  In computer simulations of dry foams and of epithelial tissues, vertex models
are often used to describe the shape and motion of individual cells. Although
these models have been widely adopted, relatively little is known about their
basic theoretical properties. For example, while fourfold vertices in real
foams are always unstable, it remains unclear whether a simplified vertex model
description has the same behavior. Here, we study vertex stability and the
dynamics of T1 topological transitions in vertex models. We show that, when all
edges have the same tension, stationary fourfold vertices in these models do
indeed always break up. In contrast, when tensions are allowed to depend on
edge orientation, fourfold vertices can become stable, as is observed in some
biological systems. More generally, our formulation of vertex stability leads
to an improved treatment of T1 transitions in simulations and paves the way for
studies of more biologically realistic models that couple topological
transitions to the dynamics of regulatory proteins.
