Kinetics of viral self-assembly: role of the single-stranded RNA antenna

Escaping Kinetic Traps Using Nonreciprocal Interactions

Kinetic traps are a notorious problem in equilibrium statistical mechanics, where temperature quenches ultimately fail to bring the system to low energy configurations. Using multifarious self-assembly as a model system, we introduce a mechanism to escape kinetic traps by utilizing nonreciprocal interactions between components. Introducing nonequilibrium effects offered by broken action-reaction symmetry in the system pushes the trajectory of the system out of arrested dynamics. The dynamics of the model is studied using tools from the physics of interfaces and defects. Our proposal can find applications in self-assembly, glassy systems, and systems with arrested dynamics to facilitate escape from local minima in rough energy landscapes.

Crystal structure of the potato leafroll virus coat protein and implications for viral assembly

Luteoviruses, poleroviruses, and enamoviruses are insect-transmitted, agricultural pathogens that infect a wide array of plants, including staple food crops. Previous cryo-electron microscopy studies of virus-like particles show that luteovirid viral capsids are built from a structural coat protein that organizes with T = 3 icosahedral symmetry. Here, we present the crystal structure of a truncated version of the coat protein monomer from potato leafroll virus at 1.80-Å resolution. In the crystal lattice, monomers pack into flat sheets that preserve the two-fold and three-fold axes of icosahedral symmetry and show minimal structural deviations when compared to the full-length subunits of the assembled virus-like particle. These observations have important implications in viral assembly and maturation and suggest that the CP N-terminus and its interactions with RNA play an important role in generating capsid curvature.

Physics of viral dynamics

Viral capsids are often regarded as inert structural units, but in actuality they display fascinating dynamics during different stages of their life cycle. With the advent of single-particle approaches and high-resolution techniques, it is now possible to scrutinize viral dynamics during and after their assembly and during the subsequent development pathway into infectious viruses. In this Review, the focus is on the dynamical properties of viruses, the different physical virology techniques that are being used to study them, and the physical concepts that have been developed to describe viral dynamics.

Revealing in real-time a multistep assembly mechanism for SV40 virus-like particles

Many viruses use their genome as template for self-assembly into an infectious particle. However, this reaction remains elusive because of the transient nature of intermediate structures. To elucidate this process, optical tweezers and acoustic force spectroscopy are used to follow viral assembly in real time. Using Simian virus 40 (SV40) virus-like particles as model system, we reveal a multistep assembly mechanism. Initially, binding of VP1 pentamers to DNA leads to a significantly decreased persistence length. Moreover, the pentamers seem able to stabilize DNA loops. Next, formation of interpentamer interactions results in intermediate structures with reduced contour length. These structures stabilize into objects that permanently decrease the contour length to a degree consistent with DNA compaction in wild-type SV40. These data indicate that a multistep mechanism leads to fully assembled cross-linked SV40 particles. SV40 is studied as drug delivery system. Our insights can help optimize packaging of therapeutic agents in these particles.

Measurements of the self-assembly kinetics of individual viral capsids around their RNA genome

Self-assembly is a process in which functional nanoscale structures build themselves, driven by Brownian motion and interactions between components. The term was originally coined to describe the formation of a viral capsid, the protein shell that protects the genome of a virus. Despite decades of study, how capsids self-assemble has remained a mystery, because there were no methods to measure the assembly kinetics of individual capsids. We surmount this obstacle using a sensitive microscopy technique based on laser interferometry. The measurements show that a small nucleus of proteins must form on the viral RNA before the capsid assembles. These results might help researchers design strategies to stop the assembly of pathogenic viruses or to build synthetic nanostructures.

Self-assembly is widely used by biological systems to build functional nanostructures, such as the protein capsids of RNA viruses. But because assembly is a collective phenomenon involving many weakly interacting subunits and a broad range of timescales, measurements of the assembly pathways have been elusive. We use interferometric scattering microscopy to measure the assembly kinetics of individual MS2 bacteriophage capsids around MS2 RNA. By recording how many coat proteins bind to each of many individual RNA strands, we find that assembly proceeds by nucleation followed by monotonic growth. Our measurements reveal the assembly pathways in quantitative detail and also show their failure modes. We use these results to critically examine models of the assembly process.

Real-Time Assembly of Viruslike Nucleocapsids Elucidated at the Single-Particle Level

While the structure of a multitude of viral particles has been resolved to atomistic detail, their assembly pathways remain largely elusive. Key unresolved issues are particle nucleation, particle growth, and the mode of genome compaction. These issues are difficult to address in bulk approaches and are effectively only accessible by the real-time tracking of assembly dynamics of individual particles. This we do here by studying the assembly into rod-shaped viruslike particles (VLPs) of artificial capsid polypeptides. Using fluorescence optical tweezers, we establish that small oligomers perform one-dimensional diffusion along the DNA. Larger oligomers are immobile and nucleate VLP growth. A multiplexed acoustic force spectroscopy approach reveals that DNA is compacted in regular steps, suggesting packaging via helical wrapping into a nucleocapsid. By reporting how real-time assembly tracking elucidates viral nucleation and growth principles, our work opens the door to a fundamental understanding of the complex assembly pathways of both VLPs and naturally evolved viruses.

Orientational phase transitions and the assembly of viral capsids

We present a Landau theory for large-l orientational phase transitions and apply it to the assembly of icosahedral viral capsids. The theory predicts two distinct types of ordering transitions. Transitions dominated by the l=6,10,12, and 18 icosahedral spherical harmonics resemble robust first-order phase transitions that are not significantly affected by chirality. The remaining transitions depend essentially on including mixed l states denoted as l=15+16 corresponding to a mixture of l=15 and l=16 spherical harmonics. The l=15+16 transition is either continuous or weakly first-order and it is strongly influenced by chirality, which suppresses spontaneous chiral symmetry breaking. The icosahedral state is in close competition with states that have tetrahedral, D_{5}, and octahedral symmetries. We present a group-theoretic method to analyze the competition between the different symmetries. The theory is applied to a variety of viral shells.

Illuminating the Reaction Pathways of Viromimetic Assembly

The coassembly of well-defined biological nanostructures relies on a delicate balance between attractive and repulsive interactions between biomolecular building blocks. Viral capsids are a prototypical example, where coat proteins exhibit not only self-interactions but also interact with the cargo they encapsulate. In nature, the balance between antagonistic and synergistic interactions has evolved to avoid kinetic trapping and polymorphism. To date, it has remained a major challenge to experimentally disentangle the complex kinetic reaction pathways that underlie successful coassembly of biomolecular building blocks in a noninvasive approach with high temporal resolution. Here we show how macromolecular force sensors, acting as a genome proxy, allow us to probe the pathways through which a viromimetic protein forms capsids. We uncover the complex multistage process of capsid assembly, which involves recruitment and complexation, followed by allosteric growth of the proteinaceous coat. Under certain conditions, the single-genome particles condense into capsids containing multiple copies of the template. Finally, we derive a theoretical model that quantitatively describes the kinetics of recruitment and growth. These results shed new light on the origins of the pathway complexity in biomolecular coassembly.

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

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.

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.

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.

Scaffold properties are a key determinant of the size and shape of self-assembled virus-derived particles

Controlling the geometry of self-assembly will enable a greater diversity of nanoparticles than now available. Viral capsid proteins, one starting point for investigating self-assembly, have evolved to form regular particles. The polyomavirus SV40 assembles from pentameric subunits and can encapsidate anionic cargos. On short ssRNA (≤814 nt), SV40 pentamers form 22 nm diameter capsids. On RNA too long to fit a T = 1 particle, pentamers forms strings of 22 nm particles and heterogeneous particles of 29-40 nm diameter. However, on dsDNA SV40 forms 50 nm particles composed of 72 pentamers. A 7.2-Å resolution cryo-EM image reconstruction of 22 nm particles shows that they are built of 12 pentamers arranged with T = 1 icosahedral symmetry. At 3-fold vertices, pentamers each contribute to a three-helix triangle. This geometry of interaction is not seen in crystal structures of T = 7 viruses and provides a structural basis for the smaller capsids. We propose that the heterogeneous particles are actually mosaics formed by combining different geometries of interaction from T = 1 capsids and virions. Assembly can be trapped in novel conformations because SV40 interpentamer contacts are relatively strong. The implication is that by virtue of their large catalog of interactions, SV40 pentamers have the ability to self-assemble on and conform to a broad range of shapes.

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.

To build a virus on a nucleic acid substrate

Many viruses package their genomes concomitant with assembly. Here, we show that this reaction can be described by three coefficients: association of capsid protein (CP) to nucleic acid (NA), KNA; CP-CP interaction, ω; and α, proportional to the work required to package NA. The value of α can vary as NA is packaged. A phase diagram of average lnα versus lnω identifies conditions where assembly is likely to fail or succeed. NA morphology can favor (lnα > 0) or impede (lnα < 0) assembly. As lnω becomes larger, capsids become more stable and assembly becomes more cooperative. Where (lnα + lnω) < 0, the CP is unable to contain the NA, so that assembly results in aberrant particles. This phase diagram is consistent with quantitative studies of cowpea chlorotic mottle virus, hepatitis B virus, and simian virus 40 assembling on ssRNA and dsDNA substrates. Thus, the formalism we develop is suitable for describing and predicting behavior of experimental studies of CP assembly on NA.

RNA encapsidation by SV40-derived nanoparticles follows a rapid two-state mechanism

Remarkably, uniform virus-like particles self-assemble in a process that appears to follow a rapid kinetic mechanism. The mechanisms by which spherical viruses assemble from hundreds of capsid proteins around nucleic acid, however, are yet unresolved. Using time-resolved small-angle X-ray scattering (TR-SAXS), we have been able to directly visualize SV40 VP1 pentamers encapsidating short RNA molecules (500mers). This assembly process yields T = 1 icosahedral particles comprised of 12 pentamers and one RNA molecule. The reaction is nearly one-third complete within 35 ms, following a two-state kinetic process with no detectable intermediates. Theoretical analysis of kinetics, using a master equation, shows that the assembly process nucleates at the RNA and continues by a cascade of elongation reactions in which one VP1 pentamer is added at a time, with a rate of approximately 10(9) M(-1) s(-1). The reaction is highly robust and faster than the predicted diffusion limit. The emerging molecular mechanism, which appears to be general to viruses that assemble around nucleic acids, implicates long-ranged electrostatic interactions. The model proposes that the growing nucleo-protein complex acts as an electrostatic antenna that attracts other capsid subunits for the encapsidation process.

Regulation of biochemical reaction rates by flexible tethers

We explore how ligand-receptor binding kinetics can be controlled by tethering the receptor to the end of a flexible polymer. The tether confines the diffusive motion of the receptor thus influencing the rate at which it captures ligands that are free in solution. We compute steady-state collision rates between ligand and receptor for this "tethered-capture" mechanism using a combination of analytic and numerical techniques. In doing so, we uncover a dimensionless control parameter, the "opacity," that determines under what conditions and to what extent a tether regulates the ligand-receptor collision rate. We compute the opacity for a number of different tethering scenarios that appear in biology and use these results to predict the affect of changing the length and flexibility of the tether on the rate at which ligands are captured from solution.

Nematic ordering of polymers in confined geometry applied to DNA packaging in viral capsids

A density functional theory of the spatial distribution and biaxial nematic order of polymers of arbitrary length and rigidity inside a spherical cavity is proposed. The local order of different chain segments is considered as an alignment to a spatially varying director field of cylindrical symmetry. The steric interactions are taken into account in the second virial approximation. Polymer density and orientational order distributions inside the spherically cavity are the principal results. It was found that short and flexible polymers were located at the center of the sphere and were orientationaly disordered. Upon increasing polymer length and/or polymer rigidity, the location of the polymer was continuously shifted toward the surface of the spherical cavity and the polymer segments became gradually more aligned. Parameters have been selected to model the behavior of genomes in spherical viral capsids.

Encapsulation of a polymer by an icosahedral virus

The coat proteins of many viruses spontaneously form icosahedral capsids around nucleic acids or other polymers. Elucidating the role of the packaged polymer in capsid formation could promote biomedical efforts to block viral replication and enable use of capsids in nanomaterials applications. To this end, we perform Brownian dynamics on a coarse-grained model that describes the dynamics of icosahedral capsid assembly around a flexible polymer. We identify several mechanisms by which the polymer plays an active role in its encapsulation, including cooperative polymer-protein motions. These mechanisms are related to experimentally controllable parameters such as polymer length, protein concentration and solution conditions. Furthermore, the simulations demonstrate that assembly mechanisms are correlated with encapsulation efficiency, and we present a phase diagram that predicts assembly outcomes as a function of experimental parameters. We anticipate that our simulation results will provide a framework for designing in vitro assembly experiments on single-stranded RNA virus capsids.

Mechanisms of capsid assembly around a polymer

Capsids of many viruses assemble around nucleic acids or other polymers. Understanding how the properties of the packaged polymer affect the assembly process could promote biomedical efforts to prevent viral assembly or nanomaterials applications that exploit assembly. To this end, we simulate on a lattice the dynamical assembly of closed, hollow shells composed of several hundred to 1000 subunits, around a flexible polymer. We find that assembly is most efficient at an optimum polymer length that scales with the surface area of the capsid; polymers that are significantly longer than optimal often lead to partial-capsids with unpackaged polymer "tails" or a competition between multiple partial-capsids attached to a single polymer. These predictions can be tested with bulk experiments in which capsid proteins assemble around homopolymeric RNA or synthetic polyelectrolytes. We also find that the polymer can increase the net rate of subunit accretion to a growing capsid both by stabilizing the addition of new subunits and by enhancing the incoming flux of subunits; the effects of these processes may be distinguishable with experiments that monitor the assembly of individual capsids.

A theory for viral capsid assembly around electrostatic cores

We develop equilibrium and kinetic theories that describe the assembly of viral capsid proteins on a charged central core, as seen in recent experiments in which brome mosaic virus capsids assemble around nanoparticles functionalized with polyelectrolyte. We model interactions between capsid proteins and nanoparticle surfaces as the interaction of polyelectrolyte brushes with opposite charge using the nonlinear Poisson Boltzmann equation. The models predict that there is a threshold density of functionalized charge, above which capsids efficiently assemble around nanoparticles, and that light scatter intensity increases rapidly at early times without the lag phase characteristic of empty capsid assembly. These predictions are consistent with and enable interpretation of preliminary experimental data. However, the models predict a stronger dependence of nanoparticle incorporation efficiency on functionalized charge density than measured in experiments and do not completely capture a logarithmic growth phase seen in experimental light scatter. These discrepancies may suggest the presence of metastable disordered states in the experimental system. In addition to discussing future experiments for nanoparticle-capsid systems, we discuss broader implications for understanding assembly around charged cores such as nucleic acids.

Nonspecific interactions in spontaneous assembly of empty versus functional single-stranded RNA viruses

We investigate the influence of salt concentration, charge on viral proteins and the length of single-stranded RNA (ssRNA) molecule on the spontaneous assembly of viruses. Only the nonspecific interactions are assumed to guide the assembly, i.e., we exclude any chemical specificity that may lock the viral proteins and ssRNA in preferred configurations. We demonstrate that the electrostatic interactions screened by the salt in the solution impose strong limits on viral composition that can be achieved by spontaneous assembly. In particular, we show that viruses whose ssRNA carries more than twice the amount of charge that is located on the viral proteins, cannot be assembled spontaneously. We find that the spatial distribution of protein charge is important for the energetics of the assembly. We also show that the pressures that act on the viruses as a result of attractive protein-ssRNA electrostatic interactions are at least an order of magnitude smaller than is the case with bacteriophage viruses that contain double-stranded DNA molecule.

Model of human immunodeficiency virus budding and self-assembly: role of the cell membrane

Budding from the plasma membrane of the host cell is an indispensable step in the life cycle of the human immunodeficiency virus (HIV), which belongs to a large family of enveloped RNA viruses, retroviruses. Unlike regular enveloped viruses, retrovirus budding happens concurrently with the self-assembly of the main retrovirus protein subunits (called Gag protein after the name of the genetic material that codes for this protein: Group-specific AntiGen) into spherical virus capsids on the cell membrane. Led by this unique budding and assembly mechanism, we study the free energy profile of retrovirus budding, taking into account the Gag-Gag attraction energy and the membrane elastic energy. We find that if the Gag-Gag attraction is strong, budding always proceeds to completion. During early stage of budding, the zenith angle of partial budded capsids, alpha , increases with time as alpha proportional t1/2. However, if the Gag-Gag attraction is weak, a metastable state of partial budding appears. The zenith angle of these partially spherical capsids is given by alpha0 approximately (tau2/kappasigma)1/4 in a linear approximation, where kappa and sigma are the bending modulus and the surface tension of the membrane, and tau is a line tension of the capsid proportional to the strength of Gag-Gag attraction. Numerically, we find alpha0<0.3pi without any approximations. Using experimental parameters, we show that HIV budding and assembly always proceed to completion in normal biological conditions. On the other hand, by changing Gag-Gag interaction strength or membrane rigidity, it is relatively easy to tune it back and forth between complete budding and partial budding. Our model agrees reasonably well with experiments observing partial budding of retroviruses including HIV.

Radial distribution of RNA genomes packaged inside spherical viruses

The problem of RNA genomes packaged inside spherical viruses is studied. The RNA-capsid attraction is assumed to be nonspecific and occurs at the inner capsid surface only. For weak attraction, RNA concentration is maximum at the center of the capsid to maximize their configurational entropy. For stronger attraction, RNA concentration peaks near the capsid surface. In the latter case, the competition between the branching of RNA secondary structure and its adsorption to the inner capsid results in the formation of a dense layer of RNA near capsid surface. The layer thickness is a slowly varying (logarithmic) function of the capsid inner radius. Consequently, the amount of RNA packaged is proportional to the capsid area (or the number of proteins) instead of its volume. The numerical profiles describe reasonably well the observed RNA concentration profiles of various viruses.

Controlling viral capsid assembly with templating

We develop coarse-grained models that describe the dynamic encapsidation of functionalized nanoparticles by viral capsid proteins. We find that some forms of cooperative interactions between protein subunits and nanoparticles can dramatically enhance rates and robustness of assembly, as compared to the spontaneous assembly of subunits into empty capsids. For large core-subunit interactions, subunits adsorb onto core surfaces en masse in a disordered manner, and then undergo a cooperative rearrangement into an ordered capsid structure. These assembly pathways are unlike any identified for empty capsid formation. Our models can be directly applied to recent experiments in which viral capsid proteins assemble around functionalized inorganic nanoparticles [Sun, Proc. Natl. Acad. Sci. U.S.A. 104, 1354 (2007)]. In addition, we discuss broader implications for understanding the dynamic encapsidation of single-stranded genomic molecules during viral replication and for developing multicomponent nanostructured materials.

Packaging of a polymer by a viral capsid: the interplay between polymer length and capsid size

We report a study of the in vitro self-assembly of virus-like particles formed by the capsid protein of cowpea chlorotic mottle virus and the anionic polymer poly(styrene sulfonate) (PSS) for five molecular masses ranging from 400 kDa to 3.4 MDa. The goal is to explore the effect on capsid size of the competition between the preferred curvature of the protein and the molecular mass of the packaged cargo. The capsid size distribution for each polymer was unimodal, but two distinct sizes were observed: 22 nm for the lower molecular masses, jumping to 27 nm at a molecular mass of 2 MDa. A model is provided for the formation of the virus-like particles that accounts for both the PSS and capsid protein self-interactions and the interactions between the protein and PSS. Our study suggests that the size of the encapsidated polymer cargo is the deciding factor for the selection of one distinct capsid size from several possible sizes with the same inherent symmetry.