Interpreting macromolecular diffraction through simulation

Abstract

This chapter discusses the use of diffraction simulators to improve experimental outcomes in macromolecular crystallography, in particular for future experiments aimed at diffuse scattering. Consequential decisions for upcoming data collection include the selection of either a synchrotron or free electron laser X-ray source, rotation geometry or serial crystallography, and fiber-coupled area detector technology vs. pixel-array detectors. The hope is that simulators will provide insights to make these choices with greater confidence. Simulation software, especially those packages focused on physics-based calculation of the diffraction, can help to predict the location, size, shape, and profile of Bragg spots and diffuse patterns in terms of an underlying physical model, including assumptions about the crystal's mosaic structure, and therefore can point to potential issues with data analysis in the early planning stages. Also, once the data are collected, simulation may offer a pathway to improve the measurement of diffraction, especially with weak data, and might help to treat problematic cases such as overlapping patterns.

Keywords: X-ray crystallography; diffraction pattern; diffuse scattering; simulated diffraction.

Fig. 1.

A general list of experimental elements for physical modeling. The first three columns list parameters relevant to all experiments, while the fourth column applies only to rotation experiments, and not to serial crystallographic data, where every shot is static with no known rotational relationship to any other shot.

Fig. 2.

A comparison of imaging detector types. (A) In a fiber-coupled area detector, X-rays are converted to visible photons in the phosphor layer (green), then coupled through a fiber-optic taper (orange) to a charge-coupled device (blue). (B) In a pixel array detector (PAD), the intensity response of one pixel as a knife edge is translated across the detector surface, exposed to an X-ray flood field. Photons are directly converted to detected charge in silicon, with most of the response confined to one pixel. (C) A simulated image models the point-spread function of a fiber-coupled detector, while (D) the corresponding simulation of a PAD shows much sharper Bragg spots. Panel A: reproduced from Holton et al. (2012), with permission from IUCr Journals. Panel B: adapted from Koerner et al. (2009), with permission from IOP Publishing.

Fig. 3

Fundamental framework for image simulation. As implied by Scheme 1, (A) the adjunct spaces representing crystal (direct) and diffraction pattern (reciprocal) are (B) related by Bragg’s law, relating the wavelength $\lambda$, the scattering angle θ, and the spatial resolution d. The red reciprocal lattice point (located a reciprocal distance 1/d from the reciprocal origin O) satisfies the diffraction condition and therefore produces a scattered ray, reaching the indicated detector pixel. Although the adjunct spaces co-rotate, their rotation origins differ in the Ewald sphere construction (B), with the crystal rotating around the sphere center, and the reciprocal lattice rotating around O. Adapted from Sauter et al., 2014.

Fig. 4.

Size and shape of the mosaic block affects the Bragg spot geometry. Each reciprocal lattice point may be thought of as a Fourier transform of the average coherently scattering block within the crystal (Scheme 2). If the block is a rectangular parallelepiped, seven unit cells on an edge (Eqn. 2), the simulated pattern is as shown in (A), with five fringes appearing between every Bragg spot center. In contrast, if the block geometry draws from a 3D Gaussian distribution with a mean width of $〈N〉$=7 cells (Eqn. 4), the generated pattern is as shown in (B). Crosshairs indicate the direct beam.

Fig. 5.

Bragg spot position changes slightly with crystal rotation. (A) As the crystal rotates on the goniometer spindle, a reciprocal lattice point (RLP) co-rotates thru the diffraction condition (the Ewald sphere). However, the ${\mathbf{s}}_1$ vector, indicating the direction of the corresponding Bragg spot, makes a slightly larger scattering angle 2θ when the RLP is inside the sphere, than when the RLP is outside. (B) The rocking curve indicates that the diffracted spot intensity reaches a maximum when the RLP intersects the Ewald sphere surface. (C) Experimental diffraction patterns collected at increasing spindle rotations reflect these effects. Panels A,B: reproduced from Sauter & Adams, 2017 with permission from the Royal Society of Chemistry. Panel C: adapted from Pflugrath, 1999.

Fig. 6.

Summary of the diffuse scattering pattern from macromolecular crystallography. (A) A section through reciprocal space, highlighting various contributions to the isotropic scattering (left) and anisotropic, or variational scattering (right). (B) Detail of the scattering focusing on one Bragg spot, as the goniometer spindle rotation brings the reciprocal point through the Ewald sphere. The blue curve is the rocking curve proper, with its sharp peak at the central position (note the logarithmic scale of intensity), while the black data originate from the long-range correlations, forming an intense halo around the tails of the Bragg peak. Adapted from Meisburger et al., 2020.

Fig. 7.

Snapshot of the simtbx.sim_view image viewer, simulating the diffuse halo pattern from calmodulin crystals (PDB entry 1CM1), which reflects a long-range correlation between unit cells, according to the model of Wall et al. (1997), equation 8. Parameters for the correlation length γ and vibrational amplitude-squared ${\sigma }^2$ may be specified by the user to empirically fit experimental data. In this case, streaked diffuse features were produced by multiplying γ by an anisotropy factor along the a−b and c unit cell directions in direct space (leaving a+b alone), and multiplying ${\sigma }^2$ by the same anisotropy factor along the a−b direction (leaving a+b and c alone). The Laue symmetry was then applied, leading to streaks along both the a*−b* and a*+b* directions in reciprocal space. In the present case, this type of anisotropy was purely an ad hoc decision allowing us to generate a simulation that resembles the data in the 1997 paper, but this exercise suggests that correct models may require similar anisotropic features in the general case. Courtesy of Michael Wall.

Fig. 8.

Simulations that differ in the crystal’s mosaic rotational disorder but that are otherwise identical. (A) Perfectly ordered mosaic blocks (rotational full width η = 0°). (B) Small mosaicity typical of ambient temperature crystals (η = 0.1°). (C) Medium-high mosaicity found with cryopreserved samples (η = 0.6°). (D) Large mosaicity, with arclets indicating high disorder (η = 1.5°). Courtesy of James Holton.

Fig. 9.

Simulations that differ in the dispersion of the incident beam but that are otherwise identical. (A) Low dispersion (ΔE/E = 0.014%) typically produced by a synchrotron monochromator equipped with a Si 111 crystal. (B) Medium dispersion (ΔE/E = 0.25%) found in an XFEL pulse. (C) Pink beam, ΔE/E = 1.6% and (D) Laue, ΔE/E = 5.1%. Courtesy of James Holton.

Scheme 1.

Initial framework for simulation

Scheme 2.

Revised simulation framework taking into account the finite size of the mosaic block.