The equilibrium structure of self-assembled protein nano-cages

Study of impacts of two types of cellular aging on the yeast bud morphogenesis

Understanding the mechanisms of the cellular aging processes is crucial for attempting to extend organismal lifespan and for studying age-related degenerative diseases. Yeast cells divide through budding, providing a classical biological model for studying cellular aging. With their powerful genetics, relatively short cell cycle, and well-established signaling pathways also found in animals, yeast cells offer valuable insights into the aging process. Recent experiments suggested the existence of two aging modes in yeast characterized by nucleolar and mitochondrial declines, respectively. By analyzing experimental data, this study shows that cells evolving into those two aging modes behave differently when they are young. While buds grow linearly in both modes, cells that consistently generate spherical buds throughout their lifespan demonstrate greater efficacy in controlling bud size and growth rate at young ages. A three-dimensional multiscale chemical-mechanical model was developed and used to suggest and test hypothesized impacts of aging on bud morphogenesis. Experimentally calibrated model simulations showed that during the early stage of budding, tubular bud shape in one aging mode could be generated by locally inserting new materials at the bud tip, a process guided by the polarized Cdc42 signal. Furthermore, the aspect ratio of the tubular bud could be stabilized during the late stage as observed in experiments in this work. The model simulation results suggest that the localization of new cell surface material insertion, regulated by chemical signal polarization, could be weakened due to cellular aging in yeast and other cell types, leading to the change and stabilization of the bud aspect ratio.

Study of Impacts of Two Types of Cellular Aging on the Yeast Bud Morphogenesis

Understanding the mechanisms of cellular aging processes is crucial for attempting to extend organismal lifespan and for studying age-related degenerative diseases. Yeast cells divide through budding, providing a classical biological model for studying cellular aging. With their powerful genetics, relatively short lifespan and well-established signaling pathways also found in animals, yeast cells offer valuable insights into the aging process. Recent experiments suggested the existence of two aging modes in yeast characterized by nucleolar and mitochondrial declines, respectively. In this study, by analyzing experimental data it was shown that cells evolving into those two aging modes behave differently when they are young. While buds grow linearly in both modes, cells that consistently generate spherical buds throughout their lifespan demonstrate greater efficacy in controlling bud size and growth rate at young ages. A three-dimensional chemical-mechanical model was developed and used to suggest and test hypothesized mechanisms of bud morphogenesis during aging. Experimentally calibrated simulations showed that tubular bud shape in one aging mode could be generated by locally inserting new materials at the bud tip guided by the polarized Cdc42 signal during the early stage of budding. Furthermore, the aspect ratio of the tubular bud could be stabilized during the late stage, as observed in experiments, through a reduction on the new cell surface material insertion or an expansion of the polarization site. Thus model simulations suggest the maintenance of new cell surface material insertion or chemical signal polarization could be weakened due to cellular aging in yeast and other cell types.

Effect of coat-protein concentration on the self-assembly of bacteriophage MS2 capsids around RNA

Self-assembly is a vital part of the life cycle of certain icosahedral RNA viruses. Furthermore, the assembly process can be harnessed to make icosahedral virus-like particles (VLPs) from coat protein and RNA in vitro. Although much previous work has explored the effects of RNA-protein interactions on the assembly products, relatively little research has explored the effects of coat-protein concentration. We mix coat protein and RNA from bacteriophage MS2, and we use a combination of gel electrophoresis, dynamic light scattering, and transmission electron microscopy to investigate the assembly products. We show that with increasing coat-protein concentration, the products transition from well-formed MS2 VLPs to "monster" particles consisting of multiple partial capsids to RNA-protein condensates consisting of large networks of RNA and partially assembled capsids. We argue that the transition from well-formed to monster particles arises because the assembly follows a nucleation-and-growth pathway in which the nucleation rate depends sensitively on the coat-protein concentration, such that at high protein concentrations, multiple nuclei can form on each RNA strand. To understand the formation of the condensates, which occurs at even higher coat-protein concentrations, we use Monte Carlo simulations with coarse-grained models of capsomers and RNA. These simulations suggest that the formation of condensates occurs by the adsorption of protein to the RNA followed by the assembly of capsids. Multiple RNA molecules can become trapped when a capsid grows from capsomers attached to two different RNA molecules or when excess protein bridges together growing capsids on different RNA molecules. Our results provide insight into an important biophysical process and could inform design rules for making VLPs for various applications.

Theoretical Studies on Assembly, Physical Stability, and Dynamics of Viruses

All matter must obey the general laws of physics and living matter is not an exception. Viruses have not only learnt how to cope with them but have managed to use them for their own survival. In this chapter, we will review some of the exciting physics that are behind viruses and discuss simple physical models that can shed some light on different aspects of the viral life cycle and viral properties. In particular, we will focus on how the structure and shape of the viral capsid, its assembly and stability, and the entry and exit of viral particles and their genomes can be explained using fundamental physics theories.

An interaction network approach predicts protein cage architectures in bionanotechnology

Protein nanoparticles play pivotal roles in many areas of bionanotechnology, including drug delivery, vaccination, and diagnostics. These technologies require control over the distinct particle morphologies that protein nanocontainers can adopt during self-assembly from their constituent protein components. The geometric construction principle of virus-derived protein cages is by now fairly well understood by analogy to viral protein shells in terms of Caspar and Klug's quasi-equivalence principle. However, many artificial, or genetically modified, protein containers exhibit varying degrees of quasi-equivalence in the interactions between identical protein subunits. They can also contain a subset of protein subunits that do not participate in interactions with other assembly units, called capsomers, leading to gaps in the particle surface. We introduce a method that exploits information on the local interactions between the capsomers to infer the geometric construction principle of these nanoparticle architectures. The predictive power of this approach is demonstrated here for a prominent system in nanotechnology, the AaLS pentamer. Our method not only rationalises hitherto discovered cage structures but also predicts geometrically viable options that have not yet been observed. The classification of nanoparticle architecture based on the geometric properties of the interaction network closes a gap in our current understanding of protein container structure and can be widely applied in protein nanotechnology, paving the way to programmable control over particle polymorphism.

Close packings of identical proteins in small spherical capsids and similar proteinaceous shells

Understanding the principles governing protein arrangement in viral capsids and structurally similar protein shells can enable the development of new antiviral strategies and the design of artificial protein cages for various applications. We study these principles within the context of the close packing problem, by analyzing dozens of small spherical shells assembled from a single type of protein. First, we use icosahedral spherical close packings containing 60T identical disks, where T ≤ 4, to rationalize the protein arrangement in twenty real icosahedral shells both satisfying and violating the paradigmatic Caspar-Klug model. We uncover a striking correspondence between the protein mass centers in the considered shells and the centers of disks in the close packings. To generalize the packing model, we consider proteins with a weak shape anisotropy and propose an interaction energy, minimization of which allows us to obtain spherical dense packings of slightly anisotropic structural units. In the case of strong anisotropy, we model the proteins as sequences of overlapping discs of different sizes, with minimum energy configuration not only resulting in packings, accurately reproducing locations and orientations of individual proteins, but also revealing that icosahedral packings that display the handedness of real capsids are energetically more favorable. Finally, by introducing effective disc charges, we rationalize the formation of inter-protein bonds in protein shells.

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.

Relaxational dynamics of the T-number conversion of virus capsids

We extend a recently proposed kinetic theory of virus capsid assembly based on Model A kinetics and study the dynamics of the interconversion of virus capsids of different sizes triggered by a quench, that is, by sudden changes in the solution conditions. The work is inspired by in vitro experiments on functionalized coat proteins of the plant virus cowpea chlorotic mottle virus, which undergo a reversible transition between two different shell sizes (T = 1 and T = 3) upon changing the acidity and salinity of the solution. We find that the relaxation dynamics are governed by two time scales that, in almost all cases, can be identified as two distinct processes. Initially, the monomers and one of the two types of capsids respond to the quench. Subsequently, the monomer concentration remains essentially constant, and the conversion between the two capsid species completes. In the intermediate stages, a long-lived metastable steady state may present itself, where the thermodynamically less stable species predominate. We conclude that a Model A based relaxational model can reasonably describe the early and intermediate stages of the conversion experiments. However, it fails to provide a good representation of the time evolution of the state of assembly of the coat proteins in the very late stages of equilibration when one of the two species disappears from the solution. It appears that explicitly incorporating the nucleation barriers to assembly and disassembly is crucial for an accurate description of the experimental findings, at least under conditions where these barriers are sufficiently large.

Voltage controlled shutter regulates channel size and motion direction of protein aperture as durable nano-electric rectifier-----An opinion in biomimetic nanoaperture

In optical devices such as camera or microscope, an aperture is used to regulate light intensity for imaging. Here we report the discovery and construction of a durable bio-aperture at nanometerscale that can regulate current at the pico-ampere scale. The nano-aperture is made of 12 identical protein subunits that form a 3.6-nm channel with a shutter and "one-way traffic" property. This shutter responds to electrical potential differences across the aperture and can be turned off for double stranded DNA translocation. This voltage enables directional control, and three-step regulation for opening and closing. The nano-aperture was constructed in vitro and purified into homogeneity. The aperture was stable at pH2-12, and a temperature of -85C-60C. When an electrical potential was held, three reproducible discrete steps of current flowing through the channel were recorded. Each step reduced 32% of the channel dimension evident by the reduction of the measured current flowing through the aperture. The current change is due to the change of the resistance of aperture size. The transition between these three distinct steps and the direction of the current was controlled via the polarity of the voltage applied across the aperture. When the C-terminal of the aperture was fused to an antigen, the antibody and antigen interaction resulted in a 32% reduction of the channel size. This phenomenon was used for disease diagnosis since the incubation of the antigen-nano-aperture with a specific cancer antibody resulted in a change of 32% of current. The purified truncated cone-shape aperture automatically self-assembled efficiently into a sheet of the tetragonal array via head-to-tail self-interaction. The nano-aperture discovery with a controllable shutter, discrete-step current regulation, formation of tetragonal sheet, and one-way current traffic provides a nanoscale electrical circuit rectifier for nanodevices and disease diagnosis.

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.

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.

Polymorphic self-assembly of helical tubules is kinetically controlled

In contrast to most self-assembling synthetic materials, which undergo unbounded growth, many biological self-assembly processes are self-limited. That is, the assembled structures have one or more finite dimensions that are much larger than the size scale of the individual monomers. In many such cases, the finite dimension is selected by a preferred curvature of the monomers, which leads to self-closure of the assembly. In this article, we study an example class of self-closing assemblies: cylindrical tubules that assemble from triangular monomers. By combining kinetic Monte Carlo simulations, free energy calculations, and simple theoretical models, we show that a range of programmable size scales can be targeted by controlling the intricate balance between the preferred curvature of the monomers and their interaction strengths. However, their assembly is kinetically controlled-the tubule morphology is essentially fixed shortly after closure, resulting in a distribution of tubule widths that is significantly broader than the equilibrium distribution. We develop a simple kinetic model based on this observation and the underlying free-energy landscape of assembling tubules that quantitatively describes the distributions. Our results are consistent with recent experimental observations of tubule assembly from triangular DNA origami monomers. The modeling framework elucidates design principles for assembling self-limited structures from synthetic components, such as artificial microtubules that have a desired width and chirality.

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.

The Dynamics of Viruslike Capsid Assembly and Disassembly

Cowpea chlorotic mottle virus (CCMV) is a widely used model for virus replication studies. A major challenge lies in distinguishing between the roles of the interaction between coat proteins and that between the coat proteins and the viral RNA in assembly and disassembly processes. Here, we report on the spontaneous and reversible size conversion of the empty capsids of a CCMV capsid protein functionalized with a hydrophobic elastin-like polypeptide which occurs following a pH jump. We monitor the concentrations of T = 3 and T = 1 capsids as a function of time and show that the time evolution of the conversion from one T number to another is not symmetric: The conversion from T = 1 to T = 3 is a factor of 10 slower than that of T = 3 to T = 1. We explain our experimental findings using a simple model based on classical nucleation theory applied to virus capsids, in which we account for the change in the free protein concentration, as the different types of shells assemble and disassemble by shedding or absorbing single protein subunits. As far as we are aware, this is the first study confirming that both the assembly and disassembly of viruslike shells can be explained through classical nucleation theory, reproducing quantitatively results from time-resolved experiments

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.

Thermodynamic Size Control in Curvature-Frustrated Tubules: Self-Limitation with Open Boundaries

We use computational modeling to investigate the assembly thermodynamics of a particle-based model for geometrically frustrated assembly, in which the local packing geometry of subunits is incompatible with uniform, strain-free large-scale assembly. The model considers discrete triangular subunits that drive assembly toward a closed, hexagonal-ordered tubule, but have geometries that locally favor negative Gaussian curvature. We use dynamical Monte Carlo simulations and enhanced sampling methods to compute the free energy landscape and corresponding self-assembly behavior as a function of experimentally accessible parameters that control assembly driving forces and the magnitude of frustration. The results determine the parameter range where finite-temperature self-limiting assembly occurs, in which the equilibrium assembly size distribution is sharply peaked around a well-defined finite size. The simulations also identify two mechanisms by which the system can escape frustration and assemble to unlimited size, and determine the particle-scale properties of subunits that suppress unbounded growth.

Local rules for the self-assembly of a non-quasi-equivalent viral capsid

The structures of many large bacteriophages, such as the P23-77 capsids, do not adhere strictly to the quasi-equivalence principle of viral architecture. Although the general architecture of the P23-77 capsids is classed as T=28d, it self-assembles from multiple copies of two types of coat protein subunits, and the resulting hexameric capsomers do not conform to the Caspar-Klug paradigm. There are two types of hexamers with distinct internal organization, that are located at specific positions in the capsid. It is an open problem which assembly mechanism can lead to such a complex capsid organization. Here we propose a simple set of local rules that can explain how such non-quasi-equivalent capsid structures can arise as a result of self-assembly.

Protein Cages: From Fundamentals to Advanced Applications

Proteins that self-assemble into polyhedral shell-like structures are useful molecular containers both in nature and in the laboratory. Here we review efforts to repurpose diverse protein cages, including viral capsids, ferritins, bacterial microcompartments, and designed capsules, as vaccines, drug delivery vehicles, targeted imaging agents, nanoreactors, templates for controlled materials synthesis, building blocks for higher-order architectures, and more. A deep understanding of the principles underlying the construction, function, and evolution of natural systems has been key to tailoring selective cargo encapsulation and interactions with both biological systems and synthetic materials through protein engineering and directed evolution. The ability to adapt and design increasingly sophisticated capsid structures and functions stands to benefit the fields of catalysis, materials science, and medicine.

Programmable polymorphism of a virus-like particle

Virus-like particles (VLPs) have significant potential as artificial vaccines and drug delivery systems. The ability to control their size has wide ranging utility but achieving such controlled polymorphism using a single protein subunit is challenging as it requires altering VLP geometry. Here we achieve size control of MS2 bacteriophage VLPs via insertion of amino acid sequences in an external loop to shift morphology to significantly larger forms. The resulting VLP size and geometry is controlled by altering the length and type of the insert. Cryo electron microscopy structures of the new VLPs, in combination with a kinetic model of their assembly, show that the abundance of wild type (T = 3), T = 4, D3 and D5 symmetrical VLPs can be biased in this way. We propose a mechanism whereby the insert leads to a change in the dynamic behavior of the capsid protein dimer, affecting the interconversion between the symmetric and asymmetric conformers and thus determining VLP size and morphology.

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.

Effect of Mechanical Properties on Multicomponent Shell Patterning

Self-organized shells are fundamental in biological compartmentalization. They protect genomic material or enclose enzymes to aid the metabolic process. Studies of crystalline shells have shown the importance of the mechanical properties of building units in the shell morphology. However, the mechanism underlying the morphology of multicomponent assemblies is still poorly understood. Here, we analyze multicomponent closed shells that have different mechanical properties. By minimizing elastic energy, we show that heterogeneous bending rigidities regulate the surface pattern into circular, spikes, and ridge shapes. Interestingly, our continuum elasticity model recovers the patterns that have been proposed in bacterial microcompartments (BMCs), which are self-organized protein shells that aid the breakdown of complex molecules and allow bacteria to survive in hostile environments. In addition, our work elucidates the principles of pattern formation that can be used to design and engineer multicomponent microcompartments with a specific surface distribution of the components.

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.

Virus Assembly Pathways: Straying Away but Not Too Far

Non-enveloped RNA viruses pervade all domains of life. In a cell, they co-assemble from viral RNA and capsid proteins. Virus-like particles can form in vitro where virtually any non-cognate polyanionic cargo can be packaged. How only viral RNA gets selected for packaging in vivo, in presence of myriad other polyanionic species, has been a puzzle. Through a combination of charge detection mass spectrometry and cryo-electron microscopy, it is determined that co-assembling brome mosaic virus (BMV) coat proteins and nucleic acid oligomers results in capsid structures and stoichiometries that differ from the icosahedral virion. These previously unknown shell structures are strained and less stable than the native one. However, they contain large native structure fragments that can be recycled to form BMV virions, should a viral genome become available. The existence of such structures suggest the possibility of a previously unknown regulatory pathway for the packaging process inside cells.

Effect of protein steric constraints on the symmetry of membrane protein polyhedra

Experiments have shown that, in an aqueous environment, lipids and membrane proteins can self-assemble into membrane protein polyhedral nanoparticles (MPPNs). MPPNs are closed, spherical vesicles composed of a lipid bilayer membrane and membrane proteins, with a polyhedral arrangement of membrane proteins. The observed symmetry and size of MPPNs can be understood from the interplay of protein-induced lipid bilayer deformations in MPPNs, topological defects in protein packing necessitated by the spherical shape of MPPNs, and thermal fluctuations in MPPN self-assembly. We explore here the effect of protein steric constraints on MPPN shape. The protein steric constraints considered here may arise from a well-defined shape of protein domains outside the membrane, entropic repulsion between membrane proteins with flexible domains outside the membrane, or binding of other molecules to membrane proteins. Calculating MPPN self-assembly diagrams under protein steric constraints we find that protein steric constraints can strongly affect MPPN self-assembly. Depending on the specific scenario considered, protein steric constraints can leave large portions of the MPPN self-assembly diagrams with no clearly defined MPPN symmetry or substantially expand the regions of MPPN self-assembly diagrams dominated by highly symmetric MPPN states, such as MPPNs with icosahedral or snub cube symmetry. Our results suggest that modification of protein steric constraints may allow the directed self-assembly of MPPNs with specified symmetry, size, and protein composition and may thus facilitate the further utilization of MPPNs for membrane protein structural analysis or targeted drug delivery.

Should Virus Capsids Assemble Perfectly? Theory and Observation of Defects

Although published structural models of viral capsids generally exhibit a high degree of regularity or symmetry, structural defects might be expected because of the fluctuating environment in which capsids assemble and the requirement of some capsids for disassembly before genome delivery. Defective structures are observed in computer simulations, and are evident in single-particle cryoelectron microscopy studies. Here, we quantify the conditions under which defects might be expected, using a statistical mechanics model allowing for ideal, defective, and vacant sites. The model displays a threshold in affinity parameters below which there is an appreciable population of defective capsids. Even when defective sites are not allowed, there is generally some population of vacancies. Analysis of single particles in cryoelectron microscopy micrographs yields a confirmatory ≳15% of defective particles. Our findings suggest structural heterogeneity in virus capsids may be under-appreciated, and also points to a nontraditional strategy for assembly inhibition.

Role of combined cell membrane and wall mechanical properties regulated by polarity signals in cell budding

Budding yeast, Saccharomyces cerevisiae, serves as a prime biological model to study mechanisms underlying asymmetric growth. Previous studies have shown that prior to bud emergence, polarization of a conserved small GTPase Cdc42 must be established on the cell membrane of a budding yeast. Additionally, such polarization contributes to the delivery of cell wall remodeling enzymes and hydrolase from cytosol through the membrane, to change the mechanical properties of the cell wall. This leads to the hypothesis that Cdc42 and its associated proteins at least indirectly regulate cell surface mechanical properties. However, how the surface mechanical properties in the emerging bud are changed and whether such change is important are not well understood. To test several hypothesised mechanisms, a novel three-dimensional coarse-grained particle-based model has been developed which describes inhomogeneous mechanical properties of the cell surface. Model simulations predict alternation of the levels of stretching and bending stiffness of the cell surface in the bud region by the polarized Cdc42 signals is essential for initiating bud formation. Model simulations also suggest that bud shape depends strongly on the distribution of the polarized signaling molecules while the neck width of the emerging bud is strongly impacted by the mechanical properties of the chitin and septin rings. Moreover, the temporal change of the bud mechanical properties is shown to affect the symmetry of the bud shape. The 3D model of asymmetric cell growth can also be used for studying viral budding and other vegetative reproduction processes performed via budding, as well as detailed studies of cell growth.

Shape selection and mis-assembly in viral capsid formation by elastic frustration

The successful assembly of a closed protein shell (or capsid) is a key step in the replication of viruses and in the production of artificial viral cages for bio/nanotechnological applications. During self-assembly, the favorable binding energy competes with the energetic cost of the growing edge and the elastic stresses generated due to the curvature of the capsid. As a result, incomplete structures such as open caps, cylindrical or ribbon-shaped shells may emerge, preventing the successful replication of viruses. Using elasticity theory and coarse-grained simulations, we analyze the conditions required for these processes to occur and their significance for empty virus self-assembly. We find that the outcome of the assembly can be recast into a universal phase diagram showing that viruses with high mechanical resistance cannot be self-assembled directly as spherical structures. The results of our study justify the need of a maturation step and suggest promising routes to hinder viral infections by inducing mis-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.

Symmetry of membrane protein polyhedra with heterogeneous protein size

In experiments on membrane protein polyhedral nanoparticles (MPPNs) [Basta et al., Proc. Natl. Acad. Sci. USA 111, 670 (2014)PNASA60027-842410.1073/pnas.1321936111], it has been observed that membrane proteins and lipids can self-assemble into closed lipid bilayer vesicles with a polyhedral arrangement of membrane proteins. In particular, MPPNs formed from the mechanosensitive channel of small conductance (MscS) were found to have the symmetry of the snub cube-a chiral, Archimedean solid-with one MscS protein located at each one of the 24 vertices of the snub cube. It is currently unknown whether MPPNs with heterogeneous protein composition maintain a high degree of symmetry. Inspired by previous work on viral capsid symmetry, we employ here computational modeling to study the symmetry of MPPNs with heterogeneous protein size. We focus on MPPNs formed from MscS proteins, which can exist in closed or open conformational states with distinct sizes. We find that, as an increasing number of closed-state MscS proteins transitions to the open conformational state of MscS, the minimum-energy MscS arrangement in MPPNs follows a strikingly regular pattern, with the dominant MPPN symmetry always being provided by the snub cube. Our results suggest that MPPNs with heterogeneous protein size can be highly symmetric, with a well-defined polyhedral ordering of membrane proteins of different sizes.

Osmotic stress and pore nucleation in charged biological nanoshells and capsids

A model system is proposed to investigate the chemical equilibrium and mechanical stability of biological spherical-like nanoshells in contact with an aqueous solution with added dissociated electrolyte of a given concentration. The ionic chemical equilibrium across the permeable shell is investigated in the framework of an accurate Density Functional Theory (DFT) that incorporates electrostatic and hardcore correlations beyond the traditional mean-field (e.g., Poisson-Boltzmann) limit. The accuracy of the theory is tested by a direct comparison with Monte Carlo (MC) simulations. A simple analytical expression is then deduced which clearly highlights the entropic, electrostatic, and self-energy contributions to the osmotic stress over the shell in terms of the calculated ionic profiles. By invoking a continuum mean-field elastic approach to account for the shell surface stress upon osmotic stretching, the mechanical equilibrium properties of the shell under a wide variety of ionic strengths and surface charges are investigated. The model is further coupled to a continuum mechanical approach similar in structure to a Classical Nucleation Theory (CNT) to address the question of mechanical stability of the shells against a pore nucleation. This allows us to construct a phase diagram which delimits the mechanical stability of capsids for different ionic strengths and shell surface charges.

Ground States of Crystalline Caps: Generalized Jellium on Curved Space

We study the ground states of crystals on spherical surfaces. These ground states consist of positive disclination defects in structures spanning from flat and weakly curved caps to closed shells. Comparing two continuum theories and one discrete-lattice simulation, we first investigate the transition between defect-free caps to single-disclination ground states and show it to be continuous and symmetry breaking. Further, we show that ground states adopt icosahedral subgroup symmetries across the full range of curvatures, even far from the closure of complete shells. While superficially similar to other models of 2D "jellium" (e.g., superconducting disks and 2D Wigner crystals), the interplay between the free edge of caps and the non-Euclidean geometry of its embedding leads to nontrivial ground state behavior that is without counterpart in planar jellium models.

Mechanical stress relaxation in molecular self-assembly

Molecular self-assembly on a curved substrate leads to the spontaneous inclusion of topological defects in the growing bidimensional crystal, unlike assembly on a flat substrate. We propose in this work a quantitative mechanism for this phenomenon by using standard thin shell elasticity. The Gaussian curvature of the substrate induces large in-plane compressive stress as the surface grows, in particular at the rim of the assembly, and the addition of a single defect relaxes this mechanical stress. We found out that the value of azimuthal stress at the rim of the assembly determines the preferred directions for defect nucleation. These results are also discussed as a function of different defect combinations, like dislocations and grain boundaries or scars. In particular, the elastic model permits us to compare quantitatively the ability of various defects to relax mechanical stress. Moreover, these findings allow us to understand the progressive building-up of the typical disclination and grain boundary pattern observed for ground states of large 2D spherical crystals.

Elasticity in curved topographies: Exact theories and linear approximations

Almost all available results in elasticity on curved topographies are obtained within either a small curvature expansion or an empirical covariant generalization that accounts for screening between Gaussian curvature and disclinations. In this paper, we present a formulation of elasticity theory in curved geometries that unifies its underlying geometric and topological content with the theory of defects. The two different linear approximations widely used in the literature are shown to arise as systematic expansions in reference and actual space. Taking the concrete example of a two-dimensional crystal, with and without a central disclination, constrained on a spherical cap, we compare the exact results with different approximations and evaluate their range of validity. We conclude with some general discussion about the universality of nonlinear elasticity.