Theory of active particle penetration through a planar elastic membrane

With the rapid advent of biomedical and biotechnological innovations, a deep understanding of the nature of interaction between nanomaterials and cell membranes, tissues, and organs, has become increasingly important. Active penetration of nanoparticles through cell membranes is a fascinating phenomenon that may have important implications in various biomedical and clinical applications. Using a fully analytical theory supplemented by particle-based computer simulations, the penetration process of an active particle through a planar two-dimensional elastic membrane is studied. The membrane is modeled as a self-assembled sheet of particles, uniformly arranged on a square lattice. A coarse-grained model is introduced to describe the mutual interactions between the membrane particles. The active penetrating particle is assumed to interact sterically with the membrane particles. State diagrams are presented to fully characterize the system behavior as functions of the relevant control parameters governing the transition between different dynamical states. Three distinct scenarios are identified. These compromise trapping of the active particle, penetration through the membrane with subsequent self-healing, in addition to penetration with permanent disruption of the membrane. The latter scenario may be accompanied by a partial fragmentation of the membrane into bunches of isolated or clustered particles and creation of a hole of a size exceeding the interaction range of the membrane components. It is further demonstrated that the capability of penetration is strongly influenced by the size of the approaching particle relative to that of the membrane particles. Accordingly, active particles with larger size are more likely to remain trapped at the membrane for the same propulsion speed. Such behavior is in line with experimental observations. Our analytical theory is based on a combination of a perturbative expansion technique and a discrete-to-continuum formulation. It well describes the system behavior in the small-deformation regime. Particularly, the theory allows to determine the membrane displacement of the particles in the trapping state. Our approach might be helpful for the prediction of the transition threshold between the trapping and penetration in real-space experiments involving motile swimming bacteria or artificial active particles.