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

2D capsid formation within an oscillatory energy landscape: orderly self-assembly depends on the interplay between a dynamic potential and intrinsic relaxation times

Multiple dissipative self-assembly protocols designed to create novel structures or to reduce kinetic traps have recently emerged. Specifically, temporal oscillations of particle interactions have been shown effective at both aims, but investigations thus far have focused on systems of simple colloids or their binary mixtures. In this work, we expand our understanding of the effect of temporally oscillating interactions to a two-dimensional coarse-grained viral capsid-like model that undergoes a self-limited assembly. This model includes multiple intrinsic relaxation times due to the internal structure of the capsid subunits and, under certain interaction regimes, proceeds via a two-step nucleation mechanism. We find that oscillations much faster than the local intrinsic relaxation times can be described via a time averaged inter-particle potential across a wide range of interaction strengths, while oscillations much slower than these relaxation times result in structures that adapt to the attraction strength of the current half-cycle. Interestingly, oscillation periods similar to these relaxation times shift the interaction window over which orderly assembly occurs by enabling error correction during the half-cycles with weaker attractions. Our results provide fundamental insights to non-equilibrium self-assembly on temporally variant energy landscapes.

Discovering optimal kinetic pathways for self-assembly using automatic differentiation

Macromolecular complexes are often composed of diverse subunits. The self-assembly of these subunits is inherently nonequilibrium and must avoid kinetic traps to achieve high yield over feasible timescales. We show how the kinetics of self-assembly benefits from diversity in subunits because it generates an expansive parameter space that naturally improves the "expressivity" of self-assembly, much like a deeper neural network. By using automatic differentiation algorithms commonly used in deep learning, we searched the parameter spaces of mass-action kinetic models to identify classes of kinetic protocols that mimic biological solutions for productive self-assembly. Our results reveal how high-yield complexes that easily become kinetically trapped in incomplete intermediates can instead be steered by internal design of rate-constants or external and active control of subunits to efficiently assemble. Internal design of a hierarchy of subunit binding rates generates self-assembly that can robustly avoid kinetic traps for all concentrations and energetics, but it places strict constraints on selection of relative rates. External control via subunit titration is more versatile, avoiding kinetic traps for any system without requiring molecular engineering of binding rates, albeit less efficiently and robustly. We derive theoretical expressions for the timescales of kinetic traps, and we demonstrate our optimization method applies not just for design but inference, extracting intersubunit binding rates from observations of yield-vs.-time for a heterotetramer. Overall, we identify optimal kinetic protocols for self-assembly as a powerful mechanism to achieve efficient and high-yield assembly in synthetic systems whether robustness or ease of "designability" is preferred.

Hierarchical assembly is more robust than egalitarian assembly in synthetic capsids

Self-assembly of complex and functional materials remains a grand challenge in soft material science. Efficient assembly depends on a delicate balance between thermodynamic and kinetic effects, requiring fine-tuning affinities and concentrations of subunits. By contrast, we introduce an assembly paradigm that allows large error-tolerance in the subunit affinity and helps avoid kinetic traps. Our combined experimental and computational approach uses a model system of triangular subunits programmed to assemble into T = 3 icosahedral capsids comprising 60 units. The experimental platform uses DNA origami to create monodisperse colloids whose three-dimensional geometry is controlled to nanometer precision, with two distinct bonds whose affinities are controlled to kBT precision, quantified in situ by static light scattering. The computational model uses a coarse-grained representation of subunits, short-ranged potentials, and Langevin dynamics. Experimental observations and modeling reveal that when the bond affinities are unequal, two distinct hierarchical assembly pathways occur, in which the subunits first form dimers in one case and pentamers in another. These hierarchical pathways produce complete capsids faster and are more robust against affinity variation than egalitarian pathways, in which all binding sites have equal strengths. This finding suggests that hierarchical assembly may be a general engineering principle for optimizing self-assembly of complex target structures.

Discovering optimal kinetic pathways for self-assembly using automatic differentiation

During self-assembly of macromolecules ranging from ribosomes to viral capsids, the formation of long-lived intermediates or kinetic traps can dramatically reduce yield of the functional products. Understanding biological mechanisms for avoiding traps and efficiently assembling is essential for designing synthetic assembly systems, but learning optimal solutions requires numerical searches in high-dimensional parameter spaces. Here, we exploit powerful automatic differentiation algorithms commonly employed by deep learning frameworks to optimize physical models of reversible self-assembly, discovering diverse solutions in the space of rate constants for 3-7 subunit complexes. We define two biologically-inspired protocols that prevent kinetic trapping through either internal design of subunit binding kinetics or external design of subunit titration in time. Our third protocol acts to recycle intermediates, mimicking energy-consuming enzymes. Preventative solutions via interface design are the most efficient and scale better with more subunits, but external control via titration or recycling are effective even for poorly evolved binding kinetics. Whilst all protocols can produce good solutions, diverse subunits always helps; these complexes access more efficient solutions when following external control protocols, and are simpler to design for internal control, as molecular interfaces do not need modification during assembly given sufficient variation in dimerization rates. Our results identify universal scaling in the cost of kinetic trapping, and provide multiple strategies for eliminating trapping and maximizing assembly yield across large parameter spaces.

Temporal control by cofactors prevents kinetic trapping in retroviral Gag lattice assembly

For retroviruses like HIV to proliferate, they must form virions shaped by the self-assembly of Gag polyproteins into a rigid lattice. This immature Gag lattice has been structurally characterized and reconstituted in vitro, revealing the sensitivity of lattice assembly to multiple cofactors. Due to this sensitivity, the energetic criterion for forming stable lattices is unknown, as are their corresponding rates. Here, we use a reaction-diffusion model designed from the cryo-ET structure of the immature Gag lattice to map a phase diagram of assembly outcomes controlled by experimentally constrained rates and free energies, over experimentally relevant timescales. We find that productive assembly of complete lattices in bulk solution is extraordinarily difficult due to the large size of this ∼3700 monomer complex. Multiple Gag lattices nucleate before growth can complete, resulting in loss of free monomers and frequent kinetic trapping. We therefore derive a time-dependent protocol to titrate or "activate" the Gag monomers slowly within the solution volume, mimicking the biological roles of cofactors. This general strategy works remarkably well, yielding productive growth of self-assembled lattices for multiple interaction strengths and binding rates. By comparing to the in vitro assembly kinetics, we can estimate bounds on rates of Gag binding to Gag and the cellular cofactor IP6. Our results show that Gag binding to IP6 can provide the additional time delay necessary to support smooth growth of the immature lattice with relatively fast assembly kinetics, mostly avoiding kinetic traps. Our work provides a foundation for predicting and disrupting formation of the immature Gag lattice via targeting specific protein-protein binding interactions.

Self-assembly coupled to liquid-liquid phase separation

Liquid condensate droplets with distinct compositions of proteins and nucleic acids are widespread in biological cells. While it is known that such droplets, or compartments, can regulate irreversible protein aggregation, their effect on reversible self-assembly remains largely unexplored. In this article, we use kinetic theory and solution thermodynamics to investigate the effect of liquid-liquid phase separation on the reversible self-assembly of structures with well-defined sizes and architectures. We find that, when assembling subunits preferentially partition into liquid compartments, robustness against kinetic traps and maximum achievable assembly rates can be significantly increased. In particular, both the range of solution conditions leading to productive assembly and the corresponding assembly rates can increase by orders of magnitude. We analyze the rate equation predictions using simple scaling estimates to identify effects of liquid-liquid phase separation as a function of relevant control parameters. These results may elucidate self-assembly processes that underlie normal cellular functions or pathogenesis, and suggest strategies for designing efficient bottom-up assembly for nanomaterials applications.

Rip it, stitch it, click it: A Chemist's guide to VLP manipulation

Viruses are some of nature's most ubiquitous self-assembled molecular containers. Evolutionary pressures have created some incredibly robust, thermally, and enzymatically resistant carriers to transport delicate genetic information safely. Virus-like particles (VLPs) are human-engineered non-infectious systems that inherit the parent virus' ability to self-assemble under controlled conditions while being non-infectious. VLPs and plant-based viral nanoparticles are becoming increasingly popular in medicine as their self-assembly properties are exploitable for applications ranging from diagnostic tools to targeted drug delivery. Understanding the basic structure and principles underlying the assembly of higher-order structures has allowed researchers to disassemble (rip it), reassemble (stitch it), and functionalize (click it) these systems on demand. This review focuses on the current toolbox of strategies developed to manipulate these systems by ripping, stitching, and clicking to create new technologies in the biomedical space.

Hierarchical Coarse-Grained Strategy for Macromolecular Self-Assembly: Application to Hepatitis B Virus-Like Particles

Macromolecular self-assembly is at the basis of many phenomena in material and life sciences that find diverse applications in technology. One example is the formation of virus-like particles (VLPs) that act as stable empty capsids used for drug delivery or vaccine fabrication. Similarly to the capsid of a virus, VLPs are protein assemblies, but their structural formation, stability, and properties are not fully understood, especially as a function of the protein modifications. In this work, we present a data-driven modeling approach for capturing macromolecular self-assembly on scales beyond traditional molecular dynamics (MD), while preserving the chemical specificity. Each macromolecule is abstracted as an anisotropic object and high-dimensional models are formulated to describe interactions between molecules and with the solvent. For this, data-driven protein-protein interaction potentials are derived using a Kriging-based strategy, built on high-throughput MD simulations. Semi-automatic supervised learning is employed in a high performance computing environment and the resulting specialized force-fields enable a significant speed-up to the micrometer and millisecond scale, while maintaining high intermolecular detail. The reported generic framework is applied for the first time to capture the formation of hepatitis B VLPs from the smallest building unit, i.e., the dimer of the core protein HBcAg. Assembly pathways and kinetics are analyzed and compared to the available experimental observations. We demonstrate that VLP self-assembly phenomena and dependencies are now possible to be simulated. The method developed can be used for the parameterization of other macromolecules, enabling a molecular understanding of processes impossible to be attained with other theoretical models.

Large self-assembled clathrin lattices spontaneously disassemble without sufficient adaptor proteins

Clathrin-coated structures must assemble on cell membranes to internalize receptors, with the clathrin protein only linked to the membrane via adaptor proteins. These structures can grow surprisingly large, containing over 20 clathrin, yet they often fail to form productive vesicles, instead aborting and disassembling. We show that clathrin structures of this size can both form and disassemble spontaneously when adaptor protein availability is low, despite high abundance of clathrin. Here, we combine recent in vitro kinetic measurements with microscopic reaction-diffusion simulations and theory to differentiate mechanisms of stable vs unstable clathrin assembly on membranes. While in vitro conditions drive assembly of robust, stable lattices, we show that concentrations, geometry, and dimensional reduction in physiologic-like conditions do not support nucleation if only the key adaptor AP-2 is included, due to its insufficient abundance. Nucleation requires a stoichiometry of adaptor to clathrin that exceeds 1:1, meaning additional adaptor types are necessary to form lattices successfully and efficiently. We show that the critical nucleus contains ~25 clathrin, remarkably similar to sizes of the transient and abortive structures observed in vivo. Lastly, we quantify the cost of bending the membrane under our curved clathrin lattices using a continuum membrane model. We find that the cost of bending the membrane could be largely offset by the energetic benefit of forming curved rather than flat structures, with numbers comparable to experiments. Our model predicts how adaptor density can tune clathrin-coated structures from the transient to the stable, showing that active energy consumption is therefore not required for lattice disassembly or remodeling during growth, which is a critical advance towards predicting productive vesicle formation.

Molecular Assembly Lines in Active Droplets

Large protein complexes are assembled from protein subunits to form a specific structure. In our theoretic work, we propose that assembly into the correct structure could be reliably achieved through an assembly line with a specific sequence of assembly steps. Using droplet interfaces to position compartment boundaries, we show that an assembly line can be self-organized by active droplets. As a consequence, assembly steps can be arranged spatially so that a specific order of assembly is achieved and incorrect assembly is strongly suppressed.

Protein Nanoparticles: Uniting the Power of Proteins with Engineering Design Approaches

Protein nanoparticles, PNPs, have played a long-standing role in food and industrial applications. More recently, their potential in nanomedicine has been more widely pursued. This review summarizes recent trends related to the preparation, application, and chemical construction of nanoparticles that use proteins as major building blocks. A particular focus has been given to emerging trends related to applications in nanomedicine, an area of research where PNPs are poised for major breakthroughs as drug delivery carriers, particle-based therapeutics or for non-viral gene therapy.

Virus self-assembly proceeds through contact-rich energy minima

Self-assembly of supramolecular complexes such as viral capsids occurs prominently in nature. Nonetheless, the mechanisms underlying these processes remain poorly understood. Here, we uncover the assembly pathway of hepatitis B virus (HBV), applying fluorescence optical tweezers and high-speed atomic force microscopy. This allows tracking the assembly process in real time with single-molecule resolution. Our results identify a specific, contact-rich pentameric arrangement of HBV capsid proteins as a key on-path assembly intermediate and reveal the energy balance of the self-assembly process. Real-time nucleic acid packaging experiments show that a free energy change of ~1.4 k_BT per condensed nucleotide is used to drive protein oligomerization. The finding that HBV assembly occurs via contact-rich energy minima has implications for our understanding of the assembly of HBV and other viruses and also for the development of new antiviral strategies and the rational design of self-assembling nanomaterials.

Speed limits of protein assembly with reversible membrane localization

Protein assembly is often studied in a three-dimensional solution, but a significant fraction of binding events involve proteins that can reversibly bind and diffuse along a two-dimensional surface. In a recent study, we quantified how proteins can exploit the reduced dimensionality of the membrane to trigger complex formation. Here, we derive a single expression for the characteristic timescale of this multi-step assembly process, where the change in dimensionality renders rates and concentrations effectively time-dependent. We find that proteins can accelerate dimer formation due to an increase in relative concentration, driving more frequent collisions, which often win out over slow-downs due to diffusion. Our model contains two protein populations that dimerize with one another and use a distinct site to bind membrane lipids, creating a complex reaction network. However, by identifying two major rate-limiting pathways to reach an equilibrium steady-state, we derive an excellent approximation for the mean first passage time when lipids are in abundant supply. Our theory highlights how the "sticking rate" or effective adsorption coefficient of the membrane is central in controlling timescales. We also derive a corrected localization rate to quantify how the geometry of the system and diffusion can reduce rates of membrane localization. We validate and test our results using kinetic and particle-based reaction-diffusion simulations. Our results establish how the speed of key assembly steps can shift by orders-of-magnitude when membrane localization is possible, which is critical to understanding mechanisms used in cells.

In Vitro Assembly of Virus-Like Particles and Their Applications

Virus-like particles (VLPs) are increasingly used for vaccine development and drug delivery. Assembly of VLPs from purified monomers in a chemically defined reaction is advantageous compared to in vivo assembly, because it avoids encapsidation of host-derived components and enables loading with added cargoes. This review provides an overview of ex cella VLP production methods focusing on capsid protein production, factors that impact the in vitro assembly, and approaches to characterize in vitro VLPs. The uses of in vitro produced VLPs as vaccines and for therapeutic delivery are also reported.

Equilibrium mechanisms of self-limiting assembly

Self-assembly is a ubiquitous process in synthetic and biological systems, broadly defined as the spontaneous organization of multiple subunits (e.g. macromolecules, particles) into ordered multi-unit structures. The vast majority of equilibrium assembly processes give rise to two states: one consisting of dispersed disassociated subunits, and the other, a bulk-condensed state of unlimited size. This review focuses on the more specialized class of self-limiting assembly, which describes equilibrium assembly processes resulting in finite-size structures. These systems pose a generic and basic question, how do thermodynamic processes involving non-covalent interactions between identical subunits "measure" and select the size of assembled structures? In this review, we begin with an introduction to the basic statistical mechanical framework for assembly thermodynamics, and use this to highlight the key physical ingredients that ensure equilibrium assembly will terminate at finite dimensions. Then, we introduce examples of self-limiting assembly systems, and classify them within this framework based on two broad categories: self-closing assemblies and open-boundary assemblies. These include well-known cases in biology and synthetic soft matter - micellization of amphiphiles and shell/tubule formation of tapered subunits - as well as less widely known classes of assemblies, such as short-range attractive/long-range repulsive systems and geometrically-frustrated assemblies. For each of these self-limiting mechanisms, we describe the physical mechanisms that select equilibrium assembly size, as well as potential limitations of finite-size selection. Finally, we discuss alternative mechanisms for finite-size assemblies, and draw contrasts with the size-control that these can achieve relative to self-limitation in equilibrium, single-species assemblies.

Physics of viral dynamics

Viral capsids are often regarded as inert structural units, but in actuality they display fascinating dynamics during different stages of their life cycle. With the advent of single-particle approaches and high-resolution techniques, it is now possible to scrutinize viral dynamics during and after their assembly and during the subsequent development pathway into infectious viruses. In this Review, the focus is on the dynamical properties of viruses, the different physical virology techniques that are being used to study them, and the physical concepts that have been developed to describe viral dynamics.

The Integrity of the Intradimer Interface of the Hepatitis B Virus Capsid Protein Dimer Regulates Capsid Self-Assembly

During the hepatitis B virus lifecycle, 120 copies of homodimeric capsid protein assemble around a copy of reverse transcriptase and viral RNA and go on to produce an infectious virion. Assembly needs to be tightly regulated by protein conformational change to ensure symmetry, fidelity, and reproducibility. Here, we show that structures at the intradimer interface regulate conformational changes at the distal interdimer interface and so regulate assembly. A pair of interacting charged residues, D78 from each monomer, conspicuously located at the top of a four-helix bundle that forms the intradimer interface, were mutated to serine to disrupt communication between the two monomers. The mutation slowed assembly and destabilized the dimer to thermal and chemical denaturation. Mutant dimers showed evidence of transient partial unfolding based on the appearance of new proteolytically sensitive sites. Though the mutant dimer was less stable, the resulting capsids were as stable as the wildtype, based on assembly and thermal denaturation studies. Cryo-EM image reconstructions of capsid indicated that the subunits adopted an "open" state more usually associated with a free dimer and that the spike tips were either disordered or highly flexible. Molecular dynamics simulations provide mechanistic explanations for these results, suggesting that D78 stabilizes helix 4a, which forms part of the intradimer interface, by capping its N-terminus and hydrogen-bonding to nearby residues, whereas the D78S mutation disrupts these interactions, leading to partial unwinding of helix 4a. This in turn weakens the connection from helix 4 and the intradimer interface to helix 5, which forms the interdimer interface.

Rapid prediction of crucial hotspot interactions for icosahedral viral capsid self-assembly by energy landscape atlasing validated by mutagenesis

Icosahedral viruses are under a micrometer in diameter, their infectious genome encapsulated by a shell assembled by a multiscale process, starting from an integer multiple of 60 viral capsid or coat protein (VP) monomers. We predict and validate inter-atomic hotspot interactions between VP monomers that are important for the assembly of 3 types of icosahedral viral capsids: Adeno Associated Virus serotype 2 (AAV2) and Minute Virus of Mice (MVM), both T = 1 single stranded DNA viruses, and Bromo Mosaic Virus (BMV), a T = 3 single stranded RNA virus. Experimental validation is by in-vitro, site-directed mutagenesis data found in literature. We combine ab-initio predictions at two scales: at the interface-scale, we predict the importance (cruciality) of an interaction for successful subassembly across each interface between symmetry-related VP monomers; and at the capsid-scale, we predict the cruciality of an interface for successful capsid assembly. At the interface-scale, we measure cruciality by changes in the capsid free-energy landscape partition function when an interaction is removed. The partition function computation uses atlases of interface subassembly landscapes, rapidly generated by a novel geometric method and curated opensource software EASAL (efficient atlasing and search of assembly landscapes). At the capsid-scale, cruciality of an interface for successful assembly of the capsid is based on combinatorial entropy. Our study goes all the way from resource-light, multiscale computational predictions of crucial hotspot inter-atomic interactions to validation using data on site-directed mutagenesis' effect on capsid assembly. By reliably and rapidly narrowing down target interactions, (no more than 1.5 hours per interface on a laptop with Intel Core i5-2500K @ 3.2 Ghz CPU and 8GB of RAM) our predictions can inform and reduce time-consuming in-vitro and in-vivo experiments, or more computationally intensive in-silico analyses.

Assembly of Capsids from Hepatitis B Virus Core Protein Progresses through Highly Populated Intermediates in the Presence and Absence of RNA

The genetic material of viruses is protected by protein shells that are assembled from a large number of subunits in a process that is efficient and robust. Many of the mechanistic details underpinning efficient assembly of virus capsids are still unknown. The assembly mechanism of hepatitis B capsids has been intensively researched using a truncated core protein lacking the C-terminal domain responsible for binding genomic RNA. To resolve the assembly intermediates of hepatitis B virus (HBV), we studied the formation of nucleocapsids and empty capsids from full-length hepatitis B core proteins, using time-resolved small-angle X-ray scattering. We developed a detailed structural model of the HBV capsid assembly process using a combination of analysis with multivariate curve resolution, structural modeling, and Bayesian ensemble inference. The detailed structural analysis supports an assembly pathway that proceeds through the formation of two highly populated intermediates, a trimer of dimers and a partially closed shell consisting of around 40 dimers. These intermediates are on-path, transient and efficiently convert into fully formed capsids. In the presence of an RNA oligo that binds specifically to the C-terminal domain the assembly proceeds via a similar mechanism to that in the absence of nucleic acids. Comparisons between truncated and full-length HBV capsid proteins reveal that the unstructured C-terminal domain has a significant impact on the assembly process and is required to obtain a more complete mechanistic understanding of HBV capsid formation. These results also illustrate how combining scattering information from different time-points during time-resolved experiments can be utilized to derive a structural model of protein self-assembly pathways.

Rapidly Forming Early Intermediate Structures Dictate the Pathway of Capsid Assembly

There are ∼10³⁰ possible intermediates on the assembly path from hepatitis B capsid protein dimers to the 120-dimer capsid. If every intermediate was tested, assembly would often get stuck in an entropic trap and essentially every capsid would follow a unique assembly path. Yet, capsids assemble rapidly with minimal trapped intermediates, a realization of the Levinthal paradox. To understand the fundamental mechanisms of capsid assembly, it is critical to resolve the early stages of the reaction. We have used time-resolved small angle X-ray scattering, which is sensitive to solute size and shape and has millisecond temporal resolution. Scattering curves were fit to a thermodynamically curated library of assembly intermediates, using the principle of maximum entropy. Maximum entropy also provides a physical rationale for the selection of species. We found that the capsid assembly pathway was exquisitely sensitive to initial assembly conditions. With the mildest conditions tested, the reaction appeared to be two-state from dimer to 120-dimer capsid with some dimers-of-dimers and trimers-of-dimers. In slightly more aggressive conditions, we observed transient accumulation of a decamer-of-dimers and the appearance of 90-dimer capsids. In conditions where there is measurable kinetic trapping, we found that highly diverse early intermediates accumulated within a fraction of a second and propagated into long-lived kinetically trapped states (≥90-mer). In all cases, intermediates between 35 and 90 subunits did not accumulate. These results are consistent with the presence of low barrier paths that connect early and late intermediates and direct the ultimate assembly path to late intermediates where assembly can be paused.

Stochastic yield catastrophes and robustness in self-assembly

A guiding principle in self-assembly is that, for high production yield, nucleation of structures must be significantly slower than their growth. However, details of the mechanism that impedes nucleation are broadly considered irrelevant. Here, we analyze self-assembly into finite-sized target structures employing mathematical modeling. We investigate two key scenarios to delay nucleation: (i) by introducing a slow activation step for the assembling constituents and, (ii) by decreasing the dimerization rate. These scenarios have widely different characteristics. While the dimerization scenario exhibits robust behavior, the activation scenario is highly sensitive to demographic fluctuations. These demographic fluctuations ultimately disfavor growth compared to nucleation and can suppress yield completely. The occurrence of this stochastic yield catastrophe does not depend on model details but is generic as soon as number fluctuations between constituents are taken into account. On a broader perspective, our results reveal that stochasticity is an important limiting factor for self-assembly and that the specific implementation of the nucleation process plays a significant role in determining the yield.

The self-assembly of a large biological molecule from small building blocks is like finishing a puzzle of magnetic pieces by shaking the box. Even though each piece of the puzzle is attracted to its correct neighbours, the limited control makes it very hard to finish the puzzle in a short amount of time.

The problem becomes even more difficult if several copies of the same puzzle are assembled in one box. If several puzzles start at the same time, the different parts might steal pieces from each other, making it impossible to successfully complete any of the puzzles. This is called a depletion trap. If the box is only shaken and there is no real control over individual pieces, these traps occur at random.

Overcoming these random depletion traps is an important challenge when assembling nanostructures and other artificial molecules designed by humans without wasting many, potentially expensive, components. Previous studies have shown that when multiple copies of the same structure are assembled simultaneously, slowing the rate of initiation increases the yield of correctly-made structures. This prevents new structures from stealing pieces from existing structures before they are fully completed.

Now, Gartner, Graf, Wilke et al. have used a mathematical model to show that changing the way initiation is delayed leads to different yields. This was especially true for small systems where fluctuations in the availability of the different pieces strongly enhanced the initiation of new structures. In these cases, the self-assembly process terminated undesirably with many incomplete structures.

Nanostructures have various applications ranging from drug delivery to robotics. These findings suggest that in order to efficiently assemble biological molecules, the concentrations of the different building blocks need to be tightly controlled. A question for further research is to investigate strategies that reduce fluctuations in the availability of the building blocks to develop more efficient assembly protocols.

Physical virology: From virus self-assembly to particle mechanics

Viruses are highly ordered supramolecular complexes that have evolved to propagate by hijacking the host cell's machinery. Although viruses are very diverse, spreading through cells of all kingdoms of life, they share common functions and properties. Next to the general interest in virology, fundamental viral mechanisms are of growing importance in other disciplines such as biomedicine and (bio)nanotechnology. However, in order to optimally make use of viruses and virus-like particles, for instance as vehicle for targeted drug delivery or as building blocks in electronics, it is essential to understand their basic chemical and physical properties and characteristics. In this context, the number of studies addressing the mechanisms governing viral properties and processes has recently grown drastically. This review summarizes a specific part of these scientific achievements, particularly addressing physical virology approaches aimed to understand the self-assembly of viruses and the mechanical properties of viral particles. Using a physicochemical perspective, we have focused on fundamental studies providing an overview of the molecular basis governing these key aspects of viral systems. This article is categorized under: Biology-Inspired Nanomaterials > Protein and Virus-Based Structures Nanotechnology Approaches to Biology > Nanoscale Systems in Biology.

Mesoscale model of the assembly and cross-linking of HPV virus-like particles

We present a novel kinetic Monte Carlo model to simulate the real process time-scale of the assembly of Human Papillomavirus (HPV) virus-like particles (VLPs) incorporating the formation of intercapsomeric disulfide bonds. The objective was to develop insights into the underlying mechanisms of HPV VLP assembly and cross-linking during in vitro production of the HPV vaccine. The model integrates actual experimental data and detailed information of VLP geometrical structure in microscopic mechanistic steps. The principal novelty of this model is in the concurrent simulation of VLP assembly and cross-linking including a variable for spatial angular arrangement of capsomeres during their assembly that affects the overall rates of VLP assembly and cross-linking. The cross-linking modeled by using the mechanistic probability rules between involved cysteine residues. The model was utilized to better understand the actual process data and check on the hypothesis related to factors affecting the rates of HPV growth and maturation.

Gaussian curvature and the budding kinetics of enveloped viruses

The formation of a membrane-enveloped virus starts with the assembly of a curved layer of capsid proteins lining the interior of the plasma membrane (PM) of the host cell. This layer develops into a spherical shell (capsid) enveloped by a lipid-rich membrane. In many cases, the budding process stalls prior to the release of the virus. Recently, Brownian dynamics simulations of a coarse-grained model system reproduced protracted pausing and stalling, which suggests that the origin of pausing/stalling is to be found in the physics of the budding process. Here, we propose that the pausing/stalling observed in the simulations can be understood as a purely kinetic phenomenon associated with the neck geometry. A geometrical potential energy barrier develops during the budding that must be overcome by capsid proteins diffusing along the membrane prior to incorporation into the capsid. The barrier is generated by a conflict between the positive Gauss curvature of the assembling capsid and the negative Gauss curvature of the neck region. A continuum theory description is proposed and is compared with the Brownian simulations of the budding of enveloped viruses.

Despite intense study, the life-cycle of the HIV-1 virus continues to pose mysteries. One of these is the fact that the assembly of an HIV-1 virus along the plasma membrane (PM) of the host cell—the budding process—stalls prior to release of the virus. Many other important viral pathogens with a surrounding lipid membrane envelope display similar stalling. Combining numerical and analytical methods, we demonstrate that the neck-like shape of the membrane that forms prior to release of the virus creates a barrier that blocks the proteins required for the assembly process from reaching the budding virus. An improved understanding of the physics of the blocking process could enable new strategies to combat enveloped viruses.

Real-Time Assembly of Viruslike Nucleocapsids Elucidated at the Single-Particle Level

While the structure of a multitude of viral particles has been resolved to atomistic detail, their assembly pathways remain largely elusive. Key unresolved issues are particle nucleation, particle growth, and the mode of genome compaction. These issues are difficult to address in bulk approaches and are effectively only accessible by the real-time tracking of assembly dynamics of individual particles. This we do here by studying the assembly into rod-shaped viruslike particles (VLPs) of artificial capsid polypeptides. Using fluorescence optical tweezers, we establish that small oligomers perform one-dimensional diffusion along the DNA. Larger oligomers are immobile and nucleate VLP growth. A multiplexed acoustic force spectroscopy approach reveals that DNA is compacted in regular steps, suggesting packaging via helical wrapping into a nucleocapsid. By reporting how real-time assembly tracking elucidates viral nucleation and growth principles, our work opens the door to a fundamental understanding of the complex assembly pathways of both VLPs and naturally evolved viruses.

Assembly Reactions of Hepatitis B Capsid Protein into Capsid Nanoparticles Follow a Narrow Path through a Complex Reaction Landscape

For many viruses, capsids (biological nanoparticles) assemble to protect genetic material and dissociate to release their cargo. To understand these contradictory properties, we analyzed capsid assembly for hepatitis B virus; an endemic pathogen with an icosahedral, 120-homodimer capsid. We used solution X-ray scattering to examine trapped and equilibrated assembly reactions. To fit experimental results, we generated a library of distinct intermediates, selected by umbrella sampling of Monte Carlo simulations. The number of possible capsid intermediates is immense, ∼10³⁰, yet assembly reactions are rapid and completed with high fidelity. If the huge number of possible intermediates were actually present, maximum entropy analysis shows that assembly reactions would be blocked by an entropic barrier, resulting in incomplete nanoparticles. When an energetic term was applied to select the stable species that dominated the reaction mixture, we found only a few hundred intermediates, mapping out a narrow path through the immense reaction landscape. This is a solution to a viral application of the Levinthal paradox. With the correct energetic term, the match between predicted intermediates and scattering data was striking. The grand canonical free energy landscape for assembly, calibrated by our experimental results, supports a detailed analysis of this complex reaction. There is a narrow range of energies that supports on-path assembly. If association energy is too weak or too strong, progressively more intermediates will be entropically blocked, spilling into paths leading to dissociation or trapped incomplete nanoparticles, respectively. These results are relevant to many viruses and provide a basis for simplifying assembly models and identifying new targets for antiviral intervention. They provide a basis for understanding and designing biological and abiological self-assembly reactions.

Computational approaches to macromolecular interactions in the cell

Structural modeling of a cell is an evolving strategic direction in computational structural biology. It takes advantage of new powerful modeling techniques, deeper understanding of fundamental principles of molecular structure and assembly, and rapid growth of the amount of structural data generated by experimental techniques. Key modeling approaches to principal types of macromolecular assemblies in a cell already exist. The main challenge, along with the further development of these modeling approaches, is putting them together in a consistent, unified whole cell model. This opinion piece addresses the fundamental aspects of modeling macromolecular assemblies in a cell, and the state-of-the-art in modeling of the principal types of such assemblies.

A New Model System for Exploring Assembly Mechanisms of the HIV-1 Immature Capsid In Vivo

The assembly of the HIV-1 immature capsid (HIC) is an essential step in the virus life cycle. In vivo, the HIC is composed of [Formula: see text] hexameric building blocks, and it takes 5-6 min to complete the assembly process. The involvement of numerous building blocks and the rapid timecourse makes it difficult to understand the HIC assembly process. In this work, we study HIC assembly in vivo by using differential equations. We first obtain a full model with 420 differential equations. Then, we reduce six addition reactions for separate building blocks to a single complex reaction. This strategy reduces the full model to 70 equations. Subsequently, the theoretical analysis of the reduced model shows that it might not be an effective way to decrease the HIC concentration at the equilibrium state by decreasing the microscopic on-rate constants. Based on experimental data, we estimate that the nucleating structure is much smaller than the HIC. We also estimate that the microscopic on-rate constant for nucleation reactions is far less than that for elongation reactions. The parametric collinearity investigation testifies the reliability of these two characteristics, which might explain why free building blocks do not readily polymerize into higher-order polymers until their concentration reaches a threshold value. These results can provide further insight into the assembly mechanisms of the HIC in vivo.

Size-Defined Cracked Vesicle Formation via Self-Assembly of Gold Nanoparticles Covered with Carboxylic Acid-Terminated Surface Ligands

The self-assembly of gold nanoparticles (GNPs) into a defined structure, particularly hollow capsule structures, provides great potential for applications in materials science and medicine. However, the complexity of the parameters for the preparation of those structures through self-assembly has limited access to critical mechanistic questions. With this in mind, we have studied GNP vesicle (GNV) formation through self-assembly by the surface modification of GNPs with low-molecular-weight ligands. Here, we successfully prepared GNVs composed of GNPs with a diameter of 30 nm by surface modification with carboxylic acid-terminated fluorinated oligo(ethylene glycol) ligands (CFLs). As the carboxylic acid has two states (protonated and deprotonated), the balance of the attraction and repulsion between GNPs covered with CFLs is tunable. Sodium carboxylate-terminated fluorinated oligo(ethylene glycol) ligands (SCFLs) provided smaller GNVs than did CFLs at 0.8 × 10¹¹ NPs/mL. Time-course study revealed that CFL-covered GNPs quickly form small aggregates and gradually grow to larger GNVs (ca. 200 nm), but no gradual growth was observed for SCFL-covered GNPs. This result indicated that the electrostatic repulsion inhibits fusion of the small GNVs. The size of the GNVs formed with the aid of CFLs was independent of the initial GNP concentration, but the extinction spectra were concentration-dependent. Electron microscopy imaging and simulations supported the defect formation in the assemblies. These results provided new insights into the vesicle formation mechanism.

Assembly Properties of Hepatitis B Virus Core Protein Mutants Correlate with Their Resistance to Assembly-Directed Antivirals

The hepatitis B virus (HBV) capsid or core protein (Cp) can self-assemble to form an icosahedral capsid. It is now being pursued as a target for small-molecule antivirals that enhance the rate and extent of its assembly to yield empty and/or aberrant capsids. These small molecules are thus called core protein allosteric modulators (CpAMs). We sought to understand the physical basis of CpAM-resistant mutants and how CpAMs might overcome them. We examined the effects of two closely related CpAMs, HAP12 and HAP13, which differ by a single atom but have drastically different antiviral activities, on the assembly of wild-type Cp and three T109 mutants (T109M, T109I, and T109S) that display a range of resistances. The T109 side chain forms part of the mouth of the CpAM binding pocket. A T109 mutant that has substantial resistance even to a highly active CpAM strongly promotes normal assembly. Conversely, a mutant that weakens assembly is more susceptible to CpAMs. In crystal and cryo-electron microscopy (cryo-EM) structures of T=4 capsids with bound CpAMs, the CpAMs preferentially fit into two of four quasi-equivalent sites. In these static representations of capsid structures, T109 does not interact with the neighboring subunit. However, all-atom molecular dynamics simulations of an intact capsid show that T109 of one of the four classes of CpAM site has a hydrophobic contact with the neighboring subunit at least 40% of the time, providing a physical explanation for the mutation's ability to affect capsid stability, assembly, and sensitivity to CpAMs.IMPORTANCE The HBV core protein and its assembly into capsids have become important targets for development of core protein allosteric modulators (CpAMs) as antivirals. Naturally occurring T109 mutants have been shown to be resistant to some of these CpAMs. We found that mutation of T109 led to changes in capsid stability and recapitulated resistance to a weak CpAM, but much less so than to a strong CpAM. Examination of HBV capsid structures, determined by cryo-EM and crystallography, could not explain how T109 mutations change capsid stability and resistance. However, by mining data from a microsecond-long all-atom molecular dynamics simulation, we found that the capsid was extraordinarily flexible and that T109 can impede entry to the CpAM binding site. In short, HBV capsids are incredibly dynamic and molecular mobility must be considered in discussions of antiviral mechanisms.

The role of the encapsulated cargo in microcompartment assembly

Bacterial microcompartments are large, roughly icosahedral shells that assemble around enzymes and reactants involved in certain metabolic pathways in bacteria. Motivated by microcompartment assembly, we use coarse-grained computational and theoretical modeling to study the factors that control the size and morphology of a protein shell assembling around hundreds to thousands of molecules. We perform dynamical simulations of shell assembly in the presence and absence of cargo over a range of interaction strengths, subunit and cargo stoichiometries, and the shell spontaneous curvature. Depending on these parameters, we find that the presence of a cargo can either increase or decrease the size of a shell relative to its intrinsic spontaneous curvature, as seen in recent experiments. These features are controlled by a balance of kinetic and thermodynamic effects, and the shell size is assembly pathway dependent. We discuss implications of these results for synthetic biology efforts to target new enzymes to microcompartment interiors.

Molecular jenga: the percolation phase transition (collapse) in virus capsids

Virus capsids are polymeric protein shells that protect the viral cargo. About half of known virus families have icosahedral capsids that self-assemble from tens to thousands of subunits. Capsid disassembly is critical to the lifecycles of many viruses yet is poorly understood. Here, we apply a graph and percolation theory to examine the effect of removing capsid subunits on capsid stability and fragmentation. Based on the structure of the icosahedral capsid of hepatitis B virus (HBV), we constructed a graph of rhombic subunits arranged with icosahedral symmetry. Though our approach neglects dependence on energetics, time, and molecular detail, it quantitatively predicts a percolation phase transition consistent with recent in vitro studies of HBV capsid dissociation. While the stability of the capsid graph followed a gradual quadratic decay, the rhombic tiling abruptly fragmented when we removed more than 25% of the 120 subunits, near the percolation threshold observed experimentally. This threshold may also affect results of capsid assembly, which also experimentally produces a preponderance of 90 mer intermediates, as the intermediate steps in these reactions are reversible and can thus resemble dissociation. Application of percolation theory to understanding capsid association and dissociation may prove a general approach to relating virus biology to the underlying biophysics of the virus particle.

Multiple Pathways in Capsid Assembly

For a three-dimensional structure to spontaneously self-assemble from many identical components, the steps on the pathway must be kinetically accessible. Many virus capsids are icosahedral and assembled from hundreds of identical proteins, but how they navigate the assembly process is poorly understood. Capsid assembly is thought to involve stepwise addition of subunits to a growing capsid fragment. Coarse-grained models suggest that the reaction occurs on a downhill energy landscape, so intermediates are expected to be fleeting. In this work, charge detection mass spectrometry (CDMS) has been used to track assembly of the hepatitis B virus (HBV) capsid in real time. The icosahedral T = 4 capsid of HBV is assembled from 120 capsid protein dimers. Our results indicate that there are multiple pathways for assembly. Under conditions that favor a modest association energy there is no accumulation of large intermediates, which indicates that available pathways include ones on a downhill energy surface. Under higher salt conditions, where subunit interactions are strengthened, around half of the products of the initial assembly reaction have masses close to the T = 4 capsid and the other half are stalled intermediates which emerge abruptly at around 90 dimers, indicating a bifurcation in the ensemble of assembly paths. When incubated at room temperature, the 90-dimer intermediates accumulate dimers and gradually shift to higher mass and merge with the capsid peak. Though free subunits are present in solution, the stalled intermediates indicate the presence of a local minima on the energy landscape. Some intermediates may result from hole closure, where the growing capsid distorts to close the hole due to the missing capsid proteins or from a species where subsequent additions are particularly labile.

Molecular dynamics study of T = 3 capsid assembly

Molecular dynamics simulation is used to model the self-assembly of polyhedral shells containing 180 trapezoidal particles that correspond to the T = 3 virus capsid. Three kinds of particle, differing only slightly in shape, are used to account for the effect of quasi-equivalence. Bond formation between particles is reversible and an explicit atomistic solvent is included. Under suitable conditions the simulations are able to produce complete shells, with the majority of unused particles remaining as monomers, and practically no other clusters. There are also no incorrectly assembled clusters. The simulations reveal details of intermediate structures along the growth pathway, information that is relevant for interpreting experiment.

Hepatitis B Virus Capsid Completion Occurs through Error Correction

Understanding capsid assembly is important because of its role in virus lifecycles and in applications to drug discovery and nanomaterial development. Many virus capsids are icosahedral, and assembly is thought to occur by the sequential addition of capsid protein subunits to a nucleus, with the final step completing the icosahedron. Almost nothing is known about the final (completion) step because the techniques usually used to study capsid assembly lack the resolution. In this work, charge detection mass spectrometry (CDMS) has been used to track the assembly of the T = 4 hepatitis B virus (HBV) capsid in real time. The initial assembly reaction occurs rapidly, on the time scale expected from low resolution measurements. However, CDMS shows that many of the particles generated in this process are defective and overgrown, containing more than the 120 capsid protein dimers needed to form a perfect T = 4 icosahedron. The defective and overgrown capsids self-correct over time to the mass expected for a perfect T = 4 capsid. Thus, completion is a distinct phase in the assembly reaction. Capsid completion does not necessarily occur by inserting the last building block into an incomplete, but otherwise perfect icosahedron. The initial assembly reaction can be predominently imperfect, and completion involves the slow correction of the accumulated errors.

Kinetic constraints on self-assembly into closed supramolecular structures

Many biological and synthetic systems exploit self-assembly to generate highly intricate closed supramolecular architectures, ranging from self-assembling cages to viral capsids. The fundamental design principles that control the structural determinants of the resulting assemblies are increasingly well-understood, but much less is known about the kinetics of such assembly phenomena and it remains a key challenge to elucidate how these systems can be engineered to assemble in an efficient manner and avoid kinetic trapping. We show here that simple scaling laws emerge from a set of kinetic equations describing the self-assembly of identical building blocks into closed supramolecular structures and that this scaling behavior provides general rules that determine efficient assembly in these systems. Using this framework, we uncover the existence of a narrow range of parameter space that supports efficient self-assembly and reveal that nature capitalizes on this behavior to direct the reliable assembly of viral capsids on biologically relevant timescales.

Quantitative computational models of molecular self-assembly in systems biology

Molecular self-assembly is the dominant form of chemical reaction in living systems, yet efforts at systems biology modeling are only beginning to appreciate the need for and challenges to accurate quantitative modeling of self-assembly. Self-assembly reactions are essential to nearly every important process in cell and molecular biology and handling them is thus a necessary step in building comprehensive models of complex cellular systems. They present exceptional challenges, however, to standard methods for simulating complex systems. While the general systems biology world is just beginning to deal with these challenges, there is an extensive literature dealing with them for more specialized self-assembly modeling. This review will examine the challenges of self-assembly modeling, nascent efforts to deal with these challenges in the systems modeling community, and some of the solutions offered in prior work on self-assembly specifically. The review concludes with some consideration of the likely role of self-assembly in the future of complex biological system models more generally.

Nanofluidic Devices with 8 Pores in Series for Real-Time, Resistive-Pulse Analysis of Hepatitis B Virus Capsid Assembly

To improve the precision of resistive-pulse measurements, we have used a focused ion beam instrument to mill nanofluidic devices with 2, 4, and 8 pores in series and compared their performance. The in-plane design facilitates the fabrication of multiple pores in series which, in turn, permits averaging of the series of pulses generated from each translocation event. The standard deviations (σ) of the pulse amplitude distributions decrease by 2.7-fold when the average amplitudes of eight pulses are compared to the amplitudes of single pulses. Similarly, standard deviations of the pore-to-pore time distributions decrease by 3.2-fold when the averages of the seven measurements from 8-pore devices are contrasted to single measurements from 2-pore devices. With signal averaging, the inherent uncertainty in the measurements decreases; consequently, the resolution (mean/σ) improves by a factor equal to the square root of the number of measurements. We took advantage of the improved size resolution of the 8-pore devices to analyze in real time the assembly of Hepatitis B Virus (HBV) capsids below the pseudocritical concentration. We observe that abundances of assembly intermediates change over time. During the first hour of the reaction, the abundance of smaller intermediates decreased, whereas the abundance of larger intermediates with sizes closer to a T = 4 capsid remained constant.

Mathematical modeling of HIV-like particle assembly in vitro

In vitro, the recombinant HIV-1 Gag protein can generate spherical particles with a diameter of 25-30 nm in a fully defined system. It has approximately 80 building blocks, and its intermediates for assembly are abundant in geometry. Accordingly, there are a large number of nonlinear equations in the classical model. Therefore, it is difficult to compute values of geometry parameters for intermediates and make the mathematical analysis using the model. In this work, we develop a new model of HIV-like particle assembly in vitro by using six-fold symmetry of HIV-like particle assembly to decrease the number of geometry parameters. This method will greatly reduce computational costs and facilitate the application of the model. Then, we prove the existence and uniqueness of the positive equilibrium solution for this model with 79 nonlinear equations. Based on this model, we derive the interesting result that concentrations of all intermediates at equilibrium are independent of three important parameters, including two microscopic on-rate constants and the size of nucleating structure. Before equilibrium, these three parameters influence the concentration variation rates of all intermediates. We also analyze the relationship between the initial concentration of building blocks and concentrations of all intermediates. Furthermore, the bounds of concentrations of free building blocks and HIV-like particles are estimated. These results will be helpful to guide HIV-like particle assembly experiments and improve our understanding of the assembly dynamics of HIV-like particles in vitro.

Imaging and Quantitation of a Succession of Transient Intermediates Reveal the Reversible Self-Assembly Pathway of a Simple Icosahedral Virus Capsid

Understanding the fundamental principles underlying supramolecular self-assembly may facilitate many developments, from novel antivirals to self-organized nanodevices. Icosahedral virus particles constitute paradigms to study self-assembly using a combination of theory and experiment. Unfortunately, assembly pathways of the structurally simplest virus capsids, those more accessible to detailed theoretical studies, have been difficult to study experimentally. We have enabled the in vitro self-assembly under close to physiological conditions of one of the simplest virus particles known, the minute virus of mice (MVM) capsid, and experimentally analyzed its pathways of assembly and disassembly. A combination of electron microscopy and high-resolution atomic force microscopy was used to structurally characterize and quantify a succession of transient assembly and disassembly intermediates. The results provided an experiment-based model for the reversible self-assembly pathway of a most simple (T = 1) icosahedral protein shell. During assembly, trimeric capsid building blocks are sequentially added to the growing capsid, with pentamers of building blocks and incomplete capsids missing one building block as conspicuous intermediates. This study provided experimental verification of many features of self-assembly of a simple T = 1 capsid predicted by molecular dynamics simulations. It also demonstrated atomic force microscopy imaging and automated analysis, in combination with electron microscopy, as a powerful single-particle approach to characterize at high resolution and quantify transient intermediates during supramolecular self-assembly/disassembly reactions. Finally, the efficient in vitro self-assembly achieved for the oncotropic, cell nucleus-targeted MVM capsid may facilitate its development as a drug-encapsidating nanoparticle for anticancer targeted drug delivery.

The Structural Biology of Hepatitis B Virus: Form and Function

Hepatitis B virus is one of the smallest human pathogens, encoded by a 3,200-bp genome with only four open reading frames. Yet the virus shows a remarkable diversity in structural features, often with the same proteins adopting several conformations. In part, this is the parsimony of viruses, where a minimal number of proteins perform a wide variety of functions. However, a more important theme is that weak interactions between components as well as components with multiple conformations that have similar stabilities lead to a highly dynamic system. In hepatitis B virus, this is manifested as a virion where the envelope proteins have multiple structures, the envelope-capsid interaction is irregular, and the capsid is a dynamic compartment that actively participates in metabolism of the encapsidated genome and carries regulated signals for intracellular trafficking.

Modeling Effects of RNA on Capsid Assembly Pathways via Coarse-Grained Stochastic Simulation

The environment of a living cell is vastly different from that of an in vitro reaction system, an issue that presents great challenges to the use of in vitro models, or computer simulations based on them, for understanding biochemistry in vivo. Virus capsids make an excellent model system for such questions because they typically have few distinct components, making them amenable to in vitro and modeling studies, yet their assembly can involve complex networks of possible reactions that cannot be resolved in detail by any current experimental technology. We previously fit kinetic simulation parameters to bulk in vitro assembly data to yield a close match between simulated and real data, and then used the simulations to study features of assembly that cannot be monitored experimentally. The present work seeks to project how assembly in these simulations fit to in vitro data would be altered by computationally adding features of the cellular environment to the system, specifically the presence of nucleic acid about which many capsids assemble. The major challenge of such work is computational: simulating fine-scale assembly pathways on the scale and in the parameter domains of real viruses is far too computationally costly to allow for explicit models of nucleic acid interaction. We bypass that limitation by applying analytical models of nucleic acid effects to adjust kinetic rate parameters learned from in vitro data to see how these adjustments, singly or in combination, might affect fine-scale assembly progress. The resulting simulations exhibit surprising behavioral complexity, with distinct effects often acting synergistically to drive efficient assembly and alter pathways relative to the in vitro model. The work demonstrates how computer simulations can help us understand how assembly might differ between the in vitro and in vivo environments and what features of the cellular environment account for these differences.

Allosteric Control of Icosahedral Capsid Assembly

During the life cycle of a virus, viral proteins and other components self-assemble to form an ordered protein shell called a capsid. This assembly process is subject to multiple competing constraints, including the need to form a thermostable shell while avoiding kinetic traps. It has been proposed that viral assembly satisfies these constraints through allosteric regulation, including the interconversion of capsid proteins among conformations with different propensities for assembly. In this article, we use computational and theoretical modeling to explore how such allostery affects the assembly of icosahedral shells. We simulate assembly under a wide range of protein concentrations, protein binding affinities, and two different mechanisms of allosteric control. We find that above a threshold strength of allosteric control, assembly becomes robust over a broad range of subunit binding affinities and concentrations, allowing the formation of highly thermostable capsids. Our results suggest that allostery can significantly shift the range of protein binding affinities that lead to successful assembly and thus should be taken into account in models that are used to estimate interaction parameters from experimental data.

Kinetics versus Thermodynamics in Virus Capsid Polymorphism

Virus coat proteins spontaneously self-assemble into empty shells in aqueous solution under the appropriate physicochemical conditions, driven by an interaction free energy per bond on the order of 2-5 times the thermal energy kBT. For this seemingly modest interaction strength, each protein building block nonetheless gains a very large binding free energy, between 10 and 20 kBT. Because of this, there is debate about whether the assembly process is reversible or irreversible. Here we discuss capsid polymorphism observed in in vitro experiments from the perspective of nucleation theory and of the thermodynamics of mass action. We specifically consider the potential contribution of a curvature free energy term to the effective interaction potential between the proteins. From these models, we propose experiments that may conclusively reveal whether virus capsid assembly into a mixture of polymorphs is a reversible or an irreversible process.

Multiscale modeling of virus replication and spread

Replication and spread of human viruses is based on the simultaneous exploitation of many different host functions, bridging multiple scales in space and time. Mathematical modeling is essential to obtain a systems-level understanding of how human viruses manage to proceed through their life cycles. Here, we review corresponding advances for viral systems of large medical relevance, such as human immunodeficiency virus-1 (HIV-1) and hepatitis C virus (HCV). We will outline how the combination of mathematical models and experimental data has advanced our quantitative knowledge about various processes of these pathogens, and how novel quantitative approaches promise to fill remaining gaps.

Anomalous approach to thermodynamic equilibrium: Structure formation of molecules after vapor deposition

We describe experiments and computer simulations of molecular deposition on a substrate in which the molecules (substituted adenine derivatives) self-assemble into ordered structures. The resulting structures depend strongly on the deposition rate (flux). In particular, there are two competing surface morphologies (α and β), which differ by their topology (interdigitated vs lamellar structure). Experimentally, the α phase dominates at both low and high flux, with the β phase being most important in the intermediate regime. A similar nonmonotonic behavior is observed on varying the substrate temperature. To understand these effects from a theoretical perspective, a lattice model is devised which reproduces qualitatively the topological features of both phases. Via extensive Monte Carlo studies we can, on the one hand, reproduce the experimental results and, on the other hand, obtain a microscopic understanding of the mechanisms behind this anomalous behavior. The results are discussed in terms of an interplay between kinetic trapping and temporal exploration of configuration space.