Universal Geometry Controls the Mechanics of 2D and 3D Models for Dense Biological Tissues

Understanding how mechanical tissue properties emerge from cellular behavior is vital for understanding the mechanisms that guide embryonic development, cancer growth, and wound healing. To study universality of mechanical properties in dense tissues, we turn to vertex models that represent biological tissues as disordered polygonal networks (polyhedral networks in 3D). Recently, a new type of rigidity transition was discovered in a family of vertex models for 2D and 3D biological tissues, controlled by a minimal average surface (perimeter in 2D) of dense disordered cellular packings. Here, we show that not only the onset of rigidity, but also the properties of vertex models away from the transition point can be understood based on the behavior of this minimal surface. In particular, we show that universal relations exist between minimal average cell surface and the fluctuations in cell surface and volume, and these relations exactly predict the behavior of both shear and bulk moduli. Our work demonstrates how universal geometrical properties of a disordered material precisely control its mechanical behavior.