research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

Journal logoFOUNDATIONS
ADVANCES
ISSN: 2053-2733

A new density-modification procedure extending the application of the recent |ρ|-based phasing algorithm to larger crystal structures

crossmark logo

aInstitut de Ciència de Materials de Barcelona, CSIC, Campus de la UAB, Bellaterra, Catalonia 08193, Spain
*Correspondence e-mail: jordi.rius@icmab.es

Edited by I. Margiolaki, University of Patras, Greece (Received 27 January 2021; accepted 10 May 2021; online 21 June 2021)

The incorporation of the new peakness-enhancing fast Fourier transform compatible ipp procedure (ipp = inner-pixel preservation) into the recently published SM 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 Patterson-type 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 unit cell (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 SM algorithm is simple, since it only includes a few easy-to-adjust parameters.

1. Introduction

The novel [{S_{M,| \rho |}}] phasing function is rooted in the ZR origin-free modulus sum function, a nearly 30 years-old direct-methods phasing function (Rius, 1993[Rius, J. (1993). Acta Cryst. A49, 406-409.]). Both mainly differ in (i) the introduction of `Fourier transform' calculations instead of the complex manipulation of `structure invariants' (Rius et al., 2007[Rius, J., Crespi, A. & Torrelles, X. (2007). Acta Cryst. A63, 131-134.]); (ii) the replacement of [{\rho }^{2}({\bf r})] by [| {\rho ({\bf r} )} |] at each point r of the unit cell by using the property that [{\rho ^2}({\bf r} )] and [| {\rho ({\bf r} )} |] are positive-definite functions with similar shape (Rius, 2020[Rius, J. (2020). Acta Cryst. A76, 489-493.]). The resulting [{S_{M,| \rho |}}] phase refinement function is defined by

[\eqalignno{{S}_{M,|\rho |}\left(\Phi \right) &= {{1}\over{V}} \sum _{K}\left(\left|{E}_{K}\right|-\langle |E|\rangle \right) \left|{C}_{-K}\left(\Phi \right)\right|&\cr & ={{1}\over{V}} \sum _{K}\left(\left|{E}_{K}\right|-\langle |E|\rangle \right) \left\{\exp\left[i{\alpha }_{K}\left(\Phi \right)\right] {C}_{-K}\left(\Phi \right)\right\}&(1)\cr}]

in which the K sum extends over all reflections (i.e. strong and weak ones), |EK| denotes the experimental structure-factor modulus with [| E |] being their average value, V is the volume of the unit cell, and Φ denotes the collectivity of φ phases involved in the computation of ρ. The CK(Φ) = |CK(Φ)| exp[iαK(Φ)] complex quantity is the Fourier transform of the |ρ(Φ)| density function in terms of the [\Phi] structure-factor phases to be refined. Their refinement is achieved by maximizing [{S_{M,| \rho |}}(\Phi )] through the iterative [{S_{M,| \rho |}}] 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[Sheldrick, G. M. & Gould, R. O. (1995). Acta Cryst. B51, 423-431.]). (Mathematically, however, nothing prevents its implementation as a full-symmetry algorithm.) As demonstrated by Rius (2020[Rius, J. (2020). Acta Cryst. A76, 489-493.]), maximizing SM,|ρ| is equivalent to minimizing the phasing residual

[{R}_{M}\left(\Phi \right) = \textstyle\int\limits _{V}\left[{\delta }_{M}\left(\Phi \right)-k \left|\rho \left(\Phi \right)\right|\right]^{2}\,{\rm d}V, \eqno(2)]

which measures the discrepancy between δM(Φ) and |ρ(Φ)|. In integral (2)[link], δM(Φ) and k are, respectively, the inverse Fourier transform of (|EK| − 〈|E|〉) exp[iαK(Φ)] and a suitable scaling constant (Rius, 2012[Rius, J. (2012). Acta Cryst. A68, 77-81.]). Since integral (2)[link] can be exactly worked out in terms of [\Phi], its minimum value should correspond (for data reaching atomic resolution) to the true solution or an equivalent, to the maximum of the correlation coefficient

[{\rm CC}_{M} = {\left\{{{{\left|{\sum }_{K}\left(\left|{E}_{K}\right|\times \left|{C}_{K}(\Phi)\right|\right)\right|}^{2}}\over{{\sum }_{K}{\left|{E}_{K}\right|}^{2}\times {\sum }_{K}{\left|{C}_{K}\left(\Phi \right)\right|}^{2} }}\right\}}^{1/2} \eqno(3)]

measuring the agreement between experimental and calculated modulus functions. CCM rapidly increases at the beginning of the iterative [{S_{M,| \rho |}}] phase refinement, 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 refinement algorithms working at atomic resolution and alternating between real- and reciprocal-space calculations is the density modification of the intermediate Fourier maps. Peak-picking is the simplest procedure which has been applied in the Shake-and-Bake approach (Weeks et al., 1993[Weeks, C. M., DeTitta, G. T., Miller, R. & Hauptman, H. A. (1993). Acta Cryst. D49, 179-181.]; Miller et al., 1993[Miller, R., DeTitta, G. T., Jones, R., Langs, D. A., Weeks, C. M. & Hauptman, H. A. (1993). Science, 259, 1430-1433.]), i.e. once the centers and heights of the N highest peaks in the map have been determined (N is the expected number of non-H atoms in the unit cell) these are used to calculate the new structure-factor estimates. For large structures, however, application of the FFT algorithm (Cooley & Tukey, 1965[Cooley, J. W. & Tukey, J. W. (1965). Math. C, 19, 297-301.]) to the Fourier map is more efficient than direct calculation of the structure factors. In the literature other density-modification 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[Burla, M. C., Camalli, M., Carrozzini, B., Cascarano, G. L., Giacovazzo, C., Polidori, G. & Spagna, R. (2000). Acta Cryst. A56, 451-457.]) and Shiono & Woolfson (1992[Shiono, M. & Woolfson, M. M. (1992). Acta Cryst. A48, 451-456.]) for a related procedure]; Caliandro et al. (2008[Caliandro, R., Carrozzini, B., Cascarano, G. L., De Caro, L., Giacovazzo, C., Mazzone, A. & Siliqi, D. (2008). J. Appl. Cryst. 41, 548-553.]) 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 density-modification scheme incorporated in ACORN2 (Dodson & Woolfson, 2009[Dodson, E. J. & Woolfson, M. M. (2009). Acta Cryst. D65, 881-891.]). Alternatively, peakness in the electron-density 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[Sheldrick, G. M. (2015). Acta Cryst. A71, 3-8.]) and allows the posterior application of the FFT algorithm. In the present work, the alternative peakness-enhancing ipp procedure (ipp = inner-pixel preservation) is described. It directly operates on the η = δMmρ product function of the SM algorithm wherein [{m_\rho }] is the mask relating [| \rho |] to [\rho] through the expression

[\left|\rho \left(\Phi \right)\right| = \rho \left(\Phi \right) {m}_{\rho }\left(\Phi \right). \eqno(4)]

According to Rius (2020[Rius, J. (2020). Acta Cryst. A76, 489-493.]), the values of mρ are 1 (for ρ > 0), 0 (for ρ between 0 and −tρσρ) and −1 (for ρ < −tρσρ) with [\sigma_\rho^2] being the variance of ρ(Φ) and tρ ≃ 2.5. Hereafter [{S_{M,| \rho |}}] will be shortened to SM for simplicity.

2. The SM phasing algorithm with enhanced peakness: the ipp procedure

The phasing residual (2)[link] can be minimized with the SM algorithm (Rius, 2020[Rius, J. (2020). Acta Cryst. A76, 489-493.]), i.e. by the iterative application of the modified tangent formula

[{\varphi }_{H}^{\rm new} = {\rm phase\ of}\ \left\{\textstyle\int _{V}{\delta }_{M}\left({\bf r},\Phi \right) {m}_{\rho }\left({\bf r},\Phi \right) \exp\left(i2\pi {\bf H}{\bf r}\right)\,{\rm d}V\right\} \eqno(5)]

which corresponds to the angular part of the Fourier transform within brackets. One characteristic of the SM algorithm is the presence of the η = δMmρ product function. To enhance the peakness of η, the simple ipp procedure based on the preservation of the inner-peak pixels has been added to SM, giving rise to the SM-ipp algorithm (Fig. 1[link]). This procedure consists of two well differentiated parts:

[Figure 1]
Figure 1
The recursive SM-ipp phase refinement algorithm with enhanced peakness: (upper right corner) φ phase estimates (either initial or updated values) are combined with experimental |E|'s to obtain ρ, |ρ| and mρ (the latter is stored). Next, the Fourier transform of |ρ| is calculated leading to new |C| and α values, and the former are used in the calculation of CCM. The new α values are combined with the experimental (|E| − 〈|E|〉) (lower left corner), and their inverse Fourier transform, δM, is calculated. In the next step, function δM is multiplied with the stored mρ mask to give the η product function. Peakness in η is enhanced by applying the ipp density-modification procedure and, finally, the Fourier transform of the modified η supplies the updated φ phases. [Initial sets of φ estimates investigated in this article are either Φrnd (random phase values) or ΦM (phase values corresponding to the Fourier coefficients of M′, i.e. the randomly shifted modulus function).]

(i) Peak search in the η product function. The lowest value of η which is accepted as a peak is fixed by the tη ση threshold ([\sigma_\eta^2] is the variance of η, and tη 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 (xo, yo, zo) 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 (xo ±1, yo ±1, zo ±1); 4 (xo, yo ±1, zo ±1); 4 (xo ±1, yo, zo ±1); 4 (xo ±1, yo ±1, zo); 2 (xo, yo, zo ±1); 2 (xo, yo ±1, zo); 2 (xo ±1, yo, zo) (Rollet, 1965[Rollet, J. S. (1965). Editor. Computing Methods in Crystallography, pp. 35-37. Oxford: Pergamon Press.]). If this is the case, the density value and the pixel coordinates of the central pixel are stored. At the end, the Nη stored peaks are ordered in decreasing strength. (Note, tη and Nη are inversely related.)

(ii) Density modification of η. If Nη > N, then for each one of the N highest-ranked η peaks, the density values of the 26+1 inner-peak pixels are preserved. The density-modification procedure finishes by setting to zero all pixels of η not having preserved density values. For NηN, the inner pixels of all Nη 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 direct methods by Sheldrick (Usón & Sheldrick, 1999[Usón, I. & Sheldrick, G. M. (1999). Curr. Opin. Struct. Biol. 9, 643-648.]). For illustrative purposes, a successful SM-ipp phase refinement obtained with starting random (rnd) phases and with tη = 3.7 is reproduced in Fig. 2[link]. It is interesting to note that only Nη(1) is smaller than N (the number 1 in parentheses indicates the iteration number).

[Figure 2]
Figure 2
SM-ipp phasing with initial random phases (Φrnd): variation of Nη and CCM with the iteration number for data set 1a7z (tη = 3.7). N is the number of non-H atoms in the unit cell.

Compared with the SM algorithm in Rius (2020[Rius, J. (2020). Acta Cryst. A76, 489-493.]) in which all reflections participate in the computation of the ρ synthesis, SM-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[link])]. Notice that the faster calculation of ρ in SM-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 SM-ipp and in SM is very similar. The SM-ipp algorithm has been programmed in a modified version of the XLENS_v1 code (Rius, 2011[Rius, J. (2011). XLENS_v1: a computer program for solving crystal structures from diffraction data by direct methods. Institut de Ciència de Materials de Barcelona, CSIC, Spain (downloadable from https://departments.icmab.es/crystallography/software).]). 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[Rius, J. (2020). Acta Cryst. A76, 489-493.]), the SM algorithm was only tested by assigning random values to the initial phases, Φrnd = {φrnd}. However, the ideal situation for any phasing algorithm is to start with phase values derived from initial ρ estimates (ρini) containing structural information. Since the M modulus synthesis is a Patterson-type synthesis (Ramachandran & Raman, 1959[Ramachandran, G. N. & Raman, S. (1959). Acta Cryst. 12, 957-964.]), it can be regarded as the sum of N weighted shifted images of the crystal structure (or its enantiomorph) (Wrinch, 1939[Wrinch, D. M. (1939). London, Edinb. Dubl. Philos. Mag. J. Sci. 27, 98-122.]; Buerger, 1950[Buerger, M. J. (1950). Acta Cryst. 3, 87-97.]). Consequently, it contains valuable structural information and can be taken as ρini. The success of the phasing process will obviously depend on the capability of the phasing algorithm to develop one incomplete shifted image of the crystal structure 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 unit cell 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 refinement trial requires a different ρini. 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 [|E_H|\exp(\varphi _H')] with [\varphi _H' = - i2\pi {\bf Hu}] and ΦM = {[\varphi _H']}. In this way each trial follows a different refinement path (in the test calculations, the sequence of u vectors is the same for all data sets). The number of selected phase refinement trials (Ntrials) is either 5, 25 or 50 depending on the success rate; the maximum number of allowed iterations per trial is always Niter(max) = 1000 (excepting 3bcj with 200).

4. Comparison of the phasing efficiencies of the SM-ipp and SM algorithms

The efficiencies of the SM-ipp and SM algorithms have been calculated for both Φrnd and ΦM. For simplicity, the various phase refinement strategies are specified by A1, A2, B1, B2, i.e. A1: Φrnd with SM-ipp; A2: Φrnd with SM; B1: ΦM with SM-ipp; B2: ΦM with SM.

The compounds participating in the test calculations are listed in Tables 1[link] and 2[link]. For those compounds in Table 1[link] only containing weak scatterers, the checked strategies are A1, A2 and B1 (Table 3[link]). In the case of compounds with medium/strong scatterers (Tables 1[link] and 2[link]), the investigated strategies are B1 and B2 (Tables 4[link], 5[link] and 6[link]). To make comparisons between strategies stricter, corresponding refinement trials started with the same set of randomly generated phase values.

Table 1
Data sets from the Protein Data Bank (PDB) used to compare the SM-ipp and SM phasing algorithms corresponding to compounds with only weak scatterers (top five) or with weak and medium scatterers (remaining)

Residues = number of residues; c = number of centerings; N = number of non-H atoms in the unit cell (PDB); M and H2O = number of medium scatterers and refined water molecules; %Sol = solvent volume percentage; dmin = minimum d spacing in Å of used reflection data; T = data collection temperature in K. (1a7y, 1ob4, 1a7z, 1alz, 2erl, 1a0m are rotating-anode or sealed-tube data sets; otherwise synchrotron data.)

PDB code Compound Residues Space group N/c M/c H2O/c %Sol dmin T
1a7y Actino D(1) 33 P1 314 44 18 0.94 133
3sbn Trichovirin(2) 30 P21 444 32 24 0.95 100
1ob4 Cephaibol A(3) 17 P21212 548 60 22 1.00 100
1a7z Actino Z3(1) 22 P212121 1228 8Cl 4 49 0.95 173
1alz Gramicidin A(4) 34 P212121 1348 4 30 1.00 120
1byz Alpha1-peptide(5) 52 P1 479 1Cl 30 27 0.91 100
2erl Er-1 pheromone(6) 40 C2 656 14S 44 20 1.00 273
1p9g Antifungal(7) 41 P21 702 20S 122 23 1.00 283
3nir Crambin(8) 48 P21 902 12S 196 31 1.00 100
1a0m Conotoxin(9) 34 I4 1144 40S 168 24 1.10 286
4lzt Lysozime(10) 129 P1 1183 10S 139 32 1.00 295
1f94 Bucandin(11) 63 C2 1232 20S 236 35 1.02 100
1hhu Balhimycin(12) 28 P21 1310 16Cl 250 22 0.89 100
3odv Kaliotoxin(13) 76 [P\bar 1] 1392 32S 180 20 1.00 100
3psm Plant defensin(14) 94 P21 1882 16S 366 45 0.98 100
3bcj Aldose reductase(15) 316 P21 7308 26S 1374 43 0.78 15
          + 3P     0.85  
(1) Schäfer et al. (1998[Schäfer, M., Sheldrick, G. M., Bahner, I. & Lackner, H. (1998). Angew. Chem. Int. Ed. 37, 2381-2384.]); (2) Gessmann et al. (2012[Gessmann, R., Axford, D., Owen, R. L., Brückner, H. & Petratos, K. (2012). Acta Cryst. D68, 109-116.]); (3) Bunkóczi et al. (2003[Bunkóczi, G., Schiell, M., Vértesy, L. & Sheldrick, G. M. (2003). J. Pept. Sci. 9, 745-752.]); (4) Burkhart et al. (1998[Burkhart, B. M., Gassman, R. M., Langs, D. A., Pangborn, W. A. & Duax, W. L. (1998). Biophys. J. 75, 2135-2146.]); (5) Privé et al. (1999[Privé, G. G., Anderson, D. H., Wesson, L., Cascio, D. & Eisenberg, D. (1999). Protein Sci. 8, 1400-1409.]); (6) Anderson et al. (1996[Anderson, D. H., Weiss, M. S. & Eisenberg, D. (1996). Acta Cryst. D52, 469-480.]); (7) Xiang et al. (2004[Xiang, Y., Huang, R. H., Liu, X. Z., Zhang, Y. & Wang, D. C. (2004). J. Struct. Biol. 148, 86-97.]); (8) Schmidt et al. (2011[Schmidt, A., Teeter, M., Weckert, E. & Lamzin, V. S. (2011). Acta Cryst. F67, 424-428.]); (9) Hu et al. (1998[Hu, S.-H., Loughnan, M., Miller, R., Weeks, C. M., Blessing, R. H., Alewood, P. F., Lewis, R. J. & Martin, J. L. (1998). Biochemistry, 37, 11425-11433.]); (10) Walsh et al. (1998[Walsh, M. A., Schneider, T. R., Sieker, L. C., Dauter, Z., Lamzin, V. S. & Wilson, K. S. (1998). Acta Cryst. D54, 522-546. ]); (11) Kuhn et al. (2000[Kuhn, P., Deacon, A. M., Comoso, S., Rajaseger, G., Kini, R. M., Usón, I. & Kolatkar, P. R. (2000). Acta Cryst. D56, 1401-1407.]); (12) Lehmann et al. (2002[Lehmann, C., Bunkóczi, G., Vértesy, L. & Sheldrick, G. M. (2002). J. Mol. Biol. 318, 723-732.]); (13) Pentelute et al. (2010[Pentelute, B. L., Mandal, K., Gates, Z. P., Sawaya, M. R., Yeates, T. O. & Kent, S. B. (2010). Chem. Commun. 46, 8174-8176.]); (14) Song et al. (2011[Song, X., Zhang, M., Zhou, Z. & Gong, W. (2011). FEBS Lett. 585, 300-306.]); (15) Zhao et al. (2008[Zhao, H. T., Hazemann, I., Mitschler, A., Carbone, V., Joachimiak, A., Ginell, S., Podjarny, A. & El-Kabbani, O. (2008). J. Med. Chem. 51, 1478-1481.]).

Table 2
PDB data sets used to test the SM-ipp and SM phasing algorithms corresponding to compounds with strong scatterers

Residues, space group, c, %Sol and dmin as in Table 1[link]. N = number of non-H atoms in the unit cell (PDB); M and S = number of medium and strong scatterers; H2O = number of refined water molecules. Data sets 2bf9, 8rxn and 1c7k measured at room temperature; otherwise at 100 K.

PDB code Compound Residues Space group N/c (M+S)/c H2O/c %Sol dmin
2bf9 aPP(1) 36 C2 768 2Zn 164 31 1.00
8rxn Rubredoxin(2) 52 P21 1010 12S+2Fe 204 35 1.00
1w3m Tsushimycin(3) 132 P1 1276 10Cl+24Ca 191 35 1.00
2ov0 Amicyanin(4) 105 P21 2060 6S+2P+2Cu 432 34 0.95
1c75 Cythochrome 553(5) 71 P212121 2660 12S+4Fe 500 38 0.97
3d1p Transferase(6) 120 C2 2702 2S+2Cl+4Se 498 29 0.95
1pwl Aldose reductase Br(7) 316 P1 3030 14S+3P+1Br 429 25 1.10
1a6m Myoglobin(8) 151 P21 3154 8S+2Fe 372 36 1.00
41au Geodin(9) 161 P21 3278 2Ca+ 6Se 740 40 0.99
1eb6 Deuterolysin(10) 177 P21 3300 12S+2Zn 518 39 1.00
1b0y H42Q(11) 85 P212121 3348 36S+16Fe 824 30 0.90
1x8q Nitro­phorin 4C(12) 184 C2 3662 10S+2Fe 720 24 0.90
2fdn Ferredoxin(13) 55 P4322 3964 128S+64Fe 768 35 1.00
3fsa Azurin(14) 125 P212121 4488 36S+4Cu 856 38 1.00
1c7k Endoprotease Zn(15) 132 P212121 4532 12S+4Ca+4Zn 464 37 1.00
3ks3 H. C. anhydrase II(16) 260 P21 5626 6S+2Zn 962 41 0.95
1heu L. A. de­hydrogenase(17) 748 P1 7618 58S+4Cd 1297 50 1.15
(1) Glover et al. (1983[Glover, I., Haneef, I., Pitts, J., Wood, S., Moss, D., Tickle, I. & Blundell, T. L. (1983). Biopolymers, 22, 293-304.]); (2) Dauter et al. (1992[Dauter, Z., Sieker, L. C. & Wilson, K. S. (1992). Acta Cryst. B48, 42-59.]); (3) Bunkóczi et al. (2005[Bunkóczi, G., Vértesy, L. & Sheldrick, G. M. (2005). Acta Cryst. D61, 1160-1164.]); (4) Carrell et al. (unpublished); (5) Benini et al. (2000[Benini, S., González, A., Rypniewski, W. R., Wilson, K. S., Van Beeumen, J. J. & Ciurli, S. (2000). Biochemistry, 39, 13115-13126.]); (6) Nocek et al. (unpublished); (7) El-Kabbani et al. (2004[El-Kabbani, O., Darmanin, C., Schneider, T. R., Hazemann, I., Ruiz, F., Oka, M., Joachimiak, A., Schulze-Briese, C., Tomizaki, T., Mitschler, A. & Podjarny, A. (2004). Proteins, 55, 805-813.]); (8) Vojtěchovský et al. (1999[Vojtěchovský, J., Chu, K., Berendzen, J., Sweet, R. M. & Schlichting, I. (1999). Biophys. J. 77, 2153-2174.]); (9) Fanfrlik et al. (2013[Fanfrlik, J., Kolar, M., Kamlar, M., Hurny, D., Ruiz, F. X., Cousido-Siah, A., Mitschler, A., Rezac, J., Munusamy, E., Lepsik, M., Matejícek, P., Veselý, J., Podjarny, A. & Hobza, P. (2013). ACS Chem. Biol. 8, 2484-2492.]); (10) McAuley et al. (2001[McAuley, K. E., Jia-Xing, Y., Dodson, E. J., Lehmbeck, J., Østergaard, P. & Wilson, K. S. (2001). Acta Cryst. D57, 1571-1578.]); (11) Parisini et al. (1999[Parisini, E., Capozzi, F., Lubini, P., Lamzin, V., Luchinat, C. & Sheldrick, G. M. (1999). Acta Cryst. D55, 1773-1784.]); (12) Kondrashov et al. (2004[Kondrashov, D. A., Roberts, S. A., Weichsel, A. & Montfort, W. R. (2004). Biochemistry, 43, 13637-13647.]); (13) Dauter et al. (1997[Dauter, Z., Wilson, K. S., Sieker, L. C., Meyer, J. & Moulis, J. M. (1997). Biochemistry, 36, 16065-16073.]); (14) Sato et al. (2009[Sato, K., Li, C., Salard, I., Thompson, A. J., Banfield, M. J. & Dennison, C. (2009). Proc. Natl Acad. Sci. USA, 106, 5616-5621.]); (15) Kurisu et al. (2000[Kurisu, G., Kai, Y. & Harada, S. (2000). J. Inorg. Biochem. 82, 225-228.]); (16) Avvaru et al. (2010[Avvaru, B. S., Kim, C. U., Sippel, K. H., Gruner, S. M., Agbandje-McKenna, M., Silverman, D. N. & McKenna, R. (2010). Biochemistry, 49, 249-251.]); (17) Meijers et al. (2001[Meijers, R., Morris, R. J., Adolph, H. W., Merli, A., Lamzin, V. S. & Cedergren-Zeppezauer, E. S. (2001). J. Biol. Chem. 276, 9316-9321.]).
†Four disordered Se positions.

Table 3
Application of the SM-ipp and SM algorithms to crystal structures only containing weak scatterers (A1, A2 and B1 phasing strategies)

The tρ parameter controlling the threshold of the mρ mask is always 2.50. N/c as in Table 1[link]; Np = number of peaks showing up in the final E map above the n σρ threshold; CCM = correlation coefficient between experimental and calculated modulus function; Niter = number of iterations to achieve convergence (n.c. = no convergence in 1000 iterations); tη is the parameter controlling the number Nη of strongest η peaks; Q = Nη(2)/N.

PDB code N/c Phasing strategy Np/c (n) CCM Niter〉 for 5, 25 or 50 trials tη Q
1a7y 314 A1 376 (1.1) 0.86 Niter〉 = 37 (25×) 4.0 1.0
    A2 363 (1.1) 0.86 Niter〉 = 105 (25×)
    B1 370 (1.1) 0.85 Niter〉 = 100 (25×) 4.0 0.9
3sbn 444 A1 449 (1.1) 0.82 Niter〉 = 112 (23×); n.c. (2×) 3.7 1.4
    A2 456 (1.1) 0.85 Niter〉 = 558 (13×); n.c. (12×)
    B1 456 (1.1) 0.82 Niter〉 = 139 (21×); n.c. (4×) 3.7 1.2
1ob4 548 A1 564 (1.1) 0.86 Niter〉 = 308 (10×); n.c. (15×) 3.7 1.1
    A2 573 (1.1) 0.87 Niter〉 = 392 (3×); n.c. (22×)
    B1 569 (1.1) 0.86 Niter〉 = 446 (8×); n.c. (17×) 3.7 1.0
1a7z 1228 A1 1279 (1.1) 0.83 Niter〉 = 133 (22×); n.c. (3×) 3.7 1.4
    A2 1372 (1.1) 0.85 Niter〉 = 334 (16×); n.c. (9×)
    B1 1281 (1.1) 0.83 Niter〉 = 338 (23×); n.c. (2×) 3.7 1.0
1alz 1348 A1 1308 (1.5) 0.84 136, 520; n.c. (48×) 3.8 1.2
    A2 n.c. (50×)
    B1 n.c. (50×) 3.8 0.9

Table 4
Application of the SM-ipp and SM algorithms to crystal structures with medium scatterers

Upper and lower lines refer to phasing strategies B1 and B2, respectively (except for 3bcj). N, M, c as in Table 1[link]; Np = number of peaks showing up in the final E map above the n σρ threshold; CCM = correlation coefficient between experimental and calculated modulus function; Niter = number of iterations to achieve convergence (n.c. = no convergence in 1000 iterations); tρ, tη = parameters controlling, respectively, the threshold of the mρ mask and the number Nη of strongest η peaks; Q = Nη(2)/N.

PDB code N/c (M/c) Np/c (n) CCM Niter for 5 or 25 trials tρ tη Q
1byz 479 (1Cl) 520 (1.1) 0.86 46, 56, 102, 134, 146 2.65 4.0 0.9
529 (1.1) 0.86 234, 356, 535; n.c. (2×)
2erl 656 (14S) 610 (1.1) 0.78 32, 77, 112, 251, 270 2.50 4.0 0.8
703 (1.1) 0.83 330, 475, 731; n.c. (2×)
1p9g 702 (20S) 695 (1.4) 0.85 24, 29, 30, 31, 43 2.60 3.8 1.1
769 (1.4) 0.86 156, 174, 279, 296, 409
3nir 902 (12S) 934 (1.1) 0.86 24, 38, 46, 98, 127 2.70 4.0 0.7
921 (1.1) 0.88 211, 333, 666, 690; n.c. (1×)
1a0m 1144 (40S) 1124 (1.3) 0.83 93, 97, 110, 158,186 2.60 3.7 1.0
1342 (1.3) 0.86 217, 344, 366, 510, 844
4lzt 1183 (10S) 1134 (1.5) 0.83 43, 47, 48, 49, 51 2.65 4.0 1.0
n.c. (5×)
1f94 1232 (20S) 1160 (1.1) 0.81 108, 110, 111, 171, 189, 353, 834, 897; n.c. (17×) 2.50 3.8 1.1
1230 (1.1) 0.83 342, 681; n.c. (23×)
1hhu 1310 (16Cl) 1360 (1.4) 0.82 Niter〉 = 117 (17×); n.c. (8×) 2.60 3.9 1.1
1378 (1.4) 0.85 Niter〉 = 426 (12×); n.c. (13×)
3odv 1392 (32S) 1480 (1.0) 0.72 18, 23, 23, 27, 36 2.50 3.5 1.2
1499 (1.0) 0.77 176, 249, 256, 290, 583
3psm 1882 (16S) 1854 (1.4) 0.78 23, 24, 27, 33, 45 2.60 4.0 0.9
2132 (1.4) 0.82 309; n.c. (4×)
3bcj 7308 (26S) 7222 (1.3) 0.81 Niter〉 = 73 (20×); n.c. (5×) 2.65 3.8 1.5
7086 (1.3) 0.82 Niter〉 = 114 (9×); n.c. (16×) 3.8 1.4
†Upper and lower lines correspond to B1 at dmin = 0.78 and 0.85 Å, respectively. Niter(max) = 200.

Table 5
Application of SM-ipp to crystal structures containing strong scatterers (S) (strategy B1)

N = number of non-H atoms in the unit cell (PDB); c = number of centerings; Np, CCM, Niter, n.c., tρ, tη and Q as in Table 3[link]. [tρ = 2.80 except for 1w3m (2.60), 3d1p (2.70), 1a6m (2.75), 41au (2.70) and 3fsa (2.70); 〈BWilson〉 is 6.8 (1.1) Å2 with the extrema being 5.3 for 2fdn and 9.1 for 1eb6.]

PDB code N/c (S/c) Np/c (n) CCM Niter for 5 trials tη Q
2bf9 768 (2Zn) 709 (1.1) 0.81 10, 11, 12, 13, 15 3.5 2.2
8rxn 1010 (2Fe) 905 (1.1) 0.83 14, 15, 17, 18, 22 3.5 1.4
1w3m 1276 (24Ca) 1275 (1.4) 0.81 30, 33, 37, 42, 80 4.0 1.1
2ov0 2060 (2Cu) 1990 (1.5) 0.84 14, 15, 16, 16, 29 4.0 0.9
1c75 2660 (4Fe) 2541 (1.4) 0.82 12, 13, 14, 16,16 4.0 1.1
3d1p 2702 (4Se) 2642 (1.1) 0.83 12, 12, 14, 15, 16 3.8 1.1
1pwl 3030 (1Br) 3123 (1.1) 0.83 32, 54, 57, 62, 149 3.8 1.2
1a6m 3154 (2Fe) 3203 (1.1) 0.83 28, 30, 31, 37, 48 4.0 0.8
41au 3278 (6Se) 3440 (1.1) 0.85 41, 69, 79; n.c. (2×) 3.8 1.0
1eb6 3300 (2Zn) 3406 (1.1) 0.82 16, 18, 23, 24, 40 4.0 0.9
1b0y 3348 (16Fe) 3360 (1.3) 0.76 38, 39, 41, 53, 79 3.5 1.5
1x8q 3662 (2Fe) 3510 (1.5) 0.83 34, 36, 58, 64, 92 4.0 1.0
2fdn 3944 (64Fe) 3832 (1.3) 0.81 21, 21, 22, 23, 26 3.8 1.0
3fsa 4488 (4Cu) 4580 (1.5) 0.83 31, 39, 40, 44, 56 4.0 1.0
1c7k 4532 (4Zn) 4548 (1.3) 0.84 80, 96, 128, 202, 399 4.0 0.9
3ks3 5626 (2Zn) 5588 (1.2) 0.83 9, 10, 10, 10, 10 3.9 1.1
1heu 7618 (4Cd) 7603 (1.1) 0.82 35, 40, 42, 45, 176 3.9 1.1
†Four Se atoms are partially disordered.

Table 6
Comparison of strategies B1 and B2 when applied to crystal structures with strong scatterers (S)

For B2, the individual Niter values are given; for B1, 〈Niter〉 corresponds to Niter values in Table 5[link]. It is evident that B1 (using ipp) performs better than B2 in all cases. N = number of non-H atoms in the unit cell (PDB); c = number of centerings; 〈Niter〉 = average number of iterations to achieve convergence (n.c. = no convergence in 1000 iterations).

PDB code N/c (S/c) B1 strategy 〈Niter B2 strategy Niter for 5 trials
2bf9 768 (2Zn) 12.2 (5×) 29, 29, 36, 39, 55
8rxn 1010 (2Fe) 17.2 (5×) 44, 53, 54, 56, 61
1w3m 1276 (24Ca) 44.4 (5×) 97, 109, 118, 124, 134
2ov0 2060 (2Cu) 18.0 (5×) 52, 54, 61, 62, 86
1c75 2660 (4Fe) 14.2 (5×) 41, 51, 52, 57, 73
3d1p 2702 (4Se) 13.8 (5×) 35, 38, 44, 45, 47
1pwl 3030 (1Br) 70.8 (5×) n.c. (5×)
1a6m 3154 (2Fe) 34.8 (5×) n.c. (5×)
41au 3278 (6Se) 63.0 (3×) 400, n.c. (4×)
1eb6 3300 (2Zn) 24.2 (5×) 69, 86, 109, 156, 289
1b0y 3348 (16Fe) 50.0 (5×) 210, 225, 243, 254,259
1x8q 3662 (2Fe) 56.8 (5×) 261, 317, 404, 961, n.c.
2fdn 3944 (64Fe) 22.6 (5×) 62, 77, 82, 93, 97
3fsa 4488 (4Cu) 42.0 (5×) 163, 288, 324, 413, n.c.
1c7k 4532 (4Zn) 181.0 (5×) n.c. (5×)
3ks3 5626 (2Zn) 9.8 (5×) 36, 42, 45, 46, 48
1heu 7618 (4Cd) 67.6 (5×) 226, 259, 273, 534, n.c.
†Four Se atoms are partially disordered.

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[link]). The first three data sets belong to small crystal structures and the last two to relatively large ones. Of these, 1a7z corresponds to a Cl-containing compound with 1228 atoms in the unit cell. In spite of the presence of Cl, it has been included in this section because the refinement 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 crystal structure of gramicidin with 1348 C, N and O atoms in the unit cell 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[link]). Compared with A2, A1 yields the smallest 〈Niter〉 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[link] for data sets 3sbn and 1a7z. In the case of gramicidin, two correct solutions are obtained with A1 (trial 21 with Niter = 136 and trial 45 with Niter = 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[link] indicates that A1 converges somewhat faster than B1 and is superior in the case of gramicidin (B1 gives no correct solutions).

[Figure 3]
Figure 3
Effect of the ipp procedure on the phasing efficiency of the SM algorithm with Φrnd. The two selected data sets belong to: (top) 3sbn (trichovirin) with 444 atoms in the unit cell; (bottom) 1a7z (Actino Z3) with 1228. True solutions obtained with/without the ipp procedure in black/gray (same starting random phase values for each pair of trials).

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[link]. 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 SM-ipp to 1f94 (bucandin) was less successful. Consequently, Ntrials was increased to 25 to estimate more reliably the success percentage (32%). This structure has large atomic disorder (BWilson = 14.3 Å2) 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 refinement accuracy can be estimated with ΔCCM, i.e. the difference between CCM values for SM-ipp and for SM. As can be clearly seen in Tables 3[link] and 4[link], ΔCCM is only slightly negative, generally between −0.02 and −0.03, which suggests that truncation of the outer-peak regions during the application of the ipp procedure is not critical.

To estimate the influence of ipp on the convergence of the phase refinement, the same tests carried out with strategy B1 were repeated with B2 (Table 4[link]). Comparison of both sets of Niter values confirms the much faster convergence of B1.

4.3. Crystal structures with strong scatterers

From Table 5[link] 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 〈Niter〉 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 symmetry-independent seleno­methio­nine Se atoms showing partial occupancies, i.e. (0.52, 0.48) and (0.31 and 0.69) (Fanfrlik et al., 2013[Fanfrlik, J., Kolar, M., Kamlar, M., Hurny, D., Ruiz, F. X., Cousido-Siah, A., Mitschler, A., Rezac, J., Munusamy, E., Lepsik, M., Matejícek, P., Veselý, J., Podjarny, A. & Hobza, P. (2013). ACS Chem. Biol. 8, 2484-2492.]). For 1pwl and 1heu, the larger 〈Niter〉 values could be ascribed to the larger dmin values (Table 2[link]). For comparison purposes, the results obtained with strategies B1 and B2 are summarized in Table 6[link]. Its inspection confirms the clear superiority of B1 over B2, especially for the larger test crystal structures.

5. Discussion

One characteristic of the SM algorithm is its mathematical simplicity, a consequence of the straightforward implementation of the modified tangent formula (5)[link]. One relevant parameter of SM is tρ which modifies the threshold value in the calculation of |ρ| through expression (4)[link]. The value of tρ mainly depends on the scattering power of the strongest scatterer present in the crystal structure. In Rius (2020[Rius, J. (2020). Acta Cryst. A76, 489-493.]), tρ 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 tρ 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 tη. The N value used in the test calculations is the sum of both protein and solvent atoms (taken from the PDB), i.e. NProt + NH2O. An idea of 〈NH2O〉 can be obtained by averaging (NProt + NH2O)/NProt over all structures with more than 700 atoms listed in Tables 1[link] and 2[link] which gives 1.22 (5), i.e.NH2O〉 ≃ 0.22 × NProt. The second parameter, tη, controls the number of η peaks above the tη ση threshold. It can be estimated from Q = Nη(2)/N. Suitable tη values are those for which Q is close to 1 or not much smaller (the ipp procedure does not use Nη peaks exceeding N). According to Tables 3[link], 4 [link] and 5[link], values of tη 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 refinement ends with a sudden increase of CCM concomitant with a marked Nη decrease.

Of interest is the comparison of the Nη(1) values obtained with strategies A1 (Φrnd) and B1 (ΦM) by using similar tη values. As was already shown in Section 2[link], Nη(1) is smaller than N for Φrnd (Fig. 2[link]). However, for ΦM (Fig. 4[link]), Nη(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 ΦM set at the end of the first iteration is always calculated with the N largest η peaks. The only exception is 1b0y. Since the unit cell of this compound contains four dominant scattering units (Fe4S4 clusters), only the 240 (= 162 − 16) strongest η peaks (mostly corresponding to Fe–Fe interactions) were used.

[Figure 4]
Figure 4
SM-ipp phasing with ΦM: variation of Nη and CCM with the iteration number for data set 3ks3 (tη = 3.9). N = number of non-H atoms in the unit cell.

For the compounds in Table 1[link] (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 ultra-high 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 SM-ipp with ultra high resolution data. With 5934 atoms in the unit cell (solvent atoms excluded) this crystal structure is in the same order of magnitude as those listed in Table 2[link] containing strong scatterers. Application of SM-ipp with ΦM (strategy B1) yields success percentages of 80%, 36% and 0% for dmin = 0.78, 0.85 and 0.90 Å, respectively (Fig. 5[link] reproduces the E map of one arbitrary successful refinement). Notice that SM-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[McCoy, A. J., Oeffner, R. D., Wrobel, A. G., Ojala, J. R. M., Tryggvason, K., Lohkamp, B. & Read, R. J. (2017). Proc. Natl Acad. Sci. USA, 114, 3637-3641.]).

[Figure 5]
Figure 5
Unit-cell content of aldose reductase (Zhao et al., 2008[Zhao, H. T., Hazemann, I., Mitschler, A., Carbone, V., Joachimiak, A., Ginell, S., Podjarny, A. & El-Kabbani, O. (2008). J. Med. Chem. 51, 1478-1481.]; data set 3bcj) showing the two unique protein chains related by the screw axis along b as obtained with the SM-ipp phasing algorithm directly from the experimental modulus synthesis (ΦM) by assuming P1 symmetry (S and light atoms are found simultaneously). Atoms with higher refined peak strength are shown in red.

A limitation of SM-ipp (when used as an ab initio phasing algorithm) arises for crystal structures belonging to high-symmetry 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, molecular replacement or anomalous dispersion techniques. In such cases SM-ipp will become the phase refinement stage of a more general two-stage strategy.

6. Conclusions

It has been shown that the introduction of the new peakness-enhancing ipp procedure in the SM phase refinement 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 SM-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 light-atom crystal structures either Φrnd or ΦM phases can be used (peak overlap in the modulus function can still be managed by SM-ipp).

(b) Starting with Φrnd is appropriate for crystal structures containing only weak scatterers (the largest N value tested is around 1500 atoms).

(c) Starting with ΦM 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 Niter(max) iterations, then phase refinement with Φrnd should be tried (with a larger Niter(max)); however, ΦM should always be the first choice.

(d) Use of ΦM 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 ΦM is their fast convergence. This allows one to reduce Niter(max) and, consequently, increase the number of explored trials.

Finally, some words regarding data completeness are in order. As already mentioned in Section 1[link], the SM algorithm relies on the validity of the RM residual (2)[link] 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), RM 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 RM to moderate-resolution data sets. This `structure-factor extrapolation' technique (Caliandro et al., 2005a[Caliandro, R., Carrozzini, B., Cascarano, G. L., De Caro, L., Giacovazzo, C. & Siliqi, D. (2005a). Acta Cryst. D61, 556-565.],b[Caliandro, R., Carrozzini, B., Cascarano, G. L., De Caro, L., Giacovazzo, C. & Siliqi, D. (2005b). Acta Cryst. D61, 1080-1087.], 2007[Caliandro, R., Carrozzini, B., Cascarano, G. L., De Caro, L., Giacovazzo, C. & Siliqi, D. (2007). J. Appl. Cryst. 40, 931-937.]; see also Jia-xing et al., 2005[Jia-xing, Y., Woolfson, M. M., Wilson, K. S. & Dodson, E. J. (2005). Acta Cryst. D61, 1465-1475.]) is particularly effective for crystal structures containing heavy atoms (Caliandro et al., 2008[Caliandro, R., Carrozzini, B., Cascarano, G. L., De Caro, L., Giacovazzo, C., Mazzone, A. & Siliqi, D. (2008). J. Appl. Cryst. 41, 548-553.]; Burla et al., 2012[Burla, M. C., Caliandro, R., Camalli, M., Carrozzini, B., Cascarano, G. L., Giacovazzo, C., Mallamo, M., Mazzone, A., Polidori, G. & Spagna, R. (2012). J. Appl. Cryst. 45, 357-361.]). The combination of SM with the extrapolation technique could represent a further source of progress.

Supporting information


Funding information

The following funding is acknowledged: MINECO and FEDER (grant No. RTI2018-098537-B-C21); Severo Ochoa Programme for Centres of Excellence in R&D (grant No. SEV-2015-0496).

References

First citationAnderson, D. H., Weiss, M. S. & Eisenberg, D. (1996). Acta Cryst. D52, 469–480.  CrossRef CAS Web of Science IUCr Journals Google Scholar
First citationAvvaru, B. S., Kim, C. U., Sippel, K. H., Gruner, S. M., Agbandje-McKenna, M., Silverman, D. N. & McKenna, R. (2010). Biochemistry, 49, 249–251.  Web of Science CrossRef CAS PubMed Google Scholar
First citationBenini, S., González, A., Rypniewski, W. R., Wilson, K. S., Van Beeumen, J. J. & Ciurli, S. (2000). Biochemistry, 39, 13115–13126.  Web of Science CrossRef PubMed CAS Google Scholar
First citationBuerger, M. J. (1950). Acta Cryst. 3, 87–97.  CrossRef CAS IUCr Journals Web of Science Google Scholar
First citationBunkóczi, G., Schiell, M., Vértesy, L. & Sheldrick, G. M. (2003). J. Pept. Sci. 9, 745–752.  Web of Science CSD CrossRef PubMed Google Scholar
First citationBunkóczi, G., Vértesy, L. & Sheldrick, G. M. (2005). Acta Cryst. D61, 1160–1164.  CrossRef IUCr Journals Google Scholar
First citationBurkhart, B. M., Gassman, R. M., Langs, D. A., Pangborn, W. A. & Duax, W. L. (1998). Biophys. J. 75, 2135–2146.  Web of Science CrossRef CAS PubMed Google Scholar
First citationBurla, M. C., Caliandro, R., Camalli, M., Carrozzini, B., Cascarano, G. L., Giacovazzo, C., Mallamo, M., Mazzone, A., Polidori, G. & Spagna, R. (2012). J. Appl. Cryst. 45, 357–361.  Web of Science CrossRef CAS IUCr Journals Google Scholar
First citationBurla, M. C., Camalli, M., Carrozzini, B., Cascarano, G. L., Giacovazzo, C., Polidori, G. & Spagna, R. (2000). Acta Cryst. A56, 451–457.  Web of Science CrossRef CAS IUCr Journals Google Scholar
First citationCaliandro, R., Carrozzini, B., Cascarano, G. L., De Caro, L., Giacovazzo, C., Mazzone, A. & Siliqi, D. (2008). J. Appl. Cryst. 41, 548–553.  Web of Science CrossRef CAS IUCr Journals Google Scholar
First citationCaliandro, R., Carrozzini, B., Cascarano, G. L., De Caro, L., Giacovazzo, C. & Siliqi, D. (2005a). Acta Cryst. D61, 556–565.  Web of Science CrossRef CAS IUCr Journals Google Scholar
First citationCaliandro, R., Carrozzini, B., Cascarano, G. L., De Caro, L., Giacovazzo, C. & Siliqi, D. (2005b). Acta Cryst. D61, 1080–1087.  Web of Science CrossRef CAS IUCr Journals Google Scholar
First citationCaliandro, R., Carrozzini, B., Cascarano, G. L., De Caro, L., Giacovazzo, C. & Siliqi, D. (2007). J. Appl. Cryst. 40, 931–937.  Web of Science CrossRef CAS IUCr Journals Google Scholar
First citationCooley, J. W. & Tukey, J. W. (1965). Math. C, 19, 297–301.  CrossRef Google Scholar
First citationDauter, Z., Sieker, L. C. & Wilson, K. S. (1992). Acta Cryst. B48, 42–59.  CrossRef CAS Web of Science IUCr Journals Google Scholar
First citationDauter, Z., Wilson, K. S., Sieker, L. C., Meyer, J. & Moulis, J. M. (1997). Biochemistry, 36, 16065–16073.  Web of Science CrossRef CAS PubMed Google Scholar
First citationDodson, E. J. & Woolfson, M. M. (2009). Acta Cryst. D65, 881–891.  Web of Science CrossRef CAS IUCr Journals Google Scholar
First citationEl-Kabbani, O., Darmanin, C., Schneider, T. R., Hazemann, I., Ruiz, F., Oka, M., Joachimiak, A., Schulze-Briese, C., Tomizaki, T., Mitschler, A. & Podjarny, A. (2004). Proteins, 55, 805–813.  Web of Science CrossRef PubMed CAS Google Scholar
First citationFanfrlik, J., Kolar, M., Kamlar, M., Hurny, D., Ruiz, F. X., Cousido-Siah, A., Mitschler, A., Rezac, J., Munusamy, E., Lepsik, M., Matejícek, P., Veselý, J., Podjarny, A. & Hobza, P. (2013). ACS Chem. Biol. 8, 2484–2492.  CAS PubMed Google Scholar
First citationGessmann, R., Axford, D., Owen, R. L., Brückner, H. & Petratos, K. (2012). Acta Cryst. D68, 109–116.  Web of Science CrossRef IUCr Journals Google Scholar
First citationGlover, I., Haneef, I., Pitts, J., Wood, S., Moss, D., Tickle, I. & Blundell, T. L. (1983). Biopolymers, 22, 293–304.  CrossRef CAS PubMed Google Scholar
First citationHu, S.-H., Loughnan, M., Miller, R., Weeks, C. M., Blessing, R. H., Alewood, P. F., Lewis, R. J. & Martin, J. L. (1998). Biochemistry, 37, 11425–11433.  Web of Science CrossRef CAS PubMed Google Scholar
First citationJia-xing, Y., Woolfson, M. M., Wilson, K. S. & Dodson, E. J. (2005). Acta Cryst. D61, 1465–1475.  Web of Science CrossRef IUCr Journals Google Scholar
First citationKondrashov, D. A., Roberts, S. A., Weichsel, A. & Montfort, W. R. (2004). Biochemistry, 43, 13637–13647.  CrossRef PubMed CAS Google Scholar
First citationKuhn, P., Deacon, A. M., Comoso, S., Rajaseger, G., Kini, R. M., Usón, I. & Kolatkar, P. R. (2000). Acta Cryst. D56, 1401–1407.  CrossRef CAS IUCr Journals Google Scholar
First citationKurisu, G., Kai, Y. & Harada, S. (2000). J. Inorg. Biochem. 82, 225–228.  Web of Science CrossRef PubMed CAS Google Scholar
First citationLehmann, C., Bunkóczi, G., Vértesy, L. & Sheldrick, G. M. (2002). J. Mol. Biol. 318, 723–732.  CrossRef PubMed CAS Google Scholar
First citationMcAuley, K. E., Jia-Xing, Y., Dodson, E. J., Lehmbeck, J., Østergaard, P. & Wilson, K. S. (2001). Acta Cryst. D57, 1571–1578.  Web of Science CrossRef CAS IUCr Journals Google Scholar
First citationMcCoy, A. J., Oeffner, R. D., Wrobel, A. G., Ojala, J. R. M., Tryggvason, K., Lohkamp, B. & Read, R. J. (2017). Proc. Natl Acad. Sci. USA, 114, 3637–3641.  Web of Science CrossRef CAS PubMed Google Scholar
First citationMeijers, R., Morris, R. J., Adolph, H. W., Merli, A., Lamzin, V. S. & Cedergren-Zeppezauer, E. S. (2001). J. Biol. Chem. 276, 9316–9321.  Web of Science CrossRef PubMed CAS Google Scholar
First citationMiller, R., DeTitta, G. T., Jones, R., Langs, D. A., Weeks, C. M. & Hauptman, H. A. (1993). Science, 259, 1430–1433.  CSD CrossRef CAS PubMed Web of Science Google Scholar
First citationParisini, E., Capozzi, F., Lubini, P., Lamzin, V., Luchinat, C. & Sheldrick, G. M. (1999). Acta Cryst. D55, 1773–1784.  Web of Science CrossRef CAS IUCr Journals Google Scholar
First citationPentelute, B. L., Mandal, K., Gates, Z. P., Sawaya, M. R., Yeates, T. O. & Kent, S. B. (2010). Chem. Commun. 46, 8174–8176.  CrossRef CAS Google Scholar
First citationPrivé, G. G., Anderson, D. H., Wesson, L., Cascio, D. & Eisenberg, D. (1999). Protein Sci. 8, 1400–1409.  Web of Science PubMed CAS Google Scholar
First citationRamachandran, G. N. & Raman, S. (1959). Acta Cryst. 12, 957–964.  CrossRef CAS IUCr Journals Web of Science Google Scholar
First citationRius, J. (1993). Acta Cryst. A49, 406–409.  CrossRef CAS Web of Science IUCr Journals Google Scholar
First citationRius, J. (2011). XLENS_v1: a computer program for solving crystal structures from diffraction data by direct methods. Institut de Ciència de Materials de Barcelona, CSIC, Spain (downloadable from https://departments.icmab.es/crystallography/software).  Google Scholar
First citationRius, J. (2012). Acta Cryst. A68, 77–81.  Web of Science CrossRef CAS IUCr Journals Google Scholar
First citationRius, J. (2020). Acta Cryst. A76, 489–493.  CrossRef IUCr Journals Google Scholar
First citationRius, J., Crespi, A. & Torrelles, X. (2007). Acta Cryst. A63, 131–134.  Web of Science CrossRef CAS IUCr Journals Google Scholar
First citationRollet, J. S. (1965). Editor. Computing Methods in Crystallography, pp. 35–37. Oxford: Pergamon Press.  Google Scholar
First citationSato, K., Li, C., Salard, I., Thompson, A. J., Banfield, M. J. & Dennison, C. (2009). Proc. Natl Acad. Sci. USA, 106, 5616–5621.  CrossRef PubMed CAS Google Scholar
First citationSchäfer, M., Sheldrick, G. M., Bahner, I. & Lackner, H. (1998). Angew. Chem. Int. Ed. 37, 2381–2384.  CrossRef CAS Google Scholar
First citationSchmidt, A., Teeter, M., Weckert, E. & Lamzin, V. S. (2011). Acta Cryst. F67, 424–428.  Web of Science CrossRef CAS IUCr Journals Google Scholar
First citationSheldrick, G. M. (2015). Acta Cryst. A71, 3–8.  Web of Science CrossRef IUCr Journals Google Scholar
First citationSheldrick, G. M. & Gould, R. O. (1995). Acta Cryst. B51, 423–431.  CrossRef CAS Web of Science IUCr Journals Google Scholar
First citationShiono, M. & Woolfson, M. M. (1992). Acta Cryst. A48, 451–456.  CrossRef CAS Web of Science IUCr Journals Google Scholar
First citationSong, X., Zhang, M., Zhou, Z. & Gong, W. (2011). FEBS Lett. 585, 300–306.  Web of Science CrossRef CAS PubMed Google Scholar
First citationUsón, I. & Sheldrick, G. M. (1999). Curr. Opin. Struct. Biol. 9, 643–648.  Web of Science CrossRef PubMed CAS Google Scholar
First citationVojtěchovský, J., Chu, K., Berendzen, J., Sweet, R. M. & Schlichting, I. (1999). Biophys. J. 77, 2153–2174.  Web of Science PubMed Google Scholar
First citationWalsh, M. A., Schneider, T. R., Sieker, L. C., Dauter, Z., Lamzin, V. S. & Wilson, K. S. (1998). Acta Cryst. D54, 522–546.   CrossRef CAS IUCr Journals Google Scholar
First citationWeeks, C. M., DeTitta, G. T., Miller, R. & Hauptman, H. A. (1993). Acta Cryst. D49, 179–181.  CrossRef CAS Web of Science IUCr Journals Google Scholar
First citationWrinch, D. M. (1939). London, Edinb. Dubl. Philos. Mag. J. Sci. 27, 98–122.  Google Scholar
First citationXiang, Y., Huang, R. H., Liu, X. Z., Zhang, Y. & Wang, D. C. (2004). J. Struct. Biol. 148, 86–97.  CrossRef PubMed CAS Google Scholar
First citationZhao, H. T., Hazemann, I., Mitschler, A., Carbone, V., Joachimiak, A., Ginell, S., Podjarny, A. & El-Kabbani, O. (2008). J. Med. Chem. 51, 1478–1481.  CrossRef PubMed CAS Google Scholar

This is an open-access article distributed under the terms of the Creative Commons Attribution (CC-BY) Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.

Journal logoFOUNDATIONS
ADVANCES
ISSN: 2053-2733
Follow Acta Cryst. A
Sign up for e-alerts
Follow Acta Cryst. on Twitter
Follow us on facebook
Sign up for RSS feeds