Icosahedral Symmetry Research Articles

Explicit description of viral capsid subunit shapes by unfolding dihedrons.

Viral capsid assembly and the design of capsid-based nanocontainers critically depend on understanding the shapes and interfaces of constituent protein subunits. However, a comprehensive framework for characterizing these features is still lacking. Here, we introduce a novel approach based on spherical tiling theory that explicitly describes the 2D shapes and interfaces of subunits in icosahedral capsids. Our method unfolds spherical dihedrons defined by icosahedral symmetry axes, enabling systematic characterization of all possible subunit geometries. Applying this framework to real T = 1 capsid structures reveals distinct interface groups within this single classification, with variations in interaction patterns around 3-fold and 5-fold symmetry axes. We validate our classification through molecular docking simulations, demonstrating its consistency with physical subunit interactions. This analysis suggests different assembly pathways for capsid nucleation. Our general framework is applicable to other triangular numbers, paving the way for broader studies in structural virology and nanomaterial design.

Comprehensive experimental datasets of quasicrystals and their approximants

AbstractQuasicrystals are solid-state materials that typically exhibit unique symmetries, such as icosahedral or decagonal diffraction symmetry. They were first discovered in 1984. Over the past four decades of quasicrystal research, around 100 stable quasicrystals have been discovered. In recent years, machine learning has been employed to explore quasicrystals with unique properties inherent to quasiperiodic systems. However, the lack of open data on quasicrystal composition, structure, and physical properties has hindered the widespread use of machine learning in quasicrystal research. This study involves a comprehensive literature review and manual data extraction to develop open datasets consisting of composition, structure types, phase diagrams, and sample fabrication processes for a wide range of stable and metastable quasicrystals and approximant crystals, as well as the temperature-dependent thermal, electrical, and magnetic properties.

Sums of distinct fifth roots of unity and the regular dodecahedron

Integer linear combinations of cube, fourth, or sixth roots of unity form lattices in the complex plane . In contrast, integer linear combinations of fifth roots of unity do not form a lattice; in fact, they are dense in . Nevertheless, the geometry for fifth roots of unity has considerable structure. Here we consider only sums of distinct fifth roots of unity, and show that 20 of these sums are orthogonal projections of the vertices of a regular dodecahedron. Pentagonal symmetry here is only to be expected, but it is a little surprising to encounter a plane projection of a polyhedron with much richer dodecahedral symmetry.

Multitwinned Ice Nanocrystals.

Multitwinned nanocrystals are commonly found in substances that preferentially adopt tetrahedral local arrangements, but not yet in water crystals. Ice nanocrystals are pivotal in cloud microphysics, and their surfaces become increasingly prominent in determining structure as crystal size decreases. Nevertheless, discussions on nanocrystal structures have predominantly centered on ice polymorphs observed in bulk: hexagonal (Ih), cubic (Ic), and stacking-disordered (Isd) ices. Here, we demonstrate, through molecular dynamics (MD) simulations, that decahedral and icosahedral nanocrystals form from liquid water droplets of a few nanometers in size without violating the ice rule. The brute force spontaneous crystallization is conducted using the mW model, and the thermodynamic stability is examined using the TIP4P/Ice model. During the crystallization process, the formation of twin boundaries precedes the emergence of centers exhibiting 5-fold and icosahedral symmetry. The free energy calculation suggests the icosahedron has comparable stability with ice Ih nanocrystal. The frequent occurrence of these unreported ice nanocrystals aligns with the fact that natural polycrystalline snow crystals predominantly display a 70.5-degree angle between the Ih c-axes of adjacent branches. Moreover, we show that the formation of multitwinned ice nanocrystals is enhanced within a fullerene, providing a potential avenue for experimental observations.

Atomic Structure and 3D Shape of a Multibranched Plasmonic Nanostar from a Single Spatially Resolved Electron Diffraction Map.

Despite the interest in improving the sensitivity of optical sensors using plasmonic nanoparticles (NPs) (rods, wires, and stars), the full structural characterization of complex shape nanostructures is challenging. Here, we derive from a single scanning transmission electron microscope diffraction map (4D-STEM) a detailed determination of both the 3D shape and atomic arrangement of an individual 6-branched AuAg nanostar (NS) with high-aspect-ratio legs. The NS core displays an icosahedral structure, and legs are decahedral rods attached along the 5-fold axes at the core apexes. The NS legs show an anomalous anisotropic spatial distribution (all close to a plane) due to an interplay between the icosahedral symmetry and the unzipping of the surfactant layer on the core. The results significantly improve our understanding of the star growth mechanism. This low dose diffraction mapping is promising for the atomic structure study of individual multidomain, multibranched, or multiphase NPs, even when constituted of beam-sensitive materials.

Structural Investigations of Cargo Molecules Inside Icosahedrally Symmetric Encapsulin by VUVCD Spectroscopic Measurements.

Prokaryotes organize intracellular compartments with protein-based organelles called encapsulins. Encapsulins with icosahedral symmetry can encapsulate specific cargo proteins mediated by targeting peptides or encapsulation-mediating domains. Encapsulins have been used in eukaryotic cells for bioengineering, vaccine development, and nanoparticle alignment. Their versatility makes them attractive for research; however, detailed structural information on encapsulins is crucial for further applied research. However, cargo proteins are randomly oriented inside the icosahedral encapsulins. The random orientation of cargo proteins presents a challenge for structural analysis that relies on averaging processes such as x-ray crystallography and cryo-electron microscopy (cryo-EM) single-particle imaging. Therefore, we aimed to accurately estimate the secondary structure content and elucidate the structure of cargo proteins inside the particle by measuring the circular dichroism (CD) spectra using vacuum ultraviolet circular dichroism (VUVCD) spectroscopy. Thus, the structure of the cargo protein inside encapsulin was evaluated. This approach could potentially set a standard for evaluating cargo proteins inside particles in future applied research on encapsulins.

Exploring the Effects of Intersubunit Interface Mutations on Virus-Like Particle Structure and Stability.

Virus-like particles (VLPs) from bacteriophage MS2 provide a platform to study protein self-assembly and create engineered systems for drug delivery. Here, we aim to understand the impact of intersubunit interface mutations on the local and global structure and function of MS2-based VLPs. In previous work, our lab identified locally supercharged double mutants [T71K/G73R] that concentrate positive charge at capsid pores, enhancing uptake into mammalian cells. To study the effects of particle size on cellular internalization, we combined these double mutants with a single point mutation [S37P] that was previously reported to switch particle geometry from T = 3 to T = 1 icosahedral symmetry. These new variants retained their enhanced cellular uptake activity and could deliver small-molecule drugs with efficacy levels similar to our first-generation capsids. Surprisingly, these engineered triple mutants exhibit increased thermostability and unexpected geometry, producing T = 3 particles instead of the anticipated T = 1 assemblies. Transmission electron microscopy revealed various capsid assembly states, including wild-type (T = 3), T = 1, and rod-like particles, that could be accessed using different combinations of these point mutations. Molecular dynamics experiments recapitulated the structural rationale in silico for the single point mutation [S37P] forming a T = 1 virus-like particle and showed that this assembly state was not favored when combined with mutations that favor rod-like architectures. Through this work, we investigated how interdimer interface dynamics influence VLP size and morphology and how these properties affect particle function in applications such as drug delivery.

Comparative Pore Structure and Dynamics for Bacterial Microcompartment Shell Protein Assemblies in Sheets or Shells.

Bacterial microcompartments (BMCs) are protein-bound organelles found in some bacteria that encapsulate enzymes for enhanced catalytic activity. These compartments spatially sequester enzymes within semipermeable shell proteins, analogous to many membrane-bound organelles. The shell proteins assemble into multimeric tiles; hexamers, trimers, and pentamers, and these tiles self-assemble into larger assemblies with icosahedral symmetry. While icosahedral shells are the predominant form in vivo, the tiles can also form nanoscale cylinders or sheets. The individual multimeric tiles feature central pores that are key to regulating transport across the protein shell. Our primary interest is to quantify pore shape changes in response to alternative component morphologies at the nanoscale. We used molecular modeling tools to develop atomically detailed models for both planar sheets of tiles and curved structures representative of the complete shells found in vivo. Subsequently, these models were animated using classical molecular dynamics simulations. From the resulting trajectories, we analyzed the overall structural stability, water accessibility to individual residues, water residence time, and pore geometry for the hexameric and trimeric protein tiles from the Haliangium ochraceum model BMC shell. These exhaustive analyses suggest no substantial variation in pore structure or solvent accessibility between the flat and curved shell geometries. We additionally compare our analysis to hydroxyl radical footprinting data to serve as a check against our simulation results, highlighting specific residues where water molecules are bound for a long time. Although with little variation in morphology or water interaction, we propose that the planar and capsular morphology can be used interchangeably when studying permeability through BMC pores.

Isolation and characterization of bacteriophages against E.coli urinary tract infection and evaluating their anti-biofilm activity and antibiotic synergy

Urinary tract infections (UTIs) by Uropathogenic Escherichia coli (UPEC) are a significant health concern, especially due to the increasing prevalence of antibiotic resistance. This study focuses on isolating and characterizing bacteriophages specific to UPEC strains isolated from UTI samples. The isolated phages were assessed for their ability to target and lyse UPEC in vitro, focusing on their efficacy in disrupting biofilms, a key virulence factor contributing to UTI recurrence and antibiotic resistance. The morphological structure observed by TEM belongs to Myoviridae, the phage exhibited icosahedral symmetry with a long non-constricting tail, the approximate measurement of the phage head was 39 nm in diameter, and the phage tail was 105.317 nm in length. One-step growth experiments showed that the latent period was approximately 20 min, followed by a rise period of 40 min, and a growth plateau was reached within 20 min and the burst size observed was 26 phages/infected bacterial cells. These phages were capable of killing cells within the biofilms, leading to a reduction in living cell counts after a single treatment. This study highlights the potential of phages to play a significant role in disrupting, inactivating, and destroying Uropathogenic Escherichia coli (UPEC) biofilms. Such findings could be instrumental in developing treatment strategies that complement antibiotics and disinfectants. The phage-antibiotic synergistic activity was compared to have the possibility to facilitate the advancement of focused and enduring alternatives to traditional antibiotic therapies for UTIs.

Label-Free Tracking of Hepatitis B Virus Core Protein Capsid Assembly in Real-Time Using Protein Charge Transfer Spectra.

Hepatitis B virions are double-shelled particles, with a diameter of 40-42 nm, consisting of a nucleocapsid called the HBV core protein (HBV Cp). It is an ordered assembly of 90-120 homodimers arranged in an icosahedral symmetry. Both the full-length HBV Cp and the first-149 residue domain, HBV Cp149, can spontaneously assemble in vitro into capsids with 120 Cp dimers (T = 4) or 90 Cp dimers (T = 3), triggered by high ionic strength of 0.25-0.5 M NaCl. The assembly disassembly of HBV Cp149 capsids are generally studied by light scattering, size-exclusion chromatography, atomic force microscopy, transmission electron microscopy, and other high-end expensive techniques. Here, we report a simple, yet robust, label-free technique exploiting protein charge transfer spectra (ProCharTS) to monitor the capsid assembly in real-time. ProCharTS absorption in the near UV-visible region (250-800 nm) arises when photoinduced electron transfer occurs from HOMO of COO- in glutamate (donor) to LUMO of NH3+ in lysine or polypeptide backbone (acceptor) of the protein. Alternatively, it can also occur from polypeptide backbone (donor) to acceptor in arginine, histidine, or lysine cation. ProCharTS is observed profusely among proximal charge clusters in folded proteins. Here, we show that, ProCharTS absorption among growing HBV capsids is amplified when HBV Cp homodimers assemble, generating new contacts among charged residues in the dimer-dimer interface. We notice a time-dependent sigmoidal increase in ProCharTS absorbance and luminescence during capsid formation in comparison to pure dimers. Additionally, a combined approach of anisotropy-based fluorescence assay is reported, where an increased fluorescence anisotropy was observed in capsids as compared to native and unfolded dimers. We conclude that ProCharTS can serve as a sensitive label-free tool for rapid tracking of capsid assembly in real-time and characterize the assembled capsids from dimers.

Point mutation in a virus-like capsid drives symmetry reduction to form tetrahedral cages

Protein capsids are a widespread form of compartmentalization in nature. Icosahedral symmetry is ubiquitous in capsids derived from spherical viruses, as this geometry maximizes the internal volume that can be enclosed within. Despite the strong preference for icosahedral symmetry, we show that simple point mutations in a virus-like capsid can drive the assembly of unique symmetry-reduced structures. Starting with the encapsulin from Myxococcus xanthus, a 180-mer bacterial capsid that adopts the well-studied viral HK97 fold, we use mass photometry and native charge detection mass spectrometry to identify a triple histidine point mutant that forms smaller dimorphic assemblies. Using cryoelectron microscopy, we determine the structures of a precedented 60-mer icosahedral assembly and an unexpected 36-mer tetrahedron that features significant geometric rearrangements around a new interaction surface between capsid protomers. We subsequently find that the tetrahedral assembly can be generated by triple-point mutation to various amino acids and that even a single histidine point mutation is sufficient to form tetrahedra. These findings represent a unique example of tetrahedral geometry when surveying all characterized encapsulins, HK97-like capsids, or indeed any virus-derived capsids reported in the Protein Data Bank, revealing the surprising plasticity of capsid self-assembly that can be accessed through minimal changes in the protein sequence.

Photoelectron Imaging Signature for Selective Formation of Icosahedral Anionic Silver Cages Encapsulating Group 5 Elements: M@Ag12- (M = V, Nb, and Ta).

An assembly of 13 atoms can form highly symmetric architectures like those belonging to D3h, Oh, D5h, and Ih point groups. Here, using photoelectron imaging spectroscopy in combination with density functional theory (DFT) calculations, we present a simple yet convincing experimental signature for the selective formation of icosahedral cages of anionic silver clusters encapsulating a dopant atom of group 5 elements: M@Ag12- (M = V, Nb, and Ta). Their photoelectron images obtained at 4 eV closely resemble one another: only a single ring is observed, which is assignable to photodetachment signals from a 5-fold degenerate superatomic 1D electronic shell in the 1S21P61D10 configuration of valence electrons. The perfect degeneracy represents an unambiguous fingerprint of an icosahedral symmetry, which would otherwise be lifted in all of the other structural isomers. DFT calculations confirm that Ih forms are the most stable and that D5h, Oh, and D3h structures are not found even in metastable states.

The Tsai vs. Bergman Cluster Stability in ZnMgSc 1/1 Periodic Approximant Crystal

AbstractQuasicrystals with icosahedral symmetry can be classified into two main groups: Bergman‐type and Tsai‐type. While these are considered as distinct phases, they share a common feature of being constructed from atomic clusters, either Tsai or Bergman. In this study, we employ Density Functional Theory to perform electronic structure calculations on the Zn84.79Mg0.86Sc14.35 periodic approximant crystal phase. Our investigation involves systematically displacing atoms from Tsai cluster sites to Bergman cluster sites, allowing us to explore modifications in the electronic structure. Our findings reveal that the stability of the phase is influenced by the interaction between Zn‐4p and Sc‐3d orbitals. We observe that the sp‐d hybridization effect may play a more crucial role rather than Hume‐Rothery stabilization, as this observation holds true regardless of the presence or absence of periodic boundary conditions. Notably, the additional atom present in the Tsai structure serves as a significant electron acceptor in low‐energy orbitals. Its absence in Bergman structures results in a composition with fewer atoms possessing high‐energy d orbitals. This discovery provides a rationale for the frequent occurrence of Bergman phases with transition metals or rare earth elements, which occupy less than 10 % of the sites in the structure.

Cryogenic electron microscopy and tomography reveal imperfect icosahedral symmetry in alphaviruses.

Alphaviruses are spherical, enveloped RNA viruses with two-layered icosahedral architecture. The structures of many alphaviruses have been studied using cryogenic electron microscopy (cryo-EM) reconstructions, which impose icosahedral symmetry on the viral particles. Using cryogenic electron tomography (cryo-ET), we revealed a polarized symmetry defect in the icosahedral lattice of Chikungunya virus (CHIKV) in situ, similar to the late budding particles, suggesting the inherent imperfect symmetry originates from the final pinch-off of assembled virions. We further demonstrated this imperfect symmetry is also present in in vitro purified CHIKV and Mayaro virus, another arthritogenic alphavirus. We employed a subparticle-based single-particle analysis protocol to circumvent the icosahedral imperfection and boosted the resolution of the structure of the CHIKV to ∼3 Å resolution, which revealed detailed molecular interactions between glycoprotein E1-E2 heterodimers in the transmembrane region and multiple lipid-like pocket factors located in a highly conserved hydrophobic pocket. This complementary use of in situ cryo-ET and single-particle cryo-EM approaches provides a more precise structural description of near-icosahedral viruses and valuable insights to guide the development of structure-based antiviral therapies against alphaviruses.

Synthesis and functionalities of FeSn12 superatom prepared by single atom introduction with a dendrimer template.

Superatoms are promising as new building block materials that can be designed by precise controlling of the constituent atoms. Stannaspherene (Sn12 2-) is a rigid cage-like cluster with icosahedral symmetry, for which one-atom encapsulation was theoretically expected and detected in the gas phase. Here, a single-atom introduction method into stannaspherene using a dendrimer template with polyvinylpyrrolidone (PVP) protection is demonstrated. This advanced solution-phase synthesis allows not only the selective doping of one atom into the cluster cage, but also enable further detail characterization of optical and magnetic properties that were not possible in the gas-phase synthesis. In other words, this liquid-phase synthesis method has enabled the adaptation of detailed analytical methods. In this study, FeSn12 was synthesized and characterized, revealing that a single Fe atom introduction in the Sn12 2- cage result in the appearance of near-infrared emission and enhancement in the magnetism.

Innovative stain-free technique for high-resolution imaging of virus particles via standard transmission electron microscopy

This research presents a groundbreaking approach in virus-related research, addressing challenges in electron microscopy (EM). This imaging technique has been crucial in exploring virus structures; however, traditional methods involve complex sample preparations and the risk of contamination. Herein, we introduce an approach that overcomes these obstacles, enabling high-resolution virus imaging without toxic staining procedures. Focusing on Begomovirus particles, an economically significant plant virus genus, our images confirm their non-enveloped structure and their twin icosahedral symmetry. Our methods involve sample collection, purification, and crystallization, followed by transmission electron microscopy - selected area electron diffraction (TEM-SAED) analysis. Notably, this study achieves 2D and 3D virus imaging through standard TEM, providing a new avenue for virus structure analysis and advancing virus-related research. Remarkable high image quality stemmed from the crystallization process, offering exciting possibilities for improving virus research and diagnosis while eliminating staining limitations.

A nuclear configuration interaction approach to study nuclear spin effects: an application to ortho- and para-3 He2 @C60.

We introduce a non-orthogonal configuration interaction approach to investigate nuclear quantum effects on energies and densities of confined fermionic nuclei. The Hamiltonian employed draws parallels between confined systems and many-electron atoms, where effective non-Coulombic potentials represent the interactions of the trapped particles. One advantage of this method is its generality, as it offers the potential to study the nuclear quantum effects of various confined species affected by effective isotropic or anisotropic potentials. As a first application, we analyze the quantum states of two 3 He atoms encapsulated in C60 . At the Hartree-Fock level, we observe the breaking of spin and spatial symmetries. To ensure wavefunctions with the correct symmetries, we mix the broken-symmetry Hartree-Fock states within the non-orthogonal configuration interaction expansion. Our proposed approach predicts singly and triply degenerate ground states for the singlet (para-3 He2 @C60 ) and triplet (ortho-3 He2 @C60 ) nuclear spin configurations, respectively. The ortho-3 He2 @C60 ground state is 5.69 cm-1 higher in energy than the para-3 He2 @C60 ground state. The nuclear densities obtained for these states exhibit the icosahedral symmetry of the C60 embedding potential. Importantly, our calculated energies for the lowest 85 states are in close agreement with perturbation theory results based on a harmonic oscillator plus rigid rotor model of 3 He2 @C60 .

Theory of density waves and organization of proteins in icosahedral virus capsids.

Understanding the physical principles underlying the structural organization of the proteinaceous viral shells is of major importance to advance antiviral strategies. Here, we develop a phenomenological thermodynamic theory, which considers structures of small and middle-size icosahedral viral shells as a result of condensation of a minimum number of protein density waves on a spherical surface. Each of these irreducible critical waves has icosahedral symmetry and can be expressed as a specific series of the spherical harmonics Ylm with the same wave number l. As we demonstrate, in small viral shells self-assembled from individual proteins, the maxima of one critical density wave determine the positions of proteins, while the spatial derivatives of the second one control the protein orientations on the shell surface. In contrast to the small shells, the middle-size ones are always formed from pentamers and hexamers (referred to as capsomers). Considering all such structures deposited in the Protein Data Bank, we unexpectedly found that the positions of capsomeres in these shells correspond to the maxima of interference patterns produced by no more than two critical waves with close wave numbers. This fact allows us to explain the observed limit size of the icosahedral shells assembled from pentamers and hexamers. We also construct nonequilibrium thermodynamic potentials describing the protein crystallization and discuss the reasons behind the specific handedness of the viral shells.

Breaking the size constraint for nano cages using annular patchy particles.

Engineering structures like nanocages, shells, and containers, by self-assembly of colloids is a challenging problem. One of the main challenges is to define the shape of the individual subunits to control the radius of the closed shell structures. In this work, we have proposed a simple model for the subunit, which comprises a spheroidal or spherical hardcore decorated with an annular patch. The self-assembly of these building blocks leads to the formation of monodispersed spherical cages (close shells) or containers (curved clusters). For a spheroid with a given bonding range, the curvature of the shell is analytically related to only the patch angle of the building blocks and independent of the shape of the subunits. This model with only one control parameter can be used to engineer cages with the desired radius, which also have been verified using thermodynamic calculations. In the phase diagram of the system, 4 phases are identified which includes gas, closed shell, partially closed (containers) shell and percolated structures. When the diameters of the spherical cages formed are small, we observe an icosahedral symmetry similar to virus capsids. We also observed that the kinetics of the cage formation is very similar to the nucleation and growth kinetics of viruses and is the key factor in determining the yield of closed shells.

Exploring dengue genome to design effective multi epitope-based peptide vaccine by immunoinformatics approach against all serotypes of dengue virus

Dengue fever is an acute tropical disease that is spread by Aedes species mosquitoes. It is a virus-mediated illness that causes headaches, fever, severe joint and muscle pain, swollen glands (lymphadenopathy), and rashes. Every year, 390 million people around the world suffer from it and ∼20 % die from cases that are left untreated. The virus is transmitted to humans by mosquito bites of infected Aedes species. When it gets into the body, its envelope protein attaches to the host cell surface receptor and initiates the viral infection cycle by the formation of a smooth coat with icosahedral symmetry. Due to its high mutation-prone nature and differences in protein sequences among serotypes, it is difficult to develop a stable vaccine suitable against all serotypes. In this study, a software-based vaccinology approach was used to design a vaccine construct against the envelope protein of all four subtypes of the dengue virus. This construct was designed by joining two epitopes of the protein. One epitope was taken from the envelope protein of DEN-2 which was conserved for DEN-1, DEN-2, and DEN-3, and another epitope from the same protein of DEN-4 serotype as the sequence composition of the protein is slightly different in these two groups of serotypes. Here, firstly 18 epitopes for dengue virus-1, 2, 3, and 12 epitopes for dengue virus-4 were retrieved based on the highest combined score of NetCTL server analysis of human leukocyte antigen (HLA) supertypes. Among them, the epitope MAILGDTAW has shown 100 % conservancy in strains of dengue virus 1, 2, 3, and KLRIKGMSY, TTAKEVALL, TPRSPSVEV, FRKGSSIGK, KTWLVHKQW have shown 100 % conservancy in dengue virus 4. In population coverage analysis, MAILGDTAW has shown 88.63 % and FRKGSSIGK has shown 90.72 % (highest among other DEN-4 epitopes) population coverage of the entire world population. These two epitopes along with their 15-mer extensions (MHCII binding peptides) were found as non-allergen and non-toxic. Finally, the two different 15-mer extensions (to get 100 % sequence conservancy) of the epitope MAILGDTAW and the 15-mer extension of the epitope FRKGSSIGK were put together using an EAAAK linker to construct the multi-epitope vaccine. In-vivo analysis of the proposed vaccine construct would aid in the development of a functional immune response that would eradicate dengue complications worldwide.