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The rheology of biological tissues is important for their function, and wewould like to better understand how single cells control global tissueproperties such as tissue fluidity. A confluent tissue can fluidize when cellsdiffuse by executing a series of cell rearrangements, or T1 transitions. In adisordered 2D vertex model, the tissue fluidizes when the T1 energy barriersdisappear as the target shape index approaches a critical value ($s^*_{0} \sim3.81$), and the shear modulus describing the linear response also vanishes atthis same critical point. However, the ordered ground states of 2D vertexmodels become linearly unstable at a lower value of the target shape index(3.72) [1,2]. We investigate whether the ground states of the 2D vertex modelare fluid-like or solid-like between 3.72 and 3.81 $-$ does the "equation ofstate" for these systems have two branches, like glassy particulate matter, oronly one? Using four-cell and many-cell numerical simulations, we demonstratethat for a hexagonal ground state, T1 energy barriers only vanish at $\sim3.81$, indicating that ordered systems have the same critical point asdisordered systems. We also develop a simple geometric argument that correctlypredicts how non-linear stabilization disappears at $s^*_{0}$ in orderedsystems.

Nonlinear analysis of the fluid-solid transition in a model for ordered biological tissues