Direct Phase Determination Research Articles

Direct phase determination in protein crystallography using iterative projection algorithms

Direct phase determination in protein crystallography using iterative projection algorithms

A general method for directly phasing diffraction data from high-solvent-content protein crystals.

A procedure is described for direct phase determination in protein crystallography, applicable to crystals with high solvent content. The procedure requires only the diffraction data and an estimate of the solvent content as input. Direct phase determination is treated as a constraint satisfaction problem, in which an image is sought that is consistent with both the diffraction data and generic constraints on the density distribution in the crystal. The problem is solved using an iterative projection algorithm, the Difference Map algorithm, which has good global convergence properties, and can locate the correct solution without any initial phase information. Computational efficiency is improved by breaking the problem down into two stages; initial approximation of the molecular envelope at low resolution, followed by subsequent phase determination using all of the data. The molecular envelope is continually updated during the phase determination step. At both stages, the algorithm is initiated with many different and random phase sets, which are evolved subject to the constraints. A clustering procedure is used to identify consistent results across multiple runs, which are then averaged to generate consensus envelopes or phase sets. The emergence of highly consistent phase sets is diagnostic of success. The effectiveness of the procedure is demonstrated by application to 42 known structures of solvent fraction 0.60-0.85. The procedure works robustly at intermediate resolutions (1.9-3.5 Å) but is strongly dependent on crystal solvent content, only working routinely with solvent fractions greater than 0.70.

An optical Fourier transform coprocessor with direct phase determination

The Fourier transform is a ubiquitous mathematical operation which arises naturally in optics. We propose and demonstrate a practical method to optically evaluate a complex-to-complex discrete Fourier transform. By implementing the Fourier transform optically we can overcome the limiting O(nlogn) complexity of fast Fourier transform algorithms. Efficiently extracting the phase from the well-known optical Fourier transform is challenging. By appropriately decomposing the input and exploiting symmetries of the Fourier transform we are able to determine the phase directly from straightforward intensity measurements, creating an optical Fourier transform with O(n) apparent complexity. Performing larger optical Fourier transforms requires higher resolution spatial light modulators, but the execution time remains unchanged. This method could unlock the potential of the optical Fourier transform to permit 2D complex-to-complex discrete Fourier transforms with a performance that is currently untenable, with applications across information processing and computational physics.

Colloquium: Understanding quantum weak values: Basics and applications

Since its introduction 25 years ago, the quantum weak value has gradually transitioned from a theoretical curiosity to a practical laboratory tool. While its utility is apparent in the recent explosion of weak value experiments, its interpretation has historically been a subject of confusion. Here a pragmatic introduction to the weak value in terms of measurable quantities is presented, along with an explanation for how it can be determined in the laboratory. Further, its application to three distinct experimental techniques is reviewed. First, as a large interaction parameter it can amplify small signals above technical background noise. Second, as a measurable complex value it enables novel techniques for direct quantum state and geometric phase determination. Third, as a conditioned average of generalized observable eigenvalues it provides a measurable window into nonclassical features of quantum mechanics. In this selective review, a single experimental configuration to discuss and clarify each of these applications is used.

Measurement of the transverse electric field profile of light by a self-referencing method with direct phase determination

We present a method for measuring the transverse electric field profile of a beam of light which allows for direct phase retrieval. The measured values correspond, within a normalization constant, to the real and imaginary parts of the electric field in a plane normal to the direction of propagation. This technique represents a self-referencing method for probing the wavefront characteristics of light.

Experimental proposal for measuring the Gouy phase of matter waves

The Schrödinger equation for an atomic beam predicts that it must have a phase anomaly near the beam waist analogous to the Gouy phase of an electromagnetic beam. We propose here a feasible experiment that allows for direct determination of this anomalous phase using Ramsey interferometry with Rydberg atoms. Possible experimental limitations are discussed, and shown to be completely under control within present-day technology. We also discuss how this finding can open the possibility of using the spatial mode wavefunctions of atoms as q-dits, since the Gouy phase is an essential ingredient for making rotations in the quantum states.

Collimation testing using wedge plate lateral shearing interferometry and Fourier fringe analysis

For checking the collimation of an optical beam Fourier fringe analysis has been incorporated into the wedge plate interferometric setup. Typical interferograms corresponding to ‘in-focus’, ‘at-focus’ and ‘out-of-focus’ positions of an optical beam have been recorded. As per the testing procedure, FFT of the recorded interferometer is computed digitally, and necessary processing for direct determination of phase is undertaken. Finally, the phase data is unwrapped and plotted as a function of pixel position along the direction perpendicular to the shear. The slope of the phase provides the information regarding collimation position of the collimator. As the collimation position is detected by the direct measurement of the phase over the whole area of the interferogram, high accuracy, reliability and precision are achieved.

Phase Analysis of Hydraulic Cements by X-Ray Powder Diffraction: Precision, Bias, and Qualification

Abstract Estimates of cement phase composition have traditionally been made using bulk chemical analysis and the Bogue calculations, developed over 75 years ago. X-ray powder diffraction analysis provides an alternative with direct phase determination, qualitatively and quantitatively. The Rietveld method to refine multi-phase X-ray powder diffraction patterns of cement uses a whole-pattern approach with determination of the best-fit set of calculated diffraction patterns based on crystal structure models. The pattern intensities calculated in the analysis are used to estimate relative phase abundances. To estimate precision and bias and to establish qualification criteria, an inter-laboratory study involving eleven laboratories was undertaken. Four cement reference specimens were prepared using NIST SRM clinkers compounded with known amounts of one or more of gypsum, bassanite, anhydrite, and calcite. Results of the study were used to estimate the inter- and intra-laboratory precision and bias of phase abundance determinations. The data were also used to establish criteria for qualification of a laboratory, using prediction intervals that reflect the collective performance characteristics of the participants. Results show an improvement in repeatability over previous cement inter-laboratory studies utilizing internal standard-based, peak area measurement methods.

Optical tunneling of single-cycle terahertz bandwidth pulses.

We report time-domain measurements with subpicosecond resolution of optical tunneling of terahertz electromagnetic pulses undergoing frustrated total internal reflection. Measurements of the transmitted electromagnetic pulses over a 3 THz bandwidth permits direct determination of frequency-dependent phase and amplitude changes in both the thin and opaque barrier limits in a single measurement. A complex frequency response function describing propagation through the barrier is developed based upon linear dispersion theory and the Fresnel coefficients at complex angles in the optical barrier. Our measurements are in excellent agreement with this theoretical model across the experimentally determined bandwidth; the model makes no assumptions about the beam path through the barrier and has no adjustable parameters. The theory is shown to satisfy the Kramers-Kronig relations, indicating causal propagation across the barrier.

Substrate-mediated multiwave resonance grazing incidence x-ray diffraction in thin films: A method for direct phase determination

Direct phase determination of surface in-plane reflection is realized for thin films on substrates by using substrate reflections as an intermediary to enhance the coherent interaction in resonant multiwave grazing incidence diffraction in thin films. The coupling of the in-plane diffracted waves at the interface between the thin film and the substrate is essential. The intensity variation due to this enhanced interaction/coupling becomes clearly visible, thus leading to unambiguous phase determination. This opens a different way for direct phase determination of surface reflections in thin films.

Polarization-resolved output analysis of X-ray multiple-wave interaction.

The polarization suppression of the interfering components in X-ray multiple-wave interaction is observed for the first time by using a polarization analyzer with an arbitrary inclination of the diffraction plane with respect to that of the investigated crystal. The condition for total suppression of the multiple-wave interaction outside the investigated crystals by a polarization analyzer is derived theoretically from the modified Born approximation. By means of the partial suppression of the strong interfering component, the increase in the visibility of multiple-wave interference is experimentally and theoretically demonstrated. The proposed experimental polarization-resolved technique provides an operational way to enhance the visibility of X-ray multiple-wave interaction outside the investigated crystals for direct phase determination.

Density histograms in protein electron crystallography—direct phase determination of bacteriorhodopsin at 6 Å resolution

Direct methods for crystallographic phase determination have been used to solve the projected structure of native glucose-embedded bacteriorhodopsin (plane group p3, a=62.4 Å) to 6 Å, without accepting any information from electron micrographs. After defining the origin and enantiomorph and adding one algebraic phase term to the basis set, expansions into the complete set of 50 unique reflections via the Sayre equation, testing four possible values of the algebraic term, could be screened with figures of merit based on the protein density histogram, where the auto-correlation function of the ideal histogram is an endpoint for the cross-correlation of experimental histogram with this ideal distribution. Initial phase determination, followed by Fourier refinement and some additional phase permutation, yields a solution with an overall mean phase error of 60.8 or 35.4° for the 18 most intense reflections.

Low-resolution direct phase determination in protein electron crystallography - breaking globular constraints.

Although the assumption of an overall globular scattering entity can be useful for determining crystallographic phases for a protein at low resolution, there is a point where this pseudoatomic model must be abandoned for further phase refinement. Using 6 A resolution electron diffraction data from aquaporin (AQP-CHIP) as an example, phases of the 16 most intense reflections from a previous direct solution (Dorset & Jap (1998). Acta Cryst. D54, 615-621) were modified with a Hadamard error-correcting code to produce potential maps very similar to the ones obtained using phases from the Fourier transform of averaged electron micrographs. The choice of the optimal phase set was made via the cross correlation of experimental with anticipated density histograms using the autocorrelation function of the latter histogram as the desired endpoint.

Direct Phase Determination of Surface In-plane Reflection of Thin films by X-ray Grazing Incident Three-wave Resonance Diffraction

Direct Phase Determination of Surface In-plane Reflection of Thin films by X-ray Grazing Incident Three-wave Resonance Diffraction

The phenomenon of polarization suppression of X-ray Umweg multiple waves in crystals

The phenomenon of the polarization suppression of X-ray Umweg multiple waves in Renninger scans [Renninger (1937). Z. Kristallogr. 97, 107-121] of crystals, showing intensity decrease due to properly chosen wavelength and polarization of incident radiation, is observed. That is, one of the participating wave components in the multiple-wave interference is reduced considerably so that the intensity of multiple diffraction is decreased. The condition for total suppression of the multiple-wave interaction in crystals is derived theoretically from the Born approximation and verified with exact dynamical calculation and experiments. Partial suppression of the strong Umweg interfered component is demonstrated using elliptically or linearly polarized synchrotron radiation. The suppressed multiple-wave intensity distribution reveals high sensitivity to X-ray reflection phase. This multiple-diffraction technique under partial polarization suppression provides an alternative way of enhancing the visibility of multiple-wave interference in crystals for direct phase determination.

Possibility of direct determination of the quantum phase of continua utilizing the phase of lasers

We show that the quantum phase of continua can be directly determined utilizing the phase of linearly/circularly polarized lasers. With this method the phase difference of continua with opposite as well as same parities can be obtained from the phase lag of the angle-resolved photoelectron signal with respect to the relative phase of lasers. For illustration, the proposed method is specifically applied to the Na atom.

Direct phase determination in protein electron crystallography: the pseudo-atom approximation.

The crystal structure of halorhodopsin is determined directly in its centrosymmetric projection using 6.0-A-resolution electron diffraction intensities, without including any previous phase information from the Fourier transform of electron micrographs. The potential distribution in the projection is assumed a priori to be an assembly of globular densities. By an appropriate dimensional re-scaling, these "globs" are then assumed to be pseudo-atoms for normalization of the observed structure factors. After this treatment, the structure is determined directly by conventional direct methods, followed by Fourier refinement, leading to a mean phase deviation of only 20 degrees (from the values originally found from the image transform) for the 45 most intense reflections.

Molecular haplotyping of genetic markers 10 kb apart by allele-specific long-range PCR.

Haplotypes, combinations of polymorphic markers in a chromosome, are critical for genome diversity research. However, their utility in population samplings is compromised by uncertain linkage phase determinations from unrelated individuals. Molecular haplotyping accomplishes direct phase determination by generation of hemizygous templates from diploid genomic samples. We report molecular haplotyping by allele-specific long-range PCR of two markers 9.5 kb apart at the CD4 locus: a bi-allelic Alu deletion and a multi-allelic repeat. We verified CD4 molecular haplotypes by classical Mendelian analysis. Molecular haplotyping should prove useful in mapping disease genes and in establishing founder effects.

Direct determination of phase from three-beam convergent-beam diffraction patterns of centrosymmetric crystals

Direct determination of phase from three-beam convergent-beam diffraction patterns of centrosymmetric crystals

Direct Phasing in Protein Electron Crystallography – Phase Extension and the Prospects forAb InitioDeterminations

Zonal diffraction amplitudes and crystallographic phases, derived from an averaged electron micrograph of two-dimensionally crystalline E. coli Omp F outer membrane porin (plane group p31m, a = 72 A), embedded in glucose, were used as a model data set to test the feasibility of direct phase extension and ab initio direct phase determination. If 17 phase terms derived from e.g. a 10 A (diffraction) resolution image are expanded to 6 A by the Sayre-Hughes equation, the unknown phases are found with reasonable accuracy (mean error 43 degrees for 25 reflections). This, however, is not the most optimal starting point. As a function of initial image resolution, the accuracy of the phase extension to 6 A is approximately a parabolic function. That is, an optimal basis resolution, found at 11 A (i.e. 14 defined reflections), produces a least mean error of 18 degrees for 28 new reflections. In addition, ab initio phase determination is possible via a multisolution technique, using a test for density flatness as a figure of merit. The success of the determination, again is sensitive to the size of the starting basis set generated from the permuted unknown reflections. If an annealing step is used to improve the basis set, the test for flatness will identify which reflections should be changed in phase. However, this figure of merit is not absolutely reliable for finding the exact value of the unknown phases.