On the interplay between two rigidity transitions in disordered semiflexible polymer-tissue networks*

Low-connectivity semiflexible polymer networks undergo a rigidity transition from floppy to rigid as shear strain is increased beyond some finite value. Collections of cells with no gaps between them (confluent tissues) also undergo a rigidity transition from fluid to solid, as a parameter characterizing the shape of an individual cell is varied. Both transitions are driven by geometric constraints, though in confluent tissue, topology also plays an important role. We present numerical studies of a vertex model of a two-dimensional confluent tissue embedded in a two-dimensional semiflexible polymer network, with mechanosensitive coupling between the two systems. We study how a rigidity transition in one system may or may not induce rigidity in the other, depending on the nature of the coupling. In addition to going beyond studying the onset of rigidity in a single network construct, the semiflexible polymer-tissue network is relevant to cancer in which cancerous tissue (a tumor) is surrounded by and interacts with an extracellular matrix, consisting primarily of cross-linked collagen, a semiflexible polymer.