Clathrin lattice reorganization: theoretical considerations

We wish to postulate a mechanism by which flat hexagonal lattices of clathrin trimers transform into coated pits. Using an established model for packing trimers into lattices, we explored the assembly process by single addition of trimers to form polygons. Subject to favorable conditions, removal of a single trimer from a hexagon could lead to the formation of a pentagon. Elimination of trimers from polygonal sheets can occur either at the center of the network or at the edges. Removal of a trimer from the center of these adjacent polygons, "hub transformation," is possible in very few instances, whereas removal from the edges of a polygonal sheet, "fringe transformation," is possible in a host of cases. These hypothetical constructs can be used effectively to explain intermediate structures actually observed in flat hexagonal lattices. The geometry of a purely hexagonal lattice seems to dictate that the first step in transformation must be a "fringe transformation," which then will allow subsequent "hub transformation" to take place leading to the introduction of pentagons into the center of the lattice and ultimately to the curvature of the clathrin lattice.     

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.     

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.     

Increasing Biomass Production on Limited Land Area Through an Optimal Planting Arrangement

Planting density is a primary consideration in silviculture; however, planting arrangement is often ignored. Most, if not all, forest plantations are arranged in rectangular or square lattices (i.e., grids). Using a simple mathematical model, we investigate the potential influence of planting arrangement on planting density, biomass yield, and rotation period by assuming that efficiently arranging trees is similar to packing congruent circles on a plane. The hexagonal lattice achieves the densest circle packing on a plane; therefore, a hexagonal or triangular lattice arrangement of stems provides the highest planting density for a given spacing. Using packing density to quantify arrangement efficiency, tree crowns in a hexagonal lattice fill approximately 90.7% of available area at initial canopy contact, while tree crowns in a square lattice fill approximately 78.5% of available area at initial canopy contact. The hexagonal lattice permits about a 15% higher density than the square lattice, which allows canopy closure to occur earlier without any change in individual tree growth. Short rotation woody crop (SRWC) systems are excellent candidates under the model’s assumptions of level stand with even-age monoculture. If belowground resources are non-limiting, a hexagonal lattice arrangement shortens rotation period and thus optimizes the biomass yield per land area over time. Higher productivity over time is central to sustainable and efficient use of limited area for bioenergy and biomass products.     

An Anionic Phospholipid Enables the Hydrophobic Surfactant Proteins to Alter Spontaneous Curvature

The hydrophobic surfactant proteins, SP-B and SP-C, greatly accelerate the adsorption of the surfactant lipids to an air/water interface. Previous studies of factors that affect curvature suggest that vesicles may adsorb via a rate-limiting structure with prominent negative curvature, in which the hydrophilic face of the lipid leaflets is concave. To determine if SP-B and SP-C might promote adsorption by inducing negative curvature, we used small-angle x-ray scattering to test whether the physiological mixture of the two proteins affects the radius of cylindrical monolayers in the inverse hexagonal phase. With dioleoyl phosphatidylethanolamine alone, the proteins had no effect on the hexagonal lattice constant, suggesting that the proteins fail to insert into the cylindrical monolayers. The surfactant lipids also contain ∼10% anionic phospholipids, which might allow incorporation of the cationic proteins. With 10% of the anionic dioleoyl phosphatidylglycerol added to dioleoyl phosphatidylethanolamine, the proteins induced a dose-related decrease in the hexagonal lattice constant. At 30°C, the reduction reached a maximum of 8% relative to the lipids alone at ∼1% (w/w) protein. Variation of NaCl concentration tested whether the effect of the protein represented a strictly electrostatic effect that screening by electrolyte would eliminate. With concentrations up to 3 M NaCl, the dose-related change in the hexagonal lattice constant decreased but persisted. Measurements at different hydrations determined the location of the pivotal plane and proved that the change in the lattice constant produced by the proteins resulted from a shift in spontaneous curvature. These results provide the most direct evidence yet that the surfactant proteins can induce negative curvature in lipid leaflets. This finding supports the model in which the proteins promote adsorption by facilitating the formation of a negatively curved, rate-limiting structure.     

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.     

Formation of crystalline arrays of chlorophyll a/b - light-harvesting protein by membrane reconstitution.

The structure of the major protein constituent of photosynthetic membranes in higher plants, the chlorophyll a/b-light harvesting complex (LHC), was studied by x-ray diffraction and electron microscopy. The LHC was purified from Triton X-100 solubilized thylakoid membranes of the pea, and contained 6 mol of chlorophylls a and b per mole of a polypeptide of 27,000 molecular weight. X-ray diffraction showed that in the presence of 10 mM MgCl2, purified LHC forms planar aggregates that stack with a period of 51 A. Within each layer, LHC molecules pack with a center-to-center distance of 85 A but without long-range order. However, when LHC is incorporated into single-walled vesicles of plant lecithin, the addition of NaCl above 10 mM, or MgCl2 above 2 mM, led to the formation of plaques of hexagonal lattices, with a lattice constant of 125 A. The large domain size and high degree of order in the plane of the membrane are evident from the sharp lattice lines observed to 7 A resolution on the equator of the x-ray pattern. Freeze-fracture electron micrographs demonstrated an aligned stacking of the lattices in adjacent membranes, resulting in crystallinity in the third dimension over short distances. Micrographs of negatively stained membranes revealed a hexagonal lattice of the same lattice constant, formed by surface-exposed parts of the LHC molecules which are probably responsible for the ordered stacking of lattices. In both the LHC aggregates and in the reconstituted membrane lattices the diffracted x-ray intensities at 10-A spacing on the equator indicate that the LHC molecule contains paralled alpha-helices or beta-sheets that are oriented perpendicular to the planar arrays.     

Assembly of bacteriophage T4 head-related structures. Assembly of polyheads in vitro.

The fine structure of Z-discs from frog, chameleon, rabbit, rat and human muscles was studied. Our data lead us to conclude that the basket-weave (woven) lattice represents the fundamental en face pattern of the vertebrate Z-disc, regardless of the manner of fixation, and we suggest that the large and small-“square” lattices are fixation artifacts. We also find that the woven lattice pattern remains essentially unchanged throughout physiological ranges of resting sarcomere length, and is not detectably altered by active contraction. A model of the vertebrate Z-line, based on anatomical and possible functional considerations, is presented. It presumes that a thin filament, as it enters the Z-line, is continuous with three curved strands which unite with other I-filaments of the same sarcomere. The I-filaments and extending strands from the opposite sarcomere are proposed to be similarly arranged, with the main Z-line substance consisting of the two sets of strands from adjacent sarcomeres. The anatomical features of the Z-line and the phenomenon of “Z-line splitting” are explained by the proposed model. In addition, a potential hexagonal structural arrangement of the Z-line is retained so that a consistent geometrical organization persists throughout the entire sarcomere. Thus, the model also presents a means of understanding the recently suggested role of the Z-line in forming new sarcomeres.     

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.     

Controlling the Orientation and Synaptic Differentiation of Myotubes with Micropatterned Substrates

Micropatterned poly(dimethylsiloxane) substrates fabricated by soft lithography led to large-scale orientation of myoblasts in culture, thereby controlling the orientation of the myotubes they formed. Fusion occurred on many chemically identical surfaces in which varying structures were arranged in square or hexagonal lattices, but only a subset of patterned surfaces yielded aligned myotubes. Remarkably, on some substrates, large populations of myotubes oriented at a reproducible acute angle to the lattice of patterned features. A simple geometrical model predicts the angle and extent of orientation based on maximizing the contact area between the myoblasts and patterned topographic surfaces. Micropatterned substrates also provided short-range cues that influenced higher-order functions such as the localization of focal adhesions and accumulation of postsynaptic acetylcholine receptors. Our results represent what we believe is a new approach for musculoskeletal tissue engineering, and our model sheds light on mechanisms of myotube alignment in vivo.     

Spontaneous-curvature theory of clathrin-coated membranes.

Clathrin-coated membranes are precursors to coated vesicles in the receptor-mediated endocytic pathway. In this paper we present a physical model for the first steps of the transformation of a clathrin-coated membrane into a coated vesicle. The theory is based on in vitro cytoplasmic acidification experiments of Heuser (J. Cell Biol. 108:401-411) that suggest the transformation proceeds by changes in the chemical environment of the clathrin lattice, wherein the chemical environment determines the amount of intrinsic, or spontaneous, curvature of the network. We show that a necessary step of the transformation, formation of free pentagons in the clathrin network, can proceed via dislocation unbinding, driven by changes in the spontaneous curvature. Dislocation unbinding is shown to favor formation of coated vesicles that are quite small compared to those predicted by the current continuum theories, which do not include the topology of the clathrin lattice.     

Low resolution structure of microtubules in solution. Synchrotron X-ray scattering and electron microscopy of taxol-induced microtubules assembled from purified tubulin in comparison with glycerol and MAP-induced microtubules.

The structure of microtubules has been characterized to 3 nm resolution employing time-resolved X-ray scattering. This has revealed detailed structural features of microtubules not observed before in solution. The polymerization of highly purified tubulin, induced by the antitumour drug taxol, has been employed as a microtubule model system. This assembly reaction requires Mg2+, is optimal at a 1:1 taxol to tubulin heterodimer molar ratio, proceeds with GTP or GDP and is intrinsically reversible. The X-ray scattering profiles are consistent with identical non-globular alpha and beta-tubulin monomers ordered within the known helical surface lattice of microtubules. Purified tubulin-taxol microtubules have a smaller mean diameter (approx. 22 nm) than those induced by microtubule associated proteins or glycerol (approx. 24 nm), but nearly identical wall substructure to the resolution of the measurements. This is because the majority of the former consist of only 12 protofilaments instead of the typical 13 protofilaments, as confirmed by electron microscopy of thin-sectioned, negatively stained and ice-embedded taxol microtubules. It may be concluded that taxol induces a slight reduction of the lateral contact curvature between tubulin monomers. The main fringe pattern observed in cryo-electron micrographs is consistent with a simple 12 protofilament 3-start skewed lattice model. Cylindrical closure of this lattice can be achieved by tilting the lattice 0.8 degrees with respect to the microtubule axis. The closure implies a discontinuity in the type of lateral contacts between the tubulin monomers (regardless of whether these are of the -alpha-beta- or the -alpha-alpha-/-beta-beta- type), which indicates that lateral contacts and the subunit specificity of taxol binding are, to a large degree, equivalent.     

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.     

Interstitial contacts in an RNA-dependent RNA polymerase lattice

Catalytic activities can be facilitated by ordered enzymatic arrays that co-localize and orient enzymes and their substrates. The purified RNA-dependent RNA polymerase from poliovirus self-assembles to form two-dimensional lattices, possibly facilitating the assembly of viral RNA replication complexes on the cytoplasmic face of intracellular membranes. Creation of a two-dimensional lattice requires at least two different molecular contacts between polymerase molecules. One set of polymerase contacts, between the “thumb” domain of one polymerase and the back of the “palm” domain of another, has been previously defined. To identify the second interface needed for lattice formation and to test its function in viral RNA synthesis, we used a hybrid approach of electron microscopic and biochemical evaluation of both wild-type and mutant viral polymerases to evaluate computationally generated models of this second interface. A unique solution satisfied all constraints and predicted a two-dimensional structure formed from antiparallel arrays of polymerase fibers that use contacts from the flexible amino-terminal region of the protein. Enzymes that contained mutations in this newly defined interface did not form lattices and altered the structure of wild-type lattices. When reconstructed into virus, mutations that disrupt lattice assembly exhibited growth defects, synthetic lethality or both, supporting the function of the oligomeric lattice in infected cells. Understanding the structure of polymerase lattices within the multimeric RNA-dependent RNA polymerase complex should facilitate antiviral drug design and provide a precedent for other positive-strand RNA viruses.     

Curvature properties of novel forms of phosphatidylcholine with branched acyl chains.

We studied the properties of a series of phosphatidylcholine molecules with branched acyl chains. These lipids have previously been shown to have marked stimulatory effects on the side-chain cleavage activity of cytochrome P450SCC (CYP11A1), an enzyme of the inner mitochondrial membrane. The synthetic lipids used were diacyl phosphatidylcholines with the decanoyl, dodecanoyl or tetradecanoyl chain having a hexyl, octyl or decyl straight chain aliphatic branch at the 2-position. All three lipids lowered the bilayer to hexagonal phase transition temperature of dielaidoyl phosphatidylethanolamine, the lipids with longer acyl chains being more effective in this regard. As pure lipids all of the forms were found by X-ray diffraction to be predominantly in the hexagonal phase (HII) over the entire temperature range of 7-75 degrees C. The properties of the HII phase were unusual with regard to the small size of the lattice spacings and the small temperature dependence of the spacings. We used tetradecane to relieve hydrocarbon packing constraints to determine the intrinsic radius of curvature of the lipid monolayer. The elastic bending modulus was measured in the presence of tetradecane by introducing an osmotic gradient across the hexagonal phase cylinders with aqueous solutions of poly(ethylene glycol). The elastic bending modulus was found to be higher than that observed with other lipids and to increase with temperature. Both the small intrinsic radius of curvature and the high elastic bending modulus indicate that the presence of these lipids in bilayer membranes will impose a high degree of negative curvature strain.     

Curvature of the caudal region is responsible for failure of neural tube closure in the curly tail (ct) mouse embryo.

Delayed closure of the posterior neuropore (PNP) occurs to a variable extent in homozygous mutant curly tail (ct) mouse embryos, and results in the development of spinal neural tube defects (NTD) in 60% of embryos. Previous studies have suggested that curvature of the body axis may delay neural tube closure in the cranial region of the mouse embryo. In order to investigate the relationship between curvature and delayed PNP closure, we measured the extent of ventral curvature of the neuropore region in ct/ct embryos with normal or delayed PNP closure. The results show significantly greater curvature in ct/ct embryos with delayed PNP closure in vivo than in their normal littermates. Reopening of the posterior neuropore in non-mutant mouse embryos, to delay neuropore closure experimentally, did not increase ventral curvature, suggesting that increased curvature in ct/ct embryos is not likely to be a secondary effect of delayed PNP closure. Experimental prevention of ventral curvature in ct/ct embryos, brought about by implantation of an eyelash tip longitudinally into the hindgut lumen, ameliorated the delay in PNP closure. We propose, therefore, that increased ventral curvature of the neuropore region of ct/ct embryos imposes a mechanical stress, which opposes neurulation and thus delays closure of the PNP. Increased ventral curvature may arise as a result of a cell proliferation imbalance, which we demonstrated previously in affected ct/ct embryos.     

Contact interactions method: a new algorithm for protein folding simulations.

Computer simulations of simple exact lattice models are an aid in the study of protein folding process; they have sometimes resulted in predictions experimentally proved. The contact interactions (CI) method is here proposed as a new algorithm for the conformational search in the low-energy regions of protein chains modeled as copolymers of hydrophobic and polar monomers configured as self-avoiding walks on square or cubic lattices. It may be regarded as an extension of the standard Monte Carlo method improved by the concept of cooperativity deriving from nonlocal contact interactions. A major difference with respect to other algorithms is that criteria for the acceptance of new conformations generated during the simulations are not based on the energy of the entire molecule, but cooling factors associated with each residue define regions of the model protein with higher or lower mobility. Nine sequences of length ranging from 20 to 64 residues were used on the square lattice and 15 sequences of length ranging from 46 to 136 residues were used on the cubic lattice. The CI algorithm proved very efficient both in two and three dimensions, and allowed us to localize energy minima not localized by other searching algorithms described in the literature. Use of this algorithm is not limited to the conformational search, because it allows the exploration of thermodynamic and kinetic behavior of model protein chains.     

Comparative studies of S-layer proteins from Bacillus stearothermophilus strains expressed during growth in continuous culture under oxygen-limited and non-oxygen-limited conditions.

The specific properties of S-layer proteins from three different Bacillus stearothermophilus strains revealing oblique, square, or hexagonal lattice symmetry were preserved during growth in continuous culture on complex medium only under oxygen-limited conditions in which glucose was used as the sole carbon source. When oxygen limitation was relieved, amino acids became metabolized, cell density increased, and different S-layer proteins from wild-type strains became rapidly replaced by a new common type of S-layer protein with an apparent subunit molecular weight of 97,000 which assembled into an identical oblique (p2) lattice type. During switching from wild-type strains to variants, patches of the S-layer lattices characteristics for wild-type strains, granular regions, and areas with oblique lattice symmetry could be observed on the surface of individual cells from all organisms. The granular regions apparently consisted of mixtures of the S-layer proteins from the wild-type strains and the newly synthesized p2 S-layer proteins from the variants. S-layer proteins from wild-type strains possessed identical N-terminal regions but led to quite different cleavage products upon peptide mapping, indicating that they are encoded by different genes. Chemical analysis including N-terminal sequencing and peptide mapping showed that the oblique S-layer lattices synthesized under increased oxygen supply were composed of identical protein species.     

Basal Body-Associated Nucleation Center for the Centrin-Based Cortical Cytoskeletal Network in Paramecium

The infraciliary lattice, a contractile cortical cytoskeletal network of Paramecium, is composed of a small number of polypeptides including centrins. Its overall pattern reflects a hierarchy of structural complexity, from assembly and bundling of microfilaments to formation of polygonal meshes arranged in a continuous network subtending the whole cell surface, with local differentiations in the shape and size of the meshes. To analyse how the geometry of this complex network is generated and maintained, we have taken two approaches. Firstly, using monoclonal antibodies raised against the purified network, we have shown that all the component polypeptides colocalize, in agreement with previous biochemical data indicating that the infraciliary lattice is formed of large complexes comprising all the component polypeptides. Secondly, by taking advantage of different experimental conditions leading to disassembly of the network, we have followed its reassembly. Cytological analysis of the process revealed 1) that the network regrows exclusively from specific infraciliary lattice organizing centers (ICLOC), precisely localized near each basal body and, 2) that the global organization is not precisely controlled by genetic information but by the basal body pattern. Finally, slight ultrastuctural differences between reassembled and control lattices suggest that the organization of the filament bundles is partly templated by that of the preexisting ones.