The Mechanical Magic of Cell Adhesion: Why Tissues "Stick Together" or "Break Apart"

*A new theoretical framework from Oxford's mathematical biology group: one equation unifies two seemingly opposite phenomena in development, regeneration, and cancer.*

1. A Puzzling Observation

Imagine a patch of cells growing outward. During normal embryonic development, these cells might form a finger, a segment of neural tube, or a smooth layer of skin—their edge tidy, like a well-trained army advancing in formation.

But under some circumstances, the same cell population looks very different: instead of a smooth front, it extends ragged "fingers," like oil spreading on frosted glass or ink branching as it drops into water. Observed under the microscope, this is called the fingering instability—a classic hallmark of cancer cells invading surrounding tissue.

The question is: why do the same cells sometimes form beautiful patterns and sometimes become dangerous invaders?

For decades, biologists studied these two phenomena—orderly morphogenesis and chaotic tissue invasion—as separate problems: the former in developmental biology, the latter in oncology. But the recent work by Falcó, Johnson, Dalwadi, and Philip Maini at Oxford's Wolfson Centre for Mathematical Biology reveals a surprising fact: they may be two sides of the same coin.

2. Adhesion: The "Glue" Between Cells

To understand the problem, we must return to the most basic social property of cells: adhesion.

Every cell's surface is covered with adhesion molecules, the most famous being E-cadherin. Think of them as tiny "hands" a cell extends; when two cells come close, these hands grip each other and hold the cells together. Adhesion strength is not fixed—during development, cells tune the expression of adhesion molecules; in cancer, adhesion is often weakened (the so-called "E-cadherin switch"), making it easier for cells to break away and metastasize.

But what role does adhesion play physically?

In 1963, biologist Malcolm Steinberg performed a famous experiment: he dissociated tissue cells from different embryos into single cells and mixed them together. Remarkably, the cells spontaneously reorganized, eventually forming structures similar to the originals. Steinberg proposed the Differential Adhesion Hypothesis (DAH): cells behave like different kinds of liquids—cells with stronger adhesion tend to cluster together, pushing weaker-adhesion cells to the periphery. Brodland and others later developed this into the more refined Differential Interfacial Tension Hypothesis (DITH), drawing an analogy between tissue rearrangement and fluid interfacial tension.

But Steinberg's experiment involved only static tissue rearrangement. What about collective behavior during growth and movement? That is the question Falcó et al. set out to answer.

3. The Story of One Equation

The standard way to handle such problems is to write a partial differential equation (PDE). It may sound abstract to describe cells with equations, but it is as natural as using the Navier–Stokes equations for water flowing in a pipe—only here the "fluid" is made of living cells.

The Falcó et al. model contains three key physical processes:

First, proliferation. Cells divide, producing new cells. This is the tissue's "thrust," like a pump continuously inflating a balloon.

Second, self-diffusion. Cells do not stay rigidly in place; they make small random movements. Interestingly, this diffusion is not ordinary Fickian diffusion—it is modulated by cell density and adhesion. When cells adhere tightly, their collective movement becomes harder, and the diffusion coefficient changes.

Third, adhesion. This is the heart of the model. Adhesion between cells produces a "pulling-back" force—if you try to pull a cell out of the group, adhesive forces drag it back. Mathematically, this force is modeled as a non-local term: the force each cell feels comes from the collective action of other cells within a surrounding neighborhood.

Schematically, the equation reads:

It looks simple, but when you actually solve the equation, interesting things happen.

4. Weak Adhesion: The Steady "Infantry Square"

Suppose adhesion is weak—cells only lightly hold hands. Numerical simulations then show that the tissue front forms a stable travelling wave solution: it advances like a flat formation. The cell density profile is a smooth monotone curve, rising from zero at the front to a saturated density behind.

Mathematically, this is a monotone travelling wave. It is stable because weak adhesion cannot create disturbances at the interface. Any small bump gets smoothed out by proliferation and diffusion—like a well-trained army where anyone drifting slightly off formation is quickly pulled back by the group's rhythm.

This corresponds to many normal physiological processes: epithelial cell migration during wound healing, intestinal villus growth, even certain types of benign tumour expansion. The tissue keeps its integrity and continuity, with no fragments shedding off.

5. Strong Adhesion: From "Square Formation" to "Fingered Invasion"

Now turn the adhesion up.

Something wonderful happens: when adhesion exceeds a critical threshold, the previously stable flat front suddenly becomes unstable. Any tiny perturbation along the front—maybe just one or two extra cells in one spot—is no longer smoothed out but amplified. Bumps appear and keep growing forward, eventually forming finger-like structures extending out of the main tissue.

It is like a group of people holding hands too tightly. Imagine climbers roped together: if the rope is too slack, everyone walks their own way; if it is just right, the team advances in formation; but if the rope is pulled too tight, one person tripping on a rock sends violent waves down the line, and the team may break into pieces.

In two dimensions, these finger structures evoke the morphology seen when cancer cells invade—tumour cells extending "fingers" from the primary tumour into surrounding healthy tissue. In 2005, Cristini and colleagues' pioneering work first showed mathematically that tumour invasion can be viewed as a morphological instability, driven by the competition between proliferation (a shape-destabilizing force) and adhesion (a shape-stabilizing force). Falcó et al. now place this picture in a more universal theoretical framework: it is not merely a feature of cancer, but a generic phenomenon in any adhesion-driven system.

6. The Life-Saving Regulator: Density-Dependent Adhesion

If strong adhesion causes tissue fragmentation and dangerous fingering invasion, how do organisms solve this problem?

Nature's answer is: regulation. Falcó et al. introduce a key mechanism—density-dependent adhesion modulation. Simply put, when cell density is high (inside the tissue), adhesion can be reduced; at lower density (near the front), adhesion stays at normal levels.

The effect is dramatic: simulations show that even with strong basal adhesion, appropriate density-dependent regulation completely suppresses the front instability, restoring tidy, coherent tissue expansion.

What biological reality does this correspond to? During normal development, cells do dynamically regulate adhesion molecule expression. For example, in collective migration of neural crest cells, the "leader cells" at the front and the "follower cells" behind have different adhesive properties. In wound healing, cells adjust their connection strengths according to local density and environmental cues. These regulatory mechanisms ensure tissues remain intact and functional while growing and moving.

In cancer, this regulation often fails. Downregulation of E-cadherin and mutations in adhesion signaling pathways rob cells of this fine balance—the result is disorderly invasion and metastasis.

7. Why Does This Matter?

The significance of this work goes far beyond "explaining known phenomena with elegant equations."

First, it provides a unified framework. Previously, developmental biologists had one set of theories for morphogenesis, and oncologists had another for cancer invasion. Falcó et al. show that both can be described by the same equation—the difference lies only in parameters (adhesion strength). Weak adhesion → orderly morphogenesis; strong adhesion + no regulation → chaotic invasion. It is like thermodynamics, where the same van der Waals equation describes both gas and liquid, depending on temperature and pressure.

Second, it points to potential therapeutic targets. If fingering invasion in cancer arises from adhesion-mediated instability, then restoring density-dependent adhesion regulation could be a treatment strategy. Studies have already shown that enhancing E-cadherin expression in certain cancers can suppress metastasis—consistent with the model's prediction that "regulating adhesion restores stability."

Third, it demonstrates the power of multiscale modelling. From individual adhesion molecules (nanometre scale), to mechanical interactions between cells (micrometre scale), to tissue-level morphology (millimetre scale)—Falcó et al. connect microscopic mechanisms to macroscopic phenomena through systematic multiscale analysis. This is one of the most exciting directions in 21st-century mathematical biology.

8. Epilogue: Equations and Life

Philip Maini, director of the Wolfson Centre for Mathematical Biology at Oxford and a Fellow of the Royal Society, is one of the most influential researchers in this field; his invited lecture at the 2010 International Congress of Mathematicians was on "aspects of tumour modelling." For decades, he and his team have been translating the mysteries of life into mathematical language—from neural crest migration to angiogenesis, from wound healing to tumour evolution.

This latest work continues that tradition. It reminds us that even the most complex phenomena of life may, at the right level of abstraction, follow simple and beautiful mathematical laws. Three terms in one equation—proliferation, diffusion, adhesion—are enough to encode whether a tissue will "grow together" or "scatter and invade."

Next time you watch cells advancing hand in hand under the microscope, consider: exactly how strong is their grip? Tighten it slightly, and could that neat square formation suddenly become stretching tentacles? Inside your body, every day, countless cells are using this precise mechanical balance to decide whether you are healing, developing, or fighting some uncontrolled growth.

The order and chaos of life may lie hidden in that invisible adhesive force.

---

Reference: Falcó C., Johnson S.W.S., Dalwadi M.P., Maini P.K. *Theory of adhesion-driven self-organisation in growing tissues*. arXiv:2604.26928 (2026).