A sheet of paper tears where tension exceeds strength. Epithelium, the cell sheet lining our organs, behaves similarly, but until recently models could not show how tears appear in it. The classical vertex model describes tissue as a network of polygons tightly packed together and forbids cells from detaching. Japanese researchers added one operation to this model that allows tissue to tear, and for the first time traced how exactly this happens.
The work is based on the idea that cells are connected at nodes, called vertices, where several contacts meet. In everyday life, cells can exchange neighbors, described by the T1 transition: like partners in a dance, they swap places without breaking the circle. But when tension at a vertex becomes too high, the connection should snap. The authors introduced the reverse operation T2-inverse: when the threshold is exceeded, the vertex splits into three, and a closed polygon forms between cells, a pseudocell that does not contribute to elasticity but represents a gap, that is, external space.
They simulated stretching a sheet of regular hexagons at constant speed, varying the speed and the presence of an initial crack. Without T2-inverse, the sheet remained intact when stretched up to 50% at both speeds. With the new operation, slow stretching (speed 0.1) did not produce tears, but fast stretching (0.5) created local tension peaks that initiated T2-inverse. This led to the formation of gaps and tissue delamination. If the sheet had a pre-existing crack, even slow stretching concentrated tension at its tip, triggering T2-inverse and natural crack propagation along a tortuous path. Without the new operation, failure proceeded along an unnaturally straight line due to T1 transitions.
The result brings us back to the sheet of paper: now we can see exactly where it will tear if tension exceeds strength. The model can be applied to study mechanical failure of epithelium, for example in wound healing or tissue engineering, where predicting the strength of cell sheets is important. The next step is to calibrate the tension threshold against experimental data and extend the model to three-dimensional tissues.
The tension threshold for triggering T2-inverse in the model is 10.0, which is 10 times the linear tension of cell contacts. This means that a very strong local stress is required for rupture.