Parameterization of the critically rigid manifolds of vertex models

Vertex models are effective in predicting mechanical response of biological tissues and designing bioinspired materials. Their rigidity transition – observed in disease and development of tissues – is well-studied, but configurations at this critical transition remain difficult to probe. Vertex models are under-constrained to first order, so their rigidity transitions are second order responses caused by a configuration’s geometry rather than its topology. Consequently, rigid configurations of these models can support arbitrarily scalable internal forces, termed ‘states of self-stress’ in simple networks. When these forces are scaled to approach zero, the resulting configurations form the critically rigid manifold. In spring networks, states of self stress can be used to generate a tractable parameterization for the complete ensemble of equilibrium configurations on this critical manifold¹. Here, we develop a new type of parameterization to describe more complex vertex models with different types of constraints on cell geometries. We then use this parametrization to search over the critically rigid manifold, allowing us to optimize tissue-scale properties via tuning of mesoscale parameters. This allows us to generate vertex model configurations that are simultaneously tuned at the edge of a rigidity transition and exhibit additional desired behaviors, serving as a potential starting point for design in complex materials.   

¹Hain et al. PRE 2025