31] PHASES-95: A program package for processing and analyzing diffraction data from macromolecules

NAD-dependent glutamate dehydrogenase from the hyperthermophilic archaeon Pyrobaculum islandicum was crystallized in the apo- and holoenzyme forms. Crystals were obtained using 2-propanol and polyethylene glycol MME 550 as precipitants for the apoenzyme and holoenzyme, respectively. The apoenzyme crystals belong to the trigonal space group P3(1)21 or its enantiomorph P3(2)21. The asymmetric unit contains three subunits; the values of the Matthews coefficient (VM) and the solvent content are 2.9 A3 Da-1 and 57%, respectively. A native data set was collected to a highest resolution limit of 4.0 A on an in-house X-ray source using a rotating-anode generator (overall Rsym of 12.3% and completeness of 97%). The holoenzyme crystals belong to the orthorhombic space group P2(1)2(1)2(1); the asymmetric unit contains one hexamer, giving a VM of 2.79 A3 Da-1 and a solvent content of 55%. Native and derivative data sets were collected. The crystals diffract to a maximum resolution of 2.8 A on the KEK-NW12 beamline at the Photon Factory and gave a data set with an overall Rsym of 7.9% and a completeness of 91%. Attempts are being made to solve the structure by the SIRAS method.

Molecular averaging, including noncrystallographic symmetry (NCS) averaging, is a powerful method for ab initio phase determination and phase improvement. Applications of the cross-crystal averaging (CCA) method have been shown to be effective for phase improvement after initial phasing by molecular replacement, isomorphous replacement, anomalous dispersion or combinations of these methods. Here, a two-step process for phase determination in the X-ray structural analysis of a new coat protein from a betanodavirus, Grouper nervous necrosis virus, is described in detail. The first step is ab initio structure determination of the T = 3 icosahedral virus-like particle using NCS averaging (NCSA). The second step involves structure determination of the protrusion domain of the viral molecule using cross-crystal averaging. In this method, molecular averaging and solvent flattening constrain the electron density in real space. To quantify these constraints, a new, simple and general indicator, free fraction (ff), is introduced, where ff is defined as the ratio of the volume of the electron density that is freely changed to the total volume of the crystal unit cell. This indicator is useful and effective to evaluate the strengths of both NCSA and CCA. Under the condition that a mask (envelope) covers the target molecule well, an ff value of less than 0.1, as a new rule of thumb, gives sufficient phasing power for the successful construction of new structures.

A general scheme for the improvement of electron-density maps is described which combines information from real and reciprocal space. The use of Sayre's equation, solvent flattening and histogram matching within this scheme has been described previously [Main (1990). Acta Cryst. A46, 372-377]. Non-crystallographic symmetry averaging, the use of a partial structure and constraints on individual structure factors have now been added. A computer program, SQUASH, is described which applies all these constraints simultaneously. Its application to the maps of several structures has been successful, particularly so when non-crystallographic symmetry is present. Uninterpretable maps have been improved to the point where a significant amount of the structure can be recognized. Applying the constraints simultaneously is more powerful than applying them all in series.

The use of single isomorphous replacement (SIR) has become less widespread due to difficulties in sample preparation and the identification of isomorphous native and derivative data sets. Non-isomorphism becomes even more problematic in serial experiments, because it adds natural inter-crystal non-isomorphism to heavy-atom-soaking-induced non-isomorphism. Here, a method that can successfully address these issues (and indeed can benefit from differences in heavy-atom occupancy) and additionally significantly simplifies the SIR experiment is presented. A single heavy-atom soak into a microcrystalline slurry is performed, followed by automated serial data collection of partial data sets. This produces a set of data collections with a gradient of heavy-atom occupancies, which are reflected in differential merging statistics. These differences can be exploited by an optimized genetic algorithm to segregate the pool of data sets into `native' and `derivative' groups, which can then be used to successfully determine phases experimentally by SIR.

The leucine zipper (LZ) is a typical member of the coiled coil family, the most common and extensively investigated of structural motifs.1 In terms of their biological functions, LZs participate in a variety of cellular processes, supplying unique protein–protein interactions via leucine–leucine zipping. The LZ domain is defined as an α-helix comprised of heptad repeats (abcdefg)n in which the residues at positions a and d are hydrophobic and mediate critical interhelical interactions, while b, c, e, f, and g are hydrophilic and form the solvent-exposed part of the coiled coil.2-4 Moreover, two or more LZ domains can intertwine in parallel or antiparallel to form a bundle of α-helices that interact via hydrophobic interactions at the inner face of the coiled coils.5-7 LZ domains were originally identified as highly conserved motifs mediating the interaction between transcriptional factors, but are now known to also be present in a number of ion channels.8 The sequences of ion channel LZ domains contain features common to the transitional LZs of transcriptional factors,2, 4 but the canonical leucine at position d is often replaced by other residues such as isoluecine, valine, or even a nonhydrophobic residue, thereby forming a modified LZ.9 Recent reports on the function of ion channel LZ domains have suggested that they participate in regulating the channel's activity by targeting modulator proteins to the channel, promoting formation of "macromolecular signaling complexes."8-16 For instance, the LZ domains of the RyR2 channel recruit adaptor molecules for PKA, PP1, and PP2A, which regulate channel activity through phosphorylation of certain residues.9-11 They also facilitate correct folding and plasma membrane trafficking of ion channels.17-19 By substituting leucine residues, it was confirmed that "leucine– leucine zipping" between the LZs of ion channels and partner proteins is required for assembly of these signal complexes.11, 15, 17 For instance, the targeting of modulator molecules to the RyR2 channel was impaired when alanine was substituted for the leucine at position d of the LZ.10 In other words, ion channels and signaling complexes form specific LZ heteromers within the macromolecular signaling complexes.20 Thus LZ domains appear to be the major binding motif mediating the interaction between intracellular proteins involved in signal transduction and ion channels. Here we describe our analysis of the 2.1 Å X-ray crystal structure of the LZ domain of the small conductance voltage-independent Ca2+-activated K+ channel (SKCa) from rat (Rattus norvegicus). Intriguingly the structure of the LZ domain is characterized by a parallel trimeric α-helix bundle that is incompatible with the tetrameric state of the functional SKCa channel. The open reading frame of the LZ domain (aa. #488–#526) of the rat SKCa channel isoform 2 (U69882) was amplified by PCR using the following primers: 5′-GGAATATCATGTATGATATG-3′ (forward) and 5′-TTAAGGAGGGCATGAAT-3′ (backward). Chromosomal DNA encoding the rat SKCa channel served as the template for the PCR amplification. After digestion with NdeI and BamHI (NEB), the amplified DNA fragment was cloned into a modified pGEX vector (Pharmacia) so that the LZ domain would be expressed with glutathione S-transferase (GST) fused at its N-terminus for affinity purification. The resultant expression plasmid was confirmed by sequencing and used to transform E. coli strain BL21(DE3). The overexpressed LZ domain was initially purified using glutathione Sepharose 4B affinity resin (Peptron), after which the GST-fused proteins were eluted with 15 mL of elution buffer containing 20 mM reduced glutathione in 50 mM Tris–HCl (pH 8.0) and 0.2M NaCl. To remove the GST moiety, the eluted proteins were incubated in the presence of 40 U of thrombin with mild rotation. The digested LZ proteins including extra four residues (GSHM) at N-terminus as the result of gene cloning were then further purified on a Superdex-75 gel filtration column (Pharmacia). LZ domains were crystallized using the hanging-drop vapor diffusion method.21 Individual hexagonal crystals were grown on siliconized cover slips by equilibrating a mixture containing 1 or 2 μL of protein [5 mg/mL protein in 20 mM Hepes (pH 7.5) and 150 mM NaCl] and 1 μL of reservoir solution against 500 μL of mother liquor [0.1M sodium citrate (pH 5.6) and 20–22% (w/v) PEG8000]. For the cryogenic experiments, 10% glycerol was determined to be a suitable cryoprotectant under the crystallization conditions. Single crystals were soaked in cryoprotection solution for 30 s, mounted on a cryoloop, and flash-frozen in liquid nitrogen. The native data set was collected at beam line BL5A at the Photon Factory. The data set was processed and scaled using the programs DENZO and SCALEPACK.22 The structure of the LZ domain was determined using the molecular replacement technique, with the GCN4 LZ serving as the search model (PDB: 1C94). The program PHASER was used to calculate both the cross-rotation and translation functions in the 10−3 Å resolution range.23 The initial model was subjected to iterative cycles of crystallographic refinement with the program CNS24 that were alternated with graphic sessions for model building using the program O.25 The crystal structure of the LZ domain was refined at 2.1 Å resolution. The asymmetric unit contains two molecules. The trimeric structure of LZ domains is formed by crystallographic threefold symmetry. During the refinement, hemihedral twinning was detected and treated within the program CNS. The twin fraction was 0.35, indicating the crystal to be only partially twinned. A twinning operator was included in the refinement, lowering Rwork and Rfree to 21.0% and 27.8%. The coordinates of the final models and experimental structural factors of the LZ domain have been deposited with the PDB codes 2PNV. The data collection and refinement statistics are summarized in Table I. SKCa LZ domains formed a three-stranded parallel helix bundle shaped like twisted circular cylinder ∼50 Å long and ∼20 Å wide [Fig. 1(A)]. The leucine, isoluecine, and phenylalanine side chains at positions a and d point to the center of the trimer. The first 35 residues of each peptide monomer include more than nine helical turns, though His523 and Ala524 are not in a helical conformation in every chain within the trimer. Leu525 and Pro526 were not visible on the electron density map. (A) Stereo view of the SKCa channel LZ trimer showing heptad residues at a (red) and d (deep blue). (B) Conserved LZ motifs of various channels in vertebrates and a typical LZ motif of a transcriptional factor (SK2, SKCachannel isoform 2; BK, big conductance Ca2+- and voltage-activated K+ channels; CaV1.1, voltage-dependent L-type Ca2+ channel α1S subunit; RyR2, Ryanodine receptor 2; GCN4, yeast transcriptional activator). Residues at position d of the heptad repeats are colored deep blue and residues at position a are colored red. (C) Helical wheel representation of the LZ trimer. Shown are residues 488–526 of SKCa channel isoform 2 (NIMYDMISDLNERSEDFEKRIVTLETKLETLIGSIHALP); the view is from the N-terminus, and residues in the first helical turns are boxed (Ile489) or circled. Heptad positions are labeled a through g. Hydrophobic interactions in the core of the helix bundle at position a and d are connected with blue dashed lines. Hydrogen pairs between Asn498 at position e and Arg500 at position g are indicated by red dashed lines across the interhelical interface. (D) Glutaraldehyde cross-linking assay of LZ domain of the SKCa channel. The purified LZ domain of rat SK2 channel (lane 1: before cross-linking assay) was incubated with 0.1% glutaraldehyde for 30 s (lane 2, 4) or 60 s (lane 3, 5). The concentration of assayed protein was 0.2 mM (lane 2, 3) or 0.6 mM (lane 4, 5). The cross-linking reaction was quenched by adding sodium dodecyl sulfate (SDS) sample buffer (60 mM Tris–HCl pH6.8, 25% glycerol, 2% SDS, 14.4 mM β-mercaptoethanol, 0.1% bromophenol) to the final 1×-working concentration. The cross-linked samples were analyzed by 15% SDS–polyacrylamide gel electrophoresis. The individual helices are smoothly bent, which permits tight contacts over the length of the trimer. The trimer interface shows knobs-into-holes packing between the helices.2, 4 The knobs formed by the side chains of one helix fit into holes formed by the spaces between side chains on the neighboring helix. The isoleucine and leucine knobs at position a pack into a hole formed by the g and a residues of the neighboring monomer. Similarly, residues at positions a and d pack into a hole formed by the d and e residues. In addition, Asn498 and Arg500 contribute to the interaction among the helices through hydrogen bonding and knobs-into-holes packing [Fig. 1(B)]. The trimer interface buries 593 Å2 of protein surface, which represents approximately 17% of the total surface area of each subunit. The residues at positions a, d, e, and g of each strand are largely buried in the α-helix bundle accounting for about 60% of total contact area. The residues at the b and c positions are somewhat buried, and the f position remains completely exposed. Ours is the first description of the three-dimensional crystal structure of an ion channel LZ domain. The overall structure of the α-helix bundle of the SKCa LZ is very similar to the LZ helix bundle in HIV-1 gp41.7 It is interesting that the trimeric parallel α-helix bundle formed by SKCa LZ domains include none of the previously reported residue substitutions.5-7, 26 Some features of the SKCa LZ heptad repeats would appear to facilitate trimer formation. Like the Val residues at position a in GCN4,4 the β-branched Ile494, Ile508 and Ile522 residues at position a should make tetramer formation unfavorable.5, 6 And the Ile508 residue in the SKCa LZ would be more likely to prevent dimer formation than the Asn residue at position a in the GCN4 LZ4 [Fig. 1(C)]. Moreover, hydrogen interactions between Asn498 and Arg500 at positions e and g, respectively, would further stabilize the trimeric structure of the SKCa LZ [dashed line in Fig. 1(C)]. On the other hand, this unique trimeric state of the SKCa LZ domain is inconsistent with the tetrameric nature of the functional SKCa channel. Notably, although X-ray crystallographic analysis confirmed that the SKCa LZ domainforms a three-helix bundle, our chemical cross-linking studies with purified SKCa LZ domain variously suggested that in solution the domain assumes monomeric to tetrameric forms [Fig. 1(D)]. This may be indicative of the important role played by ion channel LZ domains in linking the channels to LZ-containing macromolecular complexes.20 We suggest the apparent flexibility in the binding characteristics of LZ homomers is the major driving force determining specific heteromer formation. Furthermore, the instability of the homomeric states of the SKCa LZ domain in solution suggests that native channel LZs are designed to favor formation of heteromers via LZs of different proteins rather than homomeric structures. The diversity of its interactions highlights the point that within the heptad repeat, the core buried residues at positions a and d or the flanking residues at positions e and g are not sufficient, by themselves, to mediate multimerization. Future crystallization of ion channel LZ domain in complex with a regulatory protein should enable us to visualize the alignment and characteristics of the LZs within the complexes and to further our understanding of their role in modulating ion channel function.

The crystal structure of the Glu-105-->Gly mutant of catabolic ornithine transcarbamoylase (OTCase; carbamoyl phosphate + L-ornithine = orthophosphate + L-citrulline, EC 2.1.3.3) from Pseudomonas aeruginosa has been determined at 3.0-A resolution. This mutant is blocked in the active R (relaxed) state. The structure was solved by the molecular replacement method, starting from a crude molecular model built from a trimer of the catalytic subunit of another transcarbamoylase, the extensively studied aspartate transcarbamoylase (ATCase) from Escherichia coli. This model was used to generate initial low-resolution phases at 8-A resolution, which were extended to 3-A by noncrystallographic symmetry averaging. Four phase extensions were required to obtain an electron density map of very high quality from which the final model was built. The structure, including 4020 residues, has been refined to 3-A, and the current crystallographic R value is 0.216. No solvent molecules have been added to the model. The catabolic OTCase is a dodecamer composed of four trimers organized in a tetrahedral manner. Each monomer is composed of two domains. The carbamoyl phosphate binding domain shows a strong structural homology with the equivalent ATCase part. In contrast, the other domain, mainly implicated in the binding of the second substrate (ornithine for OTCase and aspartate for ATCase) is poorly conserved. The quaternary structures of these two allosteric transcarbamoylases are quite divergent: the E. coli ATCase has pseudo-32 point-group symmetry, with six catalytic and six regulatory chains; the catabolic OTCase has 23 point-group symmetry and only catalytic chains. However, both enzymes display homotropic and heterotropic cooperativity.

After a first set of protein phases is obtained with the isomorphous replacement method, the molecular replacement method, or the multiple wavelength anomalous dispersion method and an electron density map calculated, the next step is the interpretation of the map in terms of the polypeptide chain. If this is successful and the major part of the chain can indeed be followed in the electron density map, refinement of the structure can begin. However, insufficient quality of the electron density map might hamper a complete and unambiguous tracing of the polypeptide chain, increasing the risk of introducing errors in the model, which cannot be easily removed during refinement. In such a case refinement should be preceded by a process to improve the quality of the map through improvement of the protein phase angles (Podjarny et al. , 1987) . During phase improvement, all available information on the structure should be used (Brünger and Nilges, 1993). This information may be in one of the following forms: 1. The structure is partially known. 2. The protein molecules distinguish themselves as relatively high regions of electron density and their boundaries can be estimated. The electron density between them is then set to a constant value or adjusted otherwise. KeywordsStructure FactorPhase AngleAsymmetric UnitDensity ModificationPhase ImprovementThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The molecular structure of a high potential iron-sulfur protein (HiPIP) isolated from the purple photosynthetic bacterium, Ectothiorhodospira halophila strain BN9626, has been solved by x-ray diffraction analysis to a nominal resolution of 2.5 A and refined to a crystallographic R value of 18.4% including all measured x-ray data from 30.0- to 2.5-A resolution. Crystals used in the investigation contained two molecules/asymmetric unit and belonged to the space group P21 with unit cell dimensions of a = 60.00 A, b = 31.94 A, c = 40.27 A, and beta = 100.5 degrees. An interpretable electron density map, obtained by combining x-ray data from one isomorphous heavy atom derivative with non-crystallographic symmetry averaging and solvent flattening, clearly showed that this high potential iron-sulfur protein contains 71 amino acid residues, rather than 70 as originally reported. As in other bacterial ferredoxins, the [4Fe-4S] cluster adopts a cubane-like conformation and is ligated to the protein via four cysteinyl sulfur ligands. The overall secondary structure of the E. halophila HiPIP is characterized by a series of Type I and Type II turns allowing the polypeptide chain to wrap around the [4Fe-4S] prosthetic group. The hydrogen bonding pattern around the cluster is nearly identical to that originally observed in the 85-amino acid residue Chromatium vinosum HiPIP and consequently, the 240 mV difference in redox potentials between these two proteins cannot be simply attributed to hydrogen bonding patterns alone.

LSL is a lectin produced by the parasitic mushroom Laetiporus sulphureus, which exhibits hemolytic and hemagglutinating activities. Here, we report the crystal structure of LSL refined to 2.6-A resolution determined by the single isomorphous replacement method with the anomalous scatter (SIRAS) signal of a platinum derivative. The structure reveals that LSL is hexameric, which was also shown by analytical ultracentrifugation. The monomeric protein (35 kDa) consists of two distinct modules: an N-terminal lectin module and a pore-forming module. The lectin module has a beta-trefoil scaffold that bears structural similarities to those present in toxins known to interact with galactose-related carbohydrates such as the hemagglutinin component (HA1) of the progenitor toxin from Clostridium botulinum, abrin, and ricin. On the other hand, the C-terminal pore-forming module (composed of domains 2 and 3) exhibits three-dimensional structural resemblances with domains 3 and 4 of the beta-pore-forming toxin aerolysin from the Gram-negative bacterium Aeromonas hydrophila, and domains 2 and 3 from the epsilon-toxin from Clostridium perfringens. This finding reveals the existence of common structural elements within the aerolysin-like family of toxins that could be directly involved in membrane-pore formation. The crystal structures of the complexes of LSL with lactose and N-acetyllactosamine reveal two dissacharide-binding sites per subunit and permits the identification of critical residues involved in sugar binding.

A graphical user interface1 (GUI) for the crystallographic data processing program SHARP2 has been designed and built from scratch using the hypertext transfer protocol (http). The advantages of this design are as follows. The scripts are easy to write, debug and maintain. The interface is easy for beginners because it uses familiar software such as a web browser and the graphics program “O”, as well as extensive hyperlinks which provide context-sensitive help while preparing input and examining output. Results such as graphs and electron density maps can be presented graphically using the mouse, making it a productivity tool for novices and experts alike. In particular, the systematic use of the residual map (log-likelihood gradient map) speeds up and facilitates the improvement of the heavy-atom model by identification of minor sites and anisotropy. Since the software uses the client-server model, it can be run from any networked computer allowing remote control and remote viewing of crystallographic calculations on distant and nearby computers alike.

Recognition of poly(C) DNA and RNA sequences in mammalian cells is achieved by a subfamily of the KH (hnRNP K homology) domain-containing proteins known as poly(C)-binding proteins (PCBPs). To reveal the molecular basis of poly(C) sequence recognition, we have determined the crystal structure, at 1.7-A resolution, of PCBP2 KH1 in complex with a 7-nucleotide DNA sequence (5'-AACCCTA-3') corresponding to one repeat of the human C-rich strand telomeric DNA. The protein-DNA interaction is mediated by the combination of several stabilizing forces including hydrogen bonding, electrostatic interactions, van der Waals contacts, and shape complementarities. Specific recognition of the three cytosine residues is realized by a dense network of hydrogen bonds involving the side chains of two conserved lysines and one glutamic acid. The co-crystal structure also reveals a protein-protein dimerization interface of PCBP2 KH1 located on the opposite side of the protein from the DNA binding groove. Numerous stabilizing protein-protein interactions, including hydrophobic contacts, stacking of aromatic side chains, and a large number of hydrogen bonds, indicate that the protein-protein interaction interface is most likely genuine. Interaction of PCBP2 KH1 with the C-rich strand of human telomeric DNA suggests that PCBPs may participate in mechanisms involved in the regulation of telomere/telomerase functions.

Protein mass spectrometry imaging (MSI) with electrospray-basedambient ionization techniques, such as nanospray desorption electrosprayionization (nano-DESI), generates data sets in which each pixel correspondsto a mass spectrum populated by peaks corresponding to multiply chargedprotein ions. Importantly, the signal associated with each proteinis split among multiple charge states. These peaks can be transformedinto the mass domain by spectral deconvolution. When proteins areimaged under native/non-denaturing conditions to retain non-covalentinteractions, deconvolution is particularly valuable in helping interpretthe data. To improve the acquisition speed, signal-to-noise ratio,and sensitivity, native MSI is usually performed using mass resolvingpowers that do not provide isotopic resolution, and conventional algorithmsfor deconvolution of lower-resolution data are not suitable for theselarge data sets. UniDec was originally developed to enable rapid deconvolutionof complex protein mass spectra. Here, we developed an updated featureset harnessing the high-throughput module, MetaUniDec, to deconvolveeach pixel of native MSI data sets and transform m/z-domain image files to the mass domain. New toolsenable the reading, processing, and output of open format .imzML filesfor downstream analysis. Transformation of data into the mass domainalso provides greater accessibility, with mass information readilyinterpretable by users of established protein biology tools such assodium dodecyl sulfate polyacrylamide gel electrophoresis.

Pairwise alignment of protein structures is a fundamental task in structural bioinformatics. There are numerous computer programs in the public domain that produce alignments for a given pair of protein structures, but the results obtained by the various programs generally differ substantially. Hence, in the application of such programs the question arises which of the alignment programs are the most trustworthy in the sense of overall performance, and which programs provide the best result for a given pair of proteins. The major problem in comparing, evaluating and judging alignment results is that there is no clear notion of the optimality of an alignment. As a consequence, the numeric criteria and scores reported by the individual structure alignment programs are largely incomparable. Here we report on the development and application of a new approach for the evaluation of structure alignment results. The method uses the translation vector and rotation matrix to generate the superposition of two structures but discards the alignment reported by the individual programs. The optimal alignment is then generated in standardized form based on a suitably implemented dynamic programming algorithm where the length of the alignment is the single most informative parameter. We demonstrate that some of the most popular programs in protein structure research differ considerably in their overall performance. In particular, each of the programs investigated here produced in at least in one case the best and the worst alignment compared with all others. Hence, at the current state of development of structure comparison techniques, it is advisable to use several programs in parallel and to choose the optimal alignment in the way reported here. The computer software that implement the method described here is freely available at http://melolab.org/stovca.

The number of macromolecular structures deposited in the Protein Data Bank now approaches 100,000, with the vast majority of them determined by crystallographic methods. Thousands of papers describing such structures have been published in the scientific literature, and 20 Nobel Prizes in chemistry or medicine have been awarded for discoveries based on macromolecular crystallography. New hardware and software tools have made crystallography appear to be an almost routine (but still far from being analytical) technique and many structures are now being determined by scientists with very limited experience in the practical aspects of the field. However, this apparent ease is sometimes illusory and proper procedures need to be followed to maintain high standards of structure quality. In addition, many noncrystallographers may have problems with the critical evaluation and interpretation of structural results published in the scientific literature. The present review provides an outline of the technical aspects of crystallography for less experienced practitioners, as well as information that might be useful for users of macromolecular structures, aiming to show them how to interpret (but not overinterpret) the information present in the coordinate files and in their description. A discussion of the extent of information that can be gleaned from the atomic coordinates of structures solved at different resolution is provided, as well as problems and pitfalls encountered in structure determination and interpretation.

The crystallographic diffraction experiment measures Bragg intensities; crystallographic electron-density maps and other crystallographic calculations in phasing require structure-factor amplitudes. If data were measured with no errors, the structure-factor amplitudes would be trivially proportional to the square roots of the intensities. When the experimental errors are large, and especially when random errors yield negative net intensities, the conversion of intensities and their error estimates into amplitudes and associated error estimates becomes nontrivial. Although this problem has been addressed intermittently in the history of crystallographic phasing, current approaches to accounting for experimental errors in macromolecular crystallography have numerous significant defects. These have been addressed with the formulation of LLGI, a log-likelihood-gain function in terms of the Bragg intensities and their associated experimental error estimates. LLGI has the correct asymptotic behaviour for data with large experimental error, appropriately downweighting these reflections without introducing bias. LLGI abrogates the need for the conversion of intensity data to amplitudes, which is usually performed with the French and Wilson method [French & Wilson (1978), Acta Cryst. A35, 517-525], wherever likelihood target functions are required. It has general applicability for a wide variety of algorithms in macromolecular crystallography, including scaling, characterizing anisotropy and translational noncrystallographic symmetry, detecting outliers, experimental phasing, molecular replacement and refinement. Because it is impossible to reliably recover the original intensity data from amplitudes, it is suggested that crystallographers should always deposit the intensity data in the Protein Data Bank.