A formalisation of the neural assembly concept 1. Constraints on neural assembly size

Hebb proposed the concept of a neural assembly distributed across cortical tissue as a model for representation of information in the cerebral cortex. Later developments of the concept highlight the need for overlapping membership between independent assemblies, and the spread of activity throughout the assembly once it is activated above a critical level (ignition). Formalisation of the neural assembly concept, especially in relation to quantitative data from the real cortex, is at a very early stage. We consider two constraints on neural assembly size: (1) if a neural assembly is too small the fraction of its neurons that need to be active to ignite the whole assembly becomes unrealistically large; (2) if assemblies in a block of cortical tissue become too large then the block becomes ‘unsafe’, that is, unwanted spread from an active assembly to overlapping ones becomes inevitable. We consider variations in three parameters: neuronal firing threshold; connection density; and the total number of assemblies stored in the block of cortical tissue. Given biologically plausible values for these parameters we estimate maximum assembly size compatible with ignitability of individual assemblies, low probability of unwanted spread to overlapping assemblies, and safe operation of the block as a whole. Received: 7 March 1997 / Accepted in revised form: 1 July 1997     

Kinetic analysis of tubulin assembly in the presence of the microtubule-associated protein TOGp

The microtubule-associated protein TOGp, which belongs to a widely distributed protein family from yeasts to humans, is highly expressed in human tumors and brain tissue. From purified components we have determined the effect of TOGp on thermally induced tubulin association in vitro in the presence of 1 mm GTP and 3.4 m glycerol. Physicochemical parameters describing the mechanism of tubulin polymerization were deduced from the kinetic curves by application of the classical theoretical models of tubulin assembly. We have calculated from the polymerization time curves a range of parameters characteristic of nucleation, elongation, or steady state phase. In addition, the tubulin subunits turnover at microtubule ends was deduced from tubulin GTPase activity. For comparison, parallel experiments were conducted with colchicine and taxol, two drugs active on microtubules and with tau, a structural microtubule-associated protein from brain tissue. TOGp, which decreases the nucleus size and the tenth time of the reaction (the time required to produce 10% of the final amount of polymer), shortens the nucleation phase of microtubule assembly. In addition, TOGp favors microtubule formation by increasing the apparent first order rate constant of elongation. Moreover, TOGp increases the total amount of polymer by decreasing the tubulin critical concentration and by inhibiting depolymerization during the steady state of the reaction.     

Transthyretin Interferes with Aβ Amyloid Formation by Redirecting Oligomeric Nuclei into Non-Amyloid Aggregates

The pathological Aβ aggregates associated with Alzheimer's disease follow a nucleation-dependent path of formation. A nucleus represents an oligomeric assembly of Aβ peptides that acts as a template for subsequent incorporation of monomers to form a fibrillar structure. Nuclei can form de novo or via surface-catalyzed secondary nucleation, and the combined rates of elongation and nucleation control the overall rate of fibril formation. Transthyretin (TTR) obstructs Aβ fibril formation in favor of alternative non-fibrillar assemblies, but the mechanism behind this activity is not fully understood. This study shows that TTR does not significantly disturb fibril elongation; rather, it effectively interferes with the formation of oligomeric nuclei. We demonstrate that this interference can be modulated by altering the relative contribution of elongation and nucleation, and we show how TTR's effects can range from being essentially ineffective to almost complete inhibition of fibril formation without changing the concentration of TTR or monomeric Aβ.     

Conformational entropy limits the transition from nucleation to elongation in amyloid aggregation

The formation of β-sheet-rich amyloid fibrils in Alzheimer’s disease and other neurodegenerative disorders is limited by a slow nucleation event. To understand the initial formation of β-sheets from disordered peptides, we used all-atom simulations to parameterize a lattice model that treats each amino acid as a binary variable with β- and non-β-sheet states. We show that translational and conformational entropy give the nascent β-sheet an anisotropic surface tension that can be used to describe the nucleus with 2D classical nucleation theory. Since translational entropy depends on concentration, the aspect ratio of the critical β-sheet changes with protein concentration. Our model explains the transition from the nucleation phase to elongation as the point where the β-sheet core becomes large enough to overcome the conformational entropy cost to straighten the terminal molecule. At this point the β-strands in the nucleus spontaneously elongate, which results in a larger binding surface to capture new molecules. These results suggest that nucleation is relatively insensitive to sequence differences in coaggregation experiments because the nucleus only involves a small portion of the peptide.     

Polyamine Sharing between Tubulin Dimers Favours Microtubule Nucleation and Elongation via Facilitated Diffusion

We suggest for the first time that the action of multivalent cations on microtubule dynamics can result from facilitated diffusion of GTP-tubulin to the microtubule ends. Facilitated diffusion can promote microtubule assembly, because, upon encountering a growing nucleus or the microtubule wall, random GTP-tubulin sliding on their surfaces will increase the probability of association to the target sites (nucleation sites or MT ends). This is an original explanation for understanding the apparent discrepancy between the high rate of microtubule elongation and the low rate of tubulin association at the microtubule ends in the viscous cytoplasm. The mechanism of facilitated diffusion requires an attraction force between two tubulins, which can result from the sharing of multivalent counterions. Natural polyamines (putrescine, spermidine, and spermine) are present in all living cells and are potent agents to trigger tubulin self-attraction. By using an analytical model, we analyze the implication of facilitated diffusion mediated by polyamines on nucleation and elongation of microtubules. In vitro experiments using pure tubulin indicate that the promotion of microtubule assembly by polyamines is typical of facilitated diffusion. The results presented here show that polyamines can be of particular importance for the regulation of the microtubule network in vivo and provide the basis for further investigations into the effects of facilitated diffusion on cytoskeleton dynamics.     

Model-based analysis of assembly kinetics for virus capsids or other spherical polymers

The assembly of virus capsids or other spherical polymers--empty, closed structures composed of hundreds of protein subunits--is poorly understood. Assembly of a closed spherical polymer is unlike polymerization of a filament or crystal, examples of open-ended polymers. This must be considered to develop physically meaningful analyses. We have developed a model of capsid assembly, based on a cascade of low-order reactions, that allows us to calculate kinetic simulations. The behavior of this model resembles assembly kinetics observed in solution (Zlotnick, A., J. M. Johnson, P. W. Wingfield, S. J. Stahl, and D. Endres. 1999. Biochemistry. 38:14644-14652). We exhibit two examples of this general model describing assembly of dodecahedral and icosahedral capsids. Using simulations based on these examples, we demonstrate how to extract robust estimates of assembly parameters from accessible experimental data. These parameters, nucleus size, average nucleation rate, and average free energy of association can be determined from measurement of subunit and capsid as time and concentration vary. Mathematical derivations of the analyses, carried out for a general model, are provided in an Appendix. The understanding of capsid assembly developed in this paper is general; the examples provided can be readily modified to reflect different biological systems. This enhanced understanding of virus assembly will allow a more quantitative analysis of virus stability and biological or antiviral factors that affect assembly.     

A theoretical model successfully identifies features of hepatitis B virus capsid assembly.

The capsids of most spherical viruses are icosahedral, an arrangement of multiples of 60 subunits. Though it is a salient point in the life cycle of any virus, the physical chemistry of virus capsid assembly is poorly understood. We have developed general models of capsid assembly that describe the process in terms of a cascade of low order association reactions. The models predict sigmoidal assembly kinetics, where intermediates approach a low steady state concentration for the greater part of the reaction. Features of the overall reaction can be identified on the basis of the concentration dependence of assembly. In simulations, and on the basis of our understanding of the models, we find that nucleus size and the order of subsequent "elongation" reactions are reflected in the concentration dependence of the extent of the reaction and the rate of the fast phase, respectively. The reaction kinetics deduced for our models of virus assembly can be related to the assembly of any "spherical" polymer. Using light scattering and size exclusion chromatography, we observed polymerization of assembly domain dimers of hepatitis B virus (HBV) capsid protein. Empty capsids assemble at a rate that is a function of protein concentration and ionic strength. The kinetics of capsid formation were sigmoidal, where the rate of the fast phase had second-power concentration dependence. The extent of assembly had third-power concentration dependence. Simulations based on the models recapitulated the concentration dependences observed for HBV capsid assembly. These results strongly suggest that in vitro HBV assembly is nucleated by a trimer of dimers and proceeds by the addition of individual dimeric subunits. On the basis of this mechanism, we suggest that HBV capsid assembly could be an important target for antiviral therapeutics.     

Microtubules form by progressively faster tubulin accretion,not by nucleation–elongation

Microtubules are dynamic polymers that play fundamental roles in all eukaryotes. Despite their importance, how new microtubules form is poorly understood. Textbooks have focused on variations of a nucleation–elongation mechanism in which monomers rapidly equilibrate with an unstable oligomer (nucleus) that limits the rate of polymer formation; once formed, the polymer then elongates efficiently from this nucleus by monomer addition. Such models faithfully describe actin assembly, but they fail to account for how more complex polymers like hollow microtubules assemble. Here, we articulate a new model for microtubule formation that has three key features: (1) microtubules initiate via rectangular, sheet-like structures that grow faster the larger they become; (2) the dominant pathway proceeds via accretion, the stepwise addition of longitudinal or lateral layers; and (3) a “straightening penalty” to account for the energetic cost of tubulin’s curved-to-straight conformational transition. This model can quantitatively fit experimental assembly data, providing new insights into biochemical determinants and assembly pathways for microtubule nucleation.     

Assembly of the yeast prion Ure2p into protein fibrils. Thermodynamic and kinetic characterization

The [URE3] phenotype in Saccharomyces cerevisiae propagates by a prion mechanism, involving the aggregation of the normally soluble and highly helical protein Ure2. Previous data have shown that the protein spontaneously forms in vitro long, straight, insoluble fibrils at neutral pH that are similar to amyloids in that they bind Congo red and show green-yellow birefringence and have an increased resistance to proteolysis. These fibrils are not amyloids as they are devoid of a cross-beta core. Here we further document the mechanism of assembly of Ure2p into fibrils. The critical concentration for Ure2p assembly is measured, and the minimal size of the nuclei that are the precursors of Ure2p fibrils is determined. Our data indicate that the assembly process is irreversible. As a consequence, the critical concentration is very low. By analyzing the elongation rates of preformed fibrils and combining the results with single-fiber imaging experiments of a variant Ure2p labeled by fluorescent dyes, we reveal the polarity of the fibrils and differences in the elongation rates at their ends. These results bring novel insight in the process of Ure2p assembly into fibrils and the mechanism of propagation of yeast prions.     

Acroplaxome, an F-actin-keratin-containing plate, anchors the acrosome to the nucleus during shaping of the spermatid head

Nuclear shaping is a critical event during sperm development as demonstrated by the incidence of male infertility associated with abnormal sperm ad shaping. Herein, we demonstrate that mouse and rat spermatids assemble in the subacrosomal space a cytoskeletal scaffold containing F-actin and Sak57, a keratin ortholog. The cytoskeletal plate, designated acroplaxome, anchors the developing acrosome to the nuclear envelope. The acroplaxome consists of a marginal ring containing keratin 5 10-nm-thick filaments and F-actin. The ring is closely associated with the leading edge of the acrosome and to the nuclear envelope during the elongation of the spermatid head. Anchorage of the acroplaxome to the gradually shaping nucleus is not disrupted by hypotonic treatment and brief Triton X-100 extraction. By examining spermiogenesis in the azh mutant mouse, characterized by abnormal spermatid/sperm head shaping, we have determined that a deformity of the spermatid nucleus is restricted to the acroplaxome region. These findings lead to the suggestion that the acroplaxome nucleates an F-actin-keratin-containing assembly with the purpose of stabilizing and anchoring the developing acrosome during spermatid nuclear elongation. The acroplaxome may also provide a mechanical planar scaffold modulating external clutching forces generated by a stack of Sertoli cell F-actin-containing hoops encircling the elongating spermatid nucleus.     

Erythrocyte microtubule assembly in vitro. Determination of the effects of erythrocyte tau, tubulin isoforms, and tubulin oligomers on erythrocyte tubulin assembly, and comparison with brain microtubule assembly

Two tubulin variants, isolated from chicken brain and erythrocytes and known to have different peptide maps and electrophoretic properties, are demonstrated to exhibit different assembly properties in vitro: 1) erythrocyte tubulin assembles with greater efficiency (lower critical concentration, greater elongation rate) but exhibits a lower nucleation rate than brain tubulin, and 2) erythrocyte tubulin readily forms oligomers whose presence significantly retards the rate of elongation, suggesting that tubulin oligomers may also be important for determining the rate of assembly and the length of microtubules in erythrocytes. Erythrocyte tubulin isolated by cycles of in vitro assembly-disassembly is also demonstrated to contain a 67-kDa tau factor that greatly enhances microtubule nucleation but has little effect on elongation rates or critical concentration. Immunofluorescence microscopy with tau antibody indicates that tau is specifically associated with marginal band microtubules, suggesting that it may be important for determining microtubule function in vivo.     

Cytochalasin B slows but does not prevent monomer addition at the barbed end of the actin filament

We used Limulus sperm acrosomal actin bundles to examine the effect of 2 microM cytochalasin B (CB) on elongation from both the barbed and pointed ends of the actin filament. In this paper we report that 2 microM CB does not prevent monomer addition onto the barbed ends of the acrosomal actin filaments. Barbed end assembly occurred over a range of actin monomer concentrations (0.2-6 microM) in solutions containing 75 mM KCl, 5 mM MgCl2, 10 mM Imidazole, pH 7.2, and 2 microM CB. However, the elongation rates were reduced such that the rates at the barbed end were approximately the same as those at the pointed end. The association and dissociation rate constants were 8- to 10-fold smaller at the barbed end in the presence of CB along with an accompanying twofold increase in critical concentration at that end. Over the time course of experimentation there was little evidence for potentiation by CB of the nucleation step of assembly. CB did not sever actin filaments; instead its presence increased the susceptibility of actin filaments to breakage from the gentle shear forces incurred during sample preparation. Under these experimental conditions, the assembly rate constants and critical concentration at the pointed end were the same in both the presence and the absence of CB.     

Shaping of the sperm nucleus in Marsilea: A distinction between factors responsible for shape generation and shape determination

Spermiogenesis in Marsilea vestita involves the elongation of a roughly spherical nucleus into a spiral that is composed of four to five gyres. A ribbon of microtubules is associated with the outer edge of the nucleus throughout the shaping process. In order to observe nuclear morphogenesis in the absence of microtubules, developing microspores were treated with drugs that are known to affect microtubule assembly. Spermatids cultured in the presence of colchicine from the beginning of spermiogenesis do not form a microtubule ribbon. The nuclei of these cells change from a spherical to an irregular shape with elongate branches or loops. The normal spiral nucleus and elongate rod of condensed chromatin are not formed and the pattern of chromatin condensation is also abnormal. These observations indicate that in Marsilea microtubules do not provide the mechanical force for nuclear shape generation. Bulk chromatin condensation can also be eliminated as the force behind nuclear shaping, because during normal development the chromatin condenses only after nuclear shaping is well advanced. We suggest that a force-generating system is located near or is a part of the nuclear envelope. Microtubules may, however, be important in the determination of the final shape of the nucleus either by organizing or directing the force-generating system or by externally restricting or guiding the shaping nucleus. Microtubules may also function in controlling the pattern of chromatin condensation.     

The kinetics of microtubule assembly. Evidence for a two-stage nucleation mechanism

A model describing the nucleation and assembly of purified tubulin has been developed. The novel feature of this model is a two stage nucleation process to allow the explicit inclusion of the two-dimensional nature of the early stages of microtubule assembly. In actin assembly the small starting nucleus has only one site for subunit addition as the two-stranded helix is formed. In contrast, microtubule assembly begins with the formation of a small two-dimensional section of microtubule wall. The model we propose is a modification of the work of Wegner and Engel (Wegner, A., and Engel, J. (1975) Biophys. Chem. 3, 215-225) wherein we add a second stage of nucleation to directly account for lateral growth, i.e. the addition of a small number of subunits to the side of an existing sheet structure. Subsequent elongation of the sheets is treated in the usual way. The experimental system used to test this model was the Mg2+/glycerol induced assembly of purified tubulin. The computer simulation of the polymerization time courses gave a fairly good fit to experimental kinetics for our model, where the primary nucleus comprises two protofilaments, of four and three subunits, and lateral growth requires a three-subunit nucleus to initiate a new protofilament.     

Apparent cooperative assembly of the bacterial cell division protein FtsZ demonstrated by isothermal titration calorimetry

The assembly dynamics of FtsZ, a prokaryotic homolog of tubulin, are important for their role in bacterial cytokinesis. Here we used isothermal titration calorimetry (ITC) to measure the heat of FtsZ self-association under various conditions. The measurements were designed to test whether FtsZ protofilaments are assembled by an isodesmic (linear aggregates in which each bond has an identical equilibrium constant) or a cooperative (aggregates only become stable after forming a oligomeric nucleus) assembly process. The isodesmic model can fit the assembly in GDP closely but cannot fit the assembly in GTP. FtsZ-GTP without Mg(2+) exhibits an apparent critical concentration, which is indicative of cooperative assembly, near 2.9 microm. With 2.5 mm Mg(2+) (which allows FtsZ to hydrolyze GTP) the critical concentration is reduced 10-fold to approximately 0.31 microm. Both with and without Mg(2+) there is no evidence for assembly below the critical concentration, but there is an abrupt transition to full assembly above. The ITC data are highly suggestive of a cooperative assembly, although this is difficult to reconcile with the 1-subunit-thick protofilaments observed by electron microscopy.     

A reassessment of the factors affecting microtubule assembly and disassembly in vitro

Current models of microtubule assembly from pure tubulin involve a nucleation phase followed by microtubule elongation at a constant polymer number. Both the rate of microtubule nucleation and elongation are thought to be tightly influenced by the free GTP-tubulin concentration, in a law of mass action-dependent manner. However, these basic hypotheses have remained largely untested due to a lack of data reporting actual measurements of the microtubule length and number concentration during microtubule assembly.Here, we performed simultaneous measurements of the polymeric tubulin concentration, of the free GTP-tubulin concentration, and of the microtubule length and number concentration in both polymerizing and depolymerizing conditions. In agreement with previous work we find that the microtubule nucleation rate is strongly dependent on the initial GTP-tubulin concentration. But we find that microtubule nucleation persists during microtubule elongation. At any given initial tubulin-GTP concentration, the microtubule nucleation rate remains constant during polymer assembly, despite the wide variation in free GTP-tubulin concentration. We also find a remarkable constancy of the rate of microtubule elongation during assembly. Apparently, the rate of microtubule elongation is intrinsic to the polymers, insensitive to large variations of the free GTP-tubulin concentration. Finally we observe that when, following assembly, microtubules depolymerize below the free GTP-tubulin critical concentration, the rate-limiting factor for disassembly is the frequency of microtubule catastrophe. At all time-points during disassembly, the microtubule catastrophe frequency is independent of the free GTP-tubulin concentration but, as the microtubule nucleation rate, is strongly dependent on the initial free GTP-tubulin concentration. We conclude that the dynamics of both microtubule assembly and disassembly depend largely on factors other than the free GTP-tubulin concentration. We propose that intrinsic structural factors and endogenous regulators, whose concentration varies with the initial conditions, are also major determinants of these dynamics.     

Identification and characterization of protein interactions in the mammalian mRNA processing body using a novel two-hybrid assay

Components of the mRNA processing body (P-body) regulate critical steps in mRNA storage, transport, translation and degradation. At the core of the P-body is the decapping complex, which removes the 5′ cap from de-adenylated mRNAs and mediates an irreversible step in mRNA degradation. The assembly of P-bodies in Saccharomyces cerevisiae, Arabidopsis thaliana and Drosophila melanogaster has been previously described. Less is known about the assembly of mammalian P-bodies. To investigate the interactions that occur between components of mammalian P-bodies, we developed a fluorescence-based, two-hybrid assay system. The assay depends on the ability of one P-body component, fused to an exogenous nuclear localization sequence (NLS), to recruit other P-body components to the nucleus. The assay was used to investigate interactions between P-body components Ge-1, DCP2, DCP1, EDC3, RAP55, and RCK. The results of this study show that the modified two-hybrid assay can be used to identify protein interactions that occur in a macromolecular complex. The assay can also be used to efficiently detect protein interaction domains. The results provide important insights into mammalian P-body assembly and demonstrate similarities, and critical differences, between P-body assembly in mammalian cells compared with that of other species.     

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.