Nanoscale Dodecahedral and Fullerene-Type Organoboroxine and Borazine Cages from Planar Building Units

Boroxine- and borazine-cage analogs to C20, C60, and C70 were calculated and compared in terms of structure, strain indicators, and physical properties relevant to nanoscale applications. The results show C60 and C70 type cages are less strained than the smaller congener, primarily due to minimized bending in the B-arylene-B segments. The smallest cage calculated has a diameter of 2.4 nm, which increases up to 4.9 nm by either variation of the polyhedron (C20 < C60 < C70-type cage) or organic spacer elongation between boron centers. All calculated cages are porous (apertures ranging from 0.6 to 1.9 nm). Molecular electrostatic potential and Hirshfeld population analysis revealed both nucleophilic and electrophilic sites in the interior and exterior cage surfaces. HOMO-LUMO gaps range from 3.98 to 4.89 eV and 5.10-5.18 eV for the boroxine- and borazine-cages, respectively. Our findings provide insights into the design and properties of highly porous boroxine and borazine cages for nanoscience.

Potassium and Cesium Fluorinated β-Diketonates: Effect of a Cation and Terminal Substituent on Structural and Thermal Properties

As potential precursors for the synthesis of fluoroperovskites, a family of heavy alkali metal (MI = K, Cs) fluorinated β-diketonates were prepared and characterized by elemental analysis, IR, and powder-XRD. The crystal structures of the new six complexes, MI(β-dikF)(H2O)X, X = 0 or 1, were also determined. The structural diversity of this poorly explored class of complexes was discussed, including the preferred types of cation polyhedra and the ligand coordination modes, and the thermal properties of the metal β-diketonates were studied by TG-DTA in an inert (He) atmosphere. The data obtained allowed us to reveal the effect of the metal cation and the terminal substituent on the structural and thermal features of this family of complexes.

The Telluraboranes closo-TeB5Cl5 and closo-TeB11Cl11 with Exceptionally Long Body Diagonals: Synthetic and Bonding Motifs for Innovative Octahedral and Icosahedral Geometries

Six-vertex closo-TeB5Cl5 (1) and twelve-vertex closo-TeB11Cl11 (2) telluraboranes have been prepared via co-pyrolysis of B2Cl4 with TeCl4in vacuo at temperatures between 360ºC and 400°C. Both compounds are sublimable, off-white solids, and they have been characterized by one- and two-dimensional 11B-NMR and high-resolution mass spectroscopy. Both ab initio/GIAO/NMR and DFT/ZORA/NMR computations support octahedral and icosahedral geometries for 1 and 2, respectively, as expected due to their closo-electron counts. The octahedral structure of 1 has been confirmed by single-crystal X-ray diffraction on an incommensurately modulated crystal. The corresponding bonding properties have been analyzed in terms of the intrinsic bond orbital (IBO) approach. 1 is the first example of a polyhedral telluraborane with a cluster size smaller than 11 vertices.

Multifaceted Structurally Coloured Materials: Diffraction and Total Internal Reflection (TIR) from Nanoscale Surface Wrinkling

We investigate the combined effects of surface diffraction and total internal reflection (TIR) in the design of 3-dimensional materials exhibiting distinct structural colour on various facets. We employ mechanical wrinkling to introduce surface diffraction gratings (from the nano to the micron scales) on one face of an elastomeric rectangular parallelepiped-shaped slab and explore the roles, in the perceived colours, of wrinkling pattern, wavelength, the directionality of incident light and observation angles. We propose a simple model that satisfactorily accounts for all experimental observations. Employing polydimethylsiloxane (PDMS), which readily swells in the presence of various liquids and gases, we demonstrate that such multifaceted colours can respond to their environment. By coupling a right angle triangular prism with a surface grating, we demonstrate the straightforward fabrication of a so-called GRISM (GRating + prISM). Finally, using a range of examples, we outline possibilities for a predictive material design using multi-axial wrinkling patterns and more complex polyhedra.

Self-Assembly, Self-Folding, and Origami: Comparative Design Principles

Self-assembly is usually considered a parallel process while self-folding and origami are usually considered to be serial processes. We believe that these distinctions do not hold in actual experiments. Based upon our experience with 4D printing, we have developed three additional hybrid classes: (1) templated-assisted (tethered) self-assembly: e.g., when RNA is bound to viral capsomeres, the subunits are constricted in their interactions to have aspects of self-folding as well; (2) self-folding can depend upon interactions with the environment; for example, a protein synthesized on a ribosome will fold as soon as peptides enter the intracellular environment in a serial process whereas if denatured complete proteins are put into solution, parallel folding can occur simultaneously; and, (3) in turbulent environments, chaotic conditions continuously alternate processes. We have examined the 43,380 Dürer nets of dodecahedra and 43,380 Dürer nets of icosahedra and their corresponding duals: Schlegel diagrams. In order to better understand models of self-assembly of viral capsids, we have used both geometric (radius of gyration, convex hulls, angles) and topological (vertex connections, leaves, spanning trees, cutting trees, and degree distributions) perspectives to develop design principles for 4D printing experiments. Which configurations fold most rapidly? Which configurations lead to complete polyhedra most of the time? By using Hamiltonian circuits of the vertices of Dürer nets and Eulerian paths of cutting trees of polyhedra unto Schlegel diagrams, we have been able to develop a systematic sampling procedure to explore the 86,760 configurations, models of a T1 viral capsid with 60 subunits and to test alternatives with 4D printing experiments, use of Magforms^TM, and origami models to demonstrate via movies the five processes described above.

Algorithmic Design of 3D Wireframe RNA Polyhedra

We address the problem of de novo design and synthesis of nucleic acid nanostructures, a challenge that has been considered in the area of DNA nanotechnology since the 1980s and more recently in the area of RNA nanotechnology. Toward this goal, we introduce a general algorithmic design process and software pipeline for rendering 3D wireframe polyhedral nanostructures in single-stranded RNA. To initiate the pipeline, the user creates a model of the desired polyhedron using standard 3D graphic design software. As its output, the pipeline produces an RNA nucleotide sequence whose corresponding RNA primary structure can be transcribed from a DNA template and folded in the laboratory. As case examples, we design and characterize experimentally three 3D RNA nanostructures: a tetrahedron, a triangular bipyramid, and a triangular prism. The design software is openly available and also provides an export of the targeted 3D structure into the oxDNA molecular dynamics simulator for easy simulation and visualization.

Numerically stable form factor of any polygon and polyhedron

Coordinate-free expressions for the form factors of arbitrary polygons and polyhedra are derived using the divergence theorem and Stokes's theorem. Apparent singularities, all removable, are discussed in detail. Cancellation near the singularities causes a loss of precision that can be avoided by using series expansions. An important application domain is small-angle scattering by nanocrystals.

Dodecahedral structures with Mosseri-Sadoc tiles

The 3D facets of the Delone cells of the root lattice D₆ which tile the 6D Euclidean space in an alternating order are projected into 3D space. They are classified into six Mosseri-Sadoc tetrahedral tiles of edge lengths 1 and golden ratio τ = (1 + 5^1/2)/2 with faces normal to the fivefold and threefold axes. The icosahedron, dodecahedron and icosidodecahedron whose vertices are obtained from the fundamental weights of the icosahedral group are dissected in terms of six tetrahedra. A set of four tiles are composed from six fundamental tiles, the faces of which are normal to the fivefold axes of the icosahedral group. It is shown that the 3D Euclidean space can be tiled face-to-face with maximal face coverage by the composite tiles with an inflation factor τ generated by an inflation matrix. It is noted that dodecahedra with edge lengths of 1 and τ naturally occur already in the second and third order of the inflations. The 3D patches displaying fivefold, threefold and twofold symmetries are obtained in the inflated dodecahedral structures with edge lengths τⁿ with n ≥ 3. The planar tiling of the faces of the composite tiles follows the edge-to-edge matching of the Robinson triangles.

The maximum surface area polyhedron with five vertices inscribed in the sphere {\bb S}^{2}

This article focuses on the problem of analytically determining the optimal placement of five points on the unit sphere {\bb S}^{2} so that the surface area of the convex hull of the points is maximized. It is shown that the optimal polyhedron has a trigonal bipyramidal structure with two vertices placed at the north and south poles and the other three vertices forming an equilateral triangle inscribed in the equator. This result confirms a conjecture of Akkiraju, who conducted a numerical search for the maximizer. As an application to crystallography, the surface area discrepancy is considered as a measure of distortion between an observed coordination polyhedron and an ideal one. The main result yields a formula for the surface area discrepancy of any coordination polyhedron with five vertices.

Algebraic approximations of a polyhedron correlation function stemming from its chord-length distribution

An algebraic approximation, of order K, of a polyhedron correlation function (CF) can be obtained from γ''(r), its chord-length distribution (CLD), considering first, within the subinterval [D_i-1, D_i] of the full range of distances, a polynomial in the two variables (r - D_i-1)^1/2 and (D_i - r)^1/2 such that its expansions around r = D_i-1 and r = D_i simultaneously coincide with the left and right expansions of γ''(r) around D_i-1 and D_i up to the terms O(r - D_i-1)^K/2 and O(D_i - r)^K/2, respectively. Then, for each i, one integrates twice the polynomial and determines the integration constants matching the resulting integrals at the common end-points. The 3D Fourier transform of the resulting algebraic CF approximation correctly reproduces, at large q's, the asymptotic behaviour of the exact form factor up to the term O[q^-(K/2+4)]. For illustration, the procedure is applied to the cube, the tetrahedron and the octahedron.

The chord-length distribution of a polyhedron

The chord-length distribution function [γ''(r)] of any bounded polyhedron has a closed analytic expression which changes in the different subdomains of the r range. In each of these, the γ''(r) expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to R[r, Δ₁], Δ₁ being the square root of a second-degree r polynomial and R[x, y] a rational function. As r approaches δ, one of the two end points of an r subdomain, the derivative of γ''(r) can only show singularities of the forms |r - δ|^-n and |r - δ|^-m+1/2, with n and m appropriate positive integers. Finally, the explicit analytic expressions of the primitives are also reported.

Sustained Neurotrophin Release from Protein Nanoparticles Mediated by Matrix Metalloproteinases Induces the Alignment and Differentiation of Nerve Cells

The spatial and temporal availability of cytokines, and the microenvironments this creates, is critical to tissue development and homeostasis. Creating concentration gradients in vitro using soluble proteins is challenging as they do not provide a self-sustainable source. To mimic the sustained cytokine secretion seen in vivo from the extracellular matrix (ECM), we encapsulated a cargo protein into insect virus-derived proteins to form nanoparticle co-crystals and studied the release of this cargo protein mediated by matrix metalloproteinase-2 (MMP-2) and MMP-8. Specifically, when nerve growth factor (NGF), a neurotrophin, was encapsulated into nanoparticles, its release was promoted by MMPs secreted by a PC12 neuronal cell line. When these NGF nanoparticles were spotted onto a cover slip to create a uniform circular field, movement and alignment of PC12 cells via their extended axons along the periphery of the NGF nanoparticle field was observed. Neural cell differentiation was confirmed by the expression of specific markers of tau, neurofilament, and GAP-43. Connections between the extended axons and the growth cones were also observed, and expression of connexin 43 was consistent with the formation of gap junctions. Extensions and connection of very fine filopodia occurred between growth cones. Our studies indicate that crystalline protein nanoparticles can be utilized to generate a highly stable cytokine gradient microenvironment that regulates the alignment and differentiation of nerve cells. This technique greatly simplifies the creation of protein concentration gradients and may lead to therapies for neuronal injuries and disease.

Antioxidant Systems as a Response to Midgut Cellular of Bombyx mori Lineu, 1758 (Lepidoptera: Bombycidae) Infection for Baculoviruses

Bombyx mori nucleopolyhedrovirus (BmNPV) is a DNA virus that infects different tissues in Bombyx mori at immature stage. Caterpillars become infected after ingesting polyhedral occlusion bodies (POB) present in contaminated mulberry leaves and spread through the body after passing the epithelium of the midgut. As this organ is responsible for digestion, most absorption of nutrients requires an intact epithelium to maintain gastrointestinal physiology. Considering the importance of this organ in the feeding of caterpillars and in the production of quality silk threads, and because it is also the first barrier faced by the BmNPV, the study analyzed details of cytopathological events in the intestinal cells as well as evaluated the action of the antioxidant systems as a response to cellular infection. For this purpose, B. mori hybrid caterpillars of fifth instar were inoculated with a suspension of 7.8 × 107 POB ml-1 and, from the first to the eighth day post-inoculation (dpi), segments of the midgut were collected and processed for light and electronic microscopy. The nuclei of columnar cells showed polyhedric occlusion bodies in the seventh dpi and fragmentation of those cells, with peritrophic matrix disorganization. Analysis of antioxidant systems shows some moments of changes of the catalase enzymes and superoxide dismutase. Analysis of the cholinergic system revealed changes only at the beginning of the infection. Thus, the article acknowledges the antioxidant system as a barrier to stop viral infection, albeit it cannot stop infection from occurring, once a coevolutionary bond is maintained between virus and host.

Geometry and arithmetic of crystallographic sphere packings

We introduce the notion of a "crystallographic sphere packing," defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit an infinite family of conformally inequivalent crystallographic packings with all radii being reciprocals of integers. We then prove a result in the opposite direction: the "superintegral" ones exist only in finitely many "commensurability classes," all in, at most, 20 dimensions.

A structured review of baculovirus infection process: integration of mathematical models and biomolecular information on cell-virus interaction

The baculovirus expression vector system (BEVS) is an emerging tool for the production of recombinant proteins, vaccines and bio-pesticides. However, a system-level understanding of the complex infection process is important in realizing large-scale production at a lower cost. The entire baculovirus infection process is summarized as a combination of various modules and the existing mathematical models are discussed in light of these modules. This covers a systematic review of the present understanding of virus internalization, viral DNA replication, protein expression, budded virus (BV) and occlusion-derived virus (ODV) formation, few polyhedral (FP) and defective interfering particle (DIP) mutant formation, cell cycle modification and apoptosis during the viral infection process. The corresponding theoretical models are also included. Current knowledge regarding the molecular biology of the baculovirus/insect cell system is integrated with population balance and mass action kinetics models. Furthermore, the key steps for simulating cell and virus densities and their underlying features are discussed. This review may facilitate the further development and refinement of mathematical models, thereby providing the basis for enhanced control and optimization of bioreactor operation.

Crystal Engineering of Self-Assembled Porous Protein Materials in Living Cells

Crystalline porous materials have been investigated for development of important applications in molecular storage, separations, and catalysis. The potential of protein crystals is increasing as they become better understood. Protein crystals have been regarded as porous materials because they present highly ordered 3D arrangements of protein molecules with high porosity and wide range of pore sizes. However, it remains difficult to functionalize protein crystals in living cells. Here, we report that polyhedra, a natural crystalline protein assembly of polyhedrin monomer (PhM) produced in insect cells infected by cypovirus, can be engineered to extend porous networks by deleting selected amino acid residues located on the intermolecular contact region of PhM. The adsorption rates and quantities of fluorescent dyes stored within the mutant crystals are increased relative to those of the wild-type polyhedra crystal (WTPhC) under both in vitro and in vivo conditions. These results provide a strategy for designing self-assembled protein materials with applications in molecular recognition and storage of exogenous substances in living cell as well as an entry point for development of bioorthogonal chemistry and in vivo crystal structure analysis.

Concave Pd-Pt Core-Shell Nanocrystals with Ultrathin Pt Shell Feature and Enhanced Catalytic Performance

One-pot creation of unique concave Pd-Pt core-shell polyhedra has been developed for the first time using an efficient approach. Due to the concave feature and ultrathin Pt shell, the created Pd-Pt core-shell polyhedra exhibit enhanced catalytic performance in both the electrooxidation of methanol and hydrogenation of nitrobenzene, as compared with commercial Pt black and Pd black catalysts.

Electrum, the Gold-Silver Alloy, from the Bulk Scale to the Nanoscale: Synthesis, Properties, and Segregation Rules

The alloy Au-Ag system is an important noble bimetallic phase, both historically (as "Electrum") and now especially in nanotechnology, as it is applied in catalysis and nanomedicine. To comprehend the structural characteristics and the thermodynamic stability of this alloy, a knowledge of its phase diagram is required that considers explicitly its size and shape (morphology) dependence. However, as the experimental determination remains quite challenging at the nanoscale, theoretical guidance can provide significant advantages. Using a regular solution model within a nanothermodynamic approach to evaluate the size effect on all the parameters (melting temperature, melting enthalpy, and interaction parameters in both phases), the nanophase diagram is predicted. Besides an overall shift downward, there is a "tilting" effect on the solidus-liquidus curves for some particular shapes exposing the (100) and (110) facets (cube, rhombic dodecahedron, and cuboctahedron). The segregation calculation reveals the preferential presence of silver at the surface for all the polyhedral shapes considered, in excellent agreement with the latest transmission electron microscopy observations and energy dispersive spectroscopy analysis. By reviewing the nature of the surface segregated element of different bimetallic nanoalloys, two surface segregation rules, based on the melting temperatures and surface energies, are deduced. Finally, the optical properties of Au-Ag nanoparticles, calculated within the discrete dipole approximation, show the control that can be achieved in the tuning of the local surface plasmon resonance, depending of the alloy content, the chemical ordering, the morphology, the size of the nanoparticle, and the nature of the surrounding environment.

Design of Enzyme-Encapsulated Protein Containers by In Vivo Crystal Engineering

Crystalline protein assemblies of polyhedra crystal (PhC) can be utilized as solid enzyme containers for long-term storage of enzymes with retention of their enzymatic activity. The enzymes can be released from the crystals at the optimum pH for the enzymatic activity by dissolution of the crystals using in vivo crystal engineering.

Polyhedral microcrystals encapsulating bone morphogenetic protein 2 improve healing in the alveolar ridge

Atelocollagen sponges incorporating polyhedra encapsulating bone morphogenetic protein 2 (BMP-2) were implanted into lateral bone defects in the mandible. Half of the bone defects on the left side were treated with atelocollagen sponges containing 1.8 × 10(7) BMP-2 polyhedra, and half were treated with sponges containing 3.6 × 10(6) BMP-2 polyhedra. As controls, we treated the right-side bone defects in each animal with an atelocollagen sponge containing 5 µg of recombinant human BMP-2 (rhBMP-2) or 1.8 × 10(7) empty polyhedral. After a healing period of six months, whole mandibles were removed for micro-computed tomography (CT) and histological analyses. Micro-CT images showed that more bone had formed at all experimental sites than at control sites. However, the density of the new bone was not significantly higher at sites with an atelocollagen sponge containing BMP-2 polyhedra than at sites with an atelocollagen sponge containing rhBMP-2 or empty polyhedra. Histological examination confirmed that the BMP-2 polyhedra almost entirely replaced the atelocollagen sponges and connected the original bone with the regenerated bone. These results show that the BMP-2 delivery system facilitates the regeneration of new bone in the mandibular alveolar bone ridge and has an advance in the technology of bone regeneration for implant site development.

Gold-copper nano-alloy, "Tumbaga", in the era of nano: phase diagram and segregation

Gold-copper (Au-Cu) phases were employed already by pre-Columbian civilizations, essentially in decorative arts, whereas nowadays, they emerge in nanotechnology as an important catalyst. The knowledge of the phase diagram is critical to understanding the performance of a material. However, experimental determination of nanophase diagrams is rare because calorimetry remains quite challenging at the nanoscale; theoretical investigations, therefore, are welcomed. Using nanothermodynamics, this paper presents the phase diagrams of various polyhedral nanoparticles (tetrahedron, cube, octahedron, decahedron, dodecahedron, rhombic dodecahedron, truncated octahedron, cuboctahedron, and icosahedron) at sizes 4 and 10 nm. One finds, for all the shapes investigated, that the congruent melting point of these nanoparticles is shifted with respect to both size and composition (copper enrichment). Segregation reveals a gold enrichment at the surface, leading to a kind of core-shell structure, reminiscent of the historical artifacts. Finally, the most stable structures were determined to be the dodecahedron, truncated octahedron, and icosahedron with a Cu-rich core/Au-rich surface. The results of the thermodynamic approach are compared and supported by molecular-dynamics simulations and by electron-microscopy (EDX) observations.

Destabilization of gold clusters for controlled nanosynthesis: from clusters to polyhedra

A precisely controlled destabilization of gold thiolate clusters is demonstrated to grow 12 {110}-faceted gold dodecahedra with greatly enhanced catalytic capability, and reveal the growth mechanism by DFT simulations. This greatly advances our understanding of nanocrystal growth and opens a new window for controlling the dissociation of clusters to produce nanocrystals with specific shapes.

Les polyèdres viraux : armures cristallines des virus d'insectes

Viral polyhedroses are very common diseases of insects. They were first identified as leading causes of losses in the silk industry. This heterogeneous group of diseases is characterized by the formation of crystals in infected cells that are called viral polyhedra or occlusion bodies and represent the infectious form of the viruses. Polyhedra have similar role in the infectious cycle of the two groups of viruses responsible for polyhedroses, the cypoviruses - members of the Reoviridae family - and the Baculoviridae. Polyhedra embed virus particles within infected cells in a robust crystalline matrix that protects viral infectivity after release in the environment. Upon ingestion by a new host, crystals dissolve readily thereby releasing the infectious particles to initiate a new viral cycle. Owing to their unique molecular organization, these atypical infectious forms have long intrigued virologists and biochemists alike. They attracted particular interest because of the in vivo crystallization process and the contrast between rapid release upon ingestion and extreme stability. It is only recently that novel approaches and technologies allowed the structure determination of such tiny crystals by X-ray crystallography. Cypovirus and baculovirus polyhedra share the same role in the virus cycle, the same crystalline lattice with a cubic centered symmetry, and matrix proteins called polyhedrins of similar sizes. However, their building blocks differ by their folds and packing in polyhedra. The two classes of polyhedra therefore harbour distinct molecular architectures and appear to have emerged independently in the virosphere. The role of tyrosine clusters in polyhedra dissolution and the use of molecular arms to achieve in vivo crystallization may thus represent striking cases of convergent evolution. This review summarizes our understanding of viral polyhedra with an emphasis on the recent structural studies. We also provide examples of biotechnological applications entailing structure-based engineering of polyhedra as novel types of crystalline microparticules.

How baculovirus polyhedra fit square pegs into round holes to robustly package viruses

Natural protein crystals (polyhedra) armour certain viruses, allowing them to survive for years under hostile conditions. We have determined the structure of polyhedra of the baculovirus Autographa californica multiple nucleopolyhedrovirus (AcMNPV), revealing a highly symmetrical covalently cross-braced robust lattice, the subunits of which possess a flexible adaptor enabling this supra-molecular assembly to specifically entrap massive baculoviruses. Inter-subunit chemical switches modulate the controlled release of virus particles in the unusual high pH environment of the target insect's gut. Surprisingly, the polyhedrin subunits are more similar to picornavirus coat proteins than to the polyhedrin of cytoplasmic polyhedrosis virus (CPV). It is, therefore, remarkable that both AcMNPV and CPV polyhedra possess identical crystal lattices and crystal symmetry. This crystalline arrangement must be particularly well suited to the functional requirements of the polyhedra and has been either preserved or re-selected during evolution. The use of flexible adaptors to generate a powerful system for packaging irregular particles is characteristic of the AcMNPV polyhedrin and may provide a vehicle to sequester a wide range of objects such as biological nano-particles.