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.

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.

Application of the minimal principle to peptide structures

A new direct-methods procedure has been devised which consists of phase refinement via the minimal function, R(phi), alternated with Fourier summation and real space filtering. All phases are initially assigned values by computing structure factors for a randomly positioned set of atoms. These phases are then refined by using a parameter shift method to minimize R(phi). The refined phases are Fourier transformed, and a specified number of the largest peaks in the electron-density function are found and used as a new trial structure. The probability of a trial structure converging to a solution appears to depend on structural complexity and a number of refinement parameters. This procedure shows potential for providing fully automatic routine solutions for structures in the 200-400 atom range.

On the extrapolation of the magnitudes magnitude of E of the normalized structure factors E

For each fixed reciprocal-lattice vector H, the correlation coefficient, rho H, of the pair (magnitude of EK, magnitude of EH-K), where the magnitudes magnitude of EK and magnitude of EH-K are presumed to be known, is itself positively correlated with magnitude of EH. Thus, the correlation coefficients rho H serve to mediate the transfer of information from one region of reciprocal space to another. In particular, the calculated values of the correlation coefficients rho H lead to estimates of the unknown magnitudes magnitude of EH, even when the latter lie outside the sphere of the experimentally observed magnitude of E s (extrapolation). The applications show that the procedure described here to carry out this extrapolation is superior to existing methods.

Shake-and-Bake applications using simulated reference-beam data for crambin

The Shake-and-Bake method, as implemented in the computer program SnB, has been applied to simulated reference-beam data for the small protein crambin at several resolutions in the range 1.5-3.0 A. Sets of triplet invariants were generated having simulated mean triplet-phase errors from 0 to 60 degrees. Provided that these errors were no larger than 40 degrees, it was possible (at all resolutions tested) to find trial sets of individual Bragg phases with mean errors of 40-45 degrees. At 1.5 A, this could be achieved using only a single reference-beam data set. Peak picking provided useful phase constraints even at the lowest resolution tested. These results suggest that direct methods may be useful in conjunction with reference-beam data at resolutions lower than 1.2 A.

Ill-conditioned Shake-and-Bake: the trap of the false minimum

The alternation of phase refinement with the imposition of real-space constraints is the essence of the Shake-and-Bake procedure. Typically, these constraints prevent trial structures from falling into local minima. Nevertheless, P1 structures appear to migrate to false minima with significant frequency. These false minima are characterized by the presence of a large 'uranium' peak on the corresponding Fourier map. Fortunately, they can be recognized and avoided by considering the values of the minimal function both before and after the application of constraints. However, it appears that finding solutions for large P1 structures is likely also to require parameter-shift conditions different from those that have been found to work well in other space groups. In fact, these conditions often yield an unusually high percentage of solutions.

P1 Shake-and-Bake: can success be guaranteed?

The multi-trial direct-methods procedure known as Shake-and-Bake has been applied to three small proteins (alpha-1 peptide, vancomycin and lysozyme) that crystallize in space group P1. Phase refinement was accomplished through parameter-shift optimization using both the cosine and exponential forms of the minimal function. By extending error-free data to sufficiently high resolution, 100% convergence of trial structures to solution could be achieved in all three cases by using the exponential minimal function and a shift angle in the range 130-150 degrees. These results suggest optimum parameters for other P1 structures and emphasize the importance of collecting data to the highest possible resolution.

Exponential Shake-and-Bake: theoretical basis and applications

The simple cosine function used in the formulation of the traditional minimal principle and the related Shake-and-Bake algorithm is here replaced by a function of exponential type and its expected value and variance are derived. These lead to the corresponding exponential minimal principle and its associated Exponential Shake-and-Bake algorithm. Recent applications of the exponential function to several protein structures within the Shake-and-Bake framework suggest that this function leads, in general, to significant improvements in the success rate (percentage of trial structures yielding solution) of the Shake-and-Bake procedure. However, only in space group P1 is it presently possible to assign optimal values a priori for the exponential-function parameters.

The use of SnB to determine an anomalous scattering substructure

The positions of eight Se atoms in a selenomethionyl 35 kDa protein were determined at 2.0 and 2.5 A resolution using the direct-methods program SnB. Data at the selenium peak, edge and remote wavelengths were measured and processed independently. Anomalous difference E magnitudes at each wavelength were derived by two different procedures: renormalized diffE values were calculated according to the equation diffE = ¿sigma[(f + f')2 + f"2]||E+|-|E-||¿1/2/2q(sigma f"2)1/2, where q is a least-squares fitted renormalization function of sinstraight theta/lambda such that <diffE2> = 1.0; and difference E magnitudes were calculated from DeltaF2. Locally normalized E magnitudes corresponding to |FA| were also derived from the combination of the data at all three wavelengths through the use of the MADSYS program suite. Each of the independent sets of anomalous difference E magnitudes was capable of producing the correct solution, as did the E data obtained from the FA data. Higher success rates with SnB were observed for the 2.0 A peak and edge diffE data.

Incorporating tangent refinement in the Shake-and-Bake formalism

Shake-and-Bake is a direct-methods procedure in which phase refinement and Fourier refinement are alternated repetitively, unconditionally and automatically. The traditional Shake-and-Bake approach invoked a parameter-shift routine to perform phase refinement in an effort to reduce the value of minimal function. In this paper, parameter shift is replaced with the tangent formula as a means of phase refinement. This study shows that the tangent formula is more efficient than parameter shift for small structures when the number of refinement cycles and number of applications of the tangent formula per Shake-and-Bake cycle are chosen very carefully. For larger structures, including the 400 non-H-atom crambin structure, the two methods generally perform with similar efficiency. However, only parameter shift has successfully produced recognizable solutions for the difficult 317 non-H-atom structure gramicidin A.

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.

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.

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.

Crambin: a direct solution for a 400-atom structure

The crystal structure of crambin, a 46-residue protein containing the equivalent of approximately 400 fully occupied non-H-atom positions, was originally solved at 1.5 A by exploiting the anomalous scattering of its six S atoms at a single wavelength far removed from the absorption edge of sulfur. The crambin structure has now been resolved without the use of any anomalous-dispersion measurements. The technique employed was an ab initio 'shake-and-bake' method, consisting of a phase-refinement procedure based on the minimal function alternated with Fourier refinement. This method has successfully yielded solutions for a smaller molecule (28 atoms) using 1.2 A data, and a crambin solution was obtained at 1.1 A.

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.

Structure solution by minimal-function phase refinement and Fourier filtering. I. Theoretical basis

Eliminating the N atomic position vectors rj, j = 1, 2, ..., N, from the system of equations defining the normalized structure factors EH yields a system of identities that the EH's must satisfy, provided that the set of EH's is sufficiently large. Clearly, for fixed N and specified space group, this system of identities depends only on the set [H], consisting of n reciprocal-lattice vectors H, and is independent of the crystal structure, which is assumed for simplicity to consist of N identical atoms per unit cell. However, for a fixed crystal structure, the magnitudes magnitude of /EH/ are uniquely determined so that a system of identities is obtained among the corresponding phases psi H alone, which depends on the presumed known magnitudes magnitude of /EH/ and which must of necessity be satisfied. The known conditional probability distributions of triplets and quartets, given the values of certain magnitudes magnitude of /E/, lead to a function R(psi) of phases, uniquely determined by magnitudes magnitude of /E/ and having the property that RT < 1/2 < RR, where RT is the value of R(psi) when the phases are equal to their true values, no matter what the choice of origin and enantiomorph, and RR is the value of R(psi) when the phases are chosen at random. The following conjecture is therefore plausible: the global minimum of R(psi), where the phases are constrained to satisfy all identities among them that are known to exist, is attained when the phases are equal to their true values and is thus equal to RT.(ABSTRACT TRUNCATED AT 250 WORDS)

Structure solution by minimal-function phase refinement and Fourier filtering. II. Implementation and applications

The minimal function, R(psi), has been used to provide the basis for a new computer-intensive direct-methods procedure that shows potential for providing fully automatic routine solutions for structures in the 200-400 atom range. This procedure, which has been called shake-and-bake, is an iterative process in which real-space filtering is alternated with phase refinement using a technique that reduces the value of R(psi). It has been successfully tested using experimental data for a dozen known structures ranging in size from 25 to 317 atoms and crystallizing in a variety of space groups. The details of this procedure, the parameters used and the results of these applications are described.

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.

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.

On integrating the techniques of direct methods with anomalous dispersion: the one-phase structure seminvariants in the monoclinic and orthorhombic systems. III. Primitive non-centrosymmetric space groups of type 1P220. Erratum

There are two errors in the short communication by Velmurugan & Hauptman [Acta Cryst. (1989). A45, 656]. On line 9 in the Abstract, Φ 2h′2k′0 should read Φ 2h2k0 and on line 3 in Summary of final results, Φ 2h′2k′0 should read Φ 2h,2k,0.

On integrating the techniques of direct methods with anomalous dispersion: the one-phase structure seminvariants in the monoclinic and orthorhombic systems. I. Theoretical background

Conditional probability distributions are derived for the one-phase structure seminvariant in the presence of anomalous scattering for the monoclinic and orthorhombic systems. The results are presented in a compact tabular form. The applications of these results are discussed in the following paper [Velmurugan, Hauptman & Potter (1989).Acta Cryst.A45, 163-165].

On integrating the techniques of direct methods with anomalous dispersion: the one-phase structure seminvariants in the monoclinic and orthorhombic systems. II. Applications

Applications are considered of the conditional probability distributions of the one-phase structure seminvariants in the monoclinic and orthorhombic systems when anomalous scatterers are present. Test results with error-free data show accurate estimates of seminvariants, the accuracy varying with the complexity of the structure and with the number and strength of the anomalous scatterers.

Direct phase determination for quasi-kinematical electron diffraction intensity data from organic microcrystals

Direct phasing techniques using cosine estimates for three-phase and four-phase structure invariants in space group P1 are applied to zonal hk0 electron diffraction intensity data from an orthorhombic paraffin and a cephalin. Ab initio use of these kinematical phases is shown to derive the correct structures of the methylene subcell packings for these compounds in agreement with earlier determinations from Patterson maps.