A thermodynamic model to predict the minimum energy required for engulfment of linearly aggregated spherical nanoparticles by tubular vesicles

Understanding the interaction of nanoparticles (NPs) with fluid membranes is important to achieve safe and improved performance of NPs in various biomedical applications. Studies show that fluid membranes can drive linear attraction between spherical NPs adsorbed on it, and forms tubular vesicles (membrane tubes) to engulf (wrap) multiple particles. The present work shows the theoretical investigation of the minimum total energy for wrapping of linearly aggregated multiple spherical NPs by a low-tense membrane tube by developing a thermodynamic model that accounts for membrane bending energy, adhesion energy between particles and the membrane, and surface tension energies of the membrane in the low tension regime. The model illustrates that the minimum total energy (negative) decreases with an increasing number of particles undergoing wrapping because of the increasing contact area between the membrane and the particles; resulting in a large adhesion energy gain compensating for the deformation energy costs of the membrane. The large adhesion energy gain results in the spontaneous wrapping and high stability of multiple spherical particles filled in a membrane tube. In addition, the model predicts the critical and the minimum adhesion strength of particles required for the initiation and completion of the wrapping process, respectively. Furthermore, the model demonstrates the wrapped states (non-wrapped, partially wrapped, deeply wrapped, and completely wrapped state) of multiple particles in a membrane tube as a function of the wrapped area of particles. Thus, the developed thermodynamic model provides insights into the understanding of how the spontaneous and simultaneous wrapping of multiple spherical NPs takes place in a low-tense membrane tube.