🧬 biology

Theory of adhesion-driven self-organisation in growing tissues

This paper presents a unified theoretical framework demonstrating that the balance between cell-cell adhesion, self-diffusion, and proliferation governs whether growing tissues expand cohesively or fragment into spatial patterns and fingering morphologies, with density-dependent adhesion regulation identified as a key mechanism for maintaining cohesive tissue expansion.

Original authors: Carles Falcó, Samuel W. S. Johnson, Mohit P. Dalwadi, Philip K. Maini

Published 2026-04-30

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CC BY 4.0

Original authors: Carles Falcó, Samuel W. S. Johnson, Mohit P. Dalwadi, Philip K. Maini

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine a city made entirely of living cells. Sometimes, this city expands smoothly like a spreading wave of people filling a park. Other times, it breaks apart into chaotic clusters, or the edge of the city sprouts strange, finger-like projections that reach out uncontrollably.

This paper asks a simple question: Why does a group of cells sometimes move together as a smooth team, and other times fall apart into messy patterns?

The authors, a team of mathematicians from Oxford, built a "virtual city" to figure this out. They didn't just look at one rule; they created a unified theory that explains both smooth expansion and chaotic pattern-making as two sides of the same coin.

Here is the story of their discovery, broken down into everyday concepts:

1. The Three Forces at Play

The researchers identified three main forces that determine how the cell city behaves:

The "Crowd Push" (Self-Diffusion): Cells naturally bump into each other and spread out, like people in a crowded room trying to find space. This keeps things loose.
The "Glue" (Adhesion): Cells like to stick to their neighbors. Think of this as Velcro. If the Velcro is weak, they drift apart. If it's strong, they cling together.
The "Growth Engine" (Proliferation): Cells are constantly having babies (dividing), which pushes the whole group to expand into new territory.

2. The Experiment: Turning Up the "Glue"

The team ran simulations to see what happens when they change the strength of the "Glue" (adhesion).

Scenario A: Weak Glue (The Smooth Wave)
When the cells don't stick together very well, they act like a fluid. As they grow and push forward, they form a smooth, flat front. It's like a calm tide rolling onto a beach. The city expands steadily, and everyone stays in a neat line. This is what happens in healthy wound healing or normal tissue growth.

Scenario B: Strong Glue (The Chaos)
When the researchers turned up the "stickiness," things got weird.

The Breakup: Instead of a smooth line, the front of the city started to wobble.
The Fingers: The edge began to sprout "fingers" or protrusions, reaching out into empty space.
The Clumps: Behind the front, the cells didn't just fill the space; they collapsed into distinct islands or "aggregates."

The authors note that this "fingering" behavior looks very much like how cancer invades healthy tissue. In cancer, the cells lose their ability to stay in a cohesive group and instead break out into invasive tendrils.

3. The "Goldilocks" Problem

The paper explains that there is a delicate balance.

If the glue is too weak, the tissue spreads but might not hold together well.
If the glue is too strong, the tissue gets so sticky that it can't move smoothly. Instead of a wave, it fractures into spots and fingers.

It's like trying to walk through a hallway. If you are too loose, you drift. If you are too tightly holding hands with everyone else, you trip over each other and form a chaotic knot instead of moving forward.

4. The Solution: The "Smart Glue"

Here is the most interesting part. The researchers asked: How do healthy tissues avoid this chaos even when they are very sticky?

They discovered that nature uses a regulatory switch. In healthy tissues, the "stickiness" isn't constant.

When cells are far apart, they stick strongly to pull together.
But as they get crowded and fill the space (reach "confluence"), they turn off the glue.

The paper shows that if you program your virtual cells to do this (reduce adhesion when crowded), the chaos disappears. Even with strong adhesion, the tissue expands as a smooth, cohesive wave again. The "fingers" and "clumps" vanish because the cells know when to let go so they don't trip over each other.

Summary

The paper unifies two things we usually think are different: invasion (moving into new space) and pattern formation (making spots and stripes).

They show that these aren't two different biological processes. They are just different "modes" of the same machine, controlled by how sticky the cells are.

Low stickiness = Smooth invasion.
High stickiness without regulation = Chaotic fingers and spots (like in cancer).
High stickiness with regulation = Smooth invasion (like in healthy development).

By understanding this balance, the researchers provide a mathematical map for why tissues sometimes grow beautifully and other times grow dangerously out of control.

1. Problem Statement

The paper addresses a fundamental gap in understanding cellular self-organization: the distinction between cell invasion (cohesive colonization of space, e.g., wound healing) and spatial pattern formation (emergence of heterogeneity, e.g., embryonic development or cancer metastasis). Traditionally, these are modeled as distinct processes.

Invasion is often modeled via reaction-diffusion systems (e.g., Fisher-KPP), while pattern formation is typically explained by Turing mechanisms requiring at least two interacting species (activator/inhibitor).
The Challenge: Many biological systems involve only a single cell species where adhesion drives both aggregation and migration. Classical Turing mechanisms fail here due to the lack of a second species or appropriate diffusivity ratios. Furthermore, existing single-species adhesion models are often non-local, making analytical tractability difficult.
Goal: To develop a unified theoretical framework that explains how a single-species population can transition between cohesive invasion and fragmented pattern formation solely through the regulation of cell-cell adhesion and proliferation.

2. Methodology

The authors employ a multiscale mathematical approach, moving from a non-local integral model to a local partial differential equation (PDE) amenable to asymptotic analysis.

A. Model Derivation

Non-local Starting Point: They begin with a continuum model describing cell density $\rho(x,t)$ $ρ (x, t)$ incorporating:
- Self-diffusion: Density-dependent diffusion $D(\rho)$ (representing contact inhibition/volume exclusion).
- Adhesion: A non-local attractive term modeled as a convolution $(W * \rho)$ with an adhesion-based mobility $m(\rho)$ .
- Proliferation: Logistic growth $r\rho(1-\rho)$ .
Short-Range Approximation: To overcome the analytical intractability of non-local terms, they apply a matched asymptotic expansion assuming the interaction range of adhesion is small relative to the tissue size. This converts the non-local convolution into a local fourth-order PDE:
$\partial_t \rho = \nabla \cdot \left[ (d(\rho) - \omega m(\rho)) \nabla \rho - m(\rho) \nabla \Delta \rho \right] + \rho(1-\rho)$
Here, $\omega$ represents the effective adhesion strength, and the term $-\nabla \cdot (m(\rho)\nabla \Delta \rho)$ acts as a higher-order aggregation term (similar to Cahn-Hilliard equations).

B. Analytical Techniques

Linear Stability Analysis: Used to determine the stability of homogeneous steady states ( $\rho=0$ and $\rho=1$ ) and derive dispersion relations for pattern formation.
Traveling Wave Analysis: The authors analyze 1D traveling wave solutions $\rho(x-ct)$ to characterize invasion speeds and front profiles.
Matched Asymptotic Expansions: Used to derive explicit approximations for wave speeds and front shapes in limits of weak ( $\mu \to +\infty$ ) and strong ( $\mu \to -\infty$ ) adhesion.
Numerical Simulations: Finite-volume schemes (splitting conservative and non-conservative parts) validate the analytical predictions in 1D and 2D.

C. Two Model Variants

Model I (Unsaturated Adhesion): $m(\rho) = \rho$ . Adhesion scales linearly with density.
Model II (Saturated/Density-Dependent Adhesion): $m(\rho) = \rho(1-\rho)$ . Adhesion peaks at intermediate density and vanishes at confluence ( $\rho=1$ ), mimicking biological regulation.

3. Key Contributions and Results

A. Unification of Invasion and Patterning

The study demonstrates that invasion and pattern formation are not distinct mechanisms but different dynamical regimes of the same adhesion-driven system, controlled by the parameter $\mu = \alpha - \omega$ (the balance between repulsive diffusion and attractive adhesion).

B. Regime I: Weak Adhesion ( $\mu \gg 0$ )

Dynamics: The tissue exhibits robust, monotonic traveling waves.
Mechanism: Diffusion dominates adhesion. The invasion front is "pushed" by the bulk dynamics rather than "pulled" by the leading edge.
Speed: The wave speed $c$ converges to the Porous-Fisher model prediction: $c \sim \sqrt{\mu/2}$ .
Morphology: Planar, stable fronts in 2D.

C. Regime II: Intermediate Adhesion ( $-2 < \mu \leq \mu_c$ )

Transition: As adhesion increases, the front loses monotonicity.
Oscillatory Tails: In 1D, the density profile behind the front develops damped oscillations.
2D Instability: In 2D, planar fronts become unstable to transverse perturbations, leading to fingering instabilities. Small protrusions grow into fingers, resembling the dysregulated invasion seen in cancer.
Critical Threshold: A critical value $\mu_c \approx 1.18$ marks the transition from monotonic to non-monotonic fronts.

D. Regime III: Strong Adhesion ( $\mu < -2$ )

Pattern Formation: The homogeneous state $\rho=1$ becomes unstable.
Dynamics: The system forms spatially localized aggregates (spots) behind the invasion front.
Structure: The aggregates follow a Bessel function profile ( $J_0$ ) in 2D, with a characteristic length scale $\ell \sim 1/\sqrt{|\mu|}$ .
Significance: This proves that a single-species system can generate robust spatial patterns (spots) without a second morphogen, driven purely by the competition between short-range repulsion and long-range adhesion.

E. The Role of Density-Dependent Regulation (Model II)

Problem: In Model I, strong adhesion leads to fragmentation and instability.
Solution: In Model II, where adhesion vanishes near saturation ( $m(1)=0$ ), the instability is suppressed.
Result: Even with strong adhesion parameters, the tissue maintains cohesive, monotonic invasion. The front slows down but does not break into fingers or spots. This suggests that biological tissues regulate adhesion strength based on local density to ensure robust morphogenesis.

4. Significance

Theoretical Advancement: The paper provides a rigorous mathematical link between cell invasion and pattern formation, showing they emerge from a single mechanism (adhesion) rather than requiring distinct frameworks (like Turing's two-species model).
Biological Insight: It identifies adhesion strength and its density-dependent regulation as key determinants of tissue behavior.
- Unregulated strong adhesion $\rightarrow$ Cancer-like fingering and fragmentation.
- Regulated adhesion (saturating at high density) $\rightarrow$ Cohesive tissue expansion and wound healing.
Mechanistic Explanation: It offers a physical explanation for "fingering" in epithelial tissues and tumor invasion without invoking external chemical gradients or multiple cell types.
Analytical Tractability: By deriving a local approximation of non-local adhesion models, the authors enable the use of powerful asymptotic tools (traveling waves, stability analysis) that are difficult to apply to non-local integral equations.

Conclusion

Falcó et al. establish that the interplay between cell-cell adhesion, self-diffusion, and proliferation creates a spectrum of collective behaviors. The model predicts that while weak adhesion ensures smooth tissue expansion, unregulated strong adhesion leads to instability and pattern formation. Crucially, the introduction of density-dependent adhesion regulation acts as a biological "safety valve," suppressing these instabilities to restore cohesive tissue growth, providing a unified theory for understanding morphogenesis and disease progression.