Two-dimensional Folding Research Articles

The design of a novel two-dimensional deployable mechanism for large-aperture planar antennas

Large-aperture planar antennas are essential in applications like satellite communication and Earth observation. In order to accommodate them within the constrained storage space of rockets during launch, the development of high-folding-ratio mechanisms becomes imperative. However, most of the existing planar antennas rely on one-dimensional deployable mechanisms, limiting their width and impeding the development of planar antennas with larger apertures. The lack of two-dimensional deployable planar antennas is primarily due to their stringent requirements, including the need for flat reflecting surfaces, regular panel shapes to accommodate electronic devices on the panels’ rear sides and minimal inter-panel gaps. In this paper, we introduce a novel two-dimensional deployable mechanism for the development of large-aperture planar antennas, capable of simultaneously meeting these requirements. Firstly, the architecture of the proposed mechanism and its two-step deployment procedure are described; next, the kinematics on the displacement level are conducted; moreover, a case study is presented to demonstrate its folding performance and frequency response. These studies indicate that the antenna thus obtained can achieve efficient two-dimensional folding while offering a reasonable frequency response, which can be a practical solution for the development of large-aperture planar antennas.

Evidence of Long Two-Dimensional Folding Chain Structure Formation of Poly(vinylidene fluoride) in N-Methylpyrrolidone Solution: Total Form Factor Determination by Combining Multiscattering Data.

Very recently, we proposed that poly(vinylidene fluoride) (PVDF) dissolves in a long rectangular column conformation induced by the formation of a two-dimensional folding chain structure in an N-methylpyrrolidone (NMP) solution based on the results obtained from static light scattering (SLS), small- to wide-angle X-ray scattering (S-WAXS), and viscometric experiments. Small- to wide-angle neutron scattering (S-WANS) experiments were able to provide clear decisive evidence for the presence of such a two-dimensional folding chain structure in a deuterated (d)NMP solution of PVDF samples with two different weight average molar masses (Mw) of 100 and 1200 kg mol–1, even under dilute conditions at concentrations less than the overlapping concentrations due to the quite high neutron scattering contrast obtained by using (d)NMP as a solvent. An increase in Mw from 100 to 1200 kg mol–1 substantially increases the particle length and width from L = 80 to 350 nm and w2 = 5.0 to 15 nm, respectively, while the thickness, w1, is maintained at 0.3 nm. Since L is much longer than w2 irrespective of Mw, the particles formed by PVDF molecules in NMP simply behave as long rigid rods in a scattering vector (q) range covered by SLS measurements. A combination study of S-WANS and S-WAXS experiments covering a higher q range with the SLS techniques could clearly reveal increases in both the L and w2 values with increasing Mw.

Long Two-Dimensional Folding Chain Structure Formation of Poly(vinylidene fluoride) in Solutions of a Polar Solvent, N-Methylpyrrolidone

Long Two-Dimensional Folding Chain Structure Formation of Poly(vinylidene fluoride) in Solutions of a Polar Solvent, <i>N</i>-Methylpyrrolidone

A new two-dimensional folding sheet-like coordination polymer assembled from cadmium(II) and (S)-2-(benzylamino)succinic acid: synthesis, structure and properties.

A new two-dimensional (2D) coordination polymer, namely, poly[[diaqua[μ3-(S)-2-(benzylamino)succinato-κ4N,O1:O1':O4]cadmium(II)] monohydrate], {[Cd(C11H11NO4)(H2O)2]·H2O}n, has been synthesized by the solvothermal reaction of Cd(CH3COO)2·2H2O with the synthesized ligand (S)-2-(benzylamino)succinic acid (H2L). The title compound has been structurally characterized by elemental analysis, IR spectroscopy and single-crystal X-ray diffraction analysis. In the crystal structure, each CdII cation binds to three carboxylate groups from three symmetry-related L2- dianions. The tetradentate L2- ligand links three symmetry-related CdII cations into a 2D folding sheet, which can be simplified as a uninodal (3,3)-connected hcb net with the point symbol (63). In the lattice, all the folding sheets are arranged in an interdigitated fashion and aggregate into zipper-like arrays through interlayer π-π interactions. The large and nonpolar side chain may play an important role in the formation and aggregation of the 2D sheet. The thermal stability and photoluminescence properties of the title compound were investigated, and it exhibits a blue emission with a quantum yield of 8%.

A new twofold interpenetrated two-dimensional cadmium(II) coordination polymer constructed from 2,2'-(benzene-1,4-dicarboxamido)dipropionate.

In recent years, much initial interest and enthusiasm has focused on the self-assembly of coordination polymers due to the aesthetics of their crystalline architectures and their potential applications as new functional materials. As part of an exploration of chiral coordination polymers, a new twofold interpenetrated two-dimensional (2D) coordination polymer, namely, poly[[tetraaquabis[μ3-(2R,2'R)-2,2'-(benzene-1,4-dicarboxamido)dipropionato-κ5O,O':O'',O''':O'']dicadmium(II)] trihydrate], {[Cd2(C14H14N2O6)2(H2O)4]·3H2O}n, has been synthesized by the reaction of Cd(CH3COO)2·2H2O with the designed ligand (2R,2'R)-2,2'-(benzene-1,4-dicarboxamido)dipropionic acid (H2L). The compound has been structurally characterized by elemental analysis, IR spectroscopy, powder X-ray diffraction and single-crystal X-ray diffraction analysis. In the crystal structure, each CdII cation binds to three carboxylate groups from two crystallographically independent L2- dianions. Four carboxylate groups link two crystallographically independent cadmium cations into a 4,4-connected secondary building unit (SBU). The resulting SBUs are extended into a two-dimensional folding sheet via the terephthalamide moiety of the ligand as a spacer, which can be simplified as a (4,4)-connected 4,4L15 net with the point symbol (3.53.62)(32.52.62). In the lattice, two independent folding sheets interpenetrate each other to yield a double-sheet layer. The resulting 2D layers pack in parallel arrays through intermolecular hydrogen bonds and interlayer π-π interactions. The thermal stability and photoluminescence properties of the title compound have been investigated and it exhibits an enhanced fluorescence emission and a longer lifetime compared with free H2L.

Two-Dimensional Folding of Polypeptides into Molecular Nanostructures at Surfaces.

Herein we report the fabrication of molecular nanostructures on surfaces via two-dimensional (2D) folding of the nine amino acid peptide bradykinin. Soft-landing electrospray ion beam deposition in conjunction with high-resolution imaging by scanning tunneling microscopy is used to fabricate and investigate the molecular nanostructures. Subnanometer resolved images evidence the large conformational freedom of the molecules if thermal motion is inhibited and the formation of stable uniform dimers of only one specific conformation when diffusion can take place. Molecular dynamics modeling supported by density functional theory calculations give atomically precise insight into the induced-fit binding scheme when the folded dimer is formed. In the absence of solvent, we find a hierarchy of binding strength from polar to nonpolar, manifested in an inverted polar-nonpolar segregation which suppresses unspecific interactions at the rim of the nanostructure. The demonstrated 2D-folding scheme resembles many key properties of its native 3D counterpart and shows that functional, molecular nanostructures on surfaces fabricated by folding could be just as versatile and specific.

Collapse Precedes Folding in Denaturant-Dependent Assembly of Ubiquitin.

The folding of small protein ubiquitin (Ub), which plays an indispensable role in targeting proteins for degradation and DNA damage response, is complex. A number of experiments on Ub folding have reached differing conclusions regarding the relation between collapse and folding, and whether intermediates are populated. In order to resolve these vexing issues, we elucidate the denaturant-dependent thermodynamics and kinetics of Ub folding at low and neutral pH as a function of guanidinium chloride and urea using coarse-grained molecular simulations. The changes in the fraction of the folded Ub, and the radius of gyration (Rg) as a function of the denaturant concentration, [C], are in quantitative agreement with experiments. Under conditions used in experiments, Rg of the unfolded state at neutral pH changes only by ≈17% as the [GdmCl] decreases from 6 to 0 M. We predict that the extent of compaction of the unfolded state increases as temperature decreases. A two-dimensional folding landscape as a function of Rg and a measure of similarity to the folded state reveals unambiguously that the native state assembly is preceded by collapse, as discovered in fast mixing experiments on several proteins. Analyses of the folding trajectories, under mildly denaturing conditions ([GdmCl] = 1.0 M or [Urea] = 1.0 M), shows that Ub folds by collision between preformed secondary structural elements involving kinetic intermediates that are primarily stabilized by long-range contacts. Our work explains the results of small angle X-ray scattering (SAXS) experiments on Ub quantitatively, and establishes that evolved globular proteins in the unfolded ensemble are poised to collapse as the solvent conditions for the biopolymer changes from good solvent to Θ-solvent like conditions on denaturant dilution. In the process, we explain the discrepancy between SAXS and single molecule fluorescent resonant energy transfer (smFRET) experiments, which have arrived at a contradicting conclusion concerning the collapse of polypeptide chains.

In Chaotropy Lies Opportunity.

In Chaotropy Lies Opportunity.

Curved large tape springs for an ultra-thin shell deployable reflector

This article presents a study of large tape springs to be used as ribs of an ultra-thin shell space deployable reflector. The tape spring, both longitudinally and transversely curved, is made of carbon/epoxy composite and subject to two-dimensional or three-dimensional folds. The behaviour of tape springs with different dimensional parameters during folding is investigated by finite element analyses and an analytical approach. Peak moments, steady-state moments and strain levels are obtained for several parameters such as curvatures, length, thickness, subtended angle, etc. The analytical approach considers the bending strain energy during two-dimensional folding of the tape spring, and it does not capture the full behaviour due to the assumptions made for the simple solution. The results obtained by the two methods are compared where they are applicable and are in agreement for the tape springs subject to two-dimensional folds. Feasibility of the curved large tape springs as a rib-reinforcing element for the deployable reflector with 6 m diameter is investigated. A breadboard model of 90 cm diameter is manufactured and tested for packaging and deployment behaviour.

Buckling folds of a single layer embedded in matrix – Folding behavior revealed by numerical analysis

Abstract Theoretical solutions have been proposed recently for various competent layer–matrix systems, including elastic, viscous and elasto-viscous materials. Furthermore, three type fold-forms of buckling fold had been proposed. These solutions were obtained based on the most simplified, one-dimensional governing equations. Therefore, these solutions require further validation by observing the two-dimensional folding behaviors. This work utilizes numerical analyses to study the buckling and post-buckling behaviors of various layer–matrix systems. As a result, it was found that for competence contrast R ≧ 10 the fold-forms obtained by numerical simulation agree well with those theoretical solutions. Three types of fold-forms can be generated and the resulting wavelengths are also close to the predictions. The fold evolution during the post-buckling stage is explored up to high amplitudes, and the results indicate that the fold-forms can remain the same or be changed from one type to another type, depending on the types of layer–matrix system, the applied strain rates, the original fold-forms at buckling, etc. The fold behaviors from buckling to the post-buckling stage of the layer–matrix systems are presented.

Two-dimensional folding technique for enhancing Fermi surface signatures in the momentum density: Application to Compton scattering data from an Al-3 at. % Li disordered alloy

We present a technique for enhancing Fermi surface (FS) signatures in the two-dimensional (2D) distribution obtained after the 3D momentum density in a crystal is projected along a specific direction in momentum space. These results are useful for investigating fermiology via high-resolution Compton scattering and positron annihilation spectroscopies. We focus on the particular case of the (110) projection in a fcc crystal where the standard approach based on the use of the Lock-Crisp-West (LCW) folding theorem fails to give a clear FS image due to the strong overlap with FS images obtained through projection from higher Brillouin zones. We show how these superposed FS images can be disentangled by using a selected set of reciprocal lattice vectors in the folding process. The applicability of our partial folding scheme is illustrated by considering Compton spectra from an Al--3 at. % Li disordered alloy single crystal. For this purpose, high-resolution Compton profiles along nine directions in the (110) plane were measured. Corresponding highly accurate theoretical profiles in Al--3 at. % Li were computed within the local density approximation (LDA)--based Korringa-Kohn-Rostoker coherent potential approximation (KKR-CPA) first-principles framework. A good level of overall accord between theory and experiment is obtained, some expected discrepancies reflecting electron correlation effects notwithstanding, and the partial folding scheme is shown to yield a clear FS image in the (110) plane in Al--3 at. % Li.

Comparative Study of the Folding Free Energy Landscape of a Three-Stranded β-Sheet Protein with Explicit and Implicit Solvent Models

We present a molecular dynamics (MD) simulation study of the folding thermodynamics for the three-stranded β-sheet protein Betanova. The protein was explicitly described by employing an all-atom model. The solvation was accounted for by two different solvent models: explicit TIP3P water and implicit Generalized Born (GB) with an exterior dielectric of 80. An umbrella sampling technique was utilized to probe thermodynamically relevant states at different stages of folding. The generated data were combined with the weighted histogram method to produce the two-dimensional folding free energy landscape. Sampling of conformational space was carried out in explicit solvent at 275 K and in implicit solvent at 275, 350, and 400 K. The folding free energy surface of Betanova at 275 K was found to be consistent with that in explicit solvent. In particular, the two models agree with regard to the location of the global minimum, the absence of a significant barrier for folding on the folding free energy surface, and...

From ionic currents to molecular mechanisms: the structure and function of voltage-gated sodium channels.

From ionic currents to molecular mechanisms: the structure and function of voltage-gated sodium channels.

Folding Free Energy Surface of a Three-Stranded β-Sheet Protein

We present a molecular dynamics (MD) simulation study of the folding thermodynamics of the three-stranded β-sheet protein Betanova. The protein and solvent are explicitly described by employing all atom models. An umbrella sampling technique was employed to probe thermodynamically relevant states at different stages of folding. A database for the sampling was generated by conducting four high-temperature simulations. The initial conditions for the umbrella sampling were selected from this database of structures by employing hierarchical clustering. Sampling of conformational space was then carried out at 275 K and the generated data were combined with the weighted histogram method to produce the two-dimensional folding free energy landscape. We found that the folding of the protein Betanova occurs in two collapse stages. The first collapse brings the protein into a basin that contains various structures differing in their size and elements of secondary structure. At the transition state from this basin of collapsed states to the native basin, the protein adopts a native-like fold and size and forms ≈60% of native contacts. Thus the formation of native-like structure is concurrent with the secondary collapse. The overall stability of protein Betanova is found to be about 1 kcal/mol, in agreement with the experimental estimate. We found that the native side chain contacts are the primary factor in driving Betanova folding and stabilizing its native three-stranded β-sheet conformation. By contrast, hydrogen bonding is found to play a minor role in the folding of Betanova. Solvent is observed to be present in the protein core until late in folding.

Evolution of three–dimensional folds for a non–Newtonian plate in a viscous medium

We derive analytical solutions for the three-dimensional time-dependent buckling of a non-Newtonian viscous plate in a less viscous medium. For the plate we assume a power-law rheology. The principal, axes of the stretching D-ij in the homogeneously deformed ground state are parallel and orthogonal to the bounding surfaces of the plate in the flat state. In the model formulation the action of the less viscous medium is replaced by equivalent reaction forces. The reaction forces are assumed to be parallel to the normal vector of the deformed plate surfaces. As a consequence, the buckling process is driven by the differences between the in-plane stresses and out of plane stress, and not by the in-plane stresses alone as assumed in previous models. The governing differential equation is essentially an orthotropic plate equation for rate dependent material, under biaxial pre-stress, supported by a viscous medium. The differential problem is solved by means of Fourier transformation and largest growth coefficients and corresponding wavenumbers are evaluated. We discuss in detail fold evolutions for isotropic in-plane stretching (D-11 = D-22), uniaxial plane straining (D-22 = 0) and in-plane flattening (D-11 = -2D(22)). Three-dimensional plots illustrate the stages of fold evolution for random initial perturbations or initial embryonic folds with axes non-parallel to the maximum compression axis. For all situations, one dominant set of folds develops normal to D-11, although the dominant wavelength differs from the Biot dominant wavelength except when the plate has a purely Newtonian viscosity. However, in the direction parallel to D-22, there exist infinitely many modes in the vicinity of the dominant wavelength which grow only marginally slower than the one corresponding to the dominant wavelength. This means that, except for very special initial conditions, the appearance of a three-dimensional fold will always be governed by at least two wavelengths. The wavelength in the direction parallel to D-11 is the dominant wavelength, and the wavelength(s) in the direction parallel to D-22 is determined essentially by the statistics of the initial state. A comparable sensitivity to the initial geometry does not exist in the classic two-dimensional folding models. In conformity with tradition we have applied Kirchhoff's hypothesis to constrain the cross-sectional rotations of the plate. We investigate the validity of this hypothesis within the framework of Reissner's plate theory. We also include a discussion of the effects of adding elasticity into the constitutive relations and show that there exist critical ratios of the relaxation times of the plate and the embedding medium for which two dominant wavelengths develop, one at ca. 2.5 of the classical Biot dominant wavelength and the other at ca. 0.45 of this wavelength. We propose that herein lies the origin of parasitic folds well known in natural examples.

Topology of the membrane protein LamB by epitope tagging and a comparison with the X-ray model

We previously developed a genetic approach to study, with a single antibody, the topology of the outer membrane protein LamB, an Escherichia coli porin with specificity towards maltodextrins and a receptor for bacteriophage lambda. Our initial procedure consisted of inserting at random the same reporter epitope (the C3 neutralization epitope from poliovirus) into permissive sites of LamB (i.e., sites which tolerate insertions without deleterious effects on the protein activities or the cell). A specific monoclonal antibody was then used to examine the position of the inserted epitope with respect to the protein and the membrane. In the present work, we set up a site-directed procedure to insert the C3 epitope at new sites in order to distinguish between two-dimensional folding models. This allowed us to identify two new surface loops of LamB and to predict another periplasmic exposed region. The results obtained by random and directed epitope tagging are analyzed in light of the recently published X-ray structure of the LamB protein. Study of 23 hybrid LamB-C3 proteins led to the direct identification of five of the nine external loops (L4, L5, L6, L7, and L9) and led to the prediction of four periplasmic loops (I1, I4, I5, and I8) of LamB. Nine of the hybrid proteins did not lead to topological conclusions, and none led to the wrong predictions or conclusions. The comparison indicates that parts of models based on secondary structure predictions alone are not reliable and points to the importance of experimental data in the establishment of outer membrane protein topological models. The advantages and limitations of genetic foreign epitope insertion for the study of integral membrane proteins are discussed.

Random mutant generation and its utility in uncovering structural and functional features of cytochromeb inSaccharomyces cerevisiae

The generation of random mutations in the mitochondrial cytochrome b gene of Saccharomyces cerevisiae has been used as a most fruitful means of identifying subregions that play a key role in the bc1 complex mechanism, best explained by the protonmotive Q cycle originally proposed by Peter Mitchell. Selection for center i and center o inhibitor resistance mutants, in particular, has yielded much information. The combined approaches of genetics and structural predictions have led to a two-dimensional folding model for cytochrome b that is most compatible with current knowledge of the protonmotive Q cycle. A three-dimensional model is emerging from studies of distant reversions of deficient mutants. Finally, interactions between cytochrome b and the other subunits of the bc1 complex, such as the iron-sulfur protein, can be affected by a single amino acid change.

Geometrical modeling of fault-related folds: a pseudo-three-dimensional approach

Area-balanced cross-sections of ramp anticlines and tip-line folds in thin-skinned fold-thrust belts can be quantitatively constructed using plane-strain fault-bend and fault-propagation fold models. Cross-sections alone, however, are somewhat inadequate for interpreting along-strike changes in thrust-related fold geometry. Complex interplay between along-strike changes in fault slip and fault shape determines a range of geometric configurations which are difficult to visualize without the aid of quantitative three-dimensional models. We extend conventional two-dimensional folding theories into the third dimension by allowing for continuous variations in fault slip and fault shape along strike. Geometric equations relating map-view angles between fold axial-surface traces are derived for a case assuming a uniform fault geometry and an along-strike displacement gradient. The resulting folds possess no unique fold axis, but rather are characterized by multiple fold hingelines each corresponding to bends in the fault. For typical displacement gradients observed in the field, analysis of the models suggests that discernible differences exist between map patterns of fault-bend folds and fault-propagation folds, whereas little difference exists between map patterns of similar and parallel fault-bend folds. Map-view angles between axial-surface traces for parallel fault-bend folds and fault-propagation folds are a function of both ramp cutoff angle and fault slip, whereas these same angles for similar fault-bend folds are solely a manifestation of fault slip. Examples assuming uniform shortening and continuously varying fault geometry along strike are also presented. Fault geometry is varied along strike to simulate oblique and lateral ramps. Depending upon model type and fault shape parameter modified, resulting folds exhibit either cylindrical or non-cylindrical fold geometries. Closure is generated by a decrease in fold amplitude where there is a decrease in ramp cutoff angle along strike or where a fault laterally cuts up section. In either instance, closure develops without varying shortening along strike. Model map patterns resemble actual thrust-related fold geometries and can be compared to maps of real structures to help predict subsurface geology.

Inelastic Scattering Cross Sections for 20-keV Electrons in Al, Be, and Polystyrene

Measurements are reported of the inelastic scattering cross sections associated with the characteristic energy losses of 20-keV electrons transmitted through thin films of Al, Be, and polystyrene. These measurements have been made using two techniques which overcome several sources of systematic error in previous measurements. For materials which have narrow characteristic loss lines (such as Al), it is possible to establish the form of the differential cross section and the generalized oscillator strength for the \ensuremath{\approx} 15-eV Al plasmon energy loss for scattering angles between 0 and 20 mrad, the cutoff angle. The differential cross section has been used in repeated two-dimensional folding calculations to correct the intensity measurements of the multiple plasmon losses at the larger scattering angles where unambiguous intensity measurements could not be made. For materials which have broad characteristic loss peaks, such as polystyrene, it is not possible to establish the variation of generalized oscillator strength $f(q)$ with momentum transfer $q$. If it can be assumed that $f(q)\ensuremath{\approx}f(0)$ for small $q$, a cross section for a given loss can be obtained from an energy-loss spectrum measured at one scattering angle. The cross-section measurement accuracy is improved by a comparison of two energy-loss spectra, that of a standard (such as Al) and that of a material of unknown loss cross section, which have been obtained at zero angle under the same measurement conditions. Cross sections of (1.5\ifmmode\pm\else\textpm\fi{}0.4)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}18}$ ${\mathrm{cm}}^{2}$ and (1.5\ifmmode\pm\else\textpm\fi{}0.5)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}18}$ ${\mathrm{cm}}^{2}$ have been measured for the excitation of the \ensuremath{\approx} 15-eV and \ensuremath{\approx} 19-eV plasmon energy losses in Al and Be, respectively, using the former measurement technique. These values are in good agreement with the cross sections expected theoretically. The second measurement technique has been used to obtain cross sections of (2.4\ifmmode\pm\else\textpm\fi{}0.8)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}20}$ ${\mathrm{cm}}^{2}$ and (1.0\ifmmode\pm\else\textpm\fi{}0.2)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}18}$ ${\mathrm{cm}}^{2}$ for the 7-eV and 21-eV energy losses, respectively, in polystyrene. The corresponding oscillator strengths $f(0)$ are in good agreement with those deduced recently by LaVilla and Mendlowitz. Values were also obtained for the Al loss-dispersion constants, $\ensuremath{\Delta}E(\ensuremath{\theta})=14.8+(1.39\ifmmode\pm\else\textpm\fi{}0.07)\ifmmode\times\else\texttimes\fi{}{10}^{4}{\ensuremath{\theta}}^{2}+(1.2\ifmmode\pm\else\textpm\fi{}0.2)\ifmmode\times\else\texttimes\fi{}{10}^{7}{\ensuremath{\theta}}^{4}$ (energy loss $\ensuremath{\Delta}E$ in eV, scattering angle $\ensuremath{\theta}$ in radians), and for the critical wave vector for plasmon excitation ${k}_{c}=1.46\ifmmode\pm\else\textpm\fi{}0.07$ ${\mathrm{\AA{}}}^{\ensuremath{-}1}$.