Section 26.1.2.6. Analysis in three dimensions

We had three options for the collection of three-dimensional data. First, we could have used precession photographs and densitometry, as in the study of myoglobin (Kendrew et al., 1960). Second, the manually controlled three-circle diffractometer used to make some of the measurements in the 6 Å study of myoglobin (Kendrew et al., 1958) and haemoglobin (Cullis et al., 1962) was available and, third, there was the prototype linear diffractometer (Arndt & Phillips, 1961). We chose the last option, since CCFB, RHF and DCP were well experienced in using the instrument, and it offered the opportunity of measuring the protein reflections automatically and in a relatively short time compared with the other methods. Before he went on leave to MIT, ACTN had written a computer program for the University of London MERCURY computer to process the diffractometer data (North, 1964) and, on his return in September 1961, he readily accepted an invitation to join the team to continue with this and other related aspects of the work.

The design of the linear diffractometer was based directly on the reciprocal-lattice representation of the genesis of X-ray reflections. The principle is illustrated in Figs. 26.1.2.4(a) and (b), which show the familiar Ewald construction. YXO represents the direction of the incident X-ray beam with X the centre of the Ewald sphere and O the origin of the reciprocal lattice. A′OA, B′OB and C′OC are the principal axes of the reciprocal lattice, here assumed to be orthogonal. XP is the direction of the reflected X-ray beam corresponding to the reciprocal-lattice point P, which lies on the surface of the sphere of reflection. The reciprocal lattice can be rotated about the axis C′OC, and this axis can be inclined to the direction of the incident X-ray beam by rotation about the axis D′OD, which is perpendicular to the incident beam.

Figure 26.1.2.4| top | pdf | Reciprocal-space diagrams showing the direction of the incident X-ray beam, the Ewald sphere and the genesis of a reflection (a) in an equatorial plane and (b) in the equi-inclination setting. Principal reciprocal-lattice directions are shown as thick lines. They also represent the slides in the diffractometer. The rotation of the diffractometer slide system about the axis C′OC is coupled to the rotation of the crystal about the axis R′XR by gears, pulleys and steel tapes. The counter arm of the diffractometer is represented by the fixed link XP = XO. Reproduced with permission from Arndt & Phillips (1961). Copyright (1961) International Union of Crystallography.

The linear diffractometer was simply a mechanical version of this diagram. The reciprocal lattice was represented by three slides, A, B and C, which were parallel, respectively, to A′OA, B′OB and C′OC. They were mounted to rotate about the axis C′O and arranged so that the saddle P could be set to any position in space within the coordinate system that they defined. This saddle P was connected to the point X by means of a link of fixed length, XP = XO, corresponding to the radius of the sphere of reflection. The link XP always lay along the direction of the reflected X-ray beam and thus became the counter arm of the diffractometer. The crystal was mounted at X for rotation about the axis R′XR (independent of the link XP, which pivoted about an independent coaxial bearing at X). The rotation of the crystal about this axis was coupled by means of gears, pulleys and steel tapes to the rotation of the slide system about the axis C′OC. The axes R′XR and C′OC, held parallel by means of parallel linkages, could be tilted with respect to the incident X-ray beam by rotation about the axes D′OD, E′XE, as shown in Fig. 26.1.2.4(b).

The scale of the instrument clearly depended only on the length chosen for XP = XO. In the instruments used in the lysozyme work, this length, which is equivalent to one reciprocal-lattice unit, was five inches. The position of the saddle P on the three slides was controlled by means of lead screws, all of which were cut with 20 turns per inch. Hence the counters, which indicated revolutions and fractions of a revolution of the lead screws, read directly in decimal divisions of reciprocal-lattice units. The screws in slides A and B were driven by means of synchro-receiver motors, forming a synchro link with corresponding transmitters in the control panel. Slide C was set manually, together with the inclination angle μ, for the measurement of upper-level reflections in the Weissenberg equi-inclination mode, Fig. 26.1.2.4(b).

The coupling of the rotations of the crystal and reciprocal lattice about the axes R′XR and C′OC, respectively, was interrupted by two ancillary mechanisms. The first simply allowed for independent rotation of the crystal with respect to the slide system and was used for setting the reciprocal-lattice axes in the equatorial plane parallel to slides A and B, and for any fine adjustment of the crystal rotation that might be necessary during the measurement procedure. The second interruption consisted of a mechanism for oscillating the crystal about the position for any reflection while X-ray intensity measurements were made. This oscillation mechanism (Arndt & Phillips, 1961) rotated with the crystal as the diffractometer was being set to a reflection position, and then controlled the independent motion of the crystal for the measurement of the integrated intensity of the reflection. The crystal remained stationary at a given angular setting for time t, was rotated at a uniform rate over a predetermined angular range for a time 2t, remained stationary at the final angular setting for a further time t, and then returned quickly to its original setting. The correct setting for the reflection peak was at the midpoint of the rotation, which might be set to be through any angle from 1 to 5°. For initial adjustments, the motor could be arrested at this midpoint by means of a micro-switch operated by a switching disc rotating with the cam. This disc otherwise actuated contacts that started and stopped the intensity measurements.

The X-ray intensities were measured with a side-window xenon-filled proportional counter made by 20th Century Electronics, together with associated amplifiers and pulse-counting circuits (Arndt & Riley, 1952). The proportional counter had a high quantum efficiency for the measurement of Cu Kα radiation (about 80%) and, when the operating potential and pulse-height discriminating circuits were carefully set, it provided useful discrimination against radiation of other wavelengths. The output from the proportional counter and its associated circuitry was fed directly to a teleprinter, which gave both a plain-language print out and a five-hole punched paper tape for input to the computer. Each count was provided with a check digit derived by a `ring-of-three' circuit, wired in parallel with the main electronic counter. During data processing, the check digit was compared with the count modulo 3: inequality of the two numbers was taken to indicate an error in the counting circuit.

Three counts were made: the first was a background count n₁, made while the crystal was stationary on one side of the reflection position; the second was an integrated intensity count N, accumulated as the crystal rotated through the reflection; and the third was a further background count n₂. The background-corrected integrated intensity of the reflection was taken to be

At this stage of the work, measurements were made of one reciprocal-lattice level at a time in the equi-inclination mode that has no blind region at the centre. In each level, the diffractometer was driven to a reflection hkl (for example) at the limit of resolution of the data set to be collected. The diffractometer then moved in a series of equal steps along the scanning slide. At the end of each step, the oscillation mechanism took control for the measurement of intensity and background. After each measurement, a further step was taken on the scanning slide, and the process continued until a limit switch, set to the required resolution limit, was reached. The diffractometer then completed the current translation, measured the last reflection in that row, and then moved one step on the stepping slide to the next parallel row. This row was scanned, in the opposite direction, until the limit switch was reached again. In this way, the whole of a reciprocal-lattice level could be measured. In order to change to another level, the vertical slide C and the inclination angle μ had to be manually adjusted.

This account ignores two difficulties, one inherent in the design of the diffractometer, and the other specific to the lysozyme crystals. First, the instrument required a good deal of supervision, since it did not set itself very well for the measurement of low-angle reflections. Second, the crystals were not easily mounted so that the c axis, the most convenient axis for efficient data collection, since it is perpendicular to the most densely populated reciprocal-lattice planes, coincided with the crystal-rotation axis of the diffractometer.

The first problem was overcome by efficient teamwork and was much eased by the fact that RHF assumed responsibility for the MHTS derivative as part of her PhD work; the second was solved by making most of the measurements from crystals mounted to rotate about the [100] axis. These crystals were oriented so that the and axes were parallel to the horizontal slides of the diffractometer, and the measurements were made in levels of constant H by scanning along rows parallel to and stepping to adjacent rows along . A number of reflections could not be measured in this way, however, because reflections near the axis were too broad to measure, particularly in the upper reciprocal-lattice levels. This difficulty was overcome by mounting some crystals with the c axis of the tetragonal crystals perpendicular to the length of the capillary tube, with the [110] axis parallel to the tube. These specimens were then mounted on a right-angled yoke so that the capillary tube was perpendicular rather than parallel to the goniometer axis (Fig. 26.1.2.5).

Figure 26.1.2.5| top | pdf | Crystal mounting. (a) Rotation about the axis; (b) rotation about [110], preliminary to mounting; (c) rotation about the axis (elevation); and (d) rotation about the axis (plan).

Given the morphology of the crystals, with an essentially square habit bounded by {110} faces (Figs. 26.1.2.1 and 26.1.2.5), all the reflections in the hkl octant could be measured in levels with constant L values without inclining the capillaries by more than about 40° to the X-ray beam. The horizontal slides of the diffractometer were set to be parallel to the and axes of the crystal. Some quadrants of hkL reciprocal-lattice levels were then scanned along rows parallel to the axis and stepped along the axis. Enough measurements were made in this mode to cover the `blind' region in the Hkl levels and provide an appropriate number of intersecting levels for scaling all the measurements into a consistent set.

Care was taken during all these measurements to index the reflections in a right-handed system of axes. Given the transparent relationship between the slide system of the diffractometer and the crystal geometry, this was easily accomplished, and it was necessary for the subsequent use of anomalous scattering (Bijvoet, 1954) in the phase determination.

During the measurements from the native and derivative crystals mounted for rotation about [100], the variation in peak intensity of the 200 reflection with φ, the angle of rotation about the axis C′OC (Fig. 26.1.2.4a), was also recorded (Fig. 26.1.2.6). (200 is the lowest-order reflection available for this purpose in this space group.) These records were then used in the data-processing stage to correct the measurements for absorption by the method described by Furnas (1957). According to this method, the absorption suffered by the incident and reflected beams for any reflection hkl is indicated by the relative intensity of 200 when the X-ray beams are parallel to the reflecting planes hkl. This is a fair approximation for reflections at low angles. This method could not be used, however, for the measurements made from crystals mounted to rotate about the [001] axes, since a full rotation about this axis was not possible. These measurements were not corrected for absorption errors.

Figure 26.1.2.6| top | pdf | Absorption curve. Variation of relative transmission with rotation angle φ for the 200 reflection and the crystal rotating about the normal to (200). Solid line: measured curve; broken line: calculated curve, neglecting effect of mother liquor and capillary. Reproduced with permission from North et al. (1968). Copyright (1968) International Union of Crystallography.

Finally, the diffractometer was reset manually at regular intervals during data collection to measure the intensities of a number of reference reflections. These measurements were used to monitor the stability of the system and the extent of irradiation damage to the crystal, and they were also recorded on the paper-tape output for analysis by the data-processing program. An attempt was made to minimize irradiation damage by using a shutter to expose the crystal only during the measuring cycle.

The measurements N, n₁ and n₂, together with the indices of the reflections, hkl, were all printed out in plain language on a teleprinter and punched in paper tape for direct transfer to a computer (Fig. 26.1.2.7). The plain-language record was important during measurement of the low-angle reflections, when the diffractometer had to be adjusted by hand. Not all imperfections in the measurements were easily spotted at this stage, however, and ACTN's data-processing program (North, 1964) therefore incorporated systematic checks on the quality of the measurements.

Figure 26.1.2.7| top | pdf | Typical output from the linear diffractometer. (a) Indices h, k, l followed by background (n1), peak (N), background (n2) counts. (b) Listing ready for the next stage in data processing with indices * h k followed by l, background corrected peak and standard deviation. Reproduced with permission from North (1964). Copyright (1964) Institute of Physics.

The program checked for the following contingencies:

These checks were made while the diffractometer tape was being read into the computer, and a monitor output was produced simultaneously, as shown in Fig. 26.1.2.8. The checks depended in large part on the fact that the significance of an intensity measurement may be assessed in terms of counting statistics. The standard deviation of a background-corrected count, , is given by , and the ratio may be taken as an indication of the significance of the measurement. Measurements were rejected when this ratio exceeded unity. might then have been taken as zero but, following Hamilton (1955), we considered it preferable to replace by a fraction (0.33 for centric and 0.5 for acentric reflections) of the minimum background-corrected count that we should have considered acceptable. Reflections were treated in the same way whether the net count was positive or negative, but measurements were rejected if was negative and .

Figure 26.1.2.8| top | pdf | Format of monitor output in which the computer lists reflections that fail the tests for format or significance. PE1 signifies punching error, indices; PE2, punching error, measurements; PE3, failure of electronic check on counting circuits; SD, standard deviation greater than set limit; N-, net count negative; BG, backgrounds significantly different; N > H, gross counts exceed counting-loss limit. This output was from the version of the program designed to be used with the diffractometer fitted with three counters. The symbols: −, =, + refer, respectively, to reflections measured by the lower, central and upper counters. Reproduced with permission from Arndt et al. (1964). Copyright (1964) Institute of Physics.

Mis-setting of the crystal was frequently revealed by marked inequality of the background counts. Measurements were therefore rejected if the difference between the two backgrounds exceeded three or four standard deviations, that is if , where b is the appropriate constant.

After monitoring the quality of the data in this way, the program proceeded: (i) to extract background-corrected counts; (ii) to apply a correction for irradiation damage derived from any systematic variation in the intensities of the reference reflections; (iii) to sort the reflections into a specified sequence of indices; (iv) to apply Lorentz–polarization factors; and (v) to apply absorption corrections (the data for which were read separately from a specially prepared punched tape, Fig. 26.1.2.6). The outputs from this program comprised data sets from a number of individual crystals of the native protein and the three derivatives. The scale factors needed to bring the measurements from the individual crystals of each crystal species to a common scale were determined from the intensities in common rows by the use of a program written by Rollett (Rollett & Sparks, 1960). The scaling factors were applied and the data merged, a weighted-average intensity being determined when more than one estimation was available.

An attempt was made to determine the absolute scale of the measured intensities by comparison with the intensities diffracted by anthracene, a small organic crystal of known structure. This method had worked well in a determination of the absolute scale for seal myoglobin (Scouloudi, 1960), but it did not give a satisfactory result with lysozyme, mainly because of the difficulty of measuring the crystal volumes precisely enough. Accordingly, we used Wilson's (1942) method to provide an estimate of the absolute scale of the intensities, knowing very well that it does not give an accurate estimate for protein data, especially at low resolution. Nevertheless, this scale gave reasonable values for the occupancies of the heavy-atom sites.

Given the three-dimensional data to 6 Å resolution for the native crystals and the three derivatives, it was next possible to calculate three-dimensional difference Patterson maps for the derivatives using the terms as coefficients in the Fourier series. This synthesis, which is now well known in protein-structure analysis, gives a modified Patterson of the heavy-atom structure in which the heavy-atom vectors appear at reduced weight in a complex background (Blow, 1958; Phillips, 1966). Nevertheless, the maps for PdCl₄ and MHTS were readily interpreted in terms of single heavy-atom substitutions, particularly because the vectors involved were all confined to defined Harker sections (Fig. 26.1.2.9).

Figure 26.1.2.9| top | pdf | Three-dimensional syntheses for the PdCl4 and MHTS derivatives. The positions of peaks in the Harker sections are marked and numbered (Fenn, 1964).

The map of the derivative was not so satisfactory, and this led to the discovery, during refinement of the heavy-atom parameters, that the mercury occupancy declined as a function of irradiation time. Consequently, the data from two individual crystals were treated separately during the remaining stages of the analysis.

The least-squares refinement mentioned in Section 26.1.2.4 was not wholly satisfactory in that it included no provision for refining heavy-atom occupancy. Accordingly, DCP – with some help from ACTN – wrote a computer program for the MERCURY computer based on Hart's (1961) method, in which the heavy-atom positions, occupancies (O) and temperature factors (B) were refined simultaneously together with the scale factors (S) between heavy-atom and native structure factors. All the centric reflections from the [100], [110] and [001] zones were included in this refinement.

The quantity minimized for each derivative was where is the calculated heavy-atom contribution to the derivative structure factor. The method involves calculating R′ for each parameter at its current value p_n and at values and , where each of the parameters is shifted in turn, the shifts having been specified, while the other parameters are at the unshifted value. Thus for each of the parameters (x, y, z, O, B and S), four values of R′ are obtained for the shifted values plus the value of R′ for the unshifted parameters, the latter being denoted .

Let the minimum value of R′ from all the calculations be and for the parameter p_n, the minimum of the list of five values of R′, be , corresponding to the value q_n of p_n. Then, according to the method of steepest descents, the shift to be applied to the parameter p_n is

If , that is, if the unshifted value of the parameter gave the minimum value of R′, then the shift was divided by 4 for the next cycle. Otherwise the shift was kept constant. Thus the new parameters and shifts were determined for the next cycle of refinement, and the process was repeated until convergence.

This program worked well, and RJP, who was reading Candide at the time, named it Pangloss – it gave the best possible values for the heavy-atom parameters. These values, which include two separate sets for the derivative, are shown in Table 26.1.2.1. At this stage, an important check was carried out. The coordinates of the heavy-atom site in each derivative were referred to an origin at the junction of a twofold axis and a twofold screw axis. However, there are four such intersections in the unit cell and, in order to ensure that the same origin had been chosen for each derivative, the sign predictions for the centric reflections from each derivative – which were checked by hand throughout this exploratory stage – were compared. They agreed well, thus establishing that the choice of origin was the same for each derivative.