Theoretical aspects of virus capsid assembly

We then extend the equilibrium and dynamical approaches to consider the co-assembly of capsid proteins with RNA or other polyanionic cargoes in section IV.

Finally, we conclude with some of the important open questions and ways in which modeling can make a stronger connection with experiments. In the interests of thoroughly examining the capsid assembly process, this review will not consider a number of interesting topics such as the structural dynamics of complete capsids e.

Viruses consist of at least two types of components: the genome, which can be DNA or RNA and can be single-stranded or double-stranded, and a protein shell called a capsid that surrounds and protects the fragile nucleic acid. Viruses vary widely in complexity, ranging from satellite tobacco mosaic virus STMV , whose nucleotide genome encodes for two proteins including the capsid protein [ ] to the giant Megavirus, with a 1,,bp genome encoding for 1, putative proteins [ 14 ] that is larger than some bacterial genomes and encased in two capsids and a lipid bilayer.

The requirement that the viral genome be enclosed in a protective shell severely constrains its length and hence the number of protein sequences that it can encode. As first proposed by Crick and Watson [ 62 ], virus capsids are therefore comprised of numerous copies of one or a few protein sequences, which are usually arranged with a high degree of symmetry in the assembled capsid. Most viruses can be classified as rodlike or spherical, with the capsids of rodlike viruses arranged with helical symmetry around the nucleic acid, such as tobacco mosaic virus TMV , and the capsids of most spherical viruses arranged with icosahedral symmetry.

There are also important exceptions discussed below. The number of protein copies comprising a helical capsid is arbitrary and thus a helical capsid can accommodate a nucleic acid of any length. In contrast, icosahedral capsids are limited by the geometric constraint that at most 60 identical subunits can be arranged into a regular polyhedron.

However, early structural experiments indicated that many spherical capsids contain multiples of 60 proteins.

Caspar and Klug proposed geometrical arguments that describe how multiples of 60 proteins can be arranged with icosahedral symmetry, where individual proteins interact through the same interfaces but take slightly different, or quasi-equivalent, conformations [ 46 ]. Icosahedral symmetry requires exactly 12 pentamers, located at the vertices of an icosahedron inscribed within the capsid.

In brief, a structure with icosahedral symmetry is comprised of 20 identical facets. The facets are equilateral triangles and thus themselves comprise at least 3 identical asymmetric units asu. The only requirement of the asymmetric units is that they are arranged with threefold symmetry, although many capsid proteins have a roughly trapezoidal shape [ ] and it has been argued that this shape is ideal for tiling icosahedral structures [ ].

The Caspar Klug C-K classification system can be obtained starting from a hexagonal lattice as shown in Fig. An edge of the icosahedral facet is defined by starting at the origin and stepping distances h and k along each of the respective lattice vectors. There is an infinite series of such equilateral triangles corresponding to integer values of h and k.

The geometry of icosahedral lattices. To connect this construction to a capsid, note that each pentagon will comprise five proteins in identical environments and each hexagon will comprise six subunits in two different types of local environments, resulting in a total of proteins in three distinct local environments.

C Example icosahedral capsid structures. Structures are shown scaled to actual size, and the protein conformations are indicated by color. In each image the 60 pentameric subunits are colored blue. The images of capsids in C were obtained from the Viper database [ ]. The images in A and B were reprinted from J.

Biol, Johnson and Speir, , Quasi-equivalent viruses: A paradigm for protein assemblies , with permission from Elsevier. From Fig. Given the threefold symmetry of the facet, there are T distinct local environments and thus T distinct asu geometries. The asu i. This has since been found to be correct for many icosahedral viruses, with structural differences between proteins at different quasi-equivalent sites often limited to loops and N- and C-termini. However, there can also be proteins with extensive conformational changes or even different sequences at different sites.

Some examples of these structural differences are reviewed in Refs. Some icosahedral virus capsid structures deviate from the class of lattice structures shown in Fig. For example, the Polyomaviridae , e. Generalizations of the C-K classification scheme have been proposed [ — , , , , ] which can describe polyomavirus capsid shapes. Mannige and Brooks identified a relationship between hexamer shapes and capsid properties such as size [ , ].

There are also non-spherical capsids with aspects of icosahedral symmetry. For example, the mature HIV virus capsid assembles into tubular or conical shapes [ 29 , 90 — 92 ] and some bacteriophages viruses that infect bacteria have capsids which are elongated or prolate icosahedra e. The C-K classification system was extended to describe prolate icosahedra by Moody [ ]. We present some approaches to model the stability and formation of capsids that correspond to C-K structures or their generalizations in section III F.

Viral assembly most generally refers to the process by which the protein capsid s form, the nucleic acid becomes encapsulated within the capsid, membrane coats are acquired if the virus is enveloped , and any maturation steps occur. For many viruses the capsid can form spontaneously, as demonstrated in by the experiments of Fraenkel-Conrat and Williams in which the RNA and capsid protein of TMV spontaneously assembled in vitro to form infectious virions [ 88 ].

The pathway of nucleic acid encapsulation differs dramatically between viruses with single-stranded or double-stranded genomes. Viruses with single-stranded genomes the best studied of which have ssRNA genomes usually assemble spontaneously around their nucleic acid in a single step. This category includes many small spherical plant viruses e.

In many cases the RNA is required for assembly at physiological conditions, but the capsid proteins can assemble without RNA into empty shells in vitro under different ionic strengths or p H. We also can include in this group the Hepadnaviridae family of viruses e. The extreme stiffness of a double-stranded genome the persistence length of dsDNA is 50 nm and the high charge density preclude spontaneous nucleic acid encapsidation.

Of these viruses, the assembly processes have been most thoroughly investigated for dsDNA viruses, such as the tailed bacteriophages, herpes virus and adenovirus.

These viruses assemble an empty capsid, without requiring a nucleic acid at physiological conditions, and a molecular motor which inserts into one vertex of the capsid [ ]. In this review we will focus on the assembly of icosahedral viruses, first discussing the assembly of empty capsids such as occurs during the first step of bacteriophage assembly, and then co-assembly of capsid proteins with RNA, such as occurs during replication of ssRNA viruses, and finally co-assembly with other polyanions in in vitro experiments.

We will not consider the assembly of rod-like viruses. Although not yet completely understood, the assembly process for the rod-like virus TMV has been studied in great detail and has been the subject of several reviews [ 42 , 45 , ] as well as more recent modeling studies [ , ]. The kinetics for spontaneous capsid assembly in vitro have been measured with size exclusion chromatography SEC and X-ray and light scattering e.

Most frequently, the fraction of subunits in capsids or other intermediates has been monitored using size exclusion chromatography SEC and the mass-averaged molecular weight has been estimated with light scattering. The SEC experiments show that under optimal assembly conditions the only species present in detectable concentrations are either complete capsids or small oligomers which we refer to as the basic assembly unit.

The size of the basic assembly unit is virus dependent; e. Provided that intermediate concentrations remain small, the mass-averaged molecular weight and thus the light scattering closely track the fraction capsid measured by SEC.

Example light scattering measurements from Zlotnick and coworkers [ ] are shown in Fig. Light scatter is approximately proportional to the mass-averaged molecular weight of assemblages and, under conditions of productive assembly, closely tracks the fraction of subunits in capsids see text. While these bulk experiments have provided tremendous information about capsid assembly kinetics, it has been difficult to characterize assembly pathways in detail because the intermediates are transient and present only at low levels.

Complementary techniques have begun to address this limitation. Restive pulse sensing was used to track the passage of individual HBV capsids through conical nanopores in a membrane [ , ]. Furthermore, fluorescent labeling of capsid proteins [ ] and in some cases RNA has enabled measuring assembly timescales for capsids in vivo reviewed in Refs.

Even with the experimental capabilities to detect and characterize key intermediates for some viruses, theoretical and computational models are important complements to elucidate assembly pathways and mechanisms.

Each intermediate is a member of a large ensemble of structures and pathways that comprise the overall assembly process for a virus. Furthermore, assembly is driven by collective interactions that are regulated by a tightly balanced competition of forces between individual molecules. It is difficult, with experiments alone, to parse these interactions for those mechanisms and factors that critically influence large-scale properties.

With a model, one can tune each factor individually to learn its affect on the assembly process. In this way, models can be used as a predictive guide to design new experiments.

However, whether at atomistic or coarse-grained resolution, models involve important simplifications or other inaccuracies in their representation of physical systems. Thus, comparison of model predictions to experiments is essential to identify and then refine important model limitations. Iterative prediction, comparison, and model refinement can identify the key factors that govern assembly mechanisms.

Recently, approaches to systematically coarse-grain from atomistic simulations have been applied to interrogate the stability of intact viruses [ 12 , , ] or to estimate subunit positions and orientations from cryo electron microscopy images of the immature HIV capsid [ 15 ]. All-atom molecular dynamics has been applied to specific elements of the assembly reaction [ ].

As we describe below, most efforts to model capsid assembly to date have considered simplified models which retain those aspects of the physics which are hypothesized to be essential; with the validity of the hypothesis to be determined by comparison of model predictions with experiments.

We will begin our discussion of viral assembly by analyzing the formation process of an empty capsid. While this process is most relevant to viruses that first form empty capsids during assembly, ssRNA capsid proteins have also been examined with in vitro experiments in which the ionic strength and p H were adjusted to enable assembly of empty capsids. For assembly to proceed spontaneously, states with capsids must be lower in free energy than a state with only free subunits.

The assembly of disordered subunits and RNA or other components if applicable into an ordered capsid structure reduces their translational and rotational entropy, and thus must be driven by favorable interactions among subunits and any other components that overcome this penalty.

We begin here with the protein-protein interactions; the subunit-RNA interactions that promote ssRNA capsids to assemble around their genome are discussed in section IV and also reviewed in great detail by Siber, Bozic, and Podgornik [ ]. Capsid proteins are typically highly charged and possess binding interfaces that bury large hydrophobic areas. Thus, as with most protein-protein interactions [ 2 ], capsid assembly results from a combination of hydrophobic, electrostatic, van der Waals, and hydrogen bonding interactions.

Covalent interactions typically do not play a role in assembly, although they appear during subsequent maturation steps for a number of viruses e. Importantly, all of these interactions are short-ranged under assembly conditions. Van der Waals interactions and hydrogen bonds operate on length scales of a few angstroms. The hydrophobic interactions are similarly characterized by a length scale of approximately a 0.

The importance of hydrophobic interactions and the sometimes antagonistic contributions of direct electrostatic interactions have been shown by measuring the dimerization affinity of the C-terminal domain of the HIV capsid protein under an extensive series of mutations to the dimerization interface [ 64 , 65 , ]. Furthermore, Ceres and Zlotnick [ 47 ] showed that the thermodynamic stability of HBV capsids increases with both temperature and ionic strength. The increase in stability with temperature suggests that hydrophobic interactions are the dominant driving force [ 48 ].

The increase in stability with ionic strength, on the other hand, suggests that the salt screens repulsive electrostatic interactions which oppose protein association. Several models based on this hypothesis reproduce the dependence of protein-protein interaction strength on ionic strength measured in the experiments [ , , ]. However, it is worth noting that the experiments were performed on capsid protein with the highly charged C-terminal domain truncated, and it is difficult to pinpoint on the crystal structure which charges are responsible for repulsive interactions.

Ceres and Zlotnick [ 47 ] suggested that higher salt concentrations could enhance assembly by favoring a capsid protein conformation which is active for assembly. We now consider the assembly thermodynamics for subunits endowed with the interactions just described. To make the calculation analytically tractable, we assume here that there is one dominant intermediate species for each number of subunits n ; extending this assumption is conceptually straightforward.

The word subunit refers to the basic assembly unit defined in section I B 1. The total free energy F EC for a system of subunits, intermediates, and capsids in solution can be written as. A plausible model for the interaction free energy is. Where n j c is the number of new subunit-subunit contacts formed by the binding of subunit j to the intermediate, g b is the free energy for such a contact, and S degen accounts for degeneracy in the number of ways subunits can bind to or unbind from an intermediate see the s factors in Refs.

These terms are specifed by the geometry of the capsid. Here we have subsumed rotational binding entropy penalties into g b seeRef. As discussed in section II A, g b depends on temperature, ionic strength, and p H. A The assembly model for a dodecahedral capsid and the statistical weights associated with symmetries for the intermediates.

The columns list respectively the number of intermediates, the lowest energy configuration, the degeneracy for adding an additional subunit s n in Eq.

Only the first four and last two intermediates are shown; the full set are given in Ref. B The mole fractions of each intermediate calculated using Eq. Figures reprinted from J. This yields the well-known law of mass action LMA result for intermediate concentrations [ 40 , ]. Due to the constraint Eq. The result for a model dodecahedral capsid comprised of 12 pentagonal subunits is shown in Fig.

Notice that in all cases the capsid protein is almost entirely sequestered either as free subunits or in complete capsids. To emphasize the generic nature of the prediction that intermediate concentrations are negligible at equilibrium, we also consider a continuum model of an assembling shell presented by Zandi and coworkers [ ]. Each partial-capsid intermediate is described as a sphere, with a missing spherical cap.

The resulting profile for G n is shown in Fig. In all cases, the intermediate concentrations are negligible. A Depiction of the continuum model description of partial capsid intermediates considered by Zandi et al [ ].

B Interaction free energy G n as a function of intermediate size n obtained from Eq. C Predicted mole fractions using Eq. Based on the observation that intermediate concentrations are negligible at equilibrium, the equations for capsid assembly thermodynamics can be simplified considerably by neglecting all intermediates except free subunits or complete capsids, so that. In the asymptotic limits Eq. The solution to Eq. We will see however in section III that this trend does not always apply at finite but experimentally relevant timescales due to kinetic effects.

If one or a few ground state capsid geometries are known or pre-assumed , the thermodynamic calculation described above can be extended to describe capsids with larger T numbers in a straightforward manner. Recalling from section IA that icosahedral capsids comprise T different subunit conformations or in some cases protein sequences , the capsid free energy G N cap must be extended to include conformation energies and contact free energies g b which depend on the subunit conformation or species [ 75 ].

Approaches to determine the lowest free energy configuration s for a shell are discussed in section III F 1. Zlotnick and coworkers have shown that the assembly of HBV [ 47 ] can be captured by Eq. The observation that capsid assembly is driven by weak interactions of this magnitude appears to be a general rule for capsid assembly [ ], for reasons discussed in section III.

B Estimated values of g b as a function of temperature and ionic strength. Reprinted with permission from Ceres and Zlotnick, J. The conclusion that most of the interactions driving capsid assembly are weak appears to be broadly valid. However, it is important to note that Eq. We can immediately see that this condition is beyond the reach of many experiments by estimating the timescale for a single subunit to leave an assembled capsid.

Similarly, we show in section III A 1 that the approach of assembly toward equilibrium must lead to ever increasing nucleation barriers. Based on dynamical assembly simulations, our group has estimated that the values of g b could be underestimated by about k B T even for measurements taken at 24 hours due to this effect. Similarly, Singh and Zlotnick [ ] measured substantial hysteresis for the dissociation of HBV capsids under denaturant.

These observations raise the possibility that there are some steps which are irreversible at least on measurement timescales in the assembly process. Irreversible steps late in assembly or during a post-assembly maturation process make sense from the perspective of virus replication, as they would extend the period of time over which the virus can remain complete in infinite dilution and unfavorable environments.

Of course, there must be a mechanism to release the genome once the virus has infected a host. The existence of irreversible steps cannot be directly revealed by assembly data alone. It has been shown that, even if there are assembly steps which are irreversible on relevant timescales late in the assembly cascade, as long as most steps are reversible the assembly data can be fit to Eq. Similarly, comparison of the dynamical equations described in section III B to kinetics data could only reveal the presence of irreversible steps on timescales exceeding the equilibration time associated with the reversible steps e.

The experimental measurements of capsid assembly kinetics described in section I B1 provide important constraints on models of capsid assembly kinetics. At the same time, they present an important opportunity for modeling; because only some intermediates can typically be characterized, models are essential to understand detailed assembly pathways.

In this section we describe different modeling approaches which have been used to predict or understand the assembly kinetics. We begin our description of capsid assembly kinetics by defining the potential rate limiting steps and presenting scaling estimates for their timescales. While our estimates are based on simplified models, we will see in the subsequent sections that many of the predictions remain applicable when additional details are accounted for.

It was noted by Prevelige [ ] that assembly kinetics for icosahedral capsids can be described in terms nucleation and elongation or growth timescales, closely analogous to crystallization. Elongation then refers to the timescale required for a critical nucleus to assemble into a complete capsid. In contrast to crystallization, there can be a well-defined elongation timescale since capsids terminate at a particular size.

For any type of spheroidal shell, including an icosahedral capsid, the first subunits to associate create fewer interparticle contacts than those associating with larger partial capsids see Figs. Under conditions which lead to productive assembly the subunit-subunit binding free energy g b is weak see Fig. Thus the favorable free energy of these contacts is insufficient to compensate for the mixing and rotational entropy penalties incurred upon association, and the small intermediates are unstable.

In fact, the number of interactions depends on the partial capsid geometry, and thus there is an ensemble of critical nuclei with different sizes. It is often assumed that the dominant assembly pathways pass through one or a few critical nuclei with the smallest sizes and thus the assembly probability can be approximated by a single valued function of partial capsid size n i.

For the thermodynamic models of partial capsids presented in section II B the grand free energy is given by. The effect of the shell geometry on the critical nucleus size can be understood elegantly from the continuum model of Zandi et al.

The free energy forms for models which account for the icosahedral geometry of capsid structures are similar to the continuum model just described, except that the critical nucleus tends to correspond to a small polygon, which is a local minimum in the free energy since it corresponds to a local maximum in the number of subunit-subunit contacts see Fig.

Although the assumption that there is one dominant intermediate per partial capsid size is an oversimplification, simulations [ , ] and theory [ 79 , ] indicate that under many conditions assembly pathways predominantly pass through only a few nucleus structures which correspond to completion of small polygons.

Measured critical nucleus sizes under simulated conditions have ranged from subunits [ 75 , 76 , , ]. Experimentally, nucleation has also been shown to correspond to completion of polygons, such as the pentamer of dimers for CCMV [ ] shown in Fig. However, it is likely that intertwining of flexible terminal arms and other subunit conformation changes can provide additional stabilization upon polygon formation.

In the case of MS2, mass spectrometry [ ] identified two polygonal intermediates, which modeling [ ] suggested were each critical nuclei for a different assembly pathway, with the prevalence dictated by binding to RNA. Most computational simulations of icosahedral capsids to this point have not incorporated allostery. Including stabilization due to polygon- or RNA-associated allostery could enable a particular structure to remain as the predominant critical nucleus over a wider range of interaction strengths and subunit concentrations than is predicted by more basic models.

Image of the CCMV pentamer of dimers that experiments [ ] indicate is the critical nucleus. Atoms are shown in van der Waals representation and colored according to their quasi-equivalent conformation, with A monomers in blue and B monomers in red. The association of subunits after nucleation has been described as elongation or growth. In contrast to the transient intermediates found below the critical nucleus size s , intermediates in the growth phase are stable.

Simulations indicate that association usually proceeds by the sequential addition of one or a few subunits at a time, although binding of larger oligomers can be significant at high concentrations or for some subunit interaction geometries [ , , ]. Association of large oligomers can also misdirect the assembly process [ ] section III C. To facilitate the presentation of how the timescales of nucleation and growth depend on system parameters, we first consider a highly simplified assembly reaction.

It was shown that the conclusions from this simplified reaction remained valid when more sophisticated models were considered[ ]. Our reaction is given by:. Similar results can be obtained by assuming that the forward rate constant differs between the two phases [ ]. For the moment, we assume that there is an average nucleus size n nuc. This gives [ 21 ]. This equation can be understood as follows. A comparable expression is derived under a continuum limit in Ref.

However, because free subunits are depleted by assembly, the net nucleation rate never reaches this value and asymptotically approaches zero as the reaction approaches equilibrium. With f c eq as the equilibrium fraction of subunits in complete capsids, which can measured experimentally [ 47 ]. This prediction is compared to simulated assembly times in Fig.

Based on data from Ref. For larger subunit concentrations, new nuclei form faster than existing ones complete assembly, and free subunits are depleted before most capsids finish assembling. These concentrations are related to binding free energies and other parameters by. A kinetic trap arising from depletion of free subunits Eq.

Morozov, Bruinsma, and Rudnick [ ] elegantly recast a model similar to Eq. If the wave does not reach the size of a complete capsid before free subunits are depleted then the system is trapped.

While the continuum model correctly predicts the presence of the free subunit depletion trap, the computer simulations described in sections III B and III C show that productive capsid assembly reactions do not resemble a shockwave.

Because nucleation is a stochastic event, each capsid elongation process starts at a different time; i. This trap can be avoided though for reactions in which subunits assemble around RNA or nanoparticles section IV , provided that subunits are in excess.

A distinctive feature of many capsid assembly kinetics measurements is an initial lag phase before detectable assembly occurs e. Although Zlotnick and coworkers [ 80 , , ] showed that partial capsid intermediates assemble during the lag phase, it has often been assumed that the duration of the lag phase corresponds to the time required for the concentration of critical nuclei to reach steady state, in analogy to models of actin nucleation.

Because the free subunit concentration is nearly constant during the lag phase under most conditions, this relationship holds even when the assumption of constant free subunit concentration is relaxed.

To illustrate this relationship, mean elongation times and lagtimes calculated from Brownian dynamics simulations of a particle-based model see section III C and Ref [ ] are shown in Fig. This correspondence could be tested experimentally by comparing elongation times measured by single molecule experiments [ 24 , , ] with lag times measured by bulk experiments. The lag time is related to the mean elongation time.

A Completion fractions f c measured from Brownian dynamics simulations of a particle based model section III C are shown as a function of time for indicated total subunit volume fractions v T. B The duration of the lag phases from the simulations shown in A are compared to mean capsid elongation times. The crossover volume fraction v c estimated from Eq. The plotted data is from Ref. To this point in this section we have made the simplifying assumption that there is a fixed critical nucleus size.

However, Eq. In Fig. As the reaction begins far out-of-equilibrium there is a relatively small critical nucleus size and correspondingly a small nucleation barrier. Substitution of this free energy barrier into Eq. In other words the reaction only approaches equilibrium asymptotically. As noted in section II B, this effect can lead to underestimating subunit-subunit binding energies when finite-time assembly data is fit to equilibrium expressions.

Lupacchini, M. Finn, U. Lopatin, A. Zlotnick, M. Digestive and Liver Disease. Maguire, Daniel D. Loeb and Adam Zlotnick. Journal of Virology , 89 6 : Single-particle electrophoresis in Nanochannels.

Harms , Daniel G. Haywood , Andrew R. Analytical chemistry , 87 1 : Journal of Virology , 88 24 Biochemistry , 53 34 Transmission electron microscopy enables the reconstruction of the catenane and ring forms of CS2 hydrolase.

Chem Comm , Encapsidated hepatitis B virus reverse transcriptase is poised on an ordered RNA lattice. PNAS , 31 Belloni, L. Li, G. Palumbo, S. Calvo, M. Scaffold properties are a key determinant of the size and shape of self-assembled virus-derived particles. ACS Chemical Biology , 8 12 — ACS Nano , 7 10 — Sarah P. Finn and Adam Zlotnick. Structure , 21 8 Cell Host and Microbe , 14 1 — To build a virus on a nucleic acid substrate. Adam Zlotnick, J. Biophysical J , 7 Structure , 21 1 :6—8.

Genetically altering the thermo and kinetics of Hepatitis B Virus capsid assembly has profound effects on virus replication in cell culture. Journal of Virology , 87 6 Phase diagrams map the properties of anti-viral agents directed against Hepatitis B virus core assembly. Antimicrob Agents Chemother , 57 3 : Mark B. Wang, Inge J. Nolte, Jeroen J. Cornelissen and Jan C. In vitro assembly of an empty picornavirus capsid, an authentic dodecahedral virus.

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Abstract Available from publisher site using DOI. A subscription may be required. Zlotnick A. Though it is a salient point in the life cycle of any virus, the physical chemistry of virus capsid assembly is poorly understood. We have developed general models of capsid assembly that describe the process in terms of a cascade of low order association reactions.

The models predict sigmoidal assembly kinetics, where intermediates approach a low steady state concentration for the greater part of the reaction.