Date Published: January 01, 2019
Publisher: International Union of Crystallography
Author(s): Andrew J. Morgan, Kartik Ayyer, Anton Barty, Joe P. J. Chen, Tomas Ekeberg, Dominik Oberthuer, Thomas A. White, Oleksandr Yefanov, Henry N. Chapman.
This article reports on the combined use of Bragg reflections and diffuse scatter for structure determination in crystallography.
The diffraction of coherent radiation from an object onto a detector placed far from the object gives rise to smoothly varying diffraction features that are bandwidth limited by the size of the object. The detector measures the intensity, the mean-squared value of the electric field amplitude, but not the phases of the scattered radiation. If the phases were known, then one could synthesize an image of the object directly by numerical propagation of the wavefront of the coherent field from the detector back to the sample. This image would be proportional to the electron density or scattering strength of the object. However, without the phases, the numerical transformation of the measured intensities only yields a map of the pair correlations, also known as the autocorrelation of the object density, of the point scatterers in the object.
We consider the mathematical description of a crystal that is generated from a single rigid unit [with density at position ]. This rigid unit may be what is generally thought of as the asymmetric unit of the crystal, or it may be a particular molecular complex. We consider for now that there is only one repeating rigid unit, but more generally there could be several types, such as two domains of a molecule. The unit-cell density can be generated from the single rigid unit along with the crystal symmetry and the unit-cell dimensions via rotation and translation operations , where the sum is over the M symmetry-related copies of in the unit cell. Here is the rotation matrix for the mth copy and is the translation vector.
Having described the observable quantities, namely the diffraction intensities , in terms of the quantity of interest which is the rigid-unit density , we now turn to the task of recovering from for a crystal with translational disorder. We assume that all quantities in equation (2) (except of course for ) have been determined. This includes the disorder parameter σ, the internal symmetry of the unit cell (the ’s and ’s) and additionally the unit-cell parameters as well as the solvent fraction of the crystal. We cast this problem in the form of a phase problem in coherent diffractive imaging (CDI), which requires that we formulate projection operators responsible for enforcing the known constraints on the solution which are described below in Sections 4 and 5. We also describe the conditions that must be satisfied for a unique solution to exist in Section 6 and, finally, we verify that the rigid-unit density can be reconstructed from the simulated noisy diffraction intensity in Section 7.
For the data projection , we now employ a useful property of projection operators, which is that they may be defined in real or reciprocal space. This is because the Euclidean distances between vectors are preserved under a Fourier transform (Parseval’s theorem). In this example, where diffraction is measured from a single finite object, the diffraction intensities of our object are equal to the square modulus of the Fourier transform of the object density which in turn is equal to the sum of the squares of its real and imaginary components , where is the Fourier transform of . At every voxel in reciprocal space we wish to make the smallest change to the independent variables and such that . This equation describes the constraint surface at each q value as a circle of radius and the projection operator simply scales while keeping the ratio fixed: .
is more straightforward to construct; it makes the smallest change to a given estimate for the rigid-unit densities at a given iterate such that the mapped projection is consistent with our prior knowledge of the crystal. We must ensure that the rigid units are all identical copies of themselves (in different orientations), that they are arranged according to the symmetry of the crystal, that their densities do not overlap, and that they each have a given number of volume elements that deviate from the solvent density level, consistent with the solvent fraction of the crystal.
In phase retrieval the constraint ratio (Ω) is defined by the ratio of linearly independent equations to unknown quantities in the phase problem3 (Elser & Millane, 2008 ▸). If then the phase problem is certainly under-determined and there is no unique solution. For , a given solution may be unique and in some cases it can be shown that multiple solutions are pathologically rare (Bates, 1984 ▸). Thus is a necessary but not sufficient condition for a unique solution. A single isolated object has , where the lower bound corresponds to an object with a convex and centrosymmetric support, while non-convex supports have a higher constraint ratio and are easier to solve (Fienup, 1987 ▸).
Now that we have defined the crystal diffraction model, determined the required projection operators and that a unique solution may exist, we now demonstrate that our IPA is capable of solving for the electron density of a potato multicystatin crystal from simulated noisy diffraction.
Having shown that model-free phasing of diffraction from crystals with translational disorder is possible, we now consider some aspects of the application of this method to experimental data. Because Bragg peaks often yield very bright and sharp peaks on the detector, any underlying background can usually be estimated (and thus subtracted from the data) by examining the detected signal in the immediate neighbourhood of the diffraction spot. This is not true however for the continuous diffraction. In general, this method places higher demands on data collection and estimation of the background, for example due to the crystal solvent, ice formation or from the carrying medium of the crystal such as a liquid jet, aerosol or sample holder. Chapman et. al. have recently suggested a method to estimate this background (Chapman et al., 2017 ▸). Standard crystallographic methods for structure retrieval are also fairly robust with regard to missing diffraction intensity measurements. For instance, when calculating the R-free metric, some reflections are excluded when fitting the molecular model to the diffraction data (Brünger, 1992 ▸). However, in model-free phasing, missing data regions can lead to unconstrained modes in the reconstruction (Thibault et al., 2006 ▸) which can be a problem, particularly near the origin where a beamstop is often placed. For these reasons, we expect that a combination of our proposed method with model fitting and refinement may often be the more robust approach, particularly for structures where prior information is available.