ACORN — theory and practice

ACORN is a flexible and general procedure for the ab initio solution of protein structures when atomic resolution data are available. It has two components — ACORN-MR that locates a single atom or a fragment of the structure and ACORN-PHASE that refines the structure starting with the phase estimates from the fragment. ACORN-MR depends heavily on a correlation coefficient (CC) between the observed normalized structure factors and those from the fragment to find the fragment‘s correct orientation and/or position. ACORN-PHASE has three components, a Patterson-superposition sum function, real-space Sayre-equation refinement and, most importantly, Dynamic Density Modification (DDM). The concepts that underline DDM are explained by a theoretical approach. ACORN has been shown in trials to be able to solve structures with up to 5000 independent non-hydrogen atoms in the asymmetric unit. Several previously unknown structures have also been solved. At lower resolution, if anomalous scattering or isomorphous replacement data are available, then substructures can be determined using the ACORN approach. The quality of the final maps from ACORN, without further processing, enables automatic interpretation in terms of structure to be readily made.

ACORN2: new developments of the ACORN concept

The density-modification procedures incorporated in ACORN, available in the CCP4 package, have proved to be very successful in solving and refining high-resolution crystal structures from very poor starting sets. These can be calculated from a correctly positioned initial fragment containing between 1 and 8% of the scattering power of the total structure. Improvements of ACORN, reported here and incorporated in the program ACORN2, have lowered the size of the fragment required and examples are given of structures solved with only 0.25% of the scattering power in the fragment, which may be a single atom. Applications of ACORN2 to structures with space group P1 have shown the remarkable property that when the starting point is a pair of equal atoms, or even a single atom placed at the origin, the refinement process breaks the centric nature of the initial phases and converges to phases corresponding to one of the two possible enantiomorphs. Examples are given of the application of ACORN2 to the solution and/or refinement of a number of known trial structures and to the refinement of structures when phases are available either from MAD or from a molecular-replacement model.

ACORN: a review

The ACORN system was originally developed as a means of ab initio solution of protein structures when atomic resolution data were available. The first step is to obtain a starting set of phases, which must be at least slightly better than random. These may be calculated from a fragment of the structure, which can be anything from a single metal atom to a complete molecular-replacement model. A number of standard procedures are available in ACORN to orientate and position such a fragment. The fragment provides initial phases that give the first of a series of maps that are iteratively refined by a dynamic density-modification (DDM) process. Another FFT-based procedure is Sayre-equation refinement (SER), which modifies phases better to satisfy the Sayre equation. With good-quality atomic resolution data, the final outcome of applying DDM and SER is a map similar in appearance to that found from a refined structure, which is readily interpreted by automated procedures. Further development of ACORN now enables structures to be solved with less than atomic resolution data. A critical part of this development is the artificial extension of the data from the observed limit to 1 A resolution. These extended reflections are allocated unit normalized structure amplitudes and then treated in a similar way to observed reflections except that they are down-weighted in the calculation of maps. ACORN maps, especially at low resolution, tend to show C atoms less well, in particular C(alpha) atoms which fall within the first diffraction minimum of their three neighbours. Two new density-modification procedures (DDM1 and DDM2) and a density-enhancement procedure (ENH) have been devised to counter this problem. It is demonstrated that high-quality maps showing individual atoms can be produced with the new ACORN. ACORN has also been demonstrated to be very effective in refining phase sets derived from physical processes such as those using anomalous scattering or isomorphous derivative data. Future work will be directed towards applying ACORN to resolutions down to 2 A.

A modified ACORN to solve protein structures at resolutions of 1.7 A or better

ACORN has previously been shown to provide an efficient density-modification procedure for the solution of protein structures using diffraction data to better than 1.3 A. The initial phase set could be obtained from a variety of sources such as the position of a heavy atom, a set of scatterers such as S that had been positioned from anomalous dispersion measurements, a fragment or a very low homology model placed from a molecular-replacement search. Several structures solved using the early version of ACORN have been reported in the literature. Here, the effect of applying the original ACORN procedures at lower resolution is reported and new procedures that yield good-quality maps with data sets of resolution down to 1.7 A are described. These new procedures involve the artificial extension of data to atomic resolution and new density-modification processes that develop density at atomic positions that was previously suppressed. The test calculations were aimed firstly towards a proof of principle using a small fragment of a known structure to demonstrate that the procedure could generate correct density and a derived model in initially empty regions of the cell. Further tests addressed the use of more realistic starting models.

Channels in the gramicidin S-with-urea structure and their possible relation to transmembrane ion transport

The structure of membrane-active antibiotic cyclodecapeptide gramicidin S in the crystals of its complex with urea, C(60)H(92)N(12)0(10).0.(5)[(NH(2))(2)CO].7.94H(2)0, has been investigated with three-dimensional X-ray data by the automatic sequential approximation method. The crystals are trigonal, space group P3(1)21, a = 25.80(3), c= 21.49 (2) A, M(r) = 7968, calculated density = 1.088 mg m(-3), Z = 1. Conventional R factor: R1 = 0.0943, wR2 = 0.2478 [I> 2sigma(I)]. The molecule possesses an antiparallel twisted beta-structure, with turns involving the Phe-Pro peptides. The Orn side chains extend on one side of the sheet, while the non-polar Val and Leu side chains are located on the other face. One of the Orn residues (namely Orn2) is linked by an intermolecular hydrogen bond to the O atom of Phe4 residue, the other is free. The side chains of the Phe residues have trans orientation (chi(1) approximately 180 degrees ) and those of the Val, Orn, Leu residues, except those of Orn2, have the preferential gauche orientation with the chi(1) angle close to 60. Two side chains show statistical disorder and conformation of the Pro residues is C(s)-C(beta)-exo. There is half a urea molecule and also 7.94 water molecules distributed on 13 positions for each antibiotic molecule. A partially occupied and poorly ordered alcohol molecule had been identified. The gramicidin S molecules are arranged around the 3(1) axis in the form of a left-handed double spiral forming suggestive channels. The outer hydrophobic surface of the spiral is made of uncharged side radicals while the inside surface consists of the main-chain atoms, mainly O and N, and of ornithine side chains with N atoms at the ends. By changing the Orn side-chain conformation, the inner diameter of the channels may change from 3.4 to 6.3 A. Thus, ions and particles of rather large size may pass through the channel. The possibility of the creation of the gramicidin S channels in mitochondrial membranes has been noted by some biochemists. The channel complexes are close-packed in a hexagonal arrangement in the crystal. The CI(-) ions, present in abundance in the mother solution, are not found ordered in the crystals, which may indicate the absence of the charges in the terminal N atoms of the Orn residues.

Direct-space methods in phase extension and phase refinement. IV. The double-histogram method

In the conventional histogram-matching technique for phase extension and refinement for proteins a simple one-to-one transformation is made in the protein region to modify calculated density so that it will have some target histogram in addition to solvent flattening. This work describes an investigation where the density modification takes into account not only the current calculated density at a grid point but also some characteristic of the environment of the grid point within some distance R. This characteristic can be one of the local maximum density, the local minimum density or the local variance of density. The grid points are divided into ten groups, each containing the same number of grid points, for ten different ranges of value of the local characteristic. The ten groups are modified to give different histograms, each corresponding to that obtained under the same circumstances from a structure similar to the one under investigation. This process is referred to as the double-histogram matching method. Other processes which have been investigated are the weighting of structure factors when calculating maps with estimated phases and also the use of a factor to dampen the change of density and so control the refinement process. Two protein structures were used in numerical trials, RNApl [Bezborodova, Ermekbaeva, Shlyapnikov, Polyakov & Bezborodov (1988). Biokhimiya, 53, 965-973] and 2-Zn insulin [Baker, Blundell, Cutfield, Cutfield, Dodson, Dodson, Hodgkin, Hubbard, lsaacs, Reynolds, Sakabe, Sakabe & Vijayan (1988). Philos. Trans. R. Soc. London Ser. B, 319, 456--469]. Comparison of the proposed procedures with the normal histogram-matching technique without structure-factor weighting or damping gives mean phase errors reduced by up to 10 degrees with map correlation coefficients improved by as much as 0.14. Compared to the normal histogram used with weighting of structure factors and damping, the improvement due to the use of the double-histogram method is usually of order 4 degrees in mean phase error and an increase of 0.06-0.08 in the map correlation coefficient. It is concluded that the most reliable results are found with the local-maximum condition and with R in the range 0.5-0.6 A.

On the application of direct methods to oligonucleotide crystallography

The direct methods program SAYTAN was applied to simulated data at various resolutions from three oligonucleotides. Success in solving the structures was found to depend more upon the resolution of the data than upon errors in the data or the complexity of the structure. Collecting the data at a reduced temperature has little effect, unless it alters the mosaicity of the crystal or changes the resolution of the data. The presence of a heavy atom dramatically improved the phase refinement, particularly at low resolution.

Mean phase error and the map-correlation coefficient

In judging the effectiveness of methods of solving crystal structures, or in phase refinement and development, two criteria are commonly used. The first is the mean phase error, which may be weighted in some way, and the second is the map correlation coefficient which describes the similarity of a map with estimated phases to that with true phases. It is shown that these two measures are directly related and that given the individual phase errors the map correlation coefficient may be found without the need to calculate a map. Various aspects of this connection are examined, including the map correlation coefficient when weights are used for calculating maps and the conditions under which phase extension leads to maps with a higher map correlation coefficient - which involves a balance between the advantage of employing more data and the disadvantage that the extra data may have a higher average phase error.

A direct-method ab initio phasing of a protein, cupredoxin amicyanin, at 1.31 A resolution

The direct-methods program MULTAN88 has been applied successfully to redetermine the structure of a protein, cupredoxin amicyanin, containing 808 non-H atom sites, one Cu atom and 132 ordered water molecules in the asymmetric unit using data at 1.31 A resolution. Starting with initially random phases, useful phase sets selected by figures of merit could be obtained from multiple trials. The E maps corresponding to the best eight phase sets in order of combined figures of merit (CFOM2) revealed a distorted tetrahedral geometry around the Cu site. The phase estimates from the metal and a few neighbouring atoms in the initial E map corresponding to the set with the highest CFOM2 could be improved by the density-modification procedure PERP and led to an interpretable electron-density map.

A direct-method ab initio phasing of a protein, pseudoazurin, at 1.55 A resolution

The direct-method program MULTAN88 has been applied to solve a known protein, pseudoazurin, space group P6(5), unit-cell parameters a = b = 50.0 (1), c = 98.5 (3) A, with 917 protein atoms, a Cu atom and 93 solvent water molecules in the asymmetric unit (>6,000 non-H atoms in the unit cell) and data at 1.55 A resolution. One of several trials with sets of initially random phases yielded phase estimates for 1,000 reflections with a mean phase error of 68.3 degrees that was recognized as the best available solution by the figure of merit being used. Phase extension to 2,000 largest Es showed a distorted tetrahedral geometry around the Cu site. Density modification was applied to improve the phases and the quality of the maps, starting with phases calculated from the atomic positions indicated by the first map.

A flexible and efficient procedure for the solution and phase refinement of protein structures

An ab initio method is described for solving protein structures for which atomic resolution (better than 1.2 A) data are available. The problem is divided into two stages. Firstly, a substructure composed of a small percentage ( approximately 5%) of the scattering matter of the unit cell is positioned. This is used to generate a starting set of phases that are slightly better than random. Secondly, the full structure is developed from this phase set. The substructure can be a constellation of atoms that scatter anomalously, such as metal or S atoms. Alternatively, a structural fragment such as an idealized alpha-helix or a motif from some distantly related protein can be orientated and sometimes positioned by an extensive molecular-replacement search, checking the correlation coefficient between observed and calculated structure factors for the highest normalized structure-factor amplitudes |E|. The top solutions are further ranked on the correlation coefficient for all E values. The phases generated from such fragments are improved using Patterson superposition maps and Sayre-equation refinement carried out with fast Fourier transforms. Phase refinement is completed using a novel density-modification process referred to as dynamic density modification (DDM). The method is illustrated by the solution of a number of known proteins. It has proved fast and very effective, able in these tests to solve proteins of up to 5000 atoms. The resulting electron-density maps show the major part of the structures at atomic resolution and can readily be interpreted by automated procedures.

On the application of phase relationships to complex structures. XXXVI: some experiments with a small protein without heavy atoms

The direct-methods program MULTAN88 has been applied to a known protein, ribonuclease (RNAP1), containing 808 non-H atoms, including five S atoms, plus 83 ordered solvent water molecules. Phase sets with mean phase errors between 69 and 75 degrees were selected by modified figures of merit for trials with the full data at 1.17 A resolution and also with restricted data at 1.25 and 1.5 A resolution. These figures of merit had previously only been applied to protein structures containing heavy atoms, and this is the first demonstration of their usefulness with no heavy atom present. An initial set of 1091 phases from a 1.17 A trial was developed by an objective procedure to give the full structure with a residual of 0. 21, which agrees well with the published structure.

New techniques for applying anomalous-scattering and isomorphous-replacement data incorporated in ANOMIR - a general application package

A computer package ANOMIR is described which can derive phases from anomalous scattering and/or isomorphous-replacement data in any combination. For anomalous scattering it incorporates five methods of applying one-wavelength data and three methods for multiple-wavelength data including SPIN, reported here for the first time. In addition there are three procedures for multiple-wavelength data - the first modifying data for different wavelengths to make them mutually consistent, the second estimating the contributions of the anomalous scatterers alone and the third which finds anomalous differences. For single isomorphous replacement or one-wavelength anomalous scattering the phase ambiguity can be resolved by the direct method [Fan, Han, Qian & Yao (1984). Acta Cryst. A40, 489- 495] but for multiple isomorphous replacement the main method is an adaptation of the probability-curve method [Blow & Crick (1959). Acta Cryst. 12, 794-802]. A new statistical method is described for estimating the standard error in measuring magnitudes which is independent of having subsets of centric reflections. A method is described whereby the weights associated with phase estimates are used to generate probability curves, through which it is possible to combine estimates from different methods and to produce a 'best phase' and figure-of-merit for every reflection. ANOMIR procedures are also available for handling combinations of one-wavelength anomalous scattering with single- or multiple-isomorphous replacement. A final process, which is always beneficial, is a single parallel application of the tangent formula. The ANOMIR package has been designed for easy use and is controlled throughout by KEYWORDS. Results for several structures are given and compared with those found from the MLPHARE program in the CCP4 package.

Direct-space methods in phase extension and phase refinement. V. The histogram moments method

Any distribution is completely defined by its moments. It is shown that a process of phase refinement can be carried out, based on Fourier transforms, which modifies the moments of electron density, separately in the protein and solvent regions, towards target values. Tests have been carried out on two moderate-sized proteins with 800-900 atoms in the asymmetric unit, one containing heavy atoms and the other not. It has been found that refinement using the third moment about zero in the protein region is most effective and that refinement with higher moments, or in the solvent region, adds nothing useful. Two kinds of weights are necessary in the method. One is for giving a weighted mixture of new phase indications with original phase estimates from, say, multiple isomorphous replacement. The other weights are applied to the Fourier coefficients of density maps to give the best possible signal:noise ratio. These weights have been explored empirically and the best ones found are described. It is concluded that since the moments method, which changes phases in reciprocal space, is independent of other histogram-matching procedures, which change density in real space, it has something to offer in a refinement package containing several procedures.

Direct-space methods in phase extension and phase refinement. VI. PERP (phase extension and refinement program)

Several techniques for extending and refining phases for macromolecular structures have been incorporated into a program package PERP. In addition to previously employed techniques such as solvent flattening and histogram matching, PERP includes a new way of applying the Sayre equation [Refaat, Tate & Woolfson (1995). Acta Cryst. D51, 1036-1040], low-density elimination [Shiono & Woolfson (1992). Acta Cryst. A48, 451-456] and two double-histogram methods [Refaat, Tate & Woolfson (1996). Acta Cryst. D52, 252-256]. PERP is an easy-to-use package controlled by keywords and provided with default parameters that usually give near-optimum results. Examples are given of refinement, and also extension and refinement, for six known protein structures with a variety of characteristics. In each case PERP gives a very satisfactory outcome as measured by improvements in the mean-phase-error and conventional map-correlation coefficient. The main conclusion is that the several methods used in sequence give more effective extension and refinement than using any single method alone.

Direct-space methods in phase extension and phase determination. III. Phase refinement using Sayre's equation

An algorithm is described for refining a set of phases to agree with the Sayre equation. All operations are carried out using Fourier transforms with modest computer-store requirements even for very large systems. The procedure is tested with two moderate-sized proteins, one containing heavy atoms, and is found to give good refinement with data at more than atomic resolution (1.17 A) and useful, if less good, refinement when the data resolution is lower (1.5 A). It is concluded that at atomic resolution, or slightly below, the Sayre equation still has something to offer both for phase refinement and phase extension, especially if used cautiously with weighted multiple isomorphous replacement phases acting as a constraint on the phase changes. Even when the Sayre equation on its own refines phases badly, or not at all, it may still make an important contribution in conjunction with other real-space refinement procedures.

On the application of anomalous scattering in oligonucleotide crystallography

Simulated anomalous-scattering differences, at wavelengths between 1.5 and 5.5 A, were used with MULTAN to locate P atoms in an oligonucleotide hexamer. The success of the method depended heavily on the level of errors in the data. With error-free data most or all P atoms were located at all wavelengths. With noisy data, the best results were obtained by refining the phases associated with the largest values of |DeltaF|/sigma(|DeltaF|) rather than with the largest values of |DeltaF|. In this case a few of the P-atom positions could be located, with the best results occurring at wavelengths between 3.0 and 4.0 A. Further improvements were gained by reducing the values of the thermal parameters of the P atoms. MULTAN figures of merit had limited success in indicating the best phase sets, but a small improvement was gained by modifying the procedure for selecting those reflections used in the calculation of PSIZERO.

On the application of phase relationships to complex structures. XXXV. Some experiments with 2-Zn insulin

The direct-methods program SAYTAN has been applied to the known structure of 2-Zn insulin with 806 atoms, excluding solvent, in the asymmetric unit. Useful sets of phases can be obtained and selected by figures of merit for data resolutions between 1.5 and 2.25 A and these can be extended by SAYTAN to give mean phase errors of 68 degrees for more than 2000 reflections. A feature of the phases so found is that the phase errors decrease with increasing resolution - which is the opposite of the situation when phases are found by isomorphous-replacement techniques.

Direct phasing of one-wavelength anomalous-scattering data of the protein core streptavidin

The direct method [Fan, Hao, Gu, Qian, Zheng & Ke (1990). Acta Cryst. A46, 935-939] was used to break the phase ambiguity intrinsic to one-wavelength anomalous scattering data from a known protein of moderate size, core streptavidin, which was solved originally with three-wavelength anomalous diffraction data [Hendrickson, Pähler, Smith, Satow, Merritt & Phizackerley (1989). Proc. Natl Acad. Sci. USA, 86, 2190-2194]. Unlike that in the previous test with a small protein, the Fourier map calculated with the direct-method phases could not clearly reveal the moderate-sized protein structure. However, the phases can be improved step by step using Wang's solvent-flattening method, non-crystallographic symmetry averaging and the skeletonization method. The final electron-density map clearly shows most Calpha positions and some side chains and it is traceable without prior knowledge of the structure. It is concluded that the direct method is capable of breaking the OAS phase ambiguity of a moderate-sized protein at moderate resolution such as 3 A, while the combination of direct methods with macromolecular techniques may produce phases good enough for unknown protein structure to be traced.

On the application of phase relationships to complex structures. XXXIV. VFOM – a new figure of merit for protein phase sets at moderate resolution

In recent years it has been shown that direct methods are capable of solving the structures of small proteins. Mukherjee & Woolfson [Acta Cryst. (1993), D49, 9-12] have shown that useful phase sets can be produced even at 3 A resolution but that the standard figures of merit could not distinguish the better phase sets from others. They found modified forms of the standard figures of merit that could pick out better phase sets for 2 A resolution or higher. Gilmore, Henderson & Bricogne [Acta Cryst. (1991), A47, 842-846] have shown that evaluation of the log-likelihood gain, coming from entropy-maximization procedures, is also very successful in picking out good protein phases sets. A new figure of merit is described, based on the expected charactistics of an electron-density map for a protein, and comparisons are made with the other figures of merit mentioned above.

On the application of one-wavelength anomalous scattering. IV. The absolute configuration of the anomalous scatterers

An essential first step in most techniques for using anomalous-scattering data for phase determination is to determine the positions of the anomalous scatterers. This is usually done by use of the anomalous differences, either as input to a direct-methods procedure or to produce a Patterson map. If the arrangement of anomalous scatterers is noncentrosymmetric then it is also necessary to find their absolute configuration and a process is described for doing this based on the properties of the P(s) function [Okaya, Saito & Pepinsky (1955). Phys. Rev. 98, 1857-1858]. If the arrangement of anomalous scatterers is centrosymmetric then the problem does not occur.

Direct-space methods in phase extension and phase determination. II. Developments of low-density elimination

The low-density elimination method for phase extension and refinement [Shiono & Woolfson (1992). Acta Cryst. A48, 451-456] has been improved by substituting a smoother density-modification procedure for the original sharp cut-off function. In addition, better criteria have been found for limiting the number of refinement cycles, which gives a better final result for much less work. The effectiveness of the process has been illustrated by phase refinement for a protein with high-resolution (1.17 A) data containing 808 independent non-H atoms plus 83 water molecules in the asymmetric unit; the unweighted mean-phase error was reduced from 74 to 39.3 degrees. Phase extension and refinement was also demonstrated for pig 2Zn insulin starting with multiple isomorphous replacement (MIR) phases at 1.9 A and extending out to 1.5 A. There was a significant improvement of phases and the final map had a correlation coefficient of 0.540.

On the application of phase relationships to complex structures. XXXIII. The problems with large structures and low resolution

A conventional direct method, using the Sayre equation as a basis, has been shown to be capable of solving a small protein with data of 3.0 A resolution or better. An analysis of the Sayre equation, with data of various resolutions and with different lower limits of |E| for the contributors in the summation, shows that its effectiveness for phasing is independent of structural complexity but does decline as the resolution becomes worse. It is suggested that a practicable lower limit for the application of conventional direct methods is about 3.5 A. For large macromolecular structures the number of contributors to the summation in the Sayre equation becomes too large to handle and it is suggested that real-space methods should be used instead.

On the application of phase relationships to complex structures. XXXII. A small protein at low resolution

The direct-methods program SAYTAN is applied to data at various restricted resolutions for a small protein. It is shown that useful sets of phases can be obtained even down to 3 A resolution. Conventional figures of merit are not very discriminating for the phase sets developed, but modified figures of merit seem capable of selecting the better phase sets, at least for those generated from 2 A or higher resolution data.