Introduction to phasing

Abstract

When collecting X-ray diffraction data from a crystal, we measure the intensities of the diffracted waves scattered from a series of planes that we can imagine slicing through the crystal in all directions. From these intensities we derive the amplitudes of the scattered waves, but in the experiment we lose the phase information; that is, how we offset these waves when we add them together to reconstruct an image of our molecule. This is generally known as the 'phase problem'. We can only derive the phases from some knowledge of the molecular structure. In small-molecule crystallography, some basic assumptions about atomicity give rise to relationships between the amplitudes from which phase information can be extracted. In protein crystallography, these ab initio methods can only be used in the rare cases in which there are data to at least 1.2 A resolution. For the majority of cases in protein crystallography phases are derived either by using the atomic coordinates of a structurally similar protein (molecular replacement) or by finding the positions of heavy atoms that are intrinsic to the protein or that have been added (methods such as MIR, MIRAS, SIR, SIRAS, MAD, SAD or combinations of these). The pioneering work of Perutz, Kendrew, Blow, Crick and others developed the methods of isomorphous replacement: adding electron-dense atoms to the protein without disturbing the protein structure. Nowadays, methods from small-molecule crystallography can be used to find the heavy-atom substructure and the phases for the whole protein can be bootstrapped from this prior knowledge. More recently, improved X-ray sources, detectors and software have led to the routine use of anomalous scattering to obtain phase information from either incorporated selenium or intrinsic sulfurs. In the best cases, only a single set of X-ray data (SAD) is required to provide the positions of the anomalous scatters, which together with density-modification procedures can reveal the structure of the complete protein.

Figure 1

The diffraction experiment.

Figure 2

(a) The definition of a phase angle α. (b) The result of adding three waves, where the third wave is added with two different phase angles.

Figure 3

The importance of phases in carrying information. Top, the diffraction pattern, or Fourier transform (FT), of a duck and of a cat. Bottom left, a diffraction pattern derived by combining the amplitudes from the duck diffraction pattern with the phases from the cat diffraction pattern. Bottom right, the image that would give rise to this hybrid diffraction pattern. In the diffraction pattern, different colours show different phases and the brightness of the colour indicates the amplitude. Reproduced courtesy of Kevin Cowtan.

Figure 4

The process of molecular replacement.

Figure 5

Two protein diffraction patterns superimposed and shifted vertically relative to one another. One is from native bovine β-lactoglobulin and the other is from a crystal soaked in a mercury-salt solution. Note the intensity changes for certain reflections and the identical unit cells (spacing of the spots) suggesting isomorphism. (Photograph courtesy of Professor Lindsay Sawyer.)

Figure 6

Argand diagram for SIR. |F _P| is the amplitude of a reflection for the native crystal and |F _PH| is that for the derivative crystal.

Figure 7

Estimation of the native protein phase for SIR.

Figure 8

Harker construction for SIR.

Figure 9

The lack of closure.

Figure 10

Phase probability for one reflection in an SIR experiment. F _best is the centroid of the distribution. The map calculated with |F _best|exp(iα_best) [or m|F _P|exp(iα_best)〈cosΔα〉, where m is the figure of merit] has least error. m = 0.23 implies a 76° phase error, since cos(76) = 0.23.

Figure 11

(a) An uninterpretable 2.6 Å SIR electron-density map with the final C^α trace of the structure superimposed. ρ(x) = (1/V) m|F _P|exp(iα_best)× exp(−2πi h·x). (b) A small section of the map with the final structure superimposed.

Figure 12

Harker diagram for MIR with two heavy-atom derivatives.

Figure 13

Phase probability for one reflection. (a) Single derivative in an SIR experiment. (b) Three derivatives. In an MIR experiment P(α_P) ∝ Πexp(−∊_i ²/2E _i ²), where i is summed from 1 up to the number of derivatives.

Figure 14

Density-modification techniques. (a) Solvent flattening uses automated methods to define the protein–solvent boundary and then modifies the solvent electron density to be a certain fixed value. (b) Histogram matching redefines the values of electron-density points in a map so that they conform to an expected distribution of electron-density values. (c) Noncrystallographic (NCS) symmetry averaging imposes identical electron-density values to points related by local symmetry, in this case a trimer of ducks that forms the asymmetric unit. The local NCS symmetry operators relating points in duck A to ducks B and C are shown.

Figure 15

Phase improvement by density modification.

Figure 16

(a) 2.6 Å MIR electron density. (b) Electron density after solvent flattening and histogram matching in DM. The solvent envelope determined by DM is shown in green.

Figure 17

Variation in anomalous scattering signal versus incident X-ray energy in the vicinity of the K edge of selenium.

Figure 18

Breakdown of Friedel’s law when an anomalous scatterer is present. f(θ, λ) = f ₀(θ) + f′(λ) + if′′(λ). |F _hkl| ≠ |F _−h−k−l| or |F _PH(+)| ≠ |F _PH(−)|. ΔF ^± = |F _PH(+)| − |F_PH(−)| is the Bijvoet difference.

Figure 19

Harker construction for SIRAS.

Figure 20

MAD phasing. (a) Typical absorption curve for an anomalous scatterer. (b) Phase diagram. |F _P| is not measured, so one of the data sets is chosen as the ‘native’. (c) Harker construction.

Figure 21

Estimation of signal size. The expected Bijvoet ratio is r.m.s.(ΔF ^±)/r.m.s.(|F|) ≃ (N _A/2N _T)^1/2(2f′′_A/Z _eff). The expected dispersive ratio is r.m.s.(ΔF _Δλ)/r.m.s.(|F|) ≃ (N _A/2N _T)^1/2[|f′_A(λ_i) - f′_A(λ_j)|]/Z _eff, where N _A is the number of anomalous scatterers, N _T is the total number of atoms in the structure and Z _eff is the normal scattering power for all atoms (6.7 e⁻ at 2θ = 0).

Figure 22

Harker construction for SAD.

Figure 23

(a) Statistics from SHELXC showing the anomalous signal for the S-SAD example. (b) Heavy-atom sites determined by SHELXD.

Figure 24

2.1 Å electron-density map for the S-SAD example before and after density modification using SHELXE.

Figure 25

Improving phases for the S-SAD problem. (a) 2.1 Å resolution density-modified map. (b) 1.45 Å resolution phase-extended map. (c) ‘1.0 Å resolution’ free-lunch map.

Figure 26

Autotraced polyalanine model produced by SHELXE superimposed on the density-modified electron-density map at 1.45 Å resolution.

Figure 27

A SHELXE-derived 2.1 Å resolution electron-density map phased from a Hg-SAD data set with superimposed polyalanine trace produced by SHELXE. The view is down the crystallographic threefold axis.

Figure 28

Cross-crystal averaging. Two crystal forms of the same protein for which phase information to low resolution is known for one form (left) and high-resolution data exist but no phase information is known for another form (right).

Figure 29

Cross-crystal averaging of hemagglutinin–neuraminidase (HN). Left, the unit cell showing the 6.0 Å resolution MIR map derived from eight heavy-atom derivatives contoured at 2.0σ, revealing two blobs corresponding to the two molecules in the asymmetric unit. Right, a section of the 2.0 Å resolution map after phase extension and cross-crystal averaging over four non-isomorphous data sets.