Dongyao Guo

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) is assumed to be normal, with mean micro = and variance sigma(2) = <(B-)(2)>, the statistical expectation value of the Debye-Waller factor W(2) = exp(-2Bs(2)), where s = (sin theta)/lambda, is = 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%.

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.

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

A new strategy for employing three phase triples invariant estimates from Hauptmans 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.

On integrating the techniques of direct methods with anomalous dispersion. II. Statistical properties of the two-phase structure invariants.

Results of a statistical study of probabilistic estimates of two-phase structure invariants (TPSI) for Friedel pairs in the case of single-wavelength anomalous scattering are reported. Numerical analysis of the TPSI sign, magnitude and error distributions shows that the concise formula for TPSI by probability theory [Hauptman (1982). Acta Cryst. A38, 632-641; Giacovazzo (1983). Acta Cryst. A39, 585-592] has desirable statistical properties. Computational results for the known structures of cocaine methiodide (N-methylcocaine iodide) and of cytochrome c550 and its PtCl2-4 derivative show that when [E[ values are large most of the signs of the TPSI are correctly determined - for [E[ greater than 1.0, 90% or more of the TPSI signs are positive as predicted - and the errors in the estimated TPSI magnitudes do not exceed approximately 10% for [E[ greater than 1.0 in the small-molecule case or approximately 50% for [E[ greater than 1.5 in the macromolecular case. These results suggest that the theory will be useful for estimating the TPSI for unknown structures.

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.

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.

On integrating the techniques of direct methods with anomalous dispersion. III. Estimation of two-wavelength two-phase structure invariants

For diffraction data at two wavelengths from a crystal with anomalous scatterers, there are six types of two-phase structure invariants for Friedel pairs. Two of the six are single-wavelength invariants; the other four are mixed-wavelength invariants. It is shown that the latter can be estimated by a straightforward extension of results from the probabilistic direct-methods theory for the single-wavelength anomalous scattering case described in paper I [Hauptman (1982). Acta Cryst. A38, 632-641]. Statistical tests of the mixed-wavelength estimates are reported for small-molecule and macro-molecular examples.

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.

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.