Kinetics of cellular uptake of viruses and nanoparticles via clathrin-mediated endocytosis

Abstract

Several viruses exploit clathrin-mediated endocytosis to gain entry into host cells. This process is also used extensively in biomedical applications to deliver nanoparticles (NPs) to diseased cells. The internalization of these nano-objects is controlled by the assembly of a clathrin-containing protein coat on the cytoplasmic side of the plasma membrane, which drives the invagination of the membrane and the formation of a cargo-containing endocytic vesicle. Current theoretical models of receptor-mediated endocytosis of viruses and NPs do not explicitly take coat assembly into consideration. In this paper we study cellular uptake of viruses and NPs with a focus on coat assembly. We characterize the internalization process by the mean time between the binding of a particle to the membrane and its entry into the cell. Using a coarse-grained model which maps the stochastic dynamics of coat formation onto a one-dimensional random walk, we derive an analytical formula for this quantity. A study of the dependence of the mean internalization time on NP size shows that there is an upper bound above which this time becomes extremely large, and an optimal size at which it attains a minimum. Our estimates of these sizes compare well with experimental data. We also study the sensitivity of the obtained results on coat parameters to identify factors which significantly affect the internalization kinetics.

Figure 1.

Schematic diagram of NP internalization via clathrin-mediated endocytosis. A NP first binds to a specific cell surface receptor, forming a NP–receptor complex. The complex binds the coat proteins, and clathrin-coated pit (CCP) assembly begins. The CCP either grows to form a vesicle, in which case the NP is internalized, or grows only up to a certain size and subsequently disassembles. In such an event the NP fails to be internalized and the complex becomes free of coat proteins.

Figure 2.

(a) Coarse-grained model for the assembly of a clathrin-coated pit (CCP). In a real CCP the protein coat contains clathrin, adaptors and several other accessory proteins. In our model of a CCP the coat is made of identical monomeric units. The shape of the CCP is assumed to be a spherical cap, and the average area of a monomer is chosen to be the same as that occupied by a clathrin triskelion in a real CCP. (b) Kinetic scheme of the pit assembly shown in (a). Symbols n and N refer to the number of monomers in a pit and a complete vesicle, respectively. The rate constants α_n and β_n characterize the growth and decay rates of a pit of size n. k₀ is the rate at which the first monomer binds to the NP–receptor complex, and k_N is the rate of scission of a vesicle from the membrane.

Figure 3.

Relation between NP and vesicle diameters. The vesicle diameter, d_V, and the NP diameter, d_NP, are related by d_V = d_NP + 2l_b, where l_b is the length of a receptor–ligand bond.

Figure 4.

Average energy per monomer, E(N), equation (13), (solid curve) as a function of NP size. E(N) is the sum of the membrane bending energy per monomer (dashed curve) and the energy due to coat formation per monomer (dashed–dotted curve). Dashed vertical lines atd_min ≈ 46 nm and d_max ≈ 105 nm correspond to the NP sizes at which E (N) = 0. The arrow at $d_{\text{E}}^{*}\approx 68$ nm indicates the size of the NP whose carrier vesicle is energetically most stable.

Figure 5.

The probability of successful CCP assembly around a NP–receptor complex, P_s, as a function of the NP size. P_s is high in the region where E < 0, and pit assembly is energetically favorable. The maximum value of P_s is sensitive to the parameter β₁, the values of which in s⁻¹ are given near the curves. The dashed vertical lines are the same as in figure 4.

Figure 6.

Mean time for successful CCP assembly, τ_s (solid curve) and the mean waiting time, τ_w, (curves with symbols) for different values of β₁, as functions of the NP size, calculated using equations (3)–(6). In contrast to the time τ_s (only the curve for β₁ = 5s⁻¹ shown), the time τ_w changes significantly with β₁. The dashed vertical lines are the same as in figure 4. The arrow indicates the NP diameter d_s, at which τ_s is minimum.

Figure 7.

The mean internalization time, τ, given in equation (2), as a function of the NP size, for different values of β₁. The arrows point to the NP diameters corresponding to the shortest values of the mean internalization time, which we refer to as d_opt. The value of d_opt is weakly sensitive to changes in β₁, but the value of τ at d_opt varies significantly. The dashed vertical lines are the same as in figure 4.

Figure 8.

Plots of d_min, d_max (equation (16)), d_opt, τ (d_opt), and τ_s (d_opt) (calculated numerically), as functions of κ_p—the bending rigidity of the protein coat ((a) and (b)), ∈_b—effective monomer binding energy ((c) and (d), and N_p—the natural number of monomers in the coat ((e) and (f)). Among the characteristic sizes only the maximum NP size d_max shows significant variation. The time τ (d_opt) is the sum of τ_s (d_opt) and τ_w (d_opt). The weak dependence of τ_s (d_opt) on coat parameters shows that the increase in τ (d_opt) is mainly due to the increase in τ_w (d_opt). The arrows show the parameter values used in our calculations.

Figure 9.

Spherical cap model of a pit.