Direct methods – from birth to maturity

ARCIMBOLDO on coiled coils

ARCIMBOLDO solves the phase problem by combining the location of small model fragments using Phaser with density modification and autotracing using SHELXE. Mainly helical structures constitute favourable cases, which can be solved using polyalanine helical fragments as search models. Nevertheless, the solution of coiled-coil structures is often complicated by their anisotropic diffraction and apparent translational noncrystallographic symmetry. Long, straight helices have internal translational symmetry and their alignment in preferential directions gives rise to systematic overlap of Patterson vectors. This situation has to be differentiated from the translational symmetry relating different monomers. ARCIMBOLDO_LITE has been run on single workstations on a test pool of 150 coiled-coil structures with 15-635 amino acids per asymmetric unit and with diffraction data resolutions of between 0.9 and 3.0 Å. The results have been used to identify and address specific issues when solving this class of structures using ARCIMBOLDO. Features from Phaser v.2.7 onwards are essential to correct anisotropy and produce translation solutions that will pass the packing filters. As the resolution becomes worse than 2.3 Å, the helix direction may be reversed in the placed fragments. Differentiation between true solutions and pseudo-solutions, in which helix fragments were correctly positioned but in a reverse orientation, was found to be problematic at resolutions worse than 2.3 Å. Therefore, after every new fragment-placement round, complete or sparse combinations of helices in alternative directions are generated and evaluated. The final solution is once again probed by helix reversal, refinement and extension. To conclude, density modification and SHELXE autotracing incorporating helical constraints is also exploited to extend the resolution limit in the case of coiled coils and to enhance the identification of correct solutions. This study resulted in a specialized mode within ARCIMBOLDO for the solution of coiled-coil structures, which overrides the resolution limit and can be invoked from the command line (keyword coiled_coil) or ARCIMBOLDO_LITE task interface in CCP4i.

Image definition evaluation functions for X-ray crystallography: a new perspective on the phase problem

The core theme of X-ray crystallography is reconstructing the electron-density distribution of crystals under the constraints of observed diffraction data. Nevertheless, reconstruction of the electron-density distribution by straightforward Fourier synthesis is usually hindered due to the well known phase problem and the finite resolution of diffraction data. In analogy with optical imaging systems, the reconstructed electron-density map may be regarded as the image of the real electron-density distribution in crystals. Inspired by image definition evaluation functions applied in the auto-focusing process, two evaluation functions are proposed for the reconstructed electron-density images. One of them is based on the atomicity of the electron-density distribution and properties of Fourier synthesis. Tests were performed on synthetic data of known structures, and it was found that this evaluation function can distinguish the correctly reconstructed electron-density image from wrong ones when diffraction data of atomic resolution are available. An algorithm was established based on this evaluation function and applied in reconstructing the electron-density image from the synthetic data of known structures. The other evaluation function, which is based on the positivity of electron density and constrained power spectrum entropy maximization, was designed for cases where only diffraction data of rather limited resolution are available. Tests on the synthetic data indicate that this evaluation function may identify the correct phase set even for a data set with resolution as low as 3.5 Å. Though no algorithm for structure solution has been figured out based on the latter function, the results presented here provide a new perspective on the phase problem.

Optical and terahertz spectra analysis by the maximum entropy method

Phase retrieval is one of the classical problems in various fields of physics including x-ray crystallography, astronomy and spectroscopy. It arises when only an amplitude measurement on electric field can be made while both amplitude and phase of the field are needed for obtaining the desired material properties. In optical and terahertz spectroscopies, in particular, phase retrieval is a one-dimensional problem, which is considered as unsolvable in general. Nevertheless, an approach utilizing the maximum entropy principle has proven to be a feasible tool in various applications of optical, both linear and nonlinear, as well as in terahertz spectroscopies, where the one-dimensional phase retrieval problem arises. In this review, we focus on phase retrieval using the maximum entropy method in various spectroscopic applications. We review the theory behind the method and illustrate through examples why and how the method works, as well as discuss its limitations.

Busting out of crystallography's Sisyphean prison: from pencil and paper to structure solving at the press of a button: past, present and future of crystallographic software development, maintenance and distribution

The history of crystallographic computing and use of crystallographic software is one which traces the escape from the drudgery of manual human calculations to a world where the user delegates most of the travail to electronic computers. In practice, this involves practising crystallographers communicating their thoughts to the crystallographic program authors, in the hope that new procedures will be implemented within their software. Against this background, the development of small-molecule single-crystal and powder diffraction software is traced. Starting with the analogue machines and the use of Hollerith tabulators of the late 1930's, it is shown that computing developments have been science led, with new technologies being harnessed to solve pressing crystallographic problems. The development of software is also traced, with a final caution that few of the computations now performed daily are really understood by the program users. Unless a sufficient body of people continues to dismantle and re-build programs, the knowledge encoded in the old programs will become as inaccessible as the knowledge of how to build the Great Pyramid at Giza.

The charge flipping algorithm

This paper summarizes the current state of charge flipping, a recently developed algorithm ofab initiostructure determination. Its operation is based on the perturbation of large plateaus of low electron density but not directly on atomicity. Such a working principle radically differs from that of classical direct methods and offers complementary applications. The list of successful structure-solution cases includes periodic and aperiodic crystals using single-crystal and powder diffraction data measured with X-ray and neutron radiation. Apart from counting applications, the paper mainly deals with algorithmic issues: it describes and compares new variants of the iteration scheme, helps to identify and improve solutions, discusses the required data and the use of known information. Finally, it tries to foretell the future of such an alternative among well established direct methods.

Charge flipping at work: a case of pseudosymmetry

Charge flipping (CF) is an amazingly simple structure solution method that uses single-crystal X-ray diffraction data. It is truly ab initio, no preliminary information on atom types, chemical composition, or space group symmetry is required. The algorithm is iterative and alternates between real and reciprocal spaces. Its simplest version only changes (flips) the sign of the electron density below a threshold, while in reciprocal space, it prescribes the moduli of observed structure factors. In this communication, we apply the algorithm in practice. The selected example presents a whole range of difficulties: it is large, contains only light atoms, is noncentrosymmetric, and shows a particularly awkward pseudosymmetry. To solve it with any of the traditional methods requires many hours of computer time, followed by a day of expert's handwork to find missing and to remove spurious atoms. In contrast, the CF algorithm provides the complete structure in a few seconds and without human intervention. It is also remarkable that the success rate is 100%, that is, any starting point in the high-dimensional phase space leads to the solution. The treatment of translational pseudosymmetry is obviously a favorable case. Similar resistant structures with pseudosymmetries or ambiguous space groups are the practical applications where the CF method could well complement standard software procedures.

Structure determination of organic molecules from diffraction data by simulated annealing

We study simulated annealing techniques for crystal structure determination from diffraction data. We demonstrate that for this problem the efficiency of simulated annealing can be systematically improved by an iterative simulation protocol. Our approach is tested for the example of 9-(methylamino)-1 H-phenalen-1-one-1, 4-dioxan-2-yl hydroperoxide solvate (C18H19NO5).

An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images

We describe an approach to the high-resolution three-dimensional structural determination of macromolecules that utilizes ultrashort, intense x-ray pulses to record diffraction data in combination with direct phase retrieval by the oversampling technique. It is shown that a simulated molecular diffraction pattern at 2.5-A resolution accumulated from multiple copies of single rubisco biomolecules, each generated by a femtosecond-level x-ray free electron laser pulse, can be successfully phased and transformed into an accurate electron density map comparable to that obtained by more conventional methods. The phase problem is solved by using an iterative algorithm with a random phase set as an initial input. The convergence speed of the algorithm is reasonably fast, typically around a few hundred iterations. This approach and phasing method do not require any ab initio information about the molecule, do not require an extended ordered lattice array, and can tolerate high noise and some missing intensity data at the center of the diffraction pattern. With the prospects of the x-ray free electron lasers, this approach could provide a major new opportunity for the high-resolution three-dimensional structure determination of single biomolecules.

Subsurface dimerization in III-V semiconductor (001) surfaces

We present the atomic structure of the c(8 x 2) reconstructions of InSb-, InAs-, and GaAs-(001) surfaces as determined by surface x-ray diffraction using direct methods. Contrary to common belief, group III dimers are not prominent on the surface, instead subsurface dimerization of group III atoms takes place in the second bilayer, accompanied by a major rearrangement of the surface atoms above the dimers to form linear arrays. By varying the occupancies of four surface sites the (001)-c(8 x 2) reconstructions of III-V semiconductors can be described in a unified model.

Density modification for macromolecular phase improvement

Density modification provides a simple and largely automatic tool for improving phase estimates for observed structure factors. The phase information arises from a combination of the known structure factor magnitudes, the current phase estimates, and stereochemical information. The magnitudes, the current phase estimates, and stereochemical information. The addition of these phase information derived from theoretical sources renders new structures amenable to solution, and reduces the effort required to solve other structures. A diverse array of techniques which have been applied to the phase improvement problem are reviewed.

Maximum entropy methods in electron crystallography

The maximum entropy (ME) method of solving crystal structures in two or three dimensions from electron diffraction data is described. Applications to organic and inorganic molecules, membrane proteins and surface structures are outlined, and the power of the ME formalism to deal with incomplete and error prone data is demonstrated.

Stability and Three-Dimensional Aromaticity of closo-Monocarbaborane Anions, CB(n)()(-)(1)H(n)(-), and closo-Dicarboranes, C(2)B(n)()(-)(2)H(n)()

Comprehensive ab initio calculations RMP2(fc)/6-31G on the closo-monocarbaboranes, CB(n)()(-)(1)H(n)()(-) (n = 5-12), and the closo-dicarboranes, C(2)B(n)()(-)(2)H(n)() (n = 5-12), show that the relative energies of all the positional isomers agree with the qualitative connectivity considerations of Williams and with the topological charge stabilization rule of Gimarc. The reaction energies (DeltaH) of the most stable positional isomers, 1-CB(4)H(5)(-), CB(5)H(6)(-), 2-CB(6)H(7)(-), 1-CB(7)H(8)(-), 5-CB(8)H(9)(-), 1-CB(9)H(10)(-), 2-CB(10)H(11)(-), CB(11)H(12)(-), as well as 1,5-C(2)B(3)H(5), 1,6-C(2)B(4)H(6), 2,4-C(2)B(5)H(7), 1,7-C(2)B(6)H(8), 4,5-C(2)B(7)H(9), 1,10-C(2)B(8)H(10), 2,3-C(2)B(9)H(11), and 1,12-C(2)B(10)H(12) (computed using the equations, CBH(2)(-) + (n - 1)BH(increment) --> CB(n)()H(n)()(+1)(-) (n = 4-11) and C(2)H(2) + nBH(increment) --> C(2)B(n)()H(n)()(+2) (n = 3-10)), show that the stabilities of closo-CB(n)()(-)(1)H(n)()(-) and of closo-C(2)B(n)()(-)(2)H(n)() generally increase with increasing cluster size from 5 to 12 vertexes. This is a characteristic of three-dimensional aromaticity. There are variations in stabilities of individual closo-CB(n)()(-)(1)H(n)()(-) and closo-C(2)B(n)()(-)(2)H(n)() species, but these show quite similar trends. Moreover, there is rough additivity for each carbon replacement. The rather large nucleus independent chemical shifts (NICS) and the magnetic susceptibilities (chi), which correspond well with one another, also show all closo-CB(n)()(-)(1)H(n)()(-) and closo-C(2)B(n)()(-)(2)H(n)() species to exhibit "three-dimensional aromaticity". However, the aromaticity ordering based on these magnetic properties does not always agree with the relative stabilities of positional isomers of the same cluster, when other effects such as connectivity and charge considerations are important.

Applications of the maximum entropy method to powder diffraction and electron crystallography

A new multisolution phasing method based on entropy maximization and likelihood ranking, proposed for the specific purpose of increasing the accuracy and sensitivity of probabilistic phase indications compared with conventional direct methods, has been implemented and applied to a wide variety of problems. The latter comprise the determination of small crystal structures from X-ray diffraction data obtained from single crystals or from powders, and from electron diffraction data, both with and without partial phase information obtained by image processing of electron micrographs; the ranking of phase sets for a small protein; and the improvement of poor quality phases for a larger protein at medium resolution under constraint of solvent flatness. The main components of the method are (1) a tree-directed search through a space of trial phase sets; (2) the saddlepoint method for calculating joint probabilities of structure factors, using entropy maximization; (3) likelihood-based scores to rank trial phase sets and prune the search tree; (4) a statistical analysis of the scores for automatically selecting reliable phase indications. Their use is illustrated here on structure determinations from powder X-ray diffraction data and from electron diffraction data.

Computational challenges for macromolecular structure determination by X-ray crystallography and solution NMR-spectroscopy

Macromolecular structure determination by X-ray crystallography and solution NMR spectroscopy has experienced unprecedented growth during the past decade.