial tissues, the sheets of cells that line our organs and skin. Scientists have long studied how these tissues move, often comparing them to liquid crystals—materials like the liquid inside a watch or a TV screen that flow like a liquid but have a specific direction, like tiny compass needles all pointing the same way. Usually, these "compass needles" in biology are thought to point in just one or two directions (like a line or a flat plane). But what if the cells are arranged in a honeycomb pattern, where they have six-fold symmetry, like a snowflake? This creates a more complex, "hexatic" order. The big question is: if these cells are constantly generating their own energy (like tiny motors), how does that energy interact with the sticky forces holding them together? Does the tissue stay calm, or does it start to dance, swirl, and form wild patterns? Understanding this helps us grasp how tissues heal wounds, grow into shapes, or even how cancer spreads, because the way cells organize is the key to their behavior.

In this study, researchers Josep-Maria Armengol-Collado, Leonardo Puggioni, Livio N. Carenza, and Luca Giomi dive into a computer simulation of these hexagonal cell layers to see how they react when they get "active." They treat the tissue like a fluid that has its own internal engine. Their main finding is that the stability of this cellular city depends entirely on a tug-of-war between two forces: the energy the cells inject to move (like a motor revving up) and the energy lost to friction as cells stick to their neighbors (like dragging your feet in mud).

The researchers discovered a fascinating hierarchy of behaviors. If the cells stick together very strongly (high friction), the tissue remains calm and still, no matter how big the room is. It's like a crowd of people holding hands so tightly that even if everyone tries to dance, they can't move. However, if the cells become more active or the friction drops, the calm state breaks down. Instead of just flowing in one direction, the tissue spontaneously organizes itself into lanes of traffic flowing in opposite directions. Imagine a highway where cars in the left lanes drive north and cars in the right lanes drive south, all without a traffic cop telling them to do so.

The paper suggests that the width of these lanes isn't random; it is determined by a specific "crossover" scale where the energy injection matches the energy dissipation. In a narrow channel, you might get just one pair of lanes. But as the channel gets wider, the system doesn't just get a wider lane; it splits into more lanes, keeping the width of each lane roughly the same. It's like a river that, instead of getting wider, suddenly splits into a dozen smaller, parallel streams.

The story gets even more chaotic when the researchers let the tissue move in a 2D open space rather than a narrow channel. In this scenario, the neat lanes start to bend, twist, and break apart. The system enters a state of spatiotemporal chaos, which the authors describe as a new kind of "active turbulence." While it looks messy, it's not the same chaos seen in other active fluids. The patterns resemble a "herringbone" design, similar to the chaotic, swirling structures seen in certain types of malignant tumors like fibrosarcoma. The authors emphasize that while their simulations show this behavior clearly, it is a result of their specific model of hexatic order and energy balance, suggesting that the complex geometry of cells (the six-fold symmetry) plays a crucial role in creating these unique, self-organizing patterns.

Technical Summary: Hydrodynamic Stability and Pattern Formation in Hexatic Epithelial Layers

Problem Statement
The paper addresses the hydrodynamic stability and pattern formation in epithelial layers, specifically focusing on the role of hexatic order (6-fold orientational symmetry) arising from the close-packing of deformable cells. While active liquid crystal hydrodynamics has been extensively applied to polar (1-fold) and nematic (2-fold) systems to explain collective cell migration and defect dynamics, the specific hydrodynamic behavior of hexatic epithelia remains underexplored. A central challenge is understanding how the interplay between mechanical activity, lateral adhesion, and the multi-scale nature of hexatic order influences the stability of the quiescent state and the emergence of collective motion. Unlike standard active nematics, where instability is often generic, hexatic systems may exhibit a more complex balance of energy injection and dissipation across different length scales.

Methodology
The authors employ a continuum hydrodynamic model derived from the theory of passive $p$ -atic liquid crystals, extended to active hexatic systems ( $p=6$ ).

Governing Equations: The model couples the mass density, momentum density, and a rank-6 order parameter tensor ( $Q_6$ ) representing the local hexatic orientation. The active stress is modeled as $\sigma^{(a)} = \alpha_6 \nabla^{\otimes 4} \odot Q_6$ , reflecting forces generated at the cellular scale.
Dimensional Reduction: The study focuses on two regimes:
1. Channel Confinement: A 2D channel of width $L$ where the system is invariant along the longitudinal direction. This reduces the problem to coupled equations for the hexatic angle $\theta(y)$ and longitudinal velocity $v_x(y)$ .
2. Periodic Domains: A 2D periodic domain allowing for full vectorial flow and orientation dynamics.
Regularization of Dissipation: A critical methodological step involves addressing the unphysical instability of standard viscous models at small scales. The authors introduce hyperviscosity (higher-order velocity gradients, specifically $\eta_6 \nabla^4$ ) to ensure that energy dissipation scales match energy injection rates ( $\sim q^6$ ) at small scales, preventing divergence. This leads to a scale-dependent dissipation rate involving both standard shear viscosity ( $\eta_2$ ) and hyperviscosity ( $\eta_6$ ).
Analysis: The authors perform linear stability analysis on the quiescent state to derive dispersion relations. They identify critical parameters, including the Deborah number ($De$) and a crossover length scale $\ell_\times = (\eta_6/\eta_2)^{1/4}$ , which dictates the transition between viscous and hyperviscous dissipation regimes. Numerical simulations of the full nonlinear equations are used to validate analytical predictions and explore the post-transition dynamics.

Key Contributions and Results

Scale-Dependent Stability Mechanism:
The paper demonstrates that the hydrodynamic stability of hexatic epithelia is not solely determined by system size (as in active nematics) but by the balance between energy injection and dissipation across length scales.
- Generic Stability: When lateral adhesive interactions (represented by $\eta_2$ ) are strong, the system is generically stable regardless of size. The quiescent state remains stable against infinitesimal perturbations because energy is dissipated efficiently at the scale of injection.
- Instability Threshold: Instability arises only when the activity is sufficiently high relative to the dissipation, characterized by a critical Deborah number ( $De_\times > 16/3$ ) defined at the crossover scale $\ell_\times$ .
Hierarchy of Pattern-Forming Instabilities:
In confined channels, the system exhibits a cascade of instabilities leading to the formation of counter-flowing lanes:
- Spontaneous Flow Transition: Similar to active nematics, a primary instability leads to the formation of two lanes of cells flowing in opposite directions.
- Lane Proliferation: Unlike nematics, the presence of multiple inherent length scales allows for a hierarchy of further instabilities. As the channel width $L$ increases or activity rises, the system partitions into an increasing number of lanes ( $n_0 \approx 2\lfloor q_0/q_{min} \rfloor$ ).
- Fixed Lane Width: Crucially, the width of these lanes converges to a scale-independent value proportional to the crossover length $\ell_\times$ , rather than scaling with the system size. This suggests that the lane structure is an intrinsic property of the material's rheology.
Two-Dimensional Dynamics and "Active Turbulence":
In 2D periodic domains, the 1D lane structures become unstable to the proliferation of vortices.
- Low Activity: The system forms stationary, ordered bands (herringbone patterns) with a wavelength $\sim \ell_\times$ .
- High Activity: The system transitions into a state of spatiotemporal chaos. While this regime shares features with "active turbulence" observed in nematics, the authors note distinct differences. The chaotic state retains a "memory" of the linear instability, exhibiting clusters of orientation and vortical structures that resemble the herringbone patterns seen in certain malignant mesenchymal tumors (e.g., fibrosarcoma).

Significance and Claims
The authors claim this work provides the first exhaustive analysis of the hydrodynamic stability of hexatic epithelia. The primary significance lies in establishing that hexatic order introduces a multi-scale organization that fundamentally alters the stability criteria of active fluids.

Contrast with Nematics: The paper highlights a stark contrast with bulk active nematics, where any finite activity can destabilize the system if the domain is large enough. In hexatic epithelia, strong lateral adhesion (high $\eta_2$ ) can stabilize the quiescent state indefinitely, regardless of system size.
Biological Relevance: The model suggests that the mechanical stability of epithelial tissues is tunable via lateral adhesion (e.g., E-cadherin expression). The emergence of fixed-width lanes and herringbone patterns offers a theoretical framework for understanding the spatial organization of migrating cell clusters and the specific morphological patterns observed in certain pathologies, without invoking complex cell-level rules.
Theoretical Framework: The work establishes a continuum framework where the interplay between $p$ -atic order and scale-dependent dissipation governs the emergence of complex collective behaviors, moving beyond the single-order-parameter descriptions typical of polar or nematic active matter.

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