Size and shape effects on diffusion and absorption of colloidal particles near a partially absorbing sphere: Implications for uptake of nanoparticles in animal cells

Wendong Shi, Jizeng Wang, Xiaojun Fan, and Huajian Gao

Phys. Rev. E 78, 061914 – Published 16 December 2008

Abstract

A mechanics model describing how a cell membrane with diffusive mobile receptors wraps around a ligand-coated cylindrical or spherical particle has been recently developed to model the role of particle size in receptor-mediated endocytosis. The results show that particles in the size range of tens to hundreds of nanometers can enter cells even in the absence of clathrin or caveolin coats. Here we report further progress on modeling the effects of size and shape in diffusion, interaction, and absorption of finite-sized colloidal particles near a partially absorbing sphere. Our analysis indicates that, from the diffusion and interaction point of view, there exists an optimal hydrodynamic size of particles, typically in the nanometer regime, for the maximum rate of particle absorption. Such optimal size arises as a result of balance between the diffusion constant of the particles and the interaction energy between the particles and the absorbing sphere relative to the thermal energy. Particles with a smaller hydrodynamic radius have larger diffusion constant but weaker interaction with the sphere while larger particles have smaller diffusion constant but stronger interaction with the sphere. Since the hydrodynamic radius is also determined by the particle shape, an optimal hydrodynamic radius implies an optimal size as well as an optimal aspect ratio for a nonspherical particle. These results show broad agreement with experimental observations and may have general implications on interaction between nanoparticles and animal cells.

Received 9 April 2008

DOI:https://doi.org/10.1103/PhysRevE.78.061914

Authors & Affiliations

Wendong Shi, Jizeng Wang, Xiaojun Fan, and Huajian Gao^*

Department of Engineering, Brown University, Providence, Rhode Island 02912, USA

^*huajiaṉgao@brown.edu

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Vol. 78, Iss. 6 — December 2008

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Images

Figure 1
(Color online) Schematic illustrations of (a) uptake of finite-sized cylindrical particles with hydrodynamic radius $a$ by a partially absorbing sphere of radius $R$ . (b) The particle-cell interaction potential as a function of particle-cell separation for different particle sizes. (c) The depth of the primary potential well of the particle-cell interaction as a function of the normalized particle size under parameters $A = 12 zJ$ , $σ = 0.5 nm$ , $h = 10 nm$ , $δ = 5 nm$ , $R = 5 μ m$ , $k_{B} T_{0} = 4.1 zJ$ (room temperature).Reuse & Permissions
Figure 2
(Color online) Schematic illustration of particle absorption into a cell via receptor-mediated endocytosis. (a) An initially flat membrane containing diffusive receptor molecules wraps around a ligand-coated cylindrical particle. The receptor density distribution in the membrane becomes nonuniform upon ligand-receptor binding; the receptor density is depleted in the near vicinity of the binding area and induces diffusion of receptors toward the binding site. (b) The normalized wrapping time $t_{w} ∕ (B ∕ ξ_{L} D)$ versus the normalized particle length L/R with $B = 20$ , $e_{RL} = 15$ , $D_{p} = 10^{2} {nm}^{2} ∕ s$ , $ξ_{L} = 5 \times 10^{3} ∕ μ m^{2}$ , $ξ_{0} = 5 \times 10^{2} ∕ μ m^{2}$ for wrapping a cylindrical particle into an infinite membrane.Reuse & Permissions
Figure 3
(Color online) The normalized absorption rate versus the diameter and aspect ratio of cylindrical particles under the particle-cell interaction potential in Eq. (28): (a) Three-dimensional plot and (b) contour plot. The values of parameters are chosen as $γ = 0.5 pN$ , $R = 5 μ m$ , $k_{B} T_{0} = 4.1 zJ$ , $B = 20$ , $e_{RL} = 15$ , $D_{p} = 10^{2} {nm}^{2} ∕ s$ , $ξ_{L} = 5 \times 10^{3} ∕ μ m^{2}$ , and $ξ_{0} = 5 \times 10^{2} ∕ μ m^{2}$ .Reuse & Permissions
Figure 4
(Color online) Numerical results for particle absorption under the cell-particle interaction potential given in the Appendix. (a) The normalized particle absorption rate $Φ η ∕ n_{\infty} k_{B} T$ versus the normalized particle radius $a ∕ R$ . (b) The optimal particle size $a^{*}$ versus the uptake fraction $c$ . (c) The three-dimensional plot of the normalized absorption rate of cylindrical particles as a function of particle diameter and aspect ratio. (d) The contour plot of the normalized absorption rate of cylindrical particles as a function of particle diameter and aspect ratio. The chosen parameters are $A = 12 zJ$ , $σ = 0.5 nm$ , $h = 10 nm$ , $δ = 5 nm$ , $R = 5 μ m$ , $k_{B} T_{0} = 4.1 zJ$ , $η = 1 mP s$ , $B = 20$ , $e_{RL} = 15$ , $D_{p} = 10^{2} {nm}^{2} ∕ s$ , $ξ_{L} = 5 \times 10^{3} ∕ μ m^{2}$ , and $ξ_{0} = 5 \times 10^{2} ∕ μ m^{2}$ . In (a) $c = 0.5$ .Reuse & Permissions
Figure 5
(Color online) Comparison of (a) the normalized absorption rate versus the normalized particle radius and (b) the optimal particle size versus the uptake fraction $c$ , when the particle absorption time $t_{w}$ is taken from Eq. (26) and taken to be a constant equal to $20 s$ . (c) Three-dimensional plot of the normalized absorption rate of cylindrical particles with different diameters and aspect ratios and contour plot of (d) the normalized absorption rate of cylindrical particles with different diameters and aspect ratios. The values of parameters are chosen as $A = 12 zJ$ , $σ = 0.5 nm$ , $h = 10 nm$ , $δ = 5 nm$ , $R = 5 μ m$ , $k_{B} T_{0} = 4.1 zJ$ , $η = 1 mP s$ , and $β = 0.5 {nm}^{3} {ns}^{- 1}$ , and $c = 0.5$ .Reuse & Permissions
Figure 6
The effects of discrete absorber disks on the cell surface. (a) The optimal particle size versus the number of absorbers $N$ . (b) The number of absorbers, $N_{h r}$ , at which the absorption rate reaches one-half of its maximum value versus the particle radius $a$ . The values of parameters are chosen as $A = 12 zJ$ , $σ = 0.5 nm$ , $h = 10 nm$ , $δ = 5 nm$ , $L = 5 μ m$ , $k_{B} T_{0} = 4.1 zJ$ , $η = 1 mP s$ , $t_{w} = 20 ns$ , and $c = 0.5$ . In (a) $b = 150 nm$ , $b = 250 nm$ , and in (b), $b = 250 nm$ .Reuse & Permissions

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