Stochastic model of clathrin-coated pit assembly

Endocytosis gated by emergent properties of membrane-clathrin interactions

Clathrin-mediated endocytosis (CME), the major cellular entry pathway, starts when clathrin assembles on the plasma membrane into clathrin-coated pits (CCPs). Two populations of CCPs are detected within the same cell: 'productive' CCPs that invaginate and pinch off, forming clathrin-coated vesicles (CCVs) [1, 2], and 'abortive' CCPs [3, 4, 5] that prematurely disassemble. The mechanisms of gating between these two populations and their relations to the functions of dozens of early-acting endocytic accessory proteins (EAPs) [5, 6, 7, 8, 9] have remained elusive. Here, we use experimentally-guided modeling to integrate the clathrin machinery and membrane mechanics in a single dynamical system. We show that the split between the two populations is an emergent property of this system, in which a switch between an Open state and a Closed state follows from the competition between the chemical energy of the clathrin basket and the mechanical energy of membrane bending. In silico experiments revealed an abrupt transition between the two states that acutely depends on the strength of the clathrin basket. This critical strength is lowered by membrane-bending EAPs [10, 11, 12]. Thus, CME is poised to be shifted between abortive and productive events by small changes in membrane curvature and/or coat stability. This model clarifies the workings of a putative endocytic checkpoint whose existence was previously proposed based on statistical analyses of the lifetime distributions of CCPs [4, 13]. Overall, a mechanistic framework is established to elucidate the diverse and redundant functions of EAPs in regulating CME progression.

Remeshing flexible membranes under the control of free energy

Cell membranes are flexible and often undergo large-scale morphological changes during processes like mitosis, protrusion and retraction, or vesicle fusion. Mathematical modeling of cell membranes depends on a representation of the free-form surface by discrete meshes. During morphological changes, these meshes must be adjusted under the minimization of the total free energy. Current methodology for meshing is limited in one of two ways: 1) Free energy-dependent methods have no restriction on the mesh geometry. The resulting irregular meshes cause artifacts in follow-up models of morphodynamics. 2) Geometry-dependent methods maintain mesh quality but violate the physics of free energy minimization. To fill this gap, we regulate mesh geometries via a free-energy-determined remeshing process: adding and removing mesh elements upon morphological changes based on barrier crossings in a double-barrier potential between neighboring vertices in the meshes. We test the method's robustness by reproducing the morphodynamics of red blood cells and vesicle fusions; and we demonstrate the method's adaptability by simulating the formation of filopodia, lamellipodia and invaginations. Finally, we use the method to study a mechanical decoupling effect of two connected membrane tethers that has been recently observed experimentally, but has not been mechanistically explained in the context of a complete membrane surface. We propose a biophysical model that strengthens the decoupling effect and broadens the original interpretation of the experiment. The method is developed in C/Matlab and distributed via https://github.com/DanuserLab/biophysicsModels.

The morphological role of ligand inhibitors in blocking receptor- and clathrin-mediated endocytosis

Cells often internalize particles through endocytic pathways that involve the binding between cell receptors and particle ligands, which drives the cell membrane to wrap the particle into a delivery vesicle. Previous findings showed that receptor-mediated endocytosis is impossible for spherical particles smaller than a minimum size because of the energy barrier created by membrane bending. In this study, we investigate the morphological role of ligand inhibitors in blocking endocytosis, inspired by antibodies that inhibit virus ligands to prevent infection. While ligand inhibitors have the obvious effect of reducing the driving force due to adhesion, they also have a nontrivial (morphological) impact on the entropic and elastic energy of the system. We determine the necessary conditions for endocytosis by considering the additional energy barrier due to the membrane bending to wrap the inhibiting protrusions. We find that inhibitors increase the minimum radius previously reported, depending on their density and size. In addition, we extend this result to the case of clathrin-mediated endocytosis, which is the most common pathway for virus entry. The assembly of a clathrin coat with a spontaneous curvature increases the energy barrier and sets a maximum particle size (in agreement with experimental observations on spherical particles). Our investigation suggests that morphological considerations can inform the optimal design of neutralizing viral antibodies and new strategies for targeted nanomedicine.

Large self-assembled clathrin lattices spontaneously disassemble without sufficient adaptor proteins

Clathrin-coated structures must assemble on cell membranes to internalize receptors, with the clathrin protein only linked to the membrane via adaptor proteins. These structures can grow surprisingly large, containing over 20 clathrin, yet they often fail to form productive vesicles, instead aborting and disassembling. We show that clathrin structures of this size can both form and disassemble spontaneously when adaptor protein availability is low, despite high abundance of clathrin. Here, we combine recent in vitro kinetic measurements with microscopic reaction-diffusion simulations and theory to differentiate mechanisms of stable vs unstable clathrin assembly on membranes. While in vitro conditions drive assembly of robust, stable lattices, we show that concentrations, geometry, and dimensional reduction in physiologic-like conditions do not support nucleation if only the key adaptor AP-2 is included, due to its insufficient abundance. Nucleation requires a stoichiometry of adaptor to clathrin that exceeds 1:1, meaning additional adaptor types are necessary to form lattices successfully and efficiently. We show that the critical nucleus contains ~25 clathrin, remarkably similar to sizes of the transient and abortive structures observed in vivo. Lastly, we quantify the cost of bending the membrane under our curved clathrin lattices using a continuum membrane model. We find that the cost of bending the membrane could be largely offset by the energetic benefit of forming curved rather than flat structures, with numbers comparable to experiments. Our model predicts how adaptor density can tune clathrin-coated structures from the transient to the stable, showing that active energy consumption is therefore not required for lattice disassembly or remodeling during growth, which is a critical advance towards predicting productive vesicle formation.

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.

Optimality in Self-Organized Molecular Sorting

We introduce a simple physical picture to explain the process of molecular sorting, whereby specific proteins are concentrated and distilled into submicrometric lipid vesicles in eukaryotic cells. To this purpose, we formulate a model based on the coupling of spontaneous molecular aggregation with vesicle nucleation. Its implications are studied by means of a phenomenological theory describing the diffusion of molecules toward multiple sorting centers that grow due to molecule absorption and are extracted when they reach a sufficiently large size. The predictions of the theory are compared with numerical simulations of a lattice-gas realization of the model and with experimental observations. The efficiency of the distillation process is found to be optimal for intermediate aggregation rates, where the density of sorted molecules is minimal and the process obeys simple scaling laws. Quantitative measures of endocytic sorting performed in primary endothelial cells are compatible with the hypothesis that these optimal conditions are realized in living cells.

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.

NERDSS: A Nonequilibrium Simulator for Multibody Self-Assembly at the Cellular Scale

Currently, a significant barrier to building predictive models of cellular self-assembly processes is that molecular models cannot capture minutes-long dynamics that couple distinct components with active processes, whereas reaction-diffusion models cannot capture structures of molecular assembly. Here, we introduce the nonequilibrium reaction-diffusion self-assembly simulator (NERDSS), which addresses this spatiotemporal resolution gap. NERDSS integrates efficient reaction-diffusion algorithms into generalized software that operates on user-defined molecules through diffusion, binding and orientation, unbinding, chemical transformations, and spatial localization. By connecting the fast processes of binding with the slow timescales of large-scale assembly, NERDSS integrates molecular resolution with reversible formation of ordered, multisubunit complexes. NERDSS encodes models using rule-based formatting languages to facilitate model portability, usability, and reproducibility. Applying NERDSS to steps in clathrin-mediated endocytosis, we design multicomponent systems that can form lattices in solution or on the membrane, and we predict how stochastic but localized dephosphorylation of membrane lipids can drive lattice disassembly. The NERDSS simulations reveal the spatial constraints on lattice growth and the role of membrane localization and cooperativity in nucleating assembly. By modeling viral lattice assembly and recapitulating oscillations in protein expression levels for a circadian clock model, we illustrate the adaptability of NERDSS. NERDSS simulates user-defined assembly models that were previously inaccessible to existing software tools, with broad applications to predicting self-assembly in vivo and designing high-yield assemblies in vitro.

Entry modes of ellipsoidal nanoparticles on a membrane during clathrin-mediated endocytosis

The membrane wrapping and internalization of nanoparticles, such as viruses and drug nanocarriers, through clathrin-mediated endocytosis (CME) are vitally important for intracellular transport. During CME, the shape of the particle plays crucial roles in the determination of particle-membrane interactions, but much of the previous work has been focused on spherical particles. In this work, we develop a stochastic model to study the CME of ellipsoidal particles. In our model, the deformation of the membrane and wrapping of the nanoparticles are driven by the accumulation of clathrin lattices, which is stimulated by the ligand-receptor interactions. Using our model, we systematically investigate the effect of particle shape (ellipsoids with different aspect ratios) on the CME. Our results show three entry modes: tip-first, tilted, and laying-down modes, used by ellipsoidal nanoparticles for internalization depending on the aspect ratio. Certain ellipsoids are able to take multiple entry modes for internalization. Interestingly, the prolate ellipsoid with an aspect ratio of 0.42 can be internalized with a significantly reduced number of ligand-receptor bonds. Particles which can be internalized with fewer bonds are excellent candidates for transcellular drug delivery. Moreover, our results demonstrate that internalization of ellipsoids with intermediate aspect ratios is easier than that of particles with low and high aspect ratios. Our model and simulations provide critical mechanistic insights into CME of ellipsoidal particles, and represent a viable platform for optimal design of nanoparticles for targeted drug delivery applications.

Stochastic simulations of nanoparticle internalization through transferrin receptor dependent clathrin-mediated endocytosis

BACKGROUND

Receptor dependent clathrin-mediated endocytosis (CME) is one of the most important endocytic pathways for the internalization of bioparticles into cells. During CME, the ligand-receptor interactions, development of clathrin-coated pit (CCP) and membrane evolution all act together to drive the internalization of bioparticles. In this work, we develop a stochastic computational model to investigate the CME based on the Metropolis Monte Carlo simulations.

METHODS

The model is based on the combination of a stochastic particle binding model with a membrane model. The energetic costs of membrane bending, CCP formation and ligand-receptor interactions are systematically linked together.

RESULTS

We implement our model to investigate the effects of particle size, ligand density and membrane stiffness on the overall process of CME from the drug delivery perspectives. Consistent with some experiments, our results show that the intermediate particle size and ligand density favor the particle internalization. Moreover, our results show that it is easier for a particle to enter a cell with softer membrane.

CONCLUSIONS

The model presented here is able to provide mechanistic insights into CME and can be readily modified to include other important factors, such as actins. The predictions from the model will aid in the therapeutic design of intracellular/transcellular drug delivery and antiviral interventions.

Rule-based modelling provides an extendable framework for comparing candidate mechanisms underpinning clathrin polymerisation

Polymerisation of clathrin is a key process that underlies clathrin-mediated endocytosis. Clathrin-coated vesicles are responsible for cell internalization of external substances required for normal homeostasis and life –sustaining activity. There are several hypotheses describing formation of closed clathrin structures. According to one of the proposed mechanisms cage formation may start from a flat lattice buildup on the cellular membrane, which is later transformed into a curved structure. Creation of the curved surface requires rearrangement of the lattice, induced by additional molecular mechanisms. Different potential mechanisms require a modeling framework that can be easily modified to compare between them. We created an extendable rule-based model that describes polymerisation of clathrin molecules and various scenarios of cage formation. Using Global Sensitivity Analysis (GSA) we obtained parameter sets describing clathrin pentagon closure and the emergence/production and closure of large-size clathrin cages/vesicles. We were able to demonstrate that the model can reproduce budding of the clathrin cage from an initial flat array.

Clathrin polymerization exhibits high mechano-geometric sensitivity

How tension modulates cellular transport has become a topic of interest in the recent past. However, the effect of tension on clathrin assembly and vesicle growth remains less understood. Here, we use the classical Helfrich theory to predict the energetic cost that clathrin is required to pay to remodel the membrane at different stages of vesicle formation. Our study reveals that this energetic cost is highly sensitive to not only the tension in the membrane but also to the instantaneous geometry of the membrane during shape evolution. Our study predicts a sharp reduction in clathrin coat size in the intermediate tension regime (0.01-0.1 mN m-1). Remarkably, the natural propensity of the membrane to undergo bending beyond the Ω shape causes a significant decrease in the energy needed from clathrin to drive vesicle growth. Our studies in mammalian cells confirm a reduction in clathrin coat size in an increased tension environment. In addition, our findings suggest that the two apparently distinct clathrin assembly modes, namely coated pits and coated plaques, observed in experimental investigations might be a consequence of varied tensions in the plasma membrane. Overall, the mechano-geometric sensitivity revealed in this study might also be at play during the polymerization of other membrane remodeling proteins.

Kinetics of cellular uptake of viruses and nanoparticles via clathrin-mediated endocytosis

Several viruses exploit clathrin-mediated endocytosis to gain entry into host cells. This process is also used extensively in biomedical applications to deliver nanoparticles (NPs) to diseased cells. The internalization of these nano-objects is controlled by the assembly of a clathrin-containing protein coat on the cytoplasmic side of the plasma membrane, which drives the invagination of the membrane and the formation of a cargo-containing endocytic vesicle. Current theoretical models of receptor-mediated endocytosis of viruses and NPs do not explicitly take coat assembly into consideration. In this paper we study cellular uptake of viruses and NPs with a focus on coat assembly. We characterize the internalization process by the mean time between the binding of a particle to the membrane and its entry into the cell. Using a coarse-grained model which maps the stochastic dynamics of coat formation onto a one-dimensional random walk, we derive an analytical formula for this quantity. A study of the dependence of the mean internalization time on NP size shows that there is an upper bound above which this time becomes extremely large, and an optimal size at which it attains a minimum. Our estimates of these sizes compare well with experimental data. We also study the sensitivity of the obtained results on coat parameters to identify factors which significantly affect the internalization kinetics.

Imaging and modeling the dynamics of clathrin-mediated endocytosis

Clathrin-mediated endocytosis (CME) plays a central role in cellular homeostasis and is mediated by clathrin-coated pits (CCPs). Live-cell imaging has revealed a remarkable heterogeneity in CCP assembly kinetics, which can be used as an intrinsic source of mechanistic information on CCP regulation but also poses several major problems for unbiased analysis of CME dynamics. The backbone of unveiling the molecular control of CME is an imaging-based inventory of the full diversity of individual CCP behaviors, which requires detection and tracking of structural fiduciaries and regulatory proteins with an accuracy of >99.9%, despite very low signals. This level of confidence can only be achieved by combining appropriate imaging modalities with self-diagnostic computational algorithms for image analysis and data mining.

Actin and dynamin2 dynamics and interplay during clathrin-mediated endocytosis

Actin assembly influences the precise temporal and quantitative recruitment of dynamin2 to sites of clathrin-mediated endocytosis.

Clathrin-mediated endocytosis (CME) involves the recruitment of numerous proteins to sites on the plasma membrane with prescribed timing to mediate specific stages of the process. However, how choreographed recruitment and function of specific proteins during CME is achieved remains unclear. Using genome editing to express fluorescent fusion proteins at native levels and live-cell imaging with single-molecule sensitivity, we explored dynamin2 stoichiometry, dynamics, and functional interdependency with actin. Our quantitative analyses revealed heterogeneity in the timing of the early phase of CME, with transient recruitment of 2–4 molecules of dynamin2. In contrast, considerable regularity characterized the final 20 s of CME, during which ∼26 molecules of dynamin2, sufficient to make one ring around the vesicle neck, were typically recruited. Actin assembly generally preceded dynamin2 recruitment during the late phases of CME, and promoted dynamin recruitment. Collectively, our results demonstrate precise temporal and quantitative regulation of the dynamin2 recruitment influenced by actin polymerization.

Combinatoric analysis of heterogeneous stochastic self-assembly

We analyze a fully stochastic model of heterogeneous nucleation and self-assembly in a closed system with a fixed total particle number M, and a fixed number of seeds Ns. Each seed can bind a maximum of N particles. A discrete master equation for the probability distribution of the cluster sizes is derived and the corresponding cluster concentrations are found using kinetic Monte-Carlo simulations in terms of the density of seeds, the total mass, and the maximum cluster size. In the limit of slow detachment, we also find new analytic expressions and recursion relations for the cluster densities at intermediate times and at equilibrium. Our analytic and numerical findings are compared with those obtained from classical mass-action equations and the discrepancies between the two approaches analyzed.

Membrane fluctuations destabilize clathrin protein lattice order

We develop a theoretical model of a clathrin protein lattice on a flexible cell membrane. The clathrin subunit is modeled as a three-legged pinwheel with elastic deformation modes and intersubunit binding interactions. The pinwheels are constrained to lie on the surface of an elastic sheet that opposes bending deformation and is subjected to tension. Through Monte Carlo simulations, we predict the equilibrium phase behavior of clathrin lattices at various levels of tension. High membrane tensions, which correspond to suppressed membrane fluctuations, tend to stabilize large, flat crystalline structures similar to plaques that have been observed in vivo on cell membranes that are adhered to rigid surfaces. Low tensions, on the other hand, give rise to disordered, defect-ridden lattices that behave in a fluidlike manner. The principles of two-dimensional melting theory are applied to our model system to further clarify how high tensions can stabilize crystalline order on flexible membranes. These results demonstrate the importance of environmental physical cues in dictating the collective behavior of self-assembled protein structures.

AP180 couples protein retrieval to clathrin-mediated endocytosis of synaptic vesicles

How clathrin-mediated endocytosis (CME) retrieves vesicle proteins into newly formed synaptic vesicles (SVs) remains a major puzzle. Besides its roles in stimulating clathrin-coated vesicle formation and regulating SV size, the clathrin assembly protein AP180 has been identified as a key player in retrieving SV proteins. The mechanisms by which AP180 recruits SV proteins are not fully understood. Here, we show that following acute inactivation of AP180 in Drosophila, SV recycling is severely impaired at the larval neuromuscular synapse based on analyses of FM 1-43 uptake and synaptic ultrastructure. More dramatically, AP180 activity is important to maintain the integrity of SV protein complexes at the plasma membrane during endocytosis. These observations suggest that AP180 normally clusters SV proteins together during recycling. Consistent with this notion, SV protein composition and distribution are altered in AP180 mutant flies. Finally, AP180 co-immunoprecipitates with SV proteins, including the vesicular glutamate transporter and neuronal synaptobrevin. These results reveal a new mode by which AP180 couples protein retrieval to CME of SVs. AP180 is also genetically linked to Alzheimer's disease. Hence, the findings of this study may provide new mechanistic insight into the role of AP180 dysfunction in Alzheimer's disease.

Micellization model for the polymerization of clathrin baskets

A thermodynamic model is used to investigate the conditions under which clathrin triskelions form polyhedral baskets. The analysis, which is similar to classical methods used to study micelle formation, relates clathrin basket energetics to system parameters linked to triskelial rigidity, the natural curvature of an isolated triskelion, and interactions between triskelial legs in the assembled polyhedra. Mathematical theory predicts that a minimal ("critical") clathrin concentration, C(C), needs to be surpassed in order for basket polymerization to occur, and indicates how C(C), and the amount of polymerized material, depend on the chosen parameters. Analytical expressions are obtained to indicate how changes in the parameters affect the sizes of the polyhedra which arise when the total clathrin concentration exceeds C(C). A continuum analytic approximation then is used to produce numerical results that illustrate the derived dependences.

Unraveling protein-protein interactions in clathrin assemblies via atomic force spectroscopy

Atomic force microscopy (AFM), single molecule force spectroscopy (SMFS), and single particle force spectroscopy (SPFS) are used to characterize intermolecular interactions and domain structures of clathrin triskelia and clathrin-coated vesicles (CCVs). The latter are involved in receptor-mediated endocytosis (RME) and other trafficking pathways. Here, we subject individual triskelia, bovine-brain CCVs, and reconstituted clathrin-AP180 coats to AFM-SMFS and AFM-SPFS pulling experiments and apply novel analytics to extract force-extension relations from very large data sets. The spectroscopic fingerprints of these samples differ markedly, providing important new information about the mechanism of CCV uncoating. For individual triskelia, SMFS reveals a series of events associated with heavy chain alpha-helix hairpin unfolding, as well as cooperative unraveling of several hairpin domains. SPFS of clathrin assemblies exposes weaker clathrin-clathrin interactions that are indicative of inter-leg association essential for RME and intracellular trafficking. Clathrin-AP180 coats are energetically easier to unravel than the coats of CCVs, with a non-trivial dependence on force-loading rate.

Distributions of lifetime and maximum size of abortive clathrin-coated pits

Clathrin-mediated endocytosis is a complex process through which eukaryotic cells internalize nutrients, antigens, growth factors, pathogens, etc. The process occurs via the formation of invaginations on the cell membrane, called clathrin-coated pits (CCPs). Over the years, much has been learned about the mechanism of CCP assembly, but a complete understanding of the assembly process still remains elusive. In recent years, using fluorescence microscopy, studies have been done to determine the statistical properties of CCP formation. In this paper, using a recently proposed coarse-grained, stochastic model of CCP assembly [Banerjee, Berezhkovskii, and Nossal, Biophys. J. 102, 2725 (2012)], we suggest new ways of analyzing such experimental data. To be more specific, we derive analytical expressions for the distribution of maximum size of abortive CCPs, and the probability density of their lifetimes. Our results show how these functions depend on the kinetic and energetic parameters characterizing the assembly process, and therefore could be useful in extracting information about the mechanism of CCP assembly from experimental data. We find excellent agreement between our analytical results and those obtained from kinetic Monte Carlo simulations of the assembly process.