A precise packing sequence for self-assembled convex structures

Virtual indentation of the empty capsid of the minute virus of mice using a minimal coarse-grained model

A minimal coarse-grained model for T=1 viral capsids assembled from 20 protein rigid trimers has been designed by extending a previously proposed form of the interaction energy written as a sum of anisotropic pairwise interactions between the trimeric capsomers. The extension of the model has been performed to properly account for the coupling between two internal coordinates: the one that measures the intercapsomer distance and the other that gives the intercapsomer dihedral angle. The model has been able to fit with less than a 10% error the atomic force microscopy (AFM) indentation experimental data for the empty capsid of the minute virus of mice (MVM), providing in this way an admissible picture of the main mechanisms behind the capsid deformations. In this scenario, the bending of the intercapsomer dihedral angle is the angular internal coordinate that can support larger deformations away from its equilibrium values, determining important features of the AFM indentation experiments as the elastic constants along the three symmetry axes of the capsid and the critical indentations. From the value of one of the parameters of our model, we conclude that trimers in the MVM must be quite oblate tops, in excellent agreement with their known structure. The transition from the linear to the nonlinear regimes sampled in the indentation process appears to be an interesting topic for future research in physical virology.

Packing and trimer-to-dimer protein reconstruction in icosahedral viral shells with a single type of symmetrical structural unit

Understanding the principles of protein packing and the mechanisms driving morphological transformations in virus shells (capsids) during their maturation can be pivotal for the development of new antiviral strategies. Here, we study how these principles and mechanisms manifest themselves in icosahedral viral capsids assembled from identical symmetric structural units (capsomeres). To rationalize such shells, we model capsomers as symmetrical groups of identical particles interacting with a short-range potential typical of the classic Tammes problem. The capsomere particles are assumed to retain their relative positions on the vertices of planar polygons placed on the spherical shell and to interact only with the particles from other capsomeres. Minimization of the interaction energy enforces equal distances between the nearest particles belonging to neighboring capsomeres and minimizes the number of different local environments. Thus, our model implements the Caspar and Klug quasi-equivalence principle and leads to packings strikingly similar to real capsids. We then study a reconstruction of protein trimers into dimers in a Flavivirus shell during its maturation, connecting the relevant structural changes with the modifications of the electrostatic charges of proteins, wrought by the oxidative switch in the bathing solution that is essential for the process. We highlight the key role of pr peptides in the shell reconstruction and show that the highly ordered arrangement of these subunits in the dimeric state is energetically favored at a low pH level. We also discuss the electrostatic mechanisms controlling the release of pr peptides in the last irreversible step of the maturation process.

Biophysical Modeling of SARS-CoV-2 Assembly: Genome Condensation and Budding

The COVID-19 pandemic caused by the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) has spurred unprecedented and concerted worldwide research to curtail and eradicate this pathogen. SARS-CoV-2 has four structural proteins: Envelope (E), Membrane (M), Nucleocapsid (N), and Spike (S), which self-assemble along with its RNA into the infectious virus by budding from intracellular lipid membranes. In this paper, we develop a model to explore the mechanisms of RNA condensation by structural proteins, protein oligomerization and cellular membrane-protein interactions that control the budding process and the ultimate virus structure. Using molecular dynamics simulations, we have deciphered how the positively charged N proteins interact and condense the very long genomic RNA resulting in its packaging by a lipid envelope decorated with structural proteins inside a host cell. Furthermore, considering the length of RNA and the size of the virus, we find that the intrinsic curvature of M proteins is essential for virus budding. While most current research has focused on the S protein, which is responsible for viral entry, and it has been motivated by the need to develop efficacious vaccines, the development of resistance through mutations in this crucial protein makes it essential to elucidate the details of the viral life cycle to identify other drug targets for future therapy. Our simulations will provide insight into the viral life cycle through the assembly of viral particles de novo and potentially identify therapeutic targets for future drug development.

Virus Assembly Pathways Inside a Host Cell

Simple RNA viruses self-assemble spontaneously and encapsulate their genome into a shell called the capsid. This process is mainly driven by the attractive electrostatics interaction between the positive charges on capsid proteins and the negative charges on the genome. Despite its importance and many decades of intense research, how the virus selects and packages its native RNA inside the crowded environment of a host cell cytoplasm in the presence of an abundance of nonviral RNA and other anionic polymers has remained a mystery. In this paper, we perform a series of simulations to monitor the growth of viral shells and find the mechanism by which cargo-coat protein interactions can impact the structure and stability of the viral shells. We show that coat protein subunits can assemble around a globular nucleic acid core by forming nonicosahedral cages, which have been recently observed in assembly experiments involving small pieces of RNA. We find that the resulting cages are strained and can easily be split into fragments along stress lines. This suggests that such metastable nonicosahedral intermediates could be easily reassembled into the stable native icosahedral shells if the larger wild-type genome becomes available, despite the presence of a myriad of nonviral RNAs.

Minimal Design Principles for Icosahedral Virus Capsids

The geometrical structures of single- and multiple-shell icosahedral virus capsids are reproduced as the targets that minimize the cost corresponding to relatively simple design functions. Capsid subunits are first identified as building blocks at a given coarse-grained scale and then represented in these functions as point particles located on an appropriate number of concentric spherical surfaces. Minimal design cost is assigned to optimal spherical packings of the particles. The cost functions are inspired by the packings favored for the Thomson problem, which minimize the electrostatic potential energy between identical charged particles. In some cases, icosahedral symmetry constraints are incorporated as external fields acting on the particles. The simplest cost functions can be obtained by separating particles in disjoint nonequivalent sets with distinct interactions, or by introducing interacting holes (the absence of particles). These functions can be adapted to reproduce any capsid structure found in real viruses. Structures absent in Nature require significantly more complex designs. Measures of information content and complexity are assigned to both the cost functions and the capsid geometries. In terms of these measures, icosahedral structures and the corresponding cost functions are the simplest solutions.

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.

Fabrication of magnetic colloidal bicones and their switchable self-assembly

The creation and fabrication of novel shapes are of paramount importance for colloids to self-assemble into desired structured materials and in order to use them as model systems for fundamental studies. Here, we demonstrate a simple yet effective method to fabricate bicone-shaped colloids by using SiO₂/PS core/shell particles through a thermal stretching route. The shape parameters are tunable. The aspect ratio is found to be predictable, and the experimental results are consistent with the model calculations. The yields are high enough to be used for self-assembly studies. As an illustration, we investigate the phase behavior of particles when interacting with depletion forces and show that the particles can form hexagonal or non-hexagonal crystal lattices in quasi-two-dimensional space. The method also allows functional nanoparticles to be integrated into the cores, resulting in responsive colloidal bicones. Moreover, the magnetically responsive self-assembly of particles is demonstrated.

A minimal coarse-grained model for the low-frequency normal mode analysis of icosahedral viral capsids

The main goal of this work is the design of a coarse-grained theoretical model of minimal resolution for the study of the physical properties of icosahedral virus capsids within the linear-response regime. In this model the capsid is represented as an interacting many-body system whose composing elements are capsid subunits (capsomers), which are treated as three-dimensional rigid bodies. The total interaction potential energy is written as a sum of pairwise capsomer-capsomer interactions. Based on previous work [Gomez Llorente et al., Soft Matter, 2014, 10, 3560], a minimal and complete anisotropic binary interaction that includes a full Hessian matrix of independent force constants is proposed. In this interaction model, capsomers have rotational symmetry around an axis of order n > 2. The full coarse-grained model is applied to analyse the low-frequency normal-mode spectrum of icosahedral T = 1 capsids. The model performance is evaluated by fitting its predicted spectrum to the full-atom results for the Satellite Tobacco Necrosis Virus (STNV) capsid [Dykeman and Sankey, Phys. Rev. Lett., 2008, 100, 028101]. Two capsomer choices that are compatible with the capsid icosahedral symmetry are checked, namely pentamers (n = 5) and trimers (n = 3). Both subunit types provide fair fits, from which the magnitude of the coarse-grained force constants for a real virus is obtained. The model is able to uncover latent instabilities whose analysis is fully consistent with the current knowledge about the STNV capsid, which does not self-assemble in the absence of RNA and is thermally unstable. The straightforward generalisability of the model beyond the linear regime and its completeness make it a promising tool to theoretically interpret many experimental data such as those provided by the atomic force microscopy or even to better understand processes far from equilibrium such as the capsid self-assembly.

How a Virus Circumvents Energy Barriers to Form Symmetric Shells

Previous self-assembly experiments on a model icosahedral plant virus have shown that, under physiological conditions, capsid proteins initially bind to the genome through an en masse mechanism and form nucleoprotein complexes in a disordered state, which raises the question as to how virions are assembled into a highly ordered structure in the host cell. Using small-angle X-ray scattering, we find out that a disorder-order transition occurs under physiological conditions upon an increase in capsid protein concentrations. Our cryo-transmission electron microscopy reveals closed spherical shells containing in vitro transcribed viral RNA even at pH 7.5, in marked contrast with the previous observations. We use Monte Carlo simulations to explain this disorder-order transition and find that, as the shell grows, the structures of disordered intermediates in which the distribution of pentamers does not belong to the icosahedral subgroups become energetically so unfavorable that the caps can easily dissociate and reassemble, overcoming the energy barriers for the formation of perfect icosahedral shells. In addition, we monitor the growth of capsids under the condition that the nucleation and growth is the dominant pathway and show that the key for the disorder-order transition in both en masse and nucleation and growth pathways lies in the strength of elastic energy compared to the other forces in the system including protein-protein interactions and the chemical potential of free subunits. Our findings explain, at least in part, why perfect virions with icosahedral order form under different conditions including physiological ones.

100th Anniversary of Macromolecular Science Viewpoint: Block Copolymer Particles: Tuning Shape, Interfaces, and Morphology

Confined assembly of block copolymers (BCPs) is receiving increasing attention due to the ability to create unconventional morphologies that cannot be observed in the corresponding bulk systems. This effect is further driven by the simplicity and versatility of these procedures for controlling the shape of particles prepared by 3D soft confinement of BCPs in emulsions. By taking advantage of a mobile emulsion interface, the one-step formation of nonspherical BCP particles through spontaneous deformation is possible with design principles and theoretical models for controlling shape/nanostructure now being established. This Viewpoint highlights strategies for shape tuning of BCP particles, currently accessible shapes, their controllability, and potential application. The emergence of 3D soft confinement of BCPs and related theory is overviewed with a focus on current strategies, types of nonspherical shapes achieved, and structure-property relationships for nonspherical BCP particles. Finally, the applications and future perspectives for these materials are discussed.

Kinetics of empty viral capsid assembly in a minimal model

The efficient construction of a protective protein shell or capsid is one of the most crucial steps in the replication cycle of a virus. The formation of the simplest capsid typically proceeds by the spontaneous assembly of identical building blocks. This process can also be achieved in vitro even in the absence of genetic material, thus opening the door to the production of artificial viral cages for a myriad of applications. In this work, we analyze the efficiency and the kinetic peculiarities of this self-assembly process using Brownian Dynamics simulations. We use a minimal model that considers identical assembly units and is able to reproduce successfully the correct final architecture of spherical capsids. The selection of a specific size and structure is achieved by changing a single parameter that imposes an angular anisotropy on the interaction. We analyze how the geometrical constraints of the interaction affect the efficiency of the assembly. We find that the optimal conditions for an efficient assembly from a kinetic point of view strongly depart from the lowest capsid energy corresponding to the minimum of the potential energy landscape. Our work illustrates the important differences between the equilibrium and dynamic characteristics of viral self-assembly, and provides important insights on how to design specific interactions for a successful assembly of artificial viral cages.

Shape control of nanostructured cone-shaped particles by tuning the blend morphology of A-b-B diblock copolymers and C-type copolymers within emulsion droplets

An approach to blend AB-type block copolymers and C-type copolymers within the emulsion droplet is an efficient particle shape-engineering strategy.

The equilibrium structure of self-assembled protein nano-cages

Understanding how highly symmetric, robust, monodisperse protein nano-cages self-assemble can have major applications in various areas of bio-nanotechnology, such as drug delivery, biomedical imaging and gene therapy. We develop a model to investigate the assembly of protein subunits into the structures with different sizes and symmetries. Using Monte Carlo simulation, we obtain global minimum energy structures. Our results suggest that the physical properties including the spontaneous curvature, flexibility and bending rigidity of coat proteins are sufficient to predict the size, symmetry and shape selectivity of the assembly products. Further, on a thermodynamic basis, we discuss the polymorphism of nano-cages observed in assembly experiments.

Why large icosahedral viruses need scaffolding proteins

While small single-stranded viral shells encapsidate their genome spontaneously, many large viruses, such as the herpes simplex virus or infectious bursal disease virus (IBDV), typically require a template, consisting of either scaffolding proteins or an inner core. Despite the proliferation of large viruses in nature, the mechanisms by which hundreds or thousands of proteins assemble to form structures with icosahedral order (IO) is completely unknown. Using continuum elasticity theory, we study the growth of large viral shells (capsids) and show that a nonspecific template not only selects the radius of the capsid, but also leads to the error-free assembly of protein subunits into capsids with universal IO. We prove that as a spherical cap grows, there is a deep potential well at the locations of disclinations that later in the assembly process will become the vertices of an icosahedron. Furthermore, we introduce a minimal model and simulate the assembly of a viral shell around a template under nonequilibrium conditions and find a perfect match between the results of continuum elasticity theory and the numerical simulations. Besides explaining available experimental results, we provide a number of predictions. Implications for other problems in spherical crystals are also discussed.

Defects and Chirality in the Nanoparticle-Directed Assembly of Spherocylindrical Shells of Virus Coat Proteins

Virus coat proteins of small isometric plant viruses readily assemble into symmetric, icosahedral cages encapsulating noncognate cargo, provided the cargo meets a minimal set of chemical and physical requirements. While this capability has been intensely explored for certain virus-enabled nanotechnologies, additional applications require lower symmetry than that of an icosahedron. Here, we show that the coat proteins of an icosahedral virus can efficiently assemble around metal nanorods into spherocylindrical closed shells with hexagonally close-packed bodies and icosahedral caps. Comparison of chiral angles and packing defects observed by in situ atomic force microscopy with those obtained from molecular dynamics models offers insight into the mechanism of growth, and the influence of stresses associated with intrinsic curvature and assembly pathways.

The effect of RNA stiffness on the self-assembly of virus particles

Under many in vitro conditions, some small viruses spontaneously encapsidate a single stranded (ss) RNA into a protein shell called the capsid. While viral RNAs are found to be compact and highly branched because of long distance base-pairing between nucleotides, recent experiments reveal that in a head-to-head competition between an ssRNA with no secondary or higher order structure and a viral RNA, the capsid proteins preferentially encapsulate the linear polymer! In this paper, we study the impact of genome stiffness on the encapsidation free energy of the complex of RNA and capsid proteins. We show that an increase in effective chain stiffness because of base-pairing could be the reason why under certain conditions linear chains have an advantage over branched chains when it comes to encapsidation efficiency. While branching makes the genome more compact, RNA base-pairing increases the effective Kuhn length of the RNA molecule, which could result in an increase of the free energy of RNA confinement, that is, the work required to encapsidate RNA, and thus less efficient packaging.

Self-assembly of amphiphilic truncated cones to form hollow nanovesicles

To mimic the unique properties of capsid (protein shell of a virus), we performed Brownian dynamics simulations of the self-assembly of amphiphilic truncated cone particles with anisotropic interactions. The particle shape of a truncated cone in our simulations depended on the cone angle θ, truncated height h_c and particle type (A_xB_y and B_xA_yB_z). The hydrophobic A moieties and hydrophilic B moieties are responsible for attractive and repulsive interactions, respectively. By varying the particle shape, truncated cones can assemble into hollow and vesicle-like clusters with a specific cluster size N. To assemble into hollow vesicles, the truncated height h_c must be below a critical value. When h_c exceeds this critical value, malformation will occur. The dynamics shows that the vesicle formation occurs in three stages: initially the growth is slow, then rapid, and finally it slows down. The truncated height h_c has a stronger impact on the growth kinetics than the cone angle θ or the particle type. We explored how the cluster packing depended on the cooling rate and particle number as well as discussing the relationship between the cluster geometry and the interparticle interactions. Further, we also discuss possible methods to experimentally prepare the truncated cones. The results of our work deepen our understanding of the self-assembly behavior of truncated cones and our results will aid the effective design of particle building blocks for novel nanostructures.

To mimic the unique properties of capsid (protein shell of a virus), we performed Brownian dynamics simulations of the self-assembly of amphiphilic truncated cone particles with anisotropic interactions.

Tapered Bottlebrush Polymers: Cone-Shaped Nanostructures by Sequential Addition of Macromonomers

Tapered (cone-shaped) bottlebrush polymers were synthesized for the first time by ring-opening metathesis polymerization (ROMP) using a sequential-addition of macromonomers (SAM) strategy. Polystyrene macromonomers with molecular weights that increased from 1 to 10 kg mol^-1 were polymerized in sequence to high conversion, yielding tapered bottlebrush polymers that could be visualized by atomic force microscopy (AFM).

Impact of a nonuniform charge distribution on virus assembly

Many spherical viruses encapsulate their genomes in protein shells with icosahedral symmetry. This process is spontaneous and driven by electrostatic interactions between positive domains on the virus coat proteins and the negative genomes. We model the effect of the nonuniform icosahedral charge distribution from the protein shell instead using a mean-field theory. We find that this nonuniform charge distribution strongly affects the optimal genome length and that it can explain the experimentally observed phenomenon of overcharging of virus and viruslike particles.

Fabrication of large area flexible nanoplasmonic templates with flow coating

We describe the development of a custom-built two-axis flow coater for the deposition of polymeric nanosphere monolayers that could be used in the fabrication of large area nanoplasmonic films. The technique described here has the capability of depositing large areas (up to 7 in. × 10 in.) of self-assembled monolayers of polymeric nanospheres onto polyethylene terephthalate (PET) films. Here, three sets of films consisting of different diameters (ranging from 100 to 300 nm) of polymeric nanospheres were used to demonstrate the capabilities of this instrument. To improve the surface wettability of the PET substrates during wet-deposition, we enhanced the wettability by using a forced air blown-arc plasma treatment system. Both the local microstructure, as confirmed by scanning electron microscopy, describing monolayer and multilayer coverage, and the overall macroscopic uniformity of the resultant nanostructured film were optimized by controlling the relative stage to blade speed and nanosphere concentration. We also show using a smaller nanoparticle template that such monolayers can be used to form nanoplasmonic films. As this flow-coating approach is a scalable technique, large area films such as the ones described here have a variety of crucial emerging applications in areas such as energy, catalysis, and chemical sensing.

Depletion-Induced Encapsulation by Dumbbell-Shaped Patchy Colloids Stabilize Microspheres against Aggregation

In this paper, we demonstrate the stabilization of polystyrene microspheres by encapsulating them with dumbbell-shaped colloids with a sticky and a nonsticky lobe. Upon adding a depletant, an effective short ranged attraction is induced between the microspheres and the smaller, smooth lobes of the dumbbells, making those specifically sticky, whereas the interaction with the larger lobes of the dumbbells is considerably less attractive due to their rough surface, which reduces the overlap volume and leaves them nonsticky. The encapsulation of the microspheres by these rough-smooth patchy dumbbells is investigated using a combination of experiments and computer simulations, both resulting in partial coverage of the template particles. For larger microspheres, the depletion attraction is stronger, resulting in a larger fraction of dumbbells that are attached with both lobes to the surface of microspheres. We thus find a template curvature dependent orientation of the dumbbells. In the Monte Carlo simulations, the introduction of such a small, curvature dependent attraction between the rough lobes of the dumbbells resulted in an increased coverage. However, kinetic constraints imposed by the dumbbell geometry seem to prevent optimal packing of the dumbbells on the template particles under all investigated conditions in experiments and simulations. Despite the incomplete coverage, the encapsulation by dumbbell particles does prevent aggregation of the microspheres, thus acting as a colloid-sized steric stabilizer.

Planet-Satellite Micellar Superstructures Formed by ABCB Terpolymers in Solution

The occurrence and relative stability of planet-satellite nanostructures, composed of a host micelle (the planet) accompanied by a number of guest micelles (the satellites), in ABCB tetrablock terpolymer solutions are studied using the polymeric self-consistent field theory and dissipative particle dynamics simulations. The theoretical results demonstrate that the self-assembly of the ABCB tetrablock terpolymers with solvophobic A- and C-blocks and solvophilic B-blocks could lead to the formation of various planet-satellite superstructures, where the planet and satellites are composed of the A- and C-blocks, respectively. Furthermore, the number of satellites is controlled by the ratio of the two B-blocks. The arrangement of the satellites surrounding the planet resembles the solution of the well-known Thomson's problem concerning the optimum arrangement of a given number of electrons on a sphere. Besides providing a facile route to engineering novel multicompartment micelles with planet-satellite superstructures for potential advanced applications, the study strengthens the prospect that multiblock copolymers could become a useful platform for the fabrication of complex nanostructures.

A Method for Molecular Dynamics on Curved Surfaces

Dynamics simulations of constrained particles can greatly aid in understanding the temporal and spatial evolution of biological processes such as lateral transport along membranes and self-assembly of viruses. Most theoretical efforts in the field of diffusive transport have focused on solving the diffusion equation on curved surfaces, for which it is not tractable to incorporate particle interactions even though these play a crucial role in crowded systems. We show here that it is possible to take such interactions into account by combining standard constraint algorithms with the classical velocity Verlet scheme to perform molecular dynamics simulations of particles constrained to an arbitrarily curved surface. Furthermore, unlike Brownian dynamics schemes in local coordinates, our method is based on Cartesian coordinates, allowing for the reuse of many other standard tools without modifications, including parallelization through domain decomposition. We show that by applying the schemes to the Langevin equation for various surfaces, we obtain confined Brownian motion, which has direct applications to many biological and physical problems. Finally we present two practical examples that highlight the applicability of the method: 1) the influence of crowding and shape on the lateral diffusion of proteins in curved membranes; and 2) the self-assembly of a coarse-grained virus capsid protein model.

Faceted particles formed by the frustrated packing of anisotropic colloids on curved surfaces

We use computer simulations and simple theoretical models to analyze the morphologies that result when rod-like particles end-attach onto a curved surface, creating a finite-thickness monolayer aligned with the surface normal. This geometry leads to two forms of frustration, one associated with the incompatibility of hexagonal order on surfaces with Gaussian curvature, and the second reflecting the deformation of a layer with finite thickness on a surface with non-zero mean curvature. We show that the latter effect leads to a faceting mechanism. Above threshold values of inter-particle attraction strength and surface mean curvature, the adsorbed layer undergoes a transition from orientational disorder to an ordered state that is demarcated by reproducible patterns of line defects. The number of facets is controlled by the competition between line defect energy and intra-facet strain. Tuning control parameters thus leads to a rich variety of morphologies, including icosahedral particles and irregular polyhedra. In addition to suggesting a new strategy for the synthesis of aspherical particles with tunable symmetries, our results may shed light on recent experiments in which rod-like HIV GAG proteins assemble around nanoscale particles.

Hidden symmetry of small spherical viruses and organization principles in "anomalous" and double-shelled capsid nanoassemblies

We propose the principles of structural organization in spherical nanoassemblies with icosahedral symmetry constituted by asymmetric protein molecules. The approach modifies the paradigmatic geometrical Caspar and Klug (CK) model of icosahedral viral capsids and demonstrates the common origin of both the "anomalous" and conventional capsid structures. In contrast to all previous models of "anomalous" viral capsids the proposed modified model conserves the basic structural principles of the CK approach and reveals the common hidden symmetry underlying all small viral shells. We demonstrate the common genesis of the "anomalous" and conventional capsids and explain their structures in the same frame. The organization principles are derived from the group theory analysis of the positional order on the spherical surface. The relationship between the modified CK geometrical model and the theory of two-dimensional spherical crystallization is discussed. We also apply the proposed approach to complex double-shelled capsids and capsids with protruding knob-like proteins. The introduced notion of commensurability for the concentric nanoshells explains the peculiarities of their organization and helps to predict analogous, but yet undiscovered, double-shelled viral capsid nanostructures.

Physical Ingredients Controlling Stability and Structural Selection of Empty Viral Capsids

One of the crucial steps in the viral replication cycle is the self-assembly of its protein shell. Typically, each native virus adopts a unique architecture, but the coat proteins of many viruses have the capability to self-assemble in vitro into different structures by changing the assembly conditions. However, the mechanisms determining which of the possible capsid shapes and structures is selected by a virus are still not well-known. We present a coarse-grained model to analyze and understand the physical mechanisms controlling the size and structure selection in the assembly of empty viral capsids. Using this model and Monte Carlo simulations, we have characterized the phase diagram and stability of T = 1,3,4,7 and snub cube shells. In addition, we have studied the tolerance of different shells to changes in physical parameters related to ambient conditions, identifying possible strategies to induce misassembly or failure. Finally, we discuss the factors that select the shape of a capsid as spherical, faceted, elongated, or decapsidated. Our model sheds important light on the ingredients that control the assembly and stability of viral shells. This knowledge is essential to get capsids with well-defined size and structure that could be used for promising applications in medicine or bionanotechnology.

The Robust Assembly of Small Symmetric Nanoshells

Highly symmetric nanoshells are found in many biological systems, such as clathrin cages and viral shells. Many studies have shown that symmetric shells appear in nature as a result of the free-energy minimization of a generic interaction between their constituent subunits. We examine the physical basis for the formation of symmetric shells, and by using a minimal model, demonstrate that these structures can readily grow from the irreversible addition of identical subunits. Our model of nanoshell assembly shows that the spontaneous curvature regulates the size of the shell while the mechanical properties of the subunit determine the symmetry of the assembled structure. Understanding the minimum requirements for the formation of closed nanoshells is a necessary step toward engineering of nanocontainers, which will have far-reaching impact in both material science and medicine.

Methods of Monitoring Cell Fate and Tissue Growth in Three-Dimensional Scaffold-Based Strategies for In Vitro Tissue Engineering

In the field of tissue engineering, there is a need for methods that allow assessing the performance of tissue-engineered constructs noninvasively in vitro and in vivo. To date, histological analysis is the golden standard to retrieve information on tissue growth, cellular distribution, and cell fate on tissue-engineered constructs after in vitro cell culture or on explanted specimens after in vivo applications. Yet, many advances have been made to optimize imaging techniques for monitoring tissue-engineered constructs with a sub-mm or μm resolution. Many imaging modalities have first been developed for clinical applications, in which a high penetration depth has been often more important than lateral resolution. In this study, we have reviewed the current state of the art in several imaging approaches that have shown to be promising in monitoring cell fate and tissue growth upon in vitro culture. Depending on the aimed tissue type and scaffold properties, some imaging methods are more applicable than others. Optical methods are mostly suited for transparent materials such as hydrogels, whereas magnetic resonance-based methods are mostly applied to obtain contrast between hard and soft tissues regardless of their transparency. Overall, this review shows that the field of imaging in scaffold-based tissue engineering is developing at a fast pace and has the potential to overcome the limitations of destructive endpoint analysis.

Clusters of polyhedra in spherical confinement

Dense particle packing in a confining volume remains a rich, largely unexplored problem, despite applications in blood clotting, plasmonics, industrial packaging and transport, colloidal molecule design, and information storage. Here, we report densest found clusters of the Platonic solids in spherical confinement, for up to [Formula: see text] constituent polyhedral particles. We examine the interplay between anisotropic particle shape and isotropic 3D confinement. Densest clusters exhibit a wide variety of symmetry point groups and form in up to three layers at higher N. For many N values, icosahedra and dodecahedra form clusters that resemble sphere clusters. These common structures are layers of optimal spherical codes in most cases, a surprising fact given the significant faceting of the icosahedron and dodecahedron. We also investigate cluster density as a function of N for each particle shape. We find that, in contrast to what happens in bulk, polyhedra often pack less densely than spheres. We also find especially dense clusters at so-called magic numbers of constituent particles. Our results showcase the structural diversity and experimental utility of families of solutions to the packing in confinement problem.

Computer simulation of self-assembly of cone-shaped nanoparticles

Two kinds of cone-shaped particles are designed: one with a Janus structure and the other with a sandwich structure. The effects of the cone angle and particle structure (i.e. AB type and BAB type) on the kinetic pathway and assembled structures are discussed.

Self-Assembly of an Alphavirus Core-like Particle Is Distinguished by Strong Intersubunit Association Energy and Structural Defects

Weak association energy can lead to uniform nanostructures: defects can anneal due to subunit lability. What happens when strong association energy leads to particles where defects are trapped? Alphaviruses are enveloped viruses whose icosahedral nucleocapsid core can assemble independently. We used a simplest case system to study Ross River virus (RRV) core-like particle (CLP) self-assembly using purified capsid protein and a short DNA oligomer. We find that capsid protein binds the oligomer with high affinity to form an assembly competent unit (U). Subsequently, U assembles with concentration dependence into CLPs. We determined that U-U pairwise interactions are very strong (ca. -6 kcal/mol) compared to other virus assembly systems. Assembled RRV CLPs appeared morphologically uniform and cryo-EM image reconstruction with imposed icosahedral symmetry yielded a T = 4 structure. However, 2D class averages of the CLPs show that virtually every class had disordered regions. These results suggested that irregular cores may be present in RRV virions. To test this hypothesis, we determined 2D class averages of RRV virions using authentic virions or only the core from intact virions isolated by computational masking. Virion-based class averages were symmetrical, geometric, and corresponded well to projections of image reconstructions. In core-based class averages, cores and envelope proteins in many classes were disordered. These results suggest that partly disordered components are common even in ostensibly well-ordered viruses, a biological realization of a patchy particle. Biological advantages of partly disordered complexes may arise from their ease of dissociation and asymmetry.

Generic, phenomenological, on-the-fly renormalized repulsion model for self-limited organization of terminal supraparticle assemblies

Self-limited, or terminal, supraparticles have long received great interest because of their abundance in biological systems (DNA bundles and virus capsids) and their potential use in a host of applications ranging from photonics and catalysis to encapsulation for drug delivery. Moreover, soft, uniform colloidal aggregates are a promising candidate for quasicrystal and other hierarchical assemblies. In this work, we present a generic coarse-grained model that captures the formation of self-limited assemblies observed in various soft-matter systems including nanoparticles, colloids, and polyelectrolytes. Using molecular dynamics simulations, we demonstrate that the assembly process is self-limited when the repulsion between the particles is renormalized to balance their attraction during aggregation. The uniform finite-sized aggregates are further shown to be thermodynamically stable and tunable with a single dimensionless parameter. We find large aggregates self-organize internally into a core-shell morphology and exhibit anomalous uniformity when the constituent nanoparticles have a polydisperse size distribution.

The role of bond tangency and bond gap in hard sphere crystallization of chains

We report results from Monte Carlo simulations on dense packings of linear, freely-jointed chains of hard spheres of uniform size. In contrast to our past studies where bonded spheres along the chain backbone were tangent, in the present work a finite tolerance in the bond is allowed. Bond lengths are allowed to fluctuate in the interval [σ, σ + dl], where σ is the sphere diameter. We find that bond tolerance affects the phase behaviour of hard-sphere chains, especially in the close vicinity of the melting transition. First, a critical dl(crit) exists marking the threshold for crystallization, whose value decreases with increasing volume fraction. Second, bond gaps enhance the onset of phase transition by accelerating crystal nucleation and growth. Finally, bond tolerance has an effect on crystal morphologies: in the tangent limit the majority of structures correspond to stack-faulted random hexagonal close packing (rhcp). However, as bond tolerance increases a wealth of diverse structures can be observed: from single fcc (or hcp) crystallites to random hcp/fcc stackings with multiple directions. By extending the simulations over trillions of MC steps (10(12)) we are able to observe crystal-crystal transitions and perfection even for entangled polymer chains in accordance to the Ostwald's rule of stages in crystal polymorphism. Through simple geometric arguments we explain how the presence of rigid or flexible constraints affects crystallization in general atomic and particulate systems. Based on the present results, it can be concluded that proper tuning of bond gaps and of the connectivity network can be a controlling factor for the phase behaviour of model, polymer-based colloidal and granular systems.

Self-assembly of "Mickey Mouse" shaped colloids into tube-like structures: experiments and simulations

The self-assembly of anisotropic patchy particles with a triangular shape was studied by experiments and computer simulations. The colloidal particles were synthesized in a two-step seeded emulsion polymerization process, and consist of a central smooth lobe connected to two rough lobes at an angle of ∼90°, resembling the shape of a "Mickey Mouse" head. Due to the difference in overlap volume, adding an appropriate depletant induces an attractive interaction between the smooth lobes of the colloids only, while the two rough lobes act as steric constraints. The essentially planar geometry of the Mickey Mouse particles is a first geometric deviation of dumbbell shaped patchy particles. This new geometry enables the formation of one-dimensional tube-like structures rather than spherical, essentially zero-dimensional micelles. At sufficiently strong attractions, we indeed find tube-like structures with the sticky lobes at the core and the non-sticky lobes pointing out as steric constraints that limit the growth to one direction, providing the tubes with a well-defined diameter but variable length both in experiments and simulations. In the simulations, we found that the internal structure of the tubular fragments could either be straight or twisted into so-called Bernal spirals.

Modeling Viral Capsid Assembly

I present a review of the theoretical and computational methodologies that have been used to model the assembly of viral capsids. I discuss the capabilities and limitations of approaches ranging from equilibrium continuum theories to molecular dynamics simulations, and I give an overview of some of the important conclusions about virus assembly that have resulted from these modeling efforts. Topics include the assembly of empty viral shells, assembly around single-stranded nucleic acids to form viral particles, and assembly around synthetic polymers or charged nanoparticles for nanotechnology or biomedical applications. I present some examples in which modeling efforts have promoted experimental breakthroughs, as well as directions in which the connection between modeling and experiment can be strengthened.

Effect of shape on the self-assembly of faceted patchy nanoplates with irregular shape into tiling patterns

Recent reports of the synthesis and assembly of faceted nanoplates with a wide range of shapes and composition motivates the possibility of a new class of two-dimensional materials with specific patterns targeted for a host of exciting properties. Yet, studies of how nanoplate shape controls their assembly - knowledge necessary for their inverse design from target structures - has been performed for only a handful of systems. By constructing a general framework in which many known faceted nanoplates may be described in terms of four anisotropy dimensions, we discover design rules to guide future synthesis and assembly. We study via Monte Carlo simulations attractive polygons whose shape is altered systematically under the following four transformations: faceting, pinching, elongation and truncation. We report that (i) faceting leads to regular porous structures (ii) pinching stabilizes complex structures such as dodecagonal quasicrystals (iii) elongation leads to asymmetric phase behavior, where low and high aspect ratio nanoplates self-assemble completely different structures and (iv) low and high degrees of truncation transform a complex self-assembler into a disk-like assembler, providing design ideas that could lead to switchable structures. We provide important insight into how the shape and attractive interactions of a nanoplate can be exploited or designed to target specific classes of structures, including space-filling, porous, and complex tilings.

Biomolecular engineering of virus-like particles aided by computational chemistry methods

Multi-scale investigation of VLP self-assembly aided by computational methods is facilitating the design, redesign, and modification of functionalized VLPs.

Molecular dynamics simulation: a tool for exploration and discovery using simple models

Emergent phenomena share the fascinating property of not being obvious consequences of the design of the system in which they appear. This characteristic is no less relevant when attempting to simulate such phenomena, given that the outcome is not always a foregone conclusion. The present survey focuses on several simple model systems that exhibit surprisingly rich emergent behavior, all studied by molecular dynamics (MD) simulation.The examples are taken from the disparate fields of fluid dynamics, granular matter and supramolecular self-assembly. In studies of fluids modeled at the detailed microscopic level using discrete particles, the simulations demonstrate that complex hydrodynamic phenomena in rotating and convecting fluids—the Taylor–Couette and Rayleigh–Bénard instabilities—cannot only be observed within the limited length and time scales accessible to MD, but even allow quantitative agreement to be achieved. Simulation of highly counter-intuitive segregation phenomena in granular mixtures, again using MD methods, but now augmented by forces producing damping and friction, leads to results that resemble experimentally observed axial and radial segregation in the case of a rotating cylinder and to a novel form of horizontal segregation in a vertically vibrated layer. Finally, when modeling self-assembly processes analogous to the formation of the polyhedral shells that package spherical viruses, simulation of suitably shaped particles reveals the ability to produce complete, error-free assembly and leads to the important general observation that reversible growth steps contribute to the high yield. While there are limitations to the MD approach, both computational and conceptual, the results offer a tantalizing hint of the kinds of phenomena that can be explored and what might be discovered when sufficient resources are brought to bear on a problem.

A minimal representation of the self-assembly of virus capsids

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

Synthesis of Janus particles via kinetic control of phase separation in emulsion droplets

A reaction-induced phase separation strategy in emulsion droplets has been developed to fabricate non-spherical Janus particles. The final morphology of the resulting Janus particles is controlled by a reaction sequence. These Janus particles are amphiphilic and show regio-selective adsorption orientation at oil/water interfaces.

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

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

Self-assembled clusters of spheres related to spherical codes

We consider the thermodynamically driven self-assembly of spheres onto the surface of a central sphere. This assembly process forms self-limiting, or terminal, anisotropic clusters (N-clusters) with well-defined structures. We use Brownian dynamics to model the assembly of N-clusters varying in size from two to twelve outer spheres and free energy calculations to predict the expected cluster sizes and shapes as a function of temperature and inner particle diameter. We show that the arrangements of outer spheres at finite temperatures are related to spherical codes, an ideal mathematical sequence of points corresponding to the densest possible sphere packings. We demonstrate that temperature and the ratio of the diameters of the inner and outer spheres dictate cluster morphology. We present a surprising result for the equilibrium structure of a 5-cluster, for which the square pyramid arrangement is preferred over a more symmetric structure. We show this result using Brownian dynamics, a Monte Carlo simulation, and a free energy approximation. Our results suggest a promising way to assemble anisotropic building blocks from constituent colloidal spheres.

Physics of shell assembly: line tension, hole implosion, and closure catastrophe

The self-assembly of perfectly ordered closed shells is a challenging process involved in many biological and nanoscale systems. However, most of the aspects that determine their formation are still unknown. Here we investigate the growth of shells by simulating the assembly of spherical structures made of N identical subunits. Remarkably, we show that the formation and energetics of partially assembled shells are dominated by an effective line-tension that can be described in simple thermodynamic terms. In addition, we unveil two mechanisms that can prevent the correct formation of defect-free structures: "hole implosion," which leads to a premature closure of the shell; and "closure catastrophe," which causes a dramatic production of structural disorder during the later stages of the growth of big shells.

Surface roughness directed self-assembly of patchy particles into colloidal micelles

Colloidal particles with site-specific directional interactions, so called "patchy particles", are promising candidates for bottom-up assembly routes towards complex structures with rationally designed properties. Here we present an experimental realization of patchy colloidal particles based on material independent depletion interaction and surface roughness. Curved, smooth patches on rough colloids are shown to be exclusively attractive due to their different overlap volumes. We discuss in detail the case of colloids with one patch that serves as a model for molecular surfactants both with respect to their geometry and their interactions. These one-patch particles assemble into clusters that resemble surfactant micelles with the smooth and attractive sides of the colloids located at the interior. We term these clusters "colloidal micelles". Direct Monte Carlo simulations starting from a homogeneous state give rise to cluster size distributions that are in good agreement with those found in experiments. Important differences with surfactant micelles originate from the colloidal character of our model system and are investigated by simulations and addressed theoretically. Our new "patchy" model system opens up the possibility for self-assembly studies into finite-sized superstructures as well as crystals with as of yet inaccessible structures.

Facilitated self-assembly of novel dendron-based copolymers

Self-assembly of newly synthesized dendron-based amphiphilic copolymers with controlled hydrophilic-lipophilic balances has been investigated to evaluate their potential as a novel nanocarrier. The hydroxyl-terminated polyester dendron (G3) bearing a focal alkyne moiety was used to mediate the combination of poly(s-caprolactone) (PCL) with multiple polyethylene glycol (PEG) moieties. Four types of PCL-G3-mPEG with different block lengths were prepared and their structures were confirmed by (1)H NMR, FT-IR, and GPC. Critical micelle concentration (CMC) values varied from 6.50 × 10(-8) to 3.52 × 10(-7) M, which were lower than those reported for linear PCL-mPEG. TEM revealed that all PCL-G3-mPEG micelles were spherical with an average diameter of 20 nm. The drug release profile for each PCL-G3-mPEG was investigated by loading indomethacin (IMC), as a model drug, within the micelles. IMC was released in a controlled manner over 72 hours. Synthesized copolymers used in this study were also found to be non-cytotoxic at concentrations up to 100 μM. The low CMC, along with the controlled morphology, release profile and biocompatibility, all demonstrate the potential of the dendron-based micelles as a novel nanocarrier.