Recent experiments have pointed out that cellular uptake is strongly dependent on the physical dimensions of endocytosed nanoparticles and tile optimal radius of endocytosed virus-like particle coated by transferrin is around 50 nm. As the same time, the dimensions of receptor-ligand complex have strong effects on the size-dependent exclusion of proteins ill cell environments. Inspired by these experimental results, a continuum elastic model is constructed to resolve the relationship between the dinlensions of receptor-ligand complex and the optimal radius of endocytosed virus-like particle. These results demonstrate that the optimal radius of endocytosed virus-like particle depends on the dimensions of reeeptor-ligand complex and the dimension of receptor-ligand complex reduces the depletion zone.
We present an extended analytical model including the depletion effect and the dimension of ligand-receptor complex, aiming to elucidate their influences on endocytosis of spherocylindrical nanoparticles （NPs）. It is found that the dimension of ligand-receptor complex （δ） and the depletion effect interrelatedly govern the optimal conditions of NP endocytosis. The endocytosis phase diagram constructed in the space of NP radius and relative aspect ratio indicates that the endocytosis of NP is enhanced evidently by reducing the optimal radius and the threshold radius of endocytosed NP. Meanwhile, through thermodynamic and kinetic analysis of the diffusion of receptors, the dependence of diffusion length on depletion effect and the dimension of ligand-receptor complex can be identified in great detail. For small aspect ratio, diffusion length decreases with increasing concentration c of small bioparticles in cellular environment. Endocytosis speed corresponding to large radius R and high concentration c of small bioparticles strongly depends on the increasing （2r-δ）. These results may show some highlights into the conscious design of NPs for diagnostic agents and therapeutic drug delivery applications.
Recent simulations have demonstrated that bioparticle size and shape modulate the process of endocytosis, and studies have provided more quantitative information that the endocytosis efficiency of spherocylindrical bioparticles is decided by its aspect ratio. At the same time, the dimensions of the receptor-ligand complex have strong effects on the size-dependent exclusion of proteins within the cellular environment. However, these earlier theoretical works including simulations did not consider the effects of ligand-receptor complex dimension on the endocytosis process. Thus, it is necessary to resolve the effects of ligand-receptor complex dimension and determine the optimal aspect ratio of spherocylindrical bioparticles in the process of endocytosis. Accordingly, we proposed a continuum elastic model, of which the results indicate that the aspect ratio depends on the ligand-receptor complex dimension and the radius of the spherocylindrical bioparticle. This model provides a phase diagram of the aspect ratio of endocytosed spherocylindrical bioparticles, the larger aspect ratio of which appears in the phase diagram with increasing ligand density, and highlights the bioparticle design.
