article: Force generation by actin assembly shapes cellular membranes. An experimentally constrained multiscale model shows that a minimal branched actin network is sufficient to internalize endocytic pits against membrane tension. Around 200 activated Arp2/3 complexes are required for robust internalization. A newly developed molecule-counting method determined that ~200 Arp2/3 complexes assemble at sites of clathrin-mediated endocytosis in human cells. Simulations predict that actin self-organizes into a radial branched array with growing ends oriented toward the base of the pit. Long actin filaments bend between attachment sites in the coat and the base of the pit. Elastic energy stored in bent filaments, whose presence was confirmed by cryo-electron tomography, contributes to endocytic internalization. Elevated membrane tension directs more growing filaments toward the base of the pit, increasing actin nucleation and bending for increased force production. Thus, spatially constrained actin filament assembly utilizes an adaptive mechanism enabling endocytosis under varying physical constraints.
  Cells polymerize actin filaments to produce force and provide mechanical integrity for a variety of cellular processes, from cytokinesis and cell migration, to membrane reshaping and trafficking (Pollard, 2016). For each cellular process, actin filaments organize into a specific geometry that confers structural integrity and force-generation capacity. Most membrane deformation processes use branched actin networks nucleated by the Arp2/3 complex, a branched actin filament network nucleator (Carlsson, 2018; Rotty et al. , 2013). On a large (µm) length scale, branched actin networks drive the plasma membrane forward during cell migration, such that on the scale of individual actin branches, the membrane shape can be thought of as more or less constant (Keren et al. , 2008; Mueller et al. , 2017; Schaus et al. , 2007). However, on a smaller (sub-micron) length scale, branched actin networks deform many cellular membranes as part of organelle and vesicle biogenesis and function (Rottner et al. , 2017). The relationship between cellular membrane curvature and local actin assembly for each of these ‘local’ membrane-deformation processes remains relatively unexplored (Daste et al. , 2017). Clathrin-mediated endocytosis (CME) is an especially attractive process for studies of actin’s role in membrane shape changes due to the relatively complete parts list and available quantitative information about the positions, recruitment timing and biochemical function of many of the participating proteins (Arasada et al. , 2018; Idrissi et al. , 2012; Kaksonen et al. , 2005; Kaksonen et al. , 2003; Mund et al. , 2018; Picco et al. , 2015; Sochacki et al. , 2017; Taylor et al. , 2011). CME is a ubiquitous and essential cellular process by which cells take macromolecules from the extracellular space and the plasma membrane into the cell interior (Kaksonen and Roux, 2018). During CME, the plasma membrane is bent, pinched, and pulled inward in a time frame of ~60 s thereby transitioning from a flat sheet into a spherical vesicle ~100 nm in diameter. Clathrin and its adaptor proteins establish a coat that generates initial membrane curvature (Chen et al. , 1998; Pearse, 1976; Stachowiak et al. , 2012), and BAR (bin-amphiphysin-rvs) -domain proteins bind curved membranes and support further membrane curvature (Buser and Drubin, 2013; David et al. , 1996; Kishimoto et al. , 2011). During yeast endocytosis, branched actin filaments provide the force required for membrane tubule formation (Engqvist-Goldstein and Drubin, 2003; Idrissi et al. , 2012; Kukulski et al. , 2012; Picco et al. , 2018; Sun et al. , 2006; Wang and Carlsson, 2017). In metazoan cells, endocytic pits under high tension stall at a ‘U’-shaped intermediate in the absence of functional actin (Boulant et al. , 2011), implying that actin is required to generate plasma membrane shape changes late in CME (Hassinger et al. , 2017; Yarar et al. , 2005; Yoshida et al. , 2018). The molecular mechanism by which a network of polarized, branched actin filaments assembles at these sites for productive force generation is poorly understood. Actin network assembly is known to play a key role in membrane shape change in some contexts. For example, mathematical modeling (Berro et al. , 2010; Carlsson and Bayly, 2014; Dmitrieff and Nédélec, 2015; Liu et al. , 2009; Mund et al. , 2018; Wang et al. , 2016) and quantitative fluorescence imaging in yeast (Wu and Pollard, 2005; Sirotkin et al. , 2010; Berro and Pollard, 2014; Picco et al. , 2015) have established the relationship between actin filament assembly and plasma membrane shape particular to fungi, which have unique mechanical requirements due to very high (~10 atm) hydrostatic turgor pressure. However, less is known about actin organization and function in the lower force regime characteristic of metazoan cells. A multiscale modeling effort that accounts for the mechanics of single actin filaments and that is constrained by experimental measurements of actin dynamics, spatial organization of the filaments, and tension in the plasma membrane is required to gain insight into actin organization and force generation capacity. We hypothesize that in localized membrane-reshaping processes such as endocytosis, branched actin networks assemble under specific spatial ‘boundary conditions, ’ which serve as geometrical constraints dictated both by the shape of the membrane and the spatial segregation of membrane-associated proteins that interact with actin. These unique spatial boundary conditions on a curved surface, combined with the knowledge of numbers of molecules in cells and known reaction rate constants, provide the necessary information for multiscale modeling and a mechanistic framework to understand the relationship between plasma membrane mechanics and branched actin assembly and mechanics associated with CME. Using this framework, we sought to answer the following questions: How do branched actin networks assemble, organize, and produce force around an endocytic pit? How does the spatial segregation of Arp2/3 complex activators (Almeida-Souza et al. , 2018; Mund et al. , 2018) and actin-binding proteins associated with endocytic coats (Clarke and Royle, 2018; Engqvist-Goldstein et al. , 2001; Sochacki et al. , 2017) influence this organization? And finally, how do endocytic actin networks adapt to changing loads due to the stochastic environment and changes in membrane tension? To answer these questions, we combined live-cell molecule counting methods in genome-edited diploid human cells and cryo-electron tomography of intact cells with multiscale modeling of plasma membrane mechanics and actin filament dynamics. Our results show that a minimal branched actin network is sufficient to create sustained internalization of an endocytic pit against physiological membrane tension. Actin filament self-organization and bending, which arise from the spatial distribution of actin-coat attachments around the curved endocytic pit, allow the actin network to adapt to changing loads. We anticipate that the mechanistic insights gained for actin in mammalian endocytosis will also apply to a variety of local membrane-bending processes carried out by branched actin throughout the cell.
  We combined a continuum-mechanics model of the plasma membrane, an agent-based model of actin filament dynamics, quantitative fluorescence microscopy, and electron tomography in cells to determine the molecular mechanism by which branched actin networks produce force during mammalian clathrin-mediated endocytosis (Figure 1, Scheme 1). First, we used a continuum-mechanics model of the plasma membrane (Alimohamadi et al. , 2018; Hassinger et al. , 2017; Rangamani et al. , 2014) to determine the force-extension relationship for clathrin-coated pits stalled at a U-shaped intermediate under high membrane tension (Figure 1B). Here, the extension refers to the extent of pit internalization, which is a displacement in the -Z direction (Figure 1A–B). Previously, we showed that membrane curvature generation by the endocytic coat during vesicle formation could snap the membrane into a pinched ‘omega’ shape as a function of membrane tension and the curvature induced by the coat (Hassinger et al. , 2017), but we did not focus on force produced by the actin cytoskeleton. Here, we modeled the coated membrane based on the Helfrich (1973) energy and applied a linear force to the clathrin-coated pit in increasing value over successive simulations, corresponding to a simplified actin force. Simulations demonstrated that a clathrin-coated pit experiences a nearly linear force-extension relationship until an internalization of ~100 nm, at which point the pit can also adopt a pinched (or ‘omega’) shape, which requires a lower force (Figure 1C and Figure 1—video 1). We calculated the resistance to internalization as the slope of the force-extension plot for the linear regime and found that it is directly proportional to plasma membrane tension for a wide range of coat rigidities (Figure 1D). Importantly, this direct scaling between resistance to internalization and membrane tension allowed us to treat this step of endocytic pit internalization as a linear spring, with the spring constant calibrated using measurements of plasma membrane tension in mammalian cells (Diz-Muñoz et al. , 2016; Kaplan et al. , in preparation). The simple spring-like relationship uncovered above between force and endocytic pit internalization (Figure 1D) allowed us to simplify our mechanical treatment of the plasma membrane while modeling individual actin filaments and actin-binding proteins with realistic kinetics and mechanics (Figure 1E–G, Supplementary file 3). We used Cytosim (Nedelec and Foethke, 2007) to construct a filament-based model of the endocytic actin network. This agent-based model allowed us to simulate the emergent architecture and mechanical functions of branched actin for realistic endocytic geometries. We simplified the endocytic pit as a solid, impermeable structure, initially a hemisphere, attached to a flat plasma membrane corresponding to the ‘U-shaped’ intermediate (Avinoam et al. , 2015; Boulant et al. , 2011; Messa et al. , 2014; Yarar et al. , 2005; Figure 1E). The following rules were prescribed for actin filament dynamics. Initially, actin filament-nucleating proteins seed a small number of actin filaments near the endocytic pit. These randomly-oriented ‘mother filaments’ serve as templates for binding pre-activated Arp2/3 complexes, which correspond to the coincidence of Arp2/3 complex and its activator N-WASP, arranged in a ring (Almeida-Souza et al. , 2018; Mund et al. , 2018) at the base of the endocytic pit (Idrissi et al. , 2008; Kaksonen et al. , 2003; Picco et al. , 2015; Kaplan et al. , in preparation). When an active Arp2/3 complex comes in proximity with an actin filament, it can bind to the filament and nucleate the growth of a new branched filament at an ~77° angle (Blanchoin et al. , 2000). Growing actin filaments can polymerize, diffuse under thermal fluctuations, and bend under force, and their growing ends are capped stochastically. Filament growth decreases with load according to the Brownian ratchet mechanism (Mogilner and Oster, 1996; Peskin et al. , 1993). Growth of the actin network is coupled to internalization of the endocytic pit by an actin-linking protein (Hip1/Hip1R/Epsin, simplified here as Hip1R), which is embedded in the coated pit and binds to actin filaments (Clarke and Royle, 2018; Engqvist-Goldstein et al. , 2001; Engqvist-Goldstein et al. , 1999; Sochacki et al. , 2017). Importantly, most of the parameters in this model have been determined with measurements in vitro or in vivo, including the dimensions of the endocytic pit, its resistance to internalization (modeled as a spring, Figure 1D), rates of association and dissociation of different proteins, branching angles, capping rates, filament persistence length, and stall force (Supplementary file 3 and Materials and methods). Stochastic simulations of the model showed that this minimal branched actin network internalizes endocytic pits up to ~60 nm against physiological membrane tension (Figure 1F and Figure 1—video 2). In order to compare different conditions, we used two metrics -- internalization of the pit (in nm) over time (Figure 1G) and the 95th percentile of internalization (Figure 1—figure supplement 1A). Then, we evaluated the robustness of the model to different parameters by conducting a series of parameter sweeps (Figure 1—figure supplement 1). We found that the extent of internalization is robust to a wide range of parameters, including filament stiffness, stall force, and affinity between Hip1R attachments and actin filaments (Figure 1—figure supplement 1). Initiating the simulations from a cytoplasmic pool of linear actin filaments (Raz-Ben Aroush et al. , 2017) allowed for endocytosis but the timing of the onset of internalization was more variable (Figure 1—figure supplement 2). The extent of internalization was particularly sensitive to the number of available Arp2/3 complexes (Figure 1G), indicating a need for precise measurements of this molecule at mammalian endocytic sites. Motivated by our prediction that internalization rate is sensitive to the number of Arp2/3 complexes, we developed a method to count the number of molecules of endogenously GFP-tagged proteins in living mammalian cells (Figure 2). We adapted the self-assembling GFP-tagged protein nanocages developed by Hsia et al. (2016) for expression in live cells to create a fluorescence-based calibration curve relating fluorescence intensity of endogenously GFP-tagged proteins to numbers of molecules per endocytic site. Given that the nanocages are derived from bacterial glycolytic enzymes, we made point mutations known to abolish enzymatic activity of the proteins. To slow the diffusion of the intracellular nanocages and facilitate fluorescence measurements, we introduced an inducible dimerization motif to the plasma membrane by fusing the construct to FKBP and coexpressing a palmitoylated and myristoylated FRB variant (Figure 2A). The resulting two-component fusion protein transiently associated with the plasma membrane even without rapamycin analog AP21967, but the extent of association with the plasma membrane increased in a dose-dependent manner with the concentration of AP21967 (Figure 2—figure supplement 1A–B and Figure 3—video 1). We measured the fluorescence intensity of four GFP-tagged nanocages of copy number ranging from 12 to 120 copies of GFP per structure using spinning disk confocal microscopy (Figure 2B). After correcting for exposure time (Figure 2—figure supplement 1E–F), uneven illumination intensity, and local background (Materials and methods), the fluorescence intensity per spot was unitary (Figure 2C) and directly proportional to the predicted numbers of molecules per structure (R2 = 0. 996) (Figure 2D). Using this calibration curve, we measured the numbers of molecules of an E. coli flagellar motor protein eGFP-MotB, which resulted in measurements similar to previously published measurements (Figure 2—figure supplement 1G–I). Thus, we established the suitability of this method to relate fluorescence intensity of endogenously GFP-tagged proteins to numbers of molecules inside live mammalian cells. To measure the timing, frequency, and numbers of Arp2/3 complexes assembling at sites of clathrin-mediated endocytosis, we used CRISPR/Cas9-mediated genome editing to endogenously tag the C terminus of ArpC3, a subunit of the Arp2/3 complex, with the fluorescent protein tagGFP2 in human induced pluripotent stem cells (Figure 2E, Figure 2—figure supplement 2A–B). Human induced pluripotent stem cells are diploid and thus suitable for molecule-counting measurements when both alleles of the ArpC3 gene are fused to the gene for GFP. C-terminal GFP tags on ArpC3 are more functional than on other subunits of the Arp2/3 complex (Egile et al. , 2005; Picco et al. , 2015; Sirotkin et al. , 2010; Smith et al. , 2013). Cells tagged at both alleles of the ArpC3 gene had twice the fluorescence intensity of cells with a single allele tagged, suggesting direct proportionality between GFP fluorescence intensity and numbers of molecules (Figure 2—figure supplement 2C–D). These cells also endogenously express a tagRFP-T fusion with the µ2 subunit of the adaptor protein AP2, allowing us to identify sites of clathrin-mediated endocytosis (Hong et al. , 2015). We determined the relative timing of AP2 and ArpC3 appearance at endocytic sites using time-lapse TIRF imaging and automated two-color particle tracking (Dambournet et al. , 2018; Hong et al. , 2015; Figure 2F). The vast majority (81 ± 10%, n = 136) of CME events marked by AP2-RFP culminated in a burst of ArpC3-GFP fluorescence, prior to internalization of the pit, persisting until the pit internalized (Figure 2G and Figure 2—video 2). In addition, 24 ± 4% of ArpC3-GFP tracks (n = 145) did not colocalize with AP2. We hypothesize that these are sites of clathrin-independent endocytosis. Then, using spinning-disk confocal fluorescence microscopy, we compared the fluorescence intensities of ArpC3-GFP spots and GFP-tagged nanocage proteins in cells to determine the numbers of ArpC3-GFP molecules at clathrin-mediated endocytosis sites (Figure 2H). Thus, we determined that ~200 molecules of Arp2/3 complex accumulate at clathrin-mediated endocytosis sites over time (Figure 2I). Incorporating the Arp2/3 molecule number we determined experimentally into our multiscale model, we next conducted simulations of the model to investigate the spatial organization of actin and force generation capacity of the endocytic network (Figure 3). Strikingly, the actin network self-organized around the endocytic pit. This self-organized network drove the assembly of 150 ± 30 actin filaments (Figure 3—figure supplement 1A) containing 5700 ± 1100 monomers (Figure 3—figure supplement 1B). Interestingly, only a small number of actin filaments (<5) grew at any given time because the filaments became capped soon after they were nucleated (Figure 3—figure supplement 1C; Berro et al. , 2010; Rangamani et al. , 2011; Xiong et al. , 2010). Filament lengths were exponentially distributed with a final length of 90 ± 80 nm (Figure 3—figure supplement 1D–E). Actin filaments bound to 120 ± 10 Hip1R molecules in the coat (Figure 3—figure supplement 1F). The endocytic pit internalized ~60 nm in 10–15 s (Figure 3A and D). Based on the initial geometry of the endocytic pit and activated Arp2/3 complex, branched actin filaments self-organized into a radial dendritic network: the network attached to the clathrin coat by binding to Hip1R, the pointed (minus) ends localized close to the pit and the barbed (plus) ends near the base of the pit were oriented to grow toward the base of the pit (Figure 3A–C and Figure 3—video 1). The axial self-organization of this branched actin network resembles that at the leading edge of cells (Figure 3—figure supplement 1G–I; Maly and Borisy, 2001; Mueller et al. , 2017; Schaus et al. , 2007; Svitkina and Borisy, 1999), with an important difference. Because actin filament attachment sites are located on the coat of the endocytic pit, filaments radiate away from the center of the pit, such that most of the barbed ends orient radially away from the center of the pit rather than toward the coat or neck (Figure 3E). The radial orientation of barbed ends gradually increases from the center of the pit, where there is no preferred orientation, to the periphery, where the barbed end radial orientation is highest (Figure 3F). The extent of radial distribution of the filaments increases rapidly after time 0 (Figure 3G). An important consequence of this self-organization is that, based on the position of Hip1R and the Arp2/3 complex, more barbed filament ends localize near the base (10 ± 4 ends) than near the neck of the endocytic pit (1 ± 1 ends) (Figure 3H). These data result in an important prediction from our model: an actin network self-organized as described here will produce an axial force during pit internalization (Figure 3I). We predict that the radial dendritic self-organization is a powerful mechanism that makes endocytic actin networks resilient to biochemical and mechanical perturbations. Our finding that self-organized endocytic actin networks grow toward the base of the pit prompted us to explore the molecular mechanism by which actin filaments self-organize. Actin dynamics in association with the endocytic machinery can be thought of as a polymerization engine constrained by two spatial boundary conditions -- active Arp2/3 complex at the base of the pit (Almeida-Souza et al. , 2018; Idrissi et al. , 2008; Kaksonen et al. , 2003; Mund et al. , 2018; Picco et al. , 2015; Kaplan et al. , in preparation) and Hip1R/actin attachments on the curved pit surface (Clarke and Royle, 2018; Engqvist-Goldstein et al. , 2001; Engqvist-Goldstein et al. , 1999; Sochacki et al. , 2017; Figure 4A). Given that such spatial boundary conditions confer unique mechanical properties and adaptation to loads under flat geometries in vitro (Bieling et al. , 2016), we aimed to understand how the boundary conditions corresponding to the curved endocytic pit affect endocytic actin organization and internalization. We tested two different scenarios: varying the surface density of Arp2/3 complex at the base of the pit and varying Hip1R surface coverage around the pit itself. First, we tested whether the surface density of the Arp2/3 complex at the base of the pit affects endocytic internalization because recent studies in vitro and in yeast suggest that the local concentration of Arp2/3 complex activators is critical for the timing of Arp2/3 complex activation and endocytic progression (Case et al. , 2019; Sun et al. , 2017). In a series of simulations, we distributed 200 molecules of active Arp2/3 complex in a ring of increasing outer diameter to vary the surface density. Surprisingly, we found that varying the surface density of Arp2/3 complex along the base of the pit by a factor of 20 had little impact on endocytic outcome (Figure 4—figure supplement 1). We also explored whether localization of a fraction of Arp2/3 complexes at the neck of the pit provided an additional advantage for the endocytic outcome. In this scenario, we distributed 50 of the 200 molecules of the active Arp2/3 complex near the neck of the pit. We found that localizing some of the active Arp2/3 complex near the neck of the pit did not have an impact on the outcome of simulations (p>0. 5) (Figure 4—figure supplement 2D–E; Figure 4—figure supplement 2). We next conducted a series of simulations in which we varied the surface distribution of a constant number of Hip1R molecules to cover between 1% (localized to the tip of the pit) and 80% (up to the neck of the pit) of the pit (Figure 4B) and found that the surface distribution of Hip1R around the endocytic pit strongly impacted endocytic outcome (Figure 4). Simulations in each of these conditions revealed that endocytic internalization depends on the surface distribution of actin-coat attachments around the endocytic site (Figure 4C and Figure 4—video 1). Both the rate and extent of internalization increased with increasing surface area of Hip1R around the pit (Figure 4D). From a functional standpoint, increased Hip1R surface coverage around the pit drove more barbed ends toward the base of the pit (Figure 4E). This increase in Hip1R surface coverage resulted in an increase in Arp2/3 complexes bound in the endocytic actin network (Figure 4F), which in turn nucleated more actin filaments (Figure 4G). Simulations showed that a threshold of ~100 Hip1R molecules on the pit is necessary for endocytic internalization (Figure 4—figure supplement 3A). The high impact of Hip1R surface distribution on actin filament organization implies that Hip1R molecules distributed broadly around the pit allow for multivalent attachments between the pit and actin filaments, resulting in filaments being captured in an orientation conducive to force production. Further examination of the simulations revealed that the Hip1R surface distribution supports a self-organized dendritic actin network via a mechanism of stochastic self-assembly and selection for actin filaments growing toward the base of the pit (Figure 4—figure supplement 3B). Mother filaments initially bind and unbind the coat in random orientations (Figure 4—figure supplement 3B–C). Filaments growing toward the interior of the cell do not template the growth of new branched actin filaments. However, filaments growing toward the base of the pit encounter active Arp2/3 complex, which catalyzes dendritic nucleation of new actin filaments growing in a similar direction (Figure 4—figure supplement 3B and D; Carlsson, 2001). As a result, near the base of the pit, filaments increasingly orient toward the base of the pit over time (Figure 4—figure supplement 3E–F). Our observations therefore establish an important principle for actin organization in endocytosis: the positions of active Arp2/3 complexes are critical for organizing the actin network and determining the direction of force production, while the Hip1R linker distribution is critical for recruiting mother filaments that activate the Arp2/3 complex to direct filament growth to the area surrounding the base of the pit. Given that self-organized actin filaments help to nucleate new actin filaments that grow toward the base of the pit, questions remained about how these filaments contribute to and sustain force production. Closer examination of the simulations revealed that long actin filaments bend between their attachment sites in the clathrin coat and the base of the pit as a result of assembly confined by the membrane at the base of the pit (Figure 5A). We predict that these bent filaments provide a previously unrecognized means of force production by endocytic actin filaments. To test the prediction that actin filaments bend at sites of mammalian endocytosis, we used cryo-electron tomography on intact vitrified mammalian (SK-MEL-2) cells. SK-MEL-2 cells grown on electron-microscopy grids are thin at their periphery (<1 µm), which makes them suitable for electron tomography studies. Indeed, we found bent actin filaments present at sites of clathrin-mediated endocytosis, between the clathrin coat (Figure 5—figure supplement 1) and the base of the pit (Figure 5B and Figure 5—video 1), in extended ‘U’-shaped clathrin-coated pits similar to the stage modeled in our simulations (Figure 5C–D). What could be a functional consequence of such bent filaments? We hypothesized that the bent actin filaments store elastic energy that promotes endocytic internalization. We first quantified the filament bending in simulations and found that many (13 ± 3%) of the actin filaments bend further than can be accounted for by thermal fluctuations (Boal and Boal, 2012; Mogilner and Oster, 1996; Figure 5E and Figure 5—figure supplement 2A). Most (92%) of the bent filaments bent less than the minimum energy expected to sever the filaments (De La Cruz et al. , 2015b; Sept and McCammon, 2001; Figure 5E). Importantly, the bent filaments stored elastic energy by collectively continuing to bend over time, storing up to ~750 pN·nm of elastic energy -- mostly in capped filaments (Figure 5F). In the context of pit internalization, the amount of elastic energy stored was larger than the magnitude of work required to internalize endocytic pits (Figure 5F and Figure 5—figure supplement 2B). The elastic energy stored in bent filaments was ~1% of the total energy required to polymerize the endocytic actin network (Figure 5—figure supplement 2C–D), with pit internalization constituting ~0. 5% of the total energy from actin filament assembly (Figure 5—figure supplement 2E). The majority (62 ± 20%) of filament bending energy came from filaments directly bound to Hip1R, and 78 ± 25% of the bending energy came from filaments with barbed ends > 5 nm from the coat surface (Figure 5—figure supplement 2F–H). 17 ± 16% of bending energy came from filaments with barbed ends near the base of the pit (Figure 5—figure supplement 2I). For filaments near the base of the pit, the bending energy was distributed radially such that filaments with barbed ends ~ 130 nm from the center of the pit contribute the most bending energy (Figure 5—figure supplement 2J). Filament bending serves as an important functional consequence of the self-organization of actin filaments at endocytic sites (Figure 4). With high Hip1R surface coverage around the pit, filaments directed to grow toward the base of the pit bend, storing elastic energy (Figure 5—figure supplement 2K–L). This elastic energy can be harnessed gradually under thermal fluctuations to drive endocytic internalization through a ratchet mechanism (Mogilner and Oster, 1996). To test the hypothesis that energy stored in bent actin filaments can promote endocytic internalization, we conducted simulations in which the resistance from membrane tension was released at a late time point (t = 10 s, internalization ~50 nm) along with capping filament growth (Figure 5G). This scenario allowed us to test how the stored energy in the bent filaments (rather than force generated by growing filaments) can promote internalization in response to an abrupt decrease in tension. We found that once membrane tension decreases, pit internalization sharply increases (Figure 5H and Figure 5—video 2) and filament bending near the base of the pit gradually decreases by 50% with wide variance (Figure 5I). Thus, we found that in addition to generating force actively by filament growth (Figure 3), the endocytic actin network stores potential energy in the form of bent filaments that can promote internalization even after filaments have stopped growing. We next investigated how inhibiting the activity of Arp2/3 complex would affect endocytosis (Figure 6). Our simulations, conducted by varying the nucleation rate of Arp2/3 complex, predicted that inhibiting Arp2/3 complex activity stalls endocytosis (Figure 6A). Endocytosis was inhibited when Arp2/3 complex nucleation rates fell below the basal value of 1 filament per second (Beltzner and Pollard, 2008), and was insensitive to increased rates of nucleation (Figure 6B). We validated this relationship with experiments modulating Arp2/3 complex activity in cells. The small molecule inhibitor CK-666 prevents the Arp2/3 complex from nucleating actin filaments (Hetrick et al. , 2013; Nolen et al. , 2009). Treatment of SK-MEL-2 cells with CK-666 inhibited endocytic progression, as marked by the lifetimes of endogenously tagged AP2-RFP or dynamin2-GFP at endocytic sites, in a dose-dependent and time-dependent manner (Figure 6C–D and Figure 6—figure supplement 1A). Because we and others previously modeled that membrane tension plays an important role in membrane bending during the formation of an endocytic pit (Hassinger et al. , 2017; Rangamani et al. , 2014; Walani et al. , 2014) we next varied the value of membrane tension in simulations to understand the relationship between tension, actin filament bending, and actin assembly (Figure 7A). In our simulations, endocytic progression attenuated in a tension-dependent manner (Figure 7A–E), consistent with previous modeling (Hassinger et al. , 2017) and experimental observations (Boulant et al. , 2011; Ferguson et al. , 2017; Ferguson et al. , 2016; Wu et al. , 2017). However, at higher membrane tensions, endocytosis persisted better than expected for a non-adapting network, suggesting the existence of an adaptive mechanism (Figure 7E). Therefore, we sought to understand how endocytic actin networks adapt to increases in load. We found that under low tension (0. 015 pN/nm), endocytic pits internalize strongly (Figure 7B and E) and few barbed ends encounter the base of the pit (Figure 7F), with fewer Arp2/3 complexes recruited to the network (Figure 7—figure supplement 1A) and a correspondingly low filament bending energy (Figure 7H). Under >50 x higher membrane tension (1 pN/nm), endocytic internalization slowed but was not abolished (Figure 7E). For these pits, more barbed ends encountered the base of the pit (Figure 7F), binding more Arp2/3 complexes (Figure 7—figure supplement 1A) to nucleate more actin filaments (Figure 7G) and increasing the total actin filament bending energy near the base of the pit (Figure 7H). As a result, upon increasing membrane tension, the overall endocytic energy efficiency increased (Figure 7—figure supplement 1B). Thus, the self-organization of the endocytic actin network allows it to adapt to elevated membrane tension by nucleating more filaments at the base of the pit. Having established that endocytic internalization depends on two spatially confined boundary conditions -- Hip1R/actin attachments at the curved pit (Figure 4) and active Arp2/3 complex activity at the base of the pit (Figure 6) -- we next investigated how these boundary conditions alter the endocytic response to membrane tension (Figure 8A). We systematically varied membrane tension and Arp2/3 complex activity in our model to generate a phase diagram of endocytic internalization as a function of membrane tension and Arp2/3 complex activity (Figure 8B). This phase diagram shows that cells with high membrane tension are especially sensitive to changes in Arp2/3 complex nucleation rate (Kaplan et al. , in preparation), whereas cells with low membrane tension carry out endocytosis even with low Arp2/3 complex activity, consistent with experimental observations (Boulant et al. , 2011). We hypothesized that actin network self-organization arising from the broad Hip1R distribution around the pit (Figure 4) and filament bending (Figure 5) might allow for the endocytic actin network to change its organization and force-producing capacity under elevated loads (Figure 7). To test this hypothesis, we conducted simulations in which Hip1R coverage was varied for different values of plasma membrane tension (Figure 8A and Figure 7—video 1). We found that the endocytic actin network’s ability to adapt to load (Figure 7) depends on Hip1R coverage around the pit (Figure 8D–F). As the coverage of Hip1R around the pit increased, actin’s ability to adapt to changes in membrane tension also increased, as measured by the number of barbed ends near the base of the pit (Figure 8D), the binding of active Arp2/3 complex at the base of the pit (Figure 8E), subsequent nucleation of additional actin filaments (Figure 8F), and bending of actin filaments near the base of the pit (Figure 8G). We conclude that sufficient Hip1R coverage around the pit (Clarke and Royle, 2018; Sochacki et al. , 2017) allows endocytic actin filaments to orient in such a way that they can encounter more Arp2/3 complexes at the base of the pit to nucleate more actin filaments. This spatial organization allows the actin network to adapt to sustain force production under a range of opposing loads (Figure 8H).
  Understanding the relationship between actin filament assembly, actin network organization, and force generation on the plasma membrane requires iterative feedback between experimental measurements and computational modeling. An ultimate goal of this study was to relate single actin filament mechanics to force generation by the collective actin filament network in CME (Lacayo et al. , 2007). We integrated modeling and quantitative cellular measurements to show that a minimal actin network composed of actin, the Arp2/3 complex and capping protein, with linker attachments in the clathrin coat and rates constrained by cellular and biochemical measurements, is able to generate sufficient force to internalize endocytic pits against mammalian plasma membrane tension. Approximately 200 Arp2/3 complexes constitutively assemble at sites of endocytosis in human induced pluripotent stem cells. Endocytic actin filaments self-organize into a radial dendritic array, in which filaments grow toward the base of the pit. These filaments bend and store elastic energy, which supports internalization. The endocytic actin network adapts to changes in membrane tension by driving more filaments to the base of the pit and increasing filament bending, which supports a higher load and nucleates more actin filaments. Four lines of experimental evidence support our model (Figure 8—figure supplement 1). Two pieces of evidence serve as model validation based on published data and two more are based on experiments conducted in this study. Previous experiments from our lab showed that knocking down Hip1R in HeLa cells inhibited endocytosis (Engqvist-Goldstein et al. , 2004). Consistent with these observations, our simulations showed that a threshold number of actin linkers such as Hip1R is necessary for endocytic internalization (Figure 4—figure supplement 3A and Figure 8—figure supplement 1A–B). This threshold appears necessary to allow efficient filament capture by the coat and force transmission from the actin network to the coat. Second, experiments showed that capping actin filament elongation with the small molecule compound Cytochalasin inhibits CME, reflected by an increase in stalled endocytic sites marked by clathrin-RFP (Brady et al. , 2010) or slower accumulation of dynamin2-GFP at endocytic sites (Grassart et al. , 2014). We also showed in our model that capping rate is an important parameter for progression of CME; our simulations show that increasing the capping rate of actin filaments inhibits CME, presumably because increasing capping decreases the total amount of actin (Figure 1—figure supplement 1C and Figure 8—figure supplement 1C–D). In this study, our simulations predicted that actin filaments bend around endocytic pits. These bent filaments store elastic energy for subsequent force production much as a pole vaulter’s pole bends and stores energy for delayed force production. Using cryo-electron tomography of intact cells, we observed bent actin filaments at sites of endocytosis in mammalian cells (Figure 5 and Figure 8—figure supplement 1E–F). Finally, we also predicted that inhibiting Arp2/3 complex activity below its basal nucleation rate of 1 filament per second inhibits endocytosis in silico and this prediction was validated in cells using pharmacological agents (Figure 6C, D and Figure 8—figure supplement 1G–H). Without sufficient Arp2/3 complex, CME fails due to insufficient force production. Three main conclusions resulted from our study. First, we found that the spatial segregation of Arp2/3 complex activation and Hip1R linker proteins on the clathrin coat are important factors for effective force generation. Unlike actin organization at the leading edge of a migrating cell wherein only one boundary condition at the plasma membrane is sufficient to enable force-generation capacity to be inferred (Abercrombie, 1980; Bieling et al. , 2016; Mogilner and Edelstein-Keshet, 2002), in CME two boundary conditions are required -- one at the base of the pit for actin polymerization against the plasma membrane and the second on the coat of the pit for attachment of the growing actin filaments so forces are transmitted to the pit to drive internalization. In our model, we used Hip1R as a surrogate for Hip1, Hip1R, and Epsin1/2/3, which cooperatively link actin filaments to the coat (Brett et al. , 2006; Chen et al. , 1998; Messa et al. , 2014; Senetar et al. , 2004; Skruzny et al. , 2012). We conclude that based on the relative positions of Arp2/3 complex activators and actin filament linkers, the resultant self-organized actin network orients to produce force orthogonal to the base of the pit rather than producing a pinching force on the neck (Collins et al. , 2011; Hassinger et al. , 2017). Pinching forces are achieved by the spontaneous curvature of the rigid coat (Alimohamadi et al. , 2018; Foret, 2014; Hassinger et al. , 2017). Any constriction forces generated by actin polymerization at the neck would likely occur at a later stage of endocytosis than is the focus of our model, and the filaments would need to be nucleated by a spatially distinct set of Arp2/3 activating proteins around the neck, or by an interaction between other actin filaments and dynamin, but the mechanism for arranging and anchoring such a network has not been elucidated (Ma and Berro, 2018). Second, the effective anchoring of actin filaments to the surface of the pit depends on the distribution of linker proteins on the pit surface. Since these linker proteins are embedded within the clathrin coat (Clarke and Royle, 2018; Engqvist-Goldstein et al. , 2001; Sochacki et al. , 2017), their surface coverage is directly proportional to the coat coverage on the endocytic pit. This observation suggests that one possible function of a large coat is for the actin-linking proteins Hip1, Hip1R and Epsin to cover enough surface area to provide leverage for internalization. The role of linker protein coverage in force generation also has implications for the flat-to-curved transition earlier in endocytosis, when the membrane either begins to bend from a flat pre-assembled coat or continually deforms while the coat assembles (Avinoam et al. , 2015; Bucher et al. , 2018; Scott et al. , 2018; Sochacki and Taraska, 2019). In cases when the clathrin coat gradually increases in area during membrane deformation, our findings imply that actin polymerization may be ineffective until the coat reaches a threshold size (Avinoam et al. , 2015; Sun et al. , 2017), with membrane tension controlling a switch for the flat-to-curved transition (Bucher et al. , 2018; Scott et al. , 2018). Future work will investigate the relationship between coat topology and actin forces during the initiation of endocytosis. Third, we showed a significant fraction of endocytic actin filaments bend under force. We predict that the bent filaments, whose existence we confirmed by cryo-electron tomography of intact cells, contribute to successful endocytic internalization in at least two ways. First, they might contribute to the resilience of endocytosis by preventing backsliding of the pit. Second, we expect that they contribute to internalization by releasing stored elastic energy when they straighten out under thermal fluctuations, consistent with the elastic Brownian ratchet mechanism for actin-mediated force production (De La Cruz and Gardel, 2015a; Mogilner and Oster, 1996). Here, filament bending occurs to a greater extent and for a longer time than previously described for coherent flat surfaces like the leading edge, possibly due to the curved geometry of endocytic pits. Fixing the filament orientation at one end increases bending energy (De La Cruz et al. , 2015b; Fernández et al. , 2006), which is accomplished here by multivalent attachments from Hip1R. Previous studies overlooked the role of actin filament bending at endocytic sites because of the predicted short length of filaments based on population averages (Berro et al. , 2010) and the possible loss of less densely branched filaments during the preparation process for platinum-replica electron microscopy (detergent extraction or sonication-based unroofing) (Collins et al. , 2011). The load response of branched actin networks in vitro can be reversible due to filament bending (Chaudhuri et al. , 2007), or permanent from a change in filament architecture (Bieling et al. , 2016; Parekh et al. , 2005). In our simulations, some of the elastic energy from bent filaments is released as internalization increases, suggesting a reversible compression of the network to store elastic energy (Figure 5). However, a significant fraction of filament bending is retained after the spring is released, which suggests that load also changes the intrinsic structure of the network (Bieling et al. , 2016). Importantly, the results presented here demonstrate a mechanism of active adaptation by the endocytic actin network to changes in load (Figure 7). Different cell types, different locations in the same cell, and different stages of endocytosis at the same location can have different membrane tension values at different times (Shi et al. , 2018). Under flat geometries, branched actin networks adapt to load (Bieling et al. , 2016; Mueller et al. , 2017). Here, the distribution of Hip1R linkers around the pit directs more filaments to grow toward the base of the pit (Figure 4), which nucleates more filaments autocatalytically and increases filament bending (Figure 5), thereby supporting greater internalization (Figure 7). It is now important to determine whether the principles of actin filament self-organization and load adaptation identified here also apply to endocytic actin in the higher force regime characteristic of fungi. An agent-based model of endocytic actin networks in yeast predicted that barbed filament ends radiate away from the center of pit in the XY plane (Mund et al. , 2018). However, the >200 x larger force requirements in this organism lead to a different axial organization of the filaments, less filament bending, and a distinct mechanism of force production sufficient to counteract high turgor pressure. Understanding the mechanical function and load adaptation in the ‘soft’ regime studied here is likely to apply to a variety of cellular membrane bending processes employing branched actin networks, including the assembly and maturation of endosomes, lysosomes, and autophagosomes (Rottner et al. , 2017). Finally, we acknowledge that our model represents a minimal core actin machinery necessary for endocytic internalization in mammalian cells. This feature of our study was necessary so the number of free parameters could be limited. Future models will add complexity to test roles for filament crosslinking, filament severing, and myosin-I motor activity, among other factors. It is anticipated that these additional features will increase our understanding of the force-generation capability and overall efficiency of the endocytic actin network.
  We combined a continuum membrane mechanics model with filament-based simulations of actin polymerization coupled to endocytic pit internalization to develop a multiscale model of actin in mammalian endocytosis. In the continuum model, the bending of the membrane was explicitly modeled and the contributions of the actin cytoskeleton were simplified to an applied localized force, while the agent-based model simplified the membrane and explicitly modeled actin and associated binding proteins (Scheme 1). We iteratively ran simulations in each module. The results from each module improved the assumptions of the other, over successive rounds of simulation and comparison to experimental measurements. In particular, experiments focused our attention on the internalization of the U-shaped pit and its transition to omega shape for both the membrane and actin modules. Experiments and the actin module informed the location of actin forces for the membrane mechanics model. The membrane mechanics simulations in turn informed the initial pit shape and force/extension relationship for the actin module. Thus, these two modules synergistically provided collective information about how actin organization and dynamics couple to the bending and internalization of the clathrin-coated pit. Continuum mechanics modeling of the plasma membrane allows a quantitative understanding of the relationship between applied forces and the shape of the membrane (Derényi et al. , 2002; Rangamani et al. , 2013). Bending the membrane requires energy, such that pulling a tether from a flat membrane requires increasing force until the membrane adopts a tubule shape (Derényi et al. , 2002; Alimohamadi et al. , 2018). Adding a region with spontaneous curvature (corresponding to the endocytic coat) can help lower this energy barrier against moderate membrane tension (Rangamani et al. , 2013; Hassinger et al. , 2017). Forces due to actin polymerization can also help overcome the energy barrier (Hassinger et al. , 2017), but the relationship between applied actin forces and coated membrane shape has not been explored quantitatively. The following assumptions guide our model of the coated plasma membrane: We use a modified form of the Helfrich energy defined asW (H, K, θα) =κ (H−C (θα) ) 2+κ¯K A general force balance on the membrane can be written as▽⋅σ+pn=f, where is ▽⋅ surface divergence, σ is the stress vector, p is the pressure difference between the inside and outside of the volume bounded by the membrane, and f is any externally applied force per unit area on the membrane. The stress vector can be split into normal and tangential components asσα=Tα+Sαn, whereTα=Tαβaβ, Tαβ=σαβ+bμβMμα, Sα=−M; βαβ. σαβ and Mαβ can also be written asσαβ=ρ (∂F (ρ, H, K; xα) ∂aαβ+∂F (ρ, H, K; xα) ∂aβα), Mαβ=ρ2 (∂F (ρ, H, K; xα) ∂bαβ+∂F (ρ, H, K; xα) ∂bβα), Here (aαβ) = (aαβ) is the dual metric or first fundamental form, bαβ is the second fundamental form and ρ is the surface mass density. Using the first and second fundamental form, we can define H (mean curvature) and K (Gaussian curvature) asH=12aαβbαβ, K=12εαβελμbαλbβμ. where εαβ is the permutation tensor defined by ε12=−ε21=1a, ε11=ε22=0. We then define an area incompressibility constraint by rewriting the free energy density asF (ρ, H, K; xα) =F~ (H, K; xα) −γ (xα, t) ρ. where γ (xα, t) is a Lagrange multiplier field required to impose invariance of ρ on the whole of the surface. This free energy density relates to the Helfrich energy density asW=ρF~ Combining these equations, we can get the stress equationsσαβ= (λ+W) aαβ− (2HWH+2κWK) aαβ+WHb~αβ, andMαβ=12WHaαβ+Wκb~αβ, where λ=− (γ+W) Simplifying this further, we can get the shape equation (normal balance) p+f⋅n=Δ12WH+ (WK); αβb~αβ+WH (2H2−K) +2H (KWK−W) −2λH, and the tangential balance (∂W∂x|expα+λ, α) aβα=f. as. where () |exp denotes the explicit derivative respect to coordinate θα. To further simplify the equations, we define the coordinate system as axisymmetric usingr (s, θ) =r (s) er (θ) +z (s) k. We define a ψ such that r′ (s) =cos (ψ), z′ (s) =sin (ψ) and n=−sinψer (θ) +cosψk Using this, we can write the mean curvature (H) and Gaussian curvature (K) asH=12 (κv+κτ) =12 (ψ′+r−1sinψ) K=κτκv=ψ′sinψr. We also introduceL=12κr (WH) ′allowing us to formulate a system of ordinary differential equations (ODE’s) as function of arc length sr′=cosψ, z′=sinψ, rψ′=2rH−r′=−sinψ, rH′=L+rC′, L′r=pk+f⋅nκ+2H[ (H−C) 2+λκ−2 (H−C) ][H2+ (H−r−1sinψ) 2]λ′=2κ (H−C) C′−f⋅as. This can also be written as a function of membrane area usinga (s) =2π∫0sr (ξ) dξ→dads=2πr. Here, we choose to non-dimensionalize the system usingα=a2πR02, x=rR0, y=yR0, h=HR0, c=CR0, l=LR0λ∗=λR02κ0, p∗=pR03κ0f∗=fR03κ0, κ∗=κκ0giving us the system of equationsxx˙=cosψ, xy˙=sinψ, x2ψ˙=2xh−sinψ, x2h˙=l+x2c˙, l˙=p∗κ∗+f∗⋅nκ∗+2h[ (h−c) 2+λ∗κ∗]−2 (h−c) [h2+ (h−x−1sinψ) 2], λ˙∗=2κ∗ (h−c) c˙−f∗⋅asx. We define a spatially varying spontaneous curvature asc=c0∗0. 5 (1−tanh (g∗ (a−a0) ) ) where a is the non-dimensional membrane area, a0 is the non-dimensional membrane area of the protein coat, g is a constant and c0 is the coat spontaneous curvature. The parameters used for the spontaneous curvature simulations are specified in Supplementary file 1: To perform the coat pulling simulations, we applied an axial force acting downward along the protein coat and upward along the base of the pit such that the net force integrates to 0 (we do this by scaling the applied force by the area over which it is applied). This force function was defined asf=f0∗ (0. 5 ( (1−tanh (g∗ (a−a0) ) ) /a0− (tanh (g∗ (a−ain) ) −tanh (g∗ (a−aout) ) ) / (aout−ain) ) ) where a0 is the non-dimensional coat area, a is the non−dimensional membrane area, aout−ain is the area of force applied at the base of the pit. ain corresponds to an inner radius rin and aout corresponds to an outer radius rout within which the upward force is applied. These parameters are specified in Supplementary file 2. To simulate the pinched (‘omega-shaped’”) curves at high membrane tension, we provided an initial guess of an ‘omega-shaped’ membrane from a lower membrane tension. We did this because the simulations stalled at U shapes at an internalization of about 100 nm. Providing this initial guess led to solutions for membrane shape and force beyond 100 nm, as seen in Figure 1C. Further, to fully explore the space of solutions, we ran the simulations backward by starting from an ‘omega shaped’ pit at a large internalization and then decreasing the internalization. In Figure 1C, we plotted the curves for negative internalization > = the farthest internalization for the U shaped pit (generally ~100 nm). Values of membrane tension in Figure 1C are [0: 051: 0. 05: 0. 451] pN/nm, and rounded to two significant digits in the figure for clarity. We used Cytosim (Nedelec and Foethke, 2007) to model the polymerization of a branched actin network coupled to the internalization of a clathrin-coated pit. This approach simplified the pit as a bead attached to a flat boundary (the plasma membrane) by a spring. This assumption of a linear force-extension relationship was validated in Figure 1. Actin filaments and actin-binding proteins (Arp2/3 complex, Hip1R) were explicitly simulated as individual objects (agents). Cytosim calculates the forces on each segment of actin from rules such as diffusion, confinement, growth, and binding based on Brownian dynamics. We derived most parameters from experimental data in the literature, and made measurements for some measurements not available (Supplementary file 3). We varied the remaining parameters to show their effect on the outcome of the simulations. Discussion of each parameter follows below. Calibration curve: We used a combination of custom-written and publicly available image analysis software in Fiji (1. 52i) and Matlab (r2017b) to analyze the traces of fluorescence intensity per spot, for multiple Z slices and time points. To measure fluorescence intensity per spot of GFP-tagged nanocages, we wrote a toolset in Fiji to select and circular regions of interest eight pixels (1. 1 µm) in diameter, and used cross-correlation to center the regions around the intensity-based center of mass. We selected as background regions of concentric circles one pixel larger than the original region of interest. The toolset measured the fluorescence intensity per spot, which we subtracted by the area-corrected background intensity to yield fluorescence intensity per spot for the four constructs. Some analysis functions were adapted from published software (Akamatsu et al. , 2017; Epstein et al. , 2018; McCormick et al. , 2013). We measured only spots that were contained within the slices imaged and were single stationary spots. For comparison of fluorescence to eGFP-MotB, we used smaller (6-pixel) ROIs. We plotted background-subtracted fluorescence intensity per spot as a function of predicted copy number per structure to obtain a calibration curve relating fluorescence intensity per spot to numbers of molecules per structure. For the curve in Figure 3D, we combined data from three experiments with different imaging conditions by defining the average 60mer-GFP intensity per experiment as 1000 arbitrary units. Lines are linear fits through zero with r2 calculated by linear least-squares fitting. Time-lapse fluorescence quantification: We made modifications to automated MATLAB-based tracking software (Aguet et al. , 2013; Hong et al. , 2015) to track and analyze fluorescence-intensity time lapse data of genome-edited cells. The core tracking program (based on the software package µ-track) automatically identifies fluorescent spots and connects them as tracks by minimizing the linear assignment problem (Jaqaman et al. , 2008). We used stringent tracking parameters with gap size 0 and search radius 0–2. 3 pixels (248 nm). GFP and RFP tracks with high variability in the intensity/time profile were automatically rejected (Ferguson et al. , 2017) as well as tracks ≤ 3 s in duration (Dambournet et al. , 2018) and the remaining tracks were associated spatiotemporally according to a cost matrix (Hong et al. , 2015). We used two track rejection schemes. In the first, users were presented with fluorescence montages and XY coordinates of the tracks to assess the fidelity of tracking for each event (Hong et al. , 2015). In the second, tracks were automatically rejected based signal-to-noise ratio (>1. 05) and proximity to neighboring tracks (>525 nm) (Hong et al. , 2015). We checked that the manual and automatic track rejection schemes yielded similar results (lifetime distributions and intensity versus time plots) as well as to manual, kymograph-based quantification of lifetimes (below). From the above workflow (Dambournet et al. , 2018) we increased throughput by connecting all steps into an automated tracking pipeline requiring minimal user input. For SK-MEL-2 cells expressing CLTA-RFP and DNM2-GFP, we tracked regions that were not near the nucleus (which has a concentration of Golgi-derived clathrin budding) and that did not have large, bright, persistent structures containing invariant RFP and GFP signals (‘plaques, ’ which are likely sites of adhesion). Alignment method: For Figure 2G, we aligned tracks based on the time point after the peak intensity in which 50% of the fluorescence remained in the GFP channel. We normalized the fluorescence intensity to compare movies from different imaging conditions. For Figure 2I, we aligned the tracks based on the maximum intensity. For Figure 6D, we aligned the tracks based on the disappearance of the RFP signal. This code is available at the following website: https: //github. com/DrubinBarnes/Akamatsu_CME_manuscript. Manual track analysis: We wrote a Fiji toolset that generated two-color kymographs from user-defined regions of interest, and then quantified the lifetime based on the lengths of the kymographs (from user-defined regions on the kymographs). This manual analysis was used in Figure 6—figure supplement 1A and for verification of the automated tracking scheme. This code is available at the following website: https: //github. com/DrubinBarnes/Akamatsu_CME_manuscript. Nanocage particle tracking: To track the membrane-tethered nanocages in 2D in cells we used TrackMate, a plugin available in Fiji that optimizes the Linear Assignment Problem (LAP) (Jaqaman et al. , 2008). We detected spots using an estimate of 0. 5 µm and threshold of 15, using a median filter and sub-pixel localization. We used the simple LAP tracker with a maximum linking and gap closing distance of 1 µm and two frames and a minimum track length of 4 frames. Calculating numbers of molecules per endocytic site: We calculated the fluorescence intensity of background-subtracted ArpC3-GFP spots colocalized with AP2-RFP spots from single time-point images using the same background correction approach described for the calibration curve. We used the slope of the calibration curve to convert fluorescence intensity to numbers of molecules of ArpC3-GFP. Because the standard is inside cells, this standard controls for fluorescence environment and fluorescent protein folding and maturation. We used the resultant histogram of numbers of ArpC3-GFP per spot and the time-lapse fluorescence intensity data in Figure 2G to create the graph in Figure 2I of numbers of molecules of ArpC3-GFP over time.
summary: The outer membrane of a cell is a tight but elastic barrier that controls what enters or leaves the cell. Large molecules typically cannot cross this membrane unaided. Instead, to enter the cell, they must be packaged into a pocket of the membrane that is then pulled inside. This process, called endocytosis, shuttles material into a cell hundreds of times a minute. Endocytosis relies on molecular machines that assemble and disassemble at the membrane as required. One component, a protein called actin, self-assembles near the membrane into long filaments with many repeated subunits. These filaments grow against the membrane, pulling it inwards. But it was not clear how actin filaments organize in such a way that allows them to pull on the membrane with enough force – and without a template to follow. Akamatsu et al. set about identifying how actin operates during endocytosis by using computer simulations that were informed by measurements made in living cells. The simulations included information about the location of actin and other essential molecules, along with the details of how these molecules work individually and together. Akamatsu et al. also developed a method to count the numbers of molecules of a key protein at individual sites of endocytosis. High-resolution imaging was then used to create 3D pictures of actin and endocytosis in action in human cells grown in the laboratory. The analysis showed the way actin filaments arrange themselves depends on the starting positions of a few key molecules that connect to actin. Imaging confirmed that, like a pole-vaulting pole, the flexible actin filaments bend to store energy and then release it to pull the membrane inwards during endocytosis. Finally, the simulations predicted that the collection of filaments adapts its shape and size in response to the resistance of the elastic membrane. This makes the system opportunistic and adaptable to the unpredictable environment within cells.
keywords: cell biology
  physics of living systems
title: Principles of self-organization and load adaptation by the actin cytoskeleton during clathrin-mediated endocytosis