Tangent formula applications in protein crystallography: an evaluation

As a test of the usefulness of matrix methods in extending phase information of proteins, the phases of triclinic lysozyme were calculated in the range from 3"3 to 2.5 A (2000 phases) using as a starting set the phases out to 3-3 A (1500 phases) and all the amplitudes. Observed ([Fobsl) and calculated (IFmodll) amplitudes were tested separately. The agreement of the predicted phases and the original ones was studied, and was observed to depend strongly on the normalized value of the corresponding structure factor, IEI. Different methods for assessing the quality of the predicted phases were considered, and the correl~ttion between electron density maps corresponding to predicted and original phases was selected as the most indicative of usefulness in structure determination. Predictions with the two different starting sets were studied using this correlation factor. A correlation of 0.5 between the predicted and original maps was obtained for the Fmodi, 0Crnodt case, and a correlation of 0.35 for the Fobs, ~,,od~ case. Both correlations are significant.

Section: Introductionmentioning

confidence: 99%

Application of multivariate distribution theory to phase extension for a crystalline protein

Podjarny¹,

Yonath²,

Traub³

1976

“…Following others (Weinzierl et al, 1969;Coulter, 1971;Hendrickson, 1971;Main, 1990;Giacovazzo et al, 1994;Mukherjee & Woolfson, 1995), the work described here originally focused on the application of the Sayre equation (Sayre, 1952). This is very similar in spirit to the recent work of Chen & Su (2000), who employed the Sayre equation and simulated-annealing methods to solve the phase problem for small-molecule crystals (up to 126 non-H atoms).…”

Section: Introductionmentioning

confidence: 91%

Resolution of the phase-ambiguity problem in the centrosymmetric P \bar{1} space group by Monte Carlo methods

Delarue

2000

Simulated-annealing methods have been used to resolve the phase-ambiguity problem in the centrosymmetric P " 1 space group. First, an energy function based on the Sayre equation is introduced and a formal comparison with classical spin systems is drawn. The energy landscape is studied in detail and the validity of several energy criteria thoroughly tested. Classical Monte Carlo methods proved to be successful using a multistart optimization of the Sayre score, along with the additional monitoring of other energetic criteria. These involved the Terwilliger map quality index in reciprocal space in the absence of envelope information, and an envelope score if the shape of the molecule is known. The inherent phase-ambiguity problem of the P " 1 space group was therefore technically solved by Monte Carlo methods. The method should also work to resolve phase ambiguity in the SIR method of protein crystallography.

“…Phase extension and refinement has also been carried out with macromolecules with the use of (33). Applications have been made to cytochrome c by Weinzierl, Eisenberg & Dickerson (1969), to carboxypeptidase A by Reeke & Lipscomb (1969) and to sperm whale myoglobin by Coulter (1971) with rather limited success. Some improvement in the results is obtained from use of the tangent formula when information from isomorphous-replacement or anomalousdispersion techniques is closely involved in the analysis.…”

Section: Tangent Formulamentioning

confidence: 99%

Direct methods in protein crystallography

Karle¹

1989

It is pointed out that the 'direct methods' of phase determination for small-structure crystallography do not have immediate applicability to macromolecular structures. The term 'direct methods in macromolecular crystallography' is suggested to categorize a spectrum of approaches to macromolecular structure determination in which the analyses are characterized by the use of two-phase and higher-order-phase invariants. The evaluation of the invariants is generally obtained by the use of heavy-atom techniques. The results of a number of the more recent algebraic and probabilistic studies involving isomorphous replacement and anomalous dispersion thus become valid subjects for discussion here. These studies are described and suggestions are also presented concerning future applicability. Additional discussion concerns the special techniques of filtering, the use of non-crystallographic symmetry, some features of maximum entropy and attempts to apply phase-determining formulas to the refinement of macromolecular structure. It is noted that, in addition to the continuing remarkable progress in macromolecular crystallography based on the traditional applications of isomorphous replacement and anomalous dispersion, recent valuable advances have been made in the application of non-crystallographic symmetry, in particular, to virus structures and in applications of filtering. Good progress has also been reported in the application of exact linear algebra to multiple-wavelength anomalous-dispersion investigations of structures containing anomalous scatterers of only moderate scattering power.