Anomalous Dispersion Data Research Articles

Unique or essentially unique results from one-wavelength anomalous dispersion data

An experiment that is sensitive to anomalous dispersion effects will produce at one wavelength independent intensity information at a reciprocal-lattice point and its negative. These pairs of intensities are known as Bijvoet pairs. The usual analysis of the implications of Bijvoet pairs leads to the conclusion that they generate a twofold ambiguity in the evaluation of certain phase differences. In this paper, it is shown that additional information contained in the Bijvoet pairs, and not normally used in the analysis leading to the implication of twofold ambiguity, can be used to obtain unique or essentially unique values for the phase differences of interest with potentially useful accuracy. The accuracy, of course, depends upon the accuracy of the data, but a test example has shown considerable insensitivity to such errors. The analysis presented here is based on an exact algebraic analysis of the intensity equations associated with the anomalous dispersion technique. Although the theory is quite general, applying exactly to any number or type of anomalously scattering atoms at any number of wavelengths, the application here concerns the case of one type or one predominant type of anomalously scattering atoms in a one-wavelength experiment. It is noted that in the two equations associated with the Bijvoet pairs there are three unknown quantities. It is shown, however, that the two intensity data provide enough information to evaluate the three unknown quantities to good approximation in an essentially unique fashion, which, in addition, can be effected in a least-squares calculation. The phase information of interest that is obtained concerns the values of phase differences,φ 1,h n −φ 2,h n , between phases associated with the structure of nonanomalously scattering atoms and those associated with the structure of the anomalously scattering atoms, respectively, with all atoms scattering as if there were no anomalous dispersion.

Many algebraic formulas for the evaluation of triplet phase invariants from isomorphous replacement and anomalous dispersion data

An algebraic analysis is presented for the calculation of triplet phase invariants from isomorphous replacement and anomalous dispersion data. The analysis applies when there is one type or one predominant type of anomalously scattering atoms. The use of the formulas largely parallels a recent approach that is based on a General Rule for evaluating triplet phase invariants. It involves the mixing of terms from isomorphous replacement with various types of terms arising in anomalous dispersion or the mixing of various terms arising in anomalous dispersion alone. The mixing of terms gives rise to a myriad of formulas that can generate values anywhere in the range from -π to π. In the tests performed, it was found that the algebraic formulas offered an improvement in accuracy over that obtained from the General Rule. The accuracy is potentially high but depends ultimately on the reliability of the experimental data.

A general rule and many formulas for the evaluation of triplet phase invariants from isomorphous replacement and anomalous dispersion data

On the basis of a common characteristic observed in previously derived formulas for the evaluation of triplet phase invariants from either isomorphous replacement data or anomalous dispersion data, it has been found possible to combine mathematical expressions, certain differences of magnitudes, arising in the analysis of the two techniques to form a myriad of new mixed formulas. The common characteristic is that the various types of differences of magnitudes that are involved in the formulas are all definable in terms of the heavy-atom structure. The formulas involve the mixing of terms arising from several isomorphous derivatives or from a combination of such terms with various types of terms arising in anomalous dispersion or the mixing of various terms arising in anomalous dispersion alone. The evaluation of the triplet phase invariants is facilitated by the use of a simple rule, called the General Rule, that is generally applicable to the case of one predominant type of anomalous scatterer. In the case of more than one predominant type of anomalous scatterer, a slightly more complicated calculation is required and is described. Test calculations show that a very large number of invariants may be evaluated by these means with reliabilities that are potentially high, but depend, of course, on the reliability of the experimental data. A benefit from having the large variety of formulas is that triplet phase invariants can be evaluated at many points throughout the range -π to π and their reliability is enhanced because much information is obtained from only the largest differences of magnitudes.

Rules for evaluating triplet phase invariants by use of anomalous dispersion data

On the basis of some mathematical and physical characteristics of anomalous dispersion experiments, it has been possible to derive simple rules, Rano, 1, Rano, 2 and Rano, 3, that permit the selection of triplet phase invariants, in a single-wavelength experiment, that have values close to π/2, -π/2, 0 and other values in the range from -π to π. The rules, Rano, 1, Rano, 2 and Rano, 3, apply to the case of a single type of predominant anomalous scatterer. The simple generalization to more than one type of predominant anomalous scatterer is also described. Test examples show that large numbers of invariants may be evaluated by these means with reliabilities that are potentially high, but depend, of course, on the reliability of the experimental data. The only information required besides the measurements of the diffraction intensities is the chemical composition of the anomalously scattering atoms. In some cases, even this information is not required if two alternative sets of estimates of the values of the triplet phase invariants are considered.

Structure of alkaline phosphatase with zinc/magnesium cobalt or cadmium in the functional metal sites

Crystals of alkaline phosphatase (EC 3.1.3.1; M r 94,000) grown at pH 9.5 from 2.25 m-(NH 4) 2SO 4 with 5 × 10 −5 m-Zn and 10 −2 m-Mg present were analyzed by X-ray diffraction at pH 7.5 in 2.66 m-(NH 4) 2SO 4 with 10 −2 m-Zn and 10 −2 m-Mg present. The crystals are orthorhombic with a = 195.5 A ̊ , b = 168.3 A ̊ and c = 76.33 A ̊ , and the space group is I222. X-ray phases were determined by the multiple isomorphous replacement and anomalous dispersion method using K 2PtCl 4, KAu(CN) 2 and K 2OsO 4 derivatives. The electron density maps and analysis of metal binding sites reveal one molecule per asymmetric unit with an internal, non-crystallographic, 2-fold rotation axis relating the subunits. Each subunit contains a major α β domain with a seven-stranded β-sheet flanked by helices. The sheets are roughly coplanar but the general direction of the strands in each is at 20 ° to the rotation axis and thus 40 ° from each other. The helical content of the α β domain is approximately 27% of the 459 residues in the monomer and the β content is approximately 7%. The chains in a smaller domain are more convoluted and less easily characterized than in the α β domain. In both there is extensive monomer-monomer contact. Removal of the zinc and magnesium from the parent crystal produces a stable apoenzyme crystal and addition of cobalt at 10 −2 m or cadmium at 10 −2 or 5 × 10 −2 m reveals seven metal binding sites per dimer. The active centers are 32 Å apart and each is shown by anomalous dispersion data to contain two metal binding sites, A and B. The cadmium derivative refinement determined the A-B separation to be 4.9 Å. Comparison of the parent and apo structures by means of difference maps reveals the double metal site with Zn at A and probably Mg at B. A prominent, partially resolved peak centered 7 Å away is interpreted as a stabilization of the backbone in this position by the metal ion co-ordination to a side-chain. Several negative peaks within 10 Å of the metals indicate local differences between apo and native structures but no significant differences are seen in the other parts of the molecule. At 5 × 10 −2 m-Cd two metal sites (D and D′) are found 25.5 Å from the active center, on the surface of the minor domain. They are related to each other by the molecular 2-fold axis with a D-D′ distance of 25 Å. The seventh Cd site, E, is 20 Å from the active center, on the major domain, near a crystalline contact region, and devoid of any molecular symmetry mate. The apparent dissociation constants for cadmium at the A, B and D sites (and A′, B′, D′) are 3 × 10 −3 m, 1.5 × 10 −1 m and 1.3 × 10 −2 m, respectively. Thus in these conditions cadmium is seen to distribute between A and B sites when the combined stoichiometry is two metal ions per dimer.

Three-dimensional fourier synthesis of human deoxyhemoglobin at 2·5 Å resolution: I. X-ray analysis

The calculation of a 2·5 A resolution electron density map for human deoxyhemoglobin is reported as part of a project to obtain refined atomic co-ordinates for the molecule. A new procedure for weighting the calculation of phases in the method of isomorphous replacement is described which separates the effects of statistical errors from those of systematic errors. The new method is shown to improve considerably the contribution of the anomalous dispersion data to the phase calculation.

Appendix I. A comparison of alternative methods for the solution of the structure of factor V 1 a

The successful and rapid elucidation of this structure by the use of the anomalous dispersion data, as described above, does raise the question as to the efficiency of other methods of structure solution in this situation. This has prompted us to apply three such approaches to the data and the results of the comparison we then made are presented below. In view of the assumption that the ambiguity in phase angle implicit in the anomalous data could be removed by choosing values nearer to the phase contribution of the cobalt atom itself, we tried two other means of dealing with these data which did not require such an approximation. These were the β an function of Ramachandran & Raman (1956) and the P s ( u ) function of Pepinsky & Okaya (1956). At the same time a closer study was made of the heavy atom phased synthesis. The results obtained are illustrated by table 7 and figure 11 a to e .

A note on the correlation of the heavy-atom positions in different isomorphous protein crystals using both isomorphous-replacement and anomalous-dispersion data

In this note more accurate expressions are obtained for the Fourier coefficients of the iso-ano correlation function and the origin correlation function of Kartha & Parthasarathy.

The sub-millimetre dispersion, rotational line strengths and dipole moment of gaseous ammonia

Using the technique of Fourier spectrometry the refractive index spectrum of gaseous ammonia has been observed from 15 to 116 cm −1 (667 to 87 μm). The integrated absorption strengths of the rotational lines ( J + 1) ← J for J = 0 to J = 4 have been evaluated from the anomalous dispersion data. A value μ = 1·48 ± 0·06 D has been calculated for the electric dipole moment of the ammonia molecule. In an assessment of accuracy, dipole moment parameters obtained from Fourier dispersion studies of gaseous HCl, HBr and HI are also considered to show that there is no evidence for a systematic error inherent in this method for finding absorption strengths.

Combination of multiple isomorphous replacement and anomalous dispersion data for protein structure determination. III. Refinement of heavy atom positions by the least-squares method

Combination of multiple isomorphous replacement and anomalous dispersion data for protein structure determination. III. Refinement of heavy atom positions by the least-squares method

COMBINATION OF MULTIPLE ISOMORPHOUS REPLACEMENT AND ANOMALOUS DISPERSION DATA FOR PROTEIN STRUCTURE DETERMINATION. I. DETERMINATION OF HEAVY-ATOM POSITIONS IN PROTEIN DERIVATIVES.

COMBINATION OF MULTIPLE ISOMORPHOUS REPLACEMENT AND ANOMALOUS DISPERSION DATA FOR PROTEIN STRUCTURE DETERMINATION. I. DETERMINATION OF HEAVY-ATOM POSITIONS IN PROTEIN DERIVATIVES.

COMBINATION OF MULTIPLE ISOMORPHOUS REPLACEMENT AND ANOMALOUS DISPERSION DATA FOR PROTEIN STRUCTURE DETERMINATION. II. CORRELATION OF THE HEAVY-ATOM POSITIONS IN DIFFERENT ISOMORPHOUS PROTEIN CRYSTALS.

COMBINATION OF MULTIPLE ISOMORPHOUS REPLACEMENT AND ANOMALOUS DISPERSION DATA FOR PROTEIN STRUCTURE DETERMINATION. II. CORRELATION OF THE HEAVY-ATOM POSITIONS IN DIFFERENT ISOMORPHOUS PROTEIN CRYSTALS.