Ground States of Crystalline Caps: Generalized Jellium on Curved Space

Limits of economy and fidelity for programmable assembly of size-controlled triply periodic polyhedra

We propose and investigate an extension of the Caspar-Klug symmetry principles for viral capsid assembly to the programmable assembly of size-controlled triply periodic polyhedra, discrete variants of the Primitive, Diamond, and Gyroid cubic minimal surfaces. Inspired by a recent class of programmable DNA origami colloids, we demonstrate that the economy of design in these crystalline assemblies-in terms of the growth of the number of distinct particle species required with the increased size-scale (e.g., periodicity)-is comparable to viral shells. We further test the role of geometric specificity in these assemblies via dynamical assembly simulations, which show that conditions for simultaneously efficient and high-fidelity assembly require an intermediate degree of flexibility of local angles and lengths in programmed assembly. Off-target misassembly occurs via incorporation of a variant of disclination defects, generalized to the case of hyperbolic crystals. The possibility of these topological defects is a direct consequence of the very same symmetry principles that underlie the economical design, exposing a basic tradeoff between design economy and fidelity of programmable, size controlled assembly.

Flower-shaped 2D crystals grown in curved fluid vesicle membranes

The morphologies of two-dimensional (2D) crystals, nucleated, grown, and integrated within 2D elastic fluids, for instance in giant vesicle membranes, are dictated by an interplay of mechanics, permeability, and thermal contraction. Mitigation of solid strain drives the formation of crystals with vanishing Gaussian curvature (i.e., developable domain shapes) and, correspondingly, enhanced Gaussian curvature in the surrounding 2D fluid. However, upon cooling to grow the crystals, large vesicles sustain greater inflation and tension because their small area-to-volume ratio slows water permeation. As a result, more elaborate shapes, for instance, flowers with bendable but inextensible petals, form on large vesicles despite their more gradual curvature, while small vesicles harbor compact planar crystals. This size dependence runs counter to the known cumulative growth of strain energy of 2D colloidal crystals on rigid spherical templates. This interplay of intra-membrane mechanics and processing points to the scalable production of flexible molecular crystals of controllable complex shape.

Thermal stability and secondary aggregation of self-limiting, geometrically frustrated assemblies: Chain assembly of incommensurate polybricks

In geometrically frustrated assemblies, equilibrium self-limitation manifests in the form of a minimum in the free energy per subunit at a finite, multisubunit size which results from the competition between the elastic costs of frustration within an assembly and the surface energy at its boundaries. Physical realizations-from ill-fitting particle assemblies to self-twisting protein superstructures-are capable of multiple mechanisms of escaping the cumulative costs of frustration, resulting in unlimited equilibrium assembly, including elastic modes of "shape flattening" and the formation of weak, defective bonds that screen intra-assembly stresses. Here we study a model of one-dimensional chain assembly of incommensurate "polybricks" and determine its equilibrium assembly as a function of temperature, concentration, degree of shape frustration, elasticity, and interparticle binding, notably focusing on how weakly cohesive, defective bonds give rise to strongly temperature-dependent assembly. Complex assembly behavior derives from the competition between multiple distinct local minima in the free-energy landscape, including self-limiting chains, weakly bound aggregates of self-limiting chains, and strongly bound, elastically defrustrated assemblies. We show that this scenario, in general, gives rise to anomalous multiple aggregation behavior, in which disperse subunits (stable at low concentration and high temperature) first exhibit a primary aggregation transition to self-limiting chains (at intermediate concentration and temperature) which are ultimately unstable to condensation into unlimited assembly of finite-chains through weak binding beyond a secondary aggregation transition (at low temperature and high concentration). We show that window of stable self-limitation is determined both by the elastic costs of frustration in the assembly as well as energetic and entropic features of intersubunit binding.

Effect of ionic strength on the assembly of simian vacuolating virus capsid protein around poly(styrene sulfonate)

Virus-like particles (VLPs) are noninfectious nanocapsules that can be used for drug delivery or vaccine applications. VLPs can be assembled from virus capsid proteins around a condensing agent, such as RNA, DNA, or a charged polymer. Electrostatic interactions play an important role in the assembly reaction. VLPs assemble from many copies of capsid protein, with a combinatorial number of intermediates. Hence, the mechanism of the reaction is poorly understood. In this paper, we combined solution small-angle X-ray scattering (SAXS), cryo-transmission electron microscopy (TEM), and computational modeling to determine the effect of ionic strength on the assembly of Simian Vacuolating Virus 40 (SV40)-like particles. We mixed poly(styrene sulfonate) with SV40 capsid protein pentamers at different ionic strengths. We then characterized the assembly product by SAXS and cryo-TEM. To analyze the data, we performed Langevin dynamics simulations using a coarse-grained model that revealed incomplete, asymmetric VLP structures consistent with the experimental data. We found that close to physiological ionic strength, [Formula: see text] VLPs coexisted with VP1 pentamers. At lower or higher ionic strengths, incomplete particles coexisted with pentamers and [Formula: see text] particles. Including the simulated structures was essential to explain the SAXS data in a manner that is consistent with the cryo-TEM images.

Building blocks of non-Euclidean ribbons: size-controlled self-assembly via discrete frustrated particles

Geometric frustration offers a pathway to soft matter self-assembly with controllable finite sizes. While the understanding of frustration in soft matter assembly derives almost exclusively from continuum elastic descriptions, a current challenge is to understand the connection between microscopic physical properties of misfitting "building blocks" and emergent assembly behavior at the mesoscale. We present and analyze a particle-based description of what is arguably the best studied example for frustrated soft matter assembly, negative-curvature ribbon assembly, observed in both assemblies of chiral surfactants and shape-frustrated nanoparticles. Based on our particle model, known as saddle wedge monomers, we numerically test the connection between microscopic shape and interactions of the misfitting subunits and the emergent behavior at the supra-particle scale, specifically focussing on the propagation and relaxation of inter-particle strains, the emergent role of extrinsic shape on frustrated ribbons and the equilibrium regime of finite width selection. Beyond the intuitive role of shape misfit, we show that self-limitation is critically dependent on the finite range of cohesive interactions, with larger size finite assemblies requiring increasing short-range interparticle forces. Additionally, we demonstrate that non-linearities arising from discrete particle interactions alter self-limiting behavior due to both strain-softening in shape-flattened assembly and partial yielding of highly strained bonds, which in turn may give rise to states of hierarchical, multidomain assembly. Tracing the regimes of frustration-limited assembly to the specific microscopic features of misfitting particle shapes and interactions provides necessary guidance for translating the theory of size-programmable assembly into design of intentionally-frustrated colloidal particles.

Multiscale Modeling of Hepatitis B Virus Capsid Assembly and Its Dimorphism

Hepatitis B virus (HBV) is an endemic, chronic virus that leads to 800000 deaths per year. Central to the HBV lifecycle, the viral core has a protein capsid assembled from many copies of a single protein. The capsid protein adopts different (quasi-equivalent) conformations to form icosahedral capsids containing 180 or 240 proteins: T = 3 or T = 4, respectively, in Caspar-Klug nomenclature. HBV capsid assembly has become an important target for recently developed antivirals; nonetheless, the assembly pathways and mechanisms that control HBV dimorphism remain unclear. We describe computer simulations of the HBV assembly, using a coarse-grained model that has parameters learned from all-atom molecular dynamics simulations of a complete HBV capsid and yet is computationally tractable. Dynamical simulations with the resulting model reproduce experimental observations of HBV assembly pathways and products. By constructing Markov state models and employing transition path theory, we identify pathways leading to T = 3, T = 4, and other experimentally observed capsid morphologies. The analysis shows that capsid polymorphism is promoted by the low HBV capsid bending modulus, where the key factors controlling polymorphism are the conformational energy landscape and protein-protein binding affinities.

Exact Solution for Elastic Networks on Curved Surfaces

The problem of characterizing the structure of an elastic network constrained to lie on a frozen curved surface appears in many areas of science and has been addressed by many different approaches, most notably, extending linear elasticity or through effective defect interaction models. In this Letter, we show that the problem can be solved by considering nonlinear elasticity in an exact form without resorting to any approximation in terms of geometric quantities. In this way, we are able to consider different effects that have been unwieldy or not viable to include in the past, such as a finite line tension, explicit dependence on the Poisson ratio, or the determination of the particle positions for the entire lattice. Several geometries with rotational symmetry are solved explicitly. Comparison with linear elasticity reveals an agreement that extends beyond its strict range of applicability. Implications for the problem of the characterization of virus assembly are also discussed.

Geometry and physics in the deformations of crystalline caps

Elucidating the interplay of stress and geometry is a fundamental scientific question arising in multiple fields. In this work, we investigate the geometric frustration of crystalline caps confined on the sphere in both elastic and plastic regimes. Based on the revealed quasi-conformal ordering, we discover the partial but uniform screening of the substrate curvature by the induced curvature underlying the inhomogeneous lattice. This scenario is fundamentally different from the conventional screening mechanism based on topological defects. In the plastic regime, the yield of highly stressed caps leads to fractures with featured morphologies not found in planar systems. We also demonstrate the strategy of engineering stress and fractures by vacancies. These results advance our general understanding of the organization and adaptivity of the geometrically frustrated crystalline order.

Microcompartment assembly around multicomponent fluid cargoes

This article describes dynamical simulations of the assembly of an icosahedral protein shell around a bicomponent fluid cargo. Our simulations are motivated by bacterial microcompartments, which are protein shells found in bacteria that assemble around a complex of enzymes and other components involved in certain metabolic processes. The simulations demonstrate that the relative interaction strengths among the different cargo species play a key role in determining the amount of each species that is encapsulated, their spatial organization, and the nature of the shell assembly pathways. However, the shell protein–shell protein and shell protein–cargo component interactions that help drive assembly and encapsulation also influence cargo composition within certain parameter regimes. These behaviors are governed by a combination of thermodynamic and kinetic effects. In addition to elucidating how natural microcompartments encapsulate multiple components involved within reaction cascades, these results have implications for efforts in synthetic biology to colocalize alternative sets of molecules within microcompartments to accelerate specific reactions. More broadly, the results suggest that coupling between self-assembly and multicomponent liquid–liquid phase separation may play a role in the organization of the cellular cytoplasm.

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.

Two-dimensional crystalization on spheres: Crystals grow cracked

Here we study how curvature affects the structure of two-dimensional crystals growing on spheres. The mechanism of crystal growth is described by means of a Landau model in curved space that accounts for the excess of strain on crystal bonds caused by the substrate's curvature (packing frustration). In curved space elastic energy penalization strongly dictates the geometry of growing crystals. While compact faceted crystals are observed when elastic energy contribution can be neglected, cracked crystals with ribbonlike forms appear as the main mechanisms to reduce elastic frustration for highly curved systems.

Defects in conformal crystals: Discrete versus continuous disclination models

We study the relationship between topological defect formation and ground-state 2D packings in a model of repulsions in external confining potentials. Specifically we consider screened 2D Coulombic repulsions, which conveniently parameterizes the effects of interaction range, but also serves as simple physical model of confined, parallel arrays of polyelectrolyte filaments or vortices in type II superconductors. The countervailing tendencies of repulsions and confinement to, respectively, spread and concentrate particle density leads to an energetic preference for nonuniform densities in the clusters. Ground states in such systems have previously been modeled as conformal crystals, which are composed of locally equitriangular packings whose local areal densities exhibit long-range gradients. Here we assess two theoretical models that connect the preference for nonuniform density to the formation of disclination defects, one of which assumes a continuum distributions of defects, while the second considers the quantized and localized nature of disclinations in hexagonal conformal crystals. Comparing both theoretical descriptions to numerical simulations of discrete particles clusters, we study the influence of interaction range and confining potential on the topological charge, number, and distribution of defects in ground states. We show that treating disclinations as continuously distributable well captures the number of topological defects in the ground state in the regime of long-range interactions, while as interactions become shorter range, it dramatically overpredicts the growth in total defect charge. Detailed analysis of the discretized defect theory suggests that that failure of the continuous defect theory in this limit can be attributed to the asymmetry in the preferred placement of positive vs negative disclinations in the conformal crystal ground states, as well as a strongly asymmetric dependence of self-energy of disclinations on sign of topological charge.

Predicting the characteristics of defect transitions on curved surfaces

The energetically optimal position of lattice defects on intrinsically curved surfaces is a complex function of shape parameters. For open surfaces, a simple condition predicts the critical size for which a central disclination yields lower energy than a boundary disclination. In practice, this transition is modified by activation energies or more favorable intermediate defect positions. Here it is shown that these transition characteristics (continuous or discontinuous, first or second order) can also be inferred from analytical, general criteria evaluated from the surface shape. A universal scale of activation energy is found, and the criteria are generalized to predict transition order as surface shape symmetry is broken. The results give practical insight into structural transitions to disorder in many cellular materials of technological and biological importance.

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.

Mechanisms of Scaffold-Mediated Microcompartment Assembly and Size Control

This article describes a theoretical and computational study of the dynamical assembly of a protein shell around a complex consisting of many cargo molecules and long, flexible scaffold molecules. Our study is motivated by bacterial microcompartments, which are proteinaceous organelles that assemble around a condensed droplet of enzymes and reactants. As in many examples of cytoplasmic liquid-liquid phase separation, condensation of the microcompartment interior cargo is driven by flexible scaffold proteins that have weak multivalent interactions with the cargo. Our results predict that the shell size, amount of encapsulated cargo, and assembly pathways depend sensitively on properties of the scaffold, including its length and valency of scaffold-cargo interactions. Moreover, the ability of self-assembling protein shells to change their size to accommodate scaffold molecules of different lengths depends crucially on whether the spontaneous curvature radius of the protein shell is smaller or larger than a characteristic elastic length scale of the shell. Beyond natural microcompartments, these results have important implications for synthetic biology efforts to target alternative molecules for encapsulation by microcompartments or viral shells. More broadly, the results elucidate how cells exploit coupling between self-assembly and liquid-liquid phase separation to organize their interiors.

Phase transitions on non-uniformly curved surfaces: coupling between phase and location

For particles confined to two dimensions, any curvature of the surface affects the structural, kinetic and thermodynamic properties of the system. If the curvature is non-uniform, an even richer range of behaviours can emerge. Using a combination of bespoke Monte Carlo, molecular dynamics and basin-hopping methods, we show that the stable states of attractive colloids confined to non-uniformly curved surfaces are distinguished not only by the phase of matter but also by their location on the surface. Consequently, the transitions between these states involve cooperative migration of the entire colloidal assembly. We demonstrate these phenomena on toroidal and sinusoidal surfaces for model colloids with different ranges of interactions as described by the Morse potential. In all cases, the behaviour can be rationalised in terms of three universal considerations: cluster perimeter, stress, and the packing of next-nearest neighbours.

Simple, General Criterion for Onset of Disclination Disorder on Curved Surfaces

Determining the positions of lattice defects on bounded elastic surfaces with Gaussian curvature is a nontrivial task of mechanical energy optimization. We introduce a simple way to predict the onset of disclination disorder from the shape of the surface. The criterion fixes the value of a weighted integral Gaussian curvature to a universal constant and proves accurate across a great variety of shapes. It provides improved understanding of the limitations to crystalline order in many natural and engineering contexts, such as the assembly of viral capsids.

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.

Measurements of the self-assembly kinetics of individual viral capsids around their RNA genome

Self-assembly is a process in which functional nanoscale structures build themselves, driven by Brownian motion and interactions between components. The term was originally coined to describe the formation of a viral capsid, the protein shell that protects the genome of a virus. Despite decades of study, how capsids self-assemble has remained a mystery, because there were no methods to measure the assembly kinetics of individual capsids. We surmount this obstacle using a sensitive microscopy technique based on laser interferometry. The measurements show that a small nucleus of proteins must form on the viral RNA before the capsid assembles. These results might help researchers design strategies to stop the assembly of pathogenic viruses or to build synthetic nanostructures.

Self-assembly is widely used by biological systems to build functional nanostructures, such as the protein capsids of RNA viruses. But because assembly is a collective phenomenon involving many weakly interacting subunits and a broad range of timescales, measurements of the assembly pathways have been elusive. We use interferometric scattering microscopy to measure the assembly kinetics of individual MS2 bacteriophage capsids around MS2 RNA. By recording how many coat proteins bind to each of many individual RNA strands, we find that assembly proceeds by nucleation followed by monotonic growth. Our measurements reveal the assembly pathways in quantitative detail and also show their failure modes. We use these results to critically examine models of the assembly process.