🔬 condensed matter

Multiscale order, flocking and phenotypic hysteresis in the cellular Potts model of epithelia

Through large-scale Cellular Potts model simulations, this study reveals how actin-driven cytoskeletal activity and cell-cell interactions generate a rich phase diagram of epithelial organization, characterized by multiscale orientational orders (flocking, nematic, and hexatic) and a phenotypic hysteresis reminiscent of the epithelial-mesenchymal transition.

Original authors: Calvin C. Bakker, Marc Durand, François Graner, Luca Giomi

Published 2026-05-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Calvin C. Bakker, Marc Durand, François Graner, 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 made entirely of living cells. In this city, the buildings (the cells) aren't static; they are constantly moving, changing shape, and interacting with their neighbors. The paper you provided is like a massive, high-tech simulation of this city, trying to figure out how the individual behavior of the buildings creates the overall traffic patterns and city layout.

Here is the story of what the researchers discovered, explained through simple analogies.

The Simulation: A Digital City of Cells

The researchers used a computer model called the "Cellular Potts Model." Think of this as a giant digital grid where every square is a pixel. They grouped these pixels together to form "cells."

To make the simulation realistic, they gave these digital cells a "muscle" (the actin cytoskeleton). Just like a real cell pushes against its surroundings to move, these digital cells have an internal engine that tries to expand them in specific directions. The researchers could turn this engine up or down, acting like a volume knob for cell activity.

The Big Discovery: Order at Different Scales

The most surprising thing they found is that the city looks different depending on how close you are to it. They call this "Multiscale Order."

Imagine looking at a crowd of people from a helicopter versus looking at them from the street:

From the street (Small Scale): If you look at a single cell, it looks like a tiny, slightly squashed hexagon (like a honeycomb cell). It has a specific shape, and its neighbors fit together neatly.
From the helicopter (Large Scale): If you zoom out, you see that these hexagons aren't just sitting still. They are all moving in the same direction, like a flock of birds or a school of fish.

The paper shows that these two things happen at the same time but at different sizes. The cells keep their hexagonal "neighborhood" shape locally, but globally, they flow together in a coordinated stream.

The "Volume Knob" Experiment

The researchers turned up the "actin engine" (the cytoskeletal activity) to see what would happen. Here is the journey they observed:

Low Activity (The Quiet Neighborhood): When the engine is weak, the cells are a bit lazy. They wiggle around but mostly stay in their spots. They form a messy, jumbled honeycomb pattern. There is no big movement.
Medium Activity (The "Hexanematic" State): As they turn up the engine, something magical happens. The cells start to organize.
- Locally: They still look like hexagons (the "honeycomb" shape).
- Globally: They start to align. It's as if the whole neighborhood suddenly decides to walk to the park together.
- The researchers call this "Hexanematic" order. It's a hybrid state: "Hexa" for the local shape, "Nematic" for the global alignment.
High Activity (The Flocking State): If they turn the engine up even more, the local hexagonal shapes start to break down a bit, but the global movement becomes incredibly strong. The entire tissue moves as one giant, coherent unit—a "flock."

The "Hysteresis" Loop: The Memory Effect

The paper also discovered a phenomenon called phenotypic hysteresis. This is a fancy way of saying the system has a "memory" and doesn't want to go back the way it came.

Imagine you are pushing a heavy boulder up a hill:

Going Up: You have to push really hard (high activity) to get the boulder to start rolling down the other side (the cells start moving collectively).
Going Down: Once the boulder is rolling, you can stop pushing, and it keeps going. Even if you lower the "push" (reduce activity) significantly, the boulder keeps rolling. It takes even less push to keep it moving than it did to start it.

In the simulation, when they increased the cell activity, the tissue suddenly started moving. But when they decreased the activity back down, the tissue kept moving even though it was "calm" enough that it should have stopped. It was stuck in the "moving" state.

Why Does This Matter?

The authors compare this "memory" effect to a biological process called the Epithelial-Mesenchymal Transition (EMT).

Epithelial: Cells that stick together in a sheet (like skin or the lining of your gut).
Mesenchymal: Cells that break away and move individually (like cells that spread cancer).

The simulation suggests that once cells switch to this "moving" state, they might be "stuck" there even if the signal telling them to move is turned off. This helps explain why, in diseases like cancer, cells might keep spreading even after the initial trigger is gone.

Summary

In short, the paper uses a giant computer simulation to show that living tissues are like complex traffic systems. By adjusting the internal "muscle" of the cells, the tissue can switch from a static honeycomb to a flowing flock. Most importantly, once the tissue starts flowing, it has a hard time stopping, creating a "memory" of its movement that could explain how cells get "stuck" in a moving state during disease.

Technical Summary: Multiscale Order, Flocking, and Phenotypic Hysteresis in the Cellular Potts Model of Epithelia

Problem Statement
The paper addresses the fundamental question of how collective cell migration and tissue spatial organization feedback on one another in epithelial tissues. While recent studies have introduced the concept of "multiscale order" (MsO)—the coexistence of different non-equilibrium phases across length scales—the mechanisms driving the development of large-scale nematic order and the specific nature of "hexanematic" organization remain unclear. Furthermore, the relationship between directed migration (flocking) and the underlying physical mechanisms of self-alignment in tissues, as opposed to undirected intercalation-based migration, requires further elucidation. The authors aim to map the complete phase diagram of epithelial migration and correlate migratory patterns with distinct manifestations of MsO.

Methodology
The authors employ large-scale numerical simulations using the Cellular Potts Model (CPM), a lattice-based model where cells are represented as connected sets of pixels. Key methodological features include:

Model Implementation: The CPM is implemented on Graphics Processing Units (GPUs) using CUDA, allowing for simulations of unprecedented scale (up to $L=2^{13}$ grids, representing nearly 700,000 cells).
Actin Cytoskeleton Integration: The model incorporates an explicit "Act-model" where a field $\phi_i$ represents the density of actin filaments in the cell's lamellipodium. This drives tread-milling behavior, where pixels translate toward regions of high actin density, generating motility.
Hamiltonian Dynamics: The system evolves via Metropolis-Hastings Monte Carlo steps based on a Hamiltonian $H$ that includes lateral surface interactions ( $J$ ), area constraints ( $\lambda_{cell}$ ), and cytoskeletal assembly energy ( $\lambda_{cyto}$ ).
Parameter Space Exploration: The authors systematically vary the cytoskeletal activity parameters ( $\lambda_{cyto}$ and $\Phi$ ) and the effective temperature/noise ( $T$ ) across 901 distinct configurations.
Order Parameter Analysis: To characterize the phases, the authors compute three orientation fields:
- $\psi_1$ : Polar axis orientation (related to cell motion).
- $\gamma_2$ : Nematic shape order ( $p=2$ ).
- $\gamma_6$ : Hexatic shape order ( $p=6$ ).
  These are coarse-grained over length scales $\ell$ to determine scale-dependent order parameters $\Gamma_p(\ell)$ and their decay exponents $\eta_p$ .
Classification: A Random Decision Forest Classification Algorithm (RDFCA) combined with a Marching Cubes algorithm is used to map the phase boundaries in the three-dimensional parameter space.

Key Results
The simulations reveal a rich phase diagram featuring multiple types of orientational order that coexist or dominate at different length scales:

Hexanematic Order: At intermediate cytoskeletal activity, the system exhibits "hexanematic" order. At length scales comparable to the cell size ( $\ell \approx \ell_{cell}$ ), cells adopt an approximately hexagonal morphology, resulting in dominant hexatic order ( $\Gamma_6 > \Gamma_2$ ). However, at larger length scales, nematic order becomes dominant ( $\Gamma_2 > \Gamma_6$ ). This crossover occurs at a specific length scale $\ell^\times$ .
Flocking Transition: As the actin polymerization rate ( $\lambda_{cyto}$ ) increases, the system undergoes a phase transition into a long-range flocking state. This is characterized by long-range order in the polar axis ( $\eta_1 = 0$ ).
Emergence of Nematic Order: Simultaneously with the onset of flocking, quasi-long-range nematic order emerges at length scales much larger than the cell size due to the interplay of directed motion and lateral interactions.
Exponent Behavior: The study identifies a specific pathway where increasing $\lambda_{cyto}$ drives a transition where nematic order becomes fully long-range ( $\eta_2 = 0$ ) while hexatic order remains quasi-long-range ( $\eta_6 \approx 1$ ) and confined to short length scales, independent of further cytoskeletal activity.
Phenotypic Hysteresis: When noise ( $T$ ) is sufficiently low to allow crystallization, cycling the cytoskeletal activity parameter $\lambda_{cyto}$ across the melting transition yields a hysteresis loop. The system exhibits different states (motile vs. non-motile honeycomb) depending on the direction of the parameter sweep, reminiscent of the epithelial-mesenchymal transition (EMT).

Significance and Claims
The paper claims to provide a quantitative framework for understanding the feedback between collective migration and spatial organization in epithelia. By combining cell morphology and cytoskeletal activity, the authors demonstrate that:

Multiscale Organization: The complex organization of epithelia can be interpreted through the language of liquid crystal hydrodynamics, albeit with anomalous dimensions extending beyond equilibrium statistical physics bounds.
Mechanism of Flocking: The study suggests a causal link between the onset of hexanematic order and the emergence of flocking, where nematic order develops at large scales while hexatic order persists at small scales.
Phenotypic Plasticity: The observation of hysteresis in the model offers a potential mechanism for understanding phenotypic plasticity, specifically how cells might retain a mesenchymal phenotype even after the withdrawal of inducing signals, a phenomenon relevant to cancer metastasis.

The authors conclude that while their model captures the essential physics of these transitions, a complete understanding of phenotypic plasticity will require future incorporation of specific biochemical pathways beyond the actin polymerization cycle. The work serves as a bridge between active matter theory and biological observations in reconstituted cell monolayers.