Finite elasticity and interplay between curvature and rigidity in vertex models

Using a mean field approach, we study the finite mechanical response of the vertex model (VM) of biological tissue to compression and dilation and compare our analytical results to simulations. The VM is known to exhibit a transition between rigid and fluid-like or floppy states driven by geometric incompatibility: perimeter and area tension set a target shape, which may or may not be geometrically achievable and thereby engender frustration. We find that near the transition region the response to finite compression and dilation is asymmetric, with dilation yielding a higher bulk modulus. In the linear response regime where strains are arbitrarily small, the asymmetry only occurs at the transition point between solid and floppy states. The asymmetry can be understood as follows. Under compression, an initially solid VM can completely relax perimeter tension, and thereby reduce bulk and shear moduli. Conversely, an initially floppy VM can rigidify under dilation, thus increasing its bulk and shear moduli. These observations imply that re-scaling of cell area shifts the transition point between rigid and liquid states. Based on this insight, we calculate the re-scaling of cell area engendered by intrinsic curvature - namely local expansion (flat to saddle-like), and local compression (flat to spherical) - and obtain an analytical prediction for the rigidity transition in the presence of curvature. The predicted shift of the transition due to curvature is compared to simulation data from Ref. 1 and we find good agreement. Our analytical prediction of the rigidity transition for the VM on curved surfaces provides a new metric for predicting tissue rigidity from image data for curved tissues in a manner analogous to the flat case.