BACKGROUND
phagocytosis is the ancient, evolutionarily conserved process by which eukaryotic cells bind, engulf, and destroy particles and cells larger than  <dig>  μm in diameter  <cit> .
the importance of phagocytosis is derived from its two main functions:  a feeding mechanism in single-cell organisms  <cit> , and  the clearance of pathogens, apoptotic and senescent cells from our body by immune cells  <cit> .
as part of our immune defense, phagocytosis is mainly performed by professional phagocytes, including macrophages, neutrophils, and dendritic cells.
initiation of phagocytosis occurs with recognition of the target particle either directly or via an opsonising molecule.
for instance the fc portion of immunoglobulin g  is recognized by the cell-surface receptor fcγriia  <cit> .
ligand-receptor binding triggers intracellular signaling  <cit> , resulting in remodeling of the actin cytoskeleton  <cit>  and coherent growth of cell membrane around the particle to form the phagocytic cup  <cit> .
eventually, the leading edge of the growing cup closes, and a membrane vesicle enclosing the particle  moves inside the cell.
subsequently, the phagosome fuses with vesicles containing enzymes  <cit> , acids  <cit> , and oxygen radicals  <cit>  to destroy the particle.
the biochemical pathways involved in phagocytosis are complex.
dozens of cell-surface receptors contribute to the recognition of a large variety of ligand molecules and subsequent particle engulfment  <cit> .
the fcγ receptor   <cit>  and complement receptor  <dig>  of the integrin receptor family  <cit>  are the most widely studied and understood receptors involved in phagocytosis.
fcγr-mediated phagocytosis proceeds through membrane protrusions and leads to thin cups  <cit> , whereas in cr3-mediated phagocytosis, particles appear to sink into the cell  <cit> .
spreading of the cell membrane over the particle involves actin-driven cell-shape changes similar to the processes involved in cell migration and adhesion  <cit> .
specifically for fcγr, binding to an igg-opsonized particle results in receptor clustering and recruitment of small gtpases of the rho family, which, via proteins of the wasp family, activate the arp2/ <dig> complex  <cit> .
the latter promotes branching of actin filaments, leading to an increase in the number of uncapped ends and to an isotropic growth of the actin network  <cit> .
additionally, the phagocytic cup has been shown to be enriched in gelsolin  <cit> , coronin  <cit> , and other regulators of actin polymerization.
all in all, this complex signaling pathway involves 100- <dig> different types of molecules  <cit> , rendering mathematical modeling at the molecular level impossible.
despite the huge biochemical complexity, the engulfment process shows a strong dependence on simple biophysical parameters.
first, it relies on the availability of extra membrane at the phagocytic cup  <cit> , provided by delivery of membrane vesicles  <cit>  or unwrinkling of membrane folds  <cit> .
second, completion of phagocytic uptake depends on the shape of the particle and, interestingly, on the initial orientation of the particle on the cell surface  <cit> .
for instance, experiments demonstrate that elongated spheroid polystyrene particles coated with igg are more efficiently engulfed when presented to the phagocyte with their tip first.
third, a recent study by one of the authors demonstrates that the biophysical requirements for phagocytosis lead to either complete phagocytosis or stalled cups due to the presence of a mechanical bottleneck  <cit> .
interestingly, the same study shows that engulfment appears to even proceed in cells treated with  cytochalasin d, an inhibitor of actin polymerization, indicating that biochemical pathways may not always be necessary for this initial stage of phagocytosis.
the mechanism of phagocytosis is only partially understood, with key insights provided more than three decades ago.
in the 1970's, griffin and his collaborators  <cit>  demonstrated that incomplete coating of particles with ligand results in only partial uptake.
this indicated that phagocytic uptake occurs via successive zipper-like ligand-receptor binding , and not by an all-or-nothing mechanism triggered at the onset of phagocytosis.
the zipper mechanism is the underlying assumption in a number of recent modeling works in phagocytosis  <cit>  and endocytosis  <cit> , mainly addressing the influence of the cell-membrane tension and ligand-receptor bond density on engulfment.
despite the general acceptance of the zipper mechanism, many of its biophysical requirements are insufficiently understood.
questions, so far unanswered, include what the energetic requirements of the zipper mechanism are, specifically what role actin polymerization plays in its progression during phagocytosis, and also whether the zipper mechanism can explain the particle-shape dependence of phagocytosis.
previous models were unable to fully address the particle-shape dependence, as they assume rotational symmetry around the axis connecting cell and particle.
additionally, large particle-to-particle variation in cup growth  <cit>  and cell-to-cell variation in the related process of endocytosis  <cit>  point towards the importance of stochasticity during the uptake, not captured in previous deterministic approaches.
recent experiments provide new insights into the biophysical mechanism for driving the membrane around the particle, suggesting a ratchet-type mechanism.
once started, phagocytosis progresses unidirectionally and irreversibly  <cit> .
this irreversible membrane progression is further supported by the loss of lipid and protein mobility at the phagocytic cup, observed using fluorescence recovery after photobleaching   <cit> .
while several models proposed mechanisms of force generation by actin polymerization , recent experiments based on fluorescent speckle microscopy demonstrate that actin does not directly push the membrane outwards.
instead, by filling gaps provided by membrane fluctuations , actin polymerization prevents the membrane from moving backwards like a ratchet  <cit> .
the relevance of such brownian ratchets in biology has previously been emphasized  <cit> .
the question is if a ratchet mechanism, together with energetic restrictions in membrane bending and stretching, can naturally lead to phagocytic uptake and account for the shape-dependence of phagocytosis.
in this work, we propose a ratchet-like biophysical model for the zipper mechanism.
this model differs from previous works in that it is, to our knowledge, the first fully three-dimensional stochastic model of phagocytic engulfment.
specifically, thermal membrane fluctuations, assumed to play a major role in our model, provide the energy source to locally deform the membrane and to build further ligand-receptor bonds for zippering the membrane around the particle.
actin polymerization makes ligand-receptor bonds effectively irreversible, i.e.
reinforced and stabilized for a significant amount of time.
to investigate the role of actin, we compare cup progression for the regular active zipper with a passive zipper model in which ligand-receptor binding remains specific and strong but reversible due to the absence of actin polymerization.
interestingly, we find that the passive zipper also leads to engulfment of small particles, rendering phagocytosis highly robust.
however, such passive engulfment is generally slower and produces much more variable phagocytic cups than the active zipper.
furthermore, our computer simulations lead to successful phagocytic engulfment in a broad range of parameters values, including different particle sizes.
for non-spherical particles, completion of engulfment depends strongly on particle shape and orientation.
our model further predicts that cup shape invariably depends on membrane biophysical parameters, in particular surface tension and cell-volume constraint.
to test the predicted difference between the active and passive zippers, we experimentally implement the two different types of zippers using cos- <dig> fibroblasts which, after transfection with gfp-tagged fcγ receptor, phagocytoze igg-coated polystyrene particles.
specifically, we performed phagocytic assays under three different conditions:  cells expressing wild-type fcγr for the active zipper ,  cells expressing a signaling-dead mutant receptor , which specifically binds igg ligand but is unable to signal to the actin cytoskeleton  <cit> , and  cells expressing wt-fcγr and treated with cytochalasin d .
the last two conditions represent two versions of the passive zipper due to the absence of actin polymerization in phagocytic cups.
to compare with our model, we systematically analyze confocal microscopy images, and quantitatively estimate cup variability for the three different conditions using small  and large  particles.
consistently with our simulations, phagocytic cups develop more slowly and are significantly more variable in the absence of actin polymerization.
our results provide new insights into the robustness of phagocytosis, as well as the role of bacterial cell shape in host-pathogen interactions.
RESULTS
ratchet model for the zipper mechanism
our model is based on the following experimental observations.
engulfment of quasi-spherical particles by neutrophils progresses continuously without significant pause or reversal, indicating that ligand-receptor binding is essentially irreversible  <cit> .
this irreversibility is further supported by frap and single-molecule experiments, which show that lipids and proteins in phagocytic cups, as well as ligand-bound fc-receptors are immobilized in an actin-dependent fashion  <cit> .
additional support for the notion of irreversible uptake was recently determined in a related context  <cit> .
fluorescent speckle microscopy of actin flow and image analysis during cell migration show that the membrane at the leading edge protrudes first, followed by actin polymerization to fill the gap between the membrane and the actin cortex.
such actin polymerization is mainly restricted to the leading edge due to signaling by receptors and/or localization of small gtpases of the rho family  <cit> .
the role of actin polymerization in phagocytosis is hence to stabilize ligand-receptor bonds and to rectify membrane movements in a ratchet-like fashion, leading to unidirectional movement of the leading edge of the engulfing cell.
dependence of phagocytic cup shape on membrane biophysical parameters
using our model for the zipper mechanism, we have successfully simulated phagocytic engulfment in a broad range of parameter values .
figure 2a shows two different characteristic cup shapes we obtained.
low surface tension , and tight cell-volume constraint , lead to a thin phagocytic cup since a thin cup requires extra membrane but little extra volume.
in contrast, weak volume constraint and/or high surface tension produce a broad cup.
based on the parameters explored, we chose intermediate values for both surface tension and cell-volume constraint as our standard parameters  for the remainder of our simulations in order to produce realistic cup shapes .
figure 2b shows that most parameter values can be changed independently by at least one order of magnitude, without negatively affecting engulfment completion.
note that changing simultaneously several parameters may affect engulfment more drastically.
our simulations also show that cup shape depends on the kinetics of engulfment, determined by membrane fluctuations and therefore temperature .
additionally, preventing thermal fluctuations  during a simulation stops cup progression.
this indicates that in our model membrane fluctuations are indeed required to bring receptors in close contact with ligand molecules on the particle, emphasizing their important role in the ratchet mechanism.
active versus passive uptake and the role of actin
although phagocytosis generally involves active processes such as actin polymerization in the cup   <cit> , recent reports indicate that phagocytosis may still work in an actin-independent manner.
indeed, phagocytic uptake was observed despite treating phagocytes with  cytochalasin d  <cit> .
hence, ligand-receptor binding may be sufficient in guiding the cell membrane around the particle under certain conditions .
to investigate the energetic requirements of the zipper mechanism, we implemented simulations of the passive zipper.
in these simulations, ligand-receptor bonds are not stabilized by actin polymerization and can unbind at later times, i.e.
remain reversible.
hence, engulfment may still progress if the energetic cost of stretching and deforming the membrane is offset by the ligand-receptor binding energy in the presence of thermal membrane fluctuations.
particle size matters for passive, not for active zipper
experiments show that phagocytosis is relatively insensitive to particle size  <cit> .
using our model for the active zipper, we simulated engulfment of spherical particles with different radii ranging from  <dig>  to  <dig>  μm.
figure 2c shows that engulfment progresses normally for small and large particles.
hence the active zipper mechanism is sufficiently robust to allow engulfment of differently sized particles using the same set of biophysical parameters, although engulfment of large particles requires more time.
noticeably, large particles  were taken up via more regular phagocytic cups than small particles , indicating that active processes may be required for engulfment of large particles.
to confirm this observation we have simulated engulfment of large  <dig> μm-radius particles by both the active and the passive zipper, shown in additional file  <dig>  figure s <dig>  while the active zipper resulted in complete uptake of the particle, the passive zipper only engulfed a few percent of the particle's surface area.
thus the difference in phagocytic efficiency between the two zipper types was exacerbated for large particles, reflecting the importance of actin polymerization for engulfment of large particles.
experimental test of model predictions
to test our model predictions and to specifically compare active with passive engulfment for small and large particles, we transfected cos- <dig> cells with either wild-type fcγr  or a signaling-dead mutant receptor .
cells expressing the wild-type receptor are expected to perform active engulfment, whereas cells expressing the signaling-dead mutant receptor are expected to perform passive engulfment.
as a control, passive engulfment is additionally implemented by treating cells expressing wt-fcγr with  <dig>  μm of cytochalasin d , which prevents actin polymerization .
synchronized phagocytosis assays using small  and large  igg-opsonized polystyrene particles were carried out, and, after fixation, receptor localization in phagocytic cups was visualized by fluorescence confocal microscopy.
cells were imaged at different time points during phagocytosis, and at each time point, three to eight imaged cells were each engulfing simultaneously between four and twenty particles .
consequently, for each condition we analyzed at least seventy particles.
to test whether passive engulfment leads to more variable cups than active engulfment, we developed an image-analysis method illustrated in figure 4b-f. the cup shape varibility was quantified by the standard deviation of the distribution of cell-membrane  height around the particle, divided by the square root of the average membrane height.
the unit of membrane height is given by the distance  between consecutive confocal image planes .
figure 5a shows that for small particles engulfed between  <dig> and 40% of their surfaces, cup variability increases from cells transfected with wild-type receptor to cells transfected with signaling-dead mutant receptor to wt-fcγ+cytod cells.
the lowest variability, found for cells expressing wild-type receptor, is statistically significant against both passive zipper types .
this result is consistent with model predictions: figure 5a, inset shows the cup variability from simulations, revealing that the active zipper leads to significantly less variable cups.
in contrast, for the ranges of engulfment between  <dig> and 60%  and between  <dig> and 100%  we observed no noticeable difference in cup variability between the three experimental conditions, while our model consistently predicts more variable cups for the passive zipper .
this discrepancy may indicate that active processes such as contraction by myosin motor proteins become important at later stages of engulfment, limiting our model's full validity to the early events in phagocytosis .
to confirm that our results are independent of the specifics of the analysis method used, we also analyzed phagocytic cups with an alternative, albeit less accurate, method .
this method consists of determining the distribution of fcγr-gfp fluorescence intensity around the particle at its equator plane, restricting the analysis to approximately half taken up particles .
the standard deviation of this distribution provides an alternative measure of the cup variability.
we arrived at the same conclusion, confirming our result of the difference in cup variability .
finally, note that temperature-induced synchronization is imperfect and may lead to variability in cup growth  <cit> .
however, our measures of the variability in cup shape are independent of such an effect since we include all time points together in the analysis .
our experiments further show that cup shape ranges from regular to variable for all three experimental conditions, but that the frequency of different cup shapes depends on the condition.
figure 5c plots the repartition of phagocytic cups for different conditions into both regular and variable cups.
note that a cup was identified as regular if its variability was below the variability averaged over all experimental conditions.
in contrast, a cup was identified as variable if its variability was above the overall average.
this plot shows that in our experiments a regular cup is most likely produced by a cell expressing wild-type receptor, whereas a variable cup is most likely to be produced by a cytochalasin-d treated cell.
hence, the cup shape has universal features independent of biochemical details.
examples of a regular and a variable cup are provided in figures 4e and 4f, respectively.
both cups were taken from a cytochalasin-d treated cell, confirming that a regular cup can occur under any of our experimental conditions.
our model also predicts that uptake by the active zipper is significantly faster than with the passive zipper .
we experimentally tested this prediction by determining the percentage of engulfed surface area for each particle for different time points after initiation of phagocytosis, and comparing this result with our simulations, in which simulation time was matched to actual time.
figure 6a shows that cells transfected with the wild-type receptor  engulf significantly faster  than cells under the other two conditions .
this result is in quantitative accordance with our model predictions .
furthermore, we determined the time dependence of phagocytic uptake for large particles.
the active zipper, although slower for large than for small particles, still engulfs regularly, both in experiments  and simulations .
note that predicted and measured time courses are in very good agreement without rescaling the time axis of the large-particle simulation.
furthermore, figures 6c and 6d demonstrate the inability of the passive zipper to take up large particles, in both experiments and simulations.
after more than  <dig> minutes, the average engulfed surface area remains below 20%.
note that for the time points measured, the average uptake does not exceed 50%, even for wt fcγr.
this is caused by the fact that some particles are not engulfed irrespective of the condition  <cit>  , reducing the average percentage of engulfment.
the proportion of almost completely engulfed particles  beyond  <dig> minutes is represented in the inset of figure 6a, showing that completion of engulfment is possible even for cells without actin polymerization.
note that long phagocytic assays  were performed for cytochalasin-d treated cells only, explaining why the difference in complete engulfment with cells expressing the wild-type receptor is smaller than one may expect.
from these results, we conclude that functional actin polymerization is required for fast and regular engulfment in phagocytosis.
nevertheless, in line with our model predictions, cells showing deficient actin polymerization at cups are still able to uptake small particles, although more slowly and with more variable cups.
active zipper reproduces particle-shape dependence of phagocytosis
previously published experiments show that phagocytosis depends strongly on particle shape.
in particular, elongated particles  are only taken up when presented to the phagocyte with their tip first  <cit> .
to test the particle-shape dependence of phagocytosis we conducted simulations of the active zipper using particles of different shapes while varying the initial orientation of the particle on the cell surface.
figure 2d shows the uptake of a prolate spheroid for two different orientations on the cell membrane after the same elapsed simulation time.
in accordance with experimental observations  <cit> , uptake is more advanced for the spheroid particle engulfed with its tip first  than for the particle attached along its major axis .
this suggests a strong inhibitory effect of high local curvature on uptake.
in our model this is readily attributed to the energetic cost of bending the membrane around the two highly curved ends of an elongated particle placed horizontally on the cell membrane.
in line with this explanation and experiments  <cit> , the spiral-shaped particle in figure 2e is not engulfed after a time duration sufficient for the engulfment of large spherical particle of twice its volume.
hence, our simulations demonstrate that particle shape and orientation are indeed important biophysical parameters for phagocytosis.
CONCLUSIONS
in this work, we studied the biophysical requirements of the zipper mechanism, in particular the role of receptor-induced actin polymerization, and the effect of particle shape on uptake.
in our model, the underlying biophysical mechanism of the zipper is an actin-driven thermal ratchet, which renders random membrane fluctuations irreversible close to the particle .
this mechanism is supported by several recent experiments  <cit> .
previously, such brownian ratchet models were successfully applied to explain force generation by actin polymerization and motility of the pathogen listeria in hosts cells  <cit> .
our fully stochastic simulations can address for the first time the variability in particle uptake, recently noticed in phagocytic cup growth  <cit>  and completion of endocytosis  <cit> .
implementation of our model in simulations led indeed to phagocytic engulfment for a broad range of values of membrane parameters , indicating exquisit robustness of the phagocytic process.
however, phagocytic cup shape depends on parameter values, specifically on the ratio between surface tension and cell-volume constraint, as well as on the kinetics of engulfment.
cells with low surface tension and/or tight volume constraint develop thin cups , characteristic of fcγr-mediated phagocytosis  <cit> .
in contrast, cells with high surface tension produce broad cups .
the latter cup shape is more reminiscent of cr3-mediated phagocytosis, although for this type of phagocytosis particles are believed to sink into cells without protrusive cups  <cit> .
using our model we were able to address the question whether the zipper mechanism requires an active driving force, such as provided by actin polymerization.
for this purpose, we compared the regular active zipper with a passive version of the zipper.
in the passive zipper, ligand-receptor bonds are as strong as for the active zipper  but are reversible, i.e.
are not supported by actin polymerization.
we demonstrated that the passive zipper also leads to engulfment of small particles , although cup progression is slower and more variable .
our active zipper can also reproduce the independence of uptake on particle size, in line with experimental observations  <cit> .
in contrast, large particles  are poorly phagocytozed by the passive zipper.
we subsequently confirmed these predictions with experiments by transfecting cos- <dig> fibroblasts with wild-type fcγr and signaling-dead mutant y282f/y298f fcγr .
while the wild-type receptor represents the active zipper, the passive zipper is implemented through the use of signaling-dead mutant receptor or treatment with cytochalasin d. both prevent actin polymerization in the cups.
our study may indicate that ancient forms of phagocytosis were driven by physical  principles, and only later in evolution biochemical regulatory pathways were added for further support and robustness.
passive phagocytosis may also become important when energy sources are scarce.
despite the robustness of phagocytosis to particle size, there appears to be a mechanical bottleneck around half-engulfment, recently observed by imaging  <cit>  and also predicted by our model.
for slight variations in some of the parameters, our simulations of the active zipper produce either complete or significantly incomplete uptake .
indeed, when the cup grows, deforming the membrane costs more and more energy per surface area engulfed.
beyond half-engulfment, the surface tension energy is twice as high as at the beginning of engulfment due to the membrane folding back onto itself.
if this energy cannot be provided by the zipper, then cup progression stalls and the particle remains incompletely engulfed.
alternatively, the experimental data on incomplete particle uptake may be the result of particle attachment on cell-membrane areas, unable to phagocytose due to other reasons, such as unfavorable local cell-surface curvature, proximity to cell edge or nucleus, or missing proteins, lipids, and smaller molecules belonging to key signaling pathways.
further studies will be required, ideally using live-cell imaging to avoid the need for conserving the cell's internal structures by fixation  <cit> .
our model for the zipper mechanism can also explain the strong particle-shape dependence observed in phagocytosis.
experiments show that an elongated spheroid is rapidly engulfed if the particle attaches to the cell membrane with its tip, but not if the particle attaches along its major axis  <cit> .
furthermore, spiral-shaped particles are not phagocytozed  <cit> .
the strong particle-shape dependence of phagocytosis is likely of biological relevance.
on the one hand, it may increase the rate of infection of host cells by pathogenic bacteria.
indeed, recent experiments show that mycobacteria tuberculosis and marinum are efficiently taken up and later released for spreading of the infection with the bacteria's tip first  <cit> .
on the other hand, the highly curved shapes of some bacteria, e.g.
the spiral-shaped helicobacter and campylobacter species, may prevent their uptake by macrophages  <cit> , although injection of effector proteins can also be used by pathogens to hijack the immune or host cell's phagocytic response  <cit> .
furthermore, the particle-shape dependence of phagocytosis may be exploited to improve drug delivery by enclosing active drugs in particles, whose shape prevents uptake and destruction by macrophages  <cit> .
while our biophysical model for the zipper mechanism is readily accessible for analysis and interpretation, the small number of model parameters makes a direct comparison with measured parameter values difficult.
first, our membrane parameters such as surface tension and bending stiffness are an order of magnitude smaller than reported bulk membrane parameters .
this reduction is not surprising as cells regulate these parameters locally for efficient uptake  <cit> .
such regulation may include lowering of the surface tension by local membrane delivery through vesicles  <cit>  and unfolding of membrane wrinkles  <cit> , as well as changes in the lipid and protein composition in the phagocytic cup  <cit> .
second, our description of ligand-receptor interaction assumes that ligand and receptor distributions are continuous and homogeneous, while experiments indicate the formation of receptor micro-clusters  <cit> , possibly as part of lipid rafts  <cit> .
third, our mechanism for actin polymerization based on membrane fluctuations neglects the role of the motor proteins , whose role in membrane deformations has been established  <cit> .
consequently, our simulations describe well the dynamics of engulfment during the first two thirds of the uptake.
at later stages of uptake, experiments show that phagocytic cups close rapidly with a thin membrane protrusion  <cit> , while our simulations show slow cup closure.
however, taking into account myosin-driven contraction is beyond the scope of the current work.
our model may also be applicable to other biological systems in which a zipper-like mechanism is involved.
one such example is sporulation of bacillus subtilis during starvation.
after asymmetric cell division, the larger mother cell engulfs the smaller forespore for spore maturation.
interestingly, the mother cell even engulfs the forespore when the cell wall is artificially removed.
this process occurs in a fast, zipper-like fashion without known sources of energy  <cit> .
importantly, forespore engulfment is subject to high variation.
about 60% of the cells successfully complete forespore engulfment, while 40% do not at all, similar to the observation of the mechanical bottleneck in phagocytosis.
other examples of engulfment may not be driven by a zipper mechanism.
for instance, the penetration of red blood cells by the malaria plasmodium merozoite is driven by an elaborate actin machinery of the particle , which devotes all its resources to wrap the cell membrane around itself.
in stark contrast to phagocytosis, the engulfing host cell is completely passive  <cit> .

 phagocytosis is the ancient, evolutionarily conserved process by which eukaryotic cells bind, engulf, and destroy particles and cells larger than μm in diameter. the importance of phagocytosis is derived from its two main functions: a feeding mechanism in single-cell organisms, and the clearance of pathogens, apoptotic and senescent cells from our body by immune cells. as part of our immune defense, phagocytosis is mainly performed by professional phagocytes, including macrophages, neutrophils, and dendritic cells. initiation of phagocytosis occurs with recognition of the target particle either directly or via an opsonising molecule. for instance the fc portion of immunoglobulin g is recognized by the cell-surface receptor fcγriia. ligand-receptor binding triggers intracellular signaling, resulting in remodeling of the actin cytoskeleton and coherent growth of cell membrane around the particle to form the phagocytic cup. eventually, the leading edge of the growing cup closes, and a membrane vesicle enclosing the particle moves inside the cell. subsequently, the phagosome fuses with vesicles containing enzymes, acids, and oxygen radicals to destroy the particle. the biochemical pathways involved in phagocytosis are complex. dozens of cell-surface receptors contribute to the recognition of a large variety of ligand molecules and subsequent particle engulfment. the fcγ receptor and complement receptor of the integrin receptor family are the most widely studied and understood receptors involved in phagocytosis. fcγr-mediated phagocytosis proceeds through membrane protrusions and leads to thin cups, whereas in cr3-mediated phagocytosis, particles appear to sink into the cell. spreading of the cell membrane over the particle involves actin-driven cell-shape changes similar to the processes involved in cell migration and adhesion. specifically for fcγr, binding to an igg-opsonized particle results in receptor clustering and recruitment of small gtpases of the rho family, which, via proteins of the wasp family, activate the arp2/ complex. the latter promotes branching of actin filaments, leading to an increase in the number of uncapped ends and to an isotropic growth of the actin network. additionally, the phagocytic cup has been shown to be enriched in gelsolin, coronin, and other regulators of actin polymerization. all in all, this complex signaling pathway involves 100- different types of molecules, rendering mathematical modeling at the molecular level impossible. despite the huge biochemical complexity, the engulfment process shows a strong dependence on simple biophysical parameters. first, it relies on the availability of extra membrane at the phagocytic cup, provided by delivery of membrane vesicles or unwrinkling of membrane folds. second, completion of phagocytic uptake depends on the shape of the particle and, interestingly, on the initial orientation of the particle on the cell surface. for instance, experiments demonstrate that elongated spheroid polystyrene particles coated with igg are more efficiently engulfed when presented to the phagocyte with their tip first. third, a recent study by one of the authors demonstrates that the biophysical requirements for phagocytosis lead to either complete phagocytosis or stalled cups due to the presence of a mechanical bottleneck. interestingly, the same study shows that engulfment appears to even proceed in cells treated with cytochalasin d, an inhibitor of actin polymerization, indicating that biochemical pathways may not always be necessary for this initial stage of phagocytosis. the mechanism of phagocytosis is only partially understood, with key insights provided more than three decades ago. in the 1970's, griffin and his collaborators demonstrated that incomplete coating of particles with ligand results in only partial uptake. this indicated that phagocytic uptake occurs via successive zipper-like ligand-receptor binding, and not by an all-or-nothing mechanism triggered at the onset of phagocytosis. the zipper mechanism is the underlying assumption in a number of recent modeling works in phagocytosis and endocytosis, mainly addressing the influence of the cell-membrane tension and ligand-receptor bond density on engulfment. despite the general acceptance of the zipper mechanism, many of its biophysical requirements are insufficiently understood. questions, so far unanswered, include what the energetic requirements of the zipper mechanism are, specifically what role actin polymerization plays in its progression during phagocytosis, and also whether the zipper mechanism can explain the particle-shape dependence of phagocytosis. previous models were unable to fully address the particle-shape dependence, as they assume rotational symmetry around the axis connecting cell and particle. additionally, large particle-to-particle variation in cup growth and cell-to-cell variation in the related process of endocytosis point towards the importance of stochasticity during the uptake, not captured in previous deterministic approaches. recent experiments provide new insights into the biophysical mechanism for driving the membrane around the particle, suggesting a ratchet-type mechanism. once started, phagocytosis progresses unidirectionally and irreversibly. this irreversible membrane progression is further supported by the loss of lipid and protein mobility at the phagocytic cup, observed using fluorescence recovery after photobleaching. while several models proposed mechanisms of force generation by actin polymerization, recent experiments based on fluorescent speckle microscopy demonstrate that actin does not directly push the membrane outwards. instead, by filling gaps provided by membrane fluctuations, actin polymerization prevents the membrane from moving backwards like a ratchet. the relevance of such brownian ratchets in biology has previously been emphasized. the question is if a ratchet mechanism, together with energetic restrictions in membrane bending and stretching, can naturally lead to phagocytic uptake and account for the shape-dependence of phagocytosis. in this work, we propose a ratchet-like biophysical model for the zipper mechanism. this model differs from previous works in that it is, to our knowledge, the first fully three-dimensional stochastic model of phagocytic engulfment. specifically, thermal membrane fluctuations, assumed to play a major role in our model, provide the energy source to locally deform the membrane and to build further ligand-receptor bonds for zippering the membrane around the particle. actin polymerization makes ligand-receptor bonds effectively irreversible, i.e. reinforced and stabilized for a significant amount of time. to investigate the role of actin, we compare cup progression for the regular active zipper with a passive zipper model in which ligand-receptor binding remains specific and strong but reversible due to the absence of actin polymerization. interestingly, we find that the passive zipper also leads to engulfment of small particles, rendering phagocytosis highly robust. however, such passive engulfment is generally slower and produces much more variable phagocytic cups than the active zipper. furthermore, our computer simulations lead to successful phagocytic engulfment in a broad range of parameters values, including different particle sizes. for non-spherical particles, completion of engulfment depends strongly on particle shape and orientation. our model further predicts that cup shape invariably depends on membrane biophysical parameters, in particular surface tension and cell-volume constraint. to test the predicted difference between the active and passive zippers, we experimentally implement the two different types of zippers using cos- fibroblasts which, after transfection with gfp-tagged fcγ receptor, phagocytoze igg-coated polystyrene particles. specifically, we performed phagocytic assays under three different conditions: cells expressing wild-type fcγr for the active zipper, cells expressing a signaling-dead mutant receptor, which specifically binds igg ligand but is unable to signal to the actin cytoskeleton, and cells expressing wt-fcγr and treated with cytochalasin d. the last two conditions represent two versions of the passive zipper due to the absence of actin polymerization in phagocytic cups. to compare with our model, we systematically analyze confocal microscopy images, and quantitatively estimate cup variability for the three different conditions using small and large particles. consistently with our simulations, phagocytic cups develop more slowly and are significantly more variable in the absence of actin polymerization. our results provide new insights into the robustness of phagocytosis, as well as the role of bacterial cell shape in host-pathogen interactions.  ratchet model for the zipper mechanism our model is based on the following experimental observations. engulfment of quasi-spherical particles by neutrophils progresses continuously without significant pause or reversal, indicating that ligand-receptor binding is essentially irreversible. this irreversibility is further supported by frap and single-molecule experiments, which show that lipids and proteins in phagocytic cups, as well as ligand-bound fc-receptors are immobilized in an actin-dependent fashion. additional support for the notion of irreversible uptake was recently determined in a related context. fluorescent speckle microscopy of actin flow and image analysis during cell migration show that the membrane at the leading edge protrudes first, followed by actin polymerization to fill the gap between the membrane and the actin cortex. such actin polymerization is mainly restricted to the leading edge due to signaling by receptors and/or localization of small gtpases of the rho family. the role of actin polymerization in phagocytosis is hence to stabilize ligand-receptor bonds and to rectify membrane movements in a ratchet-like fashion, leading to unidirectional movement of the leading edge of the engulfing cell. dependence of phagocytic cup shape on membrane biophysical parameters using our model for the zipper mechanism, we have successfully simulated phagocytic engulfment in a broad range of parameter values. figure 2a shows two different characteristic cup shapes we obtained. low surface tension, and tight cell-volume constraint, lead to a thin phagocytic cup since a thin cup requires extra membrane but little extra volume. in contrast, weak volume constraint and/or high surface tension produce a broad cup. based on the parameters explored, we chose intermediate values for both surface tension and cell-volume constraint as our standard parameters for the remainder of our simulations in order to produce realistic cup shapes. figure 2b shows that most parameter values can be changed independently by at least one order of magnitude, without negatively affecting engulfment completion. note that changing simultaneously several parameters may affect engulfment more drastically. our simulations also show that cup shape depends on the kinetics of engulfment, determined by membrane fluctuations and therefore temperature. additionally, preventing thermal fluctuations during a simulation stops cup progression. this indicates that in our model membrane fluctuations are indeed required to bring receptors in close contact with ligand molecules on the particle, emphasizing their important role in the ratchet mechanism. active versus passive uptake and the role of actin although phagocytosis generally involves active processes such as actin polymerization in the cup, recent reports indicate that phagocytosis may still work in an actin-independent manner. indeed, phagocytic uptake was observed despite treating phagocytes with cytochalasin d. hence, ligand-receptor binding may be sufficient in guiding the cell membrane around the particle under certain conditions. to investigate the energetic requirements of the zipper mechanism, we implemented simulations of the passive zipper. in these simulations, ligand-receptor bonds are not stabilized by actin polymerization and can unbind at later times, i.e. remain reversible. hence, engulfment may still progress if the energetic cost of stretching and deforming the membrane is offset by the ligand-receptor binding energy in the presence of thermal membrane fluctuations. particle size matters for passive, not for active zipper experiments show that phagocytosis is relatively insensitive to particle size. using our model for the active zipper, we simulated engulfment of spherical particles with different radii ranging from to μm. figure 2c shows that engulfment progresses normally for small and large particles. hence the active zipper mechanism is sufficiently robust to allow engulfment of differently sized particles using the same set of biophysical parameters, although engulfment of large particles requires more time. noticeably, large particles were taken up via more regular phagocytic cups than small particles, indicating that active processes may be required for engulfment of large particles. to confirm this observation we have simulated engulfment of large μm-radius particles by both the active and the passive zipper, shown in additional file figure s while the active zipper resulted in complete uptake of the particle, the passive zipper only engulfed a few percent of the particle's surface area. thus the difference in phagocytic efficiency between the two zipper types was exacerbated for large particles, reflecting the importance of actin polymerization for engulfment of large particles. experimental test of model predictions to test our model predictions and to specifically compare active with passive engulfment for small and large particles, we transfected cos- cells with either wild-type fcγr or a signaling-dead mutant receptor. cells expressing the wild-type receptor are expected to perform active engulfment, whereas cells expressing the signaling-dead mutant receptor are expected to perform passive engulfment. as a control, passive engulfment is additionally implemented by treating cells expressing wt-fcγr with μm of cytochalasin d, which prevents actin polymerization. synchronized phagocytosis assays using small and large igg-opsonized polystyrene particles were carried out, and, after fixation, receptor localization in phagocytic cups was visualized by fluorescence confocal microscopy. cells were imaged at different time points during phagocytosis, and at each time point, three to eight imaged cells were each engulfing simultaneously between four and twenty particles. consequently, for each condition we analyzed at least seventy particles. to test whether passive engulfment leads to more variable cups than active engulfment, we developed an image-analysis method illustrated in figure 4b-f. the cup shape varibility was quantified by the standard deviation of the distribution of cell-membrane height around the particle, divided by the square root of the average membrane height. the unit of membrane height is given by the distance between consecutive confocal image planes. figure 5a shows that for small particles engulfed between and 40% of their surfaces, cup variability increases from cells transfected with wild-type receptor to cells transfected with signaling-dead mutant receptor to wt-fcγ+cytod cells. the lowest variability, found for cells expressing wild-type receptor, is statistically significant against both passive zipper types. this result is consistent with model predictions: figure 5a, inset shows the cup variability from simulations, revealing that the active zipper leads to significantly less variable cups. in contrast, for the ranges of engulfment between and 60% and between and 100% we observed no noticeable difference in cup variability between the three experimental conditions, while our model consistently predicts more variable cups for the passive zipper. this discrepancy may indicate that active processes such as contraction by myosin motor proteins become important at later stages of engulfment, limiting our model's full validity to the early events in phagocytosis. to confirm that our results are independent of the specifics of the analysis method used, we also analyzed phagocytic cups with an alternative, albeit less accurate, method. this method consists of determining the distribution of fcγr-gfp fluorescence intensity around the particle at its equator plane, restricting the analysis to approximately half taken up particles. the standard deviation of this distribution provides an alternative measure of the cup variability. we arrived at the same conclusion, confirming our result of the difference in cup variability. finally, note that temperature-induced synchronization is imperfect and may lead to variability in cup growth. however, our measures of the variability in cup shape are independent of such an effect since we include all time points together in the analysis. our experiments further show that cup shape ranges from regular to variable for all three experimental conditions, but that the frequency of different cup shapes depends on the condition. figure 5c plots the repartition of phagocytic cups for different conditions into both regular and variable cups. note that a cup was identified as regular if its variability was below the variability averaged over all experimental conditions. in contrast, a cup was identified as variable if its variability was above the overall average. this plot shows that in our experiments a regular cup is most likely produced by a cell expressing wild-type receptor, whereas a variable cup is most likely to be produced by a cytochalasin-d treated cell. hence, the cup shape has universal features independent of biochemical details. examples of a regular and a variable cup are provided in figures 4e and 4f, respectively. both cups were taken from a cytochalasin-d treated cell, confirming that a regular cup can occur under any of our experimental conditions. our model also predicts that uptake by the active zipper is significantly faster than with the passive zipper. we experimentally tested this prediction by determining the percentage of engulfed surface area for each particle for different time points after initiation of phagocytosis, and comparing this result with our simulations, in which simulation time was matched to actual time. figure 6a shows that cells transfected with the wild-type receptor engulf significantly faster than cells under the other two conditions. this result is in quantitative accordance with our model predictions. furthermore, we determined the time dependence of phagocytic uptake for large particles. the active zipper, although slower for large than for small particles, still engulfs regularly, both in experiments and simulations. note that predicted and measured time courses are in very good agreement without rescaling the time axis of the large-particle simulation. furthermore, figures 6c and 6d demonstrate the inability of the passive zipper to take up large particles, in both experiments and simulations. after more than minutes, the average engulfed surface area remains below 20%. note that for the time points measured, the average uptake does not exceed 50%, even for wt fcγr. this is caused by the fact that some particles are not engulfed irrespective of the condition, reducing the average percentage of engulfment. the proportion of almost completely engulfed particles beyond minutes is represented in the inset of figure 6a, showing that completion of engulfment is possible even for cells without actin polymerization. note that long phagocytic assays were performed for cytochalasin-d treated cells only, explaining why the difference in complete engulfment with cells expressing the wild-type receptor is smaller than one may expect. from these results, we conclude that functional actin polymerization is required for fast and regular engulfment in phagocytosis. nevertheless, in line with our model predictions, cells showing deficient actin polymerization at cups are still able to uptake small particles, although more slowly and with more variable cups. active zipper reproduces particle-shape dependence of phagocytosis previously published experiments show that phagocytosis depends strongly on particle shape. in particular, elongated particles are only taken up when presented to the phagocyte with their tip first. to test the particle-shape dependence of phagocytosis we conducted simulations of the active zipper using particles of different shapes while varying the initial orientation of the particle on the cell surface. figure 2d shows the uptake of a prolate spheroid for two different orientations on the cell membrane after the same elapsed simulation time. in accordance with experimental observations, uptake is more advanced for the spheroid particle engulfed with its tip first than for the particle attached along its major axis. this suggests a strong inhibitory effect of high local curvature on uptake. in our model this is readily attributed to the energetic cost of bending the membrane around the two highly curved ends of an elongated particle placed horizontally on the cell membrane. in line with this explanation and experiments, the spiral-shaped particle in figure 2e is not engulfed after a time duration sufficient for the engulfment of large spherical particle of twice its volume. hence, our simulations demonstrate that particle shape and orientation are indeed important biophysical parameters for phagocytosis.  in this work, we studied the biophysical requirements of the zipper mechanism, in particular the role of receptor-induced actin polymerization, and the effect of particle shape on uptake. in our model, the underlying biophysical mechanism of the zipper is an actin-driven thermal ratchet, which renders random membrane fluctuations irreversible close to the particle. this mechanism is supported by several recent experiments. previously, such brownian ratchet models were successfully applied to explain force generation by actin polymerization and motility of the pathogen listeria in hosts cells. our fully stochastic simulations can address for the first time the variability in particle uptake, recently noticed in phagocytic cup growth and completion of endocytosis. implementation of our model in simulations led indeed to phagocytic engulfment for a broad range of values of membrane parameters, indicating exquisit robustness of the phagocytic process. however, phagocytic cup shape depends on parameter values, specifically on the ratio between surface tension and cell-volume constraint, as well as on the kinetics of engulfment. cells with low surface tension and/or tight volume constraint develop thin cups, characteristic of fcγr-mediated phagocytosis. in contrast, cells with high surface tension produce broad cups. the latter cup shape is more reminiscent of cr3-mediated phagocytosis, although for this type of phagocytosis particles are believed to sink into cells without protrusive cups. using our model we were able to address the question whether the zipper mechanism requires an active driving force, such as provided by actin polymerization. for this purpose, we compared the regular active zipper with a passive version of the zipper. in the passive zipper, ligand-receptor bonds are as strong as for the active zipper but are reversible, i.e. are not supported by actin polymerization. we demonstrated that the passive zipper also leads to engulfment of small particles, although cup progression is slower and more variable. our active zipper can also reproduce the independence of uptake on particle size, in line with experimental observations. in contrast, large particles are poorly phagocytozed by the passive zipper. we subsequently confirmed these predictions with experiments by transfecting cos- fibroblasts with wild-type fcγr and signaling-dead mutant y282f/y298f fcγr. while the wild-type receptor represents the active zipper, the passive zipper is implemented through the use of signaling-dead mutant receptor or treatment with cytochalasin d. both prevent actin polymerization in the cups. our study may indicate that ancient forms of phagocytosis were driven by physical principles, and only later in evolution biochemical regulatory pathways were added for further support and robustness. passive phagocytosis may also become important when energy sources are scarce. despite the robustness of phagocytosis to particle size, there appears to be a mechanical bottleneck around half-engulfment, recently observed by imaging and also predicted by our model. for slight variations in some of the parameters, our simulations of the active zipper produce either complete or significantly incomplete uptake. indeed, when the cup grows, deforming the membrane costs more and more energy per surface area engulfed. beyond half-engulfment, the surface tension energy is twice as high as at the beginning of engulfment due to the membrane folding back onto itself. if this energy cannot be provided by the zipper, then cup progression stalls and the particle remains incompletely engulfed. alternatively, the experimental data on incomplete particle uptake may be the result of particle attachment on cell-membrane areas, unable to phagocytose due to other reasons, such as unfavorable local cell-surface curvature, proximity to cell edge or nucleus, or missing proteins, lipids, and smaller molecules belonging to key signaling pathways. further studies will be required, ideally using live-cell imaging to avoid the need for conserving the cell's internal structures by fixation. our model for the zipper mechanism can also explain the strong particle-shape dependence observed in phagocytosis. experiments show that an elongated spheroid is rapidly engulfed if the particle attaches to the cell membrane with its tip, but not if the particle attaches along its major axis. furthermore, spiral-shaped particles are not phagocytozed. the strong particle-shape dependence of phagocytosis is likely of biological relevance. on the one hand, it may increase the rate of infection of host cells by pathogenic bacteria. indeed, recent experiments show that mycobacteria tuberculosis and marinum are efficiently taken up and later released for spreading of the infection with the bacteria's tip first. on the other hand, the highly curved shapes of some bacteria, e.g. the spiral-shaped helicobacter and campylobacter species, may prevent their uptake by macrophages, although injection of effector proteins can also be used by pathogens to hijack the immune or host cell's phagocytic response. furthermore, the particle-shape dependence of phagocytosis may be exploited to improve drug delivery by enclosing active drugs in particles, whose shape prevents uptake and destruction by macrophages. while our biophysical model for the zipper mechanism is readily accessible for analysis and interpretation, the small number of model parameters makes a direct comparison with measured parameter values difficult. first, our membrane parameters such as surface tension and bending stiffness are an order of magnitude smaller than reported bulk membrane parameters. this reduction is not surprising as cells regulate these parameters locally for efficient uptake. such regulation may include lowering of the surface tension by local membrane delivery through vesicles and unfolding of membrane wrinkles, as well as changes in the lipid and protein composition in the phagocytic cup. second, our description of ligand-receptor interaction assumes that ligand and receptor distributions are continuous and homogeneous, while experiments indicate the formation of receptor micro-clusters, possibly as part of lipid rafts. third, our mechanism for actin polymerization based on membrane fluctuations neglects the role of the motor proteins, whose role in membrane deformations has been established. consequently, our simulations describe well the dynamics of engulfment during the first two thirds of the uptake. at later stages of uptake, experiments show that phagocytic cups close rapidly with a thin membrane protrusion, while our simulations show slow cup closure. however, taking into account myosin-driven contraction is beyond the scope of the current work. our model may also be applicable to other biological systems in which a zipper-like mechanism is involved. one such example is sporulation of bacillus subtilis during starvation. after asymmetric cell division, the larger mother cell engulfs the smaller forespore for spore maturation. interestingly, the mother cell even engulfs the forespore when the cell wall is artificially removed. this process occurs in a fast, zipper-like fashion without known sources of energy. importantly, forespore engulfment is subject to high variation. about 60% of the cells successfully complete forespore engulfment, while 40% do not at all, similar to the observation of the mechanical bottleneck in phagocytosis. other examples of engulfment may not be driven by a zipper mechanism. for instance, the penetration of red blood cells by the malaria plasmodium merozoite is driven by an elaborate actin machinery of the particle, which devotes all its resources to wrap the cell membrane around itself. in stark contrast to phagocytosis, the engulfing host cell is completely passive. 
furthermore, we surprisingly find that even without actin-driven force generation, engulfment proceeds in a large regime of parameter values, albeit more slowly and with highly variable phagocytic cups. based on the reconstruction of phagocytic cups from imaging data, we indeed show that cells are able to engulf small particles even without support from biological actin-driven processes. despite the signaling complexity, phagocytosis also depends strongly on biophysical parameters, such as particle shape, and the need for actin-driven force generation remains poorly understood. specifically, we compare the wild-type receptor with a mutant receptor, unable to signal to the actin cytoskeleton. the particle-shape dependence of phagocytosis makes a systematic investigation of host-pathogen interactions and an efficient design of a vehicle for drug delivery possible. highly curved shapes are not taken up, in line with recent experimental results. phagocytosis is the fundamental cellular process by which eukaryotic cells bind and engulf particles by their cell membrane. particle engulfment involves particle recognition by cell-surface receptors, signaling and remodeling of the actin cytoskeleton to guide the membrane around the particle in a zipper-like fashion. this suggests that biochemical pathways render the evolutionary ancient process of phagocytic highly robust, allowing cells to engulf even very large particles.
 phagocytosis is the fundamental cellular process by which eukaryotic cells bind and engulf particles by their cell membrane. particle engulfment involves particle recognition by cell-surface receptors, signaling and remodeling of the actin cytoskeleton to guide the membrane around the particle in a zipper-like fashion. despite the signaling complexity, phagocytosis also depends strongly on biophysical parameters, such as particle shape, and the need for actin-driven force generation remains poorly understood.  here, we propose a novel, three-dimensional and stochastic biophysical model of phagocytosis, and study the engulfment of particles of various sizes and shapes, including spiral and rod-shaped particles reminiscent of bacteria. highly curved shapes are not taken up, in line with recent experimental results. furthermore, we surprisingly find that even without actin-driven force generation, engulfment proceeds in a large regime of parameter values, albeit more slowly and with highly variable phagocytic cups. we experimentally confirm these predictions using fibroblasts, transfected with immunoreceptor fcγriia for engulfment of immunoglobulin g-opsonized particles. specifically, we compare the wild-type receptor with a mutant receptor, unable to signal to the actin cytoskeleton. based on the reconstruction of phagocytic cups from imaging data, we indeed show that cells are able to engulf small particles even without support from biological actin-driven processes.  this suggests that biochemical pathways render the evolutionary ancient process of phagocytic highly robust, allowing cells to engulf even very large particles. the particle-shape dependence of phagocytosis makes a systematic investigation of host-pathogen interactions and an efficient design of a vehicle for drug delivery possible. 