Membrane Mechanics of Endocytosis in Cells with Turgor

A Role for Cross-linking Proteins in Actin Filament Network Organization and Force Generation

The high turgor pressure across the plasma membrane of yeasts creates a requirement for substantial force production by actin polymerization and myosin motor activity for clathrin-mediated endocytosis (CME). Endocytic internalization is severely impeded in the absence of fimbrin, an actin filament crosslinking protein called Sac6 in budding yeast. Here, we combine live-cell imaging and mathematical modeling to gain new insights into the role of actin filament crosslinking proteins in force generation. Genetic manipulation showed that CME sites with more crosslinking proteins are more effective at internalization under high load. Simulations of an experimentally constrained, agent-based mathematical model recapitulate the result that endocytic networks with more double-bound fimbrin molecules internalize the plasma membrane against elevated turgor pressure more effectively. Networks with large numbers of crosslinks also have more growing actin filament barbed ends at the plasma membrane, where the addition of new actin monomers contributes to force generation and vesicle internalization. Our results provide a richer understanding of the crucial role played by actin filament crosslinking proteins during actin network force generation, highlighting the contribution of these proteins to the self-organization of the actin filament network and force generation under increased load.

Inferring biophysical properties of membranes during endocytosis using machine learning

Endocytosis is a fundamental cellular process in eukaryotic cells that facilitates the transport of molecules into the cell. With the help of fluorescence microscopy and electron tomography, researchers have accumulated extensive geometric data of membrane shapes during endocytosis. These data contain rich information about the mechanical properties of membranes, which are hard to access via experiments due to the small dimensions of the endocytic patch. In this study, we propose an approach that combines machine learning with the Helfrich theory of membranes to infer the mechanical properties of membranes during endocytosis from a dataset of membrane shapes extracted from electron tomography. Our results demonstrate that machine learning can output solutions that both match the experimental profile and satisfy the membrane shape equations derived from Helfrich theory. The learning results show that during the early stage of endocytosis, the inferred membrane tension is negative, indicating the presence of strong compressive forces at the boundary of the endocytic invagination. Our method presents a generic framework for extracting membrane information from super-resolution imaging.

Quantifying turgor pressure in budding and fission yeasts based upon osmotic properties

Walled cells, such as plants, fungi, and bacteria cells, possess a high internal hydrostatic pressure, termed turgor pressure, that drives volume growth and contributes to cell shape determination. Rigorous measurement of turgor pressure, however, remains challenging, and reliable quantitative measurements, even in budding yeast are still lacking. Here, we present a simple and robust experimental approach to access turgor pressure in yeasts based upon the determination of isotonic concentration using protoplasts as osmometers. We propose three methods to identify the isotonic condition - three-dimensional cell volume, cytoplasmic fluorophore intensity, and mobility of a cytGEMs nano-rheology probe - that all yield consistent values. Our results provide turgor pressure estimates of 1.0 ± 0.1 MPa for Schizosaccharomyces pombe, 0.49 ± 0.01 MPa for Schizosaccharomyces japonicus, 0.5 ± 0.1 MPa for Saccharomyces cerevisiae W303a and 0.31 ± 0.03 MPa for Saccharomyces cerevisiae BY4741. Large differences in turgor pressure and nano-rheology measurements between the Saccharomyces cerevisiae strains demonstrate how fundamental biophysical parameters can vary even among wild-type strains of the same species. These side-by-side measurements of turgor pressure in multiple yeast species provide critical values for quantitative studies on cellular mechanics and comparative evolution.

Force redistribution in clathrin-mediated endocytosis revealed by coiled-coil force sensors

Forces are central to countless cellular processes, yet in vivo force measurement at the molecular scale remains difficult if not impossible. During clathrin-mediated endocytosis, forces produced by the actin cytoskeleton are transmitted to the plasma membrane by a multiprotein coat for membrane deformation. However, the magnitudes of these forces remain unknown. Here, we present new in vivo force sensors that induce protein condensation under force. We measured the forces on the fission yeast Huntingtin-Interacting Protein 1 Related (HIP1R) homolog End4p, a protein that links the membrane to the actin cytoskeleton. End4p is under ~19-piconewton force near the actin cytoskeleton, ~11 piconewtons near the clathrin lattice, and ~9 piconewtons near the plasma membrane. Our results demonstrate that forces are collected and redistributed across the endocytic machinery.

Quantifying turgor pressure in budding and fission yeasts based upon osmotic properties

Walled cells, such as plants, fungi, and bacteria cells, possess a high internal hydrostatic pressure, termed turgor pressure, that drives volume growth and contributes to cell shape determination. Rigorous measurement of turgor pressure, however, remains challenging, and reliable quantitative measurements, even in budding yeast are still lacking. Here, we present a simple and robust experimental approach to access turgor pressure in yeasts based upon the determination of isotonic concentration using protoplasts as osmometers. We propose three methods to identify the isotonic condition - 3D cell volume, cytoplasmic fluorophore intensity, and mobility of a cytGEMs nano-rheology probe - that all yield consistent values. Our results provide turgor pressure estimates of 1.0 ± 0.1 MPa for S. pombe, 0.49 ± 0.01 MPa for S. japonicus, 0.5 ± 0.1 MPa for S. cerevisiae W303a and 0.31 ± 0.03 MPa for S. cerevisiae BY4741. Large differences in turgor pressure and nano-rheology measurements between the S. cerevisiae strains demonstrate how fundamental biophysical parameters can vary even among wildtype strains of the same species. These side-by-side measurements of turgor pressure in multiple yeast species provide critical values for quantitative studies on cellular mechanics and comparative evolution.

Cooperative cell-cell actin network remodeling to perform Gap junction endocytosis

BACKGROUND

The endocytosis of Gap junction plaques (GJP) requires cytoskeletal forces to internalize such large membranous structures. Actin, which partners the connexin proteins constituting Gap junctions and is located close to Annular Gap Junctions (AGJ), could be actively involved in this physiological process.

RESULTS

Electron Microscopy and Light Microscopy images, associated with time-lapse analysis and 3D reconstruction, used at high resolution and enhanced using ImageJ based software analysis, revealed that: i) actin cables, originating from Donor cells, insert on the edge of GJP and contribute to their invagination, giving rise to AGJ, whereas actin cables on the Acceptor cell side of the plaque are not modified; ii) actin cables from the Donor cell are continuous with the actin network present over the entire GJP surface. These actin cables fuse at a single point distant from the plaque, which then detaches itself from the membrane, condensing to form an actin mass during the final internalization process; iii) the Acceptor cell participates in the last step of the endocytic invagination process by forming an annular actin structure known as an actin ring.

CONCLUSIONS

Together, these data suggest that the endocytosis of GJP is an example of a unique cooperative mechanism between the Donor (the traction of its actin cables) and the Acceptor cells (forming the actin ring).

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

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

Hydrogel Microtumor Arrays to Evaluate Nanotherapeutics

Nanoparticle drug formulations have many advantages for cancer therapy due to benefits in targeting selectivity, lack of systemic toxicity, and increased drug concentration in the tumor microenvironment after delivery. However, the promise of nanomedicine is limited by preclinical models that fail to accurately assess new drugs before entering human trials. In this work a new approach to testing nanomedicine using a microtumor array formed through hydrogel micropatterning is demonstrated. This technique allows partitioning of heterogeneous cell states within a geometric pattern-where boundary regions of curvature prime the stem cell-like fraction-allowing to simultaneously probe drug uptake and efficacy in different cancer cell fractions with high reproducibility. Using melanoma cells of different metastatic potential, a relationship between stem fraction and nanoparticle uptake is discovered. Deformation cytometry reveals that the stem cell-like population exhibits a more mechanically deformable cell membrane. Since the stem fraction in a tumor is implicated in drug resistance, recurrence, and metastasis, the findings suggest that nanoparticle drug formulations are well suited for targeting this dangerous cell population in cancer therapy.

The endocytic protein machinery as an actin-driven membrane-remodeling machine

In clathrin-mediated endocytosis, a principal membrane trafficking route of all eukaryotic cells, forces are applied to invaginate the plasma membrane and form endocytic vesicles. These forces are provided by specific endocytic proteins and the polymerizing actin cytoskeleton. One of the best-studied endocytic systems is endocytosis in yeast, known for its simplicity, experimental amenability, and overall similarity to human endocytosis. Importantly, the yeast endocytic protein machinery generates and transmits tremendous force to bend the plasma membrane, making this system beneficial for mechanistic studies of cellular force-driven membrane reshaping. This review summarizes important protein players, molecular functions, applied forces, and open questions and perspectives of this robust, actin-powered membrane-remodeling protein machine.

Regulation of membrane scission in yeast endocytosisDepartment of Biochemistry and National Centre of Competence in Research, Chemical Biology, University of Geneva, Geneva, Switzerland

During clathrin-mediated endocytosis, a flat plasma membrane is shaped into an invagination that undergoes scission to form a vesicle. In mammalian cells, the force that drives the transition from invagination to vesicle is primarily provided by the GTPase dynamin that acts in concert with crescent-shaped BAR domain proteins. In yeast cells, the mechanism of endocytic scission is unclear. The yeast BAR domain protein complex Rvs161/167 (Rvs) nevertheless plays an important role in this process: deletion of Rvs dramatically reduces scission efficiency. A mechanistic understanding of the influence of Rvs on scission however, remains incomplete. We used quantitative live-cell imaging and genetic manipulation to understand the recruitment and function of Rvs and other late-stage proteins at yeast endocytic sites. We found that arrival of Rvs at endocytic sites is timed by interaction of its BAR domain with specific membrane curvature. A second domain of Rvs167 - the SH3 domain - affects localization efficiency of Rvs. We show that Myo3, one of the two type-I myosins in Saccharomyces cerevisiae, has a role in recruiting Rvs167 via the SH3 domain. Removal of the SH3 domain also affects assembly and disassembly of actin and impedes membrane invagination. Our results indicate that both BAR and SH3 domains are important for the role of Rvs as a regulator of scission. We tested other proteins implicated in vesicle formation in Saccharomyces cerevisiae, and found that neither synaptojanins nor dynamin contribute directly to membrane scission. We propose that recruitment of Rvs BAR domains delays scission and allows invaginations to grow by stabilizing them. We also propose that vesicle formation is dependent on the force exerted by the actin network.

Molecular mechanics underlying flat-to-round membrane budding in live secretory cells

Membrane budding entails forces to transform flat membrane into vesicles essential for cell survival. Accumulated studies have identified coat-proteins (e.g., clathrin) as potential budding factors. However, forces mediating many non-coated membrane buddings remain unclear. By visualizing proteins in mediating endocytic budding in live neuroendocrine cells, performing in vitro protein reconstitution and physical modeling, we discovered how non-coated-membrane budding is mediated: actin filaments and dynamin generate a pulling force transforming flat membrane into Λ-shape; subsequently, dynamin helices surround and constrict Λ-profile's base, transforming Λ- to Ω-profile, and then constrict Ω-profile's pore, converting Ω-profiles to vesicles. These mechanisms control budding speed, vesicle size and number, generating diverse endocytic modes differing in these parameters. Their impact is widespread beyond secretory cells, as the unexpectedly powerful functions of dynamin and actin, previously thought to mediate fission and overcome tension, respectively, may contribute to many dynamin/actin-dependent non-coated-membrane buddings, coated-membrane buddings, and other membrane remodeling processes.

A model of actin-driven endocytosis explains differences of endocytic motility in budding and fission yeast

A comparative study (Sun et al., 2019) showed that the abundance of proteins at sites of endocytosis in fission and budding yeast is more similar in the two species than previously thought, yet membrane invaginations in fission yeast elongate twofold faster and are nearly twice as long as in budding yeast. Here we use a three-dimensional model of a motile endocytic invagination (Nickaeen et al., 2019) to investigate factors affecting elongation of the invaginations. We found that differences in turgor pressure in the two yeast species can largely explain the paradoxical differences observed experimentally in endocytic motility.

Non-linear elastic properties of actin patches to partially rescue yeast endocytosis efficiency in the absence of the cross-linker Sac6

Clathrin mediated endocytosis is an essential and complex cellular process involving more than 60 proteins. In yeast, successful endocytosis requires counteracting a large turgor pressure. To this end, yeasts assemble actin patches, which accumulate elastic energy during their assembly. We investigated the material properties of reconstituted actin patches from a wild-type (WT) strain and a mutant strain lacking the cross-linker Sac6 (sac6Δ), which has reduced endocytosis efficiency in live cells. We hypothesized that a change in the viscous properties of the actin patches, which would dissipate more mechanical energy, could explain this reduced efficiency. There was however no significant difference in the viscosity of both types of patches. However, we discovered a significantly different non-linear elastic response. While WT patches had a constant elastic modulus at different stress values, sac6Δ patches had a lower elastic modulus at low stress, before stiffening at higher ones, up to values similar to those of WT patches. To understand the consequences of this discovery, we performed, in vivo, a precise analysis of actin patch dynamics. Our analysis reveals that a small fraction of actin patches successfully complete endocytosis in sac6Δ cells, provided that those assemble an excess of actin at the membrane compared to WT. This observation indicates that the non-linear elastic properties of actin networks in sac6Δ cells contribute to rescue endocytosis, requiring nevertheless more actin material to build-up the necessary stored elastic energy.

Biomembranes undergo complex, non-axisymmetric deformations governed by Kirchhoff-Love kinematicsand revealed by a three-dimensional computational framework

Biomembranes play a central role in various phenomena like locomotion of cells, cell-cell interactions, packaging and transport of nutrients, transmission of nerve impulses, and in maintaining organelle morphology and functionality. During these processes, the membranes undergo significant morphological changes through deformation, scission, and fusion. Modelling the underlying mechanics of such morphological changes has traditionally relied on reduced order axisymmetric representations of membrane geometry and deformation. Axisymmetric representations, while robust and extensively deployed, suffer from their inability to model-symmetry breaking deformations and structural bifurcations. To address this limitation, a three-dimensional computational mechanics framework for high fidelity modelling of biomembrane deformation is presented. The proposed framework brings together Kirchhoff–Love thin-shell kinematics, Helfrich-energy-based mechanics, and state-of-the-art numerical techniques for modelling deformation of surface geometries. Lipid bilayers are represented as spline-based surface discretizations immersed in a three-dimensional space; this enables modelling of a wide spectrum of membrane geometries, boundary conditions, and deformations that are physically admissible in a three-dimensional space. The mathematical basis of the framework and its numerical machinery are presented, and their utility is demonstrated by modelling three classical, yet non-trivial, membrane deformation problems: formation of tubular shapes and their lateral constriction, Piezo1-induced membrane footprint generation and gating response, and the budding of membranes by protein coats during endocytosis. For each problem, the full three-dimensional membrane deformation is captured, potential symmetry-breaking deformation paths identified, and various case studies of boundary and load conditions are presented. Using the endocytic vesicle budding as a case study, we also present a ‘phase diagram’ for its symmetric and broken-symmetry states.

Actin-generated force applied during endocytosis measured by Sla2-based FRET tension sensors

Mechanical forces are integral to many cellular processes, including clathrin-mediated endocytosis, a principal membrane trafficking route into the cell. During endocytosis, forces provided by endocytic proteins and the polymerizing actin cytoskeleton reshape the plasma membrane into a vesicle. Assessing force requirements of endocytic membrane remodeling is essential for understanding endocytosis. Here, we determined actin-generated force applied during endocytosis using FRET-based tension sensors inserted into the major force-transmitting protein Sla2 in yeast. We measured at least 8 pN force transmitted over Sla2 molecule, hence possibly more than 300-880 pN applied during endocytic vesicle formation. Importantly, decreasing cell turgor pressure and plasma membrane tension reduced force transmitted over the Sla2. The measurements in hypotonic conditions and mutants lacking BAR-domain membrane scaffolds then showed the limits of the endocytic force-transmitting machinery. Our study provides force values and force profiles critical for understanding the mechanics of endocytosis and potentially other key cellular membrane-remodeling processes.

Clathrin: the molecular shape shifter

Clathrin is best known for its contribution to clathrin-mediated endocytosis yet it also participates to a diverse range of cellular functions. Key to this is clathrin's ability to assemble into polyhedral lattices that include curved football or basket shapes, flat lattices or even tubular structures. In this review, we discuss clathrin structure and coated vesicle formation, how clathrin is utilised within different cellular processes including synaptic vesicle recycling, hormone desensitisation, spermiogenesis, cell migration and mitosis, and how clathrin's remarkable 'shapeshifting' ability to form diverse lattice structures might contribute to its multiple cellular functions.

Rapid adaptation of endocytosis, exocytosis and eisosomes after an acute increase in membrane tension in yeast cells

During clathrin-mediated endocytosis (CME) in eukaryotes, actin assembly is required to overcome large membrane tension and turgor pressure. However, the molecular mechanisms by which the actin machinery adapts to varying membrane tension remain unknown. In addition, how cells reduce their membrane tension when they are challenged by hypotonic shocks remains unclear. We used quantitative microscopy to demonstrate that cells rapidly reduce their membrane tension using three parallel mechanisms. In addition to using their cell wall for mechanical protection, yeast cells disassemble eisosomes to buffer moderate changes in membrane tension on a minute time scale. Meanwhile, a temporary reduction in the rate of endocytosis for 2–6 min and an increase in the rate of exocytosis for at least 5 min allow cells to add large pools of membrane to the plasma membrane. We built on these results to submit the cells to abrupt increases in membrane tension and determine that the endocytic actin machinery of fission yeast cells rapidly adapts to perform CME. Our study sheds light on the tight connection between membrane tension regulation, endocytosis, and exocytosis.

More than just a barrier: using physical models to couple membrane shape to cell function

The correct execution of many cellular processes, such as division and motility, requires the cell to adopt a specific shape. Physically, these shapes are determined by the interplay of the plasma membrane and internal cellular driving factors. While the plasma membrane defines the boundary of the cell, processes inside the cell can result in the generation of forces that deform the membrane. These processes include protein binding, the assembly of protein superstructures, and the growth and contraction of cytoskeletal networks. Due to the complexity of the cell, relating observed membrane deformations back to internal processes is a challenging problem. Here, we review cell shape changes in endocytosis, cell adhesion, cell migration and cell division and discuss how by modeling membrane deformations we can investigate the inner working principles of the cell.

Value of models for membrane budding

The budding of membranes and curvature generation is common to many forms of trafficking in cells. Clathrin-mediated endocytosis, as a prototypical example of trafficking, has been studied in great detail using a variety of experimental systems and methods. Recently, advances in experimental methods have led to great strides in insights on the molecular mechanisms and the spatiotemporal dynamics of the protein machinery associated with membrane curvature generation. These advances have been ably supported by computational models, which have given us insights into the underlying mechanical principles of clathrin-mediated endocytosis. On the other hand, targeted experimental perturbation of membranes has lagged behind that of proteins in cells. In this area, modeling is especially critical to interpret experimental measurements in a mechanistic context. Here, we discuss the contributions made by these models to our understanding of endocytosis and identify opportunities to strengthen the connections between models and experiments.

Endocytosis against high turgor pressure is made easier by partial coating and freely rotating base

During clathrin-mediated endocytosis, a patch of flat plasma membrane is deformed into a vesicle. In walled cells, such as plants and fungi, the turgor pressure is high and pushes the membrane against the cell wall, thus hindering membrane internalization. In this work, we study how a patch of membrane is deformed against turgor pressure by force and by curvature-generating proteins. We show that a large amount of force is needed to merely start deforming the membrane and an even larger force is needed to pull a membrane tube. The magnitude of these forces strongly depends on how the base of the membrane is constrained and how the membrane is coated with curvature-generating proteins. In particular, these forces can be reduced by partially, but not fully, coating the membrane patch with curvature-generating proteins. Our theoretical results show excellent agreement with experimental data.

Diffuso-kinetic membrane budding dynamics

A wide range of proteins are known to create shape transformations of biological membranes, where the remodelling is a coupling between the energetic costs from deforming the membrane, the recruitment of proteins that induce a local spontaneous curvature C₀ and the diffusion of proteins along the membrane. We propose a minimal mathematical model that accounts for these processes to describe the diffuso-kinetic dynamics of membrane budding processes. By deploying numerical simulations we map out the membrane shapes, the time for vesicle formation and the vesicle size as a function of the dimensionless kinetic recruitment parameter K₁ and the proteins sensitivity to mean curvature. We derive a time for scission that follows a power law ∼K₁^-2/3, a consequence of the interplay between the spreading of proteins by diffusion and the kinetic-limited increase of the protein density on the membrane. We also find a scaling law for the vesicle size ∼1/([small sigma, Greek, macron]_avC₀), with [small sigma, Greek, macron]_av the average protein density in the vesicle, which is confirmed in the numerical simulations. Rescaling all the membrane profiles at the time of vesicle formation highlights that the membrane adopts a self-similar shape.

Diffusion on Membrane Domes, Tubes, and Pearling Structures

Diffusion is a fundamental mechanism for protein distribution in cell membranes. These membranes often exhibit complex shapes, which range from shallow domes to elongated tubular or pearl-like structures. Shape complexity of the membrane influences the diffusive spreading of proteins and molecules. Despite the importance membrane geometry plays in these diffusive processes, it is challenging to establish the dependence between diffusion and membrane morphology. We solve the diffusion equation numerically on various static curved shapes representative for experimentally observed membrane shapes. Our results show that membrane necks become diffusion barriers. We determine the diffusive half-time, i.e., the time that is required to reduce the amount of protein in the budded region by one half, and find a quadratic relation between the diffusive half-time and the averaged mean curvature of the membrane shape, which we rationalize by a scaling law. Our findings thus help estimate the characteristic diffusive timescale based on the simple measure of membrane mean curvature.

Competing pathways for the invagination of clathrin-coated membranes

Clathrin-mediated endocytosis is the major pathway by which eukaryotic cells take up extracellular material, but it is still elusive which physical pathways are being taken during membrane invagination. From a continuum point of view, it can be driven by increases in coat stiffness, preferred curvature or line tension. Here we develop a comprehensive theoretical framework that can be solved analytically and that predicts the consequences of these different scenarios. We find that for the case of increasing stiffness or preferred curvature, curvature will be acquired gradually with growth, while for increasing line tension, the lattice must have grown to a certain size before a flat-to-curved transition can occur. At low membrane tension, the critical value for coat stiffness is 30 kBT, for preferred curvature it is 200 nm, and for line tension it is 6 pN. For high membrane tension, critical coat stiffness is 150 kBT and critical preferred curvature is 70 nm. In the mixed case when a coat with finite rigidity but increasing line tension is considered, a cup-to-sphere transition can occur for a line tension of 6 pN. The flat-to-curved and the cup-to-sphere transitions driven by line tension are both suppressed by high membrane tension.

Protein crowding mediates membrane remodeling in upstream ESCRT-induced formation of intraluminal vesicles

As part of the lysosomal degradation pathway, the endosomal sorting complexes required for transport (ESCRT-0 to -III/VPS4) sequester receptors at the endosome and simultaneously deform the membrane to generate intraluminal vesicles (ILVs). Whereas ESCRT-III/VPS4 have an established function in ILV formation, the role of upstream ESCRTs (0 to II) in membrane shape remodeling is not understood. Combining experimental measurements and electron microscopy analysis of ESCRT-III-depleted cells with a mathematical model, we show that upstream ESCRT-induced alteration of the Gaussian bending rigidity and their crowding in concert with the transmembrane cargo on the membrane induce membrane deformation and facilitate ILV formation: Upstream ESCRT-driven budding does not require ATP consumption as only a small energy barrier needs to be overcome. Our model predicts that ESCRTs do not become part of the ILV, but localize with a high density at the membrane neck, where the steep decline in the Gaussian curvature likely triggers ESCRT-III/VPS4 assembly to enable neck constriction and scission.

Nonaxisymmetric Shapes of Biological Membranes from Locally Induced Curvature

In various biological processes such as endocytosis and caveolae formation, the cell membrane is locally deformed into curved morphologies. Previous models to study membrane morphologies resulting from locally induced curvature often only consider the possibility of axisymmetric shapes-an indeed unphysical constraint. Past studies predict that the cell membrane buds at low resting tensions and stalls at a flat pit at high resting tensions. In this work, we lift the restriction to axisymmetry to study all possible membrane morphologies. Only if the resting tension of the membrane is low, we reproduce axisymmetric membrane morphologies. When the resting tension is moderate to high, we show that 1) axisymmetric membrane pits are unstable and 2) nonaxisymmetric ridge-shaped structures are energetically favorable. Furthermore, we find the interplay between intramembrane viscous flow and the rate of induced curvature affects the membrane's ability to transition into nonaxisymmetric ridges and axisymmetric buds. In particular, we show that axisymmetric buds are favored when the induced curvature is rapidly increased, whereas nonaxisymmetric ridges are favored when the curvature is slowly increased. Our results hold relevant implications for biological processes such as endocytosis and physical phenomena like phase separation in lipid bilayers.

Modeling membrane nanotube morphology: the role of heterogeneity in composition and material properties

Membrane nanotubes are dynamic structures that may connect cells over long distances. Nanotubes are typically thin cylindrical tubes, but they may occasionally have a beaded architecture along the tube. In this paper, we study the role of membrane mechanics in governing the architecture of these tubes and show that the formation of bead-like structures along the nanotubes can result from local heterogeneities in the membrane either due to protein aggregation or due to membrane composition. We present numerical results that predict how membrane properties, protein density, and local tension compete to create a phase space that governs the morphology of a nanotube. We also find that there exists a discontinuity in the energy that impedes two beads from fusing. These results suggest that the membrane-protein interaction, membrane composition, and membrane tension closely govern the tube radius, number of beads, and the bead morphology.

Evolutionarily unique mechanistic framework of clathrin-mediated endocytosis in plants

In plants, clathrin mediated endocytosis (CME) represents the major route for cargo internalisation from the cell surface. It has been assumed to operate in an evolutionary conserved manner as in yeast and animals. Here we report characterisation of ultrastructure, dynamics and mechanisms of plant CME as allowed by our advancement in electron microscopy and quantitative live imaging techniques. Arabidopsis CME appears to follow the constant curvature model and the bona fide CME population generates vesicles of a predominantly hexagonal-basket type; larger and with faster kinetics than in other models. Contrary to the existing paradigm, actin is dispensable for CME events at the plasma membrane but plays a unique role in collecting endocytic vesicles, sorting of internalised cargos and directional endosome movement that itself actively promote CME events. Internalized vesicles display a strongly delayed and sequential uncoating. These unique features highlight the independent evolution of the plant CME mechanism during the autonomous rise of multicellularity in eukaryotes.

Transverse lipid organization dictates bending fluctuations in model plasma membranes

Membrane undulations play a vital role in many biological processes, including the regulation of membrane protein activity. The asymmetric lipid composition of most biological membranes complicates theoretical description of these bending fluctuations, yet experimental data that would inform any such a theory is scarce. Here, we used neutron spin-echo (NSE) spectroscopy to measure the bending fluctuations of large unilamellar vesicles (LUV) having an asymmetric transbilayer distribution of high- and low-melting lipids. The asymmetric vesicles were prepared using cyclodextrin-mediated lipid exchange, and were composed of an outer leaflet enriched in egg sphingomyelin (ESM) and an inner leaflet enriched in 1-palmitoyl-2-oleoyl-phosphoethanolamine (POPE), which have main transition temperatures of 37 °C and 25 °C, respectively. The overall membrane bending rigidity was measured at three temperatures: 15 °C, where both lipids are in a gel state; 45 °C, where both lipids are in a fluid state; and 30 °C, where there is gel-fluid co-existence. Remarkably, the dynamics for the fluid asymmetric LUVs (aLUVs) at 30 °C and 45 °C do not follow trends predicted by their symmetric counterparts. At 30 °C, compositional asymmetry suppressed the bending fluctuations, with the asymmetric bilayer exhibiting a larger bending modulus than that of symmetric bilayers corresponding to either the outer or inner leaflet. We conclude that the compositional asymmetry and leaflet coupling influence the internal dissipation within the bilayer and result in membrane properties that cannot be directly predicted from corresponding symmetric bilayers.

Principles of self-organization and load adaptation by the actin cytoskeleton during clathrin-mediated endocytosis

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.

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.

Single-molecule turnover dynamics of actin and membrane coat proteins in clathrin-mediated endocytosis

Actin dynamics generate forces to deform the membrane and overcome the cell’s high turgor pressure during clathrin-mediated endocytosis (CME) in yeast, but precise molecular details are still unresolved. Our previous models predicted that actin filaments of the endocytic meshwork continually polymerize and disassemble, turning over multiple times during an endocytic event, similar to other actin systems. We applied single-molecule speckle tracking in live fission yeast to directly measure molecular turnover within CME sites for the first time. In contrast with the overall ~20 s lifetimes of actin and actin-associated proteins in endocytic patches, we detected single-molecule residence times around 1 to 2 s, and similarly high turnover rates of membrane-associated proteins in CME. Furthermore, we find heterogeneous behaviors in many proteins’ motions. These results indicate that endocytic proteins turn over up to five times during the formation of an endocytic vesicle, and suggest revising quantitative models of force production.

Single-molecule turnover dynamics of actin and membrane coat proteins in clathrin-mediated endocytosis

Actin dynamics generate forces to deform the membrane and overcome the cell's high turgor pressure during clathrin-mediated endocytosis (CME) in yeast, but precise molecular details are still unresolved. Our previous models predicted that actin filaments of the endocytic meshwork continually polymerize and disassemble, turning over multiple times during an endocytic event, similar to other actin systems. We applied single-molecule speckle tracking in live fission yeast to directly measure molecular turnover within CME sites for the first time. In contrast with the overall ~20 s lifetimes of actin and actin-associated proteins in endocytic patches, we detected single-molecule residence times around 1 to 2 s, and similarly high turnover rates of membrane-associated proteins in CME. Furthermore, we find heterogeneous behaviors in many proteins' motions. These results indicate that endocytic proteins turn over up to five times during the formation of an endocytic vesicle, and suggest revising quantitative models of force production.

Pulling-force generation by ensembles of polymerizing actin filaments

The process by which actin polymerization generates pulling forces in cellular processes such as endocytosis is less well understood than pushing-force generation. To clarify the basic mechanisms of pulling-force generation, we perform stochastic polymerization simulations for a square array of polymerizing semiflexible actin filaments, having different interactions with the membrane. The filaments near the array center have a strong attractive component. Filament bending and actin-network elasticity are treated explicitly. We find that the outer filaments push on the membrane and the inner filaments pull, with a net balance of forces. The total calculated pulling force is maximized when the central filaments have a very deep potential well, and the outer filaments have no well. The steady-state force is unaffected by the gel rigidity, but equilibration takes longer for softer gels. The force distributions are flat over the pulling and pushing regions. Actin polymerization is enhanced by softening the gel or reducing the filament binding to the membrane. Filament-membrane detachment can occur for softer gels, even if the total binding energy of the filaments to the membrane is 100 [Formula: see text] or more. It propagates via a stress-concentration mechanism similar to that of a brittle crack in a solid, and the breaking stress is determined by a criterion similar to that of the 'Griffith' theory of crack propagation.

Dynamins 2 and 3 control the migration of human megakaryocytes by regulating CXCR4 surface expression and ITGB1 activity

Megakaryocyte (MK) migration from the bone marrow periosteal niche toward the vascular niche is a prerequisite for proplatelet extension and release into the circulation. The mechanism for this highly coordinated process is poorly understood. Here we show that dynasore (DNSR), a small-molecule inhibitor of dynamins (DNMs), or short hairpin RNA knockdown of DNM2 and DNM3 impairs directional migration in a human MK cell line or MKs derived from cultured CD34⁺ cells. Because cell migration requires actin cytoskeletal rearrangements, we measured actin polymerization and the activity of cytoskeleton regulator RhoA and found them to be decreased after inhibition of DNM2 and DNM3. Because SDF-1α is important for hematopoiesis, we studied the expression of its receptor CXCR4 in DNSR-treated cells. CXCR4 expression on the cell surface was increased, at least partially because of slower endocytosis and internalization after SDF-1α treatment. Combined inhibition of DNM2 and DNM3 or forced expression of dominant-negative Dnm2-K44A or GTPase-defective DNM3 diminished β1 integrin (ITGB1) activity. DNSR-treated MKs showed an abnormally clustered staining pattern of Rab11, a marker of recycling endosomes. This suggests decreased recruitment of the recycling pathway in DNSR-treated cells. Altogether, we show that the GTPase activity of DNMs, which governs endocytosis and regulates cell receptor trafficking, exerts control on MK migration toward SDF-1α gradients, such as those originating from the vascular niche. DNMs play a critical role in MKs by triggering membrane-cytoskeleton rearrangements downstream of CXCR4 and integrins.

Type-I myosins promote actin polymerization to drive membrane bending in endocytosis

Clathrin-mediated endocytosis in budding yeast requires the formation of a dynamic actin network that produces the force to invaginate the plasma membrane against the intracellular turgor pressure. The type-I myosins Myo3 and Myo5 are important for endocytic membrane reshaping, but mechanistic details of their function remain scarce. Here, we studied the function of Myo3 and Myo5 during endocytosis using quantitative live-cell imaging and genetic perturbations. We show that the type-I myosins promote, in a dose-dependent way, the growth and expansion of the actin network, which controls the speed of membrane and coat internalization. We found that this myosin-activity is independent of the actin nucleation promoting activity of myosins, and cannot be compensated for by increasing actin nucleation. Our results suggest a new mechanism for type-I myosins to produce force by promoting actin filament polymerization.

Crosslinking actin networks produces compressive force

Actin has been shown to be essential for clathrin-mediated endocytosis in yeast. However, actin polymerization alone is likely insufficient to produce enough force to deform the membrane against the huge turgor pressure of yeast cells. In this paper, we used Brownian dynamics simulations to demonstrate that crosslinking of a meshwork of nonpolymerizing actin filaments is able to produce compressive forces. We show that the force can be up to several thousand pico-Newtons if the crosslinker has a high stiffness. The force decays over time as a result of crosslinker turnover, and is a result of converting chemical binding energy into elastic energy.

TORC2 controls endocytosis through plasma membrane tension

Target of rapamycin complex 2 (TORC2) is a conserved protein kinase that regulates multiple plasma membrane (PM)-related processes, including endocytosis. Direct, chemical inhibition of TORC2 arrests endocytosis but with kinetics that is relatively slow and therefore inconsistent with signaling being mediated solely through simple phosphorylation cascades. Here, we show that in addition to and independently from regulation of the phosphorylation of endocytic proteins, TORC2 also controls endocytosis by modulating PM tension. Elevated PM tension, upon TORC2 inhibition, impinges on endocytosis at two different levels by (1) severing the bonds between the PM adaptor proteins Sla2 and Ent1 and the actin cytoskeleton and (2) hindering recruitment of Rvs167, an N-BAR-containing protein important for vesicle fission to endocytosis sites. These results underline the importance of biophysical cues in the regulation of cellular and molecular processes.

Molecular mechanisms of contractile-ring constriction and membrane trafficking in cytokinesis

In this review, we discuss the molecular mechanisms of cytokinesis from plants to humans, with a focus on contribution of membrane trafficking to cytokinesis. Selection of the division site in fungi, metazoans, and plants is reviewed, as well as the assembly and constriction of a contractile ring in fungi and metazoans. We also provide an introduction to exocytosis and endocytosis, and discuss how they contribute to successful cytokinesis in eukaryotic cells. The conservation in the coordination of membrane deposition and cytoskeleton during cytokinesis in fungi, metazoans, and plants is highlighted.

Molecular mechanisms of force production in clathrin-mediated endocytosis

During clathrin-mediated endocytosis (CME), a flat patch of membrane is invaginated and pinched off to release a vesicle into the cytoplasm. In yeast CME, over 60 proteins-including a dynamic actin meshwork-self-assemble to deform the plasma membrane. Several models have been proposed for how actin and other molecules produce the forces necessary to overcome the mechanical barriers of membrane tension and turgor pressure, but the precise mechanisms and a full picture of their interplay are still not clear. In this review, we discuss the evidence for these force production models from a quantitative perspective and propose future directions for experimental and theoretical work that could clarify their various contributions.

The role of traction in membrane curvature generation

Curvature of biological membranes can be generated by a variety of molecular mechanisms including protein scaffolding, compositional heterogeneity, and cytoskeletal forces. These mechanisms have the net effect of generating tractions (force per unit length) on the bilayer that are translated into distinct shapes of the membrane. Here, we demonstrate how the local shape of the membrane can be used to infer the traction acting locally on the membrane. We show that buds and tubes, two common membrane deformations studied in trafficking processes, have different traction distributions along the membrane and that these tractions are specific to the molecular mechanism used to generate these shapes. Furthermore, we show that the magnitude of an axial force applied to the membrane as well as that of an effective line tension can be calculated from these tractions. Finally, we consider the sensitivity of these quantities with respect to uncertainties in material properties and follow with a discussion on sources of uncertainty in membrane shape.

Systematic Nanoscale Analysis of Endocytosis Links Efficient Vesicle Formation to Patterned Actin Nucleation

Clathrin-mediated endocytosis is an essential cellular function in all eukaryotes that is driven by a self-assembled macromolecular machine of over 50 different proteins in tens to hundreds of copies. How these proteins are organized to produce endocytic vesicles with high precision and efficiency is not understood. Here, we developed high-throughput superresolution microscopy to reconstruct the nanoscale structural organization of 23 endocytic proteins from over 100,000 endocytic sites in yeast. We found that proteins assemble by radially ordered recruitment according to function. WASP family proteins form a circular nanoscale template on the membrane to spatially control actin nucleation during vesicle formation. Mathematical modeling of actin polymerization showed that this WASP nano-template optimizes force generation for membrane invagination and substantially increases the efficiency of endocytosis. Such nanoscale pre-patterning of actin nucleation may represent a general design principle for directional force generation in membrane remodeling processes such as during cell migration and division.

Structural organization and energy storage in crosslinked actin assemblies

During clathrin-mediated endocytosis in yeast cells, short actin filaments (< 200nm) and crosslinking protein fimbrin assemble to drive the internalization of the plasma membrane. However, the organization of the actin meshwork during endocytosis remains largely unknown. In addition, only a small fraction of the force necessary to elongate and pinch off vesicles can be accounted for by actin polymerization alone. In this paper, we used mathematical modeling to study the self-organization of rigid actin filaments in the presence of elastic crosslinkers in conditions relevant to endocytosis. We found that actin filaments condense into either a disordered meshwork or an ordered bundle depending on filament length and the mechanical and kinetic properties of the crosslinkers. Our simulations also demonstrated that these nanometer-scale actin structures can store a large amount of elastic energy within the crosslinkers (up to 10kBT per crosslinker). This conversion of binding energy into elastic energy is the consequence of geometric constraints created by the helical pitch of the actin filaments, which results in frustrated configurations of crosslinkers attached to filaments. We propose that this stored elastic energy can be used at a later time in the endocytic process. As a proof of principle, we presented a simple mechanism for sustained torque production by ordered detachment of crosslinkers from a pair of parallel filaments.

The contributions of the actin machinery to endocytic membrane bending and vesicle formation

Branched and cross-linked actin networks mediate cellular processes that move and shape membranes. To understand how actin contributes during the different stages of endocytic membrane reshaping, we analyzed deletion mutants of yeast actin network components using a hybrid imaging approach that combines live imaging with correlative microscopy. We could thus temporally dissect the effects of different actin network perturbations, revealing distinct stages of actin-based membrane reshaping. Our data show that initiation of membrane bending requires the actin network to be physically linked to the plasma membrane and to be optimally cross-linked. Once initiated, the membrane invagination process is driven by nucleation and polymerization of new actin filaments, independent of the degree of cross-linking and unaffected by a surplus of actin network components. A key transition occurs 2 s before scission, when the filament nucleation rate drops. From that time point on, invagination growth and vesicle scission are driven by an expansion of the actin network without a proportional increase of net actin amounts. The expansion is sensitive to the amount of filamentous actin and its cross-linking. Our results suggest that the mechanism by which actin reshapes the membrane changes during the progress of endocytosis, possibly adapting to varying force requirements.

Membrane bending by actin polymerization

Actin polymerization provides driving force to aid several types of processes that involve pulling the plasma membrane into the cell, including phagocytosis, cellular entry of large viruses, and endocytosis. In endocytosis, actin polymerization is especially important under conditions of high membrane tension or high turgor pressure. Recent modeling efforts have shown how actin polymerization can give rise to a distribution of forces around the endocytic site, and explored how these forces affect the shape dynamics; experiments have revealed the structure of the endocytic machinery in increasing detail, and demonstrated key feedback interactions between actin assembly and membrane curvature. Here we provide a perspective on these findings and suggest avenues for future research.

Assaying the Contribution of Membrane Tension to Clathrin-Mediated Endocytosis

Nowadays, live fluorescent microscopes allow us to study the dynamics of cellular processes in living cells with high spatial and temporal resolution. Since the implementation of this methodology to the field of clathrin-mediated endocytosis (CME), this approach has revolutionized our molecular understanding of clathrin-driven cellular uptake. Conventional live cell microscopy approaches are used to determine the precise functions of specific proteins or lipids in orchestrating CME. Here, we will describe, in depth, the procedure to investigate the contribution of membrane tension in regulating clathrin-dependent endocytosis. We will explain two alternative methods to manipulate membrane tension while performing live fluorescence microscopy: cellular swelling through osmotic shock and cellular stretching of cells grown on stretchable silicon inserts.

A master equation approach to actin polymerization applied to endocytosis in yeast

We present a Master Equation approach to calculating polymerization dynamics and force generation by branched actin networks at membranes. The method treats the time evolution of the F-actin distribution in three dimensions, with branching included as a directional spreading term. It is validated by comparison with stochastic simulations of force generation by actin polymerization at obstacles coated with actin “nucleation promoting factors” (NPFs). The method is then used to treat the dynamics of actin polymerization and force generation during endocytosis in yeast, using a model in which NPFs form a ring around the endocytic site, centered by a spot of molecules attaching the actin network strongly to the membrane. We find that a spontaneous actin filament nucleation mechanism is required for adequate forces to drive the process, that partial inhibition of branching and polymerization lead to different characteristic responses, and that a limited range of polymerization-rate values provide effective invagination and obtain correct predictions for the effects of mutations in the active regions of the NPFs.

Endocytosis is a dynamic process by which cells internalize substances from outside the cell. Especially in yeast, endocytosis is mechanically demanding due to the high pressure difference across the cell membrane, or turgor pressure. Polymerization of a branched actin network is the major process providing the mechanical force to overcome the turgor pressure. Understanding the kinetics of the actin network, and the mechanical interaction between the actin network and the cell membrane, is thus crucial for the study of endocytosis. We develop an efficient mathematical framework for actin dynamics that can realistically incorporate these two features, thus providing a practical method for quantitatively modeling actin dynamics during endocytosis. The resulting model mechanistically reveals that spontaneous nucleation at the center of the endocytic site is required for successful endocytosis, distinguishes the roles of branching and polymerization, and predicts several other experimentally testable outcomes. The accuracy and efficiency of the method, in describing both mechanics and chemistry, render it applicable to a broad field of membrane-bending processes.

How cells engulf: a review of theoretical approaches to phagocytosis

Phagocytosis is a fascinating process whereby a cell surrounds and engulfs particles such as bacteria and dead cells. This is crucial both for single-cell organisms (as a way of acquiring nutrients) and as part of the immune system (to destroy foreign invaders). This whole process is hugely complex and involves multiple coordinated events such as membrane remodelling, receptor motion, cytoskeleton reorganisation and intracellular signalling. Because of this, phagocytosis is an excellent system for theoretical study, benefiting from biophysical approaches combined with mathematical modelling. Here, we review these theoretical approaches and discuss the recent mathematical and computational models, including models based on receptors, models focusing on the forces involved, and models employing energetic considerations. Along the way, we highlight a beautiful connection to the physics of phase transitions, consider the role of stochasticity, and examine links between phagocytosis and other types of endocytosis. We cover the recently discovered multistage nature of phagocytosis, showing that the size of the phagocytic cup grows in distinct stages, with an initial slow stage followed by a much quicker second stage starting around half engulfment. We also address the issue of target shape dependence, which is relevant to both pathogen infection and drug delivery, covering both one-dimensional and two-dimensional results. Throughout, we pay particular attention to recent experimental techniques that continue to inform the theoretical studies and provide a means to test model predictions. Finally, we discuss population models, connections to other biological processes, and how physics and modelling will continue to play a key role in future work in this area.

Forces that shape fission yeast cells

One of the major challenges of modern cell biology is to understand how cells are assembled from nanoscale components into micrometer-scale entities with a specific size and shape. Here I describe how our quest to understand the morphogenesis of the fission yeast Schizosaccharomyces pombe drove us to investigate cellular mechanics. These studies build on the view that cell shape arises from the physical properties of an elastic cell wall inflated by internal turgor pressure. Consideration of cellular mechanics provides new insights into not only mechanisms responsible for cell-shape determination and growth, but also cellular processes such as cytokinesis and endocytosis. Studies in yeast can help to illuminate approaches and mechanisms to study the mechanobiology of the cell surface in other cell types, including animal cells.

The Process of Wrapping Virus Revealed by a Force Tracing Technique and Simulations

Viral entry into the host cell is the first step of virus infection; however, its dynamic process via endocytosis remains largely elusive. Here, the force tracing technique and single particle simulation are combined to investigate the invagination of single human enterovirus 71 (HEV71, a positive single-stranded RNA virus that is associated with hand, foot, and mouth disease) via cell membranes during its host cell entry. The experimental results reveal that the HEV71 invaginates in membrane vesicles at a force of 58 ± 16 pN, a duration time of 278 ± 68 ms. The simulation further shows that the virus can reach a partially wrapped state very fast, then the upper surface of the virus is covered by the membrane traveling over a long period of time. Combining the experiment with the simulation, the mechanism of membrane wrapping of virus is uncovered, which provides new insights into how the cell is operated to initiate the endocytosis of virus.

Local Turgor Pressure Reduction via Channel Clustering

The primary drivers of yeast endocytosis are actin polymerization and curvature-generating proteins, such as clathrin and BAR domain proteins. Previous work has indicated that these factors may not be capable of generating the forces necessary to overcome turgor pressure. Thus local reduction of the turgor pressure, via localized accumulation or activation of solute channels, might facilitate endocytosis. The possible reduction in turgor pressure was calculated numerically, by solving the diffusion equation through a Legendre polynomial expansion. It was found that for a region of increased permeability having radius 45 nm, as few as 60 channels with a spacing of 10 nm could locally decrease the turgor pressure by 50%. We identified a key dimensionless parameter, p = P₁a/D, where P₁ is the increased permeability, a is the radius of the permeable region, and D is the solute diffusion coefficient. When p > 0.44, the turgor pressure is locally reduced by >50%. An approximate analytic theory was used to generate explicit formulas for the turgor pressure reduction in terms of key parameters. These findings may also be relevant to plants, where the mechanisms that allow endocytosis to proceed despite high turgor pressure are largely unknown.