From biting to engulfment: curvature-actin coupling controls phagocytosis of soft, deformable targets

TL;DR

The paper addresses how phagocytes engulf soft, deformable targets, noting that existing theoretical models often treat targets as rigid. They develop a Monte Carlo membrane model for two deformable vesicles, with Curved Membrane Complexes (CMC) that recruit actin polymerization, enabling curvature–actin coupling to drive engulfment dynamics. Three dynamical regimes emerge—biting/trogocytosis, pushing, and full engulfment—whose occurrence depends on the target's bending modulus $\kappa$ and internal pressure $p$, illustrating a unified mechanical origin for these behaviors. The authors show that increasing membrane tension via osmotic pressure produces transitions analogous to those produced by higher $\kappa$, and corroborate the findings with experiments on GUVs and lymphoma cells. The results suggest that immune cells may use mechanical cues to differentiate soft targets and offer a minimal physical framework for phagocytosis with potential to guide interventions.

Abstract

Phagocytosis is a fundamental process of the innate immune system, yet the physical determinants that govern the engulfment of soft, deformable targets remain poorly understood. Existing theoretical models typically approximate targets as rigid particles, overlooking the fact that both immune cells and many biological targets undergo significant membrane deformation during contact. Here, we develop a Monte Carlo-based membrane simulation framework to model the interactions of multiple vesicles, enabling us to explore phagocytosis-like processes in systems where both the phagocyte and the target possess flexible, thermally fluctuating membranes. We first validate our approach against established observations for the engulfment of rigid objects. We then investigate how the mechanical properties of a soft target -- specifically membrane bending rigidity govern the outcome of phagocytic interactions. Our simulations reveal three distinct mechanical regimes: (i) biting or trogocytosis, in which the phagocyte extracts a portion of the target vesicle; (ii) pushing, where the target is displaced rather than engulfed; and (iii) full engulfment, in which the target is completely internalized. Increasing membrane tension via internal pressure produces analogous transitions, demonstrating a unified mechanical origin for these behaviours. Qualitative comparison with experiments involving Giant Unilamellar Vesicles (GUVs, deformable microparticles) and lymphoma cells supports the relevance of these regimes to biological phagocytosis. Together, these results highlight how target deformability fundamentally shapes phagocytic success and suggest that immune cells may exploit mechanical cues to recognize among different classes of soft targets.

Figures (7)

• Figure 1: Interaction between two vesicles in the model: A) Two surfaces of two vesicles are shown in green and blue colors respectively. We check the distance between the vertices that belong to the two different vesicles and determine it they are interacting if the distance between them is less than the length unit of our simulation, i.e., $l_{\rm min}$. B) If the distance $|\overrightarrow{r}|$ is less than $l_{\rm min}$, we find the dot product between the vector $\overrightarrow{r}$ and the normals of all the triangles common to the vertex from the other vesicle $\hat{n}$. If the dot product is negative, the MC move is discarded as the vesicles are overlapping; otherwise, it is accepted. C) The interaction force on the vertex of interest is the vector sum of the active forces applied by the vertices on the other vesicle within the interaction range. D) If a vertex from vesicle 1 is at a distance less than the interaction range $l_{\rm min}$ from any vertex of another vesicle, then the adhesive energy between them is $-E_{\rm ad}$ for both of these vertices. Otherwise, it is zero.
• Figure 2: Two identical vesicles adhere to each other for different strengths of adhesion energy parameter $E_{\rm ad}$. A) Final configuration snapshots of the pair of passive vesicles (at time step $t=1500$), for the adhesion energy parameter $E_{\rm ad}=1,~1.2,~1.4,~1.6,~1.8,~2.0$ in units of $k_BT$. Blue denotes the bare membrane nodes, while red at the passive CMC nodes. B) The average adhesive energy $E^{\rm tot}_{\rm ad}$ per vesicle is shown as a function of timestep for $E_{\rm ad}=1,~1.2,~1.4,~1.6,~1.8,~2.0$ in units of $k_BT$ with green, blue, orange, red, brown and purple solid lines respectively. C) The final total adhesive energy $E^{\rm tot}_{\rm ad}$ (averaged over the grey shaded time window shown in (B)) is shown for six different values of $E_{\rm ad}$. The grey dashed line represents the total adhesive area in the case of $E_{\rm ad}=1~k_BT$ multiplied by the $E{\rm ad}$. It shows that the total adhesive energy increases due to the increase in adhered area, faster than the increase in $E_{\rm ad}$ (dashed line). In D) we show the total energies the pair of vesicles, averaged over the grey shaded region for six different values of $E_{\rm ad}$. E) The relative difference between the vesicles increases with $E_{\rm ad}$. F) Average volumes of the vesicles as function of $E_{\rm ad}$, and G) the corresponding volume ratio. We used $722$ vertices for each vesicle, out of which $50$ vertices represent the curved-protein complexes with intrinsic curvature $c_0=1~l_{\rm min}^{-1}$, i.e. the CMC percentage is $\rho=6.93\%$. Here, volume is not conserved.
• Figure 3: The engulfment of target vesicles of different rigidities by a cell-like vesicle with passive CMC. A) The snapshots of the shapes of the interacting vesicles with time for different bending rigidity $\kappa$ values of the target vesicle. We set the bending rigidity of the cell-like vesicle at $20 k_BT$. B) The time evolution of adhered area fraction of the target vesicle for different $\kappa$ values (see SI section S4 for details on this calculation). As $\kappa$ increases, the outcome is approaching the completely rigid $\kappa=\infty$ limit that is shown in grey. C) Time evolution of the deviation of the target vesicle from a spherical shape, for different values of $\kappa$ (color code as in (B)), measured by the relative standard deviation for the position vector of all the vertices with respect to the center of mass of the vesicle (Eq.S11). As $\kappa$ increases, the target remains spherical all the time, and it takes longer time to engulf as it requires the cell-like vesicle to extend its adhesion cup over a bigger cross-section.
• Figure 4: Effect of the bending rigidity of the target vesicle on the process of phagocytosis. The time evolution of the adhered area fraction of the target cell is shown in (A), for different bending modulus values. B) The final adhered area fraction of the target vesicle is shown as a function of the bending rigidity $\kappa$. We indicated three phases of biting, push-leave, and engulfment with red green and blue as $\kappa$ increases. The time evolution of the shapes and the snapshots are shown for three example of biting, push-leave, and engulfment in C), D) and E) panels, where the bending rigidity $\kappa$ for the target vesicle is set to $40~k_BT,~200~k_BT,~$ and $1250~k_BT$ respectively. We set the bending rigidity of the cell-like vesicle to $20k_BT$.
• Figure 5: Coupling between the target vesicle deformation and alignment of the active forces during the engulfment process. A) The active force due to an active CMC node of the cell-like vesicle is applied to a neighboring node of the target vesicle $\overrightarrow{F}$. This force is decomposed into two parts, $\overrightarrow{F}_{\parallel}$ is tangential to the surface of the target vesicle and $\overrightarrow{F}_{\perp}$ is the pushing normal force (shown in magenta and green colour, respectively). B) The time evolution of the total magnitude of the force applied to the target vesicle by the cell-like vesicle is shown, for different bending rigidities of the target vesicle (as in Fig.\ref{['fig:kappa_transition_part1']}). C) Time averaged the tangential and normal force fractions as function of the bending rigidity of the target vesicle $\kappa$ (over the times denoted by bold lines in (B)). D) The deviation of the target vesicle from a sphere is shown for three different bending rigidities $\kappa$ of the target vesicle (Eq.S11). E)-G) show the fraction of tangential force on the target in the three dynamical regimes of biting, push-leave, and engulfment. H)-j) Similarly, the fraction of normal pushing force on the target in the three dynamical phases. K)-M) Snapshots showing the force decomposed into tangential and normal components together with the deformations of the target vesicle. We show how the early deformation caused by the normal components affect the later alignment of the CMC at the leading edge of the adhesion patch. K) A very soft target vesicle can initially adhere and bend into the cell-like vesicle, during the early stages. However, the CMC then impinge against the remaining target vesicle, and end up pushing and twisting it with a significant normal component. L) In the pushing regime the deformation of the target vesicle is sufficient to prevent the CMC from aligning tangentially, and a significant normal component maintains the pushing dynamics. M) For the rigid target vesicle the CMC cluster aligns tangentially and drives efficient engulfment. The bending rigidity of the bigger cell is set to $20k_BT$.