Modeling the Solid-Fluid Transition in Ordered Biological Tissues

Biological functionality of tissues relies on its rheology. Fluidity of a 2D non-proliferating confluent tissue is contingent on cellular rearrangements which are called T1 transitions. In a 2D vertex model for disordered tissues, the tissue fluidizes when the T1 energy barriers disappear as the target shape index approaches a critical value (~3.81) [Bi 2015]. The linear response also becomes fluidlike (i.e. the shear modulus vanishes) at this same value. However, shear modulus of ordered ground states of 2D vertex models vanishes at a lower value (3.72) [Farhadifar 2007, Staple 2010]. Therefore, an interesting open question is whether the ground states of the 2D vertex model are fluid-like or solid-like between 3.72 and 3.81. In other words, 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 vanish only at ~3.81, indicating that ordered systems have the same critical point as disordered systems. We also develop a simple geometric argument that predicts the correct scaling of energy barriers with T1 edge length in these systems.