Mathematical modelling of the formation of coated vesicles. A theoretical geometrical approach.

The mechanism by which flat hexagonal lattices of clathrin trimers transform into pentagonal/hexagonal spheres remains a mystery. In light of the geometrical nature of this process we have pursued a mathematical approach to the question. Through the geometrical analysis of flat hexagonal lattices we have discovered three possible forms of transformation to introduce curvature into the centre of the lattice: hub-centre transformation; hub-edge transformation; fringe transformation. Hub-edge and fringe transformations are used first to close the lattice while introducing localized curvature at the edges of the lattice. Hub-centre transformation is used after closure to relax the severely localized curvature generated during closure. This scheme not only maximizes the size of the coated vesicle generated, but also minimizes the number of transformations, thus minimizing the energy expended.     

Deep-etch views of clathrin assemblies

Clathrin assemblies were adsorbed to mica and freeze-dried by a new procedure that yields 3-D images with much topological detail. These permitted renewed inquiry into how clathrin trimers (i.e. "triskelions") assemble into polygonal coats or baskets. Freeze-drying revealed unsuspected differences in the relative shapes and dimensions of individual trimer building blocks, as compared with the completed polygonal networks, which indicate that the assembly scheme first proposed by Crowther and Peare (1) requires modification. Specifically, the freeze-etch images display the following new features: (1) Trimer arms possess terminal scroll-shaped hooks that can open or close and thus determine their lengths. (2) When extended, trimer arms are sufficiently long to pass around three sides of the final polygonal facet. Since current views hold that the arms pass around only two sides, the remaining length, including the terminal hook, must point into the basket interior. (3) Freeze-dried trimers display bends in their arms at specific loci that determine their final distribution in the completed baskets. (4) The completed struts of the final assemblies are uniformed in the calibre, cylindrical in profile, and travel directly between the vertices of each polygon, without any sign of the slew or width-variation that is predicted by the Crowther and Pearse model. Based on this direct comparison of promoter vs product, by a single technique that can image both, we offer a modified scheme for clathrin coat assembly, in which we predict that the individual arms in each clathrin triskelion emanate from its center in a slewed manner, but the final assembled struts of the basket need not be slewed. Attempts were made to capture assembly intermediates on mica to obtain support for the scheme, but these unfortunately yielded ambiguous images of incomplete polygons with blunt projections, rather than the expected "halo" of uncommitted trimer arms. These we interpret to be "dead ends" that failed to polymerize further because they included proteolyzed components. Further assembly experiments, avoiding such hazards, are indicated.     

Clathrin assembly proteins: affinity purification and a model for coat assembly

Assembly protein (AP) preparations from bovine brain coated vesicles have been fractionated by clathrin-Sepharose affinity chromatography. Two distinct fractions that possess coat assembly activity were obtained and are termed AP-1 and AP-2. The AP-1, not retained on the resin, has principal components with molecular weights of 108,000, 100,000, 47,000, and 19,000. The AP-2, bound to the resin and eluted by Tris-HCl at a concentration that parallels the latter's effect on coat disassembly, corresponds to the active complex described previously (Zaremba, S., and J. H. Keen, 1983, J. Cell Biol., 97:1339-1347). Its composition is similar to that of the AP-1 in that it contains 100,000-, 50,000-, and 16,000-mol-wt polypeptides in equimolar amounts; minor amounts of 112,000- and 115,000-mol-wt polypeptides are also present. Both are distinct from a recently described assembly protein of larger subunit molecular weight that we term AP-3. These results indicate the existence of a family of assembly proteins within cells. On incubation with clathrin both AP-1 and AP-2 induce the formation of coat structures, those containing AP-1 slightly smaller (mean diameter = 72 nm) than those formed in the presence of AP-2 (mean diameter = 79 nm); both structures have been detected previously in coated vesicle preparations from brain. Coats formed in the presence of AP-2 consistently contain approximately one molecule each of the 100,000-, 50,000-, and 16,000-mol-wt polypeptides per clathrin trimer. By low angle laser light scattering the molecular weight of native AP-2 was determined to be approximately 343,000, indicating that it is a dimer of each of the three subunits, and implying that it is functionally bivalent in clathrin binding. A model for AP-mediated coat assembly is proposed in which a bivalent AP-2 molecule bridges the distal legs or terminal domains of two clathrin trimers that are destined to occupy adjacent vertices in the assembled coat. Binding of a second AP-2 molecule locks these two trimers in register for assembly and further addition of AP-2 to free trimer legs promotes completion of the clathrin lattice. Effects of AP binding on the angle and flexibility of the legs at the hub of the trimer (the "pucker") are suggested to account for the characteristic size distributions of coats formed under varied conditions and, more speculatively, to contribute to the transformation of flat clathrin lattices to curved coated vesicles that are thought to occur during endocytosis.     

Effects of cytoplasmic acidification on clathrin lattice morphology

Reducing the internal pH of cultured cells by several different protocols that block endocytosis is found to alter the structure of clathrin lattices on the inside of the plasma membrane. Lattices curve inward until they become almost spherical yet remain stubbornly attached to the membrane. Also, the lattices bloom empty "microcages" of clathrin around their edges. Correspondingly, broken-open cells bathed in acidified media demonstrate similar changes in clathrin lattices. Acidification accentuates the normal tendency of lattices to round up in vitro and also stimulates them to nucleate microcage formation from pure solutions of clathrin. On the other hand, several conditions that also inhibit endocytosis have been found to create, instead of unusually curved clathrin lattices with extraneous microcages, a preponderance of unusually flat lattices. These treatments include pH-"clamping" cells at neutrality with nigericin, swelling cells with hypotonic media, and sticking cells to the surface of a culture dish with soluble polylysine. Again, the unusually flat lattices in such cells display a tendency to round up and to nucleate clathrin microcage formation during subsequent in vitro acidification. This indicates that regardless of the initial curvature of clathrin lattices, they all display an ability to grow and increase their curvature in vitro, and this is enhanced by lowering ambient pH. Possibly, clathrin lattice growth and curvature in vivo may also be stimulated by a local drop in pH around clusters of membrane receptors.     

Effect of Clathrin Light Chains on the Stiffness of Clathrin Lattices and Membrane Budding

Clathrin‐dependent transport processes require the polymerization of clathrin triskelia into polygonal scaffolds. Together with adapter proteins, clathrin collects cargo and induces membrane bud formation. It is not known to what extent clathrin light chains affect the structural and functional properties of clathrin lattices and the ability of clathrin to deform membranes. To address these issues, we have developed a novel procedure for analyzing clathrin lattice formation on rigid surfaces. We found that lattices can form on adaptor‐coated convex‐, planar‐ and even shallow concave surfaces, but the rate of formation and resistance to thermal dissociation of the lattice are greatly enhanced on convex surfaces. Atomic force microscopy on planar clathrin lattices demonstrates that the stiffness of the clathrin lattice is strictly dependent on light chains. The reduced stiffness of the lattice also compromised the ability of clathrin to generate coated buds on the surface of rigid liposomal membranes.      

Topological mechanisms involved in the formation of clathrin-coated vesicles.

By examining the basic characteristics of clathrin lattices, we discover that simple topological rules impose strict constraints on clathrin lattice transformations. These constraints require that internal bond rearrangements take place in conjunction with the addition or removal of pairs of clathrin triskelions within the interior of existing clathrin lattice patches. Similar constraints also are relevant to coated-vesicle shape changes and their budding-off from pit lattices. Via specific illustrations, successive vesicles with hexagonal-barrel and other coats are shown to grow out from the interior of a initially flat clathrin-coated pit so long as free triskelions are available from cytoplasm. Concomitantly, we present mathematical derivations of several simple and useful topological equations. These equations govern the numbers of nonhexagonal clathrin lattice facets and their variations during internal shape transformations and justify the proposed mechanisms of triskelion pair insertion and removal.     

The generation of curved clathrin coats from flat plaques

Flat clathrin lattices or 'plaques' are commonly believed to be the precursors to clathrin-coated buds and vesicles. The sequence of steps carrying the flat hexagonal lattice into a highly curved polyhedral cage with exactly 12 pentagons remains elusive, however, and the large numbers of disrupted interclathrin connections in previously proposed conversion pathways make these scenarios rather unlikely. The recent notion that clathrin can make controlled small conformational transitions opens new avenues. Simulations with a self-assembling clathrin model suggest that localized conformational changes in a plaque can create sufficiently strong stresses for a dome-like fragment to break apart. The released fragment, which is strongly curved but still hexagonal, may subsequently grow into a cage by recruiting free triskelia from the cytoplasm, thus building all 12 pentagonal faces without recourse to complex topological changes. The critical assembly concentration in a slightly acidic in vitro solution is used to estimate the binding energy of a cage at 25-40 k(B) T/clathrin.     

Human immunodeficiency virus type 1 matrix protein assembles on membranes as a hexamer

The membrane-binding matrix (MA) domain of the human immunodeficiency virus type 1 (HIV-1) structural precursor Gag (PrGag) protein oligomerizes in solution as a trimer and crystallizes in three dimensions as a trimer unit. A number of models have been proposed to explain how MA trimers might align with respect to PrGag capsid (CA) N-terminal domains (NTDs), which assemble hexagonal lattices. We have examined the binding of naturally myristoylated HIV-1 matrix (MyrMA) and matrix plus capsid (MyrMACA) proteins on membranes in vitro. Unexpectedly, MyrMA and MyrMACA proteins both assembled hexagonal cage lattices on phosphatidylserine-cholesterol membranes. Membrane-bound MyrMA proteins did not organize into trimer units but, rather, organized into hexamer rings. Our results yield a model in which MA domains stack directly above NTD hexamers in immature particles, and they have implications for HIV assembly and interactions between MA and the viral membrane glycoproteins.     

Interaction of assembly protein AP-2 and its isolated subunits with clathrin

The clathrin assembly protein complex AP-2 is a multimeric subunit complex consisting of two 100-115-kDa subunits known as alpha and beta and 50- and 16-kDa subunits. The subunits have been dissociated and separated by ion-exchange chromatography in 7.5 M urea. Fractions highly enriched in either the alpha or beta subunit were obtained. The alpha fraction interacted with clathrin as evidenced by its ability to bind to preassembled clathrin cages. It also reacted with dissociated clathrin trimers under conditions that favor assembly of coat structures, but did not yield discrete clathrin polygonal lattices. The enriched beta fraction (containing small amounts of alpha) reacted with clathrin to yield intact coats with the incorporation of approximately equivalent amounts of alpha and beta subunits into the polymerized species; excess free beta subunit was unreactive. The AP-2 complex was also completely dissociated in a highly denaturing solvent, 6 M Gdn.HCl, and the constituent subunits of 100-115, 50, and 16 kDa were separated by gel filtration. In a coassembly assay with clathrin, the clathrin polymerizing activity was exclusively associated with the 100-kDa subunit fraction with stoichiometric incorporation of both alpha and beta subunits of 100 kDa into the polymerized coats, and with no requirement for 50- or 16-kDa subunits. These observations demonstrate that the assembly activity of the complex is associated with the alpha and beta subunits and suggest that both subunits, through independent interactions with clathrin, are required for expression of complete lattice assembly activity.     

Light chain C-terminal region reinforces the stability of clathrin heavy chain trimers

The self-assembly of clathrin into lattices relies on the ability of heavy chain legs to form a three-legged pinwheel structure. We investigated the role of light chains in clathrin trimerization by challenging recombinant hub (plus and minus light chain) with an anionic detergent. The binding of light chain increases the amount of detergent needed to induce detrimerization, suggesting light chains reinforced hub trimers. We also show that light chain C-terminal residues are important for enhancing the in vitro assembly of hub at low pH. We assessed how much the C-terminus of light chain contributed to the stability of the trimerization domain by adding full-length and truncated light chains to trimer-defective hub mutants, C1573S and C1573A. Adding full-length LCb to C1573S caused some retrimerization, but little activity was restored, suggesting the majority of oligomeric C1573S was nonnative. A larger percentage of monomeric C1573A could be retrimerized into an assembly-competent form by adding intact LCb. We also discovered that C-terminally deleted light chains produced a heterogeneous population of hubs that were smaller than native hubs, but were assembly active. We propose a model showing how light chains reinforce the puckered clathrin triskelion. Finally, the ability of light chains to retrimerize C1573A hub suggests that the structural role of light chain may be conserved in yeast and mammals.     

The Role of Membrane-Mediated Interactions in the Assembly and Architecture of Chemoreceptor Lattices

In vivo fluorescence microscopy and electron cryo-tomography have revealed that chemoreceptors self-assemble into extended honeycomb lattices of chemoreceptor trimers with a well-defined relative orientation of trimers. The signaling response of the observed chemoreceptor lattices is remarkable for its extreme sensitivity, which relies crucially on cooperative interactions among chemoreceptor trimers. In common with other membrane proteins, chemoreceptor trimers are expected to deform the surrounding lipid bilayer, inducing membrane-mediated anisotropic interactions between neighboring trimers. Here we introduce a biophysical model of bilayer-chemoreceptor interactions, which allows us to quantify the role of membrane-mediated interactions in the assembly and architecture of chemoreceptor lattices. We find that, even in the absence of direct protein-protein interactions, membrane-mediated interactions can yield assembly of chemoreceptor lattices at very dilute trimer concentrations. The model correctly predicts the observed honeycomb architecture of chemoreceptor lattices as well as the observed relative orientation of chemoreceptor trimers, suggests a series of “gateway” states for chemoreceptor lattice assembly, and provides a simple mechanism for the localization of large chemoreceptor lattices to the cell poles. Our model of bilayer-chemoreceptor interactions also helps to explain the observed dependence of chemotactic signaling on lipid bilayer properties. Finally, we consider the possibility that membrane-mediated interactions might contribute to cooperativity among neighboring chemoreceptor trimers.     

Structure determination of clathrin coats to subnanometer resolution by single particle cryo-electron microscopy

Clathrin triskelions can assemble into lattices of different shapes, sizes and symmetries. For many years, the structures of clathrin lattices have been studied by single particle cryo-electron microscopy, which probed the architecture of the D6 hexagonal barrel clathrin coat at the molecular level. By introducing additional image processing steps we have recently produced a density map for the D6 barrel clathrin coat at subnanometer resolution, enabling us to generate an atomic model for this lattice [Fotin, A., Cheng, Y., Sliz, P., Grigorieff, N., Harrison, S.C., Kirchhausen, T., Walz, T., 2004. Molecular model for a complete clathrin lattice from electron cryomicroscopy. Nature 432, 573-579]. We describe in detail here the image processing steps that we have added to produce a density map at this high resolution. These procedures should be generally applicable and may thus help determine the structures of other large protein assemblies to higher resolution by single particle cryo-electron microscopy.     

Transferrin receptors promote the formation of clathrin lattices

Gold conjugates have been used to quantitate human transferrin receptors (hTfnRs) on transfected chick embryo fibroblasts. No relationship could be found between the number of hTfnRs and the number of clathrin-coated pits. However, hTfnRs are also associated with flat clathrin lattices that lie outside invaginated pits. With increasing levels of receptor expression, the density of hTfnRs within flat lattices increases, and at the highest levels of expression the total area of flat lattice increases up to 3-fold. These results show that increased receptor numbers can promote clathrin lattice growth and suggest that the recruitment of receptors like hTfnRs is an essential step in lattice construction. We conclude that the process of invagination, which gives rise to coated pits, is regulated separately.     

Variation in the volume of coated vesicles isoalted from human placenta

Electron microscopic analysis of 500 negatively stained coated vesicles isolated from human placenta showed that they exist within limits in a continuous range of volumes with an unimodal distribution. Some vesicles were larger than the frequently quoted maximum size of these organelles (diameter 100nm). The ratio of hexagonal to pentagonal facets in the clathrin lattice of the vesicle wall appears to be variable. This feature may be important in morphogenesis since the mean volume of prolate vesicles is larger than that of spherical vesicles. Empty lattices had a mean volume smaller than that of lattices containing phospholipid bilayers.     

Capturing the mechanics of clathrin-mediated endocytosis

Clathrin-mediated endocytosis enables selective uptake of molecules into cells in response to changing cellular needs. It occurs through assembly of coat components around the plasma membrane that determine vesicle contents and facilitate membrane bending to form a clathrin-coated transport vesicle. In this review we discuss recent cryo-electron microscopy structures that have captured a series of events in the life cycle of a clathrin-coated vesicle. Both single particle analysis and tomography approaches have revealed details of the clathrin lattice structure itself, how AP2 may interface with clathrin within a coated vesicle and the importance of PIP2 binding for assembly of the yeast adaptors Sla2 and Ent1 on the membrane. Within cells, cryo-electron tomography of clathrin in flat lattices and high-speed AFM studies provided new insights into how clathrin morphology can adapt during CCV formation. Thus, key mechanical processes driving clathrin-mediated endocytosis have been captured through multiple techniques working in partnership.     

Clathrin domains involved in recognition by assembly protein AP-2

The domains on clathrin responsible for interaction with the plasma membrane-associated assembly protein AP-2 have been studied using a novel cage binding assay. AP-2 bound to pure clathrin cages but not to coat structures already containing AP that had been prepared by coassembly. Binding to preassembled cages also occurred in the presence of elevated Tris-HCl concentrations (greater than or equal to 200 mM) which block AP-2 interactions with free clathrin. AP-2 interactions with assembled cages could also be distinguished from AP-2 binding to clathrin trimers by sodium tripolyphosphate (NaPPPi), which binds to the alpha subunit of AP-2 (Beck, K., and Keen, J. H. (1991) J. Biol. Chem. 266, 4442-4447). At concentrations of 1-5 mM, NaPPPi blocked clathrin-triskelion binding; in contrast, interactions with cages persisted in the presence of 25 mM NaPPPi. To begin to identify the region(s) of the clathrin molecule important in recognition by AP-2, clathrin cages were proteolyzed to remove heavy chain terminal domains and portions of the distal leg as well as all of the light chains. AP-2 bound to these "clipped cages"; however, unlike the interaction with native cages, binding of AP-2 to clipped cages was sensitive to the lower concentrations of both Tris-HCl and NaPPPi which disrupt interactions of AP-2 with clathrin trimers. Reconstitution of the clipped cages with clathrin light chains did not restore resistance of AP-2 binding to Tris-HCl. We conclude that one binding site for AP-2 resides on the hub and/or proximal part of the clathrin triskelion whereas a second site is likely to involve the terminal domain and/or distal leg; the second site is manifested only in the assembled lattice structure. We suggest that these two distinct binding interactions may be mediated by the two unique large subunits within the AP-2 complex, acting sequentially during assembly.     

The appendage domain of the AP-2 subunit is not required for assembly or invagination of clathrin-coated pits

Coated pits contain a resident membrane molecule(s) that binds clathrin AP-2 with high affinity. AP-2 binding to this site is likely to be the first step in coated pit assembly because this subunit functions as a template for the polymerization of clathrin into flat polygonal lattices. Integral membrane proteins involved in receptor mediated endocytosis cluster in the newly assembled pits as they invaginate and bud from the membrane. The AP-2 subunit is a multi-domain, molecular complex that can be separated by proteolysis into a brick-shaped core and ear-like appendage domains. We have used this property to identify the domain involved in the various stages of coated pit assembly and budding. We found that the core of AP-2 is the domain that binds both to membranes and to triskelions during assembly. Triskelions are perfectly capable of forming lattices on the membrane bound cores. Clathrin lattices bound only to core domains were also able to invaginate normally. Limited proteolysis was also useful for further characterizing the AP-2 binding site. Elastase treatment of the inside membrane surface released a peptide fraction that is able to bind AP-2 in solution and prevent it from interacting with membranes. Affinity purification of binding activity yielded a collection of peptides that was dominated by a 45-kD species. This is the candidate peptide for containing the AP-2-binding site. Therefore, the appendage domain does not directly participate in any of the assembly or invagination events required for coated pit function.     

Direct x-ray observation of a single hexagonal myofilament lattice in native myofibrils of striated muscle

A striated muscle fiber consists of thousands of myofibrils with crystalline hexagonal myofilament lattices. Because the lattices are randomly oriented, the fiber gives rise to an equatorial x-ray diffraction pattern, which is essentially a rotary-averaged "powder diffraction," carrying only information about the distance between the lattice planes. We were able to record an x-ray diffraction pattern from a single myofilament lattice, very likely originating from a single myofibril from the flight muscle of a bumblebee, by orienting the incident x-ray microbeam along the myofibrillar axis (end-on diffraction). The pattern consisted of a number of hexagonally symmetrical diffraction spots whose originating lattice planes were readily identified. This also held true for some of the weak higher order reflections. The spot-like appearance of reflections implies that the lattice order is extremely well maintained for a distance of millimeters, covering up to a thousand of approximately 2.5-microm-long sarcomeres connected in series. The results open the possibility of applying the x-ray microdiffraction technique to study many other micrometer-sized assemblies of functional biomolecules in the cell.     

Cryo-electron tomography of clathrin-coated vesicles: structural implications for coat assembly

Clathrin-coated vesicles mediate vesicular traffic in cells. Three-dimensional image reconstructions of homogenous populations of in vitro assembled clathrin coats have yielded a molecular model for clathrin and its interactions with some of its partners. The intrinsic averaging required for those calculations has precluded detailed analysis of heterogeneous populations of clathrin-coated vesicles isolated from cells. We have therefore used cryo-electron tomography to study the lattice organization of individual clathrin-coated vesicles and the disposition of the captured vesicle with respect to the surrounding coat. We find a wide range of designs for the clathrin lattice, with different patterns of pentagonal, hexagonal, and occasionally heptagonal facets. Many coats, even smaller ones, enclose membrane vesicles, which are generally offset from the center of the clathrin shell. The electron density distribution between the coat and the underlying vesicle is not uniform, and the number of apparent contacts that anchor the clathrin lattice to the vesicle membrane is significantly less than the number of clathrin heavy chains in the assembly. We suggest that the eccentric position of the vesicle reflects the polarity of assembly, from initiation of coat formation to membrane pinching.     

Assembly and packing of clathrin into coats

We present a model for the packing of clathrin molecules into the characteristic hexagons and pentagons covering coated pits and vesicles. The assembly unit is a symmetrical trimer with three extended legs. Polymerization of these units occurs in seconds under suitable conditions, giving empty polyhedral cages resembling the structures around coated vesicles. Images of small, negatively stained fragments of cages, assembled directly on electron microscope grids, reveal details of the structure, which correlate well with the predicted features of the model. There is one clathrin trimer at each polyhedral vertex, and each leg of the trimer extends along two neighboring polyhedral edges. Quasi-equivalent packing in pentagons and hexagons in polyhedra of different sizes requires a variable joint at the vertex of the molecule and a hinge in each leg. The construction of clathrin coats is remarkable for the extended fibrous contacts that each molecule makes with many others. Such contacts may confer mechanical strength combined with flexibility needed when a vesicle is pinched off from the membrane.