Anomalous diffusion and effective shear modulus in a semisolid membrane

From the perspective of physical properties, the cell membrane is an exotic two-dimensional material that has a dual nature: it exhibits characteristics of fluids, i.e., lipid molecules show lateral diffusion, while also demonstrating properties of solids, evidenced by a nonzero shear modulus. We construct a model for such a semisolid membrane. Our model is a fluctuating randomly triangulated mesh with two different kinds of nodes. The solid nodes never change their neighbors, while the fluid nodes do. As the area fraction occupied by the solid nodes (Φ) is increased the motion of fluid nodes transitions from diffusion to localization via subdiffusion. Next, the solid nodes are pinned to mimic the pinning of the plasma membrane to the cytoskeleton. For the pinned membrane, there exists a range of Φ over which the model has both a nonzero shear modulus and a nonzero lateral diffusivity. The bending modulus, measured through the spectrum of height fluctuations, remains unchanged.