Langevin dynamics simulation of polymer-assisted virus-like assembly

Molecular Dynamics Simulations of Deformable Viral Capsomers

Most coarse-grained models of individual capsomers associated with viruses employ rigid building blocks that do not exhibit shape adaptation during self-assembly. We develop a coarse-grained general model of viral capsomers that incorporates their stretching and bending energies while retaining many features of the rigid-body models, including an overall trapezoidal shape with attractive interaction sites embedded in the lateral walls to favor icosahedral capsid assembly. Molecular dynamics simulations of deformable capsomers reproduce the rich self-assembly behavior associated with a general T=1 icosahedral virus system in the absence of a genome. Transitions from non-assembled configurations to icosahedral capsids to kinetically-trapped malformed structures are observed as the steric attraction between capsomers is increased. An assembly diagram in the space of capsomer-capsomer steric attraction and capsomer deformability reveals that assembling capsomers of higher deformability into capsids requires increasingly large steric attraction between capsomers. Increasing capsomer deformability can reverse incorrect capsomer-capsomer binding, facilitating transitions from malformed structures to symmetric capsids; however, making capsomers too soft inhibits assembly and yields fluid-like structures.

Self-assembly of amphiphilic truncated cones to form hollow nanovesicles

To mimic the unique properties of capsid (protein shell of a virus), we performed Brownian dynamics simulations of the self-assembly of amphiphilic truncated cone particles with anisotropic interactions. The particle shape of a truncated cone in our simulations depended on the cone angle θ, truncated height h_c and particle type (A_xB_y and B_xA_yB_z). The hydrophobic A moieties and hydrophilic B moieties are responsible for attractive and repulsive interactions, respectively. By varying the particle shape, truncated cones can assemble into hollow and vesicle-like clusters with a specific cluster size N. To assemble into hollow vesicles, the truncated height h_c must be below a critical value. When h_c exceeds this critical value, malformation will occur. The dynamics shows that the vesicle formation occurs in three stages: initially the growth is slow, then rapid, and finally it slows down. The truncated height h_c has a stronger impact on the growth kinetics than the cone angle θ or the particle type. We explored how the cluster packing depended on the cooling rate and particle number as well as discussing the relationship between the cluster geometry and the interparticle interactions. Further, we also discuss possible methods to experimentally prepare the truncated cones. The results of our work deepen our understanding of the self-assembly behavior of truncated cones and our results will aid the effective design of particle building blocks for novel nanostructures.

To mimic the unique properties of capsid (protein shell of a virus), we performed Brownian dynamics simulations of the self-assembly of amphiphilic truncated cone particles with anisotropic interactions.

50th Anniversary Perspective: A Perspective on Polyelectrolyte Solutions

From the beginning of life with the information-containing polymers until the present era of a plethora of water-based materials in health care industry and biotechnology, polyelectrolytes are ubiquitous with a broad range of structural and functional properties. The main attribute of polyelectrolyte solutions is that all molecules are strongly correlated both topologically and electrostatically in their neutralizing background of charged ions in highly polarizable solvent. These strong correlations and the necessary use of numerous variables in experiments on polyelectrolytes have presented immense challenges toward fundamental understanding of the various behaviors of charged polymeric systems. This Perspective presents the author's subjective summary of several conceptual advances and the remaining persistent challenges in the contexts of charge and size of polymers, structures in homogeneous solutions, thermodynamic instability and phase transitions, structural evolution with oppositely charged polymers, dynamics in polyelectrolyte solutions, kinetics of phase separation, mobility of charged macromolecules between compartments, and implications to biological systems.

Simulation of self-assembly of polyzwitterions into vesicles

Using the Langevin dynamics method and a coarse-grained model, we have studied the formation of vesicles by hydrophobic polymers consisting of periodically placed zwitterion side groups in dilute salt-free aqueous solutions. The zwitterions, being permanent charge dipoles, provide long-range electrostatic correlations which are interfered by the conformational entropy of the polymer. Our simulations are geared towards gaining conceptual understanding in these correlated dipolar systems, where theoretical calculations are at present formidable. A competition between hydrophobic interactions and dipole-dipole interactions leads to a series of self-assembled structures. As the spacing d between the successive zwitterion side groups decreases, single chains undergo globule → disk → worm-like structures. We have calculated the Flory-Huggins χ parameter for these systems in terms of d and monitored the radius of gyration, hydrodynamic radius, spatial correlations among hydrophobic and dipole monomers, and dipole-dipole orientational correlation functions. During the subsequent stages of self-assembly, these structures lead to larger globules and vesicles as d is decreased up to a threshold value, below which no large scale morphology forms. The vesicles form via a polynucleation mechanism whereby disk-like structures form first, followed by their subsequent merger.

Combining Protein Cages and Polymers: from Understanding Self-Assembly to Functional Materials

Protein cages, such as viruses, are well-defined biological nanostructures which are highly symmetrical and monodisperse. They are found in various shapes and sizes and can encapsulate or template non-native materials. Furthermore, the proteins can be chemically or genetically modified giving them new properties. For these reasons, these protein structures have received increasing attention in the field of polymer-protein hybrid materials over the past years, however, advances are still to be made. This Viewpoint highlights the different ways polymers and protein cages or their subunits have been combined to understand self-assembly and create functional materials.

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.

Protein permeation through an electrically tunable membrane

Protein filtration is important in many fields of science and technology such as medicine, biology, chemistry, and engineering. Recently, protein separation and filtering with nanoporous membranes has attracted interest due to the possibility of fast separation and high throughput volume. This, however, requires understanding of the protein's dynamics inside and in the vicinity of the nanopore. In this work, we utilize a Brownian dynamics approach to study the motion of the model protein insulin in the membrane-electrolyte electrostatic potential. We compare the results of the atomic model of the protein with the results of a coarse-grained and a single-bead model, and find that the coarse-grained representation of protein strikes the best balance between the accuracy of the results and the computational effort required. Contrary to common belief, we find that to adequately describe the protein, a single-bead model cannot be utilized without a significant effort to tabulate the simulation parameters. Similar to results for nanoparticle dynamics, our findings also indicate that the electric field and the electro-osmotic flow due to the applied membrane and electrolyte biases affect the capture and translocation of the biomolecule by either attracting or repelling it to or from the nanopore. Our computational model can also be applied to other types of proteins and separation 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

Dewetting dynamics of a gold film on graphene: implications for nanoparticle formation

The dynamics of dewetting of gold films on graphene surfaces is investigated using molecular dynamics simulation. The effect of temperature (973-1533 K), film diameter (30-40 nm) and film thickness (0.5-3 nm) on the dewetting mechanism, leading to the formation of nanoparticles, is reported. The dewetting behavior for films ≤5 Å is in contrast to the behavior seen for thicker films. The retraction velocity, in the order of ∼300 m s(-1) for a 1 nm film, decreases with an increase in film thickness, whereas it increases with temperature. However at no point do nanoparticles detach from the surface within the temperature range considered in this work. We further investigated the self-assembly behavior of nanoparticles on graphene at different temperatures (673-1073 K). The process of self-assembly of gold nanoparticles is favorable at lower temperatures than at higher temperatures, based on the free-energy landscape analysis. Furthermore, the shape of an assembled structure is found to change from spherical to hexagonal, with a marked propensity towards an icosahedral structure based on the bond-orientational order parameters.

Elucidating Electrostatic Self-Assembly: Molecular Parameters as Key to Thermodynamics and Nanoparticle Shape

The rational design of supramolecular nanoparticles by self-assembly is a crucial field of research due to the wide applications and the possibility of control through external triggers. Understanding the shape-determining factors is the key for tailoring nanoparticles with desired properties. Here, we show how the thermodynamics of the interaction control the shape of the nanoparticle. We highlight the connection between the molecular structure of building blocks, the interaction strength, and the nanoassembly shape. Nanoparticles are prepared by electrostatic self-assembly of cationic polyelectrolyte dendrimers of different generations and oppositely charged multivalent organic dyes relying on the combination of electrostatic and π-π interactions. Different building blocks have been used to vary interaction strength, geometric constraints, and charge ratio, providing insights into the assembly process. The nanoassembly structure has been characterized using atomic force microscopy, static light scattering, small angle neutron scattering, and UV-vis spectroscopy. We show that the isotropy/anisotropy of the nanoassemblies is related to the dye valency. Isothermal titration calorimetry has been used to investigate both dye-dye and dye-dendrimer interaction. The existence of a threshold value in entropy and enthalpy change separating isotropic and anisotropic shapes for both interactions has been demonstrated. The effects of the dye molecular structure on the interaction thermodynamics and therefore on the nanoparticle structure have been revealed using molecular modeling. The polar surface area of the dye molecule takes a key role in the dye self-interaction. This study opens the possibility for a priori shape determination knowing the building blocks structure and their interactions.

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.

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.

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.

Topological effects on capsomer-polyion co-assembly

On the basis of a T = 1 icosahedral capsid model, the capsomer-polyion co-assembly process has been investigated by molecular dynamics simulations using capsomers with different net charge and charge distribution as well as linear, branched, and hyper-branched polyions. The assembly process was characterized in terms of the time-dependent cluster size probabilities, averaged cluster size, encapsulation efficiency, and polyion extension. The kinetics of the capsid formation displayed a two-step process. The first one comprised adsorption of capsomers on the polyion, driven by their electrostatic attraction, whereas the second one involved a relocation and/or reorientation of adsorbed capsomers, which rate is reduced upon increasing electrostatic interaction. We found that increased polyion branching facilitated a more rapid encapsulation process towards a higher yield. Moreover, the hyper-branched polyions were entirely encapsulated at all polyion-capsid charge ratios considered.

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.

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.

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.

Role of electrostatics in the assembly pathway of a single-stranded RNA virus

We have recently discovered (R. D. Cadena-Nava et al., J. Virol. 86:3318-3326, 2012, doi:10.1128/JVI.06566-11) that the in vitro packaging of RNA by the capsid protein (CP) of cowpea chlorotic mottle virus is optimal when there is a significant excess of CP, specifically that complete packaging of all of the RNA in solution requires sufficient CP to provide charge matching of the N-terminal positively charged arginine-rich motifs (ARMS) of the CPs with the negatively charged phosphate backbone of the RNA. We show here that packaging results from the initial formation of a charge-matched protocapsid consisting of RNA decorated by a disordered arrangement of CPs. This protocapsid reorganizes into the final, icosahedrally symmetric nucleocapsid by displacing the excess CPs from the RNA to the exterior surface of the emerging capsid through electrostatic attraction between the ARMs of the excess CP and the negative charge density of the capsid exterior. As a test of this scenario, we prepare CP mutants with extra and missing (relative to the wild type) cationic residues and show that a correspondingly smaller and larger excess, respectively, of CP is needed for complete packaging of RNA.

IMPORTANCE

Cowpea chlorotic mottle virus (CCMV) has long been studied as a model system for the assembly of single-stranded RNA viruses. While much is known about the electrostatic interactions within the CCMV virion, relatively little is known about these interactions during assembly, i.e., within intermediate states preceding the final nucleocapsid structure. Theoretical models and coarse-grained molecular dynamics simulations suggest that viruses like CCMV assemble by the bulk adsorption of CPs onto the RNA driven by electrostatic attraction, followed by structural reorganization into the final capsid. Such a mechanism facilitates assembly by condensing the RNA for packaging while simultaneously concentrating the local density of CP for capsid nucleation. We provide experimental evidence of such a mechanism by demonstrating that efficient assembly is initiated by the formation of a disordered protocapsid complex whose stoichiometry is governed by electrostatics (charge matching of the anionic RNA and the cationic N termini of the CP).

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.

Polymer directed aggregation and dispersion of anisotropic nanoparticles

The aggregation and dispersion of two anisotropic nanoparticles (NPs), cubes and tetrahedrons, in a polymer matrix are studied in this work using coarse-grained molecular dynamics simulations. We present the phase diagrams of NP-polymer composites, depicting microscopically phase-separated, dispersed, and bridged cubes and tetrahedrons in a polymer matrix, which depend on the interaction between the NPs and polymer (εnp), along with the NPs' volume fraction (ϕ). The microscopic phase separation occurs at very low εnp, where NPs self-organize into multidimensional structures, depending on ϕ. In particular, for tetrahedrons, a cross-over from an ordered spherical aggregate to a disordered sheet-like aggregate is observed with increasing ϕ. In the case of cubes, a transition from cubic array → square column → square array (sheet) is identified with increasing ϕ. The clusters of NPs are characterized by their asphericity and principal radii of gyration. The free energy profile for a structured assembly is estimated, which clearly shows that the successful assembly of NPs is energetically favorable at a lower temperature. However, there exists an energy barrier for the successful assembly of all the NPs in the system. At intermediate εnp, a transition from a clustered state to a state comprising dispersed cubes and tetrahedrons in a polymer matrix is observed. At higher εnp, a further transition takes place, where gas-like dispersed NPs form a liquid-like aggregate via polymer layers. Therefore, the findings in this work illustrate that the effective interaction between anisotropic NPs in a polymer matrix is very diverse, which can generate multidimensional structured assemblies, with the disordered clustering, dispersion, and bridging-induced aggregation of NPs.

Hierarchical Self-Assembly of Amphiphilic Homopolymer into Unique Superstructures

Supramolecular forces influence the morphologies of self-assemblies. Herein, self-assembled structures of an amphiphilic, norbornene-derived thiobarbiturate homopolymers (p-NTB) are discussed. The newly designed hompolymer shows self-assembled rod-like structures in tetrahydrofuran (THF) solvent. Formation of the rods are governed by hydrogen bonding motifs and amphiphilicity found in the molecular architecture. The solvent polarity controls their organization into cube-like and sphere-like structures at the next length scale. Encapsulation studies of hydrophobic magnetic particles as well as drug molecules into these superstructures demonstrate a novel route to fabricate multifunctional cube-like and sphere-like aggregates.

Mechanical and assembly units of viral capsids identified via quasi-rigid domain decomposition

Key steps in a viral life-cycle, such as self-assembly of a protective protein container or in some cases also subsequent maturation events, are governed by the interplay of physico-chemical mechanisms involving various spatial and temporal scales. These salient aspects of a viral life cycle are hence well described and rationalised from a mesoscopic perspective. Accordingly, various experimental and computational efforts have been directed towards identifying the fundamental building blocks that are instrumental for the mechanical response, or constitute the assembly units, of a few specific viral shells. Motivated by these earlier studies we introduce and apply a general and efficient computational scheme for identifying the stable domains of a given viral capsid. The method is based on elastic network models and quasi-rigid domain decomposition. It is first applied to a heterogeneous set of well-characterized viruses (CCMV, MS2, STNV, STMV) for which the known mechanical or assembly domains are correctly identified. The validated method is next applied to other viral particles such as L-A, Pariacoto and polyoma viruses, whose fundamental functional domains are still unknown or debated and for which we formulate verifiable predictions. The numerical code implementing the domain decomposition strategy is made freely available.

The genetic material of viruses is packaged inside capsids constituted from a few tens to thousands of proteins. The latter can organize in multimers that serve as fundamental blocks for the viral shell assembly or that control the capsid conformational transitions and response to mechanical stress. In this work, we introduce and apply a computational scheme that identifies the fundamental protein blocks from the structural fluctuations of the capsids in thermal equilibrium. These can be derived from phenomenological elastic network models with minimal computational expenditure. Accordingly, the basic functional protein units of a capsid can be obtained from the sole input of the capsid crystal structure. The method is applied to a heterogeneous set of viruses of various size and geometries. These include well-characterised instances for validation purposes, as well as debated ones for which predictions are formulated.

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.

Micellization model for the polymerization of clathrin baskets

A thermodynamic model is used to investigate the conditions under which clathrin triskelions form polyhedral baskets. The analysis, which is similar to classical methods used to study micelle formation, relates clathrin basket energetics to system parameters linked to triskelial rigidity, the natural curvature of an isolated triskelion, and interactions between triskelial legs in the assembled polyhedra. Mathematical theory predicts that a minimal ("critical") clathrin concentration, C(C), needs to be surpassed in order for basket polymerization to occur, and indicates how C(C), and the amount of polymerized material, depend on the chosen parameters. Analytical expressions are obtained to indicate how changes in the parameters affect the sizes of the polyhedra which arise when the total clathrin concentration exceeds C(C). A continuum analytic approximation then is used to produce numerical results that illustrate the derived dependences.

Dynamics of self-assembly of model viral capsids in the presence of a fluctuating membrane

A coarse-grained computational model is used to investigate the effect of a fluctuating fluid membrane on the dynamics of patchy-particle assembly into virus capsid-like cores. Results from simulations for a broad range of parameters are presented, showing the effect of varying interaction strength, membrane stiffness, and membrane viscosity. Furthermore, the effect of hydrodynamic interactions is investigated. Attraction to a membrane may promote assembly, including for subunit interaction strengths for which it does not occur in the bulk, and may also decrease single-core assembly time. The membrane budding rate is strongly increased by hydrodynamic interactions. The membrane deformation rate is important in determining the finite-time yield. Higher rates may decrease the entropic penalty for assembly and help guide subunits toward each other but may also block partial cores from being completed. For increasing subunit interaction strength, three regimes with different effects of the membrane are identified.

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.

Building a viral capsid in the presence of genomic RNA

Virus capsid assembly has traditionally been considered as a process that can be described primarily via self-assembly of the capsid proteins, neglecting interactions with other viral or cellular components. Our recent work on several ssRNA viruses, a major class of viral pathogens containing important human, animal, and plant viruses, has shown that this protein-centric view is too simplistic. Capsid assembly for these viruses relies strongly on a number of cooperative roles played by the genomic RNA. This realization requires a new theoretical framework for the modeling and prediction of the assembly behavior of these viruses. In a seminal paper Zlotnick [J. Mol. Biol. 241, 59 (1994)] laid the foundations for the modeling of capsid assembly as a protein-only self-assembly process, illustrating his approach using the example of a dodecahedral study system. We describe here a generalized framework for modeling assembly that incorporates the regulatory functions provided by cognate protein-nucleic-acid interactions between capsid proteins and segments of the genomic RNA, called packaging signals, into the model. Using the same dodecahedron system we demonstrate, using a Gillespie-type algorithm to deal with the enhanced complexity of the problem instead of a master equation approach, that assembly kinetics and yield strongly depend on the distribution and nature of the packaging signals, highlighting the importance of the crucial roles of the RNA in this process.

Theoretical studies on assembly, physical stability and dynamics of viruses

All matter has to obey the general laws of physics and living matter is not an exception. Viruses have not only learnt how to cope with them, but have managed to use them for their own survival. In this chapter we will review some of the exciting physics behind viruses and discuss simple physical models that can shed some light on different aspects of the viral life cycle and viral properties. In particular, we will focus on how the structure and shape of the capsid, its assembly and stability, and the entry and exit of viral particles and their genomes can be understood using fundamental physics theories.

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.

Spatial simulations in systems biology: from molecules to cells

Cells are highly organized objects containing millions of molecules. Each biomolecule has a specific shape in order to interact with others in the complex machinery. Spatial dynamics emerge in this system on length and time scales which can not yet be modeled with full atomic detail. This review gives an overview of methods which can be used to simulate the complete cell at least with molecular detail, especially Brownian dynamics simulations. Such simulations require correct implementation of the diffusion-controlled reaction scheme occurring on this level. Implementations and applications of spatial simulations are presented, and finally it is discussed how the atomic level can be included for instance in multi-scale simulation methods.