Intensity statistics in the presence of translational noncrystallographic symmetry.

In the case of translational noncrystallographic symmetry (tNCS), two or more copies of a component in the asymmetric unit of the crystal are present in a similar orientation. This causes systematic modulations of the reflection intensities in the diffraction pattern, leading to problems with structure determination and refinement methods that assume, either implicitly or explicitly, that the distribution of intensities is a function only of resolution. To characterize the statistical effects of tNCS accurately, it is necessary to determine the translation relating the copies, any small rotational differences in their orientations, and the size of random coordinate differences caused by conformational differences. An algorithm to estimate these parameters and refine their values against a likelihood function is presented, and it is shown that by accounting for the statistical effects of tNCS it is possible to unmask the competing statistical effects of twinning and tNCS and to more robustly assess the crystal for the presence of twinning.

Humidity control and hydrophilic glue coating applied to mounted protein crystals improves X-ray diffraction experiments.

Protein crystals are fragile, and it is sometimes difficult to find conditions suitable for handling and cryocooling the crystals before conducting X-ray diffraction experiments. To overcome this issue, a protein crystal-mounting method has been developed that involves a water-soluble polymer and controlled humid air that can adjust the moisture content of a mounted crystal. By coating crystals with polymer glue and exposing them to controlled humid air, the crystals were stable at room temperature and were cryocooled under optimized humidity. Moreover, the glue-coated crystals reproducibly showed gradual transformations of their lattice constants in response to a change in humidity; thus, using this method, a series of isomorphous crystals can be prepared. This technique is valuable when working on fragile protein crystals, including membrane proteins, and will also be useful for multi-crystal data collection.

Sparsity-based single-shot subwavelength coherent diffractive imaging.

Coherent Diffractive Imaging (CDI) is an algorithmic imaging technique where intricate features are reconstructed from measurements of the freely diffracting intensity pattern. An important goal of such lensless imaging methods is to study the structure of molecules that cannot be crystallized. Ideally, one would want to perform CDI at the highest achievable spatial resolution and in a single-shot measurement such that it could be applied to imaging of ultrafast events. However, the resolution of current CDI techniques is limited by the diffraction limit, hence they cannot resolve features smaller than one half the wavelength of the illuminating light. Here, we present sparsity-based single-shot subwavelength resolution CDI: algorithmic reconstruction of subwavelength features from far-field intensity patterns, at a resolution several times better than the diffraction limit. This work paves the way for subwavelength CDI at ultrafast rates, and it can considerably improve the CDI resolution with X-ray free-electron lasers and high harmonics.

Using the singular value decomposition to extract 2D correlation functions from scattering patterns.

The truncated singular value decomposition (TSVD) is applied to extract the underlying 2D correlation functions from small-angle scattering patterns. The approach is tested by transforming the simulated data of ellipsoidal particles and it is shown that also in the case of anisotropic patterns (i.e. aligned ellipsoids) the derived correlation functions correspond to the theoretically predicted profiles. Furthermore, the TSVD is used to analyze the small-angle X-ray scattering patterns of colloidal dispersions of hematite spindles and magnetotactic bacteria in the presence of magnetic fields, to verify that this approach can be applied to extract model-free the scattering profiles of anisotropic scatterers from noisy data.

Improving Coarse-Grained Protein Force Fields with Small-Angle X-ray Scattering Data.

Small-angle X-ray scattering (SAXS) experiments provide valuable structural data for biomolecules in solution. We develop a highly efficient maximum entropy approach to fit SAXS data by introducing minimal biases to a coarse-grained protein force field, the associative memory, water mediated, structure, and energy model (AWSEM). We demonstrate that the resulting force field, AWSEM-SAXS, succeeds in reproducing scattering profiles and models protein structures with shapes that are in much better agreement with experimental results. Quantitative metrics further reveal a modest, but consistent, improvement in the accuracy of modeled structures when SAXS data are incorporated into the force field. Additionally, when applied to a multiconformational protein, we find that AWSEM-SAXS is able to recover the population of different protein conformations from SAXS data alone. We, therefore, conclude that the maximum entropy approach is effective in fine-tuning the force field to better characterize both protein structure and conformational fluctuation.

Jscatter, a program for evaluation and analysis of experimental data.

The aim of Jscatter is the processing of experimental data and physical models with the focus to enable the user to develop/modify their own models and use them within experimental data evaluation. The basic structures dataArray and dataList contain matrix-like data of different size including attributes to store corresponding metadata. The attributes are used in fit routines as parameters allowing multidimensional attribute dependent fitting. Several modules provide models mainly applied in neutron and X- ray scattering for small angle scattering (form factors and structure factors) and inelastic neutron scattering. The intention is to provide an environment with fit routines, data handling routines (based on NumPy arrays) and a model library to allow the user to focus onto user-written models for data analysis with the benefit of convenient documentation of scientific data evaluation in a scripting environment.

Consensus Bayesian assessment of protein molecular mass from solution X-ray scattering data.

Molecular mass (MM) is one of the key structural parameters obtained by small-angle X-ray scattering (SAXS) of proteins in solution and is used to assess the sample quality, oligomeric composition and to guide subsequent structural modelling. Concentration-dependent assessment of MM relies on a number of extra quantities (partial specific volume, calibrated intensity, accurate solute concentration) and often yields limited accuracy. Concentration-independent methods forgo these requirements being based on the relationship between structural parameters, scattering invariants and particle volume obtained directly from the data. Using a comparative analysis on 165,982 unique scattering profiles calculated from high-resolution protein structures, the performance of multiple concentration-independent MM determination methods was assessed. A Bayesian inference approach was developed affording an accuracy above that of the individual methods, and reports MM estimates together with a credibility interval. This Bayesian approach can be used in combination with concentration-dependent MM methods to further validate the MM of proteins in solution, or as a reliable stand-alone tool in instances where an accurate concentration estimate is not available.

Force field development and simulations of intrinsically disordered proteins.

Intrinsically disordered proteins (IDPs) play important roles in many physiological processes such as signal transduction and transcriptional regulation. Computer simulations that are based on empirical force fields have been increasingly used to understand the biophysics of disordered proteins. In this review, we focus on recent improvement of protein force fields, including polarizable force fields, concerning their accuracy in modeling intrinsically disordered proteins. Some recent benchmarks and applications of these force fields are also overviewed.

Characterizing a crystal from an initial native dataset.

Methods are presented for characterizing a crystal given an initial X-ray diffraction dataset. These methods can facilitate the structure determination process and illuminate the oligomeric state and symmetry of your molecule before the crystal structure is determined. Specifically, these methods include (1) calculation of Matthews coefficient to estimate the number of molecules in the asymmetric unit; (2) calculation and interpretation of a self-rotation function to evaluate the point group symmetry of the crystallized oligomer, if contained in the crystal; (3) calculation and interpretation of a native Patterson map to evaluate the presence of noncrystallographic translational symmetry; and (4) calculation of statistics to evaluate the possibility of merohedral twinning in a crystal.

The statistics of the highest E value.

In a previous publication, the Gumbel-Fisher-Tippett (GFT) extreme-value analysis has been applied to investigate the statistics of the intensity of the strongest reflection in a thin resolution shell. Here, a similar approach is applied to study the distribution, expectation value and standard deviation of the highest normalized structure-factor amplitude (E value). As before, acentric and centric reflections are treated separately, a random arrangement of scattering atoms is assumed, and E-value correlations are neglected. Under these assumptions, it is deduced that the highest E value is GFT distributed to a good approximation. Moreover, it is shown that the root of the expectation value of the highest ;normalized' intensity is not only an upper limit for the expectation value of the highest E value but also a very good estimate. Qualitatively, this can be attributed to the sharpness of the distribution of the highest E value. Although the formulas were derived with various simplifying assumptions and approximations, they turn out to be useful also for real small-molecule and protein crystal structures, for both thin and thick resolution shells. The only limitation is that low-resolution data (below 2.5 A) have to be excluded from the analysis. These results have implications for the identification of outliers in experimental diffraction data.

Serial Synchrotron X-Ray Crystallography (SSX).

Prompted by methodological advances in measurements with X-ray free electron lasers, it was realized in the last two years that traditional (or conventional) methods for data collection from crystals of macromolecular specimens can be complemented by synchrotron measurements on microcrystals that would individually not suffice for a complete data set. Measuring, processing, and merging many partial data sets of this kind requires new techniques which have since been implemented at several third-generation synchrotron facilities, and are described here. Among these, we particularly focus on the possibility of in situ measurements combined with in meso crystal preparations and data analysis with the XDS package and auxiliary programs.

Reconstruction of a yeast cell from X-ray diffraction data.

Details are provided of the algorithm used for the reconstruction of yeast cell images in the recent demonstration of diffraction microscopy by Shapiro, Thibault, Beetz, Elser, Howells, Jacobsen, Kirz, Lima, Miao, Nieman & Sayre [Proc. Natl Acad. Sci. USA (2005), 102, 15343-15346]. Two refinements of the iterative constraint-based scheme are developed to address the current experimental realities of this imaging technique, which include missing central data and noise. A constrained power operator is defined whose eigenmodes allow the identification of a small number of degrees of freedom in the reconstruction that are negligibly constrained as a result of the missing data. To achieve reproducibility in the algorithm's output, a special intervention is required for these modes. Weak incompatibility of the constraints caused by noise in both direct and Fourier space leads to residual phase fluctuations. This problem is addressed by supplementing the algorithm with an averaging method. The effect of averaging may be interpreted in terms of an effective modulation transfer function, as used in optics, to quantify the resolution. The reconstruction details are prefaced with simulations of wave propagation through a model yeast cell. These show that the yeast cell is a strong-phase-contrast object for the conditions in the experiment.

Diffraction and imaging from a beam of laser-aligned proteins: resolution limits.

The effect of the limited alignment of hydrated molecules is considered in a laser-aligned molecular beam, on diffraction patterns taken from the beam. Simulated patterns for a protein beam are inverted using the Fienup-Gerchberg-Saxton phasing algorithm, and the effect of limited alignment on the resolution of the resulting potential maps is studied. For a typical protein molecule (lysozyme) with anisotropic polarizability, it is found that up to 1 kW of continuous-wave near-infrared laser power (depending on dielectric constant), together with cooling to liquid-nitrogen temperatures, may be needed to produce sufficiently accurate alignment for direct observation of the secondary structure of proteins in the reconstructed potential or charge-density map. For a typical virus (TMV), a 50 W continuous-wave laser is adequate for subnanometre resolution at room temperature. The dependence of resolution on laser power, temperature, molecular size, shape and dielectric constant is analyzed.

Crystallization and preliminary X-ray diffraction analysis of phosphoglucose isomerase from Pyrococcus furiosus.

In several euryarchaeota, phosphoglucose isomerase (PGI) activity is catalyzed by an enzyme unrelated to the well-known family of PGI enzymes found in prokaryotes, eukaryotes and some archaea. In order to understand the mechanistic differences between the two families of enzymes we have crystallized PGI from the archaeon Pyrococcus furiosus. The crystals belong to the space group P2(1) and a complete dataset extending to 1.9 A resolution has been collected.

Initiating structural studies of Lys49-PLA2 homologues complexed with an anionic detergent, a fatty acid and a natural lipid.

Lys49-Phospholipase A2 (Lys49-PLA(2) - EC 3.1.1.4) homologues damage membranes by a Ca2+-independent mechanism which does not involve catalytic activity. Both MjTX-II from Bothrops moojeni and BthTX-I from Bothrops jararacussu are dimeric in solution and in the crystalline states, and a model for the Ca2+-independent membrane damaging mechanism has been suggested in which flexibility at the dimer interface region permits quaternary structural transitions between "open" and "closed" membrane bound dimer conformations which results in the perturbation of membrane phospholipids and disruption of the bilayer structure. With the aim of gaining insights into the structural determinants involved in protein/lipid association, we report here the crystallization and preliminary X-ray analysis of the (i) MjTX-II/SDS complex at a resolution of 2.78A, (ii) MjTX-II/STE complex at a resolution of 1.8 A and (iii) BthTX-I/DMPC complex at 2.72A. These complexes were crystallized by the hanging drop vapour-diffusion technique in (i) HEPES buffer (pH 7.5) 1.8M ammonium sulfate with 2% (w/v) polyethyleneglycol 400, in (ii) 0.6-0.8 M sodium citrate as the precipitant (pH 6.0-6.5) and in (iii) sodium citrate buffer (pH 5.8) and PEG 4000 and 20% isopropanol, respectively. Single crystals of these complexes have been obtained and X-ray diffraction data have been collected at room temperature using a R-AXIS IV imaging plate system and graphite monochromated Cu Kalpha X-ray radiation generated by a Rigaku RU300 rotating anode generator for (i) and (iii) and using using a Synchrotron Radiation Source (Laboratório Nacional de Luz Sincrotron, LNLS, Campinas, Brazil) for (ii).

Structural and energetic aspects of a new bupropion hydrochloride polymorph.

Crystalline bupropion hydrochloride [(±)1-(3-chlorophenyl)-2-[(1,1-dimethylethyl)amino]-1-propanone hydrochloride], recently characterized as form 1, was found to undergo, upon storage at RT within months, a solid-solid conversion to a new polymorphic form, hereafter named form 2, containing a markedly different molecular conformer in the solid state. This new form, available only as a polycrystalline material, has been fully characterized using structural X-ray powder diffraction methods, coupled to thermoanalytical analyses. The relative stability of the two crystalline phases (forms 1 and 2) was compared by quantum mechanics calculations including density functional methods specific for solid state molecular systems. Bupropion hydrochloride form 2 crystallizes in the orthorhombic space group Pbca with Z=8, a=27.2853(5)Å, b=8.7184(3)Å, c=12.0422(3)Å, V=2864.7(1)ų, as centrosymmetric dimers, thanks to the presence of N-H…Cl interactions, and µ2-bridging chloride ions, each connected to two protonated amine moieties.

Characterizing flexible and intrinsically unstructured biological macromolecules by SAS using the Porod-Debye law.

Unstructured proteins, RNA or DNA components provide functionally important flexibility that is key to many macromolecular assemblies throughout cell biology. As objective, quantitative experimental measures of flexibility and disorder in solution are limited, small angle scattering (SAS), and in particular small angle X-ray scattering (SAXS), provides a critical technology to assess macromolecular flexibility as well as shape and assembly. Here, we consider the Porod-Debye law as a powerful tool for detecting biopolymer flexibility in SAS experiments. We show that the Porod-Debye region fundamentally describes the nature of the scattering intensity decay by capturing the information needed for distinguishing between folded and flexible particles. Particularly for comparative SAS experiments, application of the law, as described here, can distinguish between discrete conformational changes and localized flexibility relevant to molecular recognition and interaction networks. This approach aids insightful analyses of fully and partly flexible macromolecules that is more robust and conclusive than traditional Kratky analyses. Furthermore, we demonstrate for prototypic SAXS data that the ability to calculate particle density by the Porod-Debye criteria, as shown here, provides an objective quality assurance parameter that may prove of general use for SAXS modeling and validation.

Mapping granular structure in the biological adhesive of Phragmatopoma californica using phase diverse coherent diffractive imaging.

This paper demonstrates the application of the high sensitivity, low radiation dose imaging method recently presented as phase diverse coherent diffraction imaging, to the study of biological and other weakly scattering samples. The method is applied, using X-ray illumination, to quantitative imaging of the granular precursors of underwater adhesive produced by the marine sandcastle worm, Phragmatopoma californica. We are able to observe the internal structure of the adhesive precursors in a number of states.

Characterizing the space of interatomic distance distribution functions consistent with solution scattering data.

Scattering of neutrons and X-rays from molecules in solution offers alternative approaches to the study of a wide range of macromolecular structures in their solution state without crystallization. We study one part of the problem of elucidating three-dimensional structure from solution scattering data, determining the distribution of interatomic distances, P(r), where r is the distance between two atoms in the protein molecule. This problem is known to be ill-conditioned: for a single observed diffraction pattern, there may be many consistent distance distribution functions, and there is a risk of overfitting the observed scattering data. We propose a new approach to avoiding this problem: accepting the validity of multiple alternative P(r) curves rather than seeking a single "best." We place linear constraints to ensure that a computed P(r) is consistent with the experimental data. The constraints enforce smoothness in the P(r) curve, ensure that the P(r) curve is a probability distribution, and allow for experimental error. We use these constraints to precisely describe the space of all consistent P(r) curves as a polytope of histogram values or Fourier coefficients. We develop a linear programming approach to sampling the space of consistent, realistic P(r) curves. On both experimental and simulated scattering data, our approach efficiently generates ensembles of such curves that display substantial diversity.