Simulation study of the contribution of oligomer/oligomer binding to capsid assembly kinetics

The process by which hundreds of identical capsid proteins self-assemble into icosahedral structures is complex and poorly understood. Establishing constraints on the assembly pathways is crucial to building reliable theoretical models. For example, it is currently an open question to what degree overall assembly kinetics are dominated by one or a few most efficient pathways versus the enormous number theoretically possible. The importance of this question, however, is often overlooked due to the difficulties of addressing it in either theoretical or experimental practice. We apply a computer model based on a discrete-event simulation method to evaluate the contributions of nondominant pathways to overall assembly kinetics. This is accomplished by comparing two possible assembly models: one allowing growth to proceed only by the accretion of individual assembly subunits and the other allowing the binding of sterically compatible assembly intermediates any sizes. Simulations show that the two models perform almost identically under low binding rate conditions, where growth is strongly nucleation-limited, but sharply diverge under conditions of higher association rates or coat protein concentrations. The results suggest the importance of identifying the actual binding pattern if one is to build reliable models of capsid assembly or other complex self-assembly processes.     

The Impact of Viral RNA on Assembly Pathway Selection

Many single-stranded RNA viruses self-assemble their protein containers around their genomes. The roles that the RNA plays in this assembly process have mostly been ignored, resulting in a protein-centric view of assembly that is unable to explain adequately the fidelity and speed of assembly in such viruses. Using bacteriophage MS2, we demonstrate here via a combination of mass spectrometry and kinetic modelling how viral RNA can bias assembly towards only a small number of the many possible assembly pathways, thus increasing assembly efficiency. Assembly reactions have been studied in vitro using phage coat protein dimers, the known building block of the T = 3 shell, and short RNA stem-loops based on the translational operator of the replicase cistron, a 19 nt fragment (TR). Mass spectrometry has unambiguously identified two on-pathway intermediates in such reactions that have stoichiometry consistent with formation of either a particle 3-fold or 5-fold axis. These imply that there are at least two sub-pathways to the final capsid. The flux through each pathway is controlled by the length of the RNA stem-loop triggering the assembly reaction and this effect can be understood in structural terms. The kinetics of intermediate formation have been studied and show steady-state concentrations for intermediates between starting materials and the T = 3 shell, consistent with an assembly process in which all the steps are in equilibrium. These data have been used to derive a kinetic model of the assembly reaction that in turn allows us to determine the dominant assembly pathways explicitly, and to estimate the effect of the RNA on the free energy of association between the assembling protein subunits. The results reveal that there are only a small number of dominant assembly pathways, which vary depending on the relative ratios of RNA and protein. These results suggest that the genomic RNA plays significant roles in defining the precise assembly sub-pathway followed to create the final capsid.     

Energetics of quasiequivalence: computational analysis of protein-protein interactions in icosahedral viruses.

Quaternary structure polymorphism found in quasiequivalent virus capsids provides a static framework for studying the dynamics of protein interactions. The same protein subunits are found in different structural environments within these particles, and in some cases, the molecular switching required for the polymorphic quaternary interactions is obvious from high-resolution crystallographic studies. Employing atomic resolution structures, molecular mechanics, and continuum electrostatic methods, we have computed association energies for unique subunit interfaces of three icosahedral viruses, black beetle virus, southern bean virus, and human rhinovirus 14. To quantify the chemical determinants of quasiequivalence, the energetic contributions of individual residues forming quasiequivalent interfaces were calculated and compared. The potential significance of the differences in stabilities at quasiequivalent interfaces was then explored with the combinatorial assembly approach. The analysis shows that the unique association energies computed for each virus serve as a sensitive basis set that may determine distinct intermediates and pathways of virus capsid assembly. The pathways for the quasiequivalent viruses displayed isoenergetic oligomers at specific points, suggesting that these may determine the quaternary structure polymorphism required for the assembly of a quasiequivalent particle.     

Capsid assembly of dsDNA viruses

Capsid assembly has been investigated for many dsDNA viruses. Much is known about assembly pathways for these viruses and information is now emerging about the protein/protein and protein/DNA interactions that regulate these pathways. Common themes include progressive transitions in the properties of the capsid subunits as assembly proceeds and the use of accessory factors to aid assembly. Other assembly strategies, such as packaging DNA into a preformed capsid or assembling capsids from subunit oligomers are shared among some but not all viruses of this group. Despite the many similarities, there are sufficient fundamental differences in assembly strategies that it is not possible to describe them all as variations on a single assembly theme.     

A reaction landscape identifies the intermediates critical for self-assembly of virus capsids and other polyhedral structures

The capsids of spherical viruses may contain from tens to hundreds of copies of the capsid protein(s). Despite their complexity, these particles assemble rapidly and with high fidelity. Subunit and capsid represent unique end states. However, the number of intermediate states in these reactions can be enormous-a situation analogous to the protein folding problem. Approaches to accurately model capsid assembly are still in their infancy. In this paper, we describe a sail-shaped reaction landscape, defined by the number of subunits in each species, the predicted prevalence of each species, and species stability. Prevalence can be calculated from the probability of synthesis of a given intermediate and correlates well with the appearance of intermediates in kinetics simulations. In these landscapes, we find that only those intermediates along the leading edge make a significant contribution to assembly. Although the total number of intermediates grows exponentially with capsid size, the number of leading-edge intermediates grows at a much slower rate. This result suggests that only a minute fraction of intermediates needs to be considered when describing capsid assembly.     

Direct incorporation of microtubule oligomers at high GTP concentrations

Chick brain microtubule protein consists primarily of a mixture of MAP2:tubulin oligomers and dimeric tubulin. The assembly of this protein is described by a single pseudofirst-order reaction at 20 microM GTP, but by the summation of two pseudofirst-order reactions at 1 mM GTP. The protein contains two GTP-binding species, corresponding to the tubulin dimers and the oligomers, and conditions which alter the dimer: oligomer equilibrium, affect the kinetics of microtubule assembly. The results indicate that the oligomers are only direct assembly intermediates at high GTP concentrations.     

Pathways for Virus Assembly around Nucleic Acids

Understanding the pathways by which viral capsid proteins assemble around their genomes could identify key intermediates as potential drug targets. In this work, we use computer simulations to characterize assembly over a wide range of capsid protein–protein interaction strengths and solution ionic strengths. We find that assembly pathways can be categorized into two classes, in which intermediates are either predominantly ordered or disordered. Our results suggest that estimating the protein–protein and the protein–genome binding affinities may be sufficient to predict which pathway occurs. Furthermore, the calculated phase diagrams suggest that knowledge of the dominant assembly pathway and its relationship to control parameters could identify optimal strategies to thwart or redirect assembly to block infection. Finally, analysis of simulation trajectories suggests that the two classes of assembly pathways can be distinguished in single-molecule fluorescence correlation spectroscopy or bulk time-resolved small-angle X-ray scattering experiments.     

Surveying Capsid Assembly Pathways through Simulation-Based Data Fitting

Virus capsid assembly has attracted considerable interest from the biophysical modeling community as a model system for complicated self-assembly processes. Simulation methods have proven valuable for characterizing the space of possible kinetics and mechanisms of capsid assembly, but they have so far been able to say little about the assembly kinetics or pathways of any specific virus. It is not possible to directly measure the detailed interaction rates needed to parameterize a model, and there is only a limited amount of experimental evidence available to constrain possible pathways, with almost all of it gathered from in vitro studies of purified coat proteins. In prior work, we developed methods to address this problem by using simulation-based data-fitting to learn rate parameters consistent with both structure-based rule sets and experimental light-scattering data on bulk assembly progress in vitro. We have since improved these methods and extended them to fit simulation parameters to one or more experimental light-scattering curves. Here, we apply the improved data-fitting approach to three capsid systems—human papillomavirus (HPV), hepatitis B virus (HBV), and cowpea chlorotic mottle virus (CCMV)—to assess both the range of pathway types the methods can learn and the diversity of assembly strategies in use between these viruses. The resulting fits suggest three different in vitro assembly mechanisms for the three systems, with HPV capsids fitting a model of assembly via a nonnucleation-limited pathway of accumulation of individual capsomers while HBV and CCMV capsids fit similar but subtly different models of nucleation-limited assembly through ensembles of pathways involving trimer-of-dimer intermediates. The results demonstrate the ability of such data fitting to learn very different pathway types and show some of the versatility of pathways that may exist across real viruses.     

Master equation approach to the assembly of viral capsids

The distribution of inequivalent geometries occurring during self-assembly of the major capsid protein in thermodynamic equilibrium is determined based on a master equation approach. These results are implemented to characterize the assembly of SV40 virus and to obtain information on the putative pathways controlling the progressive build-up of the SV40 capsid. The experimental testability of the predictions is assessed and an analysis of the geometries of the assembly intermediates on the dominant pathways is used to identify targets for anti-viral drug design.     

A simple, RNA-mediated allosteric switch controls the pathway to formation of a T=3 viral capsid

Using mass spectrometry we have detected both assembly intermediates and the final product, the T=3 viral capsid, during reassembly of the RNA bacteriophage MS2. Assembly is only efficient when both types of quasiequivalent coat protein dimer seen in the final capsid are present in solution. NMR experiments confirm that interconversion of these conformers is allosterically regulated by sequence-specific binding of a short RNA stem-loop. Isotope pulse-chase experiments confirm that all intermediates observed are competent for further coat protein addition, i.e., they are all on the pathway to capsid formation, and that the unit of capsid growth is a coat protein dimer. The major intermediate species are dominated by stoichiometries derived from formation of the particle threefold axis, implying that there is a defined pathway toward the T=3 shell. These results provide the first experimental evidence for a detailed mechanistic explanation of the regulation of quasiequivalent capsid assembly. They suggest a direct role for the encapsidated RNA in assembly in vivo, which is consistent with the structure of the genomic RNA within wild-type phage.     

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.     

Stochastic kinetics of reproduction of virions inside a cell

We analyze intracellular viral kinetics in the framework of the model incorporating viral genome replication, mRNA synthesis and degradation, protein synthesis and degradation, capsid assembly, and virion release from a cell. Due to the existence of the critical concentration of viral capsid proteins and other features of reproduction of virions inside a cell, the kinetics is demonstrated to exhibit three distinct initial stages. Specifically, (i) the exponential growth of the viral genome, mRNA and protein concentrations is followed by (ii) the transient stage to (iii) the steady-state regime. The formation of mature virions starts during the transient stage. Comparison of the kinetics, obtained by using the mass-action law and Monte Carlo (MC) technique, indicates that they are nearly identical during the initial exponential growth of the viral intermediates and also during the steady-state stage. The transition from the initial stage to the steady-state regime occurs however somewhat faster in the determenistic case even if the steady-state populations of virions and genomes are appreciable (e.g., about 250 and 500, respectively).     

Thermodynamics and kinetics of protein folding: an evolutionary perspective

This article appeals to an evolutionary model which postulates that primordial proteins were described by small polypeptide chains which (i) lack disulfide bridges, and (ii) display slow folding rates with multi-state kinetics, to determine relations between structural properties of proteins and their folding kinetics. We parameterize the energy landscape of proteins in terms of thermodynamic activation variables. The model studies evolutionary changes in these thermodynamic parameters, and we invoke relations between these activation variables and structural properties of the protein to predict the following correspondence between protein structure and folding kinetics. 1. Proteins with inter- and intra-chain disulfide bridges: large variability in both folding rates and stability of intermediates, multi-state kinetics. 2. Proteins which lack inter and intra-chain disulfide bridges. 2.1 Single-domain chains: fast folding rates; unstable intermediates; two-state kinetics. 2.2 Multi-domain monomers: intermediate rates; metastable intermediates; multi-state kinetics. 2.3 Multi-domain oligomers: slow rates; metastable intermediates; multi-state kinetics. The evolutionary model thus provides a kinetic characterization of one important subfamily of proteins which we describe by the following properties: Folding dynamics of single-domain proteins which lack disulfide bridges are described by two-state kinetics. Folding rate of this class of proteins is positively correlated with the thermodynamic stability of the folded state.     

Folding and processing of the capsid protein precursor P1 is kinetically retarded in neutralization site 3B mutants of poliovirus.

Poliovirus mutants in neutralizing antigenic site 3B were constructed by replacing the glutamic acid residue at amino acid 74 of capsid protein VP2 (VP2074E), using site-specific mutagenesis methods. All viable mutants display small-plaque phenotypes. Characterization of these mutants indicates that capsid assembly is perturbed. Although the defect in capsid assembly reduces the yield of mutant virus particles per cell, the resultant assembled particle is wild-type-like in structure and infectivity. Analyses of capsid assembly intermediates show a transient accumulation of the unprocessed capsid protein precursor, P1, indicating that cleavage of the mutant P1 by the 3CD protease is retarded. The mutant VP0-VP3-VP1 complex generated upon P1 cleavage appears assembly competent, forming pentamer and empty capsid assembly intermediates and infectious virion particles. Although the structure of the infectious mutant virus is virtually identical with that of the wild-type virus, the thermal stability of the mutant virus is dramatically increased over that of the wild-type virus. Thus, mutations at this residue are pleiotropic, altering the kinetics of capsid assembly and generating a virus that is more thermostable and more resistant to neutralization by the site 3B monoclonal antibodies.     

Analysis of human immunodeficiency virus type 1 Gag dimerization-induced assembly

The nucleocapsid (NC) domains of retrovirus precursor Gag (PrGag) proteins play an essential role in virus assembly. Evidence suggests that NC binding to viral RNA promotes dimerization of PrGag capsid (CA) domains, which triggers assembly of CA N-terminal domains (NTDs) into hexamer rings that are interconnected by CA C-terminal domains. To examine the influence of dimerization on human immunodeficiency virus type 1 (HIV-1) Gag protein assembly in vitro, we analyzed the assembly properties of Gag proteins in which NC domains were replaced with cysteine residues that could be linked via chemical treatment. In accordance with the model that Gag protein pairing triggers assembly, we found that cysteine cross-linking or oxidation reagents induced the assembly of virus-like particles. However, efficient assembly also was observed to be temperature dependent or required the tethering of NTDs. Our results suggest a multistep pathway for HIV-1 Gag protein assembly. In the first step, Gag protein pairing through NC-RNA interactions or C-terminal cysteine linkage fosters dimerization. Next, a conformational change converts assembly-restricted dimers or small oligomers into assembly-competent ones. At the final stage, final particle assembly occurs, possibly through a set of larger intermediates.     

Assembly models for Papovaviridae based on tiling theory

A vital constituent of a virus is its protein shell, called the viral capsid, that encapsulates and hence provides protection for the viral genome. Assembly models are developed for viral capsids built from protein building blocks that can assume different local bonding structures in the capsid. This situation occurs, for example, for viruses in the family of Papovaviridae, which are linked to cancer and are hence of particular interest for the health sector. More specifically, the viral capsids of the (pseudo-) T = 7 particles in this family consist of pentamers that exhibit two different types of bonding structures. While this scenario cannot be described mathematically in terms of Caspar-Klug theory (Caspar D L D and Klug A 1962 Cold Spring Harbor Symp. Quant. Biol. 27 1), it can be modelled via tiling theory (Twarock R 2004 J. Theor. Biol. 226 477). The latter is used to encode the local bonding environment of the building blocks in a combinatorial structure, called the assembly tree, which is a basic ingredient in the derivation of assembly models for Papovaviridae along the lines of the equilibrium approach of Zlotnick (Zlotnick A 1994 J. Mol. Biol. 241 59). A phase space formalism is introduced to characterize the changes in the assembly pathways and intermediates triggered by the variations in the association energies characterizing the bonds between the building blocks in the capsid. Furthermore, the assembly pathways and concentrations of the statistically dominant assembly intermediates are determined. The example of Simian virus 40 is discussed in detail.     

Zinc ions trigger conformational change and oligomerization of hepatitis B virus capsid protein

Assembly of virus particles in infected cells is likely to be a tightly regulated process. Previously, we found that in vitro assembly of hepatitis B virus (HBV) capsid protein is highly dependent on protein and NaCl concentration. Here we show that micromolar concentrations of Zn2+ are sufficient to initiate assembly of capsid protein, whereas other mono- and divalent cations elicited assembly only at millimolar concentrations, similar to those required for NaCl-induced assembly. Altered intrinsic protein fluorescence and highly cooperative binding of at least four Zn2+ ions (KD approximately 7 microM) indicated that binding induced a conformational change in capsid protein. At 37 degrees C, Zn2+ enhanced the initial rate of assembly and produced normal capsids, but it did not alter the extent of assembly at equilibrium. Assembly mediated by high zinc concentrations (> or =300 microM) yielded few capsids but produced a population of oligomers recognized by capsid-specific antibodies, suggesting a kinetically trapped assembly reaction. Comparison of kinetic simulations to in vitro assembly reactions leads us to suggest that kinetic trapping was due to the enhancement of the nucleation rate relative to the elongation rate. Zinc-induced HBV assembly has hallmarks of an allosterically regulated process: ligand binding at one site influences binding at other sites (cooperativity) indicating that binding is associated with conformational change, and binding of ligand alters the biological activity of assembly. We conclude that zinc binding enhances the kinetics of assembly by promoting formation of an intermediate that is readily consumed in the reaction. Free zinc ions may not be the true in vivo activator of assembly, but they provide a model for regulation of assembly.     

Microtubule nucleation from stable tubulin oligomers

Microtubule assembly from purified tubulin preparations involves both microtubule nucleation and elongation. Whereas elongation is well documented, microtubule nucleation remains poorly understood because of difficulties in isolating molecular intermediates between tubulin dimers and microtubules. Based on kinetic studies, we have previously proposed that the basic building blocks of microtubule nuclei are persistent tubulin oligomers, present at the onset of tubulin assembly. Here we have tested this model directly by isolating nucleation-competent cross-linked tubulin oligomers. We show that such oligomers are composed of 10-15 laterally associated tubulin dimers. In the presence of added free tubulin dimers, several oligomers combine to form microtubule nuclei competent for elongation. We provide evidence that these nuclei have heterogeneous structures, indicating unexpected flexibility in nucleation pathways. Our results suggest that microtubule nucleation in purified tubulin solution is mechanistically similar to that templated by gamma-tubulin ring complexes with the exception that in the absence of gamma-tubulin complexes the production of productive microtubule seeds from tubulin oligomers involves trial and error and a selection process.     

Assembly Pathway of Hepatitis B Core Virus-like Particles from Genetically Fused Dimers

Macromolecular complexes are responsible for many key biological processes. However, in most cases details of the assembly/disassembly of such complexes are unknown at the molecular level, as the low abundance and transient nature of assembly intermediates make analysis challenging. The assembly of virus capsids is an example of such a process. The hepatitis B virus capsid (core) can be composed of either 90 or 120 dimers of coat protein. Previous studies have proposed a trimer of dimers as an important intermediate species in assembly, acting to nucleate further assembly by dimer addition. Using novel genetically-fused coat protein dimers, we have been able to trap higher-order assembly intermediates and to demonstrate for the first time that both dimeric and trimeric complexes are on pathway to virus-like particle (capsid) formation.     

Multiscale simulation unravel the kinetic mechanisms of inflammasome assembly

In the innate immune system, the host defense from the invasion of external pathogens triggers the inflammatory responses. Proteins involved in the inflammatory pathways were often found to aggregate into supramolecular oligomers, called ‘inflammasome’, mostly through the homotypic interaction between their domains that belong to the death domain superfamily. Although much has been known about the formation of these helical molecular machineries, the detailed correlation between the dynamics of their assembly and the structure of each domain is still not well understood. Using the filament formed by the PYD domains of adaptor molecule ASC as a test system, we constructed a new multiscale simulation framework to study the kinetics of inflammasome assembly. We found that the filament assembly is a multi-step, but highly cooperative process. Moreover, there are three types of binding interfaces between domain subunits in the ASC^PYD filament. The multiscale simulation results suggest that dynamics of domain assembly are rooted in the primary protein sequence which defines the energetics of molecular recognition through three binding interfaces. Interface I plays a more regulatory role than the other two in mediating both the kinetics and the thermodynamics of assembly. Finally, the efficiency of our computational framework allows us to design mutants on a systematic scale and predict their impacts on filament assembly. In summary, this is, to the best of our knowledge, the first simulation method to model the spatial-temporal process of inflammasome assembly. Our work is a useful addition to a suite of existing experimental techniques to study the functions of inflammasome in innate immune system.