Parametrizing the Critically Rigid Manifold of the Vertex Model for Biological Tissues

Vertex models are quantitatively predictive of mechanical responses of tissue development and disease processes, including solid-fluid transitions that facilitate large-scale tissue flow. Also, bioinspired materials may be designed with similar fluidity tunability. We want to parametrize the set of all configurations at the rigidity transition. In under-constrained networks like the vertex model, rigidity transitions occur as a second order response, caused by a system's geometry rather than connectivity. As a result, all rigid states of the network can support a force of arbitrary magnitude, which approaches zero exactly on the critically rigid manifold, while the state remains in equilibrium. In simple networks these forces are called a "state of self stress." To generalize self stresses to vertex models, we identify a set of tractable parametrizations of all equilibrium states that can support such forces. We find a subset of critically rigid states within this ensemble via one such parametrization. We demonstrate that this parametrization lets us search rigid configurations of vertex models, allowing us to describe emergent mechanics of this ensemble of states and optimize for some desired properties. This parametrization will likely be useful in describing mechanical networks with other constraints, like torsional stiffness.