Recent experiments indicate that graphene microsheets can either undergo a near-orthogonal cutting or a parallel attachment mode of interaction with cell membranes. Here we perform theoretical analysis to characterize the deformed membrane structure and investigate how these two interaction modes are influenced by the splay, tilt, compression, tension, bending and adhesion energies of the membrane. Our analysis indicates that, driven by the membrane splay and tension energies, a two-dimensional microsheet such as graphene would adopt a near-perpendicular configuration with respect to the membrane in the transmembrane penetration mode; whereas the membrane bending and tension energies would lead to parallel attachment in the absence of cross membrane penetration. These interaction modes may have broad implications in applications involving drug delivery, cell encapsulation and protection, and measurement of dynamic cell response.