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.     

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.     

Dynamic pathways for viral capsid assembly

We develop a class of models with which we simulate the assembly of particles into T1 capsidlike objects using Newtonian dynamics. By simulating assembly for many different values of system parameters, we vary the forces that drive assembly. For some ranges of parameters, assembly is facile; for others, assembly is dynamically frustrated by kinetic traps corresponding to malformed or incompletely formed capsids. Our simulations sample many independent trajectories at various capsomer concentrations, allowing for statistically meaningful conclusions. Depending on subunit (i.e., capsomer) geometries, successful assembly proceeds by several mechanisms involving binding of intermediates of various sizes. We discuss the relationship between these mechanisms and experimental evaluations of capsid assembly processes.     

Local rules simulation of the kinetics of virus capsid self-assembly.

A computer model is described for studying the kinetics of the self-assembly of icosahedral viral capsids. Solution of this problem is crucial to an understanding of the viral life cycle, which currently cannot be adequately addressed through laboratory techniques. The abstract simulation model employed to address this is based on the local rules theory of. Proc. Natl. Acad. Sci. USA. 91:7732-7736). It is shown that the principle of local rules, generalized with a model of kinetics and other extensions, can be used to simulate complicated problems in self-assembly. This approach allows for a computationally tractable molecular dynamics-like simulation of coat protein interactions while retaining many relevant features of capsid self-assembly. Three simple simulation experiments are presented to illustrate the use of this model. These show the dependence of growth and malformation rates on the energetics of binding interactions, the tolerance of errors in binding positions, and the concentration of subunits in the examples. These experiments demonstrate a tradeoff within the model between growth rate and fidelity of assembly for the three parameters. A detailed discussion of the computational model is also provided.     

Encapsulation of a polymer by an icosahedral virus

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

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.     

Accelerated maximum likelihood parameter estimation for stochastic biochemical systems

ABSTRACT: BACKGROUND: A prerequisite for the mechanistic simulation of a biochemical system is detailed knowledge of its kinetic parameters. Despite recent experimental advances, the estimation of unknown parameter values from observed data is still a bottleneck for obtaining accurate simulation results. Many methods exist for parameter estimation in deterministic biochemical systems; methods for discrete stochastic systems are less well developed. Given the probabilistic nature of stochastic biochemical models, a natural approach is to choose parameter values that maximize the probability of the observed data with respect to the unknown parameters, a.k.a. the maximum likelihood parameter estimates (MLEs). MLE computation for all but the simplest models requires the simulation of many system trajectories that are consistent with experimental data. For models with unknown parameters, this presents a computational challenge, as the generation of consistent trajectories can be an extremely rare occurrence. RESULTS: We have developed Monte Carlo Expectation-Maximization with Modified Cross-Entropy Method (MCEM2): an accelerated method for calculating MLEs that combines advances in rare event simulation with a computationally efficient version of the Monte Carlo expectation-maximization (MCEM) algorithm. Our method requires no prior knowledge regarding parameter values, and it automatically provides a multivariate parameter uncertainty estimate. We applied the method to five stochastic systems of increasing complexity, progressing from an analytically tractable pure-birth model to a computationally demanding model of yeast-polarization. Our results demonstrate that MCEM2 substantially accelerates MLE computation on all tested models when compared to a stand-alone version of MCEM. Additionally, we show how our method identifies parameter values for certain classes of models more accurately than two recently proposed computationally efficient methods. CONCLUSIONS: This work provides a novel, accelerated version of a likelihood-based parameter estimation method that can be readily applied to stochastic biochemical systems. In addition, our results suggest opportunities for added efficiency improvements that will further enhance our ability to mechanistically simulate biological processes.     

Pathways for Virus Assembly around Nucleic Acids

Understanding the pathways by which viral capsid proteins assemble around their genomes could identify key intermediates as potential drug targets. In this work, we use computer simulations to characterize assembly over a wide range of capsid protein–protein interaction strengths and solution ionic strengths. We find that assembly pathways can be categorized into two classes, in which intermediates are either predominantly ordered or disordered. Our results suggest that estimating the protein–protein and the protein–genome binding affinities may be sufficient to predict which pathway occurs. Furthermore, the calculated phase diagrams suggest that knowledge of the dominant assembly pathway and its relationship to control parameters could identify optimal strategies to thwart or redirect assembly to block infection. Finally, analysis of simulation trajectories suggests that the two classes of assembly pathways can be distinguished in single-molecule fluorescence correlation spectroscopy or bulk time-resolved small-angle X-ray scattering experiments.     

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.     

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.     

Zinc ions trigger conformational change and oligomerization of hepatitis B virus capsid protein

Assembly of virus particles in infected cells is likely to be a tightly regulated process. Previously, we found that in vitro assembly of hepatitis B virus (HBV) capsid protein is highly dependent on protein and NaCl concentration. Here we show that micromolar concentrations of Zn2+ are sufficient to initiate assembly of capsid protein, whereas other mono- and divalent cations elicited assembly only at millimolar concentrations, similar to those required for NaCl-induced assembly. Altered intrinsic protein fluorescence and highly cooperative binding of at least four Zn2+ ions (KD approximately 7 microM) indicated that binding induced a conformational change in capsid protein. At 37 degrees C, Zn2+ enhanced the initial rate of assembly and produced normal capsids, but it did not alter the extent of assembly at equilibrium. Assembly mediated by high zinc concentrations (> or =300 microM) yielded few capsids but produced a population of oligomers recognized by capsid-specific antibodies, suggesting a kinetically trapped assembly reaction. Comparison of kinetic simulations to in vitro assembly reactions leads us to suggest that kinetic trapping was due to the enhancement of the nucleation rate relative to the elongation rate. Zinc-induced HBV assembly has hallmarks of an allosterically regulated process: ligand binding at one site influences binding at other sites (cooperativity) indicating that binding is associated with conformational change, and binding of ligand alters the biological activity of assembly. We conclude that zinc binding enhances the kinetics of assembly by promoting formation of an intermediate that is readily consumed in the reaction. Free zinc ions may not be the true in vivo activator of assembly, but they provide a model for regulation of assembly.     

Modeling virus self-assembly pathways: avoiding dynamics using geometric constraint decomposition.

We develop a model for elucidating the assembly pathways by which an icosahedral viral shell forms from 60 identical constituent protein monomers. This poorly understood process a remarkable example of macromolecular self-assembly occuring in nature and possesses many features that are desirable while engineering self-assembly at the nanoscale. The model uses static geometric and tensegrity constraints to represent the driving (weak) forces that cause a viral shell to assemble and hold it together. The goal is to answer focused questions about the structural properties of a successful assembly pathway. Pathways and their properties are carefully defined and computed using computational algebra and geometry, specifically state-of-art concepts in geometric constraint decomposition. The model is analyzable and refinable and avoids expensive dynamics. We show that it has a provably tractable and accurate computational simulation and that its predictions are roughly consistent with known information about viral shell assembly. Justifications for mathematical and biochemical assumptions are provided, and comparisons are drawn with other virus assembly models. A method for more conclusive experimental validation involving specific viruses is sketched. Overall, the paper indicates a strong and direct, mutually beneficial interplay between (a) the concepts underlying macromolecular assembly; and (b) a wide variety of established as well as novel concepts from combinatorial and computational algebra, geometry and algebraic complexity.     

The VP2/VP3 minor capsid protein of simian virus 40 promotes the in vitro assembly of the major capsid protein VP1 into particles

The SV40 capsid is composed primarily of 72 pentamers of the VP1 major capsid protein. Although the capsid also contains the minor capsid protein VP2 and its amino-terminally truncated form VP3, their roles in capsid assembly remain unknown. An in vitro assembly system was used to investigate the role of VP2 in the assembly of recombinant VP1 pentamers. Under physiological salt and pH conditions, VP1 alone remained dissociated, and at pH 5.0, it assembled into tubular structures. A stoichiometric amount of VP2 allowed the assembly of VP1 pentamers into spherical particles in a pH range of 7.0 to 4.0. Electron microscopy observation, sucrose gradient sedimentation analysis, and antibody accessibility tests showed that VP2 is incorporated into VP1 particles. The functional domains of VP2 important for VP1 binding and for enhancing VP1 assembly were further explored with a series of VP2 deletion mutants. VP3 also enhanced VP1 assembly, and a region common to VP2 and VP3 (amino acids 119-272) was required to promote VP1 pentamer assembly. These results are relevant for controlling recombinant capsid formation in vitro, which is potentially useful for the in vitro development of SV40 virus vectors.     

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.     

Rapid microtubule self-assembly kinetics

Microtubule assembly is vital for many fundamental cellular processes. Current models for microtubule assembly kinetics assume that the subunit dissociation rate from a microtubule tip is independent of free subunit concentration. Total-Internal-Reflection-Fluorescence (TIRF) microscopy experiments and data from a laser tweezers assay that measures in vitro microtubule assembly with nanometer resolution, provides evidence that the subunit dissociation rate from a microtubule tip increases as the free subunit concentration increases. These data are consistent with a two-dimensional model for microtubule assembly, and are explained by a shift in microtubule tip structure from a relatively blunt shape at low free concentrations to relatively tapered at high free concentrations. We find that because both the association and the dissociation rates increase at higher free subunit concentrations, the kinetics of microtubule assembly are an order-of-magnitude higher than currently estimated in the literature.     

A mathematical model of actin filament turnover for fitting FRAP data

A novel mathematical model of the actin dynamics in living cells under steady-state conditions has been developed for fluorescence recovery after photobleaching (FRAP) experiments. As opposed to other FRAP fitting models, which use the average lifetime of actins in filaments and the actin turnover rate as fitting parameters, our model operates with unbiased actin association/dissociation rate constants and accounts for the filament length. The mathematical formalism is based on a system of stochastic differential equations. The derived equations were validated on synthetic theoretical data generated by a stochastic simulation algorithm adapted for the simulation of FRAP experiments. Consistent with experimental findings, the results of this work showed that (1) fluorescence recovery is a function of the average filament length, (2) the F-actin turnover and the FRAP are accelerated in the presence of actin nucleating proteins, (3) the FRAP curves may exhibit both a linear and non-linear behaviour depending on the parameters of actin polymerisation, and (4) our model resulted in more accurate parameter estimations of actin dynamics as compared with other FRAP fitting models. Additionally, we provide a computational tool that integrates the model and that can be used for interpretation of FRAP data on actin cytoskeleton.     

Assembly of human immunodeficiency virus (HIV) antigens on bacteriophage T4: a novel in vitro approach to construct multicomponent HIV vaccines

Bacteriophage T4 capsid is an elongated icosahedron decorated with 155 copies of Hoc, a nonessential highly antigenic outer capsid protein. One Hoc monomer is present in the center of each major capsid protein (gp23*) hexon. We describe an in vitro assembly system which allows display of HIV antigens, p24-gag, Nef, and an engineered gp41 C-peptide trimer, on phage T4 capsid surface through Hoc-capsid interactions. In-frame fusions were constructed by splicing the human immunodeficiency virus (HIV) genes to the 5' or 3' end of the Hoc gene. The Hoc fusion proteins were expressed, purified, and displayed on hoc(-) phage particles in a defined in vitro system. Single or multiple antigens were efficiently displayed, leading to saturation of all available capsid binding sites. The displayed p24 was highly immunogenic in mice in the absence of any external adjuvant, eliciting strong p24-specific antibodies, as well as Th1 and Th2 cellular responses with a bias toward the Th2 response. The phage T4 system offers new direction and insights for HIV vaccine development with the potential to increase the breadth of both cellular and humoral immune responses.