Applying Molecular Crowding Models to Simulations of Virus Capsid Assembly In?Vitro

Virus capsid assembly has been widely studied as a biophysical system, both for its biological and medical significance and as an important model for complex self-assembly processes. No current technology can monitor assembly in detail and what information we have on assembly kinetics comes exclusively from in vitro studies. There are many differences between the intracellular environment and that of an in vitro assembly assay, however, that might be expected to alter assembly pathways. Here, we explore one specific feature characteristic of the intracellular environment and known to have large effects on macromolecular assembly processes: molecular crowding. We combine prior particle simulation methods for estimating crowding effects with coarse-grained stochastic models of capsid assembly, using the crowding models to adjust kinetics of capsid simulations to examine possible effects of crowding on assembly pathways. Simulations suggest a striking difference depending on whether or not a system uses nucleation-limited assembly, with crowding tending to promote off-pathway growth in a nonnucleation-limited model but often enhancing assembly efficiency at high crowding levels even while impeding it at lower crowding levels in a nucleation-limited model. These models may help us understand how complicated assembly systems may have evolved to function with high efficiency and fidelity in the densely crowded environment of the cell.     

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.     

Simulation study of the contribution of oligomer/oligomer binding to capsid assembly kinetics

The process by which hundreds of identical capsid proteins self-assemble into icosahedral structures is complex and poorly understood. Establishing constraints on the assembly pathways is crucial to building reliable theoretical models. For example, it is currently an open question to what degree overall assembly kinetics are dominated by one or a few most efficient pathways versus the enormous number theoretically possible. The importance of this question, however, is often overlooked due to the difficulties of addressing it in either theoretical or experimental practice. We apply a computer model based on a discrete-event simulation method to evaluate the contributions of nondominant pathways to overall assembly kinetics. This is accomplished by comparing two possible assembly models: one allowing growth to proceed only by the accretion of individual assembly subunits and the other allowing the binding of sterically compatible assembly intermediates any sizes. Simulations show that the two models perform almost identically under low binding rate conditions, where growth is strongly nucleation-limited, but sharply diverge under conditions of higher association rates or coat protein concentrations. The results suggest the importance of identifying the actual binding pattern if one is to build reliable models of capsid assembly or other complex self-assembly processes.     

Exploring the parameter space of complex self-assembly through virus capsid models

We use discrete event stochastic simulations to characterize the parameter space of a model of icosahedral viral capsid assembly as functions of monomer-monomer binding rates. The simulations reveal a parameter space characterized by three major assembly mechanisms, a standard nucleation-limited monomer-accretion pathway and two distinct hierarchical assembly pathways, as well as unproductive regions characterized by kinetically trapped species. Much of the productive parameter space also consists of border regions between these domains where hybrid pathways are likely to operate. A simpler octamer system studied for comparison reveals three analogous pathways, but is characterized by much lesser sensitivity to parameter variations in contrast to the sharp changes visible in the icosahedral model. The model suggests that modest changes in assembly conditions, consistent with expected differences between in vitro and in vivo assembly environments, could produce substantial shifts in assembly pathways. These results suggest that we must be cautious in drawing conclusions about in vivo capsid self-assembly dynamics from theoretical or in vitro models, as the nature of the basic assembly mechanisms accessible to a system can substantially differ between simple and complex model systems, between theoretical models and simulation results, and between in vitro and in vivo assembly conditions.     

Protein assembly and crowding simulations

Proteins encounter frequent molecular interactions in biological environments. Computer simulations have become an increasingly important tool in providing mechanistic insights into how such interactions in vivo relate to their biological function. The review here focuses on simulations describing protein assembly and molecular crowding effects as two important aspects that are distinguished mainly by how specific and long-lived protein contacts are. On the topic of crowding, recent simulations have provided new insights into how crowding affects protein folding and stability, modulates enzyme activity, and affects diffusive properties. Recent studies of assembly processes focus on assembly pathways, especially for virus capsids, amyloid aggregation pathways, and the role of multivalent interactions leading to phase separation. Also, discussed are technical challenges in achieving increasingly realistic simulations of interactions in cellular environments.     

Stochastic Simulation of Signal Transduction: Impact of the Cellular Architecture on Diffusion

The transduction of signals depends on the translocation of signaling molecules to specific targets. Undirected diffusion processes play a key role in the bridging of spaces between different cellular compartments. The diffusion of the molecules is, in turn, governed by the intracellular architecture. Molecular crowding and the cytoskeleton decrease macroscopic diffusion. This article shows the use of a stochastic simulation method to study the effects of the cytoskeleton structure on the mobility of macromolecules. Brownian dynamics and single particle tracking were used to simulate the diffusion process of individual molecules through a model cytoskeleton. The resulting average effective diffusion is in line with data obtained in the in vitro and in vivo experiments. It shows that the cytoskeleton structure strongly influences the diffusion of macromolecules. The simulation method used also allows the inclusion of reactions in order to model complete signaling pathways in their spatio-temporal dynamics, taking into account the effects of the cellular architecture.     

Protein Stability and Folding Kinetics in the Nucleus and Endoplasmic Reticulum of Eucaryotic Cells

We measure the stability and folding relaxation rate of phosphoglycerate kinase (PGK) Förster resonance energy transfer (FRET) constructs localized in the nucleus or in the endoplasmic reticulum (ER) of eukaryotic cells. PGK has a more compact native state in the cellular compartments than in aqueous solution. Its native FRET signature is similar to that previously observed in a carbohydrate-crowding matrix, consistent with crowding being responsible for the compact native state of PGK in the cell. PGK folds through multiple states in vitro, but its folding kinetics is more two-state-like in the ER, so the folding mechanism can be modified by intracellular compartments. The nucleus increases PGK stability and folding rate over the cytoplasm and ER, even though the density of crowders in the nucleus is no greater than in the ER or cytoplasm. Nuclear folding kinetics (and to a lesser extent, thermodynamics) vary less from cell to cell than in the cytoplasm or ER, indicating a more homogeneous crowding and chemical environment in the nucleus.     

Effect of macromolecular crowding agents on human immunodeficiency virus type 1 capsid protein assembly in vitro

Previous studies on the self-assembly of capsid protein CA of human immunodeficiency virus type 1 (HIV-1) in vitro have provided important insights on the structure and assembly of the mature HIV-1 capsid. However, CA polymerization in vitro was previously observed to occur only at very high ionic strength. Here, we have analyzed the effects on CA assembly in vitro of adding unrelated, inert macromolecules (crowding agents), aimed at mimicking the crowded (very high macromolecular effective concentration) environment within the HIV-1 virion. Crowding agents induced fast and efficient polymerization of CA even at low (close to physiological) ionic strength. The hollow cylinders thus assembled were indistinguishable in shape and dimensions from those formed in dilute protein solutions at high ionic strength. However, two important differences were noted: (i) disassembly by dilution of the capsid-like particles was undetectable at very high ionic strength, but occurred rapidly at low ionic strength in the presence of a crowding agent, and (ii) a variant CA from a presumed infectious HIV-1 with mutations at the CA dimerization interface was unable to assemble at any ionic strength in the absence of a crowding agent; in contrast, this mutation allowed efficient assembly, even at low ionic strength, when a crowding agent was used. The use of a low ionic strength and inert macromolecules to mimic the crowded environment inside the HIV-1 virion may lead to a better in vitro evaluation of the effects of conditions, mutations or/and other molecules, including potential antiviral compounds, on HIV-1 capsid assembly, stability and disassembly.     

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.     

Association Equilibrium of the HIV-1 Capsid Protein in a Crowded Medium Reveals that Hexamerization during Capsid Assembly Requires a Functional C-Domain Dimerization Interface

Polymerization of the intact capsid protein (CA) of HIV-1 into mature capsidlike particles at physiological ionic strength in vitro requires macromolecularly crowded conditions that approach those inside the virion, where the mature capsid is assembled in vivo. The capsid is organized as a hexameric lattice. CA subunits in each hexamer are connected through interfaces that involve the CA N-terminal domain (NTD); pairs of CA subunits belonging to different hexamers are connected through a different interface that involves the C-terminal domain (CTD). At physiological ionic strength in noncrowded conditions, CA subunits homodimerize through this CTD-CTD interface, but do not hexamerize through the other interfaces (those involving the NTD). Here we have investigated whether macromolecular crowding conditions are able to promote hexamerization of the isolated NTD and/or full-length CA (with an inactive CTD-CTD interface to prevent polymerization). The oligomerization state of the proteins was determined using analytical ultracentrifugation in the absence or presence of high concentrations of an inert macromolecular crowding agent. Under the same conditions that promoted efficient assembly of intact CA dimers, neither NTD nor CA with an inactive CTD-CTD interface showed any tendency to form hexamers or any other oligomer. This inability to hexamerize was observed even in macromolecularly crowded conditions. The results indicate that a functional CTD-CTD interface is strictly required for hexamerization of HIV-1 CA through the other interfaces. Together with previous results, these observations suggest that establishment of NTD-CTD interactions involved in CA hexamerization during mature HIV-1 capsid assembly requires a homodimerization-dependent conformational switching of CTD.     

NERDSS: A Nonequilibrium Simulator for Multibody Self-Assembly at the Cellular Scale

Currently, a significant barrier to building predictive models of cellular self-assembly processes is that molecular models cannot capture minutes-long dynamics that couple distinct components with active processes, whereas reaction-diffusion models cannot capture structures of molecular assembly. Here, we introduce the nonequilibrium reaction-diffusion self-assembly simulator (NERDSS), which addresses this spatiotemporal resolution gap. NERDSS integrates efficient reaction-diffusion algorithms into generalized software that operates on user-defined molecules through diffusion, binding and orientation, unbinding, chemical transformations, and spatial localization. By connecting the fast processes of binding with the slow timescales of large-scale assembly, NERDSS integrates molecular resolution with reversible formation of ordered, multisubunit complexes. NERDSS encodes models using rule-based formatting languages to facilitate model portability, usability, and reproducibility. Applying NERDSS to steps in clathrin-mediated endocytosis, we design multicomponent systems that can form lattices in solution or on the membrane, and we predict how stochastic but localized dephosphorylation of membrane lipids can drive lattice disassembly. The NERDSS simulations reveal the spatial constraints on lattice growth and the role of membrane localization and cooperativity in nucleating assembly. By modeling viral lattice assembly and recapitulating oscillations in protein expression levels for a circadian clock model, we illustrate the adaptability of NERDSS. NERDSS simulates user-defined assembly models that were previously inaccessible to existing software tools, with broad applications to predicting self-assembly in vivo and designing high-yield assemblies in vitro.     

Effects of macromolecular crowding on the inhibition of virus assembly and virus-cell receptor recognition

Biological fluids contain a very high total concentration of macromolecules that leads to volume exclusion by one molecule to another. Theory and experiment have shown that this condition, termed macromolecular crowding, can have significant effects on molecular recognition. However, the influence of molecular crowding on recognition events involving virus particles, and their inhibition by antiviral compounds, is virtually unexplored. Among these processes, capsid self-assembly during viral morphogenesis and capsid-cell receptor recognition during virus entry into cells are receiving increasing attention as targets for the development of new antiviral drugs. In this study, we have analyzed the effect of macromolecular crowding on the inhibition of these two processes by peptides. Macromolecular crowding led to a significant reduction in the inhibitory activity of: 1), a capsid-binding peptide and a small capsid protein domain that interfere with assembly of the human immunodeficiency virus capsid, and 2), a RGD-containing peptide able to block the interaction between foot-and-mouth disease virus and receptor molecules on the host cell membrane (in this case, the effect was dependent on the conditions used). The results, discussed in the light of macromolecular crowding theory, are relevant for a quantitative understanding of molecular recognition processes during virus infection and its inhibition.     

Macromolecular crowding: obvious but underappreciated

Biological macromolecules evolve and function within intracellular environments that are crowded with other macromolecules. Crowding results in surprisingly large quantitative effects on both the rates and the equilibria of interactions involving macromolecules, but such interactions are commonly studied outside the cell in uncrowded buffers. The addition of high concentrations of natural and synthetic macromolecules to such buffers enables crowding to be mimicked in vitro, and should be encouraged as a routine variable to study. The stimulation of protein aggregation by crowding might account for the existence of molecular chaperones that combat this effect. Positive results of crowding include enhancing the collapse of polypeptide chains into functional proteins, the assembly of oligomeric structures and the efficiency of action of some molecular chaperones and metabolic pathways.     

An Open Model of Actin Dendritic Nucleation

The availability of quantitative experimental data on the kinetics of actin assembly has enabled the construction of many mathematical models focused on explaining specific behaviors of this complex system. However these ad hoc models are generally not reusable or accessible by the large community of actin biologists. In this work, we present a comprehensive model that integrates and unifies much of the in vitro data on the components of the dendritic nucleation mechanism for actin dynamics. More than 300 simulations have been run based on compartmental and three-dimensional spatial versions of this model. Several key findings are highlighted, including an explanation for the sharp boundary between actin assembly and disassembly in the lamellipodia of migrating cells. Because this model, with the simulation results, is “open source”, in the sense that it is publicly available and editable through the Virtual Cell database (http://vcell.org), it can be accessed, analyzed, modified, and extended.     

Small-molecule effectors of hepatitis B virus capsid assembly give insight into virus life cycle

The relationship between the physical chemistry and biology of self-assembly is poorly understood, but it will be critical to quantitatively understand infection and for the design of antivirals that target virus genesis. Here we take advantage of heteroaryldihydropyrimidines (HAPs), which affect hepatitis B virus (HBV) assembly, to gain insight and correlate in vitro assembly with HBV replication in culture. Based on a low-resolution crystal structure of a capsid-HAP complex, a closely related series of HAPs were designed and synthesized. These differentially strengthen the association between neighboring capsid proteins, alter the kinetics of assembly, and give rise to aberrant structures incompatible with a functional capsid. The chemical nature of the HAP variants correlated well with the structure of the HAP binding pocket. The thermodynamics and kinetics of in vitro assembly had strong and predictable effects on product morphology. However, only the kinetics of in vitro assembly had a strong correlation with inhibition of HBV replication in HepG2.2.15 cells; there was at best a weak correlation between assembly thermodynamics and replication. The correlation between assembly kinetics and virus suppression implies a competition between successful assembly and misassembly, small molecule induced or otherwise. This is a predictive and testable model for the mechanism of action of assembly effectors.     

Crowding Activates ClpB and Enhances Its Association with DnaK for Efficient Protein Aggregate Reactivation

Reactivation of intracellular protein aggregates after a severe stress is mandatory for cell survival. In bacteria, this activity depends on the collaboration between the DnaK system and ClpB, which in vivo occurs in a highly crowded environment. The reactivation reaction includes two steps: extraction of unfolded monomers from the aggregate and their subsequent refolding into the native conformation. Both steps might be compromised by excluded volume conditions that would favor aggregation of unstable protein folding intermediates. Here, we have investigated whether ClpB and the DnaK system are able to compensate this unproductive effect and efficiently reactivate aggregates of three different substrate proteins under crowding conditions. To this aim, we have compared the association equilibrium, biochemical properties, stability, and chaperone activity of the disaggregase ClpB in the absence and presence of an inert macromolecular crowding agent. Our data show that crowding i), increases three to four orders of magnitude the association constant of the functional hexamer; ii), shifts the conformational equilibrium of the protein monomer toward a compact state; iii), stimulates its ATPase activity; and iv), favors association of the chaperone with substrate proteins and with aggregate-bound DnaK. These effects strongly enhance protein aggregate reactivation by the DnaK-ClpB network, highlighting the importance of volume exclusion in complex processes in which several proteins have to work in a sequential manner.     

A theoretical model successfully identifies features of hepatitis B virus capsid assembly.

The capsids of most spherical viruses are icosahedral, an arrangement of multiples of 60 subunits. 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. Features of the overall reaction can be identified on the basis of the concentration dependence of assembly. In simulations, and on the basis of our understanding of the models, we find that nucleus size and the order of subsequent "elongation" reactions are reflected in the concentration dependence of the extent of the reaction and the rate of the fast phase, respectively. The reaction kinetics deduced for our models of virus assembly can be related to the assembly of any "spherical" polymer. Using light scattering and size exclusion chromatography, we observed polymerization of assembly domain dimers of hepatitis B virus (HBV) capsid protein. Empty capsids assemble at a rate that is a function of protein concentration and ionic strength. The kinetics of capsid formation were sigmoidal, where the rate of the fast phase had second-power concentration dependence. The extent of assembly had third-power concentration dependence. Simulations based on the models recapitulated the concentration dependences observed for HBV capsid assembly. These results strongly suggest that in vitro HBV assembly is nucleated by a trimer of dimers and proceeds by the addition of individual dimeric subunits. On the basis of this mechanism, we suggest that HBV capsid assembly could be an important target for antiviral therapeutics.     

Quantification of Excluded Volume Effects on the Folding Landscape of Pseudomonas aeruginosa Apoazurin In Vitro

Proteins fold and function inside cells that are crowded with macromolecules. Here, we address the role of the resulting excluded volume effects by in vitro spectroscopic studies of Pseudomonas aeruginosa apoazurin stability (thermal and chemical perturbations) and folding kinetics (chemical perturbation) as a function of increasing levels of crowding agents dextran (sizes 20, 40, and 70 kDa) and Ficoll 70. We find that excluded volume theory derived by Minton quantitatively captures the experimental effects when crowding agents are modeled as arrays of rods. This finding demonstrates that synthetic crowding agents are useful for studies of excluded volume effects. Moreover, thermal and chemical perturbations result in free energy effects by the presence of crowding agents that are identical, which shows that the unfolded state is energetically the same regardless of method of unfolding. This also underscores the two-state approximation for apoazurin’s unfolding reaction and suggests that thermal and chemical unfolding experiments can be used in an interchangeable way. Finally, we observe increased folding speed and invariant unfolding speed for apoazurin in the presence of macromolecular crowding agents, a result that points to unfolded-state perturbations. Although the absolute magnitude of excluded volume effects on apoazurin is only on the order of 1–3 kJ/mol, differences of this scale may be biologically significant.