Direct methods: a paradox with regard to the convergence of random phase trials toward solutions

A frustrating observation, based on an R(min) variance analysis within the `shake and bake' framework of direct methods phasing, is described. The variance of R(min) can on occasion identify large subsets of phases that have a significantly lower mean phase error than the entire direct methods phase set of otherwise unsuccessful phasing trials for which the overall phase error occasionally dips below 75 or 80°. This is the first time, other than for a handful of Σ1 phase indications in optimal situations, that a priori phase estimates have been attained for large numbers of E values, prior to solving the structure. Although the a priori variance of R(min) is a useful tool for identifying such phases, the a posteriori phase refinement shifts indicated by its minimum often prevent a successful convergence to the solution. Similar efforts to encourage solution convergences in the realm of real space have also been discouraging.

Structure of gramicidin D-RbCl complex at atomic resolution from low-temperature synchrotron data: interactions of double-stranded gramicidin channel contents and cations with channel wall

Gramicidin D (gD) is a naturally occurring ionophoric antibiotic that forms membrane channels specific for monovalent cations. The crystal structure of the RbCl complex of gD has been determined at 1.14 A resolution from low-temperature (100 K) synchrotron-radiation data with a final R of 16%. The structure was refined with anisotropic temperature factors for all non-H atoms and with partial occupancies for many of them. The asymmetric unit in the crystal contains four crystallographically independent molecules that form two right-handed antiparallel double-stranded dimers. There are seven distinct rubidium-binding sites in each dimeric channel. The occupancy factors of Rb cations are between 0.11 and 0.35 and the total ion contents of the two crystallographically independent channels are 1.59 and 1.22 ions, respectively. Although each channel is 'chemically symmetrical', the side-chain conformations, the distributions of rubidium cations and their binding sites in the two independent channels are not. Cations are 'coordinated' by delocalized pi-electrons of three to five carbonyl groups that together with peptide backbone chains form the gramicidin channel walls. The water:cation ratio in the channel interior is four or five:one, and five or six waters separate Rb cations during their passage through the channel.

Ab initio structure determination and refinement of a scorpion protein toxin

The structure of toxin II from the scorpion Androctonus australis Hector has been determined ab initio by direct methods using SnB at 0.96 A resolution. For the purpose of this structure redetermination, undertaken as a test of the minimal function and the SnB program, the identity and sequence of the protein was withheld from part of the research team. A single solution obtained from 1 619 random atom trials was clearly revealed by the bimodal distribution of the final value of the minimal function associated with each individual trial. Five peptide fragments were identified from a conservative analysis of the initial E-map, and following several refinement cycles with X-PLOR, a model was built of the complete structure. At the end of the X-PLOR refinement, the sequence was compared with the published sequence and 57 of the 64 residues had been correctly identified. Two errors in sequence resulted from side chains with similar size while the rest of the errors were a result of severe disorder or high thermal motion in the side chains. Given the amino-acid sequence, it is estimated that the initial E-map could have produced a model containing 99% of all main-chain and 81% of side-chain atoms. The structure refinement was completed with PROFFT, including the contributions of protein H atoms, and converged at a residual of 0.158 for 30 609 data with F >or= 2sigma(F) in the resolution range 8.0-0.964 A. The final model consisted of 518 non-H protein atoms (36 disordered), 407 H atoms, and 129 water molecules (43 with occupancies less than unity). This total of 647 non-H atoms represents the largest light-atom structure solved to date.

Map self-validation: improved criteria to resolve the SIR or SAS phase ambiguity

A procedure was recently described that used the correlation coefficient (CC) agreement between the observed /F(h)/ and their associated unbiased 'omit map' extrapolated values /X(h)/ from an initial trial map as the basis for resolving the SIR or SAS phase ambiguity. It is noted here that a significant improvement in selectivity can be obtained if this agreement is expressed in terms of the complex-valued F(h) and X(h). A new scheme is outlined to exploit the weighted average of the two SIR or SAS phase choices. This procedure requires six FFTs per phase compared with three for the older method that randomly selected either of the two permitted phase choices from the Argand diagram as starting values. Trial calculations are encouraging for applications as low as 4 A resolution.

Improvement of SAS triple invariant estimates for macromolecular direct-methods phasing

Single-wavelength anomalous dispersion (SAS) data can in principle be phased by direct methods since a priori estimates of the three-phase structure invariants can be computed from these data. The mean phase error of the most reliable triple estimates for a small protein, however, is typically no better than 60 degrees, and does not bode well for applications to larger structures. A procedure is described that can substantially lower the error in these estimates and introduce a larger number of useful triple invariants into the phasing process. The mean phase error of the most reliable triples for a 2.5 A resolution data set from a Pt derivative of a 115-residue protein was reduced from 55 to 25 degrees by this method. It was also possible to identify a significant number of the poorest triple estimates, those with mean phase errors approaching 90 degrees, such that they could be reliably down-weighted or excluded from the phasing process.

Map self-validation: a useful discriminator of phase correctness at low resolution

A new map-validation procedure is based on the correlation-coefficient agreement between the observed structure-factor magnitudes and their extrapolated values from suitably modified electron-density maps from which they have been each in turn systematically excluded. The correlation coefficient tends to a maximum as the phase errors in a map are reduced. This principle was used to resolve the single-wavelength anomalous scattering (SAS) and single-derivative isomorphous replacement (SIR) phase ambiguity for a number of error-free trial structures. Applications employing real data sets tend to be more difficult owing to data incompleteness and errors affecting the construction of the Argand diagram.

Globbic approximation in low-resolution direct-methods phasing

Probabilistic direct-methods phasing theory, originally based on a uniform atomic distribution hypothesis, is shown to be adaptable to a non-uniform bulk-solvent-compensated globbic approximation for protein crystals at low resolution. The effective number n(g) of non-H protein atoms per polyatomic glob increases with decreasing resolution; low-resolution phases depend on the positions of only N(g) = N(a)/n(g) globs rather than N(a) atoms. Test calculations were performed with measured structure-factor data and the refined structural parameters from a protein crystal with approximately 10 000 non-H protein atoms per molecule and approximately 60% solvent volume. Low-resolution data sets with d(min) ranging from 15 to 5 A gave n(g) = ad(min) + b, with a = 1.0 A(-1) and b = -1.9 for the test case. Results of tangent-formula phase-estimation trials emphasize that completeness of the low-resolution data is critically important for probabilistic phasing.

Bulk-solvent correction in direct-methods phasing

It is shown that for crystals of large proteins at low diffraction resolution, with N approximately 10 000 independent non-H protein atoms and d(min) approximately 8 A, a simple bulk-solvent correction yields the Sayre equation in its classical form, F(h) = q summation operator(k)F(k)F(h - k). In the low-resolution protein case, the proportionality factor becomes q = 1/[(<rho(P) > - rho(S))V], where V is the unit-cell volume, rho(S) is the assumed constant electron density in the solvent regions of the crystal and <rho(P)> is the average electron density in the protein regions. The classical form of the tangent formula follows from the bulk-solvent-corrected Sayre equation and its validity at low resolution is verified in empirical calculations.

Gramicidin D conformation, dynamics and membrane ion transport

The linear pentadecapeptide antibiotic, gramicidin D, a heterogeneous mixture of six components, is a naturally occurring product of Bacillus brevis known to form ion channels in synthetic and natural membranes. The conformation of gramicidin A in the solid state, in organic solvents, and in planar lipid bilayers and the relationship between the composition and the conformation of gramicidin and its selective transport of ions across membranes has been the subject of intense investigation for over 50 years. The x-ray crystal structure and nmr solution spectroscopy agree fully with one another and reveal that entirely different conformations of gramicidin are present in uncomplexed and ion complexed forms. Precise refinements of the three-dimensional structures of naturally occurring gramicidin D in crystals obtained from methanol, ethanol, and n-propanol demonstrate the unexpected presence of stable left-handed antiparallel double-helical heterodimers that vary with the crystallization solvent. The side chains of Trp residues in the three structures exhibit sequence-specific patterns of conformational preference. Tyr substitution for Trp at position 11 appears to favor beta ribbon formation and stabilization of the antiparallel double helix. This conformation acts as a template for gramicidin folding and nucleation of the different crystal forms. The fact that a minor component in a heterogeneous mixture influences aggregation and crystal nucleation has potential applications to other systems in which anomalous behavior is exhibited by aggregation of apparently homogeneous materials, such as the enigmatic behavior of prion proteins. The crystallographically determined structures of cesium, potassium, rubidium, and hydronium ion complexes of gramicidin A are in excellent agreement with the nmr structure determination of the cesium ion gramicidin complex in a methanol chloroform mixture (50 : 50). The right-handed antiparallel double stranded double helical structures (DSDHR) also exhibit geometric features compatible with the solid-state 15N and 2H nmr data recorded for gramicidin in planar lipid bilayers and attributed to the active form of gramicidin A. The DSDHR crystal structures reveal an ion channel with a single partially solvated cation distributed over three ion binding sites. The channel lumen is relatively smooth and electrostatically negative as required for cation passage, while the exterior is electrostatically neutral, a requirement for membrane insertion. The "coordination" of the Cs+ ion is achieved by interaction with the pi orbitals of the carbonyls which do not point toward the ions. The K+ binding sites, which are similar in position to Cs+ binding sites, are shifted off center slightly toward the wall of the channel.

Progress on the direct-methods solution of macromolecular structures using single-wavelength anomalous-dispersion (SAS) data

In the past few years, a number of strategies have been outlined to resolve the SAS phase ambiguity given that unique estimates omega (h, k) of the triple invariants are available. A new least-squares method is described that can in principle resolve the phase ambiguity to determine macromolecular phases provided that omega (h, k) estimates are unbiased. Limitations of the method in practical applications are discussed. An example is given where the correct solution can be identified by use of the SAS tangent formula in the instance that traditional SAS phasing methods have lead to an incorrect heavy-atom substructure.

On 'globbicity' of low-resolution protein structures

Using Harker's [Harker (1953). Acta Cryst. 6, 731-736] idea of spherically averaged polyatomic groups or 'globs' as the units of structure suitable for analyzing low-resolution diffraction data from protein crystals, 'globbic' scattering factors have been calculated for main-chain peptide units and amino-acid side-chain groups to 3 A resolution via Debye's [Debye (1915). Ann. Phys. (Leipzig), 46, 809-823] scattering formula. It is shown that the scattering factors are insensitive to intra-globbic conformational variation and can be approximated fairly well by a single-Gaussian formula, i.e. fg(s) = Zg exp(-1.7Zgs2), where s = (sin theta)/lambda and Zg is the total electron count for the atoms of the glob. Phase errors due to the globbic approximation and their effect on electron-density maps at 3.5 A resolution have been assessed via calculations for the crambin structure; this analysis indicates that the globbic scattering factors will be useful in efforts to develop procedures for direct-methods phasing of diffraction data to approximately 3.5 A resolution from protein crystals.

Heterodimer formation and crystal nucleation of gramicidin D

The linear pentadecapeptide antibiotic gramicidin D is a heterogeneous mixture of six components. Precise refinements of three-dimensional structures of naturally occurring gramicidin D in crystals obtained from methanol, ethanol, and n-propanol demonstrate the unexpected presence of stable left-handed antiparallel double-helical heterodimers that vary with the crystallization solvent. The side chains of Trp residues in the three structures exhibit sequence-specific patterns of conformational preference. Tyr substitution for Trp at position 11 appears to favor beta ribbon formation and stabilization of the antiparallel double helix that acts as a template for gramicidin folding and nucleation of different crystal forms. The fact that a minor component in a heterogeneous mixture influences aggregation and crystal nucleation has potential applications to other systems in which anomalous behavior is exhibited by aggregation of apparently homogeneous materials, such as the enigmatic behavior of prion proteins.

The conducting form of gramicidin A is a right-handed double-stranded double helix

The linear pentadecapeptide antibiotic, gramicidin D, is a naturally occurring product of Bacillus brevis known to form ion channels in synthetic and natural membranes. The x-ray crystal structures of the right-handed double-stranded double-helical dimers (DSDH) reported here agree with 15N-NMR and CD data on the functional gramicidin D channel in lipid bilayers. These structures demonstrate single-file ion transfer through the channels. The results also indicate that previous crystal structure reports of a left-handed double-stranded double-helical dimer in complex with Cs+ and K+ salts may be in error and that our evidence points to the DSDH as the major conformer responsible for ion transport in membranes.

Reinvestigation of the use of Patterson maps to extrapolate data to higher resolution

Many years ago, Karle & Hauptman proposed that the Patterson function could be used for data extrapolation beyond the observed range of the actual measured data. Few people have subsequently attempted to exploit this interesting idea, which might suggest possible limitations of this method, even in structural applications of modest complexity. This appears not to be the case, however, but the original ideas for implementing the extrapolation can be significantly improved. New calculation protocols indicate that Patterson maps may be used to extend observed data sets from 1.0 to approximately 0.5 A resolution with reasonably good precision. Correlation coefficients between the extrapolated F(hkl)'s and their structure-computed expected values typically range between 0.40 and 0.70 across the unobserved range, even for structures containing as many as 600 non-H light atoms in the asymmetric unit. The method is equally good at extrapolating F values for small zones of data that may not have been recorded within the observed resolution range of the diffraction experiment. Furthermore, triplet phase invariants that incorporate one or two extrapolated terms are nearly as reliable as those formed using only the observed data.

The X-ray structure of the monoclinic crystal form of [D-Hyi2, L-Hyi4] meso-valinomycin

The conformation and intermolecular association of [D-Hyi2, L-Hyi4] meso-valinomycin [cyclo[-D-Val-D-Hyi-L-Val-L-Hyi-(D-Val-L-Hyi-L-Val-D-+ ++Hyi)2-], C60H102N6O18] in a crystal form obtained from ethanol solution has been determined by x-ray crystallography. Two depsipeptides and one ethanol molecule per asymmetric unit crystallize in space group P2(1) (Z = 4); a = 14.579, b = 39.795, c = 13.928 A, beta = 116.90, Rl = 0.0757. The molecular conformation is very similar to that observed in the trigonal P3(2) crystal form obtained from acetone solution [V. Z. Pletnev et al. (1991) Biopolymers, Vol. 31, pp. 409-415]. Both independent molecules in the crystal adopt a similar distorted bracelet structure with a sterically inaccessible, partially formed, ion-binding center that is stabilized by six 4-->1 type H bonds. The observed conformation accounts for the inability of the molecule to complex ions. Close examination of the three crystallographically independent molecules reveals that differences in the backbone conformation associated with solvent interaction are significantly larger than those associated with hydrophobic van der Waals interactions of crystal packing.

The crystal and molecular structure of a valinomycin analogue cyclo[(D-Val-L-Lac-L-Ala-D-Hyi)2(D-Val-L-Lac-L-Val-D-Hyi)]. H2O(C50H82N6 O18.H2O)

The crystal and molecular structure of the valinomycin analogue, cyclo[(D-Val-L-Lac-L-Ala-D-Hyi)2(D-Val-L-Lac-L-Val-D-Hyi)] has been solved by x-ray direct methods using the "Shake and Bake" procedure. The crystals, grown from a mixture of octane/CH2Cl2, belong to space group P2(1) (Z = 4) with cell parameters a = 10.29, b = 32.08, c = 18.73 A, beta = 97.05 degrees, and contain two molecules per asymmetric unit. After anisotropic refinement the standard reliability factor was Rl = 0.058. The conformations of both independent molecules is similar to that observed for isoleucinomycin, cyclo[-(D-Ile-L-Lac-L-Ile-D-Hyi)3] [V. Z. Pletnev et al. (1980) Biopolymers, Vol. 19, pp. 1517-1534]. The structure has an asymmetric conformation stabilized by six intramolecular H bonds, five bonds being of the 4-->1 type and one bond being of the 5-->1 type. One water molecule is caged in the internal cavity of each cyclodepsipeptide. This conformation could represent an intermediate state between free and complexed forms of valinomycin.

On integrating the techniques of direct methods with anomalous dispersion. IV. A simplified perturbation treatment for SAS phasing

Results from probabilistic theory for the single-wavelength anomalous-scattering (SAS) Friedel pair, two-phase structure invariants, psi H = phi H + phi-H, are used to show that the SAS three-phase structure invariants, psi HK = phi H + phi K + phi-H-K, tend to positive values that are easily estimated. Appropriate averages of the estimates provide SAS perturbation corrections in the form of positive origin shifts for the probability distribution of psi HK values and for the tangent formula. The theoretical probabilistic results are verified by empirical statistical analyses of model-calculated phases and experimentally measured structure-factor magnitudes for a small-molecule and a protein crystal structure.

Statistical Expectation Value of the Debye–Waller Factor and E(hkl) Values for Macromolecular Crystals

If the unit-cell distribution of atomic mean-square displacement parameters B = 8pi(2)<u(2)> is assumed to be normal, with mean micro = <B> and variance sigma(2) = <(B-<B >)(2)>, the statistical expectation value of the Debye-Waller factor W(2) = exp(-2Bs(2)), where s = (sin theta)/lambda, is <W(2)> = exp[-2( micro - sigma(2)s(2))s(2)]. This result has been incorporated into procedures for scaling and normalizing measured Bragg intensities to their Wilson expectation values. The procedures can determine both isotropic micro (B) and sigma(B) and anisotropic micro (U(ij)) and sigma(U(ij) distribution parameters. Tests with experimental data and refined structural models for several protein crystals show that the procedures yield reliable normalized structure-factor amplitudes for direct-methods applications, with values of R = summation operator (h)||E(o)| - |E(c)||/ summation operator (h)|E(o)| averaging approximately 5%.

The Crystal Structure of cis-Inositol Monohydrate—Un Objet Retrouvé

cis-Inositol monohydrate, C6H12O6.H2O, crystallizes in the monoclinic space group P 21/n [a 9.900(8), b 9.296(8), c 17.795(15) Ǻ, β 90.5(1)°, Z 8]. The normalized structure factors Eh have an atypical statistical distribution, and attempts to solve the structure by direct methods (triplet relationships) were unsuccessful. The structure was ultimately solved by Patterson and Fourier methods, and was refined by full-matrix least squares [Rw = 0.047 for 1665 independent reflections ≥2σ(Imin)]. The cis-inositol molecules have approximately trigonal symmetry, as expected. The difficulties encountered during the structure analysis are explained by the presence of two nearly identical molecules of high symmetry in the asymmetric unit. The independent molecules are related by translational pseudosymmetry, and their orientations are such that all the C-C and C-O bonds in the structure are approximately parallel to a small number of directions.

Molecular structure and mechanisms of action of cyclic and linear ion transport antibiotics

Ionophores are antibiotics that induce ion transport across natural and artificial membranes. The specific function of a given ionophore depends upon its selectivity and the kinetics of ion capture, transport, and release. Systematic studies of complexed and uncomplexed forms of linear and cyclic ionophores provide insight into molecular mechanisms of ion capture and release and the basis for ion selectivity. The cyclic dodecadepsipeptide valinomycin, cyclo[(-L-Val-D-Hyi-D-Val-L-Lac)3-], transports potassium ions across cellular membrane bilayers selectively. The x-ray crystallographic and nmr spectroscopic data concerning the structures of Na+, K+, and Ba+2 complexes are consistent and provide a rationale for the K+ selectivity of valinomycin. Three significantly different conformations of valinomycin are observed in anhydrous crystals, in hydrated crystals grown from dimethylsulfoxide, and in crystals grown from dioxane. Each of these conformations suggests a different mechanism of ion capture. One of the observed conformations has an elliptical structure stabilized by four 4<--1 intramolecular hydrogen bonds and two 5<--1 hydrogen bonds. Ion capture could be readily achieved by disruption of the 5<--1 hydrogen bonds to permit coordination to a potassium ion entering the cavity. The conformation found in crystals obtained from dimethyl sulfoxide is an open flower shape having three petals and three 4<--1 hydrogen bonds. Complexation could proceed by a closing up of the three petals of the flower around the desolvating ion. In the third form, water molecules reside in the central cavity of a bracelet structure having six 4<--1 hydrogen bonds. Two of these bracelets stack over one another with their valine-rich faces surrounding a dioxane molecule. The stacked molecules form a channel approximately 20 A in length, suggesting that under certain circumstances valinomycin might function as a channel. A series of analogues of valinomycin differing in ring composition and size have been synthesized and their transport properties tested. Peptide substitution and chiral variation in the dodecadepsipeptide can result in stabilization or modification of the different conformers. While contraction of the ring size results in loss of ion transport properties, expansion of the ring size permits complexation of larger ions and small positively charged molecules. Gramicidin A is a pentadecapeptide that functions as a transmembrane channel for transporting monovalent cations. Crystal structures of the cesium chloride complex and two uncomplexed forms of gramicidin A have been reported. In all three structures the gramicidin A molecule is a left-handed, antiparallel, double-stranded helical dimer. In the cesium complex the beta 7.2-helix has 6.4 residues per turn with an internal cavity large enough to accommodate cesium ions. In the uncomplexed structures the channel is 31 A long and has 5.6 amino acids per turn. Because the helix is too tightly wound to permit ion transport, ion transport would require breaking and reforming of hydrogen bonds.

[Crystalline and molecular structure of the K+-complex of meso-valinomycin, cyclo(-(D-Val-L-Hyi-L-Val-D-Hyi)3-).KAuCl4]

Crystal structure of the complex of meso-valinomycin with KAuCl4 (C60H102N6O18KAuCl4) was determined using direct X-ray diffraction analysis. The conformational state of the complex is similar to that determined earlier for free meso-valinomycin. Characteristic of it is the centrosymmetric bracelet shape stabilized by six intramolecular NH...OC hydrogen bonds of 4 --> 1 type. The K+ ion is located in an inner negatively charged octahedral cavity formed by six carbonyl oxygen atoms of ester groups. The observed differences in conformational angles of the complex and free are caused by readjustment of the geometry of the ion-binding cavity to the size of the ion bound during complexation.

Use of 'random-atom" phasing models to determine macromolecular heavy-atom replacement positions

A procedure is described by which the phase-invariant translation function may be used to provide starting set phases for the minimal function that are significantly better than those generated from random-atom coordinates. Applications to determine the heavy-atom positions for single isomorphous replacement data from macromolecular structures are very encouraging. In the case of chiral space groups, e.g. P4(1)2(1)2 versus P4(3)2(1)2, unlike Patterson functions, these methods provide the correct enantiomorph coordinates for the heavy-atom sites for whichever space group is chosen.

Crystal and molecular structure of the centrosymmetric meso-valinomycin analogue--cyclo (D-Val-D-Hyi-D-Val-L-Hyi-L-Val-D-Hyi-L-Val-L-Hyi-L-Val-D-Hyi-D-Val-L-Hy i) (C60H102N6O18)

The crystal structure of cyclo (D-Val-D-Hyi-D-Val-L-Hyi-L-Val-D-Hyi-L-Val-L-Hyi -L-Val-D-Hyi-D-Val-L-Hyi).2H2O has been solved by x-ray direct methods. The crystals (grown from a mixture of octane/CH2Cl2) are an orthorhombic, centrosymmetric space group Pbca, cell parameters a = 11.458 (2), b = 25.613 (3), c = 23.691 (3) A, Z = 4; therefore the molecule lies on a center of inversion in the cell. The atomic coordinates for the C, N, and O atoms were refined in the anisotropic thermal motion approximation (allowing for H-atom contribution to Fcal) to a standard R-factor value of 0.081. In contrast to meso-valinomycin, the analogue under study does not adopt an octahedral cage bracelet conformation. It has an unusual centrosymmetric elongated form with two type II terminal beta-bends formed by N-H ... C=O 4-->1 type intramolecular H bonds. Two symmetry-related water molecules reside in the elongated molecular cavity of the centrosymmetric depsipeptide ring.

Use of globic scattering factors for protein structures at low resolution

At 3 to 4 A resolution, the electron density of a protein may be modeled by a continuous chain of 'globs' representing the amide region of the peptide backbone and the side-chain residues. Group scattering factors are derived from a trans planar C alpha C = ONC alpha backbone segment and most favored side-chain conformer for 18 different amino acids. Trial calculations indicate that the phase error and crystallographic residual comparing the atomic and 'globic' models rapidly decrease from high to low resolution. At 3 A resolution, the phase error is approximately 80 degrees. These results indicate that the electron density of a protein composed of N amino acid residues may be adequately modeled by 2N globs at low resolution.

Efficient methods for the linearization and solution of phase-invariant equations

This paper describes a linear least-squares procedure, whereby, through quadrupole relationships, the 2 pi integers that linearize sets of unique phase-invariant estimates can be determined. It is subsequently shown that the phase solutions for these linear equations can be obtained, even for basis sets of thousands of phases, without having to either build or invert the full least-squares matrix. The final r.m.s. phase errors achieved by this method can typically be less that 5 or 10 degrees.

TDSIR phasing: direct use of phase-invariant distributions in macromolecular crystallography

A new strategy for employing three phase triples invariant estimates from Hauptman's single isomorphous replacement (SIR) and anomalous dispersion (SAS) joint probability distribution formulae is outlined which produces a single unique phase-invariant solution in the case where the positions of the heavy-atom scatterers is known. A similar but non-identical result is obtained for the phase invariants of a structure for which a molecular-replacement solution has been obtained. It is important to note that the values of the individual native/derivative phases can be determined directly from the probability distribution formulae without having to utilize the phase-invariant estimates in an active way. Elimination of the multisolution aspect of utilizing phase-invariant estimates should have important repercussions with regard to phasing macromolecular sets of derivatized data. Trial calculations based on experimentally measured 2.5 A data for three derivatives of cytochrome c550 are encouraging. The average of the three SIR maps resolves a number of structural ambiguities seen in the published multiple isomorphous replacement (MIR) map obtained from eight derivatives.

Use of the minimal function for partial structure development in direct methods

The shake-and-bake procedure, which is based on the minimal function, has been tested and shown to be extremely effective in molecular-fragment recycling applications. Correctly positioned fragments as small as 5% of the scattering power of the structure typically have a 50% chance of producing a solution in a single recycling trial. While starting models for tangent-formula recycling methods normally require an average r.m.s. displacement error of less than approximately 0.25 A from the refined structure to ensure an adequate chance of success, the shake-and-bake method often tolerates r.m.s. model errors well in excess of 0.5 A. Tests indicate that the new method can outperform traditional tangent-formula procedures in difficult structural applications involving multiple copies of pseudosymmetrically related molecules or low-resolution data.

Crystal structure of cholesteryl butanoate at 123 K

Cholesteryl butanoate has a complex crystal structure that differs from those of the three main structure type for cholesteryl esters. It contains four molecules (C31H52O2) unrelated by crystal symmetry. The molecules are packed in almost planar sheets and have molecular long axes nearly parallel. However, the molecules have different orientations about their long axes and furthermore, in a given sheet, one of the independent molecules is antiparallel to the other three. Viewed down the molecular long axes, each molecule has six nearest neighbors, but the detailed environment is different for the four independent molecules. Thus the molecular arrangement has features that are characteristic of the short-range order present in the cholesteric mesophase. The monotropic transformation from the crystalline to the cholesteric phase occurs at 98 degrees C. The crystal structure has been accurately determined using 12,146 independent X-ray reflections having sin theta/lambda < 0.63 A-1. All hydrogen atoms were located from a difference Fourier and were included in a refinement that gave R(F2) = 0.064. The C-C bond lengths have sigma = 0.003 A and C-C-C bond angles have sigma = 0.2 degrees. Conformations for the steroid ring system are similar but there are differences in the C17 side chains and the butanoate chains of the four independent molecules. Analysis of atomic m.s. displacement tensors using a segmented-body model indicates that there are internal librations involving both the C17 and butanoate chains in all molecules.

C-glycosyl bond conformation in oxazofurin: crystallographic and computational studies of the oxazole analogue of tiazofurin

Oxazofurin is the inactive oxazole analogue of the C-glycosyl thiazole antitumor agent tiazofurin. Replacement of the thiazole sulfur in tiazofurin with the oxazole oxygen in oxazofurin produces conformational effects that are examined using crystallographic and computational methods. The crystal structure of oxazofurin contains six molecules in the asymmetric unit and has been refined to a standard R value of 6.8% for all data. The six oxazofurin conformers show an average C-glycosidic torsion angle of 70(9) degrees. This value is significantly higher than the average absolute C-glycosidic torsion angle of 24(10) degrees obtained from previous thiazole nucleoside structures. Previous studies suggest that, in tiazofurin, an electrostatic interaction between a positively charged thiazole sulfur and negatively charged furanose oxygen constrains the C-glycosidic torsion angle to a relatively small value. Ab initio molecular orbital studies presented here suggest that the higher C-glycosidic angles observed in the oxazofurin structures result from a repulsive interaction between negatively charged oxazole and furanose oxygens. Thus, it is likely that differences in activity between oxazo- and tiazofurin are either (1) due directly to differences in electronic properties between the thiazole and oxazole rings or (2) due to the variation in C-glycosidic bond conformation resulting from the alteration in the charge distribution of the heterocycle.

Molecular structures of two crystalline forms of the cyclic heptapeptide antibiotic ternatin, cyclo[-beta-OH-D-Leu-D-Ile-(NMe)Ala-(NMe)Leu-Leu-(NMe)Ala-D-(NMe)Ala-]

The crystal structures of two solvated forms of ternatin, cyclo[-beta-OH-D-Leu-D-Ile-(NMe)Ala-(NMe)Leu-Leu-(NMe)Ala-D-(NMe)Ala-] are reported. The first crystallizes with two molecules of peptide and one of dioxane in the asymmetric unit: P2(1)2(1)2(1), a = 11.563(1), b = 21.863(2), c = 36.330(4) A. The second crystallizes with two molecules of peptide and one of water in the asymmetric unit: P2(1)2(1)2(1), a = 14.067(2), b = 16.695(1), c = 36.824(6) A. N-Methylation of four of the seven residues of ternatin appears to reduce the number of low-energy conformations the molecule can assume. The same H-bonded macrocyclic ring conformation is adopted by the backbone of each of the four molecules observed here. All the amino-acid side chains, with the exception of D-Ile2, have similar orientations in each of the four conformers. The heptapeptide macrocycle is characterized by: (i) a cis peptide between (NMe)Ala3 and (NMe)Leu4, (ii) a type II beta-bend, involving residues Leu5-(NMe)Ala6-D-(NMe)Ala7-beta-OH-D-Leu1, stabilized by two H-bonds, N1-->O5 and N5-->O1, between Leu5 and beta-OH-D-Leu1 residues, (iii) a third intramolecular H-bond, observed in each of the four molecules, between the hydroxyl group of beta-OH-D-Leu1 and the carbonyl oxygen of D-Ile2.

Frequency statistical method for evaluating cosine invariants of three-phase relationships

A new variation on the established procedure to evaluate three-phase structure invariants through quadrupole relationships is described. This method differs from earlier algebraic formulations in that the cosine-invariant estimates are based on a conditional observed frequency distribution of magnitude of E magnitudes for the quadrupole, rather than on the values of the magnitudes themselves. Successful applications of this method to a number of structures that ranged in size from 84 to 317 independent non-hydrogen light atoms are given.

On the application of the minimal principle to solve unknown structures

The Shake-and-Bake method of structure determination is a new direct methods phasing algorithm based on a minimum-variance, phase invariant residual, which is referred to as the minimal principle. Previously, the algorithm had been applied only to known structures. This algorithm has now been applied to two previously unknown structures that contain 105 and 110 non-hydrogen atoms, respectively. This report focuses on (i) algorithmic and parametric optimizations of Shake-and-Bake and (ii) the determination of two previously unknown structures. Traditional tangent formula phasing techniques were unable to unravel these two new structures.

Ab initio direct methods: practical advice for getting beyond the first 300 atoms

Not all crystallographic structural investigations are amenable to a phasing solution by direct methods alone. Guideline procedures are outlined which are intended to help the evaluation of whether direct-methods procedures may be expected to phase diffraction data for large molecular structures. This analysis is directed at three separate levels of inquiry: (1) How good are the primary data and can E values be derived to represent a point-atom structure. (2) How well do the data interact through phase relationships and may they be expected to produce a stable phasing solution. (3) What is the prognosis for finding recognizable solutions. Data are presented from the post-mortem analyses of a number of large, difficult-to-solve, structures to illustrate each of these points. Direct-methods practitioners are to be encouraged that crystal structures having more than 300 atoms per asymmetric unit may occasionally be determined utilizing present methodologies provided that an a priori prognosis for obtaining a solution is favorably high, adequate computational resources are available, and sufficient persistent effort is applied.

Molecular structure and mechanisms of action of cyclic and linear ion transport antibiotics

As a direct result of the vision, determination, and magnetic personality of Yuri Ovchinnikov a collaboration between the Shemyakin Institute of Bioorganic Chemistry and the Medical Foundation of Buffalo was begun in the early 1970's. The collaboration generated valuable insight into the structural basis for the capture, transport, and release of ions by ion transport antibiotics and the basis for the ion selectivity of these compounds. The collaboration produced dozens of joint publications on the structure and function of cyclic and linear ion transport antibiotics, fostered fruitful exchange visits between scientists in the two Research Institutes and has been a major source of creativity in my scientific career and those of many of my colleagues in Buffalo. This review summarizes major accomplishments of the collaboration.

Molecular structure of cyclo[-(D-Val-L-Hyi-L-Val-D-Hyi)2-] revealed by x-ray analysis

The crystal structure of a synthetic analogue of valinomycin, cyclo[-(D-Val-L-Hyi-L-Val-D-Hyi)2-] (octa-meso-valinomycin) (I) (C40H68N4O12.1.5.C4H8O2, M(r) = 937.01 + 88.10), has been determined. Crystals grown from dioxane are monoclinic, space group P2(1)/a, with cell parameters a = 21.487 (8), b = 16.836 (5), c = 16.089 (4) A, beta = 111.70 (4), and Z = 4. The atomic coordinates for nonhydrogen atoms were refined in the anisotropic thermal motion approximation. H atom positions were included in the structure factor calculations at their geometrically expected positions. Values of the standard and weighted R factors after refinement are 0.11 and 0.13, respectively. The conformation of the depsipeptide crystallized from dioxane is different from that crystallized from chloroform (II). The molecule adopts a rectangular shape with two type IV beta-turns containing a hydrogen bond and possesses pseudorotational symmetry. The side chains are located on the molecular periphery. The orientation of the carbonyl groups of the molecule is not conducive for efficient metal-ion coordination and in the observed conformation cannot behave as an ionophore. In the crystal the molecules form infinite chains parallel to the c axis, and are stabilized by two intermolecular hydrogen bonds that are shorter and have better geometry than the intramolecular hydrogen bonds. A phi/psi plot for dodecadepsipeptides with a (DLLD)3 sequence has well-defined areas for Val and Hyi residues only in cases when the crystals have been grown from nonpolar or medium-polar solvents. The phi/psi plot for octadepsipeptides crystallized from chloroform (II) shows this behavior also.(ABSTRACT TRUNCATED AT 250 WORDS)

Crystal and molecular structure of the depsipeptide ionophore hexadecaisoleucinomycin, cyclo-[(D-Ile-L-Lac-L-Ile-D-Hyi)4-] (C80H136N8O24)

The crystal structure of a synthetic depsipeptide ionophore hexadecaisoleucinomycin, cyclo [-(D-Ile-L-Lac-L-Ile-D-Hyi)4-] (C80H136N8O24), has been determined by single crystal x-ray diffraction techniques. The crystals are orthorhombic, space group P2(1)2(1)2(1), number of molecules per unit cell z = 4, and cell parameters a = 11,195, b = 17.853, c = 54.835 A. The values of the standard (R) and weighted (Rw) discrepancy factors after refinement are 0.122 and 0.135, respectively. The structure is characterized by an elongated bracelet form with a twofold axis of pseudosymmetry. It is stabilized by eight intramolecular 4----1 hydrogen bonds between the amide C = O and N - H groups. The ester carbonyls are directed toward the inside of the molecule, their oxygen atoms forming an ellipsoidal internal cavity. The side chains are located on the molecular periphery. The conformational states of hexadecaisoleucinomycin in solution are discussed in the light of the data obtained.

Crystal structure of valinomycin-monohydrate cage complexes crystallized from dioxane

Valinomycin, cyclo-[(L-Val-D-Hyv-D-Val-L-Lac)3-], was crystallized from aqueous dioxane solvent as a monohydrate complex in which water molecules were found within the ion-binding cavity of the ionophore: monoclinic P2(1), a = 14.377 (3), b = 41.554 (14), c = 14.080 (3) A, beta = 118.27 (2) degrees, Z = 4. There are two non-equivalent valinomycin-water complexes and three dioxane molecules in the asymmetric unit. The ionophore molecules adopt two similar but non-identical, octahedral, bracelet, cage conformations that are a consequence of two distinct ways in which the complexed water molecules can deform the normal octahedral coordinate geometry of the metal binding site. In the first complex the water molecule forms hydrogen donor bonds to the carbonyl oxygens of two L-valine residues on one facial side of the cavity, while in the second complex the water molecule is trigonal-planar coordinate and binds to two L-valine residues on one entrant face of the cavity plus a third D-valine residue from the opposite side of the cavity.

An efficient molecular-replacement translation function based on the evaluation of direct-methods phase invariants

Traditional molecular-replacement translation functions are based on direct- or reciprocal-space correlations between the observed diffraction amplitudes and the calculated amplitudes and phases of the symmetry-related molecular transforms of the search fragment as a function of the displacement vector. An alternative method that has been described is based on evaluating a list of phase invariants as a function of the position of the search model in the unit cell and seeking those regions which satisfy the expectation value of these invariants as predicted by probability theory. As originally formulated, this procedure required the iterative computation of the phases and the evaluation of the list of invariants as the search model was stepped over the grid points defining the asymmetric portion of the unit cell. A new computational procedure is described whereby the values of the invariants are expressed solely as a function of the displacement vector r as a Fourier series that can be evaluated by a standard fast Fourier transform (FFT) without having to compute and insert the values of the phases based on the search model at each grid point.

The direct determination of phase invariants provided by diffraction data measured at two different temperatures

A procedure is described for the determination of the crystal structure phase invariants of a compound based on diffraction data measured at two different temperatures. This temperature difference replacement (TDR) technique is shown to provide phase-invariant information from experimentally measured X-ray diffraction data for two different test structures. Although the new method does not appear to be as powerful as single-derivative isomorphous replacement (SIR) phasing, it does appear to be capable of reliably determining a limited number of negative as well as positive phase-restricted invariants for structures containing as many as 300 non-H atoms in the asymmetric unit.