Dynamic pathways for viral capsid assembly

The predictive power of synthetic nucleic acid technologies in RNA biology

CONSPECTUS: The impact of nucleic acid nanotechnology in terms of transforming motifs from biology in synthetic and translational ways is widely appreciated. But it is also emerging that the thinking and vision behind nucleic acids as construction material has broader implications, not just in nanotechnology or even synthetic biology, but can feed back into our understanding of biology itself. Physicists have treated nucleic acids as polymers and connected physical principles to biology by abstracting out the molecular interactions. In contrast, biologists delineate molecular players and pathways related to nucleic acids and how they may be networked. But in vitro nucleic acid nanotechnology has provided a valuable framework for nucleic acids by connecting its biomolecular interactions with its materials properties and thereby superarchitecture ultramanipulation that on multiple occasions has pre-empted the elucidation of how living cells themselves are exploiting these same structural concepts. This Account seeks to showcase the larger implications of certain architectural principles that have arisen from the field of structural DNA/RNA nanotechnology in biology. Here we draw connections between these principles and particular molecular phenomena within living systems that have fed in to our understanding of how the cell uses nucleic acids as construction material to achieve different functions. We illustrate this by considering a few exciting and emerging examples in biology in the context of both switchable systems and scaffolding type systems. Due to the scope of this Account, we will focus our discussion on examples of the RNA scaffold as summarized. In the context of switchable RNA architectures, the synthetic demonstration of small molecules blocking RNA translation preceded the discovery of riboswitches. In another example, it was after the description of aptazymes that the first allosteric ribozyme, glmS, was discovered. In the context of RNA architectures as structural scaffolds, there are clear parallels between DNA origami and the recently emerging molecular mechanism of heterochromatin formation by Xist RNA. Further, following the construction of well-defined 2D DNA-protein architectures, the striking observation of remarkably sculpted 2D RNA-protein hydrogel sheets in Caenorhabditis elegans speaks to the in vivo relevance of designer nucleic acid architectures. It is noteworthy that discoveries of properties in synthetic space seem to precede the uncovering of similar phenomena in vivo.

Using Markov state models to study self-assembly

Markov state models (MSMs) have been demonstrated to be a powerful method for computationally studying intramolecular processes such as protein folding and macromolecular conformational changes. In this article, we present a new approach to construct MSMs that is applicable to modeling a broad class of multi-molecular assembly reactions. Distinct structures formed during assembly are distinguished by their undirected graphs, which are defined by strong subunit interactions. Spatial inhomogeneities of free subunits are accounted for using a recently developed Gaussian-based signature. Simplifications to this state identification are also investigated. The feasibility of this approach is demonstrated on two different coarse-grained models for virus self-assembly. We find good agreement between the dynamics predicted by the MSMs and long, unbiased simulations, and that the MSMs can reduce overall simulation time by orders of magnitude.

A minimal representation of the self-assembly of virus capsids

Viruses are biological nanosystems with a capsid of protein-made capsomer units that encloses and protects the genetic material responsible for their replication. Here we show how the geometrical constraints of the capsomer-capsomer interaction in icosahedral capsids and the requirement of low frustration fix the form of the shortest and universal truncated multipolar expansion of the two-body interaction between capsomers. The structures of many of the icosahedral and related virus capsids are located as single lowest energy states of a potential energy surface built from this interaction. Our minimalist representation is consistent with other models known to produce a controllable and efficient self-assembly, and unveils relevant features of the natural design of the capsids. It promises to be very useful in physical virology and may also be of interest in fields of nanoscience and nanotechnology where similar hollow convex structures are relevant.

Recent Developments in Molecular Simulation Approaches to Study Spherical Virus Capsids

Viruses are a particularly challenging systems to study via molecular simulation methods. Virus capsids typically consist of over 100 subunit proteins and reach dimensions of over 100 nm; solvated viruses capsid systems can be over 1 million atoms in size. In this review, I will present recent developments which have attempted to overcome the significant computational expense to perform simulations which can inform experimental studies, make useful predictions about biological phenomena and calculate material properties relevant to nanotechnology design efforts.

Self-assembly at a nonequilibrium critical point

We use analytic theory and computer simulation to study patterns formed during the growth of two-component assemblies in two and three dimensions. We show that these patterns undergo a nonequilibrium phase transition, at a particular growth rate, between mixed and demixed arrangements of component types. This finding suggests that principles of nonequilibrium statistical mechanics can be used to predict the outcome of multicomponent self-assembly, and suggests an experimental route to the self-assembly of multicomponent structures of a qualitatively defined nature.

Perspective: Coarse-grained models for biomolecular systems

By focusing on essential features, while averaging over less important details, coarse-grained (CG) models provide significant computational and conceptual advantages with respect to more detailed models. Consequently, despite dramatic advances in computational methodologies and resources, CG models enjoy surging popularity and are becoming increasingly equal partners to atomically detailed models. This perspective surveys the rapidly developing landscape of CG models for biomolecular systems. In particular, this review seeks to provide a balanced, coherent, and unified presentation of several distinct approaches for developing CG models, including top-down, network-based, native-centric, knowledge-based, and bottom-up modeling strategies. The review summarizes their basic philosophies, theoretical foundations, typical applications, and recent developments. Additionally, the review identifies fundamental inter-relationships among the diverse approaches and discusses outstanding challenges in the field. When carefully applied and assessed, current CG models provide highly efficient means for investigating the biological consequences of basic physicochemical principles. Moreover, rigorous bottom-up approaches hold great promise for further improving the accuracy and scope of CG models for biomolecular systems.

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

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

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

Icosahedral capsid formation by capsomers and short polyions

Kinetical and structural aspects of the capsomer-polyion co-assembly into icosahedral viruses have been simulated by molecular dynamics using a coarse-grained model comprising cationic capsomers and short anionic polyions. Conditions were found at which the presence of polyions of a minimum length was necessary for capsomer formation. The largest yield of correctly formed capsids was obtained at which the driving force for capsid formation was relatively weak. Relatively stronger driving forces, i.e., stronger capsomer-capsomer short-range attraction and∕or stronger electrostatic interaction, lead to larger fraction of kinetically trapped structures and aberrant capsids. The intermediate formation was investigated and different evolving scenarios were found by just varying the polyion length.

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.

Exploring Energy Landscapes: Metrics, Pathways, and Normal-Mode Analysis for Rigid-Body Molecules

We present new methodology for exploring the energy landscapes of molecular systems, using angle-axis variables for the rigid-body rotational coordinates. The key ingredient is a distance measure or metric tensor, which is invariant to global translation and rotation. The metric is used to formulate a generalized nudged elastic band method for calculating pathways, and a full prescription for normal-mode analysis is described. The methodology is tested by mapping the potential energy and free energy landscape of the water octamer, described by the TIP4P potential.

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

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

Viral genome structures are optimal for capsid assembly

Understanding how virus capsids assemble around their nucleic acid (NA) genomes could promote efforts to block viral propagation or to reengineer capsids for gene therapy applications. We develop a coarse-grained model of capsid proteins and NAs with which we investigate assembly dynamics and thermodynamics. In contrast to recent theoretical models, we find that capsids spontaneously ‘overcharge’; that is, the negative charge of the NA exceeds the positive charge on capsid. When applied to specific viruses, the optimal NA lengths closely correspond to the natural genome lengths. Calculations based on linear polyelectrolytes rather than base-paired NAs underpredict the optimal length, demonstrating the importance of NA structure to capsid assembly. These results suggest that electrostatics, excluded volume, and NA tertiary structure are sufficient to predict assembly thermodynamics and that the ability of viruses to selectively encapsidate their genomic NAs can be explained, at least in part, on a thermodynamic basis.

DOI:http://dx.doi.org/10.7554/eLife.00632.001

Viruses are infectious agents made up of proteins and a genome made of DNA or RNA. Upon infecting a host cell, viruses hijack the cell’s gene expression machinery and force it to produce copies of the viral genome and proteins, which then assemble into new viruses that can eventually infect other host cells. Because assembly is an essential step in the viral life cycle, understanding how this process occurs could significantly advance the fight against viral diseases.

In many viral families, a protein shell called a capsid forms around the viral genome during the assembly process. However, capsids can also assemble around nucleic acids in solution, indicating that a host cell is not required for their formation. Since capsid proteins are positively charged, and nucleic acids are negatively charged, electrostatic interactions between the two are thought to have an important role in capsid assembly. However, it is unclear how structural features of the viral genome affect assembly, and why the negative charge on viral genomes is actually far greater than the positive charge on capsids. These questions are difficult to address experimentally because most of the intermediates that form during virus assembly are too short-lived to be imaged.

Here, Perlmutter et al. have used state of the art computational methods and advances in graphical processing units (GPUs) to produce the most realistic model of capsid assembly to date. They showed that the stability of the complex formed between the nucleic acid and the capsid depends on the length of the viral genome. Yield was highest for genomes within a certain range of lengths, and capsids that assembled around longer or shorter genomes tended to be malformed.

Perlmutter et al. also explored how structural features of the virus—including base-pairing between viral nucleic acids, and the size and charge of the capsid—determine the optimal length of the viral genome. When they included structural data from real viruses in their simulations and predicted the optimal lengths for the viral genome, the results were very similar to those seen in existing viruses. This indicates that the structure of the viral genome has been optimized to promote packaging into capsids. Understanding this relationship between structure and packaging will make it easier to develop antiviral agents that thwart or misdirect virus assembly, and could aid the redesign of viruses for use in gene therapy and drug delivery.

DOI:http://dx.doi.org/10.7554/eLife.00632.002

A new paradigm for the roles of the genome in ssRNA viruses

Recent work with RNA phages and an ssRNA plant satellite virus challenges the widely held view that the sequences and structures of genomic RNAs are unimportant for virion assembly. In the T=3 phages, RNA–coat protein interactions occur throughout the genome, defining the quasiconformers of their protein shells. In the plant virus, there are multiple packaging signals dispersed throughout the genome that overcome electrostatic barriers to protein self-assembly. Both viral coat proteins cause the solution structures of their cognate genomes to collapse into a form that is readily encapsidated in a two-stage assembly process. Such similar behavior in two structurally unrelated viral protein folds implies that this might be a conserved feature of many viral assembly reactions. These results suggest a highly defined structure for the RNA in the virions, consistent with recent structural studies. They also have implications both for subsequent genome release during infection and for the evolution of viral sequences.

Dynamics simulations for engineering macromolecular interactions

The predictable engineering of well-behaved transcriptional circuits is a central goal of synthetic biology. The artificial attachment of promoters to transcription factor genes usually results in noisy or chaotic behaviors, and such systems are unlikely to be useful in practical applications. Natural transcriptional regulation relies extensively on protein-protein interactions to insure tightly controlled behavior, but such tight control has been elusive in engineered systems. To help engineer protein-protein interactions, we have developed a molecular dynamics simulation framework that simplifies features of proteins moving by constrained Brownian motion, with the goal of performing long simulations. The behavior of a simulated protein system is determined by summation of forces that include a Brownian force, a drag force, excluded volume constraints, relative position constraints, and binding constraints that relate to experimentally determined on-rates and off-rates for chosen protein elements in a system. Proteins are abstracted as spheres. Binding surfaces are defined radially within a protein. Peptide linkers are abstracted as small protein-like spheres with rigid connections. To address whether our framework could generate useful predictions, we simulated the behavior of an engineered fusion protein consisting of two 20,000 Da proteins attached by flexible glycine/serine-type linkers. The two protein elements remained closely associated, as if constrained by a random walk in three dimensions of the peptide linker, as opposed to showing a distribution of distances expected if movement were dominated by Brownian motion of the protein domains only. We also simulated the behavior of fluorescent proteins tethered by a linker of varying length, compared the predicted Förster resonance energy transfer with previous experimental observations, and obtained a good correspondence. Finally, we simulated the binding behavior of a fusion of two ligands that could simultaneously bind to distinct cell-surface receptors, and explored the landscape of linker lengths and stiffnesses that could enhance receptor binding of one ligand when the other ligand has already bound to its receptor, thus, addressing potential mechanisms for improving targeted signal transduction proteins. These specific results have implications for the design of targeted fusion proteins and artificial transcription factors involving fusion of natural domains. More broadly, the simulation framework described here could be extended to include more detailed system features such as non-spherical protein shapes and electrostatics, without requiring detailed, computationally expensive specifications. This framework should be useful in predicting behavior of engineered protein systems including binding and dissociation reactions.

Inferring bulk self-assembly properties from simulations of small systems with multiple constituent species and small systems in the grand canonical ensemble

In this paper, we generalize a methodology [T. E. Ouldridge, A. A. Louis, and J. P. K. Doye, J. Phys.: Condens. Matter 22, 104102 (2010)] for dealing with the inference of bulk properties from small simulations of self-assembling systems of characteristic finite size. In particular, schemes for extrapolating the results of simulations of a single self-assembling object to the bulk limit are established in three cases: for assembly involving multiple particle species, for systems with one species localized in space and for simulations in the grand canonical ensemble. Furthermore, methodologies are introduced for evaluating the accuracy of these extrapolations. Example systems demonstrate that differences in cluster concentrations between simulations of a single self-assembling structure and bulk studies of the same model under identical conditions can be large, and that convergence on bulk results as system size is increased can be slow and non-trivial.

Simulations of HIV capsid protein dimerization reveal the effect of chemistry and topography on the mechanism of hydrophobic protein association

Recent work has shown that the hydrophobic protein surfaces in aqueous solution sit near a drying transition. The tendency for these surfaces to expel water from their vicinity leads to self-assembly of macromolecular complexes. In this article, we show with a realistic model for a biologically pertinent system how this phenomenon appears at the molecular level. We focus on the association of the C-terminal domain (CA-C) of the human immunodeficiency virus capsid protein. By combining all-atom simulations with specialized sampling techniques, we measure the water density distribution during the approach of two CA-C proteins as a function of separation and amino acid sequence in the interfacial region. The simulations demonstrate that CA-C protein-protein interactions sit at the edge of a dewetting transition and that this mesoscopic manifestation of the underlying liquid-vapor phase transition can be readily manipulated by biology or protein engineering to significantly affect association behavior. Although the wild-type protein remains wet until contact, we identify a set of in silico mutations, in which three hydrophilic amino acids are replaced with nonpolar residues, that leads to dewetting before association. The existence of dewetting depends on the size and relative locations of substituted residues separated by nanometer length scales, indicating long-range cooperativity and a sensitivity to surface topography. These observations identify important details that are missing from descriptions of protein association based on buried hydrophobic surface area.

Theoretical studies on assembly, physical stability and dynamics of viruses

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

Stochastic dynamics of virus capsid formation: direct versus hierarchical self-assembly

BACKGROUND

In order to replicate within their cellular host, many viruses have developed self-assembly strategies for their capsids which are sufficiently robust as to be reconstituted in vitro. Mathematical models for virus self-assembly usually assume that the bonds leading to cluster formation have constant reactivity over the time course of assembly (direct assembly). In some cases, however, binding sites between the capsomers have been reported to be activated during the self-assembly process (hierarchical assembly).

RESULTS

In order to study possible advantages of such hierarchical schemes for icosahedral virus capsid assembly, we use Brownian dynamics simulations of a patchy particle model that allows us to switch binding sites on and off during assembly. For T1 viruses, we implement a hierarchical assembly scheme where inter-capsomer bonds become active only if a complete pentamer has been assembled. We find direct assembly to be favorable for reversible bonds allowing for repeated structural reorganizations, while hierarchical assembly is favorable for strong bonds with small dissociation rate, as this situation is less prone to kinetic trapping. However, at the same time it is more vulnerable to monomer starvation during the final phase. Increasing the number of initial monomers does have only a weak effect on these general features. The differences between the two assembly schemes become more pronounced for more complex virus geometries, as shown here for T3 viruses, which assemble through homogeneous pentamers and heterogeneous hexamers in the hierarchical scheme. In order to complement the simulations for this more complicated case, we introduce a master equation approach that agrees well with the simulation results.

CONCLUSIONS

Our analysis shows for which molecular parameters hierarchical assembly schemes can outperform direct ones and suggests that viruses with high bond stability might prefer hierarchical assembly schemes. These insights increase our physical understanding of an essential biological process, with many interesting potential applications in medicine and materials science.

Assembly, stability and dynamics of virus capsids

Most viruses use a hollow protein shell, the capsid, to enclose the viral genome. Virus capsids are large, symmetric oligomers made of many copies of one or a few types of protein subunits. Self-assembly of a viral capsid is a complex oligomerization process that proceeds along a pathway regulated by ordered interactions between the participating protein subunits, and that involves a series of (usually transient) assembly intermediates. Assembly of many virus capsids requires the assistance of scaffolding proteins or the viral nucleic acid, which interact with the capsid subunits to promote and direct the process. Once assembled, many capsids undergo a maturation reaction that involves covalent modification and/or conformational rearrangements, which may increase the stability of the particle. The final, mature capsid is a relatively robust protein complex able to protect the viral genome from physicochemical aggressions; however, it is also a metastable, dynamic structure poised to undergo controlled conformational transitions required to perform biologically critical functions during virus entry into cells, intracellular trafficking, and viral genome uncoating. This article provides an updated general overview on structural, biophysical and biochemical aspects of the assembly, stability and dynamics of virus capsids.

Molecular dynamics simulation of reversibly self-assembling shells in solution using trapezoidal particles

The self-assembly of polyhedral shells, each constructed from 60 trapezoidal particles, is simulated using molecular dynamics. The spatial organization of the component particles in this shell is similar to the capsomer proteins forming the capsid of a T=1 virus. Growth occurs in the presence of an atomistic solvent and, under suitable conditions, achieves a high yield of complete shells. The simulations provide details of the structure and lifetime of the particle clusters that appear as intermediate states along the growth pathway, and the nature of the transitions between them. In certain respects the growth of size-60 shells from trapezoidal particles resembles the growth of icosahedral shells from triangular particles studied previously, with reversible bonding playing a major role in avoiding incorrect assembly, although the details differ due to particle shape and bond organization. The strong preference for maximal bonding exhibited by the triangular particle clusters is also apparent for trapezoidal particles, but this is now confined to early growth and is less pronounced as shells approach completion along a variety of pathways.

Influence of fluctuating membranes on self-assembly of patchy colloids

A coarse-grained computational model is used to investigate the effect of a fluid membrane on patchy-particle assembly into biologically relevant structures motivated by viral cores and clathrin. For cores, we demonstrate a nonmonotonic dependence of the promotion of assembly on membrane stiffness. If the membrane is significantly deformable, cores are enveloped in buds, although this effect is suppressed for very flexible membranes. In the less deformable regime, we observe no marked enhancement for cores, even for strong adhesion to the surface. For clathrinlike particles, we again observe the formation of buds, whose morphology depends on membrane flexibility.

Unlocking internal prestress from protein nanoshells

The capsids of icosahedral viruses are closed shells assembled from a hexagonal lattice of proteins with fivefold angular defects located at the icosahedral vertices. Elasticity theory predicts that these disclinations are subject to an internal compressive prestress, which provides an explanation for the link between size and shape of capsids. Using a combination of experiment and elasticity theory we investigate the question of whether macromolecular assemblies are subject to residual prestress, due to basic geometric incompatibility of the subunits. Here we report the first direct experimental test of the theory: by controlled removal of protein pentamers from the icosahedral vertices, we measure the mechanical response of so-called "whiffle ball" capsids of herpes simplex virus, and demonstrate the signature of internal prestress locked into wild-type capsids during assembly.

Early stages of the HIV-1 capsid protein lattice formation

The early stages in the formation of the HIV-1 capsid (CA) protein lattice are investigated. The underlying coarse-grained (CG) model is parameterized directly from experimental data and examined under various native contact interaction strengths, CA dimer interfacial configurations, and local surface curvatures. The mechanism of early contiguous mature-style CA p6 lattice formation is explored, and a trimer-of-dimers structure is found to be crucial for CA lattice production. Quasi-equivalent generation of both the pentamer and hexamer components of the HIV-1 viral CA is also demonstrated, and the formation of pentamers is shown to be highly sensitive to local curvature, supporting the view that such inclusions in high-curvature regions allow closure of the viral CA surface. The complicated behavior of CA lattice self-assembly is shown to be reducible to a relatively simple function of the trimer-of-dimers behavior.

Surveying capsid assembly pathways through simulation-based data fitting

Virus capsid assembly has attracted considerable interest from the biophysical modeling community as a model system for complicated self-assembly processes. Simulation methods have proven valuable for characterizing the space of possible kinetics and mechanisms of capsid assembly, but they have so far been able to say little about the assembly kinetics or pathways of any specific virus. It is not possible to directly measure the detailed interaction rates needed to parameterize a model, and there is only a limited amount of experimental evidence available to constrain possible pathways, with almost all of it gathered from in vitro studies of purified coat proteins. In prior work, we developed methods to address this problem by using simulation-based data-fitting to learn rate parameters consistent with both structure-based rule sets and experimental light-scattering data on bulk assembly progress in vitro. We have since improved these methods and extended them to fit simulation parameters to one or more experimental light-scattering curves. Here, we apply the improved data-fitting approach to three capsid systems-human papillomavirus (HPV), hepatitis B virus (HBV), and cowpea chlorotic mottle virus (CCMV)-to assess both the range of pathway types the methods can learn and the diversity of assembly strategies in use between these viruses. The resulting fits suggest three different in vitro assembly mechanisms for the three systems, with HPV capsids fitting a model of assembly via a nonnucleation-limited pathway of accumulation of individual capsomers while HBV and CCMV capsids fit similar but subtly different models of nucleation-limited assembly through ensembles of pathways involving trimer-of-dimer intermediates. The results demonstrate the ability of such data fitting to learn very different pathway types and show some of the versatility of pathways that may exist across real viruses.

Langevin dynamics simulation of polymer-assisted virus-like assembly

Starting from a coarse grained representation of the building units of the minute virus of mice and a flexible polyelectrolyte molecule, we have explored the mechanism of assembly into icosahedral structures with the help of Langevin dynamics simulations and the parallel tempering technique. Regular icosahedra with appropriate symmetry form only in a narrow range of temperature and polymer length. Within this region of parameters where successful assembly would proceed, we have systematically investigated the growth kinetics. The assembly of icosahedra is found to follow the classical nucleation and growth mechanism in the absence of the polymer, with the three regimes of nucleation, linear growth, and slowing down in the later stage. The calculated average nucleation time obeys the laws expected from the classical nucleation theory. The linear growth rate is found to obey the laws of secondary nucleation as in the case of lamellar growth in polymer crystallization. The same mechanism is seen in the simulations of the assembly of icosahedra in the presence of the polymer as well. The polymer reduces the nucleation barrier significantly by enhancing the local concentration of subunits via adsorbing them on their backbone. The details of growth in the presence of the polymer are also found to be consistent with the classical nucleation theory, despite the smallness of the assembled structures.

Mechanical disassembly of single virus particles reveals kinetic intermediates predicted by theory

New experimental approaches are required to detect the elusive transient intermediates predicted by simulations of virus assembly or disassembly. Here, an atomic force microscope (AFM) was used to mechanically induce partial disassembly of single icosahedral T=1 capsids and virions of the minute virus of mice. The kinetic intermediates formed were imaged by AFM. The results revealed that induced disassembly of single minute-virus-of-mice particles is frequently initiated by loss of one of the 20 equivalent capsomers (trimers of capsid protein subunits) leading to a stable, nearly complete particle that does not readily lose further capsomers. With lower frequency, a fairly stable, three-fourths-complete capsid lacking one pentamer of capsomers and a free, stable pentamer were obtained. The intermediates most frequently identified (capsids missing one capsomer, capsids missing one pentamer of capsomers, and free pentamers of capsomers) had been predicted in theoretical studies of reversible capsid assembly based on thermodynamic-kinetic models, molecular dynamics, or oligomerization energies. We conclude that mechanical manipulation and imaging of simple virus particles by AFM can be used to experimentally identify kinetic intermediates predicted by simulations of assembly or disassembly.

A trimer of dimers is the basic building block for human immunodeficiency virus-1 capsid assembly

Human immunodeficiency virus-1 (HIV-1) capsid protein (CA) has become a target of antiviral drug design in recent years. The recognition that binding of small molecules to the CA protein can result in the perturbation of capsid assembly or disassembly has led to mathematical modeling of the process. Although a number of capsid assembly models have been developed using biophysical parameters of the CA protein obtained experimentally, there is currently no model of CA polymerization that can be practically used to analyze in vitro CA polymerization data to facilitate drug discovery. Herein, we describe an equilibrium model of CA polymerization for the kinetic analysis of in vitro assembly of CA into polymer tubes. This new mathematical model has been used to assess whether a triangular trimer of dimers rather than a hexagonal hexamer can be the basic capsomere building block of CA polymer. The model allowed us to quantify for the first time the affinity for each of the four crucial interfaces involved in the polymerization process and indicated that the trimerization of CA dimers is a relatively slow step in CA polymerization in vitro. For wild-type CA, these four interfaces include the interface between two monomers of a CA dimer (K(D) = 6.6 μM), the interface between any two dimers within a CA trimer of dimers (K(D) = 32 nM), and two types of interfaces between neighboring trimers of dimers, either within the same ring around the perimeter of the polymer tube (K(D) = 438 nM) or from two adjacent rings (K(D) = 147 nM). A comparative analysis of the interface dissociation constants between wild-type and two mutant CA proteins, cross-linked hexamer (A14C/E45C/W184A/M185A) and A14C/E45C, yielded results that are consistent with the trimer of dimers with a triangular geometry being the capsomere building block involved in CA polymer growth. This work provides additional insights into the mechanism of HIV-1 CA assembly and may prove useful in elucidating how small molecule CA binding agents may disturb this essential step in the HIV-1 life cycle.

Differential assembly of Hepatitis B Virus core protein on single- and double-stranded nucleic acid suggest the dsDNA-filled core is spring-loaded

Hepatitis B Virus (HBV) cores assemble on viral RNA, which is reverse transcribed within the core to the partially dsDNA genome of mature HBV. However, constraining dsDNA, a stiff polymer, within a core necessarily requires far greater capsid stability than constraining ssRNA. We hypothesized that, unlike ssRNA, dsDNA would be a poor substrate for assembly. We examined titrations of ssDNA and dsDNA with purified HBV core protein, Cp183, by EMSA, EM, DLS, and etheno-DNA fluorescence. Cp183 bound ssDNA with high affinity to form virus-like capsids. However, Cp183 bound dsDNA poorly, forming a mixture of irregular complexes. Nonetheless, we observed some normal cores in dsDNA assembly reactions, indicating that the energy required to bend DNA could be similar to the protein-protein association energy. This similarity of energies suggests that dsDNA stresses mature HBV cores, in agreement with calculation, which may be the basis for the virus maturation signal and DNA release.

Analyzing mechanisms and microscopic reversibility of self-assembly

We use computer simulations to investigate self-assembly in a system of model chaperonin proteins, and in an Ising lattice gas. We discuss the mechanisms responsible for rapid and efficient assembly in these systems, and we use measurements of dynamical activity and assembly progress to compare their propensities for kinetic trapping. We use the analytic solution of a simple minimal model to illustrate the key features associated with such trapping, paying particular attention to the number of ways that particles can misbind. We discuss the relevance of our results for the design and control of self-assembly in general.

Re-entrant phase behavior for systems with competition between phase separation and self-assembly

In patchy particle systems where there is a competition between the self-assembly of finite clusters and liquid-vapor phase separation, re-entrant phase behavior can be observed, with the system passing from a monomeric vapor phase to a region of liquid-vapor phase coexistence and then to a vapor phase of clusters as the temperature is decreased at constant density. Here, we present a classical statistical mechanical approach to the determination of the complete phase diagram of such a system. We model the system as a van der Waals fluid, but one where the monomers can assemble into monodisperse clusters that have no attractive interactions with any of the other species. The resulting phase diagrams show a clear region of re-entrance. However, for the most physically reasonable parameter values of the model, this behavior is restricted to a certain range of density, with phase separation still persisting at high densities.

On the effect of thermodynamic equilibrium on the assembly efficiency of complex multi-layered virus-like particles (VLP): the case of rotavirus VLP

Previous studies have reported the production of malformed virus-like-particles (VLP) in recombinant host systems. Here we computationally investigate the case of a large triple-layered rotavirus VLP (RLP). In vitro assembly, disassembly and reassembly data provides strong evidence of microscopic reversibility of RLP assembly. Light scattering experimental data also evidences a slow and reversible assembly untypical of kinetic traps, thus further strengthening the fidelity of a thermodynamically controlled assembly. In silico analysis further reveals that under favourable conditions particles distribution is dominated by structural subunits and completely built icosahedra, while other intermediates are present only at residual concentrations. Except for harshly unfavourable conditions, assembly yield is maximised when proteins are provided in the same VLP protein mass composition. The assembly yield decreases abruptly due to thermodynamic equilibrium when the VLP protein mass composition is not obeyed. The latter effect is more pronounced the higher the Gibbs free energy of subunit association is and the more complex the particle is. Overall this study shows that the correct formation of complex multi-layered VLPs is restricted to a narrow range of association energies and protein concentrations, thus the choice of the host system is critical for successful assembly. Likewise, the dynamic control of intracellular protein expression rates becomes very important to minimize wasted proteins.

Quantifying reversibility in a phase-separating lattice gas: an analogy with self-assembly

We present dynamic measurements of a lattice gas during phase separation, which we use as an analogy for self-assembly of equilibrium ordered structures. We use two approaches to quantify the degree of reversibility of this process: First, we count events in which bonds are made and broken; second, we use correlation-response measurements and fluctuation-dissipation ratios to probe reversibility during different time intervals. We show how correlation and response functions can be related directly to microscopic (ir)reversibility and we discuss the time dependence and observable dependence of these measurements, including the role of fast and slow degrees of freedom during assembly.

Mechanisms of kinetic trapping in self-assembly and phase transformation

In self-assembly processes, kinetic trapping effects often hinder the formation of thermodynamically stable ordered states. In a model of viral capsid assembly and in the phase transformation of a lattice gas, we show how simulations in a self-assembling steady state can be used to identify two distinct mechanisms of kinetic trapping. We argue that one of these mechanisms can be adequately captured by kinetic rate equations, while the other involves a breakdown of theories that rely on cluster size as a reaction coordinate. We discuss how these observations might be useful in designing and optimising self-assembly reactions.

Simulated self-assembly of the HIV-1 capsid: protein shape and native contacts are sufficient for two-dimensional lattice formation

We report Monte Carlo simulations of the initial stages of self-assembly of the HIV-1 capsid protein (CA), using a coarse-grained representation that mimics the CA backbone structure and intermolecular contacts observed experimentally. A simple representation of N-terminal domain/N-terminal domain and N-terminal domain/C-terminal domain interactions, coupled with the correct protein shape, is sufficient to drive formation of an ordered lattice with the correct hexagonal symmetry in two dimensions. We derive an approximate concentration/temperature phase diagram for lattice formation, and we investigate the pathway by which the lattice develops from initially separated CA dimers. Within this model, lattice formation occurs in two stages: 1), condensation of CA dimers into disordered clusters; and 2), nucleation of the lattice by the appearance of one hexamer unit within a cluster. Trimers of CA dimers are important early intermediates, and pentamers are metastable within clusters. Introduction of a preformed hexamer at the beginning of a Monte Carlo run does not directly seed lattice formation, but does facilitate the formation of large clusters. We discuss possible connections between these simulations and experimental observations concerning CA assembly within HIV-1 and in vitro.

Energies and pressures in viruses: contribution of nonspecific electrostatic interactions

We summarize some aspects of electrostatic interactions in the context of viruses. A simplified but, within well defined limitations, reliable approach is used to derive expressions for electrostatic energies and the corresponding osmotic pressures in single-stranded RNA viruses and double-stranded DNA bacteriophages. The two types of viruses differ crucially in the spatial distribution of their genome charge which leads to essential differences in their free energies, depending on the capsid size and total charge in a quite different fashion. Differences in the free energies are trailed by the corresponding characteristics and variations in the osmotic pressure between the inside of the virus and the external bathing solution.

A theoretical model for the dynamic structure of hepatitis B nucleocapsid

The genomic material of hepatitis B virus (HBV) is confined within a fenestrated nucleocapsid consisting of 240 identical copies of the capsid protein, which has a rigid core and a positively charged and highly flexible C-terminal domain (CTD). Although previous mutagenesis studies have demonstrated the importance of the CTD in viral RNA packaging and reverse transcription, the microscopic structure of the CTD and its interaction with encapsidated nucleic acids at various stages of viral maturation remain poorly understood. Here, we present a theoretical analysis of the radial distributions of the CTD chains and nucleic acids in the hepatitis B virus nucleocapsid at the beginning and final stages of viral reverse transcription based on classical density functional theory and a coarse-gained model for the pertinent biomolecules. We find that a significant portion of the CTD is exposed at the surface of the RNA-containing immature nucleocapsid and that the CTD is mostly confined within the DNA-containing mature nucleocapsid. Large accumulation of cations is predicted inside both immature and mature nucleocapsids. The theoretical results provide new insights into the molecular mechanism of CTD regulation of viral reverse transcription and nucleocapsid trafficking during various stages of the viral replication processes.

Multiscale simulation of microbe structure and dynamics

A multiscale mathematical and computational approach is developed that captures the hierarchical organization of a microbe. It is found that a natural perspective for understanding a microbe is in terms of a hierarchy of variables at various levels of resolution. This hierarchy starts with the N -atom description and terminates with order parameters characterizing a whole microbe. This conceptual framework is used to guide the analysis of the Liouville equation for the probability density of the positions and momenta of the N atoms constituting the microbe and its environment. Using multiscale mathematical techniques, we derive equations for the co-evolution of the order parameters and the probability density of the N-atom state. This approach yields a rigorous way to transfer information between variables on different space-time scales. It elucidates the interplay between equilibrium and far-from-equilibrium processes underlying microbial behavior. It also provides framework for using coarse-grained nanocharacterization data to guide microbial simulation. It enables a methodical search for free-energy minimizing structures, many of which are typically supported by the set of macromolecules and membranes constituting a given microbe. This suite of capabilities provides a natural framework for arriving at a fundamental understanding of microbial behavior, the analysis of nanocharacterization data, and the computer-aided design of nanostructures for biotechnical and medical purposes. Selected features of the methodology are demonstrated using our multiscale bionanosystem simulator DeductiveMultiscaleSimulator. Systems used to demonstrate the approach are structural transitions in the cowpea chlorotic mosaic virus, RNA of satellite tobacco mosaic virus, virus-like particles related to human papillomavirus, and iron-binding protein lactoferrin.

Design principles for self-assembly with short-range interactions

Recent experimental advances have opened up the possibility of equilibrium self-assembly of functionalized nanoblocks with a high degree of controllable specific interactions. Here, we propose design principles for selecting the short-range interactions between self-assembling components to maximize yield. We illustrate the approach with an example from colloidal engineering. We construct an optimal set of local interactions for eight colloidal particles (coated, e.g., with DNA strands) to assemble into a particular polytetrahedral cluster. Maximum yield is attained when the interactions between the colloids follow the design rules: All energetically favorable interactions have the same strength, as do all unfavorable ones, and the number of components and energies fall within the proposed range. In general, it might be necessary to use more component than strictly required for enforcing the ground state configuration. The results motivate design strategies for engineering components that can reliably self-assemble.

Controlling forces and pathways in self-assembly using viruses and DNA

The ability of both viruses and DNA to self-assemble in solution has continues to enable numerous applications at the nanoscale. Here we review the relevant interactions dictating the assembly of these structures, as well as discussing how they can be exploited experimentally. Because self-assembly is a process, we discuss various strategies for achieving spatial and temporal control. Finally, we highlight a few examples of recent advances that exploit the features of these nanostructures.

Modeling the self-assembly of the cellulosome enzyme complex

Most bacteria use free enzymes to degrade plant cell walls in nature. However, some bacteria have adopted a different strategy wherein enzymes can either be free or tethered on a protein scaffold forming a complex called a cellulosome. The study of the structure and mechanism of these large macromolecular complexes is an active and ongoing research topic, with the goal of finding ways to improve biomass conversion using cellulosomes. Several mechanisms involved in cellulosome formation remain unknown, including how cellulosomal enzymes assemble on the scaffoldin and what governs the population of cellulosomes created during self-assembly. Here, we present a coarse-grained model to study the self-assembly of cellulosomes. The model captures most of the physical characteristics of three cellulosomal enzymes (Cel5B, CelS, and CbhA) and the scaffoldin (CipA) from Clostridium thermocellum. The protein structures are represented by beads connected by restraints to mimic the flexibility and shapes of these proteins. From a large simulation set, the assembly of cellulosomal enzyme complexes is shown to be dominated by their shape and modularity. The multimodular enzyme, CbhA, binds statistically more frequently to the scaffoldin than CelS or Cel5B. The enhanced binding is attributed to the flexible nature and multimodularity of this enzyme, providing a longer residence time around the scaffoldin. The characterization of the factors influencing the cellulosome assembly process may enable new strategies to create designers cellulosomes.