research papers
A new densitymodification procedure extending the application of the recent ρbased phasing algorithm to larger crystal structures
^{a}Institut de Ciència de Materials de Barcelona, CSIC, Campus de la UAB, Bellaterra, Catalonia 08193, Spain
^{*}Correspondence email: jordi.rius@icmab.es
The incorporation of the new peaknessenhancing fast Fourier transform compatible ipp procedure (ipp = innerpixel preservation) into the recently published S_{M} algorithm based on ρ [Rius (2020). Acta Cryst A76, 489–493] improves its phasing efficiency for larger crystal structures with atomic resolution data. Its effectiveness is clearly demonstrated via a collection of test crystal structures (taken from the Protein Data Bank) either starting from random phase values or by using the randomly shifted modulus function (a Pattersontype synthesis) as initial ρ estimate. It has been found that in the presence of medium scatterers (e.g. S or Cl atoms) crystal structures with 1500 × c atoms in the (c = number of centerings) can be routinely solved. In the presence of strong scatterers like Fe, Cu or Zn atoms this number increases to around 5000 × c atoms. The implementation of this strengthened S_{M} algorithm is simple, since it only includes a few easytoadjust parameters.
Keywords: S_{M} phasing algorithm; ipp procedure; ρbased phasing residual; direct methods; originfree modulus sum function; structure solution.
1. Introduction
The novel phasing function is rooted in the Z_{R} originfree modulus sum function, a nearly 30 yearsold directmethods phasing function (Rius, 1993). Both mainly differ in (i) the introduction of `Fourier transform' calculations instead of the complex manipulation of `structure invariants' (Rius et al., 2007); (ii) the replacement of by at each point r of the by using the property that and are positivedefinite functions with similar shape (Rius, 2020). The resulting phase function is defined by
in which the K sum extends over all reflections (i.e. strong and weak ones), E_{K} denotes the experimental structurefactor modulus with being their average value, V is the volume of the and Φ denotes the collectivity of φ phases involved in the computation of ρ. The C_{K}(Φ) = C_{K}(Φ) exp[iα_{K}(Φ)] complex quantity is the Fourier transform of the ρ(Φ) density function in terms of the structurefactor phases to be refined. Their is achieved by maximizing through the iterative fast Fourier transform (FFT) algorithm. This algorithm has been developed in P1, since this symmetry is advantageous to ab initio phase refinements (Sheldrick & Gould, 1995). (Mathematically, however, nothing prevents its implementation as a fullsymmetry algorithm.) As demonstrated by Rius (2020), maximizing S_{M}_{,ρ} is equivalent to minimizing the phasing residual
which measures the discrepancy between δ_{M}(Φ) and ρ(Φ). In integral (2), δ_{M}(Φ) and k are, respectively, the inverse Fourier transform of (E_{K} − 〈E〉) exp[iα_{K}(Φ)] and a suitable scaling constant (Rius, 2012). Since integral (2) can be exactly worked out in terms of , its minimum value should correspond (for data reaching atomic resolution) to the true solution or an equivalent, to the maximum of the correlation coefficient
measuring the agreement between experimental and calculated modulus functions. CC_{M} rapidly increases at the beginning of the iterative phase gradually stabilizes as it progresses and suddenly increases at the end (normally by 0.035–0.045 in just a few cycles) indicating that convergence has been attained.
One common feature of most iterative phase ShakeandBake approach (Weeks et al., 1993; Miller et al., 1993), i.e. once the centers and heights of the N highest peaks in the map have been determined (N is the expected number of nonH atoms in the unit cell) these are used to calculate the new structurefactor estimates. For large structures, however, application of the FFT algorithm (Cooley & Tukey, 1965) to the Fourier map is more efficient than direct calculation of the structure factors. In the literature other densitymodification procedures can be found, e.g. in SIR2000 the density fraction above a 2.0–2.5% threshold is kept in each map inversion, the rest set to zero [Burla et al. (2000) and Shiono & Woolfson (1992) for a related procedure]; Caliandro et al. (2008) have later shown the convenience of increasing this threshold when the resolution of the data is poorer than atomic. Also highly effective but more complicated is the densitymodification scheme incorporated in ACORN2 (Dodson & Woolfson, 2009). Alternatively, peakness in the electrondensity function can be enhanced by multiplying it with a mask having unit Gaussians only at the previously determined peak positions (the rest being zero). This modification is part of Sheldrick's intrinsic phasing procedure (Sheldrick, 2015) and allows the posterior application of the FFT algorithm. In the present work, the alternative peaknessenhancing ipp procedure (ipp = innerpixel preservation) is described. It directly operates on the η = δ_{M}m_{ρ} product function of the S_{M} algorithm wherein is the mask relating to through the expression
algorithms working at atomic resolution and alternating between real and reciprocalspace calculations is the density modification of the intermediate Fourier maps. Peakpicking is the simplest procedure which has been applied in theAccording to Rius (2020), the values of m_{ρ} are 1 (for ρ > 0), 0 (for ρ between 0 and −t_{ρ}σ_{ρ}) and −1 (for ρ < −t_{ρ}σ_{ρ}) with being the variance of ρ(Φ) and t_{ρ} ≃ 2.5. Hereafter will be shortened to S_{M} for simplicity.
2. The S_{M} phasing algorithm with enhanced peakness: the ipp procedure
The phasing residual (2) can be minimized with the S_{M} algorithm (Rius, 2020), i.e. by the iterative application of the modified tangent formula
which corresponds to the angular part of the Fourier transform within brackets. One characteristic of the S_{M} algorithm is the presence of the η = δ_{M}m_{ρ} product function. To enhance the peakness of η, the simple ipp procedure based on the preservation of the innerpeak pixels has been added to S_{M}, giving rise to the S_{M}ipp algorithm (Fig. 1). This procedure consists of two well differentiated parts:
(i) Peak search in the η product function. The lowest value of η which is accepted as a peak is fixed by the t_{η} σ_{η} threshold ( is the variance of η, and t_{η} a parameter allowing tuning of the threshold and normally ranging between 3.5 and 4.0). The η peaks are searched by looking for the density values of all 26 nearest grid points around a given central pixel (satisfying the above threshold criterion). This (x_{o}, y_{o}, z_{o}) central pixel is considered a η peak if its density value is larger than the values of all its 26 nearest neighbor pixels, i.e. 8 (x_{o} ±1, y_{o} ±1, z_{o} ±1); 4 (x_{o}, y_{o} ±1, z_{o} ±1); 4 (x_{o} ±1, y_{o}, z_{o} ±1); 4 (x_{o} ±1, y_{o} ±1, z_{o}); 2 (x_{o}, y_{o}, z_{o} ±1); 2 (x_{o}, y_{o} ±1, z_{o}); 2 (x_{o} ±1, y_{o}, z_{o}) (Rollet, 1965). If this is the case, the density value and the pixel coordinates of the central pixel are stored. At the end, the N_{η} stored peaks are ordered in decreasing strength. (Note, t_{η} and N_{η} are inversely related.)
(ii) Density modification of η. If N_{η} > N, then for each one of the N highestranked η peaks, the density values of the 26+1 innerpeak pixels are preserved. The densitymodification procedure finishes by setting to zero all pixels of η not having preserved density values. For N_{η} ≤ N, the inner pixels of all N_{η} peaks will have preserved density values. The Fourier transform of the modified η yields the new φ estimates.
Notice that accurate peak center positions are not necessary for the application of the ipp procedure; consequently, no peak interpolation is needed. Notice, also, that it is compatible with the `random omit maps' strategy introduced in by Sheldrick (Usón & Sheldrick, 1999). For illustrative purposes, a successful S_{M}ipp phase obtained with starting random (rnd) phases and with t_{η} = 3.7 is reproduced in Fig. 2. It is interesting to note that only N_{η}(1) is smaller than N (the number 1 in parentheses indicates the iteration number).
Compared with the S_{M} algorithm in Rius (2020) in which all reflections participate in the computation of the ρ synthesis, S_{M}ipp works better if ρ is calculated with only those H reflections for which E ≥ E_{min} with E_{min} ≃ 1.0, i.e. Φ only includes the large and moderate E values [however, the calculation of the δ synthesis remains unchanged, i.e. it extends to all K reflections (Fig. 1)]. Notice that the faster calculation of ρ in S_{M}ipp counteracts the extra computing time due to ipp. Concerning this point, a test performed on data set 1pwl showed that the duration of one iteration in S_{M}ipp and in S_{M} is very similar. The S_{M}ipp algorithm has been programmed in a modified version of the XLENS_v1 code (Rius, 2011). In the test calculations, N always includes, besides the number of protein atoms, the number of solvent ones, i.e. water molecules.
3. The modulus function as initial estimate of ρ
It is clear that the phasing process not only depends on the phasing algorithm but also on the starting phase values. In Rius (2020), the S_{M} algorithm was only tested by assigning random values to the initial phases, Φ_{rnd} = {φ_{rnd}}. However, the ideal situation for any phasing algorithm is to start with phase values derived from initial ρ estimates (ρ_{ini}) containing structural information. Since the M modulus synthesis is a Pattersontype synthesis (Ramachandran & Raman, 1959), it can be regarded as the sum of N weighted shifted images of the (or its enantiomorph) (Wrinch, 1939; Buerger, 1950). Consequently, it contains valuable structural information and can be taken as ρ_{ini}. The success of the phasing process will obviously depend on the capability of the phasing algorithm to develop one incomplete shifted image of the while (gradually) suppressing the rest (working in P1 allows selection of one arbitrary image). The phasing process is greatly facilitated by the presence of a reduced number of strong scatterers in the with their corresponding images standing out from the rest [this justifies the separate treatment in the test calculations of compounds with weak, medium (atoms with Z < 19) and strong scatterers (Z ≥ 19)]. In multisolution phasing methods, each phase trial requires a different ρ_{ini}. This can be achieved by shifting the experimental M by a randomly generated u = OO′ vector to obtain the correspondingly shifted M′ function (O and O′ are the respective origins). The Fourier coefficients of M′ are with and Φ_{M′} = {}. In this way each trial follows a different path (in the test calculations, the sequence of u vectors is the same for all data sets). The number of selected phase trials (N_{trials}) is either 5, 25 or 50 depending on the success rate; the maximum number of allowed iterations per trial is always N_{iter(max)} = 1000 (excepting 3bcj with 200).
4. Comparison of the phasing efficiencies of the S_{M}ipp and S_{M} algorithms
The efficiencies of the S_{M}ipp and S_{M} algorithms have been calculated for both Φ_{rnd} and Φ_{M′}. For simplicity, the various phase strategies are specified by A1, A2, B1, B2, i.e. A1: Φ_{rnd} with S_{M}ipp; A2: Φ_{rnd} with S_{M}; B1: Φ_{M′} with S_{M}ipp; B2: Φ_{M′} with S_{M}.
The compounds participating in the test calculations are listed in Tables 1 and 2. For those compounds in Table 1 only containing weak scatterers, the checked strategies are A1, A2 and B1 (Table 3). In the case of compounds with medium/strong scatterers (Tables 1 and 2), the investigated strategies are B1 and B2 (Tables 4, 5 and 6). To make comparisons between strategies stricter, corresponding trials started with the same set of randomly generated phase values.





4.1. Compounds with only weak scatterers
The data sets used in the tests of crystal structures with only weak scatterers are 1a7y, 3sbn, 1ob4, 1a7z and 1alz (Table 1). The first three data sets belong to small crystal structures and the last two to relatively large ones. Of these, 1a7z corresponds to a Clcontaining compound with 1228 atoms in the In spite of the presence of Cl, it has been included in this section because the protocol deposited in the Protein Data Bank (PDB) indicates that one Cl is partially occupied and the other has a rather large B value, so that their scattering powers are considerably reduced. The last data set (1alz) corresponds to the notoriously difficult of gramicidin with 1348 C, N and O atoms in the and with nearly 25% of the atoms showing positional disorder.
Of the two A1 and A2 phasing strategies, the best one is A1 (Table 3). Compared with A2, A1 yields the smallest 〈N_{iter}〉 values and the largest number of successful trials for all five tested data sets, i.e. the correct solutions are found much faster when ipp is applied. The faster convergence of A1 is illustrated in Fig. 3 for data sets 3sbn and 1a7z. In the case of gramicidin, two correct solutions are obtained with A1 (trial 21 with N_{iter} = 136 and trial 45 with N_{iter} = 520) which represents one solution every 2.5 h using a desk computer (3.4 GHz); however, with A2 no correct solution was found. Regarding the A1 and B1 strategies, inspection of Table 3 indicates that A1 converges somewhat faster than B1 and is superior in the case of gramicidin (B1 gives no correct solutions).
4.2. Crystal structures with only medium scatterers
The application of strategy B1 to ten compounds containing medium scatterers (1byz, 2erl, 1p9g, 3nir, 1a0m, 4lzt, lf94, 1hhu, 3odv and 3psm) is summarized in Table 4. In most cases (nine out of ten) phase refinements performed smoothly, i.e. all five trials converged. Of these nine cases, only 1a0m (conotoxin) required more iterations. The acquisition of the conotoxin data with a Cu rotating anode at room temperature (outermost shell is 1.10–1.14 Å) surely contributes to the different behavior of this data set. In contrast to the nine preceding cases, application of S_{M}ipp to 1f94 (bucandin) was less successful. Consequently, N_{trials} was increased to 25 to estimate more reliably the success percentage (32%). This structure has large atomic disorder (B_{Wilson} = 14.3 Å^{2}) which is reflected in the large fraction of unobserved data in the 1.06–1.02 Å interval, i.e. 0.50 with I > 2σ(I). The influence of ipp on the phase accuracy can be estimated with ΔCC_{M}, i.e. the difference between CC_{M} values for S_{M}ipp and for S_{M}. As can be clearly seen in Tables 3 and 4, ΔCC_{M} is only slightly negative, generally between −0.02 and −0.03, which suggests that truncation of the outerpeak regions during the application of the ipp procedure is not critical.
To estimate the influence of ipp on the convergence of the phase the same tests carried out with strategy B1 were repeated with B2 (Table 4). Comparison of both sets of N_{iter} values confirms the much faster convergence of B1.
4.3. Crystal structures with strong scatterers
From Table 5 it follows that for compounds with heavy atoms of the first transition series, application of the B1 strategy allows the routine determination (in a reduced number of iterations) of crystal structures with N up to ≃5000 × c (c = number of centerings) provided that the data are of good quality and that at least the scattering power of one of the heaviest atoms is not weakened. The resulting 〈N_{iter}〉 values go from 10 to 60 except for data sets 41au, 1pwl, 1heu and 1c7k for which it is larger. In the case of 41au the increase can be related to two of the three symmetryindependent selenomethionine Se atoms showing partial occupancies, i.e. (0.52, 0.48) and (0.31 and 0.69) (Fanfrlik et al., 2013). For 1pwl and 1heu, the larger 〈N_{iter}〉 values could be ascribed to the larger d_{min} values (Table 2). For comparison purposes, the results obtained with strategies B1 and B2 are summarized in Table 6. Its inspection confirms the clear superiority of B1 over B2, especially for the larger test crystal structures.
5. Discussion
One characteristic of the S_{M} algorithm is its mathematical simplicity, a consequence of the straightforward implementation of the modified tangent formula (5). One relevant parameter of S_{M} is t_{ρ} which modifies the threshold value in the calculation of ρ through expression (4). The value of t_{ρ} mainly depends on the scattering power of the strongest scatterer present in the In Rius (2020), t_{ρ} was found to be close to 2.5. In the current work, the test examples extend to a larger variety of structures in which the strongest scatterer can be weak, medium or strong. Respective t_{ρ} values giving satisfactory results have been found to be ≃2.5, ≃2.6 and ≃2.8.
Regarding the ipp procedure, its application requires the approximate knowledge of N and the estimation of t_{η}. The N value used in the test calculations is the sum of both protein and solvent atoms (taken from the PDB), i.e. N_{Prot} + N_{H2O}. An idea of 〈N_{H2O}〉 can be obtained by averaging (N_{Prot} + N_{H2O})/N_{Prot} over all structures with more than 700 atoms listed in Tables 1 and 2 which gives 1.22 (5), i.e. 〈N_{H2O}〉 ≃ 0.22 × N_{Prot}. The second parameter, t_{η}, controls the number of η peaks above the t_{η} σ_{η} threshold. It can be estimated from Q = N_{η}(2)/N. Suitable t_{η} values are those for which Q is close to 1 or not much smaller (the ipp procedure does not use N_{η} peaks exceeding N). According to Tables 3, 4 and 5, values of t_{η} from 3.5 to 4.0 give Q values ranging from 1.5 to 0.7. Whatever the initial phase values may be, a successful ends with a sudden increase of CC_{M} concomitant with a marked N_{η} decrease.
Of interest is the comparison of the N_{η}(1) values obtained with strategies A1 (Φ_{rnd}) and B1 (Φ_{M′}) by using similar t_{η} values. As was already shown in Section 2, N_{η}(1) is smaller than N for Φ_{rnd} (Fig. 2). However, for Φ_{M′} (Fig. 4), N_{η}(1) is much larger than N, since here η essentially corresponds to the shifted modulus function with weakened origin peak. In the test calculations, the Φ_{M′} set at the end of the first iteration is always calculated with the N largest η peaks. The only exception is 1b0y. Since the of this compound contains four dominant scattering units (Fe_{4}S_{4} clusters), only the 240 (= 16^{2} − 16) strongest η peaks (mostly corresponding to Fe–Fe interactions) were used.
For the compounds in Table 1 (except for 3bcj), the average strength of the S/Cl peaks in the Fourier map is 30 (5) a.u. (a.u. = arbitrary units). For 3bcj, however, the strength increases to 59 a.u. The explanation for the much larger peak strength has to be sought in the ultrahigh resolution of the experimental data favored by its lower measurement temperature (15 K compared with the usual 100 K). This test structure was selected to check the phasing capability of S_{M}ipp with ultra high resolution data. With 5934 atoms in the (solvent atoms excluded) this is in the same order of magnitude as those listed in Table 2 containing strong scatterers. Application of S_{M}ipp with Φ_{M′} (strategy B1) yields success percentages of 80%, 36% and 0% for d_{min} = 0.78, 0.85 and 0.90 Å, respectively (Fig. 5 reproduces the E map of one arbitrary successful refinement). Notice that S_{M}ipp solves here the protein structure in one stage, i.e. it is not necessary to first locate single S atoms as, e.g., done by McCoy et al. (2017).
A limitation of S_{M}ipp (when used as an ab initio phasing algorithm) arises for crystal structures belonging to highsymmetry point groups and having large asymmetric units, since then N becomes exceedingly large. Normally, the usual way to cope with such situations is to derive the initial Φ from a larger structure model by using, among others, or techniques. In such cases S_{M}ipp will become the phase stage of a more general twostage strategy.
6. Conclusions
It has been shown that the introduction of the new peaknessenhancing ipp procedure in the S_{M} phase algorithm significantly improves the algorithm efficiency for diffraction data at atomic resolution and, consequently, has been incorporated as the default option. For ab initio structure determinations with S_{M}ipp, the proper choice of the type of starting phases is important. Regarding this point, the following rules could be established on the basis of the test calculations:
(a) For very small lightatom crystal structures either Φ_{rnd} or Φ_{M′} phases can be used (peak overlap in the modulus function can still be managed by S_{M}ipp).
(b) Starting with Φ_{rnd} is appropriate for crystal structures containing only weak scatterers (the largest N value tested is around 1500 atoms).
(c) Starting with Φ_{M′} is the best option for crystal structures with medium scatterers like S or Cl (largest N for routine determinations is 1500 × c). If no trial converges in N_{iter(max)} iterations, then phase with Φ_{rnd} should be tried (with a larger N_{iter(max)}); however, Φ_{M′} should always be the first choice.
(d) Use of Φ_{M′} is the best choice for crystal structures with strong scatterers. For metals belonging to the first transition series like Fe, Cu and Zn, the largest N value for routine determinations has been estimated to be about 5000 × c atoms (tests performed on data sets collected at ≃100 K). One characteristic of successful phase refinements starting with Φ_{M′} is their fast convergence. This allows one to reduce N_{iter(max)} and, consequently, increase the number of explored trials.
Finally, some words regarding data completeness are in order. As already mentioned in Section 1, the S_{M} algorithm relies on the validity of the R_{M} residual (2) which assumes that δ and ρ are proportional (which is satisfied for data sets reaching atomic resolution as is the case with the test calculations described in this work). If the intensities of the outer reflection shells are unobserved (a common situation for protein crystals), R_{M} is no longer strictly fulfilled. Extrapolating the structure factors of unobserved reflections beyond the experimental resolution limit, e.g. by Fourier inversion of a suitably modified map, could be a solution for extending the applicability range of R_{M} to moderateresolution data sets. This `structurefactor extrapolation' technique (Caliandro et al., 2005a,b, 2007; see also Jiaxing et al., 2005) is particularly effective for crystal structures containing heavy atoms (Caliandro et al., 2008; Burla et al., 2012). The combination of S_{M} with the extrapolation technique could represent a further source of progress.
Supporting information
The output of the test calculations A1_A2_B1_weak, B1_medium, B2_medium, B1_strong and B2_strong. DOI: https://doi.org//10.1107/S2053273321004915/ik5001sup1.zip
Funding information
The following funding is acknowledged: MINECO and FEDER (grant No. RTI2018098537BC21); Severo Ochoa Programme for Centres of Excellence in R&D (grant No. SEV20150496).
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