Steric Strain Research Articles

Unified picture for the conformation and stabilization of the O-glycosidic linkage in glycopeptide model structures

Glycoproteins play a central role in the immune response. In this study, we focus on the core of the common O-linked mucin-type glycopeptides. It has been observed that glycosylation stabilizes the protein in a stiffened, extended structure. We provide a unified picture for the conformation and stabilization of the O-glycosidic linkage using O-(2-acetylamino-2-deoxy-α- or -β-d-galacto- or -mannopyranosyl)-N-acetyl-l-serinamide model structures. We have calculated equilibrium geometries of the model structures with the B3LYP/6-31G(2df,p) method suggested in the Gaussian-4 theory. According to the relative energies, we confirm that the GalNAc-Ser linkage is more stable than its mannose analogues. The natural preference for the α-GalNAc-Ser over the β-GalNAc-Ser anomers can be explained by entropic effects. We explored the hydrogen bonding patterns on the carbohydrate unit calculating highly accurate dRPA@PBE0.25 and dRPA75 energies and found that in some cases, the acetamido group can be fixed by hydrogen bonding from the (O3Carb)H atom, but in most of the cases, it can rotate more freely. The torsion angles in the glycosidic linkage show that the linkage is stiffened more in the α-anomers and the most in the α-GalNAc-Ser structure because of the steric strains in the axial position and by two or three intramolecular hydrogen bonds. We also found that although, in our gas-phase model geometries, the peptide backbone prefers to be in a γ L-turn, a structural water molecule can stabilize a polyproline II helix of a proline-rich sequence, a β-sheet, or more likely random coils.

A Modular, DNA-Based Beacon for Single-Step Fluorescence Detection of Antibodies and Other Proteins.

A versatile platform for the one-step fluorescence detection of both monovalent and multivalent proteins has been developed. This system is based on a conformation-switching stem-loop DNA scaffold that presents a small-molecule, polypeptide, or nucleic-acid recognition element on each of its two stem strands. The steric strain associated with the binding of one (multivalent) or two (monovalent) target molecules to these elements opens the stem, enhancing the emission of an attached fluorophore/quencher pair. The sensors respond rapidly (<10 min) and selectively, enabling the facile detection of specific proteins even in complex samples, such as blood serum. The versatility of the platform was demonstrated by detecting five bivalent proteins (four antibodies and the chemokine platelet-derived growth factor) and two monovalent proteins (a Fab fragment and the transcription factor TBP) with low nanomolar detection limits and no detectable cross-reactivity.

A Modular, DNA‐Based Beacon for Single‐Step Fluorescence Detection of Antibodies and Other Proteins

AbstractA versatile platform for the one‐step fluorescence detection of both monovalent and multivalent proteins has been developed. This system is based on a conformation‐switching stem–loop DNA scaffold that presents a small‐molecule, polypeptide, or nucleic‐acid recognition element on each of its two stem strands. The steric strain associated with the binding of one (multivalent) or two (monovalent) target molecules to these elements opens the stem, enhancing the emission of an attached fluorophore/quencher pair. The sensors respond rapidly (&lt;10 min) and selectively, enabling the facile detection of specific proteins even in complex samples, such as blood serum. The versatility of the platform was demonstrated by detecting five bivalent proteins (four antibodies and the chemokine platelet‐derived growth factor) and two monovalent proteins (a Fab fragment and the transcription factor TBP) with low nanomolar detection limits and no detectable cross‐reactivity.

Synthesis, spectroscopy, electrochemistry, and coordination chemistry of substituted phosphine sulfides and selenides

Phosphines and phosphine chalcogenides are important ligands for various metal systems, especially catalysts. Based on previously synthesized phosphine oxide ligands, we describe here the synthesis and characterization of a series of compounds having the general formula, EPPh2Ar, E=S, Se; Ar=1-naphthyl, 9-phenanthryl, 9-anthracenyl, 1-pyrenyl. Reaction of EPPh2(9-phenanthryl), E=S, Se, with HAuCl4 yields ArPh2PEAuCl. Similar to the phosphine oxide derivative, the anthracenyl compounds, EPPh2(9-anthracenyl), showed a nonplanar conformation within the anthracene ring as observed using X-ray crystallography. The difference in fluorescence spectrum relative to the parent aryl species is attributed to this deformation. The deviation is probably due to steric strain within the phosphine ligand due to its position on the anthracenyl ring since DFT calculations suggest that the 1-anthracenyl ligand should be planar. Each compound has been characterized using 1H, 13C, 31P, and (where applicable) 77Se NMR, and IR spectroscopy, with structure determination by X-ray crystallography. Finally, the electrochemistry of the unoxidized phosphines is reported, showing one-electron reductions in the polycyclic aromatic hydrocarbons.

Mechanism‐Guided Engineering of ω‐Transaminase to Accelerate Reductive Amination of Ketones

AbstractAsymmetric reductive amination of ketones using ω‐transaminases (ω‐TAs) offers a promising alternative to the chemocatalytic synthesis of chiral amines. One fundamental challenge to the biocatalytic strategy is the very low enzyme activities for most ketones compared with native substrates (i.e., &lt;1% relative to pyruvate). Here we have demonstrated that a single point mutation in the active site of the (S)‐selective ω‐TA from Ochrobactrum anthropi could induce a remarkable acceleration of the amination reaction without any loss in stereoselectivity and enzyme stability. Molecular modeling of quinonoid intermediates, alanine scanning mutagenesis and kinetic analysis revealed that the W58 residue acted as a steric barrier to binding and catalytic turnover of ketone substrates. Removal of the steric strain by W58L substitution, which was selected by partial saturation mutagenesis, led to dramatic activity improvements for structurally diverse ketones (e.g., 340‐fold increase in kcat/KM for acetophenone). The W58L mutant afforded an efficient synthesis of enantiopure amines (i.e., &gt;99% ee) using isopropylamine as an amino donor.magnified image

Hydrogen Bond Controlled Anti‐Aza‐Michael Addition: Diastereoselective Synthesis of Cyclobutene‐Containing Amino Acid Derivatives

AbstractThe chemical reactivity encoded in a highly substituted cyclobutane platform, which contains the (E,E)‐dimethyl 2,2′‐(cyclobutane‐1,2‐diylidene)diacetate structural motif, has been explored. The cyclobutane platform features a C2 axis of symmetry as well as a dense and interconnected ring functionalization pattern that is defined by two allylic/benzylic stereogenic oxygen‐containing quaternary centers with a 1,2‐trans configuration and two exocyclic acrylate chains. The reactivity profile of the cyclobutane platform is defined by two important kinetic barriers (steric strain and antiaromaticity) and two structure‐biased chemical processes: (1) the thermally‐driven [3,3] sigmatropic rearrangement between one of the two equivalent aryloates and the corresponding allylic acrylate chain and (2) the allylic nucleophilic substitution (SN2′ reaction) that involves a tertiary aryloate and its exocyclic double bond (anti‐Michael addition). The reaction of platform 3a with secondary amines delivered the corresponding cyclobutene amino acid derivatives 15a–15f in excellent yields (up to ≥95 %) and high diastereoselectivities (up to 99:1). Computational studies are described to rationalize the observed diastereoselectivity.

Effect of strain on gas-phase basicity of (E )-1-methyl-2-(1-methyl-2-adamantylidene)adamantane

Steric strain, caused by forced deformation of double bond geometry, has significant impact on alkene reactivity, as was shown by a study of the gas-phase basicity of (E)-1-methyl-2-(1-methyl-2-adamantylidene)adamantane. The Gibbs free energy of the strain effect in this compound was 42 kJ/mol. This analysis was made based on adamantylideneadamantane as a congeneric reference compound with a planar double bond. It was concluded that gas-phase basicity could serve as a valuable alternative indicator for quantification of steric strain in alkenes. Copyright © 2015 John Wiley & Sons, Ltd.

Porphyrin substituent regiochemistry, conformation and packing – the case of 5,10-diphenylporphyrin

Abstract 5,10-Disubstituted porphyrins are more recent additions to the family of meso-substituted porphyrins. A crystallographic comparison of 5,10-diphenylporphyrin with the regioisomeric 5,15-disubstituted system reveals striking differences in their conformation. In the free base porphyrins the former uses mainly out-of-plane distortion to alleviate steric strain while in-plane core elongation predominates in the latter. In contrast, the structure of the Cu(II) complex is planar and forms strong π-π aggregates with very small lateral shifts. Macroscopically, the packing is similar to that of porphyrin sponges of the 5,10,15,20-tetraphenylporphyrin type.

Theoretical insights into the functioning of metallopeptidases and their synthetic analogues.

CONSPECTUS: The selective hydrolysis of a peptide or amide bond (-(O═)C-NH-) by a synthetic metallopeptidase is required in a wide range of biological, biotechnological, and industrial applications. In nature, highly specialized enzymes known as proteases and peptidases are used to accomplish this daunting task. Currently, many peptide bond cleaving enzymes and synthetic reagents have been utilized to achieve efficient peptide hydrolysis. However, they possess some serious limitations. To overcome these inadequacies, a variety of metal complexes have been developed that mimic the activities of natural enzymes (metallopeptidases). However, in comparison to metallopeptidases, the hydrolytic reactions facilitated by their existing synthetic analogues are considerably slower and occur with lower catalytic turnover. This could be due to the following reasons: (1) they lack chemical properties of amino acid residues found within enzyme active sites; (2) they contain a higher metal coordination number compared with naturally occurring enzymes; and (3) they do not have access to second coordination shell residues that provide substantial rate enhancements in enzymes. Additionally, the critical structural and mechanistic information required for the development of the next generation of synthetic metallopeptidases cannot be readily obtained through existing experimental techniques. This is because most experimental techniques cannot follow the individual chemical steps in the catalytic cycle due to the fast rate of enzymes. They are also limited by the fact that the diamagnetic d(10) Zn(II) center is silent to electronic, electron spin resonance, and (67)Zn NMR spectroscopies. Therefore, we have employed molecular dynamics (MD), quantum mechanics (QM), and hybrid quantum mechanics/molecular mechanics (QM/MM) techniques to derive this information. In particular, the role of the metal ions, ligands, and microenvironment in the functioning of mono- and binuclear metal center containing enzymes such as insulin degrading enzyme (IDE) and bovine lens leucine aminopeptidase (BILAP), respectively, and their synthetic analogues have been investigated. Our results suggested that in the functioning of IDE, the chemical nature of the peptide bond played a role in the energetics of the reaction and the peptide bond cleavage occurred in the rate-limiting step of the mechanism. In the cocatalytic mechanism used by BILAP, one metal center polarized the scissile peptide bond through the formation of a bond between the metal and the carbonyl group of the substrate, while the second metal center delivered the hydroxyl nucleophile. The Zn(N3) [Zn(His, His, His)] core of matrix metalloproteinase was better than the Zn(N2O) [Zn(His, His, Glu)] core of IDE for peptide hydrolysis. Due to the synergistic interaction between the two metal centers, the binuclear metal center containing Pd2(μ-OH)([18]aneN6)](4+) complex was found to be ∼100 times faster than the mononuclear [Pd(H2O)4](2+) complex. A successful small-molecule synthetic analogue of a mononuclear metallopeptidase must contain a metal with a strong Lewis acidity capable of reducing the pKa of its water ligand to less than 7. Ideally, the metal center should include three ligands with low basicity. The steric effects or strain exerted by the microenvironment could be used to weaken the metal-ligand interactions and increase the activity of the metallopeptidase.

Tuning the painter's palette: subtle steric effects on spectra and colour in conjugated electrochromic polymers

A series of vibrantly coloured π-conjugated electrochromic polymers (ECPs) were designed and synthesized with the goal of extracting structure–property relationships from subtle changes in steric strain or relaxation.

Steric C-N bond activation on the dimeric macrocycle [{P(μ-NR)}2(μ-NR)]2.

Dimeric cyclophosphazanes [{P(μ-NR)}2(μ-NR)]2 [R = (t)Bu ( 1) and iPr ( 3)] were oxidized with elemental selenium. During these reactions an unexpected C–N bond cleavage and N–H bond formation occurred. Compound 1 produced P4(μ-N(t)Bu)3(μ-NH)3Se4 ( 2) where three tBu groups were lost in the form of isobutylene. In contrast, during the oxidation of the less sterically hindered 3, the resulting product, P4(μ-N(i)Pr)5(μ-NH)Se4 ( 4), showed only one substituent loss. Theoretical studies confirmed the steric nature of the driving force underlying the different outcomes.

Theoretical Study of &amp;sigma;-&amp;pi; Conjugated Semiconductors Based on cis- and trans-Cyclopropane Derivatives

The high strain cyclopropane as σ block model inserted in organic semiconductors to tune the structures and optical properties was investigated by the density functional theory method. The band gaps and absorption peaks of 1,2-disubstituted cyclopropanes were between the corresponding alkanes and olefins. The cis-isomers had higher steric strain between two substituents and lower symmetry than the trans-isomers in ground-state structures. The electronic transitions (S0→S1) were mainly on the aryl parts, which were stronger coherence with surrounding atoms in cis-isomers, while the trans-isomers were on the cyclopropyl. Furthermore, 1,2-disubstituted cyclopropanes (1A, 1B, 2A, 4B) could respond to the charge stimulation, which are potential materials for organic electronics and mechatronics.

Distributed curvature and stability of fullerenes.

Energies of non-planar conjugated π systems are typically described qualitatively in terms of the balance of π stabilisation and the steric strain associated with geometric curvature. Curvature also has a purely graph-theoretical description: combinatorial curvature at a vertex of a polyhedral graph is defined as one minus half the vertex degree plus the sum of reciprocal sizes of the faces meeting at that vertex. Prisms and antiprisms have positive combinatorial vertex curvature at every vertex. Excluding these two infinite families, we call any other polyhedron with everywhere positive combinatorial curvature a PCC polyhedron. Cubic PCC polyhedra are initially common, but must eventually die out with increasing vertex count; the largest example constructed so far has 132 vertices. The fullerenes Cn have cubic polyhedral molecular graphs with n vertices, 12 pentagonal and (n/2 - 10) hexagonal faces. We show that there are exactly 39 PCC fullerenes, all in the range 20 ≤n≤ 60. In this range, there is only partial correlation between PCC status and stability as defined by minimum pentagon adjacency. The sum of vertex curvatures is 2 for any polyhedron; for fullerenes the sum of squared vertex curvatures is linearly related to the number of pentagon adjacencies and hence is a direct measure of relative stability of the lower (n≤ 60) fullerenes. For n≥ 62, non-PCC fullerenes with a minimum number of pentagon adjacencies minimise mean-square curvature. For n≥ 70, minimum mean-square curvature implies isolation of pentagons, which is the strongest indicator of stability for a bare fullerene.

Synthesis of imidazo[1,2-a]pyridin-chromones by a MW assisted Groebke–Blackburn–Bienaymé process

A series of nine novel imidazo[1,2-a]pyridin-chromones and nine novel imidazo[1,2-a]pyridin-6-fluorochromones were synthesized in good to excellent yields by a MW assisted Groebke–Blackburn–Bienaymé reaction (GBB) under mild thermal and acidic conditions. Additionally, optimized geometries for the two non-prepared 5-bromoimidazo[1,2-a]pyridin-chromones were calculated at the TPSSh/6-311G(d) level of theory in order to understand the role of the steric strain in the GBB process.

A spin-crossover catecholato–iron(III) system: Synthesis, crystal structure and magnetic properties

A new mononuclear Fe(III)–catecholate complex, which exhibits an incomplete spin transition without hysteresis, had been synthesized and structurally characterized. Analyzed from the structural characteristics, the tripodal ligand must play the important role in affecting the spin crossover (SCO) behavior. The larger steric strain effect and weak cooperativity existed in the crystal lattice are inclined to favor the incomplete transition.

Probing ternary complex equilibria of crown ether ligands by time-resolved fluorescence spectroscopy.

Ternary complex formation with solvent molecules and other adventitious ligands may compromise the performance of metal-ion-selective fluorescent probes. As Ca(II) can accommodate more than 6 donors in the first coordination sphere, commonly used crown ether ligands are prone to ternary complex formation with this cation. The steric strain imposed by auxiliary ligands, however, may result in an ensemble of rapidly equilibrating coordination species with varying degrees of interaction between the cation and the specific donor atoms mediating the fluorescence response, thus diminishing the change in fluorescence properties upon Ca(II) binding. To explore the influence of ligand architecture on these equilibria, we tethered two structurally distinct aza-15-crown-5 ligands to pyrazoline fluorophores as reporters. Due to ultrafast photoinduced electron-transfer (PET) quenching of the fluorophore by the ligand moiety, the fluorescence decay profile directly reflects the species composition in the ground state. By adjusting the PET driving force through electronic tuning of the pyrazoline fluorophores, we were able to differentiate between species with only subtle variations in PET donor abilities. Concluding from a global analysis of the corresponding fluorescence decay profiles, the coordination species composition was indeed strongly dependent on the ligand architecture. Altogether, the combination of time-resolved fluorescence spectroscopy with selective tuning of the PET driving force represents an effective analytical tool to study dynamic coordination equilibria and thus to optimize ligand architectures for the design of high-contrast cation-responsive fluorescence switches.

Solid-state transformation of nanocrystalline phyllomanganate into tectomanganate: influence of initial layer and interlayer structure.

In surficial environments, the fate of many elements is influenced by their interactions with the phyllomanganate vernadite, a nano-sized and turbostratic variety of birnessite. To advance our understanding of the surface reactivity of vernadite as a function of pH, synthetic vernadite (δ-MnO2) was equilibrated at pH ranging from 3 to 10 and characterized structurally using chemical methods, thermogravimetry and modelling of powder X-ray diffraction (XRD) patterns. With decreasing pH, the number of vacant layer sites increases in the octahedral layers of δ-MnO2 (from 0.14 per layer octahedron at pH 10 to 0.17 at pH 3), whereas the number of layer Mn(3+) is, within errors, equal to 0.12 per layer octahedron over the whole pH range. Vacant layer sites are capped by interlayer Mn(3+) sorbed as triple corner-sharing surface complexes (TC sites). The increasing number of interlayer Mn(3+) with decreasing pH (from 0.075 per layer octahedron at pH 10 to 0.175 at pH 3) results in the decrease of the average Mn oxidation degree (from 3.80 ± 0.01 at pH 10 to 3.70 ± 0.01 at pH 3) and in the lowering of the Na/Mn ratio (from 27.66 ± 0.20 at pH 10 to 6.99 ± 0.16 at pH 3). In addition, in-plane unit-cell parameters are negatively correlated to the number of interlayer Mn at TC sites and decrease with decreasing pH (from b = 2.842 Å at pH 10 to b = 2.834 Å at pH 3), layer symmetry being systematically hexagonal with a = b × 3(1/2). Finally, modelling of X-ray diffraction (XRD) patterns indicates that crystallite size in the ab plane and along the c* axis decreases with decreasing pH, ranging respectively from 7 nm to 6 nm, and from 1.2 nm to 1.0 nm (pH 10 and 3, respectively). Following their characterization, dry samples were sealed in polystyrene vials, kept in the dark, and re-analysed 4 and 8 years later. With ageing time and despite the dry state, layer Mn(3+) extensively migrates to the interlayer most likely to minimize steric strains resulting from the Jahn-Teller distortion of Mn(3+) octahedra. When the number of interlayer Mn(3+) at TC sites resulting from this migration reaches the maximum value of ∼ 1/3 per layer octahedron, interlayer species from adjacent layers share their coordination sphere, resulting in cryptomelane-like tunnel structure fragments (with a 2 × 2 tunnel size) with a significantly improved layer stacking order.

Variations of bistricyclic aromatic enes: mono-bridged tetraarylethene naphthologs

The syntheses, molecular and crystal structures, NMR spectroscopic study, and DFT computational study of naphthologs of mono-bridged (X = –, O, S, Se, and Te) tetraarylethene (BAE-1s) 11–25 with α,α-, β,β-, and α,β-dinaphthalenyl substituents have been reported. The BAE-1s have been prepared by Barton–Kellog twofold extrusion from the respective chalcogenothiones and diazomethylenebisnaphthylenes. Complete assignments of 1H- and 13C-NMR spectra of 11–25 have been made through 2-dimensional correlation spectroscopy (DQF-COSY, HSQC, HMBC, and NOESY). The corresponding intermediates, thiiranes 33–47, have been also isolated (except 38), and their molecular and crystal structures have been determined. The molecular structures of BAE-1s 12–15, 20, and 22–25 adopted folded-twisted conformations with considerably folded (φ = 30°–57°) tricyclic moieties. The α,α- and α,β-dinaphthalenyl derivatives are more overcrowded than β,β-dinaphthalenyl derivatives. The relief of the steric strain due to the overcrowding around C9 = C9′ caused by the presence of naphthalenyl substituents was achieved by their twisting around the single bonds that connect the α-naphthalenyl and β-naphthalenyl moieties to C9′. The 1H-NMR spectra have shown shielding of H2, H7 of 11–25 and the pronounced deshielding of H8′, H8″ of α,α-dinaphthalenyl-substituted BAE-1s 13–15 in contrast to β,β-dinaphthalenyl-substituted BAE-1s 16–20. The upfield shifts of H2, H7 suggested conformations in which these hydrogens are located above the planes of the opposing naphthalene rings. DFT calculations of 11–20 have been performed at B3LYP/6-31G(d) and B3LYP/SDD. The results have shown that the global minima of BAE-1s without a chalcogen bridge 11 and 16 are twisted (–sc,–sc)-C 2-t conformations. The global minima of BAE-1s with a chalcogen bridge are folded-twisted (–sc,–ac)-C 1-ft conformations for α,α-dinaphthalenyl-substituted BAE-1s 12–15 and either anti- or syn-(–sc,ac)-C 1-ft conformations for β,β-dinaphthalenyl-substituted BAE-1s 17–20. The pronounced differences between the α,α-dinaphthalenyl and the β,β-dinaphthalenyl derivatives are noted. Dispersion-corrected B3LYP calculations stabilize significantly the α,α-dinaphthalenyl derivatives versus the β,β-dinaphthalenyl derivatives. The geometrical parameters of BAEs-1 11–15 and 20, derived from their molecular X-ray structures and from their B3LYP-optimized geometries are in a good agreement.

Synthesis of High Molecular Weight Polymethacrylates with Polyhedral Oligomeric Silsesquioxane Moieties by Atom Transfer Radical Polymerization.

A new polyhedral oligomeric silsesquioxane (POSS) methacrylate monomer, 1-(3-(methacryloyloxy)propyl)dimethylsiloxy-3,5,7,9,11,13,15-hepta(isobutyl)pentacyclo-[9.5.1.13,9.15,15.17,13]octasiloxane ((i-Bu)7POSS-OSiMe2-MA, 1), with a flexible spacer between the cubic POSS cage and methacrylate group was synthesized to reduce steric strain and thus achieve polymethacrylates (poly(POSS-MA)s) with high molecular weight (MW). Atom transfer radical polymerization (ATRP) of 1 at high monomer concentration (1 M, corresponding to ca. 85 wt % of 1) led to polymers with the absolute number-average MW, determined by multiangle laser light scattering, Mn,MALLS = 2 350 000 (and apparent MW, measured by gel permeation chromatography with linear poly(methyl methacrylate) (PMMA) standards, Mn,GPC = 550 000). Optimization of the reaction conditions, including the ATRP catalyst, targeted degrees of polymerization, monomer concentrations, as well as a monomer feeding, resulting in the first well-defined high MW polymers with POSS moieties.

Mean bond-length variation in crystal structures: a bond-valence approach.

The distortion theorem of the bond-valence theory predicts that the mean bond length 〈D〉 increases with increasing deviation of the individual bond lengths from their mean value according to the equation 〈D〉 = (D' + ΔD), where D' is the length found in a polyhedron having equivalent bonds and ΔD is the bond distortion. For a given atom, D' is expected to be similar from one structure to another, whereas 〈D〉 should vary as a function of ΔD. However, in several crystal structures 〈D〉 significantly varies without any relevant contribution from ΔD. In accordance with bond-valence theory, 〈D〉 variation is described here by a new equation: 〈D〉 = (DRU + ΔDtop + ΔDiso + ΔDaniso + ΔDelec), where DRU is a constant related to the type of cation and coordination environment, ΔDtop is the topological distortion related to the way the atoms are linked, ΔDiso is an isotropic effect of compression (or stretching) in the bonds produced by steric strain and represents the same increase (or decrease) in all the bond lengths in the coordination sphere, ΔDaniso is the distortion produced by compression and stretching of bonds in the same coordination sphere, ΔDelec is the distortion produced by electronic effects. If present, ΔDelec can be combined with ΔDaniso because they lead to the same kind of distortions in line with the distortion theorem. Each D-index, in the new equation, corresponds to an algebraic expression containing experimental and theoretical bond valences. On the basis of this study, the ΔD index defined in bond valence theory is a result of both the bond topology and the distortion theorem (ΔD = ΔDtop + ΔDaniso + ΔDelec), and D' is a result of the compression, or stretching, of bonds (D' = DRU + ΔDiso). The deficiencies present in the bond-valence theory in explaining mean bond-length variations can therefore be overcome, and the observed variations of 〈D〉 in crystal structures can be described by a self-consistent model.