Most matured animal viruses enter their host cells via receptor-mediated endocytosis,^[1] in which the specific binding of a viral protein to some receptor protein on the outer cell membrane triggers the internalization of the virus inside an endosome. Inspired by the high specificity and efficiency of the above process, extensive effort has been devoted to develop biomimetic nanoparticle (NP)-based therapeutics. The design of synthetic nanoparticles (NPs) for biomedical and clinical applications, such as drug delivery and biomedical imaging,^[2–5] requires a thorough understanding of the physical mechanisms of receptor-mediated endocytosis of the NPs. Since the endocytosis are driven by the NP-membrane adhesion while penalized by the cell membrane deformation,^[6] both NPs properties^[7] and cell surface mechanics, such as membrane curvature,^[8] play important roles in determining endocytosis of NPs. Experimental studies have demonstrated particle size and shape as important factors in cellular uptake of nanomaterials, and there is an optimal size about 25 nm corresponding to the minimal endocytic time.^[9,10] Several analytical models^[11–14] based on thermodynamics and dynamics also endeavor to pursue the underlying mechanism on the receptor-mediated endocytosis, and predict that the optimal size for endocytosis is around 25 nm–30 nm.

However, few existing theoretical models concerning the endocytosis of single NP ever considered the influences of the depletion interaction as well as the dimension of receptor-ligand complex. The depletion effect has been demonstrated experimentally^[15] that larger spheres among small colloidal particles inside lipid bilayer vesicles are to be pinned to the vesicle surface due to excluded-volume effect, which might be also significant factor in the cellular uptake of NPs that evidenced by experiments. Recent experiments have demonstrated that the dimension of receptor-ligand complex ranges from 10 nm to 42 nm and has strong effect on the size-dependent exclusion of proteins.^[16] Size differences of membrane proteins can drastically alter their organization at membrane interfaces formed at cell-cell junctions, with as little as a 5nm increase in non-binding protein size driving its exclusion from the interface.^[17] At the same time, a preliminary work including depletion effect, dimension of ligand-receptor complex has pointed out that the optimal radius of endocytosed NPs depends on the dimension of ligand-receptor complex^[18] and a simulation work also demonstrated the effect of receptor length.^[19] All of which highlight necessity that the depletion effect, dimension of ligand-receptor complex and their combined effect should be considered during the endocytosis process of NPs and detected further.

In this article, an extended analytical model including the depletion effect and the dimension of ligand-receptor complex is presented, aiming to elucidate their influences on endocytosis of spherocylindrical NPs. Based on thermodynamic analysis and kinetic diffusion of receptors, we revealed that the depletion effect and the dimension of ligand-receptor complex interrelatedly govern the optimal conditions of NPs endocytosis. Furthermore, through thermodynamic and kinetic analysis of the diffusion of receptors, the endocytosis phase diagram in the space of NP radius and relative aspect ratio were constructed. The results may provide a guidance to the design of NPs for diagnostic agents and therapeutic drug delivery applications.

The engulfment of NP endocytosed by a cell is a complex process. Cell membrane deformation, including membrane bending and stretching at the wrapping area, the elastic energy of cell membrane including cytoskeleton, the diffusion of receptor towards the wrapping area from the remote area on the cell, and depletion effect determined by the dimension of ligand-receptor complex and radius of small particles in cellular environment are involved in this process. While the clathrin protein is restricted, the endocytosis can also be realized by ligand-receptor interaction,^[20] so effect of clathrin protein was not included in current work.

2.1. Bending, stretching energy, and elastic energy of cell membrane

The bending and stretching energy of the cell membrane treated as an elastic sheet can be given by the Canham-Helfrich energy,^[21] as shown in Fig. 1, 1 where c₁ and c₂ are two principal curvatures of the bound membrane surface, c₀ is the spontaneous curvature (during the endocytosis, the NP is much smaller than the cell, so that the topology state of cell does not change, c₀ is considered to be zero in the following sections), κ is the bending modulus of the membrane and λ is the tensile stress on the membrane caused by the endocytosis (in our calculations, the parameters , , where is the Boltzmann constant and T is the absolute temperature). As shown by Fig. 1, the NP is sketched as a cylindrical rod with hemispherical caps. While the dimension of ligand-receptor complex (δ) was considered, the bending energy caused by a full wrapping can be expressed as 2 Where and denote the surface of cylindrical rod and hemispherical caps of the spherocylindrical particle. So that the bending and stretching energy per area can be defined as 3 where m is aspect ratio defined as , L and R correspond to the length of cylindrical part and the radius of the hemispherical part, respectively. A₀ is the cross-sectional area of one receptor which will be taken as the unit area hereafter (in our calculations, the parameter A₀ = 15 nm × 15 nm).

Fig. 1.

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	Fig. 1. (color online) Receptor-mediated endocytosis of NP. , R, h, and δ represent diffusion length, radius of NP, engulfment of NP, and the dimension of ligand-receptor complex, respectively.

Under the condition that both NP and cell membrane are uniform and isotropic, the interaction between NP and cell membrane can be modelled as the interaction between a sphere and biomembrane, a cylindrical rod with biomembrane. The elastic energy for the interaction between a sphere and cell membrane can be expressed as ,^[22] and its extended version for the interaction between a cylindrical rod with cell membrane .^[23] Once the dimension of ligand-receptor complex is considered, the total elastic energy can be written as 4 where h is the engulfment depth, is related to the Young’s modulus and Poisson ratio of spherocylindrical particle and cell membrane. σ_n and ε_n (n = 1,2) are the passion ratio and Young’s modulus, and 1 and 2 represent NP and cell membrane, respectively. For the Young’s modulus of the biomembrane is much less than that of virus-like particle, which leads to So that D is only determined by σ₂ and ε₂, and while the typical experimental data is assigned to σ₂ and ε₂, σ₂ is taken to be 0.5 and ε₂ is on the order of 10 kPa.

By analyzing the dependence of endocytosis speed ( ) on the radius of NPs, phase diagrams are constructed on R–m plane as shown by Fig. 2. Wherein phase diagram figure 2(a) is constructed under the conditions that the depletion effect, the dimension of ligand-receptor complex and membrane tension are not considered, namely, the parameters c, δ, and λ are set to be zero. The solid dark line and white line represent the theoretical optimal NP size and the threshold NP size for endocytosis, respectively. The optimal radius for spherical NP with aspect ratio m = 0 is 25 nm, which is well consistent with previous experimental and theoretical results^[9,11,28] that the NPs with radius about 25 nm have a maximal endocytosis rate. Effect of NP size on endocytosis speed can also be identified from the phase diagram, the color bar indicates the values of endocytosis speed , the high and low value of correspond to the fast and slow endocytosis, respectively. It is impossible for the NPs to be endocytosed in the dark blue region.

Fig. 2.

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Fig. 2. (color online) The endocytosis phase diagram in the space of radius and aspect ratio m of NP based on Eq. (10). The color bar indicates the values of endocytosis speed , the endocytosis in the dark blue region is impossible. The solid dark line and white line represent the theoretical optimal NP size and the threshold NP size for endocytosis, respectively. (a) Depletion effect, the dimension of ligand-receptor complex and membrane tension are not considered, its corresponding parameters, c, δ, and λ, are set to be zero. (b) The corresponding parameters utilized in Fig. 2(b) are listed as , δ = 6 nm, and .

After the depletion effect, the dimension of ligand-receptor complex and membrane tension are considered, they make the phase diagram Fig. 2(b) different from the phase diagram Fig. 2(a). The optimal NP size and the threshold NP size for endocytosis as compared to the one in Fig. 2(a) scrolled down. As an example, the optimal radius for NP with aspect ratio m = 0 is reduced from ∼ 25 nm to ∼ 17 nm.

During its endocytosis process, the bending energy density is proportional to 1/R, and the local energy balance between adhesion energy and bending energy determine a optimal radius of NP, . α is the local adhesion energy density, which is dependent on the strength of ligand-receptor interaction and depletion effect. The introduction of depletion effect make the local adhesion energy density increased and leads to the endocytosis of NPs with small optimal radius, so that the dark blue region impossible for NPs to be endocytosed is also reduced evidently.

As demonstrated by the phase diagram Fig. 2(b), the dimension of ligand-receptor complex and depletion effect have obvious effect on the endocytosis speed. It is necessary to elucidate the effect of dimension of ligand-receptor complex and depletion effect on the endocytosis speed through energetic analysis.

Figure 3 indicates that the values of diffusion length as the function of aspect ratio m at fixed R (15 nm), c ( ), and r (10 nm) (a), and the function of R at fixed m (5), c ( ), and r (10 nm) (b), respectively, change with the ligand-receptor complex δ. The curves in Fig. 3(a) have similar characters that with increasing aspect ratio m, the value of drops sharply to a bottom and then increases quickly. whereas for the dimension of ligand-receptor complex , the value of increases gradually with increasing aspect ratio m. One can note that the sum of the fixed R and the dividing dimension of ligand-receptor complex δ between two features are equivalent to 25 nm (the optimal radius of spherical particle in Fig. 2(a) (m = 0)), which suggests that the ligand-receptor complex assists in the endocytosis of small NPs. As the same time, figure 3(b) provides direct evidences that the optimal radius of NP decreases gradually with increasing ligand-receptor complex δ.

Fig. 3.

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	Fig. 3. (color online) The values of diffusion length as the function of aspect ratio m (at fixed R (15 nm), c ( ), and r (10 nm)) (a), and the function of R (at fixed m (4), c ( ), and r (10 nm)) (b), respectively, change with the ligand-receptor complex δ.

Figure 4 demonstrates that the values of diffusion length as the function of aspect ratio m (at fixed R (15 nm), δ (2 nm), and r (10 nm)) (a), and the function of R (at fixed m (4), δ (6 nm) and r (10 nm)) (b), respectively, change with the concentration c of small bioparticles in cellular environment. For the summation of fixed R and the ligand-receptor complex δ less than 25nm, the diffusion length as the function of aspect ratio m in Fig. 4(a) shares similar features with the curves in Fig. 3(a), but the effect of the concentration c of small bioparticle in cellular environment on the diffusion length is dependent on the aspect ratio m. For small aspect ratio m, the minimum value of at different concentration c of small bioparticles is distinguishable and decreases gradually with increasing concentration c. Whereas for larger m, the value of overlaps together and increases gradually with increasing m. It is because the bending energy density decreases with increasing m (see Eq. (3)) and the adhesion energy including binding energy and depletion effect is always larger than the membrane deformation cost. Therefore, endocytosis of NPs under these conditions is always energetically possible with increasing m. Moreover, depletion effect is insufficient to endocytosis of NPs with larger aspect ratio,^[29] which suggests that the binding energy in the adhesion energy is the dominant over the depletion effect, so that the value of is nominally dependent on the depletion effect and overlaps together for larger m.

Fig. 4.

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	Fig. 4. (color online) The values of diffusion length as the function of aspect ratio m (at fixed R (15 nm), δ (2 nm), and r (10 nm)) (a), and the function of R (at fixed m (4), δ (6 nm), and r (10 nm)) (b), respectively, change with the concentration c of small bioparticles in cellular environment.

The curves in Fig. 4(b) demonstrate that the values of diffusion length as the function of R (at fixed m (4), δ (6 nm), and r (10 nm)) change with the concentration c of small bioparticles in cellular environment. As compared to the diffusion length corresponding to small R, the one corresponding to larger R is strongly dependent on the concentration c of small bioparticles in cellular environment and decreases with the increasing concentration c.

These phenomena can be elucidated by re-expressing Fig. 4(b) by phase diagram in the space of radius R and the concentration c as shown in Fig. 5. From which, some important information about depletion effect on endocytosis speed can be revealed. The key result is that the values of endocytosis speed ( ) corresponding to large radius R and high concentration c of small bioparticles take on evident change and strongly depend on the increasing corresponding to Figs. 5(a)–5(c). Increasing leads to enhanced depletion effect (see Eq. (5)), so that the endocytosis speed can be increased.

Fig. 5.

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	Fig. 5. (color online) The re-expression of Fig. 4(b) by the endocytosis phase diagram constructed in the space of radius and the concentration c of small particle in cell environment. ( ) corresponding to panels (a)–(c) increases gradually (10 nm (a), 12 nm (b), and 14 nm (c)). The color bar indicates the values of endocytosis speed ( ), the endocytosis in the dark blue region is impossible.

Nps designed for diagnostic agents and drug delivery is covered by protein corona as ligand, binding with receptor on cell membrane as a complex. It has been demonstrated that the dimension of ligand-receptor complex and the depletion effect have evident effect on NP endocytosis. The extended analytical model including the depletion effect and the dimension of ligand-receptor complex is utilized to elucidate their influences on endocytosis of spherocylindrical NPs. The dimension of ligand-receptor complex and the depletion effect interrelatedly govern the optimal conditions of NP endocytosis. As demonstrated by the endocytosis phase diagram constructed in the space of NP radius and relative aspect ratio, 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. The values of endocytosis speed corresponding to large radius R and high concentration c of small bioparticles strongly depend on the increasing . These results highlight the importance of the dimension of ligand-receptor complex and the size of small particle in cellular environment, on which more attention should be paid during the process of designing the NPs for diagnostic agents and therapeutic drug delivery applications.