A theoretical model of efficient phagocytosis driven by curved membrane proteins and active cytoskeleton forces

Abstract

Phagocytosis is the process of engulfment and internalization of comparatively large particles by cells, and plays a central role in the functioning of our immune system. We study the process of phagocytosis by considering a simplified coarse grained model of a three-dimensional vesicle, having a uniform adhesion interaction with a rigid particle, and containing curved membrane-bound protein complexes or curved membrane nano-domains, which in turn recruit active cytoskeletal forces. Complete engulfment is achieved when the bending energy cost of the vesicle is balanced by the gain in the adhesion energy. The presence of curved (convex) proteins reduces the bending energy cost by self-organizing with a higher density at the highly curved leading edge of the engulfing membrane, which forms the circular rim of the phagocytic cup that wraps around the particle. This allows the engulfment to occur at much smaller adhesion strength. When the curved membrane-bound protein complexes locally recruit actin polymerization machinery, which leads to outward forces being exerted on the membrane, we found that engulfment is achieved more quickly and at a lower protein density. We consider spherical and non-spherical particles and found that non-spherical particles are more difficult to engulf in comparison to the spherical particles of the same surface area. For non-spherical particles, the engulfment time crucially depends on the initial orientation of the particles with respect to the vesicle. Our model offers a mechanism for the spontaneous self-organization of the actin cytoskeleton at the phagocytic cup, in good agreement with recent high-resolution experimental observations.

Fig. 1

Schematic representation of our model. The vesicle is formed by a closed triangulated surface, having N vertices connected to neighbours by bonds. The red vertices on the surface of the vesicle represent the curved membrane protein complexes, while the blue vertices represent the bare membrane. A zoomed version of a small section of the vesicle surface is shown in the inset. The vesicle is kept in contact with a particle, having attractive interaction between them. We consider spherical as well as non-spherical particles, such as spheroid, spherocylinder, dumb-bell etc.

Fig. 2

Sphere engulfment by a vesicle with passive proteins. (a) Phase diagram in the E_ad − ρ plane, where E_ad is the adhesion strength and ρ is the density of curved membrane proteins. Background color is showing the adhered area fraction. The red dashed line is approximately separating the fully engulfed state and the partially engulfed state. The snapshots are shown for E_ad = 0.60, 0.80, 1.0, 1.20 (in units of k_BT) and ρ = 1.6 %, 4.8 %, 9.6 %, 12.8 %, 16%, 19.2 %. (b) Comparison of adhesion energy gain (dashed horizontal line) and bending energy cost (ΔW_b, points) per adhered node (both having units of k_BT) as a function of the engulfment fraction, for a fixed adhesion strength E_ad = 1.0 k_BT and various values of ρ. Blue symbols denote ρ = 0.8%, green circles for ρ = 2.4% and red triangles for ρ = 4.0%. The origin of the large error bars is due to the large fluctuations in the bending energy at the leading edge of the membrane, as well as having a relatively small ensemble of simulations for each case. (c) Mean cluster size (mean number of proteins in a cluster) as a function of the engulfed area fraction, for both partial and complete engulfment cases. The colors codes are same as fig. 2b. Inset shows the increase in the density of curved proteins at the rim of the phagocytic cup during the engulfment process, for ρ = 4.0%. (d) Examples of cross-sections of the vesicle membrane around the particle, at the end of the engulfment process, for different protein densities. We use E_ad = 1.0 k_BT here. Other parameters are: The total number of vertices (nodes) in the vesicle N = 3127, the radius of the spherical particle R = 10 l_min, the bending rigidity κ = 20 k_BT, the protein-protein attraction strength w = 1 k_BT, and the spontaneous curvature of curved (convex) membrane proteins $C_0=1.0l_{min}^{-1}$.

Fig. 3

Sphere engulfment by a vesicle with active proteins. (a) Phase diagram in the E_ad − F plane for a low concentration of curved proteins ρ = 1.6%, where F represents the protrusive force due to actin polymerization that is coupled with the curved membrane proteins. Background color is showing the adhered area fraction. The dashed yellow line approximately separates the fully engulfed (above the line) from the partially-engulfed states. The snapshots are shown for E_ad = 0.60, 0.80, 1.0, 1.20 (in units of k_BT) and F = 0.20, 0.40, 0.60, 0.80, 1.0 (in units of k_BT/l_min). (b) Comparison of bending energy cost with the adhesion energy gain (both having units of k_BT per adhered node) for different values of F. The blue boxes, green circles and the red triangles are for F = 0.1, 0.4 and 2 (in units of k_BT/l_min) respectively. The red dashed line is showing the combined effective energy gain per adhered node, including the adhesion and the work done by the active forces (for F = 2 k_BT/l_min). (c) The mean cluster size as a function of engulfed area fraction, for the same cases shown in (b) (same color code). Lower snap-shots correspond to F = 0.1, 0.4, and top snap-shots to F = 2. (d) Snap-shots of the vesicle (F = 2 k_BT/l_min) indicating the contribution of the active forces to the the engulfment. Arrows show the tangential component of the individual forces (small arrows), and the direction of the total force (large arrow), for proteins that are close to the spherical surface. For (b-d), we use E_ad = 1 k_BT and ρ = 1.6 %. (e) Mean cluster size for large E_ad and ρ. Green circles are for ρ = 6.4%, F = 1.0 k_BT/l_min, and blue boxes are for ρ = 4.8 %, F = 2.0 k_BT/l_min. We use E_ad = 1.5 k_BT for both the cases.

Fig. 4

Comparison of the actin organization at the rim of the phagocytic cup, observed in simulations (a) and experiments (b). In both the in vitro experiments and the simulations we find that the actin ring is highly fragmented, in the form of dispersed “teeth” (yellow arrows) or in more continuous “arcs” (red arrows) along the leading edge. “Arcs” often connect to form a complete ring around the leading-edge rim during the final stage of the engulfment process, both in the simulations and the experiments. The actin aggregate becomes more cohesive in the final stages of the engulfment, before it disperses after the engulfment is complete. (a) Snapshots from a simulation, using E_ad = 1.50 k_BT, ρ = 6.4 % and F = 1.0 k_BT/l_min. (b) Time-lapse sequences of maximum intensity projection images illustrating engulfment of Immunoglobulin-G-coated polystyrene beads (7 μm diameter) by (upper) RAW264.7 macrophage-like cell line and by (lower) murine bone-marrow derived macrophages, imaged by lattice-light sheet microscopy. In (b), cells were transfected with mEmerald-Lifeact to label F-actin. Scale bar 5 μm, time is indicated in min:sec.

Fig. 5

Engulfment dynamics for spherical particles. (a) Different growth behaviour of adhered fraction with time for protein-free, passive and active cases. We scale the area axis by maximum adhered area (A_max) and time axis by the engulfment time (t_eng). Blue cross symbols are for protein free case with E_ad = 1.30 k_BT; green star symbols for passive proteins with E_ad = 1.0 k_BT and ρ = 4.0 %; red boxes for E_ad = 1.0 k_BT, ρ = 1.6% and F = 0.40 k_BT/l_min; yellow circles for E_ad = 1.0 k_BT, ρ = 1.6% and F = 2.0 k_BT/l_min; black triangles for E_ad = 1.50 k_BT, ρ = 6.4 % and F = 1.0 k_BT/l_min. (b) Engulfment time as a function of F, showing the non-monotonic behavior: below and above a critical force (green and blue shaded areas respectively) there is partial engulfment. (c-d) Distributions of the engulfment time for F = 1, 2 k_BT/l_min, respectively.

Fig. 6

Mean engulfment time for non-spherical particles. The vertical dashed line is separating oblate (right) and prolate (left) shapes. Here, for the passive case, we use: E_ad = 1.0 k_BT, ρ = 4.8%, and for the active case: E_ad = 1.0 k_BT, ρ = 1.6%, and F = 1.0 k_BT/l_min. The surface area of all the shapes is constant, and equal to that of a sphere of radius 10 l_min. The insets demonstrate the initial conditions of either starting from the top (poles) of the shapes (along their axis of rotational symmetry), or from the side (equator).