Section 16.1.8. Applying dual-space programs successfully

When slightly heavier atoms such as sulfur are present, it is possible to start the Shake-and-Bake recycling procedure from a set of atomic positions that are consistent with the Patterson function. For large structures, the vectors between such atoms will correspond to Patterson densities around or even below the noise level, so classical methods of locating the positions of these atoms unambiguously from the Patterson are unlikely to succeed. Nevertheless, the Patterson function can still be used to filter sets of starting atoms. This filter is currently implemented as follows in SHELXD. First, a sharpened Patterson function (Sheldrick et al., 1993) is calculated, and the top 200 (for example) non-Harker peaks further than a given minimum distance from the origin are selected, in turn, as two-atom translation-search fragments, one such fragment being employed per solution attempt. For each of a large number of random translations, all unique Patterson vectors involving the two atoms and their symmetry equivalents are found and sorted in order of increasing Patterson density. The sum of the smallest third of these values is used as a figure of merit (PMF). Tests showed that although the globally highest PMF for a given two-atom search fragment may not correspond to correct atomic positions, nevertheless, by limiting the number of trials, some correct solutions may still be found. After all the vectors have been used as search fragments (e.g. after 200 attempts), the procedure is repeated starting again with the first vector. The two atoms may be used to generate further atoms using a full Patterson superposition minimum function or a weighted difference synthesis (in the current version of SHELXD, a combination of the two is used).

In the case of the small protein BPTI (Schneider, 1998), 15 300 attempts based on 100 different search vectors led to four final solutions with mean phase error less than 18°, although none of the globally highest PMF values for any of the search vectors corresponded to correct solutions. Table 16.1.8.2 shows the effect of using different two-atom search fragments for hirustasin, a previously unsolved 55-amino-acid protein containing five disulfide bridges first solved using SHELXD (Usón et al., 1999). It is not clear why some search fragments perform so much better than others; surprisingly, one of the more effective search vectors deviates considerably (1.69 Å) from the nearest true S–S vector.

The frequent imposition of real-space constraints appears to keep dual-space methods from producing most of the false minima that plague practitioners of conventional direct methods. Translated molecules have not been observed (so far), and traditionally problematic structures with polycyclic ring systems and long aliphatic chains are readily solved (McCourt et al., 1996, 1997). False minima of the type that occur primarily in space groups lacking translational symmetry and are characterized by a single large `uranium' peak do occur frequently in P1 and occasionally in other space groups. Triclinic hen egg-white lysozyme exhibits this phenomenon regardless of whether parameter-shift or tangent-formula phase refinement is employed. An example from another space group (C222) is provided by the Se substructure data for AdoHcy hydrolase. In this case, many trials converge to false minima if the feature in the SnB program that eliminates peaks at special positions is not utilized.

The problem with false minima is most serious if they have a `better' value of the figure of merit being used for diagnostic purposes than do the true solutions. Fortunately, this is not the case with the uranium `solutions', which can be distinguished on the basis of the minimal function [equation (16.1.4.2)] or the correlation coefficient [equation (16.1.6.1)]. However, it would be inefficient to compute the latter in each dual-space cycle since it requires that essentially all reflections be used. To be an effective discriminator, the figure of merit must be computed using the phases calculated from the point-atom model, not from the phases directly after refinement. Phase refinement can and does produce sets of phases, such as the uranium phases, which do not correspond to physical reality. Hence, it should not be surprising that such phase sets might appear `better' than the true phases and could lead to an erroneous choice for the best trial. Peak picking, followed by a structure-factor calculation in which the peaks are sensibly weighted, converts the phase set back to physically allowed values. If the value of the minimal function computed from the refined or unconstrained phases is denoted by and the value of the minimal function computed using the constrained phases resulting from the atomic model is denoted by , then a function defined by can be used to distinguish false minima from other nonsolutions as well as the true solutions. Once a trial falls into a false minimum, it never escapes. Therefore, the R ratio can be used, within SnB, as a criterion for early termination of unproductive trials. Based on data for several P1 structures, it appears that termination of trials with R ratio values exceeding 0.2 will eliminate most false minima without risking rejection of any potential solutions. In the case of triclinic lysozyme, false minima can be recognized, on average, by cycle 25. Since the default recommendation would be for 1000 cycles, a substantial saving in CPU time is realized by using the R ratio early-termination test. It should be noted that SHELXD optionally allows early termination of trials if the second peak is less than a specified fraction (e.g. 40%) of the height of the first. Generally, but not always, the R-ratio and peak-ratio tests eliminate the same trials.

Recognizing false minima is, of course, only part of the battle. It is also necessary to find a real solution, and essentially 100% of the triclinic lysozyme trials were found to be false minima when the standard parameter-shift conditions of two 90° shifts were used. In fact, significant numbers of solutions occur only when single-shift angles in the range 140–170° are used (Fig. 16.1.8.1), and there is a surprisingly high success rate (percentage of trial structures that go to solutions) over a narrow range of angles centred about 157.5°. It is also not surprising that there is a correlated decrease in the percentage of false minima in the range 140–150°. This suggests that a fruitful strategy for structures that exhibit a large percentage of false minima would be the following. Run 100 or so trials at each of several shift angles in the range 90–180°, find the smallest angle which gives nearly zero false minima, and then use this angle as a single shift for many trials. Balhimycin is an example of a large non-P1 structure that also requires a parameter shift of around 154° to obtain a solution using the minimal function.

Figure 16.1.8.1| top | pdf | Success rates for triclinic lysozyme are strongly influenced by the size of the parameter-shift angle. Each point represents a minimum of 256 trials.

The importance of the presence of several atoms heavier than oxygen for increasing the chance of obtaining a solution by SnB at resolutions less than 1.2 Å was noticed for truncated data from vancomycin and the 289-atom structure of conotoxin EpI (Weeks & Miller, 1999b). The results of SHELXD application to hirustasin are consistent with this (Usón et al., 1999). The 55-amino-acid protein hirustasin could be solved by SHELXD using either 1.2 Å low-temperature data or 1.4 Å room-temperature data; however, as shown in Fig. 16.1.8.2(a), the mean phase error (MPE) is significantly better for the 1.2 Å data over the whole resolution range. The MPE is determined primarily by the data-to-parameter ratio, which is reflected in the smaller number of reliable triplet invariants at lower resolution. Although small-molecule interpretation based on peak positions worked well for the 1.2 Å solution (overall ), standard protein chain tracing was required for the 1.4 Å solution (overall ). As is clear from the corresponding electron-density map (Fig. 16.1.8.2b), the Shake-and-Bake procedure produces easily interpreted protein density even when bonded atoms are barely resolved from each other. The hirustasin structure was also determined with SHELXD using 1.55 Å truncated data, and this endeavour currently holds the record for the lowest-resolution successful application of Shake-and-Bake.

Figure 16.1.8.2| top | pdf | (a) Mean phase error as a function of resolution for the two independent ab initio SHELXD solutions of the previously unsolved protein hirustasin. Either the 1.2 Å or the 1.4 Å native data set led to solution of the structure. (b) Part of the hirustasin molecule from the 1.4 Å room-temperature data after one round of B-value refinement with fixed coordinates.

The relative effects of accuracy, completeness and resolution on Shake-and-Bake success rates using SnB for three large P1 structures were studied by computing error-free data using the known atomic coordinates. The results of these studies, presented in Table 16.1.8.3, show that experimental error contributed nothing of consequence to the low success rates for vancomycin and lysozyme. However, completing the vancomycin data up to the maximum measured resolution of 0.97 Å resulted in a substantial increase in success rate which was further improved to an astounding success rate of 80% when the data were expanded to 0.85 Å.

On account of overload problems, the experimental vancomycin data did not include any data at 10 Å resolution or lower. A total of 4000 reflections were phased in the dual-space loop in the process of solving this structure with the experimental data. Some of these data were then replaced with the largest error-free magnitudes chosen from the missing reflections at several different resolution limits. The results in Table 16.1.8.4 show a tenfold increase in success rate when only 200 of the largest missing magnitudes were supplied, and it made no difference whether these reflections had a maximum resolution of 2.8 Å or were chosen randomly from the whole 0.97 Å sphere. The moral of this story is that, when collecting data for Shake-and-Bake, it pays to take a second pass using a shorter exposure to fill-in the low-resolution data.

Variations in the computational details of the dual-space loop can make major differences in the efficacy of SnB and SHELXD. Recently, several strategies were combined in SHELXD and applied to a 148-atom P1 test structure (Karle et al., 1989) with the results shown in Fig. 16.1.8.3. The CPU time requirements of parameter-shift (PS) and tangent-formula expansion (TE) are similar, both being slower than no phase refinement (NR). In real space, the random-omit-map strategy (RO) was slightly faster than simple peak picking (PP) because fewer atoms were used in the structure-factor calculations. Both of these procedures were much faster than iterative peaklist optimization (PO). The original SHELXD algorithm (TE + PO) performs quite well in comparison with the SnB algorithm (PS + PP) in terms of the percentage of correct solutions, but less well when the efficiency is compared in terms of CPU time per solution. Surprising, the two strategies involving random omit maps (PS + RO and TE + RO), which had been calculated to give reference curves, are much more effective than the other algorithms, especially in terms of CPU efficiency. Indeed these two runs appear to approach a 100% success rate as the number of cycles becomes large. The combination of random omit maps and Karle-type tangent expansion appears to be even more effective (Fig. 16.1.8.4) for gramicidin A, a structure (Langs, 1988). It should be noted that conventional direct methods incorporating the tangent formula tend to perform better in than in P1, perhaps because there is less risk of a uranium-atom pseudosolution.

Figure 16.1.8.3| top | pdf | (a) Success rates and (b) cost effectiveness for several dual-space strategies as applied to a 148-atom P1 structure. The phase-refinement strategies are: (PS) parameter-shift reduction of the minimal-function value, (TE) Karle-type tangent expansion (holding the top 40% highest fixed) and (NR) no phase refinement but Sim (1959) weights applied in the E map (these depend on and so cannot be employed after phase refinement). The real-space strategies are: (PP) simple peak picking using peaks, (PO) peaklist optimization (reducing peaks to ), and (RO) random omit maps (also reducing peaks to ). A total of about 10 000 trials of 400 internal loop cycles each were used to construct this diagram.

Figure 16.1.8.4| top | pdf | Success rates for the 317-atom structure of gramicidin A.

Subsequent tests using SHELXD on several other structures have shown that the use of random omit maps is much more effective than picking the same final number of peaks from the top of the peak list. However, it should be stressed that it is the combination TE + RO that is particularly effective. A possible special case is when a very small number of atoms is sought (e.g. Se atoms from MAD data). Preliminary tests indicate that peaklist optimization (PO) is competitive in such cases because the CPU time penalty associated with it is much smaller than when many atoms are involved.

With hindsight, it is possible to understand why the random omit maps provide such an efficient search algorithm. In macromolecular structure refinement, it is standard practice to omit parts of the model that do not fit the current electron density well, to perform some refinement or simulated annealing (Hodel et al., 1992) on the rest of the model to reduce memory effects, and then to calculate a new weighted electron-density map (omit map). If the original features reappear in the new density, they were probably correct; in other cases the omit map may enable a new and better interpretation. Thus, random omit maps should not lead to the loss of an essentially correct solution, but enable efficient searching in other cases. It is also interesting to note that the results presented in Figs. 16.1.8.3 and 16.1.8.4 show that it is possible, albeit much less efficiently, to solve both structures using random omit maps without the use of any phase relationships based on probability theory (curves NR + RO).

The results shown in Table 16.1.8.4 and Fig. 16.1.8.3 indicate that success rates in space group P1 can be anomalously high. This suggests that it might be advantageous to expand all structures to P1 and then to locate the symmetry elements afterwards. However, this is more computationally expensive than performing the whole procedure in the true space group, and in practice such a strategy is only competitive in low-symmetry space groups such as , C2 or (Chang et al., 1997). Expansion to P1 also offers some opportunities for starting from `slightly better than random' phases. One possibility, successfully demonstrated by Sheldrick & Gould (1995), is to use a rotation search for a small fragment (e.g. a short piece of α-helix) to generate many sets of starting phases; after expansion to P1 the translational search usually required for molecular replacement is not needed. Various Patterson superposition minimum functions (Sheldrick & Gould, 1995; Pavelčík, 1994) can also provide an excellent start for phase determination for data expanded to P1. Drendel et al. (1995) were successful in solving small organic structures ab initio by a Fourier recycling method using data expanded to P1 without the use of probability theory.

It has been known for some time that conventional direct methods can be a valuable tool for locating the positions of heavy-atom substructures using isomorphous (Wilson, 1978) and anomalous (Mukherjee et al., 1989) difference structure factors. Experience has shown that successful substructure applications are highly dependent on the accuracy of the difference magnitudes. As the technology for producing selenomethionine-substituted proteins and collecting accurate multiple-wavelength (MAD) data has improved (Hendrickson & Ogata, 1997; Smith, 1998), there has been an increased need to locate many selenium sites. For larger structures (e.g. more than about 30 Se atoms), automated Patterson interpretation methods can be expected to run into difficulties since the number of unique peaks to be analysed increases with the square of the number of atoms. Experimentally measured difference data are an approximation to the data for the hypothetical substructure, and it is reasonable to expect that conventional direct methods might run into difficulties sooner when applied to such data. Dual-space direct methods provide a more robust foundation for handling such data, which are often extremely noisy. Dual-space methods also have the added advantage that the expected number of Se atoms, N_u, which is usually known, can be exploited directly by picking the top N_u peaks. Successful applications require great care in data processing, especially if the values resulting from a MAD experiment are to be used.

All successful applications of SnB to previously unknown SeMet data sets, as reported in Table 16.1.8.1, actually involved the use of peak-wavelength anomalous difference data . The amount of data available for substructure problems is much larger than for full-structure problems with a comparable number of atoms to be located. Consequently, the user can afford to be stringent in eliminating data with uncertain measurements. Guidelines for rejecting uncertain data have been suggested (Smith et al., 1998). Consideration should be limited to those data pairs [i.e., isomorphous pairs and anomalous pairs ] for which and where typically and . The final choice of maximum resolution to be used should be based on inspection of the spherical shell averages versus . The purpose of this precaution is to avoid spuriously large values for high-resolution data pairs measured with large uncertainties due to imperfect isomorphism or general fall-off of scattering intensity with increasing scattering angle. Only those for which (typically ) should be deemed sufficiently reliable for subsequent phasing. The probability of very large difference 's (e.g. ) is remote, and data sets that appear to have many such measurements should be examined critically for measurement errors. If a few such data remain even after the adoption of rigorous rejection criteria, it may be best to eliminate them individually. A later paper (Blessing & Smith, 1999) elaborates further data-selection criteria.

On the other hand, it is also important that the phase:invariant ratio be maintained at 1:10 in order to ensure that the phases are overdetermined. Since the largest 's for the substructure cell are more widely separated than they are in a true small-molecule cell, the relative number of possible triplets involving the largest reciprocal-lattice vectors may turn out to be too small. Consequently, a relatively small number of substructure phases (e.g. 10N_u) may not have a sufficient number (i.e., 100N_u) of invariants. Since the number of triplets increases rapidly with the number of reflections considered, the appropriate action in such cases is to increase the number of reflections as suggested in Table 16.1.7.1. This will typically produce the desired overdetermination.

It is rare for Se atoms to be closer to each other than 5 Å, and the application of SnB to AdoHcy data truncated to 4 and 5 Å has been successful. Success rates were less for lower-resolution data, but the CPU time required per trial was also reduced, primarily because much smaller Fourier grids were necessary. Consequently, there was no net increase in the CPU time needed to find a solution.

A special version of SHELXD is being developed that makes extensive use of the Patterson function both in generating starting atoms and in providing an independent figure of merit. It has already successfully located the anomalous scatterers in a number of structures using MAD data or simple anomalous differences. A recent example was the unexpected location of 17 anomalous scatterers (sulfur atoms and chloride ions) from the 1.5 Å-wavelength anomalous differences of tetragonal HEW lysozyme (Dauter et al., 1999).