Macromolecular diffractive imaging using imperfect crystals

Abstract

The three-dimensional structures of macromolecules and their complexes are mainly elucidated by X-ray protein crystallography. A major limitation of this method is access to high-quality crystals, which is necessary to ensure X-ray diffraction extends to sufficiently large scattering angles and hence yields information of sufficiently high resolution with which to solve the crystal structure. The observation that crystals with reduced unit-cell volumes and tighter macromolecular packing often produce higher-resolution Bragg peaks suggests that crystallographic resolution for some macromolecules may be limited not by their heterogeneity, but by a deviation of strict positional ordering of the crystalline lattice. Such displacements of molecules from the ideal lattice give rise to a continuous diffraction pattern that is equal to the incoherent sum of diffraction from rigid individual molecular complexes aligned along several discrete crystallographic orientations and that, consequently, contains more information than Bragg peaks alone. Although such continuous diffraction patterns have long been observed--and are of interest as a source of information about the dynamics of proteins--they have not been used for structure determination. Here we show for crystals of the integral membrane protein complex photosystem II that lattice disorder increases the information content and the resolution of the diffraction pattern well beyond the 4.5-ångström limit of measurable Bragg peaks, which allows us to phase the pattern directly. Using the molecular envelope conventionally determined at 4.5 ångströms as a constraint, we obtain a static image of the photosystem II dimer at a resolution of 3.5 ångströms. This result shows that continuous diffraction can be used to overcome what have long been supposed to be the resolution limits of macromolecular crystallography, using a method that exploits commonly encountered imperfect crystals and enables model-free phasing.

Extended Data Fig. 1. Continuous diffraction exhibits Wilson statistics

Histogram of merged continuous intensities in a q range of 0.22 Å^-1 to 0.25 Å^-1. One can see that, above a background level of around 1 photon / pixel / pulse, the logarithm of the histogram follows a linear trend with negative slope characteristic of the exponential distribution predicted by Wilson statistics.

Extended Data Fig. 2. Data quality and resolution of Bragg diffraction

Plot of the reduced Pearson correlation coefficient, CC* (ref. 41) as an estimate of the consistency of the integrated Bragg intensities determined from 25,585 indexed patterns. A value of CC* = 0.5 is reached at q = 0.23 Å^-1, or a resolution of 4.3 Å.

Extended Data Fig. 3. Strongest continuous diffraction occurs with strongest Bragg diffraction

a, Two-dimensional histogram of patterns, sorted by the integrated counts in the continuous component of the diffraction pattern (in a q range of 0.22 Å^-1 to 0.34 Å^-1) and the integrated signal in all detected Bragg peaks, for all 25,585 indexed patterns. We chose 2,848 patterns with the strongest continuous diffraction signal above 17 X-ray counts (purple line) in the q region to generate the 3D continuous pattern shown in Fig. 2c. The featureless background due to scattering from the solvent contributes 10 X-ray counts (blue line); the measurement from the liquid jet without crystals is shown in b. Two representative patterns with speckle counts above the mean solvent background but not above the threshold (c), and one of the strongest 2,848 patterns (d).

Extended Data Fig. 4. Patterson function of a disordered lattice

A distorted lattice (left, with ideal positions in gray), with vectors connecting all lattice points having difference vectors a_n−a_k=(1,1). On the right, the arrows from the left are translated in two ways. Upper right: the heads and tails are both displaced from their ideal positions. Bottom right: the tails are lined up, resulting in the distribution of head positions forming a broader Gaussian. In the limit of a large crystal, the resulting distribution is just the autocorrelation of the displacement distribution. This process can be repeated for all difference vectors, leading to Eqn. (3).

Extended Data Fig. 5. Model refinement is improved at low resolution

A plot of the metric R-free, as a function of resolution shell q, shows a marked improvement of the model refined against the 3.5 Å diffractive image. Here R-free is calculated only using Bragg intensities (which were excluded from the refinement) for resolution below 4.5 Å.

Extended Data Fig. 6. Central sections of the 3D full-pattern diffraction volume

a–c, Bragg intensities in planes normal to the three orthogonal reciprocal-space axes, q_z, q_y, and q_x, respectively, which were arbitrarily chosen to be parallel to the c*, b*, and a* axes of the PS II crystal. d–f, full-pattern diffraction intensities in central sections normal to the same three orthogonal axes, obtained from 2,848 strongest snapshot diffraction patterns. g–i, continuous diffraction intensities calculated for a single PSII dimer, from the model refined from the 4.5 Å Bragg data, for the same set of orthogonal planes as a–c and d–f. The intensities were calculated from the incoherent sum of the squared modulus of the 3D molecular transform of a single (uncrystallised) PSII dimer in each of the four orientations of the 222 point group. All panels are plotted on the same scale, with the experimental data (d–f) extending to q = 0.33 Å^-1 at the centre edge. The agreement between each of d–f with the corresponding panel g–i is further evidence that the rigid structural unit is the photosystem II dimer.

Extended Data Fig. 7. Real space orthogonal slices

a–c, slices of the F_o electron density map, plotted as a grey-scale, of a single photosystem II dimer obtained from the 4.5 Å Bragg intensities following model refinement of that data. Each slice is one pixel thick (1.5 Å) and is normal to the z, y, and x real-space axes, respectively (conjugate to q_z, q_y, and q_x). d–f, slices through the 3D real-space support constraint used to for iterative phasing. The support was generated by blurring the 4.5 Å resolution electron density map by 2.2 Å and then thresholding to achieve a binary mask. g–i, slices through the 3.5 Å resolution image obtained by iterative phasing of the continuous diffraction, using the support constraint illustrated in d–f and Fig. 3d.

Extended Data Fig. 8. The continuous pattern is consistent with diffraction from a rigid object in crystallographic orientations

a, When the continuous diffraction intensities are substituted for intensities averaged over all orientations (i.e. constant on surfaces of constant q), iterative phasing using the support constraint of Fig. 3d fails, as indicated by this plot of the Fourier shell coefficient as a function of resolution for solutions (green) obtained from two independent phasing trials. The blue curve is the same FSC shown in Fig. 3e for comparison. The iterations never converged, so a PRTF for the control could not be generated. b, Plot of the correlation between the measured diffraction and that calculated from the determined electron density of a dimer, symmetrised by the four crystallographic orientations.

Extended Data Fig. 9. Electron density maps (2mFo-DFc) of regions of the PSII dimer

(a-f) Electron density maps obtained utilising the Bragg diffraction (with the model shaded green, left), the Bragg and continuous diffraction (model shaded blue, middle), and computed from pseudo-crystal refinement (model shaded brown, right). The maps are rendered at various density levels relative to the standard deviation of the overall density. a and b, Non-heme iron of chain A coordinated by four Histidin from chain A contoured at a, 1.5σ and b, 4σ. Neighboring Tyr and Lys residues are displayed as well. c, Part of an α–helix (chain T), showing that the side chains (e.g. Arg and Phe) are fitting better in the electron density when applying our new method (maps contoured at 1.5σ). d, helices from chains Y and Z (maps contoured at 1.5σ). The density map shows more details and better agreement to the model when applying the analysis using the Bragg and continuous diffraction. e, Detailed view of a section of chain Z, map contoured at 1.25σ. Utilising only the Bragg diffraction, no electron density is visible at this level around the side chains of Trp, Lys and Arg. Again, the model fits better into the map, when using the continuous diffraction. f, The primary electron donor chlorophyll dimer (P680) and part of the transmembrane helix of chain C, contoured at 1.5σ. g, Matrix of Pearson correlation coefficients of electron density maps obtained from the Umena model (PDB code 3WU2) the model refined from Bragg diffraction, and that obtained from the continuous diffraction.

Figure 1. Lattice disorder reveals the continuous molecular transform

Diffraction from an arrangement of objects in a perfect lattice a results in regularly-spaced Bragg peaks through constructive interference b. Translational disorder of the objects at a length scale σ, c, disrupts Bragg interference beyond a reciprocal lattice resolution length d = 1/q = 2π σ, d. The loss of correlation gives rise to the incoherent sum of the Fraunhofer diffraction patterns of individual objects, which increases with q in balance with the diminishing Bragg intensities. Here, σ was chosen to be 4% of the lattice spacing, leading to reduction of Bragg peaks at the 4^th order.

Figure 2. Molecular coherent diffraction

An X-ray FEL snapshot “still” diffraction pattern of a photosystem II microcrystal, a, shows a weak speckle structure beyond the extent of Bragg peaks which is enhanced in this figure by limiting the displayed pixel values. b, Structure factors obtained from Bragg-peak counts from 25,585 still patterns, displayed as a precession-style pattern of the [001] zone axis. c, A rendering of the entire 3D diffraction volume assembled from the 2,848 strongest patterns. d, A central section of the diffraction volume normal to the [100] axis. Speckles are clearly observed beyond the extent of Bragg diffraction to the edge of the detector.

Figure 3

An unbiased size estimate of the rigid structural unit is obtained by a Fourier transform of the continuous diffraction intensities, yielding the autocorrelation function (3D pair-distribution function. a, Projection of the experimentally-determined 3D autocorrelation along the crystal c axis. The equivalent projections through the autocorrelation functions calculated from the 4.5 Å model of photosystem II dimer, b, and the monomer, c, after applying the point group symmetries of the crystal. The extent of the rigid structural unit matches the size and shape of the PSII dimer. A loose support, d, generated by thresholding and dilating the 4.5 Å resolution structure was used as the support constraint for iterative phasing. The Fourier shell correlation (FSC) and phase retrieval transfer function (PRTF), e, indicates a resolution of 3.5 Å.

Figure 4

Improvement in electron density resolution and quality is obtained by directly phasing the continuous transform by the method of coherent diffractive imaging. a, Electron density map of the PS II dimer after refinement using structure factors obtained only from Bragg peaks at 4.5 Å resolution. b, Electron density obtained by iterative phase retrieval on the continuous diffraction data using the support constraint (molecular envelope) of Fig. 3d at 3.5 Å resolution. c and d, electron density maps (2Fo-DFc) of two regions of the PSII dimer, utilising only the Bragg diffraction (with the model shaded green) and the Bragg and continuous diffraction (model shaded blue). c, Two antenna Chlorophylls in the antenna protein PsbC with their His ligands. d, The heme group of PsbE/F.