Shape-driven confluent rigidity transition in curved biological tissues.

Collective cell motions underlie structure formation during embryonic development. Tissues exhibit emergent multicellular characteristics such as jamming, rigidity transitions, and glassy dynamics, but there remain questions about how those tissue-scale dynamics derive from local cell-level properties. Specifically, there has been little consideration of the interplay between local tissue geometry and cellular properties influencing larger-scale tissue behaviors. Here, we consider a simple two-dimensional computational vertex model for confluent tissue monolayers, which exhibits a rigidity phase transition controlled by the shape index (ratio of perimeter to square root area) of cells, on surfaces of constant curvature. We show that the critical point for the rigidity transition is a function of curvature such that positively curved systems are likely to be in a less rigid, more fluid, phase. Likewise, negatively curved systems (saddles) are likely to be in a more rigid, less fluid, phase. A phase diagram we generate for the curvature and shape index constitutes a testable prediction from the model. The curvature dependence is interesting because it suggests a natural explanation for more dynamic tissue remodeling and facile growth in regions of higher surface curvature. Conversely, we would predict stability at the base of saddle-shaped budding structures without invoking the need for biochemical or other physical differences. This concept has potential ramifications for our understanding of morphogenesis of budding and branching structures.