Chapter 16.1. Ab initio phasing

Given the individual normalized structure-factor magnitudes and the atomic scattering factors which allow for the possibility of anomalous scattering, then greatest-lower-bound estimates of SIR difference-E magnitudes are where is a least-squares-fitted empirical renormalization scaling function, dependent on , that imposes the condition and serves to define and .

Given Friedel pairs of normalized structure-factor magnitudes and the atomic scattering factors, then the greatest-lower-bound estimates of SAS difference 's are where, again, q is an empirical renormalization scaling function that imposes the condition .

The magnitude-dependent entities that constitute the foundation of direct methods are linear combinations of phases called structure invariants. The term `structure invariant' stems from the fact that the values of these quantities are independent of the choice of origin. The most useful of the structure invariants are the three-phase or triplet invariants, the conditional probability distribution (Cochran, 1955), given , of which is where and N is the number of atoms, here presumed to be identical, in the asymmetric unit of the corresponding primitive unit cell. This distribution is illustrated in Fig. 16.1.3.1. The expected value of the cosine of a particular triplet, , is given by the ratio of modified Bessel functions, .

Figure 16.1.3.1| top | pdf | The conditional probability distribution of the three-phase structure invariants.

Estimates of the invariant values are most reliable when the normalized structure-factor magnitudes (, and ) are large and the number of atoms in the unit cell, N, is small. This is the primary reason why direct phasing is more difficult for macromolecules than it is for small molecules. Four-phase or quartet invariants have proven helpful in small-molecule structure determination, particularly when used passively as the basis for a figure of merit (DeTitta et al., 1975). However, the reliability of these invariants, as given by their conditional probability distribution (Hauptman, 1975), is proportional to , and they have not as yet been shown to be useful for macromolecular phasing. The reliability of higher-order invariants decreases even more rapidly as structure size increases.

Successful crystal structure determination requires that sufficient phases be found such that a Fourier map computed using the corresponding structure factors will reveal the atomic positions. It is particularly important that the biggest terms (i.e., largest ) be included in the Fourier series. Thus, the first step in the phasing process is to sort the reflections in decreasing order according to their values and to choose the number of large reflections that are to be phased. The second step is to generate the possible invariants involving these intense reflections and then to sort them in decreasing order according to their values. Those invariants with the largest values are retained in sufficient number to achieve the desired overdetermination. Ab initio phase determination by direct methods requires not only a set of invariants, the average values of the cosines of which are presumed to be known, but also a set of starting phases. Therefore, the third step in the phasing process is the assignment of initial phase values. If enough pairs of phases, and , are known, the structure invariants can then be used to generate further phases which, in turn, can be used to evaluate still more phases. Repeated iterations will permit most reflections with large to be phased.

Depending on the space group, a small number of phases can be assigned arbitrarily in order to fix the origin position and, in noncentrosymmetric space groups, the enantiomorph. However, except for the simplest structures, these reflections provide an inadequate foundation for further phase development. Consequently, a `multisolution' or multi-trial approach (Germain & Woolfson, 1968) is normally taken in which other reflections are each assigned many different starting values in the hope that one or more of the resultant phase combinations will lead to a solution. Solutions, if they occur, must be identified on the basis of some suitable figure of merit. Although phases can be evaluated sequentially, the order determined by a so-called convergence map (Germain et al., 1970), it has become standard in recent years to use a random-number generator to assign initial values to all available phases from the outset (Baggio et al., 1978; Yao, 1981). A variant of this procedure is to use the random-number generator to assign initial coordinates to the atoms in the trial structures and then to obtain initial phases from a structure-factor calculation.

The tangent formula, (Karle & Hauptman, 1956), is the relationship used in conventional direct-methods programs to compute given a sufficient number of pairs () of known phases. It can also be used within the phase-refinement portion of the dual-space Shake-and-Bake procedure (Weeks, Hauptman et al., 1994; Sheldrick & Gould, 1995). The variance associated with depends on and, in practice, the estimate is only reliable for and for structures with a limited number of atoms (N). If equation (16.1.4.1) is used to redetermine previously known phases, the phasing process is referred to as tangent-formula refinement; if only new phases are determined, the phasing process is tangent expansion.

The tangent formula can be derived using the assumption of equal resolved atoms. Nevertheless, it suffers from the disadvantage that, in space groups without translational symmetry, it is perfectly fulfilled by a false solution with all phases equal to zero, thereby giving rise to the so-called `uranium-atom' solution with one dominant peak in the corresponding Fourier synthesis. In conventional direct-methods programs, the tangent formula is often modified in various ways to include (explicitly or implicitly) information from the so-called `negative' quartet invariants (Schenk, 1974; Hauptman, 1974; Giacovazzo, 1976) that are dependent on the smallest as well as the largest E magnitudes. Such modified tangent formulas do indeed largely overcome the problem of pseudosymmetric solutions for small N, but because of the dependence of quartet-term probabilities on , they are little more effective than the normal tangent formula for large N.

Constrained minimization of an objective function like the minimal function, (Debaerdemaeker & Woolfson, 1983; Hauptman, 1991; DeTitta et al., 1994), provides an alternative approach to phase refinement or phase expansion. is a measure of the mean-square difference between the values of the triplets calculated using a particular set of phases and the expected values of the same triplets as given by the ratio of modified Bessel functions. The minimal function is expected to have a constrained global minimum when the phases are equal to their correct values for some choice of origin and enantiomorph (the minimal principle). Experimentation has thus far confirmed that, when the minimal function is used actively in the phasing process and solutions are produced, the final trial structure corresponding to the smallest value of is a solution provided that is calculated directly from the atomic positions before the phase-refinement step (Weeks, DeTitta et al., 1994). Therefore, is also an extremely useful figure of merit. The minimal function can also include contributions from higher-order (e.g. quartet) invariants, although their use is not as imperative as with the tangent formula because the minimal function does not have a minimum when all phases are zero. In practice, quartets are rarely used in the minimal function because they increase the CPU time while adding little useful information for large structures. The cosine function in equation (16.1.4.2) can also be replaced by other functions of the phases giving rise to alternative minimal functions. In particular, an exponential expression has been found to give superior results for several P1 structures (Hauptman et al., 1999).

In principle, any minimization technique could be used to minimize by varying the phases. So far, a seemingly simple algorithm, known as parameter shift (Bhuiya & Stanley, 1963), has proven to be quite powerful and efficient as an optimization method when used within the Shake-and-Bake context to reduce the value of the minimal function. For example, a typical phase-refinement stage consists of three iterations or scans through the reflection list, with each phase being shifted a maximum of two times by 90° in either the positive or negative direction during each iteration. The refined value for each phase is selected, in turn, through a process which involves evaluating the minimal function using the original phase and each of its shifted values (Weeks, DeTitta et al., 1994). The phase value that results in the lowest minimal-function value is chosen at each step. Refined phases are used immediately in the subsequent refinement of other phases. It should be noted that the parameter-shift routine is similar to that used in ψ-map refinement (White & Woolfson, 1975) and XMY (Debaerdemaeker & Woolfson, 1989).

In its simplest form, peak picking consists of simply selecting the top E-map peaks where is the number of unique non-H atoms in the asymmetric unit. This is adequate for true small-molecule structures. It has also been shown to work well for heavy-atom or anomalously scattering substructures where is taken to be the number of expected substructure atoms (Smith et al., 1998; Turner et al., 1998). For larger structures (), it is likely to be better to select about peaks, thereby taking into account the probable presence of some atoms that, owing to high thermal motion or disorder, will not be visible during the early stages of a structure determination. Furthermore, a recent study (Weeks & Miller, 1999b) has shown that structures in the 250–1000-atom range which contain a half dozen or more moderately heavy atoms (i.e., S, Cl, Fe) are more easily solved if only peaks are selected. The only chemical information used at this stage is a minimum inter-peak distance, generally taken to be 1.0 Å. For substructure applications, a larger minimum distance (e.g. 3 Å) is more appropriate.

An alternative approach to peak picking is to select approximately peaks as potential atoms and then eliminate some of them, one by one, while maximizing a suitable figure of merit such as The top peaks are used as potential atoms to compute . The atom that leaves the highest value of P is then eliminated. Typically, this procedure, which has been termed iterative peaklist optimization (Sheldrick & Gould, 1995), is repeated until only atoms remain. Use of equation (16.1.5.1) may be regarded as a reciprocal-space method of maximizing the fit to the origin-removed sharpened Patterson function, and it is used for this purpose in molecular replacement (Beurskens, 1981). Subject to various approximations, maximum-likelihood considerations also indicate that it is an appropriate function to maximize (Bricogne, 1998). Iterative peaklist optimization provides a higher percentage of solutions than simple peak picking, but it suffers from the disadvantage of requiring much more CPU time.

A third peak-picking strategy also involves selecting approximately of the top peaks and eliminating some, but, in this case, the deleted peaks are chosen at random. Typically, one-third of the potential atoms are removed, and the remaining atoms are used to compute . By analogy to the common practice in macromolecular crystallography of omitting part of a structure from a Fourier calculation in hopes of finding an improved position for the deleted fragment, this version of peak picking is described as making a random omit map. This procedure is a little faster than simply picking atoms because fewer atoms are used in the structure-factor calculation. More important is the fact that, like iterative peaklist optimization, it has the potential for being a more efficient search algorithm.

A flowchart for the generic Shake-and-Bake algorithm, which provides the foundation for both programs, is presented in Fig. 16.1.7.1. It contains two refinement loops embedded in the trial-structure loop. The first of these loops (steps 5–9) is a dual-space phase-improvement loop entered by all trial structures, and the second (steps 11–14) is a real-space Fourier-refinement loop entered only by those trial structures that are currently judged to be the best on the basis of some figure of merit. These loops have been called the internal and external loops, respectively, in previous descriptions of the SHELXD program (e.g. Sheldrick & Gould, 1995; Sheldrick, 1997, 1998). Currently, the major algorithmic differences between the programs are the following:

All of the major parameters of the Shake-and-Bake procedure (i.e., the numbers of refinement cycles, phases, triplet invariant relationships and peaks selected) are a function of structure size and can be expressed in terms of , the number of unique non-H atoms in the asymmetric unit. These parameters have been fine-tuned in a series of tests using data for both small and large molecules (Weeks, DeTitta et al., 1994; Chang et al., 1997; Weeks & Miller, 1999b). Default (recommended) parameter values used in the SnB program are summarized in Table 16.1.7.1. At resolutions in the 1.1–1.4 Å range, recalcitrant data sets can sometimes be made to yield solutions if (1) the phase:invariant ratio is increased from 1:10 to values ranging between 1:20 and 1:50 or (2) the number of dual-space refinement cycles is doubled or tripled. The presence of moderately heavy atoms (e.g. S, C, Fe) greatly increases the probability of success at resolutions less than 1.2 Å; in general, the higher the fraction of such atoms the more the resolution requirement can be relaxed, provided that these atoms have low B values. Thus, disulfide bridges are much more helpful than methionine sulfur atoms because they tend to have lower B values. Parameter recommendations for substructures are based on an analysis of the peak-wavelength anomalous-difference data for S-adenosylhomocysteine (AdoHcy) hydrolase (Turner et al., 1998). Parameter shift with a maximum of two 90° steps [indicated by the shorthand notation PS(90°, 2)] is the default phase-refinement mode. However, some structures (especially large P1 structures) may respond better to a single larger shift [e.g. PS(157.5°, 1)] (Deacon et al., 1998). This seems to reduce the frequency of false minima (see Section 16.1.8.2).

Table 16.1.7.1| top | pdf | Recommended parameter values for the SnB program Values are expressed in terms of Nu, the number of unique non-H atoms (solvent atoms are typically ignored). Full-structure recommendations are for data sets measured to 1.1 Å resolution or better. Only heavy atoms or anomalous scatterers are counted for substructures. Parameter Full structures Substructures Phases 10Nu 30Nu Triplet invariants 100Nu 300Nu Peaks (with S, Cl) 0.4Nu Nu Peaks (no `heavy') 0.8Nu   Cycles Nu/2 if or if with S, Cl etc.; Nu otherwise 2Nu (minimum 20)

In general, the parameter values used in SHELXD are similar to those used in SnB. However, the combination of random omit maps with tangent extension has been found to be the most effective strategy within the context of SHELXD. Consequently, it is used as the default operational mode (see Section 16.1.8.4 for details).

On account of the intensive nature of the computations involved, SnB and SHELXD are designed to run unattended for long periods while also providing ways for the user to check the status of jobs in progress. The progress of current SnB jobs can be followed by monitoring a figure-of-merit histogram for the trial structures that have been processed (Fig. 16.1.7.2). A clear bimodal distribution of figure-of-merit values is a strong indication that a solution has, in fact, been found. However, not all solutions are so obvious, and it sometimes pays to inspect the best trial even when the histogram is unimodal. The course of a typical solution as a function of SnB cycle is contrasted with that of a nonsolution in Fig. 16.1.7.3. Minimal-function values for a solution usually decrease abruptly over the course of just a few cycles, and a tool is provided within SnB that allows the user to visually inspect the trace of minimal-function values for the best trial completed so far. Fig. 16.1.7.3 shows that the abrupt decrease in minimal-function values corresponds to a simultaneous abrupt increase in the number of peaks close to true atomic positions. In this example, a second abrupt increase in correct peaks occurs when Fourier refinement is started.

Figure 16.1.7.2| top | pdf | A histogram of figure-of-merit values (minimal function) for 378 scorpion toxin II trials. This bimodal histogram suggests that ten trials are solutions.

Figure 16.1.7.3| top | pdf | Tracing the history of a solution and a nonsolution trial for scorpion toxin II as a function of Shake-and-Bake cycle. (a) Minimal-function figure of merit, and (b) number of peaks closer than 0.5 Å to true atomic positions. Simple peak picking (200 or 0.4Nu peaks) was used for 500 (Nu) cycles, and 500 peaks (Nu) were then selected for an additional 50 (0.1Nu) dual-space cycles. The solution (which had the lowest minimal-function value) was then subjected to 50 cycles of Fourier refinement.

Since the correlation coefficient is a relatively absolute figure of merit (given atomic resolution, values greater than 65% almost invariably correspond to correct solutions), it is usually clear when SHELXD has solved a structure. The current version of SHELXD includes an option for calculating it using the full data every 10 or 20 internal loop cycles, and jumping to the external loop if the value is high enough. Recalculating it every cycle would be computationally less efficient overall.

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).

The integration of traditional direct methods with isomorphous replacement was initiated by Hauptman (1982a), who studied the conditional probability distribution of triplet invariants comprised jointly of native and derivative phases assuming as known the six magnitudes associated with reciprocal-lattice vectors H, K and . It was shown that many triplets, whose true values were near either 0 or π, could be identified and reliably estimated. Later it was shown that cosine estimates could be obtained anywhere in the range −1 to +1 (Fortier et al., 1985). In a series of six recent papers, Giacovazzo and collaborators utilized a combined direct-methods/isomorphous-replacement approach, with limited success, to devise procedures for the ab initio solution of the phase problem for macromolecules (Giacovazzo, Siliqi & Ralph, 1994; Giacovazzo, Siliqi & Spagna, 1994; Giacovazzo, Siliqi & Zanotti, 1995; Giacovazzo & Platas, 1995; Giacovazzo, Siliqi & Platas, 1995; Giacovazzo et al., 1996). Their methods depend only on diffraction data for a pair of isomorphous structures and do not require any prior structural knowledge. Hu & Liu (1997) have generalized the earlier work to obtain the conditional distribution of the general (n-phase) structure invariant when diffraction data are available for any number (m) of isomorphous structures. Finally, it has been shown that, provided the heavy-atom substructure is known, Hauptman's triplet distribution leads to unique values for the triplets and the individual phases (Langs et al., 1995).

In a manner analogous to the SIR case, Hauptman (1982b) derived the conditional probability distribution for triplet invariants given six magnitudes in the presence of anomalous dispersion. It was shown that unique estimates, lying anywhere in the whole interval 0–2π, could be obtained for the triplet values. This result was unanticipated since all earlier work had led to the conclusion that a twofold ambiguity in the value of an individual phase was intrinsic to the SAS approach. Later, it was demonstrated how the probabilistic estimates led to individual phases by means of a system of SAS tangent equations (Hauptman, 1996). Although the initial application of this tangent-based approach to the previously known macromomycin structure (750 non-H protein atoms plus 150 solvent molecules) was encouraging, it has not yet been applied to unknown macromolecules.

The conditional probability distributions of the quartet invariants, in both the SIR and SAS cases, have been derived based on corresponding difference structure factors rather than on the individual structure factors themselves (Kyriakidis et al., 1996). Fan and his collaborators (Fan et al., 1984; Fan & Gu, 1985; Fan et al., 1990; Sha et al., 1995; Zheng et al., 1996) have also extensively studied the use of direct methods in the SAS case. Applications to the known small protein avian pancreatic polypeptide at 2 Å revealed the essential features of the molecule. The direct-methods approach was used to break the phase ambiguity for core streptavidin and azurin II (proteins of moderate size) using SAS data at 3 Å. Although the direct-methods maps in these cases did not reveal the structures, the phases were good enough to serve as successful starting points for solvent flattening.

Recent experimental work in the field of multiple-beam diffraction provides grounds for hope that a generally applicable solution to the problem of obtaining individual invariant values can be found. It has been shown that triplet invariants can be measured for lysozyme with a mean error of approximately 20° (Weckert et al., 1993; Weckert & Hümmer, 1997). In addition, direct methods strengthened by simulated triplet invariants have been used to redetermine the structure of BPTI at resolutions as low as 2.0 Å (Mathiesen & Mo, 1997, 1998). Currently, the one-at-a-time methods used to measure triplet phases seriously limit practical applications, but faster methods of data collection have been proposed (Shen, 1998). If the means can, in fact, be found for measuring significant numbers of triplet phases quickly and accurately, dual-space direct methods may become routinely applicable to much lower resolution data than is currently possible.