Adjacent Amide Groups Research Articles

Hydrogen-Bond-Driven Peptide Nanotube Formation: A DFT Study.

DFT calculations were carried out to examine geometries and binding energies of H-bond-driven peptide nanotubes. A bolaamphiphile molecule, consisting of two N-α amido glycylglycine head groups linked by either one CH2 group or seven CH2 groups, is used as a building block for nanotube self-assembly. In addition to hydrogen bonds between adjacent carboxy or amide groups, nanotube formation is also driven by weak C-H· · ·O hydrogen bonds between a methylene group and the carboxy OH group, and between a methylene group and an amide O=C group. The intratubular O-H· · ·O=C hydrogen bonds account for approximately a third of the binding energies. Binding energies calculated with the wB97XD/DGDZVP method show that the hydrocarbon chains play a stabilizing role in nanotube self-assembly. The shortest nanotube has the length of a single monomer and a diameter than increases with the number of monomers. Lengthening of the tubular structure occurs through intertubular O-H· · ·O=C hydrogen bonds. The average intertubular O-H· · ·O=C hydrogen bond binding energy is estimated to change with the size of the nanotubes, decreasing slightly towards some plateau value near 15 kcal/mol according to the wB97XD/DGDZVP method.

Nucleation Retardation of Polyamide 6,6 by Tuning Hydrogen Bonding Reorganization via Nigrosine for High-Speed Injection Molding Applications

The mechanism of regulating hydrogen bond organization for polyamide 6,6 (PA 6,6) by nigrosine was investigated by temperature-dependent FTIR and in situ synchrotron X-ray diffraction during cooling/heating processes. The intermolecular interaction between amide groups and nigrosine was verified by the FTIR spectroscopy method. Upon melt cooling, the formation of H-bonds is promoted by nigrosine at the initial stage (regime I) with increased enthalpy change (ΔH1), which results from denser molecular coils as the interaction with nigrosine increases the ability to form H-bonds between adjacent amide groups, while at a lower temperature (regime II), the ΔH1 decreases with increasing nigrosine as nigrosine restrains the mobility of PA 6,6 chains that retards the formation of H-bonds between nonadjacent amide groups. Moreover, the ordering arrangement of H-bonds into 2-D hydrogen bond sheets (H-sheets) is hindered for both the regime I′/II′ stages with decreased ΔH2 due to the decreased mobility of chain segments, which retards the nucleation of PA 6,6 crystallization. The crystal perfection index (CPI) was investigated by WAXD, and it was found to decrease with the increase of nigrosine. Combined with microstructure analysis, we proposed that partial nigrosine can insert between H-sheets, which therefore tunes the Brill transition behavior of PA 6,6. Overall, the regulation mechanism of nigrosine on PA 6,6 crystallization has been established via interaction with amide groups to tune the organization of H-sheets. The retardation effects on crystallization can improve the flowability of PA 6,6 for high-speed injection molding applications.

Tuning the Emission of a Nonconventional Aggregation-Induced Emission Polymer via Silicon-Bridged Twisted Intramolecular Charge Transfer for Targeted Delivery and Visualized Drug Release.

The design of tunable luminescent biomaterials with large Stokes shifts is usually pursued by a twisted intramolecular charge transfer (TICT) effect with switchable emission colors in response to various external stimuli. However, such a strategy is usually realized in conjugated molecules containing benzene or its derivatives and consequently suffers from poor biocompatibility. In this work, a hyperbranched polysiloxane (HBPSi)-based non-conjugated fluorescent polymer with TICT and aggregation-induced emission (AIE) features is developed, and its luminescent properties, fluorescence mechanism, and potential applications are investigated. Initially, the non-conjugated HBPSi exhibits remarkable AIE characteristics due to the formation of through-space conjugation. With the introduction of the sulfur atom, a non-conjugated D-A type AIE material, HBPSi-Cys, that exhibits a dual-state emission with a large Stokes shift of 213 nm, is obtained. The correlation of the lower-energy emission band with solvent polarity suggests the existence of the TICT state. TICT and AIE characteristics direct different properties of HBPSi-Cys, with TICT regulating solvatochromic emission wavelengths and AIE manipulating the emission intensity with a compensation effect. Density functional theory calculations reveal that the non-conjugated D-A structure in HBPSi-Cys was formed across the silicon bridge, with auxochromic sulfhydryl groups and adjacent amide groups as acceptor units and amine and hydroxyl groups as donor units. Additionally, the AIE-active HBPSi could be utilized as a fluorescent probe for the analysis of metal ions. After grafting the AS1411 aptamer to HBPSi-Cys as the recognition motif, HBPSi-Apt possesses excellent targeted bioimaging, drug loading, pH/GSH dual-responsive drug release, and visualized drug delivery performance. This work provides a new way to design functional AIE polymers with tunable optical properties, and the synthesized HBPSi-Cys shows great potential as a smart fluorescent biomaterial.

Dipole Cooperativity and Polarization Frustration Determine the Secondary Structure Distribution of Short Alanine Peptides in Water.

The physical driving forces for secondary structure preferences of hydrated alanine peptide are investigated with B3LYP-D3(BJ) and the adaptive force matching (AFM) method. The AFM fit to the DFT surface, ALA2022, provides excellent agreement with the nuclear magnetic resonance scalar coupling constants from experiments. In turn, the model is used to gain insight into the physical driving forces behind secondary structure preferences of hydrated peptides. DFT calculations with and without the Conductor-like Screening Model (COSMO) show that the α helix is stabilized by solvent polarization due to dipole cooperativity. The two adjacent amide groups in β strand form a near-planar trapezoid that is not much larger than the size of water molecules. When the finite size of a water molecule is considered, the stabilization from solvent polarization for such a trapezoid is frustrated. Water molecules cannot find orientations to properly stabilize all four polar regions close to each other with such an awkward arrangement. This leads to quite substantial reduction in polarization stabilization. Although the polyproline II (PP-II) conformation is very similar to the β strand, the small twist in the backbone angles allowed much improved polarization stabilization. The improved polarization, when combined with favorable intrapeptide interactions, leads to the PP-II to be lowest in free energy. Other factors, such as the entropic TΔS and the ϕ, ψ coupling terms, are also studied but are found to play only a minor role. The insight shown in this work helps to better understand the structure of globular and intrinsic disordered proteins and facilitate future force field development.

Supramolecular Helical Assemblies of Dirhodium(II) Paddlewheels with 1,4-Diazabicyclo[2.2.2]octane: A Remarkable Substituent Effect on the Helical Sense Preference and Amplification of the Helical Handedness Excess of Metallo-Supramolecular Helical Polymers.

We report unique coordination-driven supramolecular helical assemblies of a series of dirhodium(II) tetracarboxylate paddlewheels bearing chiral phenyl- or methyl-substituted amide-bound m-terphenyl residues with triethylene glycol monomethyl ether (TEG) or n-dodecyl tails through a 1:1 complexation with 1,4-diazabicyclo[2.2.2]octane (DABCO). The chiral dirhodium complexes with DABCO in CHCl3/n-hexane (1:1) form one-handed helical coordination polymers with a controlled propeller chirality at the m-terphenyl groups, which are stabilized by intermolecular hydrogen-bonding networks between the adjacent amide groups at the periphery mainly via a cooperative nucleation-elongation mechanism as supported by circular dichroism (CD), vibrational CD, and variable-temperature (VT) absorption and CD analyses. The VT visible-absorption titrations revealed the temperature-dependent changes in the degree of polymerization. The columnar supramolecular helical structures were elucidated by X-ray diffraction and atomic force microscopy. The helix sense of the homopolymer carrying the bulky phenyl and n-dodecyl substituents is opposite those of other chiral homopolymers despite having the same absolute configuration at the pendants. A remarkably strong "sergeants and soldiers" (S&S) effect was observed in most of the chiral/achiral copolymers, while the copolymers of the bulky chiral phenyl-substituted dirhodium complexes with n-dodecyl chains displayed an "abnormal" S&S effect accompanied by an inversion of the helix sense, which could be switched to a "normal" S&S effect by changing the solvent composition. A nonracemic dirhodium complex of 20% enantiomeric excess bearing the less bulky chiral methyl substituents with n-dodecyl chains assembled with DABCO to form an almost one-handed helix (the "majority rule" (MR) effect), whereas the three other nonracemic copolymers showed a weak MR effect.

Effect of amide distribution in polyamide isomers on their melting points

ABSTRACTA new insight has been obtained relating the different structures of linear aliphatic homopolyamides to their melting points. A new equation is derived for structures based on monomers with an even number of methylene units. In this equation, the amide distribution along the polymer chain is defined by a newly introduced parameter K which indicates the lowest number of methylene units between adjacent amide groups. For K ≥ 4 a generic empirical relation is defined, describing the observed melting points of these polyamides. The trend in H‐bond densities in the melt obtained by modeling corresponds well with the trend in observed melting points. © 2017 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2018, 135, 45837.

Novelty of Dynamic Process in the Synthesis of Biocompatible Silica Nanotubes by Biomimetic Glycyldodecylamide as a Soft Template.

A dynamic process in the synthesis of silica nanotubes (SNTs) by utilizing glycyldodecylamide (GDA) as a soft template was thoroughly investigated. The morphological evolution from GDA to SNTs was deeply explored to elucidate the formation mechanism for optimizing the synthesis procedure. Various analytical tools, namely, XRD, FTIR, SEM, TEM, Z-potential, and N2 adsorption/desorption isotherms, were employed during the synthesis procedure. The interactive structure of GDA was also investigated using TEM-EDX as a function of aging time. These studies revealed the stepwise morphology of nanograin, nanofiber, curved plate, and nanotube in the ethanol/water solution when aged at room temperature. The supramolecular GDA molded the vesicle type nanostructure which was surrounded by silica and facilitated the formation of uniform SNTs. The stimulus for GDA to be curved into a vesicle was the intermolecular hydrogen bonding between adjacent amide groups of the template molecules. This was illustrated by FTIR spectra of GDA-silica intermediate by detecting the transition of amide I peak from 1678 to 1635 cm-1. The effect of hydrogen bonding became stronger when the sample was aged.

Discovery of bisamide-heterocycles as inhibitors of scavenger receptor BI (SR-BI)-mediated lipid uptake.

A new series of potent inhibitors of cellular lipid uptake from HDL particles mediated by scavenger receptor, class B, type I (SR-BI) was identified. The series was identified via a high-throughput screen of the National Institutes of Health Molecular Libraries Small Molecule Repository (NIH MLSMR) that measured the transfer of the fluorescent lipid DiI from HDL particles to CHO cells overexpressing SR-BI. The series is characterized by a linear peptidomimetic scaffold with two adjacent amide groups, as well as an aryl-substituted heterocycle. Analogs of the initial hit were rapidly prepared via Ugi 4-component reaction, and select enantiopure compounds were prepared via a stepwise sequence. Structure-activity relationship (SAR) studies suggest an oxygenated arene is preferred at the western end of the molecule, as well as highly lipophilic substituents on the central and eastern nitrogens. Compound 5e, with (R)-stereochemistry at the central carbon, was designated as probe ML279. Mechanistic studies indicate that ML279 stabilizes the interaction of HDL particles with SR-BI, and its effect is reversible. It shows good potency (IC50=17 nM), is non-toxic, plasma stable, and has improved solubility over our alternative probe ML278.

Epoxy-terminated self-assembled monolayers containing internal urea or amide groups.

We report the synthesis of new coupling agents with internal amide or urea groups possessing an epoxy-terminal group and trimethoxysilyl-anchoring group. The structural characterizations of the corresponding self-assembled monolayers (SAMs) were performed by polarization modulation infrared reflection adsorption spectroscopy (PM-IRRAS). The molecular assembly is mainly based on the intermolecular hydrogen-bonding between adjacent amide or urea groups in the monolayers. Because of the steric hindrance of amide or urea groups, the distance between the alkyl chains is too large to establish van der Waals interactions, inducing their disorder. The reactivity of the epoxy-terminal groups was successfully investigated through reaction with a fluorescent probe. We show that SAMs containing internal urea or amide groups exhibited a higher density of accessible epoxide groups than the corresponding long-chain (C22) glycidyl-terminated SAM.

Crystal structure of (4S)-aminoproline: conformational insight into a pH-responsive proline derivative.

The crystal structure of a (4S)-configured ammoniumproline derivative is presented. The trifluoroacetic acid salt of the acetylated methylester of (4S)-aminoproline (Ac-(4S)Amp-OMe) crystallized as the trans conformer with a C(4)-endo ring pucker. The high-resolution structure shows typical parameters for a transannular H-bond and an n → π* interaction between the adjacent amide and ester groups. The structure corroborates previous findings of solution phase studies on the importance of these two interactions in acidic and neutral aqueous environments for the conformation of this pH-responsive proline derivative.

4-Carbamoylpyridin-1-ium 2,2,2-trichloroacetate

In the asymmetric unit of the title salt, C6H7N2O+·C2Cl3O2 −, there are two crystallographic independent ion pairs. The amide groups of the 4-carbamoylpyridin-1-ium ions are slightly twisted out of the plane of the aromatic ring with C—C—C—N torsion angles of 8.8 (9)° and 4.6 (8)°. In the crystal, the 4-carbamoylpyridin-1-ium ion is N—H⋯O hydrogen bonded to the trichloro­acetate ion via the pyridinium unit and amide group. Layers parallel to the ac plane are formed due to the N—H⋯O hydrogen bonding of the adjacent amide groups of 4-carbamoylpyridin-1-ium ions. Weak C—H⋯O inter­actions also occur.

The influence of dipole moments on the mechanism of electron transfer through helical peptides

The life time of aromatic radical cations is limited by reactions like β-elimination, dimerization, and addition to the solvent. Here we show that the attachment of such a radical cation to the C-terminal end of an α-/3(10)-helical peptide further reduces its life time by two orders of magnitude. For PPII-helical peptides, such an effect is only observed if the peptide contains an adjacent electron donor like tyrosine, which enables electron transfer (ET) through the peptide. In order to explain the special role of α-/3(10)-helical peptides, it is assumed that the aromatic radical cation injects a positive charge into an adjacent amide group. This is in accord with quantum chemical calculations and electrochemical experiments in the literature showing a decrease in the amide redox potentials caused by the dipole moments of long α-/3(10)-helical peptides. Rate measurements are in accord with a mechanism for a multi-step ET through α-/3(10)-helical peptides that uses the amide groups or H-bonds as stepping stones.

Solvent Dependence of Trialanine Conformers

Determining the conformational propensities of amino acid residues in short peptides is increasingly recognized as pivotal for obtaining a reliable picture of the unfolded states of peptides and proteins. In this context, ample experimental evidence indicates that alanine does not exhibit a statistical coil behavior, but rather shows a strong preference for sampling the polyproline II (PPII) region of the Ramachadran plot. Solvation in water has been proposed as the main reason for PPII stabilization. The mechanisms involved in the respective peptide-solvent interactions are still debated. Results from DFT-calculations on dialanine suggest that water stabilizes PPII conformers by forming hydrogen bond bridges between adjacent carbonyl and amide groups. Other models have invoked electrostatic interactions. To explore the solvent dependence of PPII stabilization, we used UV-CD and H-NMR spectroscopy to determine the conformations of cationic trialanine in water, as well as in binary water/glycerol water/ethanol mixtures. All spectra were recorded as a function of temperature ranging between 0°C and 90°C. The resultant dichrosim at 216nm and 3J(HαHN) coupling constants were subjected to a simplified two state PPII↔β strand thermodynamic analysis. The experimental results indicate that both alcohol co-solvents substantially lower the PPII propensity of alanine in each respective binary mixture. Glycerol seems to be most effective in replacing water in the solvation shell, causing large enthalpic decreases between the two conformers; from ΔH=18.15 kJ/mol in water, to ΔH=12.42 kJ/mol with only a 0.012 mole fraction admixture of glycerol.

New Approaches to 12‐Coordination: Structural Consequences of Steric Stress, Lanthanoid Contraction and Hydrogen Bonding

AbstractThe anionic dinitrile ligand dicyanonitrosomethanide (dcnm), C(CN)2(NO)–, and the anion resulting from its addition product with water, carbamoylcyanonitrosomethanide (ccnm), C(CN)(CONH2)(NO)–, have been incorporated into lanthanoid complexes and display unusual η2(N,O) nitroso coordination modes. (Et4N)3[Ln(ccnm)6] (1Ln; 1Ln = 1La, 1Ce, 1Pr, 1Nd, 1Sm) and (Me4N)3[Ln(ccnm)6] (2Ln; 2Ln = 2La, 2Ce, 2Pr, 2Nd) are systems containing 12‐coordinate homoleptic trianionic lanthanoidate complexes. The nitroso groups of the ccnm ligands form three‐membered ring chelates with the lanthanoid metal centre, with the asymmetry of the nitroso η2 interactions dependent upon the intramolecular N–H···O=N hydrogen bonding. Additional intermolecular hydrogen bonding interactions exist between adjacent amide and nitrile groups giving rise to 3D α‐Po and 6,8‐connected (412.63)(420.68) networks in 1Ln and 2Ln, respectively. The compounds (Me4N)3[Ln(dcnm)6] (3Ln; 3Ln = 3La, 3Ce, 3Nd, 3Sm) also contain a 12‐coordinate trianionic lanthanoidate complex with the nitroso group exhibiting a highly symmetrical η2 interaction. The sterically crowded environments of [Ln(18‐crown‐6)(dcnm)3] (4Ln; 4Ln = 4La, 4Ce, 4Pr, 4Nd) result in a shift towards a more asymmetric η2 bonding of the nitroso group with decrease in the Ln3+ radius. There is a corresponding increase of the Ln–O–N angle, and one ligand is η1(O) binding in 4Nd. The dcnm ligands in the discrete complexes [La(phen)3(dcnm)(3–x)Clx], x ≈ 0.25 (5) (phen = 1,10‐phenanthroline), (Et4N)[Ce(phen)2(dcnm)4] (6a/b, 6c) and [Ce(phen)2(dcnm)Cl2H2O] (7) display a variety of coordination modes. Complex 5 has 1D chains formed by π–π stacking of adjacent phen co‐ligands. Complexes 6 contain the monoanionic complex [Ce(phen)2(dcnm)4]– with two geometric isomers present in the crystal structure of 6a/b. Complex 7 forms extended 1D chains via hydrogen bonding between coordinated water and chloride atoms and an extensive array of face‐to‐face π interactions.

Thermal Properties of Poly(ester amide)s with Isolated, Two Adjacent and Three Adjacent Amide Groups within a Polyester Chain

AbstractDifferential scanning calorimetry (DSC) was employed to investigate the thermal properties of segmented aliphatic poly(ester amide)s (PEA)s. Three different series of PEAs were synthesized, each containing amide blocks of defined lengths which are randomly placed in the poly(butylene adipate) (PBA) backbone. Thus, the defined amide blocks consist of isolated amide groups, two adjacent amide groups or three adjacent amide groups. The PEAs show glass transition temperatures between −63 and 1 °C, melting temperatures between 28 and 183 °C and crystallization temperatures lie between −0.2 and 148 °C. The types of groups participating in the crystallization process and the length of the amide segments have an influence on the morphology of these polymers.magnified image

Layer-by-layer multilayer films linked with reversible boronate ester bonds with glucose-sensitivity under physiological conditions

Layer-by-layer (LBL) multilayer films were fabricated from poly(vinylalcohol) (PVA) and poly[acrylamide-co-3-(acrylamido)phenylboronic acid] [P(AAm-AAPBA)] by the use of covalent phenylboronate ester bonding as the driving force. The film thickness increases with decreasing pH and increasing ionic strength of the assembly solutions. Phenylboronate ester bonds in the film were confirmed from the IR marker mode at 1730 cm−1. Because the phenylboronate ester bonding is reversible, the PVA/P(AAm-AAPBA) films disassemble gradually when immersed in aqueous solutions. The disassembly rate increases with increasing pH and decreasing ionic strength of the aqueous solutions. Furthermore, the disassembly of the film is accelerated by the addition of glucose, which competes with PVA for phenylboronic acid groups and weakens the interaction between the two polymers. Interestingly, the PVA/P(AAm-AAPBA) film presents glucose-sensitive behavior even under physiological conditions. The enhanced glucose-sensitive behavior at physiological pH may originate from the stabilization of phenylboronate ester by the adjacent amide group. These glucose-sensitive LBL films hold promise for self-regulated insulin release.

Surface‐enhanced Raman scattering on colloid gels originated from low molecular weight gelator

AbstractSurface‐enhanced Raman scattering (SERS) on silver and gold colloid gels formed by a low molecular weight organic gelator, bis‐(S‐phenylalanine) oxalyl amide, was obtained. Strong Raman signals dominate in the SERS spectra of hydrogels containing silver nanoparticles prepared by citrate and borohydride reduction methods, whereas broad bands of low intensity are detected in the spectra of gold colloid gels. Resemblance between Raman spectrum of the crystalline substance and the SERS spectra of the silver nanoparticle–hydrogel composites implies the electromagnetic nature of the signal enhancement. A change in Raman intensity of the benzene and amide II bands caused by an increase in temperature and concentration indicates that the gelling molecules are strongly attached through the benzene moieties to the metal nanoparticles while participating in gel formation by intermolecular hydrogen bonding between the adjacent oxalyl amide groups. Transmission electron microscopy reveals a dense gel structure in the close vicinity of the enhancing metal particles for both silver colloid gels. Copyright © 2008 John Wiley & Sons, Ltd.

Peptide−Heterocycle Hybrid Molecules: Solid-Phase Synthesis of a 400-Member Library of N-Terminal 2-Iminohydantoin Peptides

A method for the heterocyclic modification of the N-terminus of a peptide is described. Reaction of the N-terminal amino group of solid-supported peptides with arylisothiocyanates generates a thiourea intermediate, as in the first step of Edman degradation. Treatment of the resin-supported peptide-thioureas with Mukaiyama's reagent (2-chloro-1-methylpyridinium iodide) generates an electrophilic carbodiimide functionality, which undergoes rapid intramolecular trapping by the adjacent amide group to give a 2-iminohydantoin ring at the N-terminus of the peptide. The dehydrothiolation step in the presence of Mukaiyama's reagent prevents Edman degradation from occurring, in essence leading to a "diversion" of the Edman degradation. Cleavage from the resin then releases the hybrid molecules incorporating a 2-iminohydantoin ring conjugated onto a peptidic fragment. A 400-member library of the iminohydantoin-peptide hybrids was synthesized using this approach, starting from a chlorotrityl resin-supported tripeptides.

Asymmetric Reduction of Ketones Catalyzed by a Polymeric Chiral Ru Complex

Two polystyrene-supported chiral ligands 1 and 2, based on (1S,2S)-N-(p-toluenesulfonyl)-1,2-diphenylethylenediamine (TsDPEN), were prepared from sodium 4-hydroxybenzenesulfonate dihydrate in seven and eight steps, respectively. The catalytic properties of their Ru complexes prepared in situ were examined for the asymmetric transfer hydrogenation of aromatic ketones with formic acid-triethylamine azeotrope as the hydrogen source to exhibit the high catalytic activities and enantioselectivities. This reduction protocol also displayed excellent chemoselectivities: the keto-carbonyl group was reduced selectively under these reaction conditions whereas the adjacent ester, amide, or cyano groups remained intact. The immobilized catalyst was easily recovered by simple filtration and subsequent washing with dichloromethane, and was reused without a significant drop in enantioselectivity. The resulting optically active alcohols are known to be appropriate intermediates for the synthesis of (S)-fluoxetine, which is an important antidepressant.

Two-Dimensional Infrared Spectroscopy of Isotopomers of an Alanine Rich α-Helix

The two-dimensional infrared spectra of a series of doubly isotopically substituted 25-residue α-helices were measured with femtosecond three pulse infrared time domain interferometry. The insertion of 13C16O and 13C18O labels at known residues on the helix permitted the vibrational couplings between different amide I‘ modes separated by one, two, and three residues to be measured. The 2D IR signal of one residue in 25 was readily studied, confirming this approach is applicable to labeled proteins. We identified the couplings between each pair of isotopomer levels and between them and the helix exciton band states: the 2D IR spectra proved that the amide vibrations of the α-helix are delocalized. Cross-peaks, originating from the coupling of the isotopomer pairs, were systematically analyzed. Besides the separated pair modeling and second-order perturbation theory estimates, the experimental results were compared in detail with a full matrix diagonalization simulation based on averaged Hamiltonian matrices that represent the amide I‘ vibrator's one- and two-exciton states. The main features of the 2D IR spectra could be predicted by this modeling. The experimental results were in good agreement with a set of couplings that were derived from transition charge−transition charge interactions for all but the nearest neighbors, for which the coupling is more influenced by through-bond interactions between the adjacent amide groups. The possible ranges of the magnitudes of the three largest coupling constants β12, β13, and β14 were explored by various approaches to be within a few cm-1 accuracy of a preferred set of absolute values and their associated error bars: |β12| = 8.5 ± 1.8, |β13| = 5.4 ± 1.0, and |β14| = 6.6 ± 0.8 cm-1. The signs were independently indicated to be β12 > 0, β13 < 0, and β14 < 0.