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Viscous vertex model for active epithelial tissues

This paper introduces a rotationally invariant viscous vertex model that incorporates both cortical and bulk dissipation to describe free-floating epithelial tissues, enabling the extraction of effective shear viscosity and revealing how viscosity regulates cell-shape textures and stabilizes topological defects under active conditions.

Original authors: Shao-Zhen Lin, Sham Tlili, Jean-François Rupprecht

Published 2026-03-31

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CC BY 4.0

Original authors: Shao-Zhen Lin, Sham Tlili, Jean-François Rupprecht

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 city made entirely of living, breathing cells. In this city, the buildings (cells) are packed so tightly that they touch each other on all sides, forming a continuous sheet, like a honeycomb or a mosaic. This is what biologists call an epithelial tissue.

For a long time, scientists modeled how these tissues move and change shape by treating the cells like rigid tiles sliding on a rough floor. They assumed the main resistance to movement came from the "friction" between the cells and the ground they sit on (like the extracellular matrix).

However, this paper introduces a new way of looking at the problem. The authors argue that in many real-life scenarios—like a floating embryo in a womb or a cluster of cells growing in a petri dish (an organoid)—there is no "floor." The tissue is free-floating. In these cases, the resistance to movement doesn't come from the ground; it comes from the internal stickiness of the cells themselves.

Here is a simple breakdown of their new model, using everyday analogies:

1. The "Honey" vs. The "Floor"

Think of a standard model as a crowd of people trying to walk across a sticky carpet. The friction comes from their shoes dragging on the carpet.

The new model (the Viscous Vertex Model) is like a crowd of people trying to walk through a giant vat of honey. Even if the floor is perfectly smooth (zero friction), it's still hard to move because the honey itself is thick and sticky.

The "Honey" inside the cells: The authors realized that the inside of a cell (the cytoplasm) and the skin of the cell (the cortex) act like thick fluids. When cells squeeze past each other or change shape, they have to drag this internal fluid with them. This internal "stickiness" is called viscosity.

2. The Two Types of "Stickiness"

The paper identifies two specific places where this internal stickiness happens:

Junctional Viscosity (The Glue): Imagine the walls between two houses. If you try to stretch the wall, the "glue" holding it resists. In cells, the junctions where they touch each other have a viscous resistance.
Bulk Viscosity (The Core): Imagine the center of a house. If the house tries to expand or shrink, the air and furniture inside resist that change. In cells, the center of the cell resists moving relative to its edges.

The authors built a mathematical framework that accounts for both of these types of stickiness simultaneously.

3. The "Floating City" Problem

Here is the tricky part: If you remove the floor (friction) from a crowd of people, and they are just floating in space, how do you stop the whole group from spinning out of control or drifting away?

The Old Problem: In previous math models, if you turned off the floor friction, the equations broke. It was like trying to calculate the speed of a car with no wheels; the math became "singular" (undefined) because the whole system could spin or slide forever without any resistance.
The New Solution: The authors used a clever mathematical trick (called Lagrange multipliers) to act like invisible tethers. These tethers don't stop the cells from moving; they just ensure that the total movement of the group stays balanced (conserving momentum). This allows them to simulate tissues that are truly floating in space, like a jellyfish embryo or a growing organoid, without the math crashing.

4. What Happens When You Pull?

The researchers tested their model by simulating what happens when you pull on a single cell in a sheet of cells.

Without Internal Stickiness (Old Model): If you pull a cell, the stress relaxes instantly. It's like pulling a rubber band that snaps back immediately.
With Internal Stickiness (New Model): If you pull a cell, the "honey" inside resists. The cells around the pulled one stretch out and stay stretched for a while. The tissue behaves like a thick, slow-moving fluid. The more viscous the cells, the more they stretch and the harder it is to pull them.

5. The "Traffic Jam" of Cells

The paper also looked at "active" tissues—tissues where cells are alive and pushing/pulling on their own (like a crowd of people trying to move in a specific direction).

Low Viscosity (Runny Honey): The cells move chaotically. They crash into each other, creating a lot of "traffic jams" and chaotic swirls (called topological defects). It's like a busy intersection with no traffic lights.
High Viscosity (Thick Honey): The internal stickiness acts like a calming force. It forces the cells to line up and move together in large, coordinated groups. The chaotic traffic jams disappear, replaced by smooth, flowing rivers of cells.

Why Does This Matter?

This model is a bridge between two worlds:

Microscopic: It looks at individual cells and their specific shapes.
Macroscopic: It explains how the whole tissue behaves like a fluid or a solid.

By accounting for the internal "stickiness" of cells, this model allows scientists to accurately simulate free-floating tissues (like developing embryos or lab-grown organoids) that were previously impossible to model correctly. It helps us understand how our bodies grow, how wounds heal, and how tissues flow without needing to be glued to a surface.

In short: The authors realized that cells aren't just sliding on a floor; they are swimming in their own thick, internal honey. By modeling this "honey," they finally cracked the code for simulating how living tissues float, flow, and organize themselves in the wild.

1. Problem Statement

Epithelial tissues and dense cell assemblies behave as active viscoelastic materials. While standard vertex models effectively describe tissue mechanics using elastic energies (area and perimeter) and interfacial tensions, they typically model dissipation solely through substrate friction (local friction between vertices and a static extracellular matrix).

However, in many biological contexts—such as free-floating embryonic tissues, organoids, and suspended monolayers—substrate friction is negligible. In these systems, internal viscous forces (cortical and bulk viscosities) dominate dissipation. Existing models often fail to:

Treat junctional (interfacial) and bulk viscosities on equal footing.
Maintain rotational invariance (frame independence) when defining viscous forces.
Remain mathematically well-posed in the limit of zero substrate friction (where the system becomes singular due to unconstrained global translation/rotation).
Provide a clear link between microscopic cell viscosities and macroscopic tissue rheology.

2. Methodology

The authors develop a rotationally invariant viscous vertex model that incorporates two distinct sources of dissipation:

Cell-Cell Interfacial Viscosity ( $\eta^{(s)}$ ): Dissipation arising from the relative motion of vertices along cell edges.
Cell Bulk Viscosity ( $\eta^{(b)}$ ): Dissipation arising from the relative motion between cell vertices and the geometric center of the cell.

Key Formulation Steps:

Dissipation Rates: The model defines dissipation rates ( $S_{interface}$ and $S_{bulk}$ ) based on the square of elongation rates projected along edges and vertex-center links, respectively. This leads to viscous tensions proportional to velocity differences: $\Lambda^{(viscous)} \propto \mathbf{t} \cdot (\mathbf{v}_j - \mathbf{v}_i)$ .
Rotational Invariance: Unlike alternative formulations that simply damp relative velocities ( $\dot{\mathbf{r}}_i - \dot{\mathbf{r}}_j$ ), this model projects velocities onto the unit tangent vectors of the edges. This ensures that global rigid rotations do not generate spurious dissipation, satisfying material objectivity.
Matrix Formalism: The force balance equations are assembled into a linear system $C \cdot \mathbf{v} = \mathbf{F}(t)$ , where $C$ is a friction-viscosity coefficient matrix combining substrate friction, interfacial viscosity, and bulk viscosity.
Handling Zero Friction: To address the singularity of matrix $C$ when substrate friction $\gamma \to 0$ , the authors employ a Lagrange multiplier formalism. They augment the system into a saddle-point matrix $C_{ext}$ by imposing global kinematic constraints (conservation of total linear and angular momentum). This regularizes the system, allowing simulations at strictly zero friction without numerical divergence.
Rheology Protocol: A "slab-shear" protocol is introduced to extract an effective, coarse-grained tissue viscosity by applying shear strain rates and measuring the resulting viscous stress.

3. Key Contributions

Unified Viscous Framework: A mathematically rigorous extension of the vertex model that simultaneously accounts for junctional and bulk viscosities while preserving rotational symmetry.
Zero-Friction Regularization: A robust numerical scheme using Lagrange multipliers to simulate free-floating tissues (organoids) where substrate friction is absent, overcoming the singularity issues of previous models.
Analytical and Numerical Rheology: Derivation of an analytical expression for short-time tissue viscosity in hexagonal packings and numerical validation in disordered packings.
Active Matter Dynamics: Application of the model to both polar (traction-driven) and nematic (stress-driven) active tissues to study the interplay between viscosity, cell shape, and topological defects.

4. Results

A. Coarse-Grained Tissue Viscosity

Short-Time Viscosity: In the absence of cell rearrangements (T1 transitions), the short-time tissue viscosity $\eta_{tissue}^{(ST)}$ $η_{t i ss u e}^{(S T)}$ scales linearly with the sum of junctional ( $\eta_s$ $η_{s}$ ) and bulk ( $\eta_b$ $η_{b}$ ) viscosities.
- Analytical Result: For a hexagonal lattice, $\eta_{tissue}^{(ST)} = \frac{\sqrt{3}}{4}\eta$ (assuming $\eta_s = \eta_b = \eta$ ).
Long-Time Viscosity: Under sustained shear allowing cell rearrangements, the long-time viscosity also scales linearly with cell viscosities but is generally lower than the short-time value due to stress relaxation via T1 transitions.
Shear Banding: In the presence of high substrate friction, shear flow localizes near the boundary (shear banding). In the zero-friction limit, this banding vanishes, and the flow profile becomes uniform, resembling a Newtonian fluid.

B. Active Tissue Dynamics

The model was applied to two active scenarios:

Polar Active Traction (Leading Edge Migration):
- Low viscosity leads to turbulent-like flows with many motile topological defects.
- Increasing viscosity enhances spatial correlation of cell orientations, reduces the number of defects, and promotes a global coherent rotation mode (swirling) with well-defined $+1/2$ and $-1/2$ defects.
Nematic Active Stresses (Shape-Dependent):
- Low viscosity results in elongated cells, multicellular rosettes, and high defect density.
- High viscosity stabilizes well-defined topological defects and promotes long-range spatial correlations in cell orientation, matching experimental observations in active nematic tissues.
- Viscosity acts as a control parameter, driving a crossover from "active turbulence" to coherent, ordered motion.

C. Numerical Stability

The condition number of the extended matrix $C_{ext}$ scales as $N_c^{2.5}$ (where $N_c$ is the number of cells), remaining numerically tractable for systems up to $\sim 6 \times 10^5$ cells.
The framework successfully simulates free-floating tissues without artificial friction, capturing floppy modes (zero-dissipation deformations) that are suppressed only when bulk viscosity is non-zero.

5. Significance

This work provides a critical bridge between cell-based discrete models and continuum active nematic theories.

Biological Relevance: It offers a physically accurate tool for simulating free-floating biological systems (e.g., organoids, embryonic tissues) where substrate friction is negligible, a regime previously difficult to model.
Mechanistic Insight: It demonstrates that internal cellular viscosity is not just a damping factor but a fundamental regulator of tissue morphology, defect density, and flow coherence.
Methodological Advance: The Lagrange multiplier approach for zero-friction limits provides a generalizable framework for implementing various boundary conditions (fixed, sliding, driven) in overdamped vertex models, ensuring mathematical well-posedness and numerical stability.

In summary, the paper establishes a robust, rotationally invariant viscous vertex model that resolves long-standing issues in simulating frictionless active tissues, revealing how microscopic viscosities dictate macroscopic rheological and dynamical properties.