Dynamics of particle uptake at cell membranes

Receptor-mediated endocytosis requires that the energy of adhesion overcomes the deformation energy of the plasma membrane. The resulting driving force is balanced by dissipative forces, leading to deterministic dynamical equations. While the shape of the free membrane does not play an important role for tensed and loose membranes, in the intermediate regime it leads to an important energy barrier. Here we show that this barrier is similar to but different from an effective line tension and suggest a simple analytical approximation for it. We then explore the rich dynamics of uptake for particles of different shapes and present the corresponding dynamical state diagrams. We also extend our model to include stochastic fluctuations, which facilitate uptake and lead to corresponding changes in the phase diagrams.

Figure 1

During receptor-mediated endocytosis, the particle goes from unwrapped through partially wrapped to fully wrapped. While membrane shape is fixed by particle shape in the region of the adhered membrane (red part of the contour), the shape of the free membrane follows from an energy minimization (gray part of the contour).

Figure 2

Parametrization of membrane shape for a spherical particle. The adhered part of the membrane is shown in red, the free part of the membrane is shown in gray, and the particle is shown in blue. $R$ is the particle radius, $\theta$ is the uptake angle, and $s$ the contour length of the free membrane. Tangent angle $\psi$, radial distance $r$, and height $z$ are functions of $s$.

Figure 3

(a) The bending and tension energy of the free membrane relative to the total bending and tension energy, i.e., of the adhered and free parts of the membrane for different values of $\lambda /R$. To calculate the energy of the free parts we use Eq. (A3) in the case of $\lambda /R<1$ and Eq. (A4) in the case of $\lambda /R>1$ and the numerical solution for $\lambda =R$. The inset shows the bending and tension energy of the adhered and free parts of the membrane for $\lambda /R=0.1$. (b) The same as in (a) but for the phenomenological description of the free membrane given in Eq. (9).

Figure 4

Energies of the free membrane for (a) $\lambda /R=0.1$, (b) $\lambda /R=0.3$, and (c) $\lambda /R=1.0$. The numerically calculated total energy of the free membrane is shown in green, the line tension approximation $E^{\zeta }$ is shown in blue, the phenomenological approximation $E_{\mathrm{free}}^{\mathrm{pheno}}$ from Eq. (9) is shown in orange, and the analytical limits for tense [from Eq. (A3) 26] and loose [from Eq. (A4) 26] membranes are shown as dashed curves in red and cyan, respectively. (Top) Numerically calculated shapes of the membrane.

Figure 5

Shapes considered here: cylinders in (a) normal orientation (rocket mode) and (b) parallel orientation (submarine mode), (c) spherical particles, spherocylinders in (d) normal orientation, and (e) parallel orientation. In a deterministic model, the particle or virus is homogeneously covered with ligands (blue) that can adhere to the cell membrane. The adhered area $A_{\mathrm{ad}}$ is marked in red.

Figure 6

Cylindrical uptake. (a) Shape of the uptake potential for parallel orientation of the cylinder. Full uptake is possible for reduced membrane tension ${\alpha }^{\parallel }<1/2$ (uptake angle $\theta =\pi$ is a boundary minimum, green solid curve). For ${\alpha }^{\parallel }>1/2$, only partial uptake can take place (minimum for finite uptake angle $\theta <\pi$, green dashed). (b) Nondimensionalized uptake times of parallel (green) and normal cylinder (blue) as a function of ${\alpha }^{\parallel }$ at equal radius but different aspect ratios (i.e., different volume). For the normal cylinder the uptake time increases with aspect ratio, whereas it stays constant for the parallel cylinder. (c) Dynamical state diagram of normal cylindrical uptake. Orange region: full uptake, green region: no uptake. (d) Dynamical state diagram of parallel cylindrical uptake. Orange region: full uptake, blue region: partial uptake. Above the dashed red line the parallel cylinder and below the dashed red line the normal cylinder is taken up fastest.

Figure 7

Nondimensional uptake times for normal cylinder (blue), parallel cylinder (green), and sphere (red) for $\lambda /R=0.3$ and $\lambda /R=1.0$ as a function of ${\alpha }^{\parallel }$ and for equal volume at equal radius.

Figure 8

Uptake time as a function of spherical volume for the sphere (solid red) and spherocylinders with different radii in normal (solid blue) and parallel (solid green) orientation. The dashed blue line marks the uptake times of normal spherocylindrical particles at equal volume. Starting from the red curve, the length of the cylindrical part increases while reducing the particle radius.

Figure 9

(a) The bending and tension energy of the free membrane of a cylindrical particle in parallel orientation relative to the total bending and tension energy, i.e., of the adhered and free parts of the membrane for different values of $\lambda /R$. (b) Similar to (a) but now for the phenomenological description of the free membrane parts.

Figure 10

Deterministic uptake dynamics of a cylinder in parallel orientation including the exact solution for the free membrane. (a) Uptake potential, leading to uptake (orange) and partial uptake (blue). Parameters are given in (b). (b) Phase portrait ($\dot{\theta }$ vs $\theta$) for different parameter values corresponding to uptake (orange) and partial uptake (blue). (c) Dynamical state diagram of the final steady states as a function of ${\beta }^{\parallel }$ vs ${\alpha }^{\parallel }$. Below the red curve the cylinder is taken up completely (orange region), and above we find only partial uptake (blue region).

Figure 11

Deterministic uptake dynamics of a cylinder in parallel orientation including the free membrane by the phenomenological description. (a) Uptake potential for different parameter values corresponding to uptake (orange) and partial uptake (blue). (b) Phase portrait ($\dot{\theta }$ vs $\theta$). Parameters are given in (a). (c) Dynamical state diagram of the final steady states as a function of ${\beta }^{\parallel }$ vs ${\alpha }^{\parallel }$. Below the red curve the cylinder is taken up completely (orange region), and above we find only partial uptake (blue region).

Figure 12

Deterministic uptake dynamics of spherical particles with line tension. (a) Uptake potential, leading to uptake (orange), partial uptake (blue), and no uptake (green). Parameters are given in (b). (b) Phase portrait ($\dot{\theta }$ vs $\theta$) for different parameter values corresponding to uptake (orange), partial uptake (blue), and no uptake (green). (c) Dynamical state diagram of the final steady states as a function of $\beta$ vs $\alpha$. Below the red curve the sphere is taken up only partially (blue region), and above we find either uptake (orange region) or no uptake (green region).

Figure 13

Deterministic uptake dynamics of spherical particles with the phenomenological contribution of the free membrane. (a) Uptake potential, leading to uptake (orange) and partial uptake (blue). (b) Phase portrait ($\dot{\theta }$ vs $\theta$) for different parameter values corresponding to uptake (orange) and partial uptake (blue). Parameters as in (a). (c) Dynamical state diagram of the final uptake states as a function of $\beta$ vs $\alpha$. Full uptake is achieved in the orange region and partial uptake in the blue region.

Figure 14

Steady states as a function of the reduced membrane tension $\alpha$, in the cases of (a) no line tension ($\beta =0$), (b) for a line tension with $\beta =0.6$, and (c) including the phenomenological description of the free membrane, Eq. (9), for $\beta =0.3$. Stable steady states are marked in solid, unstable ones as dashed. The arrows display the flow of the system. As usual, regions of full uptake are marked in orange, partial uptake in blue, and no uptake in green. Note that for (a) there is only one stable steady state and the dynamics is simple. Line tension (b) and the free membrane effects (c) introduce new steady states undergoing saddle-node bifurcations (see text).

Figure 15

Steady states for line tension or free membrane effects as a function of reduced line tension $\beta$. Stable steady states are marked in solid, unstable ones as dashed. The arrows display the flow of the system. As usual, regions of full uptake are marked in orange and partial uptake in blue. (a) Case of line tension and $\alpha =0.8$. The partial uptake for small $\beta$ is due to the saddle node, whose stable branch prevents the system from reaching full uptake. Increasing $\beta$ (and assuming the unstable branch can be overcome by fluctuations) induces full uptake. (b) Phenomenological treatment for the free membrane with $\alpha =0.6$. Here the system displays partial uptake, full uptake, and again partial uptake, i.e., a “reentry” of partial uptake occurs. (c) Same as (b) but for $\alpha =0.8$ showing only partial uptake for increasing $\beta$.

Figure 16

Modeling particle uptake as a discrete stochastic process. The particle (blue) is covered with ligands (small disks) that stochastically bind to cell surface receptors (half-circles) with rate $g_N$, leading to an advancement of the adhered membrane area, or unbind with rate $r_N$. Note that axial symmetry is assumed.

Figure 17

(a) Simulated mean uptake times (symbols) for a normally oriented cylinder (blue triangles), a parallel-oriented cylinder (green lozenges), a sphere (red circles), a sphere including the phenomenological approximation for the free membrane (orange pentagons), and a sphere with line tension (cyan stars) as function of radius at equal volume and radius for the parameter values in Table 1. In addition, the numerically computed mean uptake times calculated by Eq. (44) are shown as solid lines for the different shapes in the corresponding colors. (b) For the normal cylinder, the parallel cylinder, and the sphere the deterministic uptake times are shown as dashed lines in corresponding colors (for vanishing line tension or free membrane) and compared to the numerically calculated mean uptake times (solid lines).

Figure 18

Stochastic uptake dynamics of spherical particles with line tension. (a) Uptake trajectory of a spherical particle with $\alpha =0.3$ and $\beta =0.3$ for $N={10}^2$ time steps. (b)–(d) Occupation probability during uptake with $N_{\max }=20$ receptors for different parameter combinations for an uptake trajectory of $N={10}^6$ time steps. The receptor with largest occupation probability is shown in red. (b) Uptake, (c) no uptake, (d) partial uptake. (e) Dynamical state diagram of stochastic uptake showing the maximum occupation probability of a trajectory of $N={10}^6$ time steps. The white lines correspond to the state boundaries of the deterministic model; cf. Fig. 12.

Figure 19

Dynamical state diagram of stochastic uptake of spherical particles using the phenomenological description of free membrane effects showing the maximum occupation probability of a trajectory of $N={10}^6$ time steps. The white line corresponds to the state boundary of the deterministic model; cf. Fig. 13.