Dynamic pathways for viral capsid assembly

Trapping of hepatitis B virus capsid assembly intermediates by phenylpropenamide assembly accelerators

Understanding the biological self-assembly process of virus capsids is key to understanding the viral life cycle, as well as serving as a platform for the design of assembly-based antiviral drugs. Here we identify and characterize the phenylpropenamide family of small molecules, known to have antiviral activity in vivo, as assembly effectors of the hepatitis B virus (HBV) capsid. We have found two representative phenylpropenamides to be assembly accelerators, increasing the rate of assembly with only modest increases in the stability of the HBV capsids; these data provide a physical-chemical basis for their antiviral activity. Unlike previously described HBV assembly effectors, the phenylpropenamides do not misdirect assembly; rather, the accelerated reactions proceed on-path to produce morphologically normal capsids. However, capsid assembly in the presence of phenylpropenamides is characterized by kinetic trapping of assembly intermediates. These traps resolve under conditions close to physiological, but we found that trapped intermediates persist under conditions that favor phenylpropenamide binding and strong core protein-protein interactions. The phenylpropenamides serve as chemical probes of the HBV capsid assembly pathway by trapping on-path assembly intermediates, illustrating the governing influence of reaction kinetics on capsid assembly.

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.

A parameter estimation technique for stochastic self-assembly systems and its application to human papillomavirus self-assembly

Virus capsid assembly has been a key model system for studies of complex self-assembly but it does pose some significant challenges for modeling studies. One important limitation is the difficulty of determining accurate rate parameters. The large size and rapid assembly of typical viruses make it infeasible to directly measure coat protein binding rates or deduce them from the relatively indirect experimental measures available. In this work, we develop a computational strategy to deduce coat-coat binding rate parameters for viral capsid assembly systems by fitting stochastic simulation trajectories to experimental measures of assembly progress. Our method combines quadratic response surface and quasi-gradient descent approximations to deal with the high computational cost of simulations, stochastic noise in simulation trajectories and limitations of the available experimental data. The approach is demonstrated on a light scattering trajectory for a human papillomavirus (HPV) in vitro assembly system, showing that the method can provide rate parameters that produce accurate curve fits and are in good concordance with prior analysis of the data. These fits provide an insight into potential assembly mechanisms of the in vitro system and give a basis for exploring how these mechanisms might vary between in vitro and in vivo assembly conditions.

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.

Control of pathways and yields of protein crystallization through the interplay of nonspecific and specific attractions

We use computer simulation to study crystal-forming model proteins equipped with interactions that are both orientationally specific and nonspecific. Distinct dynamical pathways of crystal formation can be selected by tuning the strengths of these interactions. When the nonspecific interaction is strong, liquidlike clustering can precede crystallization; when it is weak, growth can proceed via ordered nuclei. Crystal yields are in certain parameter regimes enhanced by the nonspecific interaction, even though it promotes association without local crystalline order. Our results suggest that equipping nanoscale components with weak nonspecific interactions (such as depletion attractions) can alter both their dynamical pathway of assembly and optimize the yield of the resulting material.

Understanding the concentration dependence of viral capsid assembly kinetics--the origin of the lag time and identifying the critical nucleus size

The kinetics for the assembly of viral proteins into a population of capsids can be measured in vitro with size exclusion chromatography or dynamic light scattering, but extracting mechanistic information from these studies is challenging. For example, it is not straightforward to determine the critical nucleus size or the elongation time (the time required for a nucleus to grow to completion). In this work, we study theoretical and computational models for capsid assembly to show that the critical nucleus size can be determined from the concentration dependence of the assembly half-life and that the elongation time is revealed by the length of the lag phase. Furthermore, we find that the system becomes kinetically trapped when nucleation becomes fast compared to elongation. Implications of this constraint for determining elongation mechanisms from experimental assembly data are discussed.

Thermodynamics of nanospheres encapsulated in virus capsids

We investigate the thermodynamics of complexation of functionalized charged nanospheres with viral proteins. The physics of this problem is governed not only by electrostatic interaction between the proteins and the nanosphere cores (screened by salt ions), but also by configurational degrees of freedom of the charged protein N tails. We approach the problem by constructing an appropriate complexation free-energy functional. On the basis of both numerical and analytical studies of this functional we construct the phase diagram for the assembly which contains the information on the assembled structures that appear in the thermodynamical equilibrium, depending on the size and surface charge density of the nanosphere cores. We show that both the nanosphere core charge and its radius determine the size of the capsid that forms around the core.

Extracting bulk properties of self-assembling systems from small simulations

For systems that self-assemble into finite-sized objects, it is sometimes convenient to compute the thermodynamics for a small system where a single assembly can form. However, we show that in the canonical ensemble the use of small systems can lead to significant finite-size effects due to the suppression of concentration fluctuations. We introduce methods for estimating the bulk yields from simulations of small systems and for following the convergence of yields with system size, under the assumptions that the various species behave ideally.

Modelling the self-assembly of virus capsids

We use computer simulations to study a model, first proposed by Wales (2005 Phil. Trans. R. Soc. A 363 357), for the reversible and monodisperse self-assembly of simple icosahedral virus capsid structures. The success and efficiency of assembly as a function of thermodynamic and geometric factors can be qualitatively related to the potential energy landscape structure of the assembling system. Even though the model is strongly coarse-grained, it exhibits a number of features also observed in experiments, such as sigmoidal assembly dynamics, hysteresis in capsid formation and numerous kinetic traps. We also investigate the effect of macromolecular crowding on the assembly dynamics. Crowding agents generally reduce capsid yields at optimal conditions for non-crowded assembly, but may increase yields for parameter regimes away from the optimum. Finally, we generalize the model to a larger triangulation number T = 3, and observe assembly dynamics more complex than that seen for the original T = 1 model.

Self-assembly of three-legged patchy particles into polyhedral cages

The self-assembly of rigid three-legged building blocks into polyhedral cages is investigated by patchy particle simulations. A four-site anisotropic interaction potential is introduced to make pairs of overlapping legs bind in an anti-parallel fashion, thereby forming the edges of a polyhedron of pentagons and hexagons. A torsional potential, reflecting an asymmetry or polarity in the legs' binding potential, proves crucial for the successful formation of closed fullerene-like cages. Self-assembly proceeds by a nucleation-and-growth mechanism, with a high success rate of cage closure. The size distribution of the self-assembled buckyballs is largely determined by the pucker angle of the particle. Nature explores a similar building block, the clathrin triskelion, to regulate vesicle formation at the cell membrane during endocytosis.

Studies of reversible capsid shell growth

Results from molecular dynamics simulations of simple, structured particles capable of self-assembling into polyhedral shells are described. The analysis focuses on the growth histories of individual shells in the presence of an explicit solvent and the nature of the events along their growth pathways; the results provide further evidence of the importance of reversibility in the assembly process. The underlying goal of this approach is the modeling of virus capsid growth, a phenomenon at the submicroscopic scale that, despite its importance, is little understood.

Steric and electrostatic complementarity in the assembly of two-dimensional virus arrays

A highly ordered assembly of biological molecules provides a powerful means to study the organizational principles of objects at the nanoscale. Two-dimensional cowpea mosaic virus arrays were assembled in an ordered manner on mica using osmotic depletion effects and a drop-and-dry method. The packing of the virus array was controlled systematically from rhombic packing to hexagonal packing by modulating the concentrations of poly(ethylene glycol) surfactant in the virus solutions. The orientation and packing symmetry of the virus arrays were found to be tuned by the concentrations of surfactants in the sample solutions. A phenomenological model for the present system is proposed to explain the assembly array morphology under the influence of the surfactant. Steric and electrostatic complementarity of neighboring virus capsids is found to be the key factors in controlling the symmetry of packing.

Coarse-grained protein-protein stiffnesses and dynamics from all-atom simulations

Large protein assemblies, such as virus capsids, may be coarse-grained as a set of rigid units linked by generalized (rotational and stretching) harmonic springs. We present an ab initio method to obtain the elastic parameters and overdamped dynamics for these springs from all-atom molecular-dynamics simulations of one pair of units at a time. The computed relaxation times of this pair give a consistency check for the simulation, and we can also find the corrective force needed to null systematic drifts. As a first application we predict the stiffness of an HIV capsid layer and the relaxation time for its breathing mode.

Hydrogen bond migration between molecular sites observed with ultrafast 2D IR chemical exchange spectroscopy

Hydrogen-bonded complexes between phenol and phenylacetylene are studied using ultrafast two-dimensional infrared (2D IR) chemical exchange spectroscopy. Phenylacetylene has two possible pi hydrogen bonding acceptor sites (phenyl or acetylene) that compete for hydrogen bond donors in solution at room temperature. The OD stretch frequency of deuterated phenol is sensitive to which acceptor site it is bound. The appearance of off-diagonal peaks between the two vibrational frequencies in the 2D IR spectrum reports on the exchange process between the two competitive hydrogen-bonding sites of phenol-phenylacetylene complexes in the neat phenylacetylene solvent. The chemical exchange process occurs in approximately 5 ps and is assigned to direct hydrogen bond migration along the phenylacetylene molecule. Other nonmigration mechanisms are ruled out by performing 2D IR experiments on phenol dissolved in the phenylacetylene/carbon tetrachloride mixed solvent. The observation of direct hydrogen bond migration can have implications for macromolecular systems.

Self-assembly of monodisperse clusters: Dependence on target geometry

We apply a simple model system of patchy particles to study monodisperse self-assembly using the Platonic solids as target structures. We find marked differences between the assembly behaviors of the different systems. Tetrahedra, octahedral, and icosahedra assemble easily, while cubes are more challenging and dodecahedra do not assemble. We relate these differences to the kinetics and thermodynamics of assembly, with the formation of large disordered aggregates a particular important competitor to correct assembly. In particular, the free energy landscapes of those targets that are easy to assemble are funnel-like, whereas for the dodecahedral system the landscape is relatively flat with little driving force to facilitate escape from disordered aggregates.

Conformational changes in the hepatitis B virus core protein are consistent with a role for allostery in virus assembly

In infected cells, virus components must be organized at the right place and time to ensure assembly of infectious virions. From a different perspective, assembly must be prevented until all components are available. Hypothetically, this can be achieved by allosterically controlling assembly. Consistent with this hypothesis, here we show that the structure of the hepatitis B virus (HBV) core protein dimer, which can spontaneously self-assemble, is incompatible with capsid assembly. Systematic differences between core protein dimer and capsid conformations demonstrate linkage between the intradimer interface and interdimer contact surface. These structures also provide explanations for the capsid-dimer selectivity of some antibodies and the activities of assembly effectors. Solution studies suggest that the assembly-inactive state is more accurately an ensemble of conformations. Simulations show that allostery supports controlled assembly and results in capsids that are resistant to dissociation. We propose that allostery, as demonstrated in HBV, is common to most self-assembling viruses.

Emergent complexity from simple anisotropic building blocks: shells, tubes, and spirals

We describe a remarkably simple, generic, coarse-grained model involving anisotropic interactions, and characterize the global minima for clusters as a function of various parameters. Appropriate choices for the anisotropic interactions can reproduce a wide variety of complex morphologies as global minima, including spheroidal shells, tubular, helical and even head-tail morphologies, elucidating the physical principles that drive the assembly of these mesoscopic structures. Our model captures several experimental observations, such as the existence of competing morphologies, capsid polymorphism, and the effect of scaffolding proteins on capsid assembly.

On the origin of cells and viruses: primordial virus world scenario

It is proposed that the precellular stage of biological evolution unraveled within networks of inorganic compartments that harbored a diverse mix of virus‐like genetic elements. This stage of evolution might makes up the Last Universal Cellular Ancestor (LUCA) that more appropriately could be denoted Last Universal Cellular Ancestral State (LUCAS). Such a scenario recapitulates the ideas of J. B. S. Haldane sketched in his classic 1928 essay. However, unlike in Haldane's day, considerable support for this scenario exits today: lack of homology between core DNA replication system components in archaea and bacteria, distinct membrane chemistries and enzymes of lipid biosynthesis in archaea and bacteria, spread of several viral hallmark genes among diverse groups of viruses, and the extant archaeal and bacterial chromosomes appear to be shaped by accretion of diverse, smaller replicons. Under the viral model of precellular evolution, the key components of cells originated as components of virus‐like entities. The two surviving types of cellular life forms, archaea and bacteria, might have emerged from the LUCAS independently, along with, probably, numerous forms now extinct.

Viral assembly: a molecular modeling perspective

Icosahedral viruses are among the smallest and simplest of biological systems. The investigation of their structures represented the first step toward the establishment of molecular biophysics, over half a century ago. Many research groups are now pursuing investigations of viral assembly, a process that could offer new opportunities for the design of antiviral drugs and novel nanoparticles. A variety of experimental, theoretical and computational methods have been brought to bear on the study of virus structure and assembly. In this Perspective we review the contributions of theoretical and computational approaches to our understanding of the structure, energetics, thermodynamics and assembly of DNA bacteriophage and single-stranded icosahedral RNA viruses.

Hierarchical self-assembly of asymmetric amphiphatic spherical colloidal particles

From dumbbells to fcc crystals, we study the self-assembly pathway of amphiphatic spherical colloidal particles as a function of the size of the hydrophobic region using molecular-dynamics simulations. Specifically, we analyze how local interparticle interactions correlate to the final self-assembled aggregate and how they affect the dynamical pathway of structure formation. We present a detailed diagram separating the many phases that we find for different sizes of the hydrophobic area and uncover a narrow region where particles self-assemble into hollow faceted cages that could potentially find interesting engineering applications.

Electrostatic regulation of genome packaging in human hepatitis B virus

Hepatitis B virus (HBV) is a contagious human pathogen causing liver diseases such as cirrhosis and hepatocellular carcinoma. An essential step during HBV replication is packaging of a pregenomic (pg) RNA within the capsid of core antigens (HBcAgs) that each contains a flexible C-terminal tail rich in arginine residues. Mutagenesis experiments suggest that pgRNA encapsidation hinges on its strong electrostatic interaction with oppositely charged C-terminal tails of the HBcAgs, and that the net charge of the capsid and C-terminal tails determines the genome size and nucleocapsid stability. Here, we elucidate the biophysical basis for electrostatic regulation of pgRNA packaging in HBV by using a coarse-grained molecular model that explicitly accounts for all nonspecific interactions among key components within the nucleocapsid. We find that for mutants with variant C-terminal length, an optimal genome size minimizes an appropriately defined thermodynamic free energy. The thermodynamic driving force of RNA packaging arises from a combination of electrostatic interactions and molecular excluded-volume effects. The theoretical predictions of the RNA length and nucleocapsid internal structure are in good agreement with available experiments for the wild-type HBV and mutants with truncated HBcAg C-termini.

Invariant polymorphism in virus capsid assembly

Directed self-assembly of designed viral capsids holds significant potential for applications in materials science and medicine. However, the complexity of preparing these systems for assembly and the difficulty of quantitative experimental measurements on the assembly process have limited access to critical mechanistic questions that dictate the final product yields and isomorphic forms. Molecular simulations provide a means of elucidating self-assembly of viral proteins into icosahedral capsids and are the focus of the present study. Using geometrically realistic coarse-grained models with specialized molecular dynamics methods, we delineate conditions of temperature and coat protein concentration that lead to the spontaneous self-assembly of T = 1 and T = 3 icosahedral capsids. In addition to the primary product of icosahedral capsids, we observe a ubiquitous presence of nonicosahedral yet highly symmetric and enclosed aberrant capsules in both T = 1 and T = 3 systems. This polymorphism in assembly products recapitulates the scope and morphology of particle types that have been observed in mis-assembly experiments of virus capsids. Moreover, we find that this structural polymorphism in the end point structures is an inherent property of the coat proteins and arises from condition-dependent kinetic mechanisms that are independent of the elemental mechanisms of capsid growth (as long as the building blocks of the coat proteins are all monomeric, dimeric, or trimeric) and the capsid T number. The kinetic mechanisms responsible for self-assembly of icosahedral capsids and aberrant capsules are deciphered; the self-assembly of icosahedral capsids requires a high level of assembly fidelity, whereas self-assembly of nonicosahedral capsules is a consequence of an off-pathway mechanism that is prevalent under nonoptimal conditions of temperature or protein concentration during assembly. The latter case involves kinetically trapped dislocations of pentamer-templated proteins with hexameric organization. These findings provide insights into the complex processes that govern viral capsid assembly and suggest some features of the assembly process that can be exploited to control the assembly of icosahedral capsids and nonicosahedral capsules.

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.

The self-assembly of DNA Holliday junctions studied with a minimal model

In this paper, we explore the feasibility of using coarse-grained models to simulate the self-assembly of DNA nanostructures. We introduce a simple model of DNA where each nucleotide is represented by two interaction sites corresponding to the sugar-phosphate backbone and the base. Using this model, we are able to simulate the self-assembly of both DNA duplexes and Holliday junctions from single-stranded DNA. We find that assembly is most successful in the temperature window below the melting temperatures of the target structure and above the melting temperature of misbonded aggregates. Furthermore, in the case of the Holliday junction, we show how a hierarchical assembly mechanism reduces the possibility of becoming trapped in misbonded configurations. The model is also able to reproduce the relative melting temperatures of different structures accurately and allows strand displacement to occur.

The role of collective motion in examples of coarsening and self-assembly

The simplest prescription for building a patterned structure from its constituents is to add particles, one at a time, to an appropriate template. However, self-organizing molecular and colloidal systems in nature can evolve in much more hierarchical ways. Specifically, constituents (or clusters of constituents) may aggregate to form clusters (or clusters of clusters) that serve as building blocks for later stages of assembly. Here we evaluate the character and consequences of such collective motion in a set of prototypical assembly processes. We do so using computer simulations in which a system's capacity for hierarchical dynamics can be controlled systematically. By explicitly allowing or suppressing collective motion, we quantify its effects. We find that coarsening within a two dimensional attractive lattice gas (and an analogous off-lattice model in three dimensions) is naturally dominated by collective motion over a broad range of temperatures and densities. Under such circumstances, cluster mobility inhibits the development of uniform coexisting phases, especially when macroscopic segregation is strongly favored by thermodynamics. By contrast, the assembly of model viral capsids is not frustrated but is instead facilitated by collective moves, which promote the orderly binding of intermediates consisting of several monomers.

Mechanical properties of the icosahedral shell of southern bean mosaic virus: a molecular dynamics study

The mechanical properties of viral shells are crucial for viral assembly and infection. To study their distribution and heterogeneity on the viral surface, we performed atomistic force-probe molecular dynamics simulations of the complete shell of southern bean mosaic virus, a prototypical T = 3 virus, in explicit solvent. The simulation system comprised more than 4,500,000 atoms. To facilitate direct comparison with atomic-force microscopy (AFM) measurements, a Lennard-Jones sphere was used as a model of the AFM tip, and was pushed with different velocities toward the capsid protein at 19 different positions on the viral surface. A detailed picture of the spatial distribution of elastic constants and yielding forces was obtained that can explain corresponding heterogeneities observed in previous AFM experiments. Our simulations reveal three different deformation regimes: a prelinear regime of outer surface atom rearrangements, a linear regime of elastic capsid deformation, and a rearrangement regime that describes irreversible structural changes and the transition from elastic to plastic deformation. For both yielding forces and elastic constants, a logarithmic velocity dependency is evident over nearly two decades, the explanation for which requires including nonequilibrium effects within the established theory of enforced barrier crossing.

Energy landscapes for shells assembled from pentagonal and hexagonal pyramids

We present new rigid body potentials that should favour efficient self-assembly of pentagonal and hexagonal pyramids into icosahedral shells over a wide range of temperature. By adding an extra repulsive site opposite the existing apex sites of the pyramids considered in a previously published model, frustrated energy landscapes are transformed into systems identified with self-assembling properties. The extra interaction may be considered analogous to a hydrophobic-hydrophilic repulsion, as in micelle formation.

The impact of conformational fluctuations on self-assembly: cooperative aggregation of archaeal chaperonin proteins

Protein complexes called rosettasomes self-assemble in solution to form large-scale filamentous and planar structures. The relative abundance of these aggregates varies abruptly with environmental conditions and sample composition. Our simulations of a model of patchy nanoparticles can reproduce this sharp crossover, but only if particles are allowed to switch between two internal states favoring different geometries of local binding. These results demonstrate how local conformational adaptivity can fundamentally influence the cooperativity of pattern-forming dynamics.

The thermodynamics of virus capsid assembly

Virus capsid assembly is a critical step in the viral life cycle. The underlying basis of capsid stability is key to understanding this process. Capsid subunits interact with weak individual contact energies to form a globally stable icosahedral lattice; this structure is ideal for encapsidating the viral genome and host partners and protecting its contents upon secretion, yet the unique properties of its assembly and inter-subunit contacts allow the capsid to dissociate upon entering a new host cell. The stability of the capsid can be analyzed by treating capsid assembly as an equilibrium polymerization reaction, modified from the traditional polymer model to account for the fact that a separate nucleus is formed for each individual capsid. From the concentrations of reactants and products in an equilibrated assembly reaction, it is possible to extract the thermodynamic parameters of assembly for a wide array of icosahedral viruses using well-characterized biochemical and biophysical methods. In this chapter we describe this basic analysis and provide examples of thermodynamic assembly data for several different icosahedral viruses. These data provide new insights into the assembly mechanisms of spherical virus capsids, as well as into the biology of the viral life cycle.

Size regulation of ss-RNA viruses

While a monodisperse size distribution is common within one kind of spherical virus, the size of viral shells varies from one type of virus to another. In this article, we investigate the physical mechanisms underlying the size selection among spherical viruses. In particular, we study the effect of genome length and genome and protein concentrations on the size of spherical viral capsids in the absence of spontaneous curvature and bending energy. We find that the coat proteins could well adjust the size of the shell to the size of their genome, which in turn depends on the number of charges on it. Furthermore, we find that different stoichiometric mixtures of proteins and genome can produce virus particles of various sizes, consistent with in vitro experiments.

Generalized structural polymorphism in self-assembled viral particles

The protein shells, called capsids, of nearly all spherical viruses adopt icosahedral symmetry; however, self-assembly of such empty structures often occurs with multiple misassembly steps resulting in the formation of aberrant structures. Using simple models that represent the coat proteins preassembled in the two different predetermined species that are common motifs of viral capsids (i.e., pentameric and hexameric capsomers), we perform molecular dynamics simulations of the spontaneous self-assembly of viral capsids of different sizes containing T = 1,3,4,7,9,12,13,16, and 19 proteins in their icosahedral repeating unit. We observe, in addition to icosahedral capsids, a variety of nonicosahedral yet highly ordered and enclosed capsules. Such structural polymorphism is demonstrated to be an inherent property of the coat proteins, independent of the capsid complexity and the elementary kinetic mechanisms. Moreover, there exist two distinctive classes of polymorphic structures: aberrant capsules that are larger than their respective icosahedral capsids, in T = 1-7 systems; and capsules that are smaller than their respective icosahedral capsids when T = 7-19. Different kinetic mechanisms responsible for self-assembly of those classes of aberrant structures are deciphered, providing insights into the control of the self-assembly of icosahedral capsids.

Mechanisms of size control and polymorphism in viral capsid assembly

We simulate the assembly dynamics of icosahedral capsids from subunits that interconvert between different conformations (or quasi-equivalent states). The simulations identify mechanisms by which subunits form empty capsids with only one morphology but adaptively assemble into different icosahedral morphologies around nanoparticle cargoes with varying sizes, as seen in recent experiments with brome mosaic virus (BMV) capsid proteins. Adaptive cargo encapsidation requires moderate cargo-subunit interaction strengths; stronger interactions frustrate assembly by stabilizing intermediates with incommensurate curvature. We compare simulation results to experiments with cowpea chlorotic mottle virus empty capsids and BMV capsids assembled on functionalized nanoparticles and suggest new cargo encapsidation experiments. Finally, we find that both empty and templated capsids maintain the precise spatial ordering of subunit conformations seen in the crystal structure even if interactions that preserve this arrangement are favored by as little as the thermal energy, consistent with experimental observations that different subunit conformations are highly similar.

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.

Role of reversibility in viral capsid growth: a paradigm for self-assembly

Self-assembly at submicroscopic scales is an important but little understood phenomenon. A prominent example is virus capsid growth, whose underlying behavior can be modeled using simple particles that assemble into polyhedral shells. Molecular dynamics simulation of shell formation in the presence of an atomistic solvent provides new insight into the self-assembly mechanism, notably that growth proceeds via a cascade of strongly reversible steps and, despite the large variety of possible intermediates, only a small fraction of highly bonded forms appear on the pathway.

Burst analysis spectroscopy: a versatile single-particle approach for studying distributions of protein aggregates and fluorescent assemblies

Many essential cellular functions depend on the assembly and disassembly of macromolecular complexes. The size, form, and distribution of these assemblies can be heterogeneous and complex, rendering their detailed characterization difficult. Here we describe a simple non-correlation-based method capable of directly measuring population distributions at very low sample concentrations. Specifically, we exploit the highest signal-to-noise light bursts from single fluorescent particles transiting a confocal excitation spot to recursively determine the brightness and size distribution of complex mixtures of fluorescent objects. We refer to this method as burst analysis spectroscopy (BAS) and demonstrate the sensitivity of this technique by examining the free-solution, time-resolved distribution of assembled protein aggregates by using two fluorescently labeled proteins: the aggregation-prone, chaperonin-dependent, folding model protein ribulose-bisphosphate carboxylase/oxygenase (RuBisCO), and an amyloidogenic fragment of the yeast prion protein Sup35. We find that the assembly kinetics of both proteins display complex multimodal behavior not readily quantifiable with other methods.

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.

Tilable nature of virus capsids and the role of topological constraints in natural capsid design

Virus capsids are highly specific assemblies that are formed from a large number of often chemically identical capsid subunits. In the present paper we ask to what extent these structures can be viewed as mathematically tilable objects using a single two-dimensional tile. We find that spherical viruses from a large number of families-eight out of the twelve studied-qualitatively possess properties that allow their representation as two-dimensional monohedral tilings of a bound surface, where each tile represents a subunit. This we did by characterizing the extent to which individual spherical capsids display subunit-subunit (1) holes, (2) overlaps, and (3) gross structural variability. All capsids with T numbers greater than 1 from the Protein Data Bank, with homogeneous protein composition, were used in the study. These monohedral tilings, called canonical capsids due to their platonic (mathematical) form, offer a mathematical segue into the structural and dynamical understanding of not one, but a large number of virus capsids. From our data, it appears as though one may only break the long-standing rules of quasiequivalence by the introduction of subunit-subunit structural variability, holes, and gross overlaps into the shell. To explore the utility of canonical capsids in understanding structural aspects of such assemblies, we used graph theory and discrete geometry to enumerate the types of shapes that the tiles (and hence the subunits) must possess. We show that topology restricts the shape of the face to a limited number of five-sided prototiles, one of which is the "bisected trapezoid" that is a platonic representation of the most ubiquitous capsid subunit shape seen in nature (the trapezoidal jelly-roll motif). This motif is found in a majority of seemingly unrelated virus families that share little to no host, size, or amino acid sequence similarity. This suggests that topological constraints may exhibit dominant roles in the natural design of biological assemblies, while having little effect on amino acid sequence similarity.