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

Abstract: The rheology of biological tissues is important for their function, and we would like to better understand how single cells control global tissue properties such as tissue fluidity. A confluent tissue can fluidize when cells diffuse by executing a series of cell rearrangements, or T1 transitions. In a disordered 2D vertex model, the tissue fluidizes when the T1 energy barriers disappear as the target shape index approaches a critical value ($s^*_{0} \sim 3.81$), and the shear modulus describing the linear response also vanishes at this same critical point. However, the ordered ground states of 2D vertex models 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 model are fluid-like or solid-like between 3.72 and 3.81 $-$ does the "equation of state" for these systems have two branches, like glassy particulate matter, or only one? Using four-cell and many-cell numerical simulations, we demonstrate that for a hexagonal ground state, T1 energy barriers only vanish at $\sim 3.81$, indicating that ordered systems have the same critical point as disordered systems. We also develop a simple geometric argument that correctly predicts how non-linear stabilization disappears at $s^*_{0}$ in ordered systems.