Dynamic pathways for viral capsid assembly

A Thermodynamic Model for Genome Packaging in Hepatitis B Virus

Understanding the fundamentals of genome packaging in viral capsids is important for finding effective antiviral strategies and for utilizing benign viral particles for gene therapy. While the structure of encapsidated genomic materials has been routinely characterized with experimental techniques such as cryo-electron microscopy and x-ray diffraction, much less is known about the molecular driving forces underlying genome assembly in an intracellular environment and its in vivo interactions with the capsid proteins. Here we study the thermodynamic basis of the pregenomic RNA encapsidation in human Hepatitis B virus in vivo using a coarse-grained molecular model that captures the essential components of nonspecific intermolecular interactions. The thermodynamic model is used to examine how the electrostatic interaction between the packaged RNA and the highly charged C-terminal domains (CTD) of capsid proteins regulate the nucleocapsid formation. The theoretical model predicts optimal RNA content in Hepatitis B virus nucleocapsids with different CTD lengths in good agreement with mutagenesis measurements, confirming the predominant role of electrostatic interactions and molecular excluded-volume effects in genome packaging. We find that the amount of encapsidated RNA is not linearly correlated with the net charge of CTD tails as suggested by earlier theoretical studies. Our thermodynamic analysis of the nucleocapsid structure and stability indicates that ∼10% of the CTD residues are free from complexation with RNA, resulting in partially exposed CTD tails. The thermodynamic model also predicts the free energy of complex formation between macromolecules, which corroborates experimental results for the impact of CTD truncation on the nucleocapsid stability.

The Structural Biology of Hepatitis B Virus: Form and Function

Hepatitis B virus is one of the smallest human pathogens, encoded by a 3,200-bp genome with only four open reading frames. Yet the virus shows a remarkable diversity in structural features, often with the same proteins adopting several conformations. In part, this is the parsimony of viruses, where a minimal number of proteins perform a wide variety of functions. However, a more important theme is that weak interactions between components as well as components with multiple conformations that have similar stabilities lead to a highly dynamic system. In hepatitis B virus, this is manifested as a virion where the envelope proteins have multiple structures, the envelope-capsid interaction is irregular, and the capsid is a dynamic compartment that actively participates in metabolism of the encapsidated genome and carries regulated signals for intracellular trafficking.

Computational virology: From the inside out

Viruses typically pack their genetic material within a protein capsid. Enveloped viruses also have an outer membrane made up of a lipid bilayer and membrane-spanning glycoproteins. X-ray diffraction and cryoelectron microscopy provide high resolution static views of viral structure. Molecular dynamics (MD) simulations may be used to provide dynamic insights into the structures of viruses and their components. There have been a number of simulations of viral capsids and (in some cases) of the inner core of RNA or DNA packaged within them. These simulations have generally focussed on the structural integrity and stability of the capsid and/or on the influence of the nucleic acid core on capsid stability. More recently there have been a number of simulation studies of enveloped viruses, including HIV-1, influenza A, and dengue virus. These have addressed the dynamic behaviour of the capsid, the matrix, and/or of the outer envelope. Analysis of the dynamics of the lipid bilayer components of the envelopes of influenza A and of dengue virus reveals a degree of biophysical robustness, which may contribute to the stability of virus particles in different environments. Significant computational challenges need to be addressed to aid simulation of complex viruses and their membranes, including the need to integrate structural data from a range of sources to enable us to move towards simulations of intact virions. This article is part of a Special Issue entitled: Membrane Proteins edited by J.C. Gumbart and Sergei Noskov.

Theoretical Study of the Initial Stages of Self-Assembly of a Carboxysome's Facet

Bacterial microcompartments, BMCs, are organelles that exist within wide variety of bacteria and act as nanofactories. Among the different types of known BMCs, the carboxysome has been studied the most. The carboxysome plays an important role in the light-independent part of the photosynthesis process, where its icosahedral-like proteinaceous shell acts as a membrane that controls the transport of metabolites. Although a structural model exists for the carboxysome shell, it remains largely unknown how the shell proteins self-assemble. Understanding the self-assembly process can provide insights into how the shell affects the carboxysome's function and how it can be modified to create new functionalities, such as artificial nanoreactors and artificial protein membranes. Here, we describe a theoretical framework that employs Monte Carlo simulations with a coarse-grain potential that reproduces well the atomistic potential of mean force; employing this framework, we are able to capture the initial stages of the 2D self-assembly of CcmK2 hexamers, a major protein-shell component of the carboxysome's facet. The simulations reveal that CcmK2 hexamers self-assemble into clusters that resemble what was seen experimentally in 2D layers. Further analysis of the simulation results suggests that the 2D self-assembly of carboxysome's facets is driven by a nucleation-growth process, which in turn could play an important role in the hierarchical self-assembly of BMC shells in general.

The Robust Assembly of Small Symmetric Nanoshells

Highly symmetric nanoshells are found in many biological systems, such as clathrin cages and viral shells. Many studies have shown that symmetric shells appear in nature as a result of the free-energy minimization of a generic interaction between their constituent subunits. We examine the physical basis for the formation of symmetric shells, and by using a minimal model, demonstrate that these structures can readily grow from the irreversible addition of identical subunits. Our model of nanoshell assembly shows that the spontaneous curvature regulates the size of the shell while the mechanical properties of the subunit determine the symmetry of the assembled structure. Understanding the minimum requirements for the formation of closed nanoshells is a necessary step toward engineering of nanocontainers, which will have far-reaching impact in both material science and medicine.

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

Allosteric Control of Icosahedral Capsid Assembly

During the life cycle of a virus, viral proteins and other components self-assemble to form an ordered protein shell called a capsid. This assembly process is subject to multiple competing constraints, including the need to form a thermostable shell while avoiding kinetic traps. It has been proposed that viral assembly satisfies these constraints through allosteric regulation, including the interconversion of capsid proteins among conformations with different propensities for assembly. In this article, we use computational and theoretical modeling to explore how such allostery affects the assembly of icosahedral shells. We simulate assembly under a wide range of protein concentrations, protein binding affinities, and two different mechanisms of allosteric control. We find that above a threshold strength of allosteric control, assembly becomes robust over a broad range of subunit binding affinities and concentrations, allowing the formation of highly thermostable capsids. Our results suggest that allostery can significantly shift the range of protein binding affinities that lead to successful assembly and thus should be taken into account in models that are used to estimate interaction parameters from experimental data.

HIV Capsid Assembly, Mechanism, and Structure

The HIV genome materials are encaged by a proteinaceous shell called the capsid, constructed from ∼1000-1500 copies of the capsid proteins. Because its stability and integrity are critical to the normal life cycle and infectivity of the virus, the HIV capsid is a promising antiviral drug target. In this paper, we review the studies shaping our understanding of the structure and dynamics of the capsid proteins and various forms of their assemblies, as well as the assembly mechanism.

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.

Multiscale modeling of virus replication and spread

Replication and spread of human viruses is based on the simultaneous exploitation of many different host functions, bridging multiple scales in space and time. Mathematical modeling is essential to obtain a systems-level understanding of how human viruses manage to proceed through their life cycles. Here, we review corresponding advances for viral systems of large medical relevance, such as human immunodeficiency virus-1 (HIV-1) and hepatitis C virus (HCV). We will outline how the combination of mathematical models and experimental data has advanced our quantitative knowledge about various processes of these pathogens, and how novel quantitative approaches promise to fill remaining gaps.

Supracolloidal fullerene-like cages: design principles and formation mechanisms

A vast collection of fascinating supracolloidal fullerene-like cages has been achievedviathe self-assembly of soft three-patch particles designed to mimic non-planar sp²hybridized carbon atoms in fullerenes, through the rational design of patch configuration, size, and interaction.

Elucidating dominant pathways of the nano-particle self-assembly process

Self-assembly processes play a key role in the fabrication of functional nano-structures with wide application in drug delivery and micro-reactors.

Computer simulation of self-assembly of cone-shaped nanoparticles

Two kinds of cone-shaped particles are designed: one with a Janus structure and the other with a sandwich structure. The effects of the cone angle and particle structure (i.e. AB type and BAB type) on the kinetic pathway and assembled structures are discussed.

Crystallization and arrest mechanisms of model colloids

We performed dynamic simulations of spheres with short-range attractive interactions for many values of interaction strength and range. Fast crystallization occurs in a localized region of this parameter space, but the character of crystallization pathways is not uniform within this region. Pathways range from one-step, in which a crystal nucleates directly from a gas, to two-step, in which substantial liquid-like clusters form and only subsequently become crystalline. Crystallization can fail because of slow nucleation from either gas or liquid, or because of dynamic arrest caused by strong interactions. Arrested states are characterized by the formation of networks of face-sharing tetrahedra that can be detected by a local common neighbor analysis.

Tabulation as a high-resolution alternative to coarse-graining protein interactions: Initial application to virus capsid subunits

Traditional coarse-graining based on a reduced number of interaction sites often entails a significant sacrifice of chemical accuracy. As an alternative, we present a method for simulating large systems composed of interacting macromolecules using an energy tabulation strategy previously devised for small rigid molecules or molecular fragments [S. Lettieri and D. M. Zuckerman, J. Comput. Chem. 33, 268-275 (2012); J. Spiriti and D. M. Zuckerman, J. Chem. Theory Comput. 10, 5161-5177 (2014)]. We treat proteins as rigid and construct distance and orientation-dependent tables of the interaction energy between them. Arbitrarily detailed interactions may be incorporated into the tables, but as a proof-of-principle, we tabulate a simple α-carbon Gō-like model for interactions between dimeric subunits of the hepatitis B viral capsid. This model is significantly more structurally realistic than previous models used in capsid assembly studies. We are able to increase the speed of Monte Carlo simulations by a factor of up to 6700 compared to simulations without tables, with only minimal further loss in accuracy. To obtain further enhancement of sampling, we combine tabulation with the weighted ensemble (WE) method, in which multiple parallel simulations are occasionally replicated or pruned in order to sample targeted regions of a reaction coordinate space. In the initial study reported here, WE is able to yield pathways of the final ∼25% of the assembly process.

Fast, Label-Free Tracking of Single Viruses and Weakly Scattering Nanoparticles in a Nanofluidic Optical Fiber

High-speed tracking of single particles is a gateway to understanding physical, chemical, and biological processes at the nanoscale. It is also a major experimental challenge, particularly for small, nanometer-scale particles. Although methods such as confocal or fluorescence microscopy offer both high spatial resolution and high signal-to-background ratios, the fluorescence emission lifetime limits the measurement speed, while photobleaching and thermal diffusion limit the duration of measurements. Here we present a tracking method based on elastic light scattering that enables long-duration measurements of nanoparticle dynamics at rates of thousands of frames per second. We contain the particles within a single-mode silica fiber having a subwavelength, nanofluidic channel and illuminate them using the fiber's strongly confined optical mode. The diffusing particles in this cylindrical geometry are continuously illuminated inside the collection focal plane. We show that the method can track unlabeled dielectric particles as small as 20 nm as well as individual cowpea chlorotic mottle virus (CCMV) virions-26 nm in size and 4.6 megadaltons in mass-at rates of over 3 kHz for durations of tens of seconds. Our setup is easily incorporated into common optical microscopes and extends their detection range to nanometer-scale particles and macromolecules. The ease-of-use and performance of this technique support its potential for widespread applications in medical diagnostics and micro total analysis systems.

Rotational diffusion affects the dynamical self-assembly pathways of patchy particles

Predicting the self-assembly kinetics of particles with anisotropic interactions, such as colloidal patchy particles or proteins with multiple binding sites, is important for the design of novel high-tech materials, as well as for understanding biological systems, e.g., viruses or regulatory networks. Often stochastic in nature, such self-assembly processes are fundamentally governed by rotational and translational diffusion. Whereas the rotational diffusion constant of particles is usually considered to be coupled to the translational diffusion via the Stokes-Einstein relation, in the past decade it has become clear that they can be independently altered by molecular crowding agents or via external fields. Because virus capsids naturally assemble in crowded environments such as the cell cytoplasm but also in aqueous solution in vitro, it is important to investigate how varying the rotational diffusion with respect to transitional diffusion alters the kinetic pathways of self-assembly. Kinetic trapping in malformed or intermediate structures often impedes a direct simulation approach of a kinetic network by dramatically slowing down the relaxation to the designed ground state. However, using recently developed path-sampling techniques, we can sample and analyze the entire self-assembly kinetic network of simple patchy particle systems. For assembly of a designed cluster of patchy particles we find that changing the rotational diffusion does not change the equilibrium constants, but significantly affects the dynamical pathways, and enhances (suppresses) the overall relaxation process and the yield of the target structure, by avoiding (encountering) frustrated states. Besides insight, this finding provides a design principle for improved control of nanoparticle self-assembly.

Hierarchical assembly may be a way to make large information-rich structures

Self-assembly in the laboratory can now yield 'information-rich' nanostructures in which each component is of a distinct type and has a defined spatial position. Ensuring the thermodynamic stability of such structures requires inter-component interaction energies to increase logarithmically with structure size, in order to counter the entropy gained upon mixing component types in solution. However, self-assembly in the presence of strong interactions results in general in kinetic trapping, so suggesting a limit to the size of an (equilibrium) structure that can be self-assembled from distinguishable components. Here we study numerically a two-dimensional hierarchical assembly scheme already considered in experiment. We show that this scheme is immune to the kinetic traps associated with strong 'native' interactions (interactions designed to stabilize the intended structure), and so, in principle, offers a way to make large information-rich structures. In this scheme the size of an assembled structure scales exponentially with the stage of assembly, and assembly can continue as long as random motion is able to bring structures into contact. The resulting superstructure could provide a template for building in the third dimension. The chief drawback of this scheme is that it is particularly susceptible to kinetic traps that result from 'non-native' interactions (interactions not required to stabilize the intended structure); the scale on which such a scheme can be realized therefore depends upon how effectively this latter kind of interaction can be suppressed.

Design strategies for self-assembly of discrete targets

Both biological and artificial self-assembly processes can take place by a range of different schemes, from the successive addition of identical building blocks to hierarchical sequences of intermediates, all the way to the fully addressable limit in which each component is unique. In this paper, we introduce an idealized model of cubic particles with patterned faces that allows self-assembly strategies to be compared and tested. We consider a simple octameric target, starting with the minimal requirements for successful self-assembly and comparing the benefits and limitations of more sophisticated hierarchical and addressable schemes. Simulations are performed using a hybrid dynamical Monte Carlo protocol that allows self-assembling clusters to rearrange internally while still providing Stokes-Einstein-like diffusion of aggregates of different sizes. Our simulations explicitly capture the thermodynamic, dynamic, and steric challenges typically faced by self-assembly processes, including competition between multiple partially completed structures. Self-assembly pathways are extracted from the simulation trajectories by a fully extendable scheme for identifying structural fragments, which are then assembled into history diagrams for successfully completed target structures. For the simple target, a one-component assembly scheme is most efficient and robust overall, but hierarchical and addressable strategies can have an advantage under some conditions if high yield is a priority.

Assembly and Release of Hepatitis B Virus

The hepatitis B virus (HBV) core protein is a dynamic and versatile protein that directs many viral processes. During capsid assembly, core protein allosteric changes ensure efficient formation of a stable capsid that assembles while packaging viral RNA-polymerase complex. Reverse transcription of the RNA genome as well as transport of the capsid to multiple cellular compartments are directed by dynamic phosphorylation and structural changes of core protein. Subsequently, interactions of the capsid with the surface proteins and/or host proteins trigger envelopment and release of the viral capsids or the transport to the nucleus. Held together by many weak protein-protein interactions, the viral capsid is an extraordinary metastable machine that is stable enough to persist in the cellular and extracellular environment but dissociates to allow release of the viral genome at the right time during infection.

Machine learning assembly landscapes from particle tracking data

Bottom-up self-assembly offers a powerful route for the fabrication of novel structural and functional materials. Rational engineering of self-assembling systems requires understanding of the accessible aggregation states and the structural assembly pathways. In this work, we apply nonlinear machine learning to experimental particle tracking data to infer low-dimensional assembly landscapes mapping the morphology, stability, and assembly pathways of accessible aggregates as a function of experimental conditions. To the best of our knowledge, this represents the first time that collective order parameters and assembly landscapes have been inferred directly from experimental data. We apply this technique to the nonequilibrium self-assembly of metallodielectric Janus colloids in an oscillating electric field, and quantify the impact of field strength, oscillation frequency, and salt concentration on the dominant assembly pathways and terminal aggregates. This combined computational and experimental framework furnishes new understanding of self-assembling systems, and quantitatively informs rational engineering of experimental conditions to drive assembly along desired aggregation pathways.

Self-Assembly of an Alphavirus Core-like Particle Is Distinguished by Strong Intersubunit Association Energy and Structural Defects

Weak association energy can lead to uniform nanostructures: defects can anneal due to subunit lability. What happens when strong association energy leads to particles where defects are trapped? Alphaviruses are enveloped viruses whose icosahedral nucleocapsid core can assemble independently. We used a simplest case system to study Ross River virus (RRV) core-like particle (CLP) self-assembly using purified capsid protein and a short DNA oligomer. We find that capsid protein binds the oligomer with high affinity to form an assembly competent unit (U). Subsequently, U assembles with concentration dependence into CLPs. We determined that U-U pairwise interactions are very strong (ca. -6 kcal/mol) compared to other virus assembly systems. Assembled RRV CLPs appeared morphologically uniform and cryo-EM image reconstruction with imposed icosahedral symmetry yielded a T = 4 structure. However, 2D class averages of the CLPs show that virtually every class had disordered regions. These results suggested that irregular cores may be present in RRV virions. To test this hypothesis, we determined 2D class averages of RRV virions using authentic virions or only the core from intact virions isolated by computational masking. Virion-based class averages were symmetrical, geometric, and corresponded well to projections of image reconstructions. In core-based class averages, cores and envelope proteins in many classes were disordered. These results suggest that partly disordered components are common even in ostensibly well-ordered viruses, a biological realization of a patchy particle. Biological advantages of partly disordered complexes may arise from their ease of dissociation and asymmetry.

Reconstruction of the Disassembly Pathway of an Icosahedral Viral Capsid and Shape Determination of Two Successive Intermediates

Viral capsids derived from an icosahedral plant virus widely used in physical and nanotechnological investigations were fully dissociated into dimers by a rapid change of pH. The process was probed in vitro at high spatiotemporal resolution by time-resolved small-angle X-ray scattering using a high brilliance synchrotron source. A powerful custom-made global fitting algorithm allowed us to reconstruct the most likely pathway parametrized by a set of stoichiometric coefficients and to determine the shape of two successive intermediates by ab initio calculations. None of these two unexpected intermediates was previously identified in self-assembly experiments, which suggests that the disassembly pathway is not a mirror image of the assembly pathway. These findings shed new light on the mechanisms and the reversibility of the assembly/disassembly of natural and synthetic virus-based systems. They also demonstrate that both the structure and dynamics of an increasing number of intermediate species become accessible to experiments.

Mechanism of polymorphism and curvature of HIV capsid assemblies probed by 3D simulations with a novel coarse grain model

BACKGROUND

During the maturation process, HIV capsid proteins self-assemble into polymorphic capsids. The strong polymorphism precludes high resolution structural characterization under in vivo conditions. In spite of the determination of structural models for various in vitro assemblies of HIV capsid proteins, the assembly mechanism is still not well-understood.

METHODS

We report 3D simulations of HIV capsid proteins by a novel coarse grain model that captures the backbone of the rigid segments in the protein accurately. The effects of protein dynamics on assembly are emulated by a static ensemble of subunits in conformations derived from molecular dynamics simulation.

RESULTS

We show that HIV capsid proteins robustly assemble into hexameric lattices in a range of conditions where trimers of dimeric subunits are the dominant oligomeric intermediates. Variations of hexameric lattice curvatures are observed in simulations with subunits of variable inter-domain orientations mimicking the conformation distribution in solution. Simulations with subunits based on pentameric structural models lead to assembly of sharp curved structures resembling the tips of authentic HIV capsids, along a distinct pathway populated by tetramers and pentamers with the characteristic quasi-equivalency of viral capsids.

CONCLUSIONS

Our results suggest that the polymorphism assembly is triggered by the inter-domain dynamics of HIV capsid proteins in solution. The assembly of highly curved structures arises from proteins in conformation with a highly specific inter-domain orientation.

SIGNIFICANCE

Our work proposes a mechanism of HIV capsid assembly based on available structural data, which can be readily verified. Our model can be applied to other large biomolecular assemblies.

Self-assembly and crystallisation of indented colloids at a planar wall

We report experimental and simulation studies of the structure of a monolayer of indented ("lock and key") colloids, on a planar surface. On adding a non-absorbing polymer with prescribed radius and volume fraction, depletion interactions are induced between the colloids, with controlled range and strength. For spherical particles, this leads to crystallisation, but the indented colloids crystallise less easily than spheres, in both simulation and experiment. Nevertheless, simulations show that indented colloids do form plastic (rotator) crystals. We discuss the conditions under which this occurs, and the possibilities of lower-symmetry crystal states. We also comment on the kinetic accessibility of these states.

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.

Early stage P22 viral capsid self-assembly mediated by scaffolding protein: atom-resolved model and molecular dynamics simulation

Molecular dynamics simulation of an atom-resolved bacteriophage P22 capsid model is used to delineate the underlying mechanism of early stage P22 self-assembly. A dimer formed by the C-terminal fragment of scaffolding protein with a new conformation is demonstrated to catalyze capsomer (hexamer and pentamer) aggregation efficiently. Effects of scaffolding protein/coat protein binding patterns and scaffolding protein concentration on efficiency, fidelity, and capsid curvature of P22 self-assembly are identified.

Roles of dynamical symmetry breaking in driving oblate-prolate transitions of atomic clusters

This paper explores the driving mechanisms for structural transitions of atomic clusters between oblate and prolate isomers. We employ the hyperspherical coordinates to investigate structural dynamics of a seven-atom cluster at a coarse-grained level in terms of the dynamics of three gyration radii and three principal axes, which characterize overall mass distributions of the cluster. Dynamics of gyration radii is governed by two kinds of forces. One is the potential force originating from the interactions between atoms. The other is the dynamical forces called the internal centrifugal forces, which originate from twisting and shearing motions of the system. The internal centrifugal force arising from twisting motions has an effect of breaking the symmetry between two gyration radii. As a result, in an oblate isomer, activation of the internal centrifugal force that has the effect of breaking the symmetry between the two largest gyration radii is crucial in triggering structural transitions into prolate isomers. In a prolate isomer, on the other hand, activation of the internal centrifugal force that has the effect of breaking the symmetry between the two smallest gyration radii is crucial in triggering structural transitions into oblate isomers. Activation of a twisting motion that switches the movement patterns of three principal axes is also important for the onset of structural transitions between oblate and prolate isomers. Based on these trigger mechanisms, we finally show that selective activations of specific gyration radii and twisting motions, depending on the isomer of the cluster, can effectively induce structural transitions of the cluster. The results presented here could provide further insights into the control of molecular reactions.

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.

Emergent rhombus tilings from molecular interactions with M-fold rotational symmetry

We show that model molecules with particular rotational symmetries can self-assemble into network structures equivalent to rhombus tilings. This assembly happens in an emergent way, in the sense that molecules spontaneously select irregular fourfold local coordination from a larger set of possible local binding geometries. The existence of such networks can be rationalized by simple geometrical arguments, but the same arguments do not guarantee a network's spontaneous self-assembly. This class of structures must in certain regimes of parameter space be able to reconfigure into networks equivalent to triangular tilings.

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.

Design of a reversible inversed pH-responsive caged protein

Controlling the self-assembly behavior of caged proteins expands their potential applications in nanotechnology. While the structure of a caged E2 protein from pyruvate dehydrogenase is inert to any pH change, the incorporation of switchable GALA peptide that undergoes a coil-to-helix transition at acidic pH modulates its self-assembly property. By substituting the native α-helix at the C-terminus of the E2 protein with the GALA peptide, we report the first engineered caged protein with reversible inversed pH-responsive behavior. The redesigned caged E2 protein assumes an assembly profile that is distinct from the native state; it disassembles at pH 7.0 and self-assembles at pH 4.0 in a reversible manner. This unique reversible pH trigger suggests the applicability of the self-assembly control on other multi-subunit macromolecules.

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.

Biomolecular engineering of virus-like particles aided by computational chemistry methods

Multi-scale investigation of VLP self-assembly aided by computational methods is facilitating the design, redesign, and modification of functionalized VLPs.

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.

On global minimizers of repulsive-attractive power-law interaction energies

We consider the minimization of the repulsive-attractive power-law interaction energies that occur in many biological and physical situations. We show the existence of global minimizers in the discrete setting and obtain bounds for their supports independently of the number of Dirac deltas in a certain range of exponents. These global discrete minimizers correspond to the stable spatial profiles of flock patterns in swarming models. Global minimizers of the continuum problem are obtained by compactness. We also illustrate our results through numerical simulations.

An assembly funnel makes biomolecular complex assembly efficient

Like protein folding and crystallization, the self-assembly of complexes is a fundamental form of biomolecular organization. While the number of methods for creating synthetic complexes is growing rapidly, most require empirical tuning of assembly conditions and/or produce low yields. We use coarse-grained simulations of the assembly kinetics of complexes to identify generic limitations on yields that arise because of the many simultaneous interactions allowed between the components and intermediates of a complex. Efficient assembly occurs when nucleation is fast and growth pathways are few, i.e. when there is an assembly "funnel". For typical complexes, an assembly funnel occurs in a narrow window of conditions whose location is highly complex specific. However, by redesigning the components this window can be drastically broadened, so that complexes can form quickly across many conditions. The generality of this approach suggests assembly funnel design as a foundational strategy for robust biomolecular complex synthesis.

The hepatitis B virus core protein intradimer interface modulates capsid assembly and stability

During the hepatitis B virus (HBV) life cycle, capsid assembly and disassembly must ensure correct packaging and release of the viral genome. Here we show that changes in the dynamics of the core protein play an important role in regulating these processes. The HBV capsid assembles from 120 copies of the core protein homodimer. Each monomer contains a conserved cysteine at position 61 that can form an intradimer disulfide that we use as a marker for dimer conformational states. We show that dimers in the context of capsids form intradimer disulfides relatively rapidly. Surprisingly, compared to reduced dimers, fully oxidized dimers assembled slower and into capsids that were morphologically similar but less stable. We hypothesize that oxidized protein adopts a geometry (or constellation of geometries) that is unfavorable for capsid assembly, resulting in weaker dimer-dimer interactions as well as slower assembly kinetics. Our results suggest that structural flexibility at the core protein intradimer interface is essential for regulating capsid assembly and stability. We further suggest that capsid destabilization by the C61-C61 disulfide has a regulatory function to support capsid disassembly and release of the viral genome.

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.

Building polyhedra by self-assembly: theory and experiment

We investigate the utility of a mathematical framework based on discrete geometry to model biological and synthetic self-assembly. Our primary biological example is the self-assembly of icosahedral viruses; our synthetic example is surface-tension-driven self-folding polyhedra. In both instances, the process of self-assembly is modeled by decomposing the polyhedron into a set of partially formed intermediate states. The set of all intermediates is called the configuration space, pathways of assembly are modeled as paths in the configuration space, and the kinetics and yield of assembly are modeled by rate equations, Markov chains, or cost functions on the configuration space. We review an interesting interplay between biological function and mathematical structure in viruses in light of this framework. We discuss in particular: (i) tiling theory as a coarse-grained description of all-atom models; (ii) the building game-a growth model for the formation of polyhedra; and (iii) the application of these models to the self-assembly of the bacteriophage MS2. We then use a similar framework to model self-folding polyhedra. We use a discrete folding algorithm to compute a configuration space that idealizes surface-tension-driven self-folding and analyze pathways of assembly and dominant intermediates. These computations are then compared with experimental observations of a self-folding dodecahedron with side 300 μm. In both models, despite a combinatorial explosion in the size of the configuration space, a few pathways and intermediates dominate self-assembly. For self-folding polyhedra, the dominant intermediates have fewer degrees of freedom than comparable intermediates, and are thus more rigid. The concentration of assembly pathways on a few intermediates with distinguished geometric properties is biologically and physically important, and suggests deeper mathematical structure.

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.

Patterns without patches: hierarchical self-assembly of complex structures from simple building blocks

Nanoparticles with "sticky patches" have long been proposed as building blocks for the self-assembly of complex structures. The synthetic realizability of such patchy particles, however, greatly lags behind predictions of patterns they could form. Using computer simulations, we show that structures of the same genre can be obtained from a solution of simple isotropic spheres, with control only over their sizes and a small number of binding affinities. In a first step, finite clusters of well-defined structure and composition emerge from natural dynamics with high yield. In effect a kind of patchy particle, these clusters can further assemble into a variety of complex superstructures, including filamentous networks, ordered sheets, and highly porous crystals.