# Theoretical model of efficient phagocytosis driven by curved membrane
proteins and active cytoskeleton forces

Raj Kumar<EMAIL_ADDRESS>Sarah R Barger2 Samo Penič3
Aleš Iglič3 Mira Krendel4 Nils C Gauthier5 Nir S<EMAIL_ADDRESS>1Department of Chemical and Biological Physics, Weizmann Institute of Science,
Rehovot 7610001, Israel  2Molecular, Cellular, Developmental Biology, Yale
University, New Haven, USA  3Laboratory of Physics, Faculty of Electrical
Engineering, University of Ljubljana, Ljubljana, Slovenia  4Department of Cell
and Developmental Biology, State University of New York Upstate Medical
University, Syracuse, USA  5IFOM, FIRC Institute of Molecular Oncology, Milan,
Italy

###### Abstract

Phagocytosis is the process of engulfment and internalization of comparatively
large particles by the cell, that 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 uniform
adhesion interaction with a rigid particle, in the presence of curved membrane
proteins and 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 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 proteins exert outwards protrusive
forces, representing actin polymerization, at the leading edge, we find that
engulfment is achieved more quickly and at lower protein density. We consider
spherical as well as non-spherical particles, and find 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 upon 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.

##  I Introduction

Phagocytosis, also termed ‘cell eating’ Cannon and Swanson (1992); Kumari et
al. (2010); Flannagan et al. (2012); Ellinger and Pietschmann (2016);
Mylvaganam et al. (2021), is a cellular process by which cells internalize
extracellular particles that are relatively large ($0.5-20\mu$m). Phagocytosis
plays an important role for the immune system, enabling immune cells to
destroy foreign elements and clear dead cells and other debris Richards and
Endres (2017). This process is also important for diagnostic and therapeutic
approaches to cancer, for example, using specialized engulfed particles Imani
et al. (2017). Despite its biological importance, it is still not understood
how the actin cytoskeleton is coordinated spatio-temporally during the
phagocytosis process.

Phagocytosis (and endocytosis) in general requires receptors on the cell
membrane that bind to the surface of the target particles Gao et al. (2005).
Binding-driven engulfment, which does not involve the forces of the
cytoskeleton, was modelled in Richards and Endres (2014); Gao et al. (2005);
Richards and Endres (2016); Frey et al. (2019); van Zon et al. (2009);
Khosravanizadeh et al. (2019), with purely diffusive dynamics of the receptors
Gao et al. (2005); Richards and Endres (2016) as well as with active drift
Richards and Endres (2014); van Zon et al. (2009). Engulfment of charged
particles was previously modelled, and shown to depend on the lateral
segregation of mobile charged membrane components (lipids and proteins),
leading to a discontinuous wrapping transition Fošnarič et al. (2009). The
effect of signalling molecules was also considered in Richards and Endres
(2014), where using drift and diffusion the model successfully recovered the
two stages of the process: an initial slow engulfment followed by fast
engulfment of the second half of the spherical object, comparing to
experiments of neutrophil phagocytosis of polystyrene beads Richards and
Endres (2014). Phagocytosis is, however, an active process which is known to
involve the forces of actin polymerization that push the engulfing membrane
forward Herant et al. (2006). In subsequent models, detailed description of
active forces Herant et al. (2006, 2011); Heinrich (2015) and also the bending
and elastic energies of the membrane and cortex are considered Bahrami (2013);
Bahrami et al. (2014); Dmitrieff and Nédélec (2015); Liu et al. (2006); Tollis
et al. (2010). The cytoskeletal components of phagocytosis were investigated
during engulfment of artificially produced particles van Zon et al. (2009);
Herant et al. (2005); Champion and Mitragotri (2006); Spanke et al. (2020), or
during bacterial engulfment by the immune cell Eierhoff et al. (2014); Patel
and Galán (2005); Rottner et al. (2005a). In van Zon et al. (2009), it was
found that the engulfment process either stalls before half engulfment of a
spherical object, or leads to complete engulfment. In Herant et al. (2005);
Heinrich and Rawicz (2005), the experiment was performed using micro-pipettes
to hold both the immune cell and the target particle, allowing the measurement
of the cortical tension. Recent experiments provide high-resolution details of
the organization of the actin cytoskeleton at the rim of the phagocytic cup
Barger et al. (2019); Vorselen et al. (2021); Ostrowski et al. (2019),
challenging the theoretical modelling of this process.

In reality, however, the shape of the engulfed particles, such as bacteria or
viruses, may be highly non-spherical, which motivated several experiments with
artificial particles of various geometries Champion and Mitragotri (2006);
Paul et al. (2013); Sharma et al. (2010); Doshi and Mitragotri (2010). In
Champion and Mitragotri (2006) it was found that the engulfment time for a
non-spherical shapes can be five times larger than for a spherical particle
Paul et al. (2013). In Sharma et al. (2010) it was observed that an oblate
spheroid was engulfed more easily compared to spheres, while spheres were
easier to engulf compared to prolate spheroid, rod-shaped particles, or
needles Lu et al. (2010). Note that the surface area was not kept constant for
the particles with the different geometries. In theoretical studies, several
non-spherical shapes were modelled, such as ellipsoids, rod-like particles,
and capped cylinder Dasgupta et al. (2013, 2014); Vácha et al. (2011),
however, in these studies the effect of active forces have not been
considered.

From these studies we know that the phagocytosis process involves highly
complex dynamics of cytoskeleton rearrangement, membrane shape deformations
and proteins aggregations Richards and Endres (2017). There is at present no
theoretical model that explains the dynamics of the self-organization of the
membrane and actin cytoskeleton, including the active forces it exerts, during
the engulfment process. Here, we address this problem, using a coarse-grained
model of a three-dimensional vesicle, having attractive interaction with the
target particle. Our model vesicle contains curvature-sensitive proteins, that
diffuse on its membrane, and which also recruit actin polymerization, thereby
inducing active protrusive forces Fošnarič et al. (2019); Sadhu et al. (2021).
This modelling approach can be further motivated by the observations of curved
membrane proteins that are also related to actin recruitment at the leading
edge of the cell membrane Linkner et al. (2014); Begemann et al. (2019);
Pipathsouk et al. (2021). Using this model we are able to expose the role of
curved membrane proteins, coupled with adhesion and active cytoskeletal
forces, during the phagocytosis process.

##  II Results

We simulate the dynamics of a three-dimensional vesicle, which is described by
a closed surface having $N$ vertices, each of them connected to its neighbours
with bonds, forming a dynamically triangulated, self-avoiding network, with
the topology of a sphere Sadhu et al. (2021); Fošnarič et al. (2019). The
details of the model are given in the Materials and Methods section (also see
SI sec. S1). We first study the case where the phagocytic particle is
spherical, and then generalize it for non-spherical particles as well.

###  II.1 Spherical particles - Passive curved proteins

As a first validation of our model, we simulated the engulfment process of a
spherical particle by a protein-free vesicle (see SI secs. S2-S3, Figs.S2-S5;
Supplementary Movies-S1,S2). In agreement with previous studies Lipowsky and
Döbereiner (1998) we find an optimal size for full engulfment of the particle
in this system; very small particles can not be engulfed due to the high
bending energy, while very large particles require larger vesicles in order to
be fully engulfed. An optimal radius of the spherical particle was found to be
$R\sim 10~{}l_{min}$ (where $l_{min}$ is the minimal allowed length of an edge
on the triangulated surface). The protein-free case can be compared to a
simplified analytical model (see SI sec. S3), and used to calibrate the
relevant parameter regime.

Figure 1: Sphere engulfment by a vesicle with passive proteins. (a) Phase
diagram in the $E_{ad}-\rho$ plane. The red dashed line is separating the
fully engulfed state (above the line) and the partially engulfed state. The
snapshots are shown for $E_{ad}=0.60,0.80,1.0,1.20$ (in units of $k_{B}T$) and
$\rho=1.6~{}\%,4.8~{}\%,9.6~{}\%,12.8~{}\%,16\%,19.2~{}\%$. (b) Comparison of
adhesion energy gain (dashed horizontal line) and bending energy cost
($\partial W_{b}/\partial A_{ad}$, points) per adhered node (both having units
of $k_{B}T$) as function of the engulfment fraction, for a fixed adhesion
strength $E_{ad}=1.0~{}k_{B}T$ and various values of $\rho$. Blue symbols
denote $\rho=0.8~{}\%$, green circles for $\rho=2.4~{}\%$ and red triangles
for $\rho=4.0~{}\%$. (c) Mean cluster size (mean number of proteins in a
cluster) as function of the engulfed area fraction, for both partial and
complete engulfment cases. The colors codes are same as fig. 1b.

Next, we consider vesicles with passive ($F=0$, where $F$ is the active
protrusive force exerted by the proteins), curved proteins. In Fig. 1a, we
show the phase diagram in the $\rho-E_{ad}$ plane (where $E_{ad}$ is the
adhesion energy per adhered node, and $\rho$ the areal density of the proteins
on the vesicle membrane). For small $E_{ad}$, below $\sim 0.60~{}k_{B}T$, we
only obtain partial engulfment, with the engulfment area increasing with
$\rho$ but not reaching full engulfment. For intermediate
$E_{ad}(0.80-1.0~{}k_{B}T)$, there is an incomplete engulfment for small
$\rho$ (Movie-S3), but the engulfment is complete for larger $\rho$
(Movie-S4). This is the interesting regime, where engulfment is facilitated by
the presence of the curved proteins.

Note that we calculate the adhered fraction of the spherical particle
($A_{ad}/A_{max}$), using for $A_{max}$ the maximum adhered area. This may be
larger than the true surface area of the particle, since we have a finite
width of the adhesive interactions, and within this width the vesicle may
fluctuate (for more details see SI, sec. S2).

Increasing $\rho$ reduces the bending energy cost of the engulfment by the
spontaneous aggregation of the curved proteins at the sharp leading edge of
the engulfing membrane, and allows the vesicle to adhere more and engulf the
particle. This is the same mechanism that allows passive curved proteins to
drive vesicle spreading on flat adhesive substrates Sadhu et al. (2021).

The engulfment process can be understood by considering the competition
between adhesion and bending energy cost of the vesicle as the engulfment
proceeds. In Fig. 1b we compare the adhesion energy gain ($E_{ad}$) and
bending energy cost ($\partial W_{b}/\partial A_{ad}$), per additional adhered
membrane node, as function of the adhered fraction of the spherical particle.
The bending energy cost increases with adhered fraction, and for small $\rho$
the engulfment process stops where the bending energy cost becomes higher than
the adhesion energy gain, per node. In Fig. 1b we show two cases of partial
engulfment, for $\rho=0.8~{}\%,2.4~{}\%$ (blue and green points), where the
bending energy cost becomes larger than the adhesion energy gain, and the
system reaches steady-state at $A_{ad}/A_{max}\sim 0.2,0.35$ respectively. At
these low densities the curved proteins fail to form large aggregates at the
leading-edge of the engulfing membrane, as shown by the mean cluster size
(Fig. 1c), and therefore do not lower the bending energy cost sufficiently
(see insets of Fig. 1c for the adhered fractions that are smaller than 0.5).

Complete engulfment is achieved for higher densities, where the adhesion
energy gain is larger than the bending energy cost. For $\rho=4.0~{}\%$ the
bending energy cost is always smaller than the adhesion energy gain, and the
system proceeds to complete engulfment (red points in Fig. 1b). This is driven
by the formation of a large protein cluster along the leading-edge of the
engulfing membrane (Fig. 1c and insets for adhered fractions that are larger
than 0.5).

Note that in addition to the adhesion energy gain, there is also the gain in
protein-protein binding energy, which further helps drive the membrane over
the bending energy barrier, but this is a small contribution due to the small
size of the ring-like clusters of proteins (Fig. 1c). Following the completion
of the engulfment, this ring-like cluster of curved proteins disperses
spontaneously. Since the topology of the membrane is fixed, we do not allow
for membrane fusion, and there remains a tiny open membrane neck that connects
the engulfed sphere with the space outside the vesicle.

For very large $\rho$ (Fig. 1a), however, the wrapping is again incomplete.
The large number of curved proteins stabilize the cluster at the membrane rim
and prevent the hole from closing up (Movie-S5). For large $E_{ad}(\geq
1.2~{}k_{B}T)$, the adhesion interaction between vesicle and particle is
strong enough to drive complete engulfment even in the absence of curved
proteins. The red dotted line in Fig. 1a encloses the region within which we
find complete engulfment. The shape of this complete-engulfment transition
line can be qualitatively reproduced by a simplified analytical model (SI sec.
S4; Figs. S7, S8).

Finally, we also calculated the engulfment time as function of $\rho$ (see SI
sec. S5, Fig. S9), where we find that it is decreasing with increasing $\rho$.
The engulfment time diverges for $\rho\sim 2.25~{}\%$, below which there is no
engulfment (for $E_{ad}=1.0~{}k_{B}T$).

###  II.2 Spherical particles - Active curved proteins

Next, we study the engulfment by a vesicle containing active curved proteins
(Fig. 2). In order to expose most clearly the role of activity during the
engulfment process, we start with a low concentration of curved proteins
($\rho=1.6~{}\%$), such that their passive effect is weak (Fig. 1a). In Fig.
2a, we show the engulfment phase diagram in the $F-E_{ad}$ plane. We find that
the active forces enable full engulfment even at low adhesion strengths, where
the passive curved proteins ($F=0$) are not sufficient to drive full
engulfment. For small $E_{ad}$, the engulfment is incomplete even for large
$F$, while for intermediate $E_{ad}$ the engulfment becomes complete as $F$
increases (Movie-S6,S7,S8). For large values of $E_{ad}$, the engulfment is
achieved even for the passive system.

Figure 2: Sphere engulfment by a vesicle with active proteins. (a) Phase
diagram in the $E_{ad}-F$ plane for a low concentration of curved proteins
$\rho=1.6~{}\%$. The dashed yellow line is the transition from the fully
engulfed (above the line) to partially-engulfed states. The snapshots are
shown for $E_{ad}=0.60,0.80,1.0,1.20$ (in units of $k_{B}T$) and
$F=0.20,0.40,0.60,0.80,1.0$ (in units of $k_{B}T/l_{min}$). (b) Comparison of
bending energy cost with the adhesion energy gain (both having units of
$k_{B}T$ 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
$k_{B}T/l_{min}$) 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 $F=2~{}k_{B}T/l_{min}$). (c) The mean cluster
size as 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 ($F=2~{}k_{B}T/l_{min}$)
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 $E_{ad}=1~{}k_{B}T$ and $\rho=1.6~{}\%$.
(e) Mean cluster size for large $E_{ad}$ and $\rho$. Green circles are for
$\rho=6.4\%$, $F=1.0~{}k_{B}T/l_{min}$, and blue boxes are for
$\rho=4.8~{}\%$, $F=2.0~{}k_{B}T/l_{min}$. We use $E_{ad}=1.5~{}k_{B}T$ for
both the cases.

As for the passive case, we analyze the engulfment process by plotting the
adhesion energy gain and the bending energy cost per adhered node, as function
of $A_{ad}/A_{max}$. In Fig. 2b we compare both the case of complete and
incomplete engulfment, as the force is increased. As in the passive case,
incomplete engulfment occurs when the bending energy cost per node becomes
larger than $E_{ad}$ (blue boxes in Fig. 2b). For a higher force
($F=0.40~{}k_{B}T/l_{min}$), the active force enhances the aggregation of the
proteins around the rim and thereby lowers the bending energy cost, resulting
in complete engulfment (green circles in Fig. 2b).

Even higher forces ($F=2~{}k_{B}T/l_{min}$), produce large shape fluctuations
of the vesicle, and the bending energy cost increases. However, a complete
engulfment is achieved nevertheless (red triangles in Fig. 2b, and top insets
in Fig. 2c). In this case, the engulfment is enabled by the active forces that
directly push the membrane over the spherical surface, and complete engulfment
is achieved even when the bending energy cost per node goes above $E_{ad}$. In
order to extract the contribution due to active forces in this case, we define
a region close to the particle (phagocytic cup or rim) and assume that only
the forces from these proteins are helping the vesicle to engulf the particle.
We then calculate the magnitude of active forces in the tangential direction
to the surface of the particle (Fig. 2d). We then divide this total force
$F_{total}$ by the length of the leading rim of the membrane, to extract the
effective active work done per adhered node (see SI sec. S6, Fig. S10 for
details). We add this active work to the adhesion energy gain per adhered node
(red dashed line in Fig. 2b), which is shown to be sufficient to offset the
bending energy cost per node.

Note that in the active case the complete engulfment can proceed at much lower
densities of proteins, compared to the passive case, such that there are not
necessarily enough of them to complete a ring-like cluster at the leading
edge. An arc-like cluster of active proteins, that forms at an earlier stage
of the engulfment process (red triangles in Fig. 2c), can be sufficient to
drive a lamellipodia-like protrusion that spreads over the spherical particle
to complete the engulfment (Fig. 2(c,d)). This is further shown in Fig. 2e,
for larger protein densities (Movie-S9,S10). The spherical particle is
transparent to ensure the visibility of the protein clusters along the rim of
the phagocytic cup. Due to the higher density the rim clusters form at lower
adhered fraction (compared to Fig. 2c), but they are still highly fragmented
and do not merge to form a complete circular ring until very late stages of
the engulfment (Fig. 2e).

Many of the spatio-temporal features that we found for the curved active
proteins in our simulations match very well the recent high-resolution
experimental images of the actin organization along the leading edge of the
engulfing membrane protrusion during phagocytosis (Fig. 3, Movie-S11,S12)
Vorselen et al. (2021). In both the experiments and the simulations we find
that the actin ring is highly fragmented, in the form of dispersed ”teeth” and
”arcs” along the leading edge. The actin aggregate becomes more cohesive in
the final stages of the engulfment, before it disperses after the engulfment
is complete (Also see SI sec. S7, Movie-S13).

###  II.3 Spherical particles - Engulfment dynamics

The engulfment dynamics for the different systems are shown in Fig. 4a. For
the protein-free and passive proteins systems, driven by adhesion alone, we
find in our simulations the typical ”two-stage” behavior: an initial slow
growth up to adhered area fraction of $\sim 0.5$, and then a very fast stage
to complete engulfment. A similar behavior is also observed for the active
system in the regime of low protein densities. This dynamics was observed
experimentally, and for the passive engulfment system was also motivated by
theoretical modelling Richards and Endres (2014); Fošnarič et al. (2009).

However, for the active system at high protein density, where the rim clusters
form at an early stage of the engulfment process (Fig. 2e), we find a much
more uniform dynamics with steady growth of the adhered fraction (Fig. 4a).
This agrees qualitatively with the observations in Barger et al. (2019);
Vorselen et al. (2021), which indeed correspond to this regime (Figs. 2e,3).

Figure 3: Comparison of the actin organization at the rim of the phagocytic
cup, observed in simulations (a) and experiments (b,c). In both the
experiments and the simulations we find that the actin ring is highly
fragmented, in the form of dispersed ”teeth” and ”arcs” along the leading edge
(yellow arrows in (b,c)). 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 $E_{ad}=1.50~{}k_{B}T$,
$\rho=6.4~{}\%$ and $F=1.0~{}k_{B}T/l_{min}$. (b,c) Time-lapse sequences of
maximum intensity projection images illustrating engulfment of Immunoglobulin-
G-coated polystyrene beads ($7~{}\mu m$ diameter) by RAW264.7 macrophage-like
cell line (b) and by murine bone-marrow derived macrophages (c), imaged by
lattice-light sheet microscopy. In b and c, cells were transfected with
mEmerald-Lifeact to label F-actin. Scale bar $5~{}\mu m$, time is indicated in
min:sec.

In Fig. 4a the engulfment dynamics are shown as function of normalized time,
so the actual engulfment duration is not presented. The dependence of the mean
engulfment time on the active force is found to be non-monotonic (Fig. 4b).
Below a critical force, in the regime of low $E_{ad}$ and low $\rho$, where
the passive system does not fully engulf (Fig. 2a), there is no complete
engulfment (green shaded area). Above this regime, increasing force induces
complete engulfment with faster and more robust engulfment as $F$ increases.
The engulfment time distribution in this regime is relatively narrow (Fig.
4c).

However, increasing $F$ beyond some value, which is set by the relation of the
adhesion energy and the active work, leads to increasing both the mean value
and the variability of the engulfment time. This occurs due to active forces
stretching the membrane away from the adhered particle (top insets in Fig.
2c), which gives rise to long ”waiting times” during which the engulfment does
not progress. Following this ”waiting time”, once the protein aggregate forms
at the membrane rim (Fig. 2c), the engulfment proceeds rapidly as seen in Fig.
4a (magenta circles). This variability is manifested in the very broad
distribution of engulfment times (Fig. 4d).

At even larger forces, the stretching of the membrane sideways by the active
forces prevents engulfment (blue shaded area in Fig. 4b, and SI sec S8, Fig.
S12(a), Movie-S14). Large membrane tension is indeed known to hinder and
prevent engulfment in cells Masters et al. (2013). In this regime of large
$F$, if we further increase $\rho$, we reach the regime where a free vesicle
will form a flat pancake-like shape, where all the proteins are aggregated at
the rim Fošnarič et al. (2019). In this regime the vesicle can engulf the
particle if this process is faster than the timescale of the formation of the
pancake-like shape (SI sec. S8, Fig. S12(b); Movie-S15). For very large
forces, we find again a non-engulfed regime (similar to the blue shaded area
in Fig. 4b, SI Fig. S12(c), Movie-S16).

Figure 4: 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 $E_{ad}=1.30~{}k_{B}T$; green star symbols for passive proteins
with $E_{ad}=1.0~{}k_{B}T$ and $\rho=4.0~{}\%$; red boxes for
$E_{ad}=1.0~{}k_{B}T$, $\rho=1.6~{}\%$ and $F=0.40~{}k_{B}T/l_{min}$; magenta
circles for $E_{ad}=1.0~{}k_{B}T$, $\rho=1.6~{}\%$ and
$F=2.0~{}k_{B}T/l_{min}$; black triangles for $E_{ad}=1.50~{}k_{B}T$,
$\rho=6.4~{}\%$ and $F=1.0~{}k_{B}T/l_{min}$. (b) Engulfment time as 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 $F=1,2~{}k_{B}T/l_{min}$,
respectively.

###  II.4 Engulfment of non-spherical particles

Immune cells often engulf highly non-spherical objects, such as bacteria .
Furthermore, studies with different particle shapes indicated a strong shape-
dependence of the engulfment time and success rate Champion and Mitragotri
(2006); van Zon et al. (2009); Richards and Endres (2016). We therefore study
the engulfment of non-spherical particles by our vesicles that contain passive
or active curved proteins.

We first consider spheroids. We keep the surface area constant, i.e. same as
for the sphere of radius $10~{}l_{min}$. We use prolate shapes
($R_{x}=R_{y}<R_{z}$) and oblate shapes ($R_{x}=R_{y}>R_{z}$), such that the
aspect ratio $R_{x}/R_{z}<1$ for prolate, and $R_{x}/R_{z}>1$ for oblate.

In Fig. 5, we show the mean engulfment time for oblate and prolate spheroids,
for both the cases when the vesicle is initially in contact from the top (the
poles of the shape) or from the side (along the equator), as shown in the
insets. For both passive and active curved proteins, we find that the
engulfment times increase as the shape deviates from a sphere. This is in
general agreement with previous experimental observations Champion and
Mitragotri (2006), and can be intuitively attributed to the higher curvature
of the non-spherical shapes, which increase the bending energy cost of their
engulfment.

For the particles with small aspect ratio ($R_{x}/R_{z}$ close to unity), the
engulfment time for all the cases are comparable. However, for the particles
with large aspect ratios, there are significant differences in the engulfment
times, between the passive and active systems. For passive curved proteins
(blue circles and red boxes), we find that the engulfment times increase
sharply with the deviation from spherical shape. Furthermore, we find that
initialising from the side (red boxes) results in relatively shorter
engulfment times, especially for the oblate shapes. For an oblate particle
with large aspect ratio (for $R_{x}=R_{y}>11~{}l_{min}$), the particle is
engulfed from side, while it is not engulfed from the top (Fig. 5). For
prolate shapes with large aspect ratios, however, the engulfment times seem
comparable for both initial conditions.

By comparison, for active curved proteins the engulfment time is lower than
for the passive case, even with a lower protein density, and the active curved
proteins enable complete engulfment over a larger range of aspect ratios. We
also note that for the active curved proteins it is the prolate shape that
exhibits a dependence of the engulfment time on the initial conditions, where
starting from the side (magenta boxes) leads to faster engulfment. For the
oblate shapes, however, the engulfment time is largely independent of the
initial conditions, opposite to the behavior of the passive curved proteins.

For a non-spherical particle with high aspect ratio, due to the highly curved
poles (in the prolate) or equator (oblate) regions, the engulfment process is
slower and the variability in engulfment times increases. The slower
engulfment allows the vesicle to reorient on the surface of the particle,
driven by both passive energy minimization and active work. We show the
details of this reorientation process in the SI sec. S9 (Movie-S17-S22).

When the vesicle is initially at the top of a prolate shape with high aspect
ratio, the vesicle reorients itself always to the side and engulfs the
particle. For passive curved proteins, this reorientation process is faster
and thereby the engulfment time when started from top or from side are
comparable (blue circles and red boxes in Fig. 5 for $R_{x}/R_{z}<1$). For
active curved proteins, this reorientation process is slower because of large
shape fluctuations of the vesicle, and thus the engulfment time when started
from the side is smaller than from the top (magenta boxes in Fig. 5 for
$R_{x}/R_{z}<1$).

For the oblate shapes we find a similar reorientation process when the vesicle
starts the adhesion from the highly curved sides. In this geometry, the
passive vesicle can engulf from side without any reorientation, which is very
fast, or it reorients to the top of the oblate shape and ends up with no
engulfment. Therefore, when we calculate the engulfment time we only average
over those realizations where the particle was engulfed without any
reorientation (red boxes for $R_{x}>11$). For the active case, the vesicle
reorients in every realization and engulfs from the top. This reorientation
process is very fast and the engulfment time is similar for starting either
from the side or from the top (green circles and magenta boxes in Fig. 5 for
$R_{x}/R_{z}>1$).

Figure 5: 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: $E_{ad}=1.0~{}k_{B}T$, $\rho=4.8~{}\%$, and for the
active case: $E_{ad}=1.0~{}k_{B}T$, $\rho=1.6~{}\%$, and
$F=1.0~{}k_{B}T/l_{min}$. 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).

We further explore more extreme non-spherical shapes, such as elongated
sphero-cylinders and dumb-bell shapes in the SI (Sec. S10, Movie-S23-S26). The
elongated particles resemble the shapes of different bacteria or fungi such as
E. coli, Bacillus subtilis or budding yeast Mölller et al. (2012); Richards
and Endres (2016); Clarke et al. (2010).

##  III Discussion

In this study we demonstrated that curved (convex) membrane proteins, both
passive and active, can strongly affect the engulfment process of rigid
particles by an adhesive vesicle. The engulfment process that we calculate,
especially using active curved proteins, exhibits many features that are
observed during phagocytosis of rigid particles and pathogens by living cells.
We therefore propose that our model provides a mechanism for the self-
organization of the membrane and the actin cytoskeleton in the phagocytic cup.

We find that the curved proteins spontaneously aggregate around the highly
curved rim of the phagocytic cup, and enable the engulfment to proceed
efficiently by reducing the bending energy costs. The active forces in our
model, which represent the protrusive forces due to the recruited actin
polymerization, further facilitate the engulfment, due to several effects that
we have identified: the protrusive force at the membrane leading edge promotes
the aggregation of the curved proteins (reducing the bending energy cost), as
well as directly pushing the membrane over the engulfed particle. In addition,
large forces can stretch the membrane to wrap around sharp corners, which
enables engulfment of non-spherical particles. Compared to the passive curved
proteins, the curved active proteins drive more efficient engulfment at lower
adhesion energy, lower curved protein density, and of highly non-spherical
particles. These results of our model highlight the central role that the
recruitment of actin plays during phagocytosis, which is usually abolished by
inhibition of actin polymerization (see for example Prashar et al. (2013)).

Our model predicts that the actin aggregation at the rim of the phagocytic cup
is typically fragmented and does not form a complete circular ring.
Nevertheless, it is still highly effective in driving the engulfment.
Furthermore, the actin aggregate forms a more cohesive ring in the final
stages of the engulfment, before it spontaneously disperses as the engulfment
is complete. These features of our model are verified and observed in recent
high-resolution imaging of the actin during phagocytosis Vorselen et al.
(2021) (Fig. 3).

The model we presented currently lacks the contractile forces that arise at
the phagocytic cup due to myosin-II activity Vorselen et al. (2021). The
contractile forces play an important role during the engulfment of soft
objects, such as cells Tsai and Discher (2008); Araki et al. (2003); Barger et
al. (2020), but they are dispensable for the phagocytosis of more rigid
particles Prashar et al. (2013); Vorselen et al. (2021). Contribution of
myosin-II-mediated contractility may be more or less pronounced depending on
the particle size and stiffness and, in addition, could influence
mechanosensitive signaling within phagocytes rather than directly drive
phagocytic cup extension and closure. Our results should therefore be
important even without this feature, which we plan to add to our model in the
future, especially when modelling the engulfment of flexible particles.

To conclude, we demonstrate that curved membrane proteins can play a key role
during phagocytosis. Convex-shaped proteins or a curved complex of proteins,
that furthermore recruit actin polymerization, spontaneously aggregate at the
leading edge of the membrane, and drive robust engulfment. This is similar to
the role of these proteins that was recently found at the leading edge of the
lamellipodia, in experimentsBegemann et al. (2019); Pipathsouk et al. (2021)
and theory Sadhu et al. (2021). We also show that concave-shaped proteins can
further enhance the engulfment process by reducing the bending energy cost of
the membrane that adheres to the particle’s surface (SI sec. S11, Fig. S18,
Movie-S27). Our results should motivate future experimental studies aimed at
the identification and characterization of the different curved membrane
components that participate in phagocytosis Flannagan et al. (2012);
Bohdanowicz and Grinstein (2013); Linkner et al. (2014).

##  IV Materials and Methods

###  IV.1 Theoretical modelling

The membrane shape and protein positions are simulated using a Monte-Carlo
(MC) procedure Sadhu et al. (2021); Fošnarič et al. (2019). The vesicle
energy, which is calculated per each MC move, has the following contributions:
(1) The bending energy, which depends on the bending rigidity $\kappa$, local
mean curvature $H$ and the spontaneous curvature $C_{0}$ (which we take to be
nonzero for the nodes occupied by the membrane proteins), (2) The attractive
protein-protein interaction energy between nearest-neighbors, of strength $w$,
(3) The magnitude of the force of the active force $F$, pushing towards the
outward normal vector of the vertex that contains a protein, and finally, (4)
The adhesion energy of magnitude $E_{ad}$ per node that is within close
proximity to the external substrate Penič et al. (2015); Fošnarič et al.
(2019); Sadhu et al. (2021). For more details, please see SI sec. S1.

Unless specified, we use a vesicle of total number of vertices, $N=3127$
(radius $\sim 20~{}l_{min}$), where $l_{min}$ is the unit of length in our
model, and defines the minimum bond length. The bending rigidity
$\kappa=20~{}k_{B}T$, the protein-protein attraction strength $w=1~{}k_{B}T$,
and $\rho=N_{c}/N$ is the protein density, with $N_{c}$ vertices occupied by
curved (convex) membrane proteins having spontaneous curvature:
$C_{0}=1.0~{}l_{min}^{-1}$. Note that we do not conserve the vesicle volume,
although this could be maintained using an osmotic pressure term Fošnarič et
al. (2019). The membrane area is very well conserved.

###  IV.2 Imaging of Macrophage Phagocytosis

Culturing of RAW264.7 macrophages (ATCC) and generation of murine bone-marrow
derived macrophages is described in Barger et al. (2019). Macrophages were
transfected with mEmerald-Lifeact (Addgene, $\\#54148$) to label F-actin using
the Neon electroporation system, according to manufacturer’s instructions, and
allowed to recover for $24~{}hr$. Flash Red polystyrene beads (Bangs
Laboratories Inc., $7~{}\mu m$ diameter) were washed three times in sterile
PBS and opsonized overnight at $4^{\circ}$C in $3~{}mg/mL$ mouse IgG
(Invitrogen). To remove excess antibody, beads were washed three times with
PBS and resuspended in sterile PBS. Beads were applied to macrophages, plated
on $5~{}mm$ round coverslips, in a $37^{\circ}$C-heated, water-coupled bath in
imaging medium (FluoroBrite (Thermo Scientific), $0-5~{}\%$ FBS, Pen/Strep)
prior to imaging acquisition. Imaging was performed using a lattice light-
sheet microscope operated and maintained by the Advanced Imaging Center at the
Howard Hughes Medical Institute Janelia Research Campus (Ashburn, VA).
$488,~{}560$, or $642~{}nm$ diode lasers (MPB Communications) were operated
between $40$ and $60~{}mW$ initial power, with $20-50~{}\%$ acousto-optic
tunable filter transmittance. The microscope was equipped with a Special
Optics $0.65~{}NA/3.75~{}mm$ water dipping lens, excitation objective, and a
Nikon CFI Apo LWD $25\times 1.1~{}NA$ water dipping collection objective,
which used a $500~{}mm$ focal length tube lens. Images were acquired with a
Hamamatsu Orca Flash $4.0$ V2 sCMOS cameras in custom-written LabView
Software. Post-image deskewing and deconvolution was performed using HHMI
Janelia custom software and $10$ iterations of the Richardson-Lucy algorithm.

##  V Acknowledgment

We acknowledge useful comments by Orion Weiner. N.S.G. is the incumbent of the
Lee and William Abramowitz Professorial Chair of Biophysics, and acknowledges
support by the Ben May Center for Theory and Computation. A.I. and S.P. were
supported by the Slovenian Research Agency (ARRS) through the Grant No.
J3-3066 and Programme No. P2-0232. LLSM imaging was performed at the Advanced
Imaging Center (AIC)-Howard Hughes Medical Institute (HHMI) Janelia Research
Campus. The AIC is jointly funded by the Gordon and Betty Moore Foundation and
the Howard Hughes Medical Institute.

##  VI Supplementary movies

All the supplementary movies are available online in the link:
https://app.box.com/s/ske4vjlp6ci3vu7mz5izfp2iebxrmgm4

##  S-0 Supplementary informations

###  S-0.1 Model and simulation details

We consider a three-dimensional vesicle, which is described by a closed
surface having $N$ vertices, each of them connected to its neighbours with
bonds, to form a dynamically triangulated, self-avoiding network, with the
topology of a sphere Sadhu et al. (2021); Fošnarič et al. (2019). The vesicle
is kept in contact with a particle, with which the vesicle has an attractive
interaction, as shown in Fig. S-1. The vesicle energy has four following
contributions: The bending energy is given by,

$W_{b}=\frac{\kappa}{2}\int_{A}(C_{1}+C_{2}-C_{0})^{2}dA,$ (S-0)

where $\kappa$ is the bending rigidity, $C_{1}$, $C_{2}$ are the principal
curvatures and $C_{0}$ is the spontaneous curvature. The nearest neighbour
protein-protein direct interaction energy is given by,

$W_{d}=-w\sum_{i<j}{\cal H}(r_{0}-r_{ij}),$ (S-0)

where, ${\cal H}$ is the Heaviside step function,
$r_{ij}=|\overrightarrow{r}_{j}-\overrightarrow{r}_{i}|$ is the distance
between proteins, $\overrightarrow{r}_{i},\overrightarrow{r}_{j}$ are the
position vectors for $i,j-th$ proteins, and $r_{0}$ is the range of
attraction, $w$ is the strength of attraction. The range of attraction is such
that only the proteins that are in the neighbouring vertices can attract each
other. The active energy is given by,

$W_{F}=-F\sum_{i}\hat{n_{i}}.\overrightarrow{r}_{i},$ (S-0)

where, $F$ is the magnitude of the force representing protrusive force due to
actin polymerization, $\hat{n_{i}}$ is the outward normal vector of the vertex
that contains a protein and $\overrightarrow{r}_{i}$ is the position vector of
the protein. Finally, the adhesion energy is given by,

$W_{A}=-\sum_{i^{\prime}}E_{ad},$ (S-0)

where $E_{ad}$ is the adhesion strength, and the sum runs over all the
vertices that are adhered to the phagocytic particle Penič et al. (2015);
Fošnarič et al. (2019); Sadhu et al. (2021). By ‘adhered vertices’, we mean
all such vertices, whose perpendicular distance from the surface of the
particle are less than $l_{min}$. Thus, the total energy of the vesicle-
particle system is given by,

$W=W_{b}+W_{d}+W_{F}+W_{A}$ (S-0)

We update the vesicle with mainly two moves, (1) vertex movement and (2) Bond
flip. In a vertex movement, a vertex is randomly chosen and attempt to move by
a random length and direction, with the maximum possible distance restricted
by $0.15~{}l_{min}$. In a bond flip move, a single bond is chosen, which is a
common side of two neighbouring triangles, and this bond is cut and
reestablished between the other two unconnected vertices Penič et al. (2015);
Fošnarič et al. (2019); Sadhu et al. (2021). The maximum bond length is
restricted to $l_{max}=1.7~{}l_{min}$. The particle is assumed to be static
and does not have any independent dynamics. We update the system using
Metropolis algorithm, where any movement that increases the energy of the
system by an amount $\Delta W$ occurs with rate $exp(-\Delta W/k_{B}T)$,
otherwise it occurs with rate unity. Note that our model only captures the
wrapping and engulfment events, while the other upcoming processes, such as
the membrane fission, and the movement of the engulfed particle inside the
vesicle are not captured in the present model.

Figure S-1: Schematic representation of our model. 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.

###  S-1.2 Results for protein-free vesicle

Here, we discuss the engulfment of spherical particles by a protein-free
vesicle. In Fig. S-2a, we show the phase diagram in the $R-E_{ad}$ plane. The
background color is showing the adhered fraction $A_{ad}/A_{max}$, where
$A_{ad}$ is the adhered area and $A_{max}$ is the maximum adhered area. The
value of $A_{max}$ is in general larger than the surface area of the particle,
since we have a finite width of adhesive interaction ($l_{min}$), within which
the vesicle can fluctuate. This value may also be different for different
parameter regime, but the difference is almost negligible. In the calculation
of adhered area fraction (background color of Fig. S-2a) we implement the
following method: For a complete engulfment, we assume $A_{max}$ to be the
maximum adhered area for a given $E_{ad}$ with different values of $\rho$, and
divide all the adhered area by that value of $A_{max}$ to get the adhered area
fraction. For a partial engulfment, we choose the $A_{max}$ corresponding to a
higher $E_{ad}$ closest to the original value of $E_{ad}$, for which we have a
complete engulfment. We use similar methods for calculating the adhered area
fraction for the passive and active cases as well.

We note that smaller particles are difficult to engulf, as the bending energy
cost becomes higher than the adhesion energy gain (Movie-S2). So, in order to
engulf a small particle, $E_{ad}$ should be increased to large values. On the
other hand, larger particle is engulfed at smaller $E_{ad}$ (Movie-S1). Since
the size of the vesicle is finite, a much larger particle will not be engulfed
by the vesicle, rather the vesicle will spread over it. So, there is an
optimal size of the particle, that requires the smaller $E_{ad}$ to engulf. We
consider this optimal radius to be $R=10~{}l_{min}$ and use this value in the
main paper.

For a given values of $E_{ad}$ we choose two particles with different sizes
$R$, such that for one value, the engulfment is partial, while for the other
value, the engulfment is complete. We then compare the adhesion energy gain
and bending energy cost for both the cases in Fig. S-2(b). Similar to the
previous cases (passive and active vesicle), here also, we can explain the
engulfment process by the competition between adhesion and bending energy
costs.

We also measure the engulfment time for a given $R(=10~{}l_{min})$ and for
various $E_{ad}$. The engulfment time shows a divergence as we reach the
critical $E_{ad}$ (below which engulfment is incomplete) (Fig. S-2c). We also
measure the curvature of the vesicle at the leading edge (rim) with the
adhered fraction $A_{ad}/A_{max}$ as the engulfment proceeds. We note that the
curvature at the leading edge increases linearly with $A_{ad}/A_{max}$ (Fig.
S-2d). We use this linear behaviour of rim curvature in our analytical
calculations.

Figure S-2: Engulfment of spherical particles by a protein-free vesicle. (a)
Phase-diagram in the $R-E_{ad}$ plane. The background color is showing the
adhered fraction. (b) Comparison of adhesion energy gain and bending energy
cost (both in units of $k_{B}T$) for partial and complete engulfment. Blue
boxes are for $R=8~{}l_{min}$ and green circles are for $R=10~{}l_{min}$. We
use $E_{ad}=1.30~{}k_{B}T$ here. (c) Engulfment time with $E_{ad}$ showing
divergence near the critical $E_{ad}~{}(\sim 1.20~{}k_{B}T)$ below which there
is no engulfment. Here, we use $R=10~{}l_{min}$. The unit of time is $5\times
10^{5}$ M C steps. (d) Mean curvature at the rim (the leading edge of the
vesicle) with adhered fraction $A_{ad}/A_{max}$ for a complete engulfment. The
parameters are same as in Fig. S-2b, green circles.

###  S-2.3 Analytical model for protein-free vesicle

Here, we present our simplified analytical (parametrical) model for protein
free case. We consider a three-dimensional vesicle formed by a closed two-
dimensional membrane, adhering to a spherical particle. We assume the system
to be symmetric, such that we can only show a two-dimensional projection of
the system on a plane (Fig. S-3). The vesicle area is $A=4\pi R_{0}^{2}$, and
the particle radius is $R$, such that $R<R_{0}$. As the vesicle engulfs the
particle, the adhered fraction increases. The adhered area can be written as,
$A_{ad}=2\pi xR$, where $x$ is the thickness of the spherical cap of the
particle that is adhered, as shown in Fig. S-3. The leading edge of the
membrane (rim) is highly curved, and the shape is assumed to be similar to a
half-torus with one principal curvature $1/r$ and the other $1/h$. We assume
$r\ll h$ and hence the other principal curvature is neglected. The area of
this region can be written as $A_{rim}=2\pi^{2}rh$, where
$h=\sqrt{2Rx-x^{2}}$.

Figure S-3: Schematic diagram of our analytical model. A spherical particle
with radius $R$ is being engulfed by a cell membrane of size $R_{0}$. We show
the projection of the system on a plane.

The bending energy of the adhered section of the membrane is,

$W_{b1}=\frac{\kappa}{2}A_{ad}(2/R)^{2}$ (S-3)

and, the bending energy of the rim section is,

$W_{b2}=\frac{\kappa}{2}A_{rim}(1/r)^{2}$ (S-3)

where, the mean curvature at the rim has the form, $1/2r=1/R_{0}+\alpha x/R$
(from simulation data in Fig. S-2d, where $\alpha$ is a fitting parameter),
where $\alpha$ a constant and $A_{ad}/A\sim x/2R$, such that at $x=0$, mean
curvature becomes $1/R_{0}$, which is the mean curvature of the unadhered
spherical vesicle.

We assume that the rest part of the unadhered vesicle have the same mean
curvature as before, $i.e.,$ $1/R_{0}$, thus, the bending energy of this
section will be,

$W_{b3}=\frac{\kappa}{2}(A-A_{rim}-A_{ad})(2/R_{0})^{2}$ (S-3)

The total bending is thus given as,

$W_{b}=W_{b1}+W_{b2}+W_{b3}$ (S-3)

We can now compare the adhesion energy gain and bending energy cost, as the
engulfment proceeds, and can calculate the transition line.

####  S-3.3.1 Graphical solution for the transition line in the $R-E_{ad}$
plane

For a given value of $E_{ad}$, we plot the adhesion energy gain ($E_{ad}$) and
the bending energy cost $\partial W_{b}/\partial A_{ad}$ with adhered area
fraction, for few values of $R$ (Fig. S-4). For very small $R$, the adhesion
strength is not enough to overcome the bending energy cost and the particle
will not be engulfed (Fig. S-4a). For a critical value of $R$, the adhesion
strength will just be sufficient to overcome the bending energy cost, and the
particle will be engulfed, as shown in Fig. S-4b. If we further increase $R$,
the adhesion energy gain will be always higher than the bending energy cost,
and the particle will easily be engulfed (Fig. S-4c).

Figure S-4: Comparison of adhesion energy gain (red line) and bending energy
cost (green line) for few values of $R$ ($=8,10,12~{}l_{min}$) and given
$E_{ad}=0.62~{}k_{B}T/l_{min}^{2}$. Other parameters are,
$R_{0}=20~{}l_{min}$, $\alpha=0.065$, $\kappa=20~{}k_{B}T$.

We calculate the transition line by finding the line corresponding to $E_{ad}$
which is just touching the peak of bending energy cost $\partial
W_{b}/\partial A_{ad}$. We show this transition line in Fig. S-5, and also
compare with the simulation line and the classical result from Ref. Lipowsky
and Döbereiner (1998). For the analytical model, we define $E_{ad}$ as the
adhesion energy per unit area, while in our simulation, we define it as the
adhesion energy per vertex. While comparing the simulation and analytical
results, we properly scale $E_{ad}$ such that they become consistent with each
other.

Figure S-5: Transition line for engulfment in the $R-E_{ad}$ plane and
comparison with simulations. In our analytical calculations, we define
$E_{ad}$ as the adhesion energy per unit area, while in simulation, we define
it as the adhesion energy per adhered node. So, while comparing, we scale
$E_{ad}$ properly, so that the definition remains consistent with each other.
The green circles are simulation result. The analytical prediction for the
critical value of $R$ with $E_{ad}$ (blue circles), which is close to the
classical prediction of $E_{ad}\sim 1/R^{2}$ Lipowsky and Döbereiner (1998),
shown by black solid line. For simulation data, the parameters are same as in
Fig. S-2a, and for analytical results, parameters are same as Fig. S-4.

###  S-5.4 Extension of the analytical model for passive curved (convex)
proteins case

We here extend our analytical model for the passive curved protein case, with
spontaneous curvature $c_{0}=1.0~{}l_{min}^{-1}$. Let $\rho_{0}$ be the
average protein density on the membrane, $\rho_{r}$ be the density at the rim
and $\rho_{s}$ be the density on the rest of the unadhered part of the
vesicle. We assume there are no proteins on the membrane that is adhered to
the spherical substrate. Thus,

$A\rho_{0}=A_{rim}\rho_{r}+(A-A_{rim}-A_{ad})\rho_{s}$ (S-5)

The bending energy of the adhered section is given by,

$W_{b1}=\frac{\kappa}{2}A_{ad}(2/R)^{2}$ (S-5)

The bending energy of the rim is,

$W_{b2}=\frac{\kappa}{2}A_{rim}(1/r-c_{0}\rho_{r})^{2}$ (S-5)

where, the mean curvature at the rim, $1/2r=1/R_{0}+\alpha x/R$ and
$\rho_{r}=\rho_{0}+\beta x/R$ is the density at the rim (validated from
simulation, Fig. S-6), where $\alpha=0.093$ and $\beta=0.085$ (determined from
Fig. S-6). Assuming that the rest part of the unadhered vesicle have the same
curvature of $1/R_{0}$, the bending energy of this part of the vesicle is
given by,

$W_{b3}=\frac{\kappa}{2}(A-A_{rim}-A_{ad})(2/R_{0}-c_{0}\rho_{s})^{2}$ (S-5)

Thus, the total bending energy cost is given by,

$W_{b}=W_{b1}+W_{b2}+W_{b3}$ (S-5)

####  S-5.4.1 Mean curvature and protein density at the rim (from simulation)

For the passive case with small protein density, we determine the mean
curvature of the vesicle at the rim and plot it with adhered fraction of the
particle, $A_{ad}/A_{max}$ in Fig. S-6a. Similar to the protein-free case,
here we note that the rim curvature seems to increase linearly, with a little
higher slop than the protein-free case. We also determine the protein density
at the rim, and plot it with adhered fraction of the particle,
$A_{ad}/A_{max}$ (Fig. S-6b). The rim density also seems to vary linearly with
the adhered fraction.

Figure S-6: Mean curvature and protein density at the highly curved rim for
passive case. (a) The mean curvature at the rim with adhered fraction of the
particle $A_{ad}/A_{max}$. (b) The protein density at the rim with
$A_{ad}/A_{max}$. The parameter used here are, $R=10~{}l_{min}$,
$E_{ad}=1.04~{}k_{B}T$, and $\rho=3.2~{}\%$.

####  S-6.4.2 Graphical solution for the engulfment transition line in the
$\rho-E_{ad}$ plane

Similar to the protein-free case, here also, we calculate the transition line
in the $\rho-E_{ad}$ plane by using graphical method. We compare the adhesion
energy gain and bending energy cost with adhered area fraction, for few values
of protein density $\rho$ and a given $R=10~{}l_{min}$ and given
$E_{ad}=0.39~{}k_{B}T/l_{min}^{2}$ in Fig. S-7. For a small $\rho$, the
vesicle will remain unadhered (Fig. S-7a). As we increase $\rho$, the adhesion
energy gain will just become sufficient to overcome the bending energy cost,
which is the transition value (Fig. S-7b). If we further increase $\rho$, the
vesicle will be easily engulfed (Fig. S-7c), but for very high $\rho$ the
vesicle will not be fully engulfed rather will stop before full engulfment
(Fig. S-7d). The analytic phase transition line is shown in Fig. S-8, compared
to the simulation results (Fig. 1(a), main paper). The analytic model captures
correctly the qualitative features of the engulfment transition line.

Figure S-7: Comparison of adhesion energy gain (red line) and bending energy
cost (green line) for few values of $\rho$ ($=5~{}\%,7~{}\%,9~{}\%,13.5~{}\%$)
and given $E_{ad}=0.39~{}k_{B}T/l_{min}^{2}$. Other parameters are,
$R=10~{}l_{min}$, $R_{0}=20~{}l_{min}$, $\alpha=0.093$, $\beta=0.085$,
$\kappa=20~{}k_{B}T$. Figure S-8: Transition line for engulfment in the $\rho-
E_{ad}$ plane and comparison with simulations. The red circles are simulation
result, and the blue line is the analytical prediction. For the simulation
results, the parameters are the same as in Fig. 1(a) of the main text. For
analytical results, parameters are the same as in Fig. S-7.

###  S-8.5 Variation of engulfment time with $\rho$ for passive case

We measure the engulfment time for a given $E_{ad}$ and various values of
$\rho$ in Fig. S-9. We note that the engulfment time diverges as we reach the
critical $\rho$ ($\sim 2.25~{}\%$).

Figure S-9: Engulfment time with $\rho$ for passive case. The engulfment time
diverges for critical $\rho\sim 2.25~{}\%$, below which there is no
engulfment. Here, we use $R=10.0l_{min}$ and $E_{ad}=1.0~{}k_{B}T$.

###  S-9.6 Calculation of contributions due to active forces

In this section, we estimate the contribution of active forces in the
engulfment process by calculating the work done by the active forces. Let us
assume a force of magnitude $F$ is acting on the membrane in the tangential
direction of the surface of the particle (perpendicular direction of the
radial vector of the particle) due to which the membrane extends to an
infinitesimally small width $dx$ (see Fig. S-10a). The circumferential length
of the small extended strip $dl=2\pi h$. Thus, the work done by the active
force in this process is, $dW_{F}=Fdx$, and the increase in the adhered area
$dA_{ad}=dx~{}dl=2\pi hdx$. Thus, the active energy gain, or the active energy
per unit adhered area is given by,

$\partial W_{F}/\partial A_{ad}=F/dl=F/2\pi h$ (S-9)

In our simulation, we choose a region close to the spherical particle (the
region in between two spheres of radius $R$ and $R+R_{c}$, where we assume
$R_{c}=3~{}l_{min}$), and assume that all the proteins inside this region is
contributing in the engulfment process. We then calculate the component of
force of all the proteins directed in the perpendicular direction of the
radial vector of the particle, as shown by arrows in Fig. S-10b. The large
arrow is showing the direction of net contribution. The rim length $dl$ can be
estimated from the adhered fraction of area, and thereby the contribution due
to active forces can easily be estimated.

Figure S-10: Calculation of contribution due to active forces in the
engulfment process. (a) A schematic diagram for the calculation of active work
done in the engulfment process. (b) A typical snapshot of vesicle engulfing a
spherical particle forming an arc-like protrusion. The arrows are showing the
direction of force which is perpendicular to the radial direction of the
particle. The parameters used here are $E_{ad}=1.0~{}k_{B}T$, $\rho=1.6~{}\%$
and $F=2.0~{}k_{B}T/l_{min}$.

###  S-10.7 Engulfment by RAW 264.7 macrophages

Here, we show another dataset of the polystyrene particle engulfment by a RAW
264.7 macrophage using particle functionalized with immunoglobulin G (Fig.
S-11).

Figure S-11: Engulfment of Immunoglobulin G-coated polystyrene particle by RAW
264.7 macrophage. Frames from a lattice light sheet movie (maximum intensity
projections) of a cell transfected with mEmerald-Lifeact are shown. Time is
indicated in min:sec, scale bar $5~{}\mu m$

###  S-11.8 Configurations for large force (F)

For a given $E_{ad}$ and $\rho$, if we increase $F$ to very large value, at
some large value of $F$, the vesicle will deadhere from the particle, and will
end up with no engulfment. In Fig. S-12a, we show such an example for
$E_{ad}=1.0~{}k_{B}T$ and $F=3.0~{}k_{B}T/l_{min}$ for very small protein
density ($=1.6~{}\%$). We note that vesicle gets deadhered from the surface of
the particle with time, and finally ends up with no engulfment (Movie-S14).

In the regime of large $F$, if we further increase $\rho$, a free vesicle
(without adhesive spherical particle) can transform into a pancake-like shape
Fošnarič et al. (2019). In this regime, if we allow the vesicle to engulf the
particle, it can engulf the particle if the force is not too large (Fig.
S-12b). Since the engulfment process here is faster than the pancake
transition process, the vesicle finally engulfs the particle before it
transform into a pancake-like shape (Movie-S15). However, if we further
increase $F$ to even large value, we will again reach to a regime, where we do
not have any engulfment (Fig. S-12c, Movie-S16).

Figure S-12: Engulfment of spherical particles by the vesicle with large
force. (a) Configuration For very large force but small protein density. The
engulfment becomes difficult and the particle ends up with no engulfment in
this case. Here, we use $E_{ad}=1.0~{}k_{B}T$, $\rho=1.6~{}\%$ and
$F=3.0~{}k_{B}T/l_{min}$. (b) Configurations for a complete engulfment with
very large $\rho$ and $F$. Here, we use $E_{ad}=1.0~{}k_{B}T$, $\rho=9.6~{}\%$
and $F=3.50~{}k_{B}T/l_{min}$. (c) Configurations for no engulfment with very
large $\rho$ and even larger $F$. Here, we use $E_{ad}=1.0~{}k_{B}T$,
$\rho=9.6~{}\%$ and $F=4.0~{}k_{B}T/l_{min}$.

###  S-12.9 Reorientation of vesicle with respect to non-spherical particles
of high aspect ratio

In Fig. S-13a we show the reorientation process, first for a vesicle with
passive curved proteins, that is initially in contact with a prolate particle
from the top. Since, the top of a prolate shape with high aspect ratio is
highly curved, the vesicle rotates and moves to the side of the particle, and
finally engulfs it (Movie-S17). We quantify this rotation by the angular
motion ($\theta$) of the center of mass of the vesicle with respect to the
particle (see S-13a, second inset). The angle becomes close to $\pi/2$ when
the vesicle rotates completely from top to side, and then it engulfs the
particle. The reorientation process in this case is fast, thereby, the total
engulfment times for either top or side initial conditions are very similar
(blue circles and red boxes in Fig. 5, main paper, for $R_{x}/R_{z}<1$).

The reorientation of the vesicle from its initial position may not always be
beneficial for engulfment. For an oblate particle with high aspect ratio, we
note that the passive vesicle can engulf the particle from side quite easily
(Fig. S-13b; Movie-S18). However, we frequently observed that an initial
fluctuation caused the vesicle to slide over to the flat top regions, where it
maximizes its adhesion energy at low bending cost. However, in this position
the vesicle has to overcome a large bending energy barrier along the entire
sharp equator of the oblate shape. For $R_{x}=R_{y}>11~{}l_{min}$ this barrier
leads to no engulfment from the top orientation, and therefore reorienting to
the top from the side will end up with no engulfment (Fig. S-13c; Movie-S19)
van Zon et al. (2009). In this case, we calculate the engulfment time by
averaging only those realizations that end up with complete engulfment, while
we discard the non-engulfed cases that shifted to the top position.

In Fig. S-13d we compare the bending energy cost and adhesion energy gain
($E_{ad}$, dashed line) per adhered node, for the vesicles with passive
proteins of Fig. S-13(a-c). Similar to the engulfment of spheres (Fig. 1(b),
main paper), we note that for the partial engulfment the bending energy cost
increases above the adhesion strength, while for the complete engulfment the
bending energy cost per node remains lower than $E_{ad}$.

For vesicles with active proteins we find similar reorientation of the vesicle
with respect to its initial position on the particle. In Fig. S-13e we show
such a case for a prolate particle with high aspect ratio. Similar to the
passive case (Fig. S-13a), the vesicle rotates to the side, to avoid the
highly curved pole, and then engulfs the particle from the side (Movie-S20).
Since the active case is more dynamic, the vesicle sometime takes more time to
reach the side orientation, and thereby the engulfment time is larger than
when it is started from side. We also show similar observation for an oblate
particle with high aspect ratio, where an active vesicle rotates from the side
of an oblate particle to the top position (similar to the passive case Fig.
S-13c) and then engulfs it (Fig. S-13f, Movie-S21). For an oblate shape, the
engulfment is easier from the side (as shown for the passive case Fig. S-13b),
but since the active vesicle is very dynamic, it does not engulf from side,
but rather it reorients to the top in every realization and finally engulfs.
This reorientation is however very fast, as the vesicle gains large adhesion
energy in this process, thereby the engulfment time starting from either top
or side are comparable (green circles and magenta boxes in Fig. 5, main paper,
for $R_{x}/R_{z}>1$).

We are therefore presented with a puzzle: what allows the active proteins to
robustly engulf the oblate particle from the top orientation, whereas the
passive vesicle faced a bending energy barrier that stalled the engulfment.
This puzzle is further emphasized when we compare the bending energy cost and
adhesion energy gain ($E_{ad}$) per adhered node, for both the prolate and
oblate cases in Fig. S-13g. We note that for the oblate shape the bending
energy cost can increase above the adhesion strength, as for the passive
system (Fig. S-13d), but nevertheless the engulfment process does not stop. In
this case, the active forces provide a mechanism to engulf the particle which
we describe below.

For an oblate shape with high aspect ratios, we find that the active forces do
not induce the formation of an arc-like aggregate of proteins at the rim of
the phgocytic cup, as happens during the engulfment of the spherical particle
(Figs. 2(c-e), main paper). The active forces rather stretch the vesicle
sideway, perpendicular to the particle surface, and this allows the stretched
membrane afterwards to bend over the highly curved equator region of the
oblate shape, leading to complete engulfment (Fig. S-13f). On the other hand,
the passive vesicle is not stretched and remains spherical-like and fails to
engulf the particle even for a much larger protein density (Movie-S22). This
stretching of the vesicle by the active forces is quantified by measuring the
volume of the vesicle, which becomes much lower than that of the passive
vesicle as the adhered fraction increases (Fig. S-13h). We also show the
cross-section view of both the passive and active vesicles for
$A_{ad}/A_{max}\sim 0.45$ in the inset of Fig. S-13h, where the engulfment by
the passive vesicle is stalled. Since the vesicle gets stretched for the
active case, there is a significant bending energy contribution which arises
far away from the surface of the particle, while for the passive vesicle the
bending is dominated by the highly curved rim along the vesicle-particle
leading edge (Fig. S-14). This indicates that the bending energy cost shown in
Fig. S-13g also has a significant contribution due to a membrane that is far
away from the particles, and is therefore not directly affected by the
progression of the membrane-particle adhesion. The sideways stretching by the
active forces is demonstrated by the force components plotted in Fig. S-13i.
At the time where the vesicle is poised at the highly curved equator of the
oblate shape, the vesicle is getting stretched by forces that are directed
radially outward ($F_{r}$), while the engulfment is proceeding along the $y$
direction. We note that the radial component $F_{r}$ is much larger that the
$F_{y}$ component. We therefore identify here another mechanism by which
active forces enable the engulfment to overcome bending energy barriers, which
is qualitatively different from the mechanism described in Fig. 2, main paper.
In Fig. S-15, we show some configurations close to the engulfment time to show
how the stretching of the vesicle finally leads to full engulfment of the
oblate particle.

Figure S-13: Reorientation of the vesicle with respect to its initial position
on the surface of spheroid particles of high aspect ratio. The reorientation
of the vesicle is quantified in (a-c) by the angular motion of the center of
mass with respect to particle’s surface (denoted in the inset of (a)). (a) A
vesicle with passive proteins initiated from the top of a prolate spheroid,
rotates over time, and finally completes the engulfment from the side. Here,
we use $R_{x}=R_{y}=8.4~{}l_{min}$, $R_{z}=14.16~{}l_{min}$. (b-c) A passive
vesicle is initiated from the side of an oblate shape and fully engulfs from
side in (b), while it rotates to the top of the oblate shape and stalls around
half-engulfment in (c). Here, we use $R_{x}=R_{y}=11.4~{}l_{min}$,
$R_{z}=7.96~{}l_{min}$. (d) Comparison of bending energy cost $\partial
W_{b}/\partial A_{ad}$ and adhesion gain ($E_{ad}$, dashed line) per adhered
node (both having units of $k_{B}T$), for (a-c). (e) A vesicle with active
curved proteins is initiated at the top of a prolate shape, rotates and
engulfs from side. Here, we use $R_{x}=R_{y}=7.6~{}l_{min}$,
$R_{z}=16.29~{}l_{min}$. (f) Active vesicle initiated at the side of an oblate
shape finally engulfs from the top. Here, we use $R_{x}=R_{y}=12.0~{}l_{min}$,
$R_{z}=6.77~{}l_{min}$. (g) Comparison of bending energy cost and adhesion
gain ($E_{ad}$, dashed line) per adhered node, for (e-f). (h) Variation of the
vesicle volume (in units of $10^{4}~{}l_{min}^{3}$) for both active and
passive proteins, initiated from the side of an oblate shape. The inset is
showing the cross section of typical snapshots for both case for
$A_{ad}/A_{max}\sim 0.45$. (i) Component of the active force along different
direction for the active vesicle of (h), with $A_{ad}/A_{max}\sim 0.45$. For
(h-i), we use $R_{x}=R_{y}=12.0~{}l_{min}$, $R_{z}=6.77~{}l_{min}$. For the
passive case we use $\rho=4.8~{}\%$ and for the active case, $\rho=1.6~{}\%$,
$F=1.0~{}k_{B}T/l_{min}$. We used $E_{ad}=1.0~{}k_{B}T$ for all the plots.

####  S-13.9.1 Engulfment of oblate particles: Active vesicle shows high mean
curvature far away from the particle

In Fig. S-13h, we compare the active and passive cases for the engulfment of
an oblate particle with high aspect ratio, when the adhered area fraction
$A_{ad}/A_{max}\sim 0.45$. Since the active vesicle gets stretched by the
active forces, and thereby the bending energy cost will have dominant
contribution far away from the surface of the particle. In order to quantify
this, we measure the mean curvature of the vesicle section, which is having a
perpendicular distance from the surface of the particle in between $r$ and
$r-1$ and plot it as a function of $r$ for both the active and passive cases
in Fig. S-14. As expected, the active vesicle have mean curvature larger than
passive far away from the surface of the particle and it drops slowly with
$r$. On the other hand, the passive vesicle has larger contribution only for
small $r$ and it drops faster with $r$.

Figure S-14: Mean curvature at a distance $r$ from the surface of an oblate
particle for $A_{ad}/A_{max}\sim 0.45$. Here, we use
$R_{x}=R_{y}=12.0~{}l_{min}$, $R_{z}=6.77~{}l_{min}$ for the oblate particle.
For passive case, we use $\rho=4.8\%$. For active case, we use $\rho=1.6~{}\%$
and $F=1.0~{}k_{B}T/l_{min}$. We use $E_{ad}=1.0~{}k_{B}T$ for both the cases.

####  S-14.9.2 Configurations for the engulfment of oblate shape by active
vesicle close to the engulfment time

Here, we show the configurations for the engulfment of an oblate shape by an
active vesicle, discussed in Fig. S-13(h-i). Here, we zoom into the time close
to the engulfment to observe how the full engulfment proceeds.

Figure S-15: Configurations for the engulfment of an oblate by an active
vesicle close to the engulfment time.

###  S-15.10 Engulfment of bacteria-like particles

Bacterial pathogens are often engulfed by phagocytic cells, as well as by
cells which they invade Rottner et al. (2005b). We therefore investigate the
engulfment of a sphero-cylindrical shape, resembling the shapes of different
bacterias such as E. coli or Bacillus subtilis Mölller et al. (2012); Richards
and Endres (2016). This shape is an extreme case of a prolate spheroid, with
an aspect ratio of $1:5$. We use here a larger vesicle size ($N=6127$), since
the large aspect ratio of the particle requires a larger membrane area for
engulfment.

Figure S-16: Engulfment of a sphero-cylindrical shape. The configurations are
shown as function of the simulation time for a vesicle with active curved
proteins, when initiated from the side (top panel), or from the top (bottom
panel). The dashed vertical line in the lower panel shows the time at which we
switch off the active forces of proteins that located above the horizontal
dashed line shown for $t=230$. The vesicle parameter values are $N=6127$,
$E_{ad}=1.5~{}k_{B}T$, $F=1.5~{}k_{B}T/l_{min}$, $\rho=1.6~{}\%$, while for
the sphero-cylindrical shape: total length $l=60~{}l_{min}$ and radius
$r=6~{}l_{min}$, such that the aspect ratio is $1:5$.

When the initial contact between the vesicle and the particle is from the
side, the engulfment proceeds smoothly (Fig. S-16, upper panel, Movie-S23).
However, when initiated from the top, the initial engulfment stage is faster
but later on the engulfment slows down and almost stalls (Fig. S-16, lower
panel, before the dashed vertical line). This type of stalling is also
observed in experiments Richards and Endres (2016). In our simulations the
stalling arises due to membrane tension, caused by active proteins that are
located away from the engulfed article, and pull the membrane in the opposite
the direction. We demonstrate this by turning off the active forces for all
proteins that are located above the horizontal dashed line shown for $t=230$,
i.e. all the proteins that are on the opposite side of the phagocytic cup are
made passive. The dashed vertical line in Fig. S-16 lower panel shows the time
at which we turn off these active proteins, and we note that as a result the
engulfment process proceeds quickly afterwards to completion (Movie-S27).

We can compare these simulation results to observed bacterial phagocytosis. In
Mölller et al. (2012). it was found that the velocity of engulfment was faster
when the initial contact was from the pole of the bacteria. We find a similar
behavior, as long as there is no restriction from membrane tension. Such
tension can stall the engulfment in our simulations (Fig. S-16 lower panel),
and indeed highly filamentous bacteria are found to avoid engulfment by immune
cells Prashar et al. (2013), which we can attribute to the restriction of
finite membrane area and the resulting membrane tension. The critical role of
membrane tension during phagocytosis was previously shown experimentally
Masters et al. (2013).

We also explore the engulfment of dumb-bell like shapes, resembling the shape
of fungi such as budding yeast Clarke et al. (2010). A similar effect of
membrane tension is also observed in our simulations during the engulfment
such dumb-bell shaped particle (Fig. S-17, Movie-S25,S26). We found that while
passive curved proteins can give rise to a smooth engulfment that does not
stall (Fig. S-17a), the active vesicle can easily stall after engulfment of
one lobe of the particle (i.e. when the leading edge is at the narrow neck)
(Fig. S-17b). Releasing the membrane tension induced by the active proteins at
the back of the vesicle, can release the vesicle and allow full engulfment.
Similar hourglass shapes were previously shown theoretically to be susceptible
to stalled engulfment Richards and Endres (2016), as well as in experiments
observing phagocytosis of dividing yeast cells Clarke et al. (2010); Dieckmann
et al. (2010) or highly deformable particles Vorselen et al. (2021). We also
show the mean cluster size and the adhered fraction for both the passive and
the active case in Fig. S-17(c-d).

Figure S-17: Engulfment of dumb-bell shaped particle. (a) Configurations with
time for passive case. (b) Configurations with time for active case. The
vertical dashed line is showing the time, when we remove the force from the
proteins at the back. The horizontal dashed line on the configuration
corresponding to $t=180$ shows the coordinate above which we switch off the
active forces of all the proteins. (c) Adhered area fraction with time. (d)
Mean cluster size with time. Blue boxes are for active case and green circles
are for passive case. The parameter values are $N=6127$,
$E_{ad}=1.5~{}k_{B}T$, $\rho=1.6~{}\%$. The radius of the spherical beads are
$10~{}l_{min}$ each, and the overlapped region $l_{p}=2~{}l_{min}$, such that
total length $L=4R-l_{p}=38~{}l_{min}$. For active case, we use
$F=1.5~{}k_{B}T/l_{min}$.

###  S-17.11 Results for multicomponent membrane with proteins of different
intrinsic curvature

Here, we show our results for multicomponent membrane with proteins of
different intrinsic curvature. We consider convex ($c_{0}=1.0~{}l_{min}^{-1}$)
as well as concave ($c_{0}=-1.0~{}l_{min}^{-1}$) proteins. We note that
addition of concave proteins makes the engulfment easier in the parameter
regime where only convex proteins fails to engulf the particle. In Fig. S-18,
we show the engulfment by passive proteins with small $E_{ad}$
($=0.80~{}k_{B}T$) and convex protein density $\rho_{p}=3.2~{}\%$ such that
there is no engulfment, and compare with the case where in addition, we have
concave proteins of density $\rho_{n}=3.2~{}\%$. The convex (concave) proteins
have attractive interaction only with the convex (concave) one. We note that
the concave proteins aggregate close to the particle, and reduces the bending
energy there. This allows the vesicle to adhere more and engulf the particle
(Movie-S24).

Figure S-18: Results for multicomponent membrane with proteins of different
intrinsic curvature. Here, we show the bending energy cost with adhered area
fraction, and compare it with the adhesion strength $E_{ad}$. Blue boxes are
for convex proteins, and green circles are for multiple proteins. Left insets
are for convex proteins and right insets are for multiple curvature proteins.
For multiple curvature case, red regions are denoting convex proteins, and
blue regions are denoted concave proteins. Here, we use $R=10.0~{}l_{min}$,
$E_{ad}=0.80~{}k_{B}T$, and $\rho_{p}=3.2~{}\%$ for convex protein case. For
multiple proteins case, we have in addition $\rho_{n}=3.2~{}\%$.

##  S-18 Supplementary movies

All the supplementary movies are available online in the link:
https://app.box.com/s/ske4vjlp6ci3vu7mz5izfp2iebxrmgm4

  * •

Movie-S1 Complete engulfment of a spherical particle of larger radius by a
protein-free vesicle. Here, we use $R=10~{}l_{min}$ and
$E_{ad}=1.30~{}k_{B}T$.

  * •

Movie-S2 Partial engulfment of a spherical particle of smaller radius by a
protein-free vesicle. We use here $R=8~{}l_{min}$ and $E_{ad}=1.30~{}k_{B}T$.

  * •

Movie-S3 Partial engulfment of a spherical particle by a passive vesicle with
small protein density. We use $R=10~{}l_{min}$, $\rho=2.4~{}\%$ and
$E_{ad}=1.0~{}k_{B}T$.

  * •

Movie-S4 Complete engulfment of a spherical particle by a passive vesicle with
medium protein density. Here, we use $R=10~{}l_{min}$, $\rho=4.0~{}\%$ and
$E_{ad}=1.0~{}k_{B}T$.

  * •

Movie-S5 Partial engulfment of a spherical particle by a passive vesicle with
very high protein density. Here, we sue $R=10~{}l_{min}$, $\rho=19.2\%$ and
$E_{ad}=1.0~{}k_{B}T$.

  * •

Movie-S6 Partial engulfment of a spherical particle by an active vesicle with
small force. The parameters are: $R=10~{}l_{min}$, $\rho=1.6~{}\%$,
$E_{ad}=1.0~{}k_{B}T$ and $F=0.10~{}k_{B}T/l_{min}$.

  * •

Movie-S7 Complete engulfment of a spherical particle by an active vesicle with
medium force. Here, we use $R=10~{}l_{min}$, $\rho=1.6~{}\%$,
$E_{ad}=1.0~{}k_{B}T$ and $F=0.40~{}k_{B}T/l_{min}$.

  * •

Movie-S8 Complete engulfment of a spherical particle by an active vesicle with
large force. We use here $R=10~{}l_{min}$, $\rho=1.6~{}\%$,
$E_{ad}=1.0~{}k_{B}T$ and $F=2.0~{}k_{B}T/l_{min}$.

  * •

Movie-S9 Engulfment of spherical particle by an active vesicle with large
$\rho$ and small $F$. Here, we show fragmented protein clusters on the
phagocytic cup at the early stage of engulfment. We use here $R=10~{}l_{min}$,
$E_{ad}=1.5~{}k_{B}T$, $\rho=6.4~{}\%$ and $F=1.0~{}k_{B}T/l_{min}$.

  * •

Movie-S10 Engulfment of spherical particle by an active vesicle with large
$\rho$ and large $F$. Here, we use $R=10~{}l_{min}$, $E_{ad}=1.5~{}k_{B}T$,
$\rho=4.8~{}\%$ and $F=2.0~{}k_{B}T/l_{min}$.

  * •

Movie-S11 Phagocytosis of Immunoglobulin-G-coated polystyrene bead (blue) by
RAW264.7 macrophage transfected with mEmerald-Lifeact (orange). Maximum
intensity projection of lattice light-sheet imaging. Scale bar, $5~{}\mu m$.
Time stamp: min:sec.

  * •

Movie-S12 Phagocytosis of Immunoglobulin-G-coated polystyrene bead (maroon) by
murine bone-marrow derived macrophage transfected with mEmerald-Lifeact
(gray). Maximum intensity projection of lattice light-sheet imaging. Scale
bar, $5~{}\mu m$. Time stamp: min:sec.

  * •

Movie-S13 Phagocytosis of Immunoglobulin-G-coated polystyrene bead (maroon) by
RAW264.7 macrophage transfected with mEmerald-Lifeact (gray). Maximum
intensity projection of lattice light-sheet imaging. Scale bar, $5~{}\mu m$.
Time stamp: min:sec.

  * •

Movie-S14 No engulfment of a spherical particle by an active vesicle with very
large force. The movie shows a case, where particle is not engulfed by the
active vesicle. Here, we use $R=10~{}l_{min}$, $\rho=1.6~{}\%$,
$E_{ad}=1.0~{}k_{B}T$ and $F=3.0~{}k_{B}T/l_{min}$.

  * •

Movie-S15 Complete engulfment of a spherical particle by an active vesicle
with very large force and large protein density. Here, we use
$R=10~{}l_{min}$, $\rho=9.6~{}\%$, $E_{ad}=1.0~{}k_{B}T$ and
$F=3.50~{}k_{B}T/l_{min}$.

  * •

Movie-S16 No engulfment of a spherical particle by an active vesicle with even
larger force and large protein density. We use here $R=10~{}l_{min}$,
$\rho=9.6~{}\%$, $E_{ad}=1.0~{}k_{B}T$ and $F=4.0~{}k_{B}T/l_{min}$.

  * •

Movie-S17 Complete engulfment of a prolate spheroid of high aspect ratio by a
passive vesicle. The vesicle is initially started from the top and then
rotates to the side and engulfs the particle. Here, we use
$R_{x}=R_{y}=8.4~{}l_{min}$, $R_{z}=14.16~{}l_{min}$, $\rho=4.8~{}\%$ and
$E_{ad}=1.0~{}k_{B}T$.

  * •

Movie-S18 Complete engulfment of an oblate spheroid of high aspect ratio by a
passive vesicle. The vesicle is initially started from the side, it engulfs
the particle without reorienting itself to the other side. Here, we use
$R_{x}=R_{y}=11.4~{}l_{min}$, $R_{z}=7.96~{}l_{min}$, $\rho=4.8~{}\%$ and
$E_{ad}=1.0~{}k_{B}T$.

  * •

Movie-S19 Partial engulfment of an oblate spheroid of high aspect ratio by a
passive vesicle. The vesicle is initially started from the side, and it
rotates to the top and ends up with no engulfment. The parameters are same as
the previous movie.

  * •

Movie-S20 Complete engulfment of a prolate spheroid of high aspect ratio by an
active vesicle. The vesicle is initially started from the top, and then
reorients itself to the side and engulfs the particle. Here, we use
$R_{x}=R_{y}=8.4~{}l_{min}$, $R_{z}=14.16~{}l_{min}$, $\rho=1.6~{}\%$,
$E_{ad}=1.0~{}k_{B}T$ and $F=1.0~{}k_{B}T/l_{min}$.

  * •

Movie-S21 Complete engulfment of an oblate spheroid of high aspect ratio by an
active vesicle. The vesicle is initially started from the side, and rotates to
the top and engulfs the particle. Here, we use $R_{x}=R_{y}=12.0~{}l_{min}$,
$R_{z}=6.77~{}l_{min}$, $\rho=1.6~{}\%$, $E_{ad}=1.0~{}k_{B}T$ and
$F=1.0~{}k_{B}T/l_{min}$.

  * •

Movie-S22 Comparison of active and passive engulfment of an oblate shaped
particle with high aspect ratio, initiated from side. For passive case the
engulfment is partial while for active case, the engulfment is complete. Here,
we use $R_{x}=R_{y}=12.0~{}l_{min}$, $R_{z}=6.77~{}l_{min}$,
$E_{ad}=1.0~{}k_{B}T$. For active case we use $\rho=1.6~{}\%$ and
$F=1.0~{}k_{B}T/l_{min}$. For passive case, we use $\rho=4.8~{}\%$

  * •

Movie-S23 Engulfment of a sphero-cylinder by an active vesicle, when initiated
from the top (from the spherical cap). Here, we use total number of vertices
on the vesicle, $N=6127$ radius of spherical cap $r=6~{}l_{min}$, total length
$l=60~{}l_{min}$, $\rho=1.6~{}\%$, $E_{ad}=1.50~{}k_{B}T$ and
$F=1.50~{}k_{B}T/l_{min}$.

  * •

Movie-S24 Engulfment of a sphero-cylinder by an active vesicle, when initiated
from the side (from the cylindrical part). The parameters are same as the
previous case.

  * •

Movie-S25 Engulfment of a dumb-bell shaped particle by a passive vesicle.
Here, we use $N=6127$ radius of spherical beads $R=10~{}l_{min}$, thickness of
the overlapped region of the two beads $l_{p}=2~{}l_{min}$, $\rho=1.6~{}\%$,
$E_{ad}=1.50~{}k_{B}T$.

  * •

Movie-S26 Engulfment of a dumb-bell shaped particle by an active vesicle. The
movie shows the engulfment of a dumb-bell shaped particle by an active vesicle
with $F=1.50~{}k_{B}T/l_{min}$. Other parameters are same as the previous
case.

  * •

Movie-S27 Engulfment of a spherical particle by a passive vesicle with
multiple curvature proteins. The movie shows a complete engulfment of
spherical particle of radius $R=10.0~{}l_{min}$ by a passive vesicle, with
$E_{ad}=1.0~{}k_{B}T$, convex protein density $\rho_{p}=3.2~{}\%$ and concave
protein density $\rho_{n}=3.2~{}\%$.

## References

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