Steric interactions between mobile ligands facilitate complete wrapping in passive endocytosis

Receptor-mediated endocytosis is an ubiquitous process through which cells internalize biological or synthetic nanoscale objects, including viruses, unicellular parasites, and nanomedical vectors for drug or gene delivery. In passive endocytosis the cell plasma membrane wraps around the “invader” particle driven by ligand-receptor complexation. By means of theory and numerical simulations, here we demonstrate how particles decorated by freely diffusing and nonmutually interacting (ideal) ligands are significantly more difficult to wrap than those where ligands are either immobile or interact sterically with each other. Our model rationalizes the relationship between uptake mechanism and structural details of the invader, such as ligand size, mobility, and ligand-receptor affinity, providing a comprehensive picture of pathogen endocytosis and helping the rational design of efficient drug delivery vectors.

Figure 1

(a) Schematic of the invader-host interaction. The invader is decorated with either fixed (left) or mobile (right) ligands, interacting with mobile receptors on the host surface. (b) Schematic of the Monte Carlo system used to compute adhesion free energy with nonideal mobile ligands. Ligands, receptors, and dimers are modeled as hard disks of diameter $\alpha$. The contact and outer regions are simulated as squares with periodic boundary conditions. Ligands are exchanged between the two regions via semi-grand canonical moves, while receptors are exchanged with an ideal reservoir through grand canonical moves. Dimerization is controlled by a reaction move [42]. See Appendix pp2-s1 for the acceptance rules of all moves.

Figure 2

The influence of ligand mobility and steric interactions on the adhesion free energy. (a) Adhesion free energy as a function of the contact-area fraction calculated for fixed and mobile ideal ligands [solid lines, Eqs. (2) and (3)] and nonideal mobile ligands [dotted lines with empty symbols, Eq. (4)]. Symbols connected by thin lines show the results of MC simulations. For the ideal cases we use $K_{2\mathrm{D}}^{\left(\mathrm{eq}\right)}{\rho }_R^{\left(0\right)}=5.5066\times {10}^3$. In the presence of steric interactions ($\varphi >0$) we increase the reservoir receptor density ${\rho }_R^{\left(0\right)}$ to maintain a constant ${\rho }_R=N_{\mathrm{L}}/S_{\mathrm{Tot}}$. For $\varphi =0.01,0.02,0.05,0.1,0.2,0.4$ we scale ${\rho }_R^{\left(0\right)}$ [and thus $K_{2\mathrm{D}}^{\left(\mathrm{eq}\right)}{\rho }_{\mathrm{R}}^{\left(0\right)}$] by a factor $1.04,1.09,1.24,1.57,2.84,20.5$, as calculated by dedicated MC simulations. In all cases we use $N_{\mathrm{L}}=500$. (b) Deviation of the nonideal adhesion free energy from the ideal case.

Figure 3

Mobile, thin ligands suppress complete wrapping of invader particles. The colormaps show the penetration depth $h\in [0,2a]$ for spherical invaders of radius $a$ as a function of the dimerization constant $K_{2\mathrm{D}}^{\left(\mathrm{eq}\right)}$ (in units of ${\rho }_R^{-1}$), and the system parameters $N_L$, $\tilde{\kappa }=\kappa /\left[k_{\mathrm{B}}T\right]$, $\tilde{\gamma }=\gamma /\left[k_{\mathrm{B}}T{a}^{-1}\right]$, $\tilde{\sigma }=\sigma /\left[k_{\mathrm{B}}T{a}^{-2}\right]$. Regions corresponding to fully wrapped and unwrapped particles are marked by a circle and a square, respectively. When not varied, the system parameters are fixed to the values marked by the dotted lines. As for Fig. 2, we keep a constant ${\rho }_R=N_L/S_{\mathrm{Tot}}$ in all calculations. For a typical invader size $a=50\mathrm{nm}$ and $T={37}^{\circ }\mathrm{C}$, these correspond to $\kappa =8.6\times {10}^{-20}$ J, $\sigma =8.5\times {10}^{-6}\mathrm{J}{\mathrm{m}}^{-2}$, $\gamma =3.4\times {10}^{-13}\mathrm{J}{\mathrm{m}}^{-1}$, $N_{\mathrm{L}}=500$. Green dashed lines and white dot-dash lines indicate, respectively, the number of ligands on typical influenza A (375) and HIV (73) virions [52, 53].

Figure 4

Penetration depth for prolate invaders as a function of $K_{2\mathrm{D}}^{\left(\mathrm{eq}\right)}$ and the system parameters (see the caption of Fig. 3 for definitions). The invader is taken with the semi-major axis $a$ perpendicular to the host surface and semi-minor axis equal to $b=a/2$. When not varied, the system parameters are fixed to the values marked by the dotted lines and, if $a=50\mathrm{nm}$ and $T={37}^{\circ }\mathrm{C}$, correspond to the values reported in the caption of Fig. 3. Regions corresponding to fully wrapped and unwrapped particles are marked by a circle and a square, respectively.

Figure 5

Equilibrium wrapping degree of ellipsoidal particles at varying number of ligands $N_{\mathrm{L}}$ and bending rigidity $\tilde{\kappa }$. The shape of the invader is described by $(x^2+y^2)/a^2+{\left(\right|z|/b)}^2=1$ with $b/a=0.8$ (a) and $b/a=1.75$ (b). Please refer to the caption of Fig. 3 for the definition of the adimensional system parameters. When not varied, the bending rigidity has been taken equal to $\tilde{\kappa }=20$, and the number of ligands to $N_{\mathrm{L}}=500$ (dotted lines). The membrane tension is set to $\tilde{\sigma }=16$ by Dasgupta et al. [45]. Regions corresponding to fully wrapped and unwrapped particles are marked by a circle and a square, respectively.

Figure 6

Equilibrium wrapping degree of rod-like particles at varying number of ligands $N_{\mathrm{L}}$ and bending rigidity $\tilde{\kappa }$. The shape of the invader is described by the equation ${[(x^2+y^2)/a^2]}^{(n/2)}+{\left(\right|z|/b)}^n=1$ with $n=4$ and $b/a=1.5$. Please refer to the caption of Fig. 3 for the definition of the adimensional system parameters. When not varied, the bending rigidity has been taken equal to $\tilde{\kappa }=20$, and the number of ligands to $N_{\mathrm{L}}=500$ (dotted lines); the membrane tension is set to $\tilde{\sigma }=16$ by Dasgupta et al. [45]. Regions corresponding to fully wrapped and unwrapped particles are marked by a circle and a square, respectively.