Three-phase Structure Invariants Research Articles

Direct-method phasing of anomalous diffraction from proteins

Abstract In dealing with anomalous diffraction data from proteins, the probability distribution of three-phase structure invariants provides additional information, which can be used to resolve the phase ambiguity intrinsic in single-wavelength anomalous diffraction (SAD) enabling solution of the protein structure. Based on this, direct methods have also been used in phasing multi-wavelength anomalous diffraction (MAD) data leading to better result than that from the conventional MAD phasing.

Direct methods and isomorphous replacement. The triplet invariant estimate when heavy atoms are located.

The probabilistic theory of the three-phase structure invariants for isomorphous pairs has been generalized to the case in which a heavy-atom structure model is available. The rigorous method of joint probability distributions has been applied: it is able to handle errors in measurements and in the heavy-atom structure model, as well as the lack of isomorphism. The conclusive formulas have been successfully applied to experimental data.

On integrating direct methods and isomorphous-replacement techniques: triplet estimation and treatment of errors

The method of joint probability distribution functions has been generalized in order to include and treat different sources of error. The probability distributions of the isomorphous pairs (E(p), E(d)) and of the two triples (E(ph), E(pk), E(ph+k), E(dh), E(dk), E(dh+k)) are obtained, on the assumption that the lack of isomorphism and the errors in measurements cumulate on the E(d) variables. The conditional distributions of the two-phase and the three-phase structure invariants are derived, showing how the reliability of the probabilistic estimates depends on the errors.

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.

Use of quantitative convergent-beam electron diffraction in materials science.

Methods for quantitative convergent-beam electron diffraction are outlined and some results of our applications of convergent-beam electron diffraction are shown, with emphasis on quantitative analysis of crystal structures in materials science. Examples of thickness measurements and determination of lattice parameters are presented. Measurements of low-order structure factors to obtain information on bonding charge-density distributions are reviewed, with examples from TiAl intermetallics. For non-centrosymmetric crystals, a method to determine three-phase structure invariants is given. Determination of polarity is also discussed.

Three-phase structure invariants and structure factors determined with the quantitative convergent-beam electron diffraction method

Quantitative convergent-beam electron diffraction is used to determine structure factors and three-phase structure invariants. The refinements are based on centre-disc intensities only. An algorithm for param-eter-sensitive pixel sampling of experimental intensities is implemented in the refinement procedure to increase sensitivity and computer speed. Typical three-beam effects are illustrated for the centrosymmetric case. The modified refinement method is applied to determine amplitudes and three-phase structure invariants in noncentrosymmetric InP. The accuracy of the results is shown to depend on the choice of the initial parameters in the refinement. Even unrealistic starting assumptions and incorrect temperature factor lead to stable results for the structure invariant. The examples show that the accuracy varies from 1 to 10 degrees in the electron three-phase invariants determined and from 0.5 to 5% for the amplitudes. Individual phases could not be determined in the present case owing to spatial intensity correlations between phase-sensitive pixels. However, for the three-phase structure invariant, stable solutions were found.

Probability distribution of the four-phase structure invariants of isomorphously related structure factors and its applications.

The probability distribution of the four-phase structure invariants (4PSIs) involving four pairs of structure factors is derived by integrating the direct methods with isomorphous replacement (IR). A simple expression of the reliability parameter for 16 types of invariant is given in the case of a native protein and a heavy-atom derivative. Test calculations on a protein and its heavy-atom derivative using experimental diffraction data show that the reliability for 4PSI estimates is comparable with that for the three-phase structure invariants (3PSIs), and that a large-modulus invariants method can be used to improve the accuracy.

Program to generate and evaluate three-phase structure-invariant ω estimates from single-wavelength anomalous dispersion data

A direct-methods computer program for evaluating three-phase structure invariants for single-wavelength anomalous dispersion data is described. The entire program was coded in 570 lines of Fortran, is highly automated and requires a minimal amount of user-supplied input data.

General Expression for Probabilistic Examination of Multiphase Structure Invariants in the Case of a Native Protein and Multiple Derivatives. Application to Estimates of the Three-Phase Structure Invariants

Concise probabilistic formulae with definite crystallographic implications are obtained from the distribution for eight three-phase structure invariants (3PSIs) in the case of a native protein and a heavy-atom derivative [Hauptman (1982). Acta Cryst. A38, 289–294] and from the distribution for 27 3PSIs in the case of a native and two derivatives [Fortier, Weeks & Hauptman (1984). Acta Cryst. A40, 646–651]. The main results of the probabilistic formulae for the four-phase structure invariants are presented and compared with those for the 3PSIs. The analysis directly leads to a general formula of probabilistic estimation for the n-phase structure invariants in the case of a native and m derivatives. The factors affecting the estimated accuracy of the 3PSIs are examined using the diffraction data from a moderate-sized protein. A method to estimate a set of the large-modulus invariants, each corresponding to one of the eight 3PSIs, that has the largest [\Delta] values and relatively large structure-factor moduli between the native and derivative is suggested, which remarkably improves the accuracy, and thus a phasing procedure making full use of all eight 3PSIs is proposed.

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.

The SAS Maximal Principle: a New Approach to the Phase Problem

The phase problem, when single-wavelength anomalous scattering (SAS) diffraction data are available, is formulated as a problem in global optimization. Although the objective function has a myriad of local maxima, its global maxima, never more than two, are readily accessible and easily identified by virtue of their isolation. The ability to determine the global maxima of the objective function represents the latest and most successful attempt to go directly from the known probabilistic estimates of the three-phase structure invariants to the values of the individual phases. The relationship between the maxima of the objective function and the solutions of the newly formulated system of SAS tangent equations plays a key role in this development.

Estimation of the one-phase structure seminvariants in the single isomorphous replacement case: an application of Hauptman's distribution

A method for estimating the one-phase structure seminvariants (OPSSs) having values of 0 or π has been proposed on the basis of the probabilistic theory of the three-phase structure invariants for a pair of isomorphous structures [Hauptman (1982). Acta Cryst. A38, 289–294]. The test calculations using error-free diffraction data of protein cytochrome c550 and its PtCl2−4 derivative show that reliable estimates of a number of the OPSSs can be obtained. The reliability of the estimation increases with the increase of the differences between diffraction intensities of the native protein and its heavy-atom derivative. A means to estimate the parameters of the distribution from the diffraction ratio is suggested.

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 sign ambiguity of triplet phases in nonsystematic many-beam effects in CBED patterns

The possibility to determine not only the magnitude but also the sign of three-phase structure invariants from nonsystematic many-beam effects in convergent-beam electron diffraction (CBED) patterns is discussed. From the full dynamical many-beam intensity expression it is clear that it is a principal difference between equivalent three-beam cases of opposite sign of the triplet phases. However, the difference and thus the ability to distinguish between the two cases depends strongly both on the relative magnitude of the structure factors involved and the specimen thickness for which the actual CBED discs are obtained. The largest differences are obtained for a weakly coupled three-beam case where the intensity in the line of the primary reflection, which in this case coincides with the kinematical two-beam position, has a distinct maximum or minimum at the three-beam condition depending on the sign of the triplet phase. In a strong coupling case where the intensity in the primary-reflection line near the three-beam condition is split into two individual segments, the differences are generally less and are not so obvious and quantitative measurements are necessary to distinguish the two cases of opposite sign of the triplet phases. Calculated examples with respect to a nonsystematic three-beam example in the noncentrosymmetric InP are given.

Phasing macromolecular structures via structure-invariant algebra.

Owing to the breakdown of Friedel's law when anomalous scatterers are present, unique values of the three-phase structure invariants in the whole range from 0 to 2pi are determined by measured values of diffraction intensities alone. Two methods are described for going from presumed known values of these invariants to the values of the individual phases. The first, dependent on a scheme for resolving the 2pi ambiguity in the estimate omega(HK) of the triplet phi(H) + phi(K) + phi(-H-K), solves by least squares the resulting redundant system of linear equations phi(H) + phi(K) + phi(-H-K) = omega(HK). The second attempts to minimize the weighted sum of squares of differences between the true values of the cosine and sine invariants and their estimates. The latter method is closely related to one based on the 'minimal principle' which determines the values of a large set of phases as the constrained global minimum of a function of all the phases in the set. Both methods work in the sense that they yield values of the individual phases substantially better than the values of the initial estimates of the triplets. However, the second method proves to be superior to the first but requires, in addition to estimates of the triplets, initial estimates of the values of the individual phases.

TELS: least-squares solution of the structure-invariant equations

Procedures are described to extract the values of individual phases from estimated structure invariants. The linear-equation and least-squares methods are used as two separate techniques in these procedures. The linear-equation method uses a linearly independent set of equations, with arbitrarily assigned integers, which are sufficient in number to solve for values of an equal number of phases. The least-squares method uses a set of overdetermined equations in which an `integer problem' has to be considered. The assembly of these two techniques with a novel integer-trial-and-error method shows a remarkable ability to overcome the `integer problem'. As a test of the whole procedure, phases were extracted from three-phase structure invariants calculated from the theoretical data for the platinum chloride derivative of cytochrome c550.

About symmetry-inconsistent three-phase structure invariants

In a recent paper [Han & Langs (1988). Acta Cryst. A44, 563-566], the 230 space groups were examined to identify conditions which permit symmetry-inconsistent triplets. A careful analysis of the paper shows that results are incomplete, several wrong statements have been made and that a great deal of literature on the subject has been missed. In the present paper new conditions allowing the existence of symmetry-consistent and -inconsistent triplets are also given.

Direct methods: the identification of space-group-specific inconsistent three-phase structure invariants.

Certain space groups often permit the generation of pairs of triple relationships involving the same three parent reflections in different symmetry forms, giving rise to two equally probable invariant estimates which, because of the space-group symmetry, must disagree by an a priori known phase shift. The 230 space groups have been examined to identify those which permit inconsistent triples, and the complete list which describes the forms of the pair of triples and their phase inconsistency is given.

Determination of structure-factor phase invariants and effective structure factors in non-centrosymmetric crystals

Many-beam diffraction effects in non-centrosymmetric crystals have been studied with emphasis on three-beam interactions and determination of three-phase structure invariants in electron diffraction experiments. The effective structure factor has been determined both by numerical many-beam calculations and from the second Bethe approximation. The dependence of this factor on the phase invariant, the excitation errors and the magnitude of the structure factors involved has been discussed in detail. From the values of the effective structure factors at symmetrical positions on each side of a three-beam condition an asymmetry ratio is introduced. By a comparison of the observed variation in this ratio with theoretical profiles, it has been shown that the magnitude of three-phase invariants can be determined in the non-centrosymmetric case. This method may in principle be applied in any type of electron or X-ray three-beam experiments where variations in the effective structure factor are projected out. An example from electron channelling patterns is given.

Combining direct methods with isomorphous replacement or anomalous scattering data. VI. Incorporation of heavy-atom information into Hauptman's distributions

Heavy-atom information has been incorporated into Hauptman's distributions of three-phase structure invariants for both isomorphous replacement and anomalous scattering cases. Reliable estimates of individual phases can be obtained by introducing the phase doublet expression ϕH = ϕ′H ± |ΔϕH|. A test calculation with error-free data of insulin showed results better than previous methods.