Modelling the self-assembly of virus capsids

Effect of ionic strength on the assembly of simian vacuolating virus capsid protein around poly(styrene sulfonate)

Virus-like particles (VLPs) are noninfectious nanocapsules that can be used for drug delivery or vaccine applications. VLPs can be assembled from virus capsid proteins around a condensing agent, such as RNA, DNA, or a charged polymer. Electrostatic interactions play an important role in the assembly reaction. VLPs assemble from many copies of capsid protein, with a combinatorial number of intermediates. Hence, the mechanism of the reaction is poorly understood. In this paper, we combined solution small-angle X-ray scattering (SAXS), cryo-transmission electron microscopy (TEM), and computational modeling to determine the effect of ionic strength on the assembly of Simian Vacuolating Virus 40 (SV40)-like particles. We mixed poly(styrene sulfonate) with SV40 capsid protein pentamers at different ionic strengths. We then characterized the assembly product by SAXS and cryo-TEM. To analyze the data, we performed Langevin dynamics simulations using a coarse-grained model that revealed incomplete, asymmetric VLP structures consistent with the experimental data. We found that close to physiological ionic strength, [Formula: see text] VLPs coexisted with VP1 pentamers. At lower or higher ionic strengths, incomplete particles coexisted with pentamers and [Formula: see text] particles. Including the simulated structures was essential to explain the SAXS data in a manner that is consistent with the cryo-TEM images.

Designing the Self-Assembly of Arbitrary Shapes Using Minimal Complexity Building Blocks

The design space for self-assembled multicomponent objects ranges from a solution in which every building block is unique to one with the minimum number of distinct building blocks that unambiguously define the target structure. We develop a pipeline to explore the design spaces for a set of structures of various sizes and complexities. To understand the implications of the different solutions, we analyze their assembly dynamics using patchy particle simulations and study the influence of the number of distinct building blocks, and the angular and spatial tolerances on their interactions, on the kinetics and yield of the target assembly. We show that the resource-saving solution with a minimum number of distinct blocks can often assemble just as well (or faster) than designs where each building block is unique. We further use our methods to design multifarious structures, where building blocks are shared between different target structures. Finally, we use coarse-grained DNA simulations to investigate the realization of multicomponent shapes using DNA nanostructures as building blocks.

Hierarchical Coarse-Grained Strategy for Macromolecular Self-Assembly: Application to Hepatitis B Virus-Like Particles

Macromolecular self-assembly is at the basis of many phenomena in material and life sciences that find diverse applications in technology. One example is the formation of virus-like particles (VLPs) that act as stable empty capsids used for drug delivery or vaccine fabrication. Similarly to the capsid of a virus, VLPs are protein assemblies, but their structural formation, stability, and properties are not fully understood, especially as a function of the protein modifications. In this work, we present a data-driven modeling approach for capturing macromolecular self-assembly on scales beyond traditional molecular dynamics (MD), while preserving the chemical specificity. Each macromolecule is abstracted as an anisotropic object and high-dimensional models are formulated to describe interactions between molecules and with the solvent. For this, data-driven protein-protein interaction potentials are derived using a Kriging-based strategy, built on high-throughput MD simulations. Semi-automatic supervised learning is employed in a high performance computing environment and the resulting specialized force-fields enable a significant speed-up to the micrometer and millisecond scale, while maintaining high intermolecular detail. The reported generic framework is applied for the first time to capture the formation of hepatitis B VLPs from the smallest building unit, i.e., the dimer of the core protein HBcAg. Assembly pathways and kinetics are analyzed and compared to the available experimental observations. We demonstrate that VLP self-assembly phenomena and dependencies are now possible to be simulated. The method developed can be used for the parameterization of other macromolecules, enabling a molecular understanding of processes impossible to be attained with other theoretical models.

Microcompartment assembly around multicomponent fluid cargoes

This article describes dynamical simulations of the assembly of an icosahedral protein shell around a bicomponent fluid cargo. Our simulations are motivated by bacterial microcompartments, which are protein shells found in bacteria that assemble around a complex of enzymes and other components involved in certain metabolic processes. The simulations demonstrate that the relative interaction strengths among the different cargo species play a key role in determining the amount of each species that is encapsulated, their spatial organization, and the nature of the shell assembly pathways. However, the shell protein–shell protein and shell protein–cargo component interactions that help drive assembly and encapsulation also influence cargo composition within certain parameter regimes. These behaviors are governed by a combination of thermodynamic and kinetic effects. In addition to elucidating how natural microcompartments encapsulate multiple components involved within reaction cascades, these results have implications for efforts in synthetic biology to colocalize alternative sets of molecules within microcompartments to accelerate specific reactions. More broadly, the results suggest that coupling between self-assembly and multicomponent liquid–liquid phase separation may play a role in the organization of the cellular cytoplasm.

Contact-Based Analysis of Aggregation of Intrinsically Disordered Proteins

We review the contact-based description of aggregation of intrinsically disordered proteins in coarse-grained and all-atom models. We consider polyglutamines and polyalanines at various concentrations of the peptides. We also study associations of two chains of α-synuclein and up to 20 chains of a 12-residue-long segment of protein tau. We demonstrate that the total number of two-chain association events (in an aggregate that comprises at least two chains) provides a useful measure of the propensity to aggregate. This measure is consistent, for instance, with the previously reported mass spectroscopy data. The distribution of the number of association events is given essentially by a power law as a function of the duration of these events. The corresponding exponent depends on the protein and the temperature but not on the concentration of the proteins.

Derangement model of ligand-receptor binding

We introduce a derangement model of ligand-receptor binding that allows us to quantitatively frame the question “How can ligands seek out and bind to their optimal receptor sites in a sea of other competing ligands and suboptimal receptor sites?” To answer the question, we first derive a formula to count the number of partial generalized derangements in a list, thus extending the derangement result of Gillis and Even. We then compute the general partition function for the ligand-receptor system and derive the equilibrium expressions for the average number of bound ligands and the average number of optimally bound ligands. A visual model of squares assembling onto a grid allows us to easily identify fully optimal bound states. Equilibrium simulations of the system reveal its extremes to be one of two types, qualitatively distinguished by whether optimal ligand-receptor binding is the dominant form of binding at all temperatures and quantitatively distinguished by the relative values of two critical temperatures. One of those system types (termed “search-limited,” as it was in previous work) does not exhibit kinetic traps and we thus infer that biomolecular systems where optimal ligand-receptor binding is functionally important are likely to be search-limited.

Cargo Release from Nonenveloped Viruses and Virus-like Nanoparticles: Capsid Rupture or Pore Formation

Virus-like nanoparticles are protein shells similar to wild-type viruses, and both aim to deliver their content into a cell. Unfortunately, the release mechanism of their cargo/genome remains elusive. Pores on the symmetry axes were proposed to enable the slow release of the viral genome. In contrast, cryo-EM images showed that capsids of nonenveloped RNA viruses can crack open and rapidly release the genome. We combined in vitro cryo-EM observations of the genome release of three viruses with coarse-grained simulations of generic virus-like nanoparticles to investigate the cargo/genome release pathways. Simulations provided details on both slow and rapid release pathways, including the success rates of individual releases. Moreover, the simulated structures from the rapid release pathway were in agreement with the experiment. Slow release occurred when interactions between capsid subunits were long-ranged, and the cargo/genome was noncompact. In contrast, rapid release was preferred when the interaction range was short and/or the cargo/genome was compact. These findings indicate a design strategy of virus-like nanoparticles for drug delivery.

Viral structural proteins as targets for antivirals

Viral structural proteins are emerging as effective targets for new antivirals. In a viral lifecycle, the capsid must assemble, disassemble, and respond to host proteins, all at the right time and place. These reactions work within a narrow range of conditions, making them susceptible to small molecule interference. In at least three specific viruses, this approach has had met with preliminary success. In rhinovirus and poliovirus, compounds like pleconaril bind capsid and block RNA release. Bevirimat binds to Gag protein in HIV, inhibiting maturation. In Hepatitis B virus, core protein allosteric modulators (CpAMs) promote spontaneous assembly of capsid protein leading to empty and aberrant particles. Despite the biological diversity between viruses and the chemical diversity between antiviral molecules, we observe common features in these antivirals' mechanisms of action. These approaches work by stabilizing protein-protein interactions.

Assembly and Stability of Simian Virus 40 Polymorphs

Understanding viral assembly pathways is of critical importance to biology, medicine, and nanotechology. Here, we study the assembly path of a system with various structures, the simian vacuolating virus 40 (SV40) polymorphs. We simulate the templated assembly process of VP1 pentamers, which are the constituents of SV40, into icosahedal shells made of N = 12 pentamers (T = 1). The simulations include connections formed between pentamers by C-terminal flexible lateral units, termed here "C-terminal ligands", which are shown to control assembly behavior and shell dynamics. The model also incorporates electrostatic attractions between the N-terminal peptide strands (ligands) and the negatively charged cargo, allowing for agreement with experiments of RNA templated assembly at various pH and ionic conditions. During viral assembly, pentamers bound to any template increase its effective size due to the length and flexibility of the C-terminal ligands, which can connect to other VP1 pentamers and recruit them to a partially completed capsid. All closed shells formed other than the T = 1 feature the ability to dynamically rearrange and are thus termed "pseudo-closed". The N = 13 shell can even spontaneously "self-correct" by losing a pentamer and become a T = 1 capsid when the template size fluctuates. Bound pentamers recruiting additional pentamers to dynamically rearranging capsids allow closed shells to continue growing via the pseudo-closed growth mechanism, for which experimental evidence already exists. Overall, we show that the C-terminal ligands control the dynamic assembly paths of SV40 polymorphs.

Shape selection and mis-assembly in viral capsid formation by elastic frustration

The successful assembly of a closed protein shell (or capsid) is a key step in the replication of viruses and in the production of artificial viral cages for bio/nanotechnological applications. During self-assembly, the favorable binding energy competes with the energetic cost of the growing edge and the elastic stresses generated due to the curvature of the capsid. As a result, incomplete structures such as open caps, cylindrical or ribbon-shaped shells may emerge, preventing the successful replication of viruses. Using elasticity theory and coarse-grained simulations, we analyze the conditions required for these processes to occur and their significance for empty virus self-assembly. We find that the outcome of the assembly can be recast into a universal phase diagram showing that viruses with high mechanical resistance cannot be self-assembled directly as spherical structures. The results of our study justify the need of a maturation step and suggest promising routes to hinder viral infections by inducing mis-assembly.

A Multiscale Model for the Self-Assembly of Coat Proteins in Bacteriophage MS2

The self-assembly of viral capsids is an essential step to the formation of infectious viruses. Elucidating the kinetic mechanisms of how a capsid or virus-like particle assembles could advance our knowledge about the viral lifecycle, as well as the general principles in self-assembly of biomaterials. However, current understanding of capsid assembly remains incomplete for many viruses due to the fact that the transient intermediates along the assembling pathways are experimentally difficult to be detected. In this paper, we constructed a new multiscale computational framework to simulate the self-assembly of virus-like particles. We applied our method to the coat proteins of bacteriophage MS2 as a specific model system. This virus-like particle of bacteriophage MS2 has a unique feature that its 90 sequence-identical dimers can be classified into two structurally various groups: one is the symmetric CC dimer, and the other is the asymmetric AB dimer. The homotypic interactions between AB dimers result in a 5-fold symmetric contact, while the heterotypic interactions between AB and CC dimers result in 6-fold symmetric contact. We found that the assembly can be described as a physical process of phase transition that is regulated by various factors such as concentration and specific stoichiometry between AB and CC dimers. Our simulations also demonstrate that heterotypic and homotypic interfaces play distinctive roles in modulating the assembling kinetics. The interaction between AB and CC dimers is much more dynamic than that between two AB dimers. We therefore suggest that the alternate growth of viral capsid through the heterotypic dimer interactions dominates the assembling pathways. This is, to the best of our knowledge, the first multiscale model to simulate the assembling process of coat proteins in bacteriophage MS2. The generality of this approach opens the door to its further applications in assembly of other viral capsids, virus-like particles, and novel drug delivery systems.

Self-assembly of a dimer system

In the self-assembly process which drives the formation of cellular membranes, micelles, and capsids, a collection of separated subunits spontaneously binds together to form functional and more ordered structures. In this work, we study the statistical physics of self-assembly in a simpler scenario: the formation of dimers from a system of monomers. The properties of the model allow us to frame the microstate counting as a combinatorial problem whose solution leads to an exact partition function. From the associated equilibrium conditions, we find that such dimer systems come in two types: "search-limited" and "combinatorics-limited," only the former of which has states where partial assembly can be dominated by correct contacts. Using estimates of biophysical quantities in systems of single-stranded DNA dimerization, transcription factor and DNA interactions, and protein-protein interactions, we find that all of these systems appear to be of the search-limited type, i.e., their fully correct dimerization regimes are more limited by the ability of monomers to find one another in the constituent volume than by the combinatorial disadvantage of correct dimers. We derive the parameter requirements for fully correct dimerization and find that rather than the ratio of particle number and volume (i.e., number density) being the relevant quantity, it is the product of particle diversity and volume that is constrained. Ultimately, this work contributes to an understanding of self-assembly by using the simple case of a system of dimers to analytically study the combinatorics of assembly.

A New Model System for Exploring Assembly Mechanisms of the HIV-1 Immature Capsid In Vivo

The assembly of the HIV-1 immature capsid (HIC) is an essential step in the virus life cycle. In vivo, the HIC is composed of [Formula: see text] hexameric building blocks, and it takes 5-6 min to complete the assembly process. The involvement of numerous building blocks and the rapid timecourse makes it difficult to understand the HIC assembly process. In this work, we study HIC assembly in vivo by using differential equations. We first obtain a full model with 420 differential equations. Then, we reduce six addition reactions for separate building blocks to a single complex reaction. This strategy reduces the full model to 70 equations. Subsequently, the theoretical analysis of the reduced model shows that it might not be an effective way to decrease the HIC concentration at the equilibrium state by decreasing the microscopic on-rate constants. Based on experimental data, we estimate that the nucleating structure is much smaller than the HIC. We also estimate that the microscopic on-rate constant for nucleation reactions is far less than that for elongation reactions. The parametric collinearity investigation testifies the reliability of these two characteristics, which might explain why free building blocks do not readily polymerize into higher-order polymers until their concentration reaches a threshold value. These results can provide further insight into the assembly mechanisms of the HIC in vivo.

A kinetics study of ligand substitution reaction on dinuclear platinum complexes: Stochastic versus deterministic approach

The kinetics on a basic ligand substitution reaction on dinuclear platinum complexes [Pt(PEt₃ )₂ PhPt(PEt₃ )₂ ]²⁺ and [Pt(PEt₃ )₂ PhCOPhPt(PEt₃ )₂ ]²⁺ , with the ligands pyridine and 3-chloropyridine, is studied. This is a fundamental step in a self-assembly, and the time evolution has been observed with a new experimental technique, QASAP (quantitative analysis of self-assembly process), which is recently developed by Hiraoka's group. As a result of numerical calculations based on master equation, we succeed in specifying the reaction rate constants with a simple reaction model. In addition, the time evolutions of all the intermediate components produced and consumed in chemical reaction are revealed, including those unobserved in the experiments. The convergence behavior of the existence ratios of specific chemical species calculated with the stochastic algorithm method is compared with those obtained from deterministic formalism based on rate equations, revealing a clear dependence on the number of constituent molecules. © 2018 Wiley Periodicals, Inc.

Modeling the Self-Assembly of Protein Complexes through a Rigid-Body Rotational Reaction-Diffusion Algorithm

The reaction-diffusion equations provide a powerful framework for modeling nonequilibrium, cell-scale dynamics over the long time scales that are inaccessible by traditional molecular modeling approaches. Single-particle reaction-diffusion offers the highest resolution technique for tracking such dynamics, but it has not been applied to the study of protein self-assembly due to its treatment of reactive species as single-point particles. Here, we develop a relatively simple but accurate approach for building rigid structure and rotation into single-particle reaction-diffusion methods, providing a rate-based method for studying protein self-assembly. Our simplifying assumption is that reactive collisions can be evaluated purely on the basis of the separations between the sites, and not their orientations. The challenge of evaluating reaction probabilities can then be performed using well-known equations based on translational diffusion in both 3D and 2D, by employing an effective diffusion constant we derive here. We show how our approach reproduces both the kinetics of association, which is altered by rotational diffusion, and the equilibrium of reversible association, which is not. Importantly, the macroscopic kinetics of association can be predicted on the basis of the microscopic parameters of our structurally resolved model, allowing for critical comparisons with theory and other rate-based simulations. We demonstrate this method for efficient, rate-based simulations of self-assembly of clathrin trimers, highlighting how formation of regular lattices impacts the kinetics of association.

The role of the encapsulated cargo in microcompartment assembly

Bacterial microcompartments are large, roughly icosahedral shells that assemble around enzymes and reactants involved in certain metabolic pathways in bacteria. Motivated by microcompartment assembly, we use coarse-grained computational and theoretical modeling to study the factors that control the size and morphology of a protein shell assembling around hundreds to thousands of molecules. We perform dynamical simulations of shell assembly in the presence and absence of cargo over a range of interaction strengths, subunit and cargo stoichiometries, and the shell spontaneous curvature. Depending on these parameters, we find that the presence of a cargo can either increase or decrease the size of a shell relative to its intrinsic spontaneous curvature, as seen in recent experiments. These features are controlled by a balance of kinetic and thermodynamic effects, and the shell size is assembly pathway dependent. We discuss implications of these results for synthetic biology efforts to target new enzymes to microcompartment interiors.

Multiple Pathways in Capsid Assembly

For a three-dimensional structure to spontaneously self-assemble from many identical components, the steps on the pathway must be kinetically accessible. Many virus capsids are icosahedral and assembled from hundreds of identical proteins, but how they navigate the assembly process is poorly understood. Capsid assembly is thought to involve stepwise addition of subunits to a growing capsid fragment. Coarse-grained models suggest that the reaction occurs on a downhill energy landscape, so intermediates are expected to be fleeting. In this work, charge detection mass spectrometry (CDMS) has been used to track assembly of the hepatitis B virus (HBV) capsid in real time. The icosahedral T = 4 capsid of HBV is assembled from 120 capsid protein dimers. Our results indicate that there are multiple pathways for assembly. Under conditions that favor a modest association energy there is no accumulation of large intermediates, which indicates that available pathways include ones on a downhill energy surface. Under higher salt conditions, where subunit interactions are strengthened, around half of the products of the initial assembly reaction have masses close to the T = 4 capsid and the other half are stalled intermediates which emerge abruptly at around 90 dimers, indicating a bifurcation in the ensemble of assembly paths. When incubated at room temperature, the 90-dimer intermediates accumulate dimers and gradually shift to higher mass and merge with the capsid peak. Though free subunits are present in solution, the stalled intermediates indicate the presence of a local minima on the energy landscape. Some intermediates may result from hole closure, where the growing capsid distorts to close the hole due to the missing capsid proteins or from a species where subsequent additions are particularly labile.

Molecular dynamics study of T = 3 capsid assembly

Molecular dynamics simulation is used to model the self-assembly of polyhedral shells containing 180 trapezoidal particles that correspond to the T = 3 virus capsid. Three kinds of particle, differing only slightly in shape, are used to account for the effect of quasi-equivalence. Bond formation between particles is reversible and an explicit atomistic solvent is included. Under suitable conditions the simulations are able to produce complete shells, with the majority of unused particles remaining as monomers, and practically no other clusters. There are also no incorrectly assembled clusters. The simulations reveal details of intermediate structures along the growth pathway, information that is relevant for interpreting experiment.

Single-molecule techniques in biophysics: a review of the progress in methods and applications

Single-molecule biophysics has transformed our understanding of biology, but also of the physics of life. More exotic than simple soft matter, biomatter lives far from thermal equilibrium, covering multiple lengths from the nanoscale of single molecules to up to several orders of magnitude higher in cells, tissues and organisms. Biomolecules are often characterized by underlying instability: multiple metastable free energy states exist, separated by levels of just a few multiples of the thermal energy scale k _B T, where k _B is the Boltzmann constant and T absolute temperature, implying complex inter-conversion kinetics in the relatively hot, wet environment of active biological matter. A key benefit of single-molecule biophysics techniques is their ability to probe heterogeneity of free energy states across a molecular population, too challenging in general for conventional ensemble average approaches. Parallel developments in experimental and computational techniques have catalysed the birth of multiplexed, correlative techniques to tackle previously intractable biological questions. Experimentally, progress has been driven by improvements in sensitivity and speed of detectors, and the stability and efficiency of light sources, probes and microfluidics. We discuss the motivation and requirements for these recent experiments, including the underpinning mathematics. These methods are broadly divided into tools which detect molecules and those which manipulate them. For the former we discuss the progress of super-resolution microscopy, transformative for addressing many longstanding questions in the life sciences, and for the latter we include progress in 'force spectroscopy' techniques that mechanically perturb molecules. We also consider in silico progress of single-molecule computational physics, and how simulation and experimentation may be drawn together to give a more complete understanding. Increasingly, combinatorial techniques are now used, including correlative atomic force microscopy and fluorescence imaging, to probe questions closer to native physiological behaviour. We identify the trade-offs, limitations and applications of these techniques, and discuss exciting new directions.

The importance of thermodynamics for molecular systems, and the importance of molecular systems for thermodynamics

Improved understanding of molecular systems has only emphasised the sophistication of networks within the cell. Simultaneously, the advance of nucleic acid nanotechnology, a platform within which reactions can be exquisitely controlled, has made the development of artificial architectures and devices possible. Vital to this progress has been a solid foundation in the thermodynamics of molecular systems. In this pedagogical review and perspective, we discuss how thermodynamics determines both the overall potential of molecular networks, and the minute details of design. We then argue that, in turn, the need to understand molecular systems is helping to drive the development of theories of thermodynamics at the microscopic scale.

Self-assembly of model proteins into virus capsids

We consider self-assembly of proteins into a virus capsid by the methods of molecular dynamics. The capsid corresponds either to SPMV or CCMV and is studied with and without the RNA molecule inside. The proteins are flexible and described by the structure-based coarse-grained model augmented by electrostatic interactions. Previous studies of the capsid self-assembly involved solid objects of a supramolecular scale, e.g. corresponding to capsomeres, with engineered couplings and stochastic movements. In our approach, a single capsid is dissociated by an application of a high temperature for a variable period and then the system is cooled down to allow for self-assembly. The restoration of the capsid proceeds to various extent, depending on the nature of the dissociated state, but is rarely complete because some proteins depart too far unless the process takes place in a confined space.

Kinetic constraints on self-assembly into closed supramolecular structures

Many biological and synthetic systems exploit self-assembly to generate highly intricate closed supramolecular architectures, ranging from self-assembling cages to viral capsids. The fundamental design principles that control the structural determinants of the resulting assemblies are increasingly well-understood, but much less is known about the kinetics of such assembly phenomena and it remains a key challenge to elucidate how these systems can be engineered to assemble in an efficient manner and avoid kinetic trapping. We show here that simple scaling laws emerge from a set of kinetic equations describing the self-assembly of identical building blocks into closed supramolecular structures and that this scaling behavior provides general rules that determine efficient assembly in these systems. Using this framework, we uncover the existence of a narrow range of parameter space that supports efficient self-assembly and reveal that nature capitalizes on this behavior to direct the reliable assembly of viral capsids on biologically relevant timescales.

Why patchy diffusion-limited aggregation belongs to the directed-percolation universality class

We present a possible link between nonequilibrium phase transition observed in patchy diffusion-limited aggregation (DLA) [M. J. Kartha and A. Sayeed, Phys. Lett. A 380, 2791 (2016)10.1016/j.physleta.2016.06.036] and directed bond percolation (DP). A system of directed percolation with patchy particles (patchy DP) in which the bond connectivity is established depending on patch size p is analyzed. It is observed that patchy DP starting from a single seed shows a nonequilibrium phase transition. Below a critical value of the patch size p_{c}, the system reaches an absorbing state above which is a fluctuating active state as observed in the DP system. The value of this nonuniversal parameter p_{c} is observed to be slightly higher than the value observed in patchy DLA. Close to the critical value, the order parameter P(∞)∼(p-p_{c})^{β} where β=0.272±0.010, which is consistent with the directed-percolation universality class. Therefore the intrinsic nature of patchy DP is responsible for the phase transition in patchy DLA. This study reveals that the estimated critical value of patch size p_{c}=0.80625±0.00020 in patchy DP is different from the critical bond probability p_{c}=0.6447 in the DP system. This elucidates that the bond probability in DP is not equivalent to the patch probability of a particular site. Our work also gives an insight into the problem related with formation of an extended network of pentagon subunits in connection with the virus capsid.

Modeling Effects of RNA on Capsid Assembly Pathways via Coarse-Grained Stochastic Simulation

The environment of a living cell is vastly different from that of an in vitro reaction system, an issue that presents great challenges to the use of in vitro models, or computer simulations based on them, for understanding biochemistry in vivo. Virus capsids make an excellent model system for such questions because they typically have few distinct components, making them amenable to in vitro and modeling studies, yet their assembly can involve complex networks of possible reactions that cannot be resolved in detail by any current experimental technology. We previously fit kinetic simulation parameters to bulk in vitro assembly data to yield a close match between simulated and real data, and then used the simulations to study features of assembly that cannot be monitored experimentally. The present work seeks to project how assembly in these simulations fit to in vitro data would be altered by computationally adding features of the cellular environment to the system, specifically the presence of nucleic acid about which many capsids assemble. The major challenge of such work is computational: simulating fine-scale assembly pathways on the scale and in the parameter domains of real viruses is far too computationally costly to allow for explicit models of nucleic acid interaction. We bypass that limitation by applying analytical models of nucleic acid effects to adjust kinetic rate parameters learned from in vitro data to see how these adjustments, singly or in combination, might affect fine-scale assembly progress. The resulting simulations exhibit surprising behavioral complexity, with distinct effects often acting synergistically to drive efficient assembly and alter pathways relative to the in vitro model. The work demonstrates how computer simulations can help us understand how assembly might differ between the in vitro and in vivo environments and what features of the cellular environment account for these differences.

Coarse-grained simulation reveals key features of HIV-1 capsid self-assembly

The maturation of HIV-1 viral particles is essential for viral infectivity. During maturation, many copies of the capsid protein (CA) self-assemble into a capsid shell to enclose the viral RNA. The mechanistic details of the initiation and early stages of capsid assembly remain to be delineated. We present coarse-grained simulations of capsid assembly under various conditions, considering not only capsid lattice self-assembly but also the potential disassembly of capsid upon delivery to the cytoplasm of a target cell. The effects of CA concentration, molecular crowding, and the conformational variability of CA are described, with results indicating that capsid nucleation and growth is a multi-stage process requiring well-defined metastable intermediates. Generation of the mature capsid lattice is sensitive to local conditions, with relatively subtle changes in CA concentration and molecular crowding influencing self-assembly and the ensemble of structural morphologies.

Significant morphological changes occur during the conversion of the immature HIV virion into a mature infectious form. Here the authors use coarse-grained molecular dynamics simulations to model HIV-1 capsid self-assembly and disassembly events that suggests several metastable capsid intermediates sensitive to local conditions.

Many-molecule encapsulation by an icosahedral shell

We computationally study how an icosahedral shell assembles around hundreds of molecules. Such a process occurs during the formation of the carboxysome, a bacterial microcompartment that assembles around many copies of the enzymes ribulose 1,5-bisphosphate carboxylase/ oxygenase and carbonic anhydrase to facilitate carbon fixation in cyanobacteria. Our simulations identify two classes of assembly pathways leading to encapsulation of many-molecule cargoes. In one, shell assembly proceeds concomitantly with cargo condensation. In the other, the cargo first forms a dense globule; then, shell proteins assemble around and bud from the condensed cargo complex. Although the model is simplified, the simulations predict intermediates and closure mechanisms not accessible in experiments, and show how assembly can be tuned between these two pathways by modulating protein interactions. In addition to elucidating assembly pathways and critical control parameters for microcompartment assembly, our results may guide the reengineering of viruses as nanoreactors that self-assemble around their reactants.

DOI:http://dx.doi.org/10.7554/eLife.14078.001

Bacterial microcompartments are protein shells that are found inside bacteria and enclose enzymes and other chemicals required for certain biological reactions. For example, the carboxysome is a type of microcompartment that enables the bacteria to convert the products of photosynthesis into sugars. During the formation of a microcompartment, the outer protein shell assembles around hundreds of enzymes and chemicals. This formation process is tightly controlled and involves multiple interactions between the shell proteins and the cargo – the enzymes and other reaction ingredients – they will enclose. Understanding how to control which enzymes are encapsulated within microcompartments could help researchers to re-engineer the microcompartments so that they contain drugs or other useful products.

Recent studies have used microscopy to visualize how microcompartments are assembled. However, most of the intermediate structures that form during assembly are too small and short-lived to be seen. It has therefore not been possible to explore in detail how shell proteins collect the necessary cargo and then assemble into an ordered shell with the cargo on the inside. Experiments alone are probably not enough to understand the process, especially since microcompartment assembly can currently only be studied within live cells or cellular extract. Within these complex environments it is difficult to determine the effect of any individual factor on the overall assembly process.

Perlmutter, Mohajerani and Hagan have now taken a different approach by developing computational and theoretical models to explore how microcompartments assemble. Computer simulations showed that microcompartments could assemble by two pathways. In one pathway, the protein shell and cargo coalesce at the same time. In the other pathway, the cargo molecules first assemble into a large disordered complex, with the shell proteins attached on the outside. The shell proteins then assemble, carving out a piece of the cargo complex. The simulations showed that many factors affect how the shell assembles, such as the strengths of the interactions between the shell proteins and the cargo. They also identified a factor that controls how much cargo ends up inside the assembled shell.

Perlmutter, Mohajerani and Hagan found that, in addition to revealing how microcompartments may assemble within their natural setting, the simulations provided guidance on how to re-engineer microcompartments to assemble around other components. This would enable researchers to create customizable compartments that self-assemble within bacteria or other host organisms, for example to carry out carbon fixation or make biofuels.

A future challenge will be to investigate other aspects of microcompartment assembly, such as the factors that control the size of these compartments.

DOI:http://dx.doi.org/10.7554/eLife.14078.002

Peptides as models for the structure and function of viral capsid proteins: Insights on dengue virus capsid

The structural organization of viral particles is among the most astonishing examples of molecular self-assembly in nature, involving proteins, nucleic acids, and, sometimes, lipids. Proper assembly is essential to produce well structured infectious virions. A great variety of structural arrangements can be found in viral particles. Nucleocapsids, for instance, may display highly ordered geometric shapes or consist in macroscopically amorphous packs of the viral genome. Alphavirus and flavivirus are viral genera that exemplify these extreme cases, the former comprising viral particles structured with a T = 4 icosahedral symmetry, whereas flavivirus capsids have no regular geometry. Dengue virus is a member of flavivirus genus and is used in this article to illustrate how viral protein-derived peptides can be used advantageously over full-length proteins to unravel the foundations of viral supramolecular assemblies. Membrane- and viral RNA-binding data of capsid protein-derived dengue virus peptides are used to explain the amorphous organization of the viral capsid. Our results combine bioinformatic and spectroscopic approaches using two- or three-component peptide and/or nucleic acid and/or lipid systems.

The Role of Packaging Sites in Efficient and Specific Virus Assembly

During the life cycle of many single-stranded RNA viruses, including many human pathogens, a protein shell called the capsid spontaneously assembles around the viral genome. Understanding the mechanisms by which capsid proteins selectively assemble around the viral RNA amidst diverse host RNAs is a key question in virology. In one proposed mechanism, short sequences (packaging sites) within the genomic RNA promote rapid and efficient assembly through specific interactions with the capsid proteins. In this work, we develop a coarse-grained particle-based computational model for capsid proteins and RNA that represents protein-RNA interactions arising both from nonspecific electrostatics and from specific packaging site interactions. Using Brownian dynamics simulations, we explore how the efficiency and specificity of assembly depend on solution conditions (which control protein-protein and nonspecific protein-RNA interactions) and the strength and number of packaging sites. We identify distinct regions in parameter space in which packaging sites lead to highly specific assembly via different mechanisms and others in which packaging sites lead to kinetic traps. We relate these computational predictions to in vitro assays for specificity in which cognate viral RNAs compete against non-cognate RNAs for assembly by capsid proteins.

Mechanisms of virus assembly

Viruses are nanoscale entities containing a nucleic acid genome encased in a protein shell called a capsid and in some cases are surrounded by a lipid bilayer membrane. This review summarizes the physics that govern the processes by which capsids assemble within their host cells and in vitro. We describe the thermodynamics and kinetics for the assembly of protein subunits into icosahedral capsid shells and how these are modified in cases in which the capsid assembles around a nucleic acid or on a lipid bilayer. We present experimental and theoretical techniques used to characterize capsid assembly, and we highlight aspects of virus assembly that are likely to receive significant attention in the near future.

Modeling Viral Capsid Assembly

I present a review of the theoretical and computational methodologies that have been used to model the assembly of viral capsids. I discuss the capabilities and limitations of approaches ranging from equilibrium continuum theories to molecular dynamics simulations, and I give an overview of some of the important conclusions about virus assembly that have resulted from these modeling efforts. Topics include the assembly of empty viral shells, assembly around single-stranded nucleic acids to form viral particles, and assembly around synthetic polymers or charged nanoparticles for nanotechnology or biomedical applications. I present some examples in which modeling efforts have promoted experimental breakthroughs, as well as directions in which the connection between modeling and experiment can be strengthened.

Simulations show that virus assembly and budding are facilitated by membrane microdomains

For many viruses, assembly and budding occur simultaneously during virion formation. Understanding the mechanisms underlying this process could promote biomedical efforts to block viral propagation and enable use of capsids in nanomaterials applications. To this end, we have performed molecular dynamics simulations on a coarse-grained model that describes virus assembly on a fluctuating lipid membrane. Our simulations show that the membrane can promote association of adsorbed subunits through dimensional reduction, but it also introduces thermodynamic and kinetic effects that can inhibit complete assembly. We find several mechanisms by which membrane microdomains, such as lipid rafts, reduce these effects, and thus, enhance assembly. We show how these predicted mechanisms can be experimentally tested. Furthermore, the simulations demonstrate that assembly and budding depend crucially on the system dynamics via multiple timescales related to membrane deformation, protein diffusion, association, and adsorption onto the membrane.

Molecular dynamics simulation: a tool for exploration and discovery using simple models

Emergent phenomena share the fascinating property of not being obvious consequences of the design of the system in which they appear. This characteristic is no less relevant when attempting to simulate such phenomena, given that the outcome is not always a foregone conclusion. The present survey focuses on several simple model systems that exhibit surprisingly rich emergent behavior, all studied by molecular dynamics (MD) simulation.The examples are taken from the disparate fields of fluid dynamics, granular matter and supramolecular self-assembly. In studies of fluids modeled at the detailed microscopic level using discrete particles, the simulations demonstrate that complex hydrodynamic phenomena in rotating and convecting fluids—the Taylor–Couette and Rayleigh–Bénard instabilities—cannot only be observed within the limited length and time scales accessible to MD, but even allow quantitative agreement to be achieved. Simulation of highly counter-intuitive segregation phenomena in granular mixtures, again using MD methods, but now augmented by forces producing damping and friction, leads to results that resemble experimentally observed axial and radial segregation in the case of a rotating cylinder and to a novel form of horizontal segregation in a vertically vibrated layer. Finally, when modeling self-assembly processes analogous to the formation of the polyhedral shells that package spherical viruses, simulation of suitably shaped particles reveals the ability to produce complete, error-free assembly and leads to the important general observation that reversible growth steps contribute to the high yield. While there are limitations to the MD approach, both computational and conceptual, the results offer a tantalizing hint of the kinds of phenomena that can be explored and what might be discovered when sufficient resources are brought to bear on a problem.

Applying molecular crowding models to simulations of virus capsid assembly in vitro

Virus capsid assembly has been widely studied as a biophysical system, both for its biological and medical significance and as an important model for complex self-assembly processes. No current technology can monitor assembly in detail and what information we have on assembly kinetics comes exclusively from in vitro studies. There are many differences between the intracellular environment and that of an in vitro assembly assay, however, that might be expected to alter assembly pathways. Here, we explore one specific feature characteristic of the intracellular environment and known to have large effects on macromolecular assembly processes: molecular crowding. We combine prior particle simulation methods for estimating crowding effects with coarse-grained stochastic models of capsid assembly, using the crowding models to adjust kinetics of capsid simulations to examine possible effects of crowding on assembly pathways. Simulations suggest a striking difference depending on whether or not a system uses nucleation-limited assembly, with crowding tending to promote off-pathway growth in a nonnucleation-limited model but often enhancing assembly efficiency at high crowding levels even while impeding it at lower crowding levels in a nucleation-limited model. These models may help us understand how complicated assembly systems may have evolved to function with high efficiency and fidelity in the densely crowded environment of the cell.

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.

Using Markov state models to study self-assembly

Markov state models (MSMs) have been demonstrated to be a powerful method for computationally studying intramolecular processes such as protein folding and macromolecular conformational changes. In this article, we present a new approach to construct MSMs that is applicable to modeling a broad class of multi-molecular assembly reactions. Distinct structures formed during assembly are distinguished by their undirected graphs, which are defined by strong subunit interactions. Spatial inhomogeneities of free subunits are accounted for using a recently developed Gaussian-based signature. Simplifications to this state identification are also investigated. The feasibility of this approach is demonstrated on two different coarse-grained models for virus self-assembly. We find good agreement between the dynamics predicted by the MSMs and long, unbiased simulations, and that the MSMs can reduce overall simulation time by orders of magnitude.

Combinatoric analysis of heterogeneous stochastic self-assembly

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

Icosahedral capsid formation by capsomers and short polyions

Kinetical and structural aspects of the capsomer-polyion co-assembly into icosahedral viruses have been simulated by molecular dynamics using a coarse-grained model comprising cationic capsomers and short anionic polyions. Conditions were found at which the presence of polyions of a minimum length was necessary for capsomer formation. The largest yield of correctly formed capsids was obtained at which the driving force for capsid formation was relatively weak. Relatively stronger driving forces, i.e., stronger capsomer-capsomer short-range attraction and∕or stronger electrostatic interaction, lead to larger fraction of kinetically trapped structures and aberrant capsids. The intermediate formation was investigated and different evolving scenarios were found by just varying the polyion length.

Viral genome structures are optimal for capsid assembly

Understanding how virus capsids assemble around their nucleic acid (NA) genomes could promote efforts to block viral propagation or to reengineer capsids for gene therapy applications. We develop a coarse-grained model of capsid proteins and NAs with which we investigate assembly dynamics and thermodynamics. In contrast to recent theoretical models, we find that capsids spontaneously ‘overcharge’; that is, the negative charge of the NA exceeds the positive charge on capsid. When applied to specific viruses, the optimal NA lengths closely correspond to the natural genome lengths. Calculations based on linear polyelectrolytes rather than base-paired NAs underpredict the optimal length, demonstrating the importance of NA structure to capsid assembly. These results suggest that electrostatics, excluded volume, and NA tertiary structure are sufficient to predict assembly thermodynamics and that the ability of viruses to selectively encapsidate their genomic NAs can be explained, at least in part, on a thermodynamic basis.

DOI:http://dx.doi.org/10.7554/eLife.00632.001

Viruses are infectious agents made up of proteins and a genome made of DNA or RNA. Upon infecting a host cell, viruses hijack the cell’s gene expression machinery and force it to produce copies of the viral genome and proteins, which then assemble into new viruses that can eventually infect other host cells. Because assembly is an essential step in the viral life cycle, understanding how this process occurs could significantly advance the fight against viral diseases.

In many viral families, a protein shell called a capsid forms around the viral genome during the assembly process. However, capsids can also assemble around nucleic acids in solution, indicating that a host cell is not required for their formation. Since capsid proteins are positively charged, and nucleic acids are negatively charged, electrostatic interactions between the two are thought to have an important role in capsid assembly. However, it is unclear how structural features of the viral genome affect assembly, and why the negative charge on viral genomes is actually far greater than the positive charge on capsids. These questions are difficult to address experimentally because most of the intermediates that form during virus assembly are too short-lived to be imaged.

Here, Perlmutter et al. have used state of the art computational methods and advances in graphical processing units (GPUs) to produce the most realistic model of capsid assembly to date. They showed that the stability of the complex formed between the nucleic acid and the capsid depends on the length of the viral genome. Yield was highest for genomes within a certain range of lengths, and capsids that assembled around longer or shorter genomes tended to be malformed.

Perlmutter et al. also explored how structural features of the virus—including base-pairing between viral nucleic acids, and the size and charge of the capsid—determine the optimal length of the viral genome. When they included structural data from real viruses in their simulations and predicted the optimal lengths for the viral genome, the results were very similar to those seen in existing viruses. This indicates that the structure of the viral genome has been optimized to promote packaging into capsids. Understanding this relationship between structure and packaging will make it easier to develop antiviral agents that thwart or misdirect virus assembly, and could aid the redesign of viruses for use in gene therapy and drug delivery.

DOI:http://dx.doi.org/10.7554/eLife.00632.002

Inferring bulk self-assembly properties from simulations of small systems with multiple constituent species and small systems in the grand canonical ensemble

In this paper, we generalize a methodology [T. E. Ouldridge, A. A. Louis, and J. P. K. Doye, J. Phys.: Condens. Matter 22, 104102 (2010)] for dealing with the inference of bulk properties from small simulations of self-assembling systems of characteristic finite size. In particular, schemes for extrapolating the results of simulations of a single self-assembling object to the bulk limit are established in three cases: for assembly involving multiple particle species, for systems with one species localized in space and for simulations in the grand canonical ensemble. Furthermore, methodologies are introduced for evaluating the accuracy of these extrapolations. Example systems demonstrate that differences in cluster concentrations between simulations of a single self-assembling structure and bulk studies of the same model under identical conditions can be large, and that convergence on bulk results as system size is increased can be slow and non-trivial.

Stochastic dynamics of virus capsid formation: direct versus hierarchical self-assembly

BACKGROUND

In order to replicate within their cellular host, many viruses have developed self-assembly strategies for their capsids which are sufficiently robust as to be reconstituted in vitro. Mathematical models for virus self-assembly usually assume that the bonds leading to cluster formation have constant reactivity over the time course of assembly (direct assembly). In some cases, however, binding sites between the capsomers have been reported to be activated during the self-assembly process (hierarchical assembly).

RESULTS

In order to study possible advantages of such hierarchical schemes for icosahedral virus capsid assembly, we use Brownian dynamics simulations of a patchy particle model that allows us to switch binding sites on and off during assembly. For T1 viruses, we implement a hierarchical assembly scheme where inter-capsomer bonds become active only if a complete pentamer has been assembled. We find direct assembly to be favorable for reversible bonds allowing for repeated structural reorganizations, while hierarchical assembly is favorable for strong bonds with small dissociation rate, as this situation is less prone to kinetic trapping. However, at the same time it is more vulnerable to monomer starvation during the final phase. Increasing the number of initial monomers does have only a weak effect on these general features. The differences between the two assembly schemes become more pronounced for more complex virus geometries, as shown here for T3 viruses, which assemble through homogeneous pentamers and heterogeneous hexamers in the hierarchical scheme. In order to complement the simulations for this more complicated case, we introduce a master equation approach that agrees well with the simulation results.

CONCLUSIONS

Our analysis shows for which molecular parameters hierarchical assembly schemes can outperform direct ones and suggests that viruses with high bond stability might prefer hierarchical assembly schemes. These insights increase our physical understanding of an essential biological process, with many interesting potential applications in medicine and materials science.

Assembly, stability and dynamics of virus capsids

Most viruses use a hollow protein shell, the capsid, to enclose the viral genome. Virus capsids are large, symmetric oligomers made of many copies of one or a few types of protein subunits. Self-assembly of a viral capsid is a complex oligomerization process that proceeds along a pathway regulated by ordered interactions between the participating protein subunits, and that involves a series of (usually transient) assembly intermediates. Assembly of many virus capsids requires the assistance of scaffolding proteins or the viral nucleic acid, which interact with the capsid subunits to promote and direct the process. Once assembled, many capsids undergo a maturation reaction that involves covalent modification and/or conformational rearrangements, which may increase the stability of the particle. The final, mature capsid is a relatively robust protein complex able to protect the viral genome from physicochemical aggressions; however, it is also a metastable, dynamic structure poised to undergo controlled conformational transitions required to perform biologically critical functions during virus entry into cells, intracellular trafficking, and viral genome uncoating. This article provides an updated general overview on structural, biophysical and biochemical aspects of the assembly, stability and dynamics of virus capsids.

Molecular dynamics simulation of reversibly self-assembling shells in solution using trapezoidal particles

The self-assembly of polyhedral shells, each constructed from 60 trapezoidal particles, is simulated using molecular dynamics. The spatial organization of the component particles in this shell is similar to the capsomer proteins forming the capsid of a T=1 virus. Growth occurs in the presence of an atomistic solvent and, under suitable conditions, achieves a high yield of complete shells. The simulations provide details of the structure and lifetime of the particle clusters that appear as intermediate states along the growth pathway, and the nature of the transitions between them. In certain respects the growth of size-60 shells from trapezoidal particles resembles the growth of icosahedral shells from triangular particles studied previously, with reversible bonding playing a major role in avoiding incorrect assembly, although the details differ due to particle shape and bond organization. The strong preference for maximal bonding exhibited by the triangular particle clusters is also apparent for trapezoidal particles, but this is now confined to early growth and is less pronounced as shells approach completion along a variety of pathways.