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

The incorporation of the new peakness-enhancing fast Fourier transform compatible ipp procedure (ipp = inner-pixel 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 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 S_M algorithm is simple, since it only includes a few easy-to-adjust parameters.

1. Introduction

The novel phasing function is rooted in the Z_R origin-free modulus sum function, a nearly 30 years-old direct-methods 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 unit cell by using the property that and are positive-definite functions with similar shape (Rius, 2020). The resulting phase refinement function is defined by

in which the K sum extends over all reflections (i.e. strong and weak ones), |E_K| denotes the experimental structure-factor modulus with 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 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 structure-factor phases to be refined. Their refinement 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 full-symmetry 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 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; 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 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) 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) 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 density-modification scheme incorporated in ACORN2 (Dodson & Woolfson, 2009). 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) 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 = δ_Mm_ρ product function of the S_M algorithm wherein is the mask relating to through the expression

According to Rius (2020), the values of m_ρ are 1 (for ρ > 0), 0 (for ρ between 0 and −t_ρσ_ρ) and −1 (for ρ < −t_ρσ_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_Mm_ρ 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 S_M, giving rise to the S_M-ipp algorithm (Fig. 1). This procedure consists of two well differentiated parts:

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 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 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 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 direct methods by Sheldrick (Usón & Sheldrick, 1999). For illustrative purposes, a successful S_M-ipp phase refinement 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).

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 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 Patterson-type synthesis (Ramachandran & Raman, 1959), it can be regarded as the sum of N weighted shifted images of the crystal structure (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 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 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 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 (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 refinement 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 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 S_M-ipp and S_M 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; d_min = 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		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); (2) Gessmann et al. (2012); (3) Bunkóczi et al. (2003); (4) Burkhart et al. (1998); (5) Privé et al. (1999); (6) Anderson et al. (1996); (7) Xiang et al. (2004); (8) Schmidt et al. (2011); (9) Hu et al. (1998); (10) Walsh et al. (1998); (11) Kuhn et al. (2000); (12) Lehmann et al. (2002); (13) Pentelute et al. (2010); (14) Song et al. (2011); (15) Zhao et al. (2008).

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

Residues, space group, c, %Sol and d_min as in Table 1. 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); (2) Dauter et al. (1992); (3) Bunkóczi et al. (2005); (4) Carrell et al. (unpublished); (5) Benini et al. (2000); (6) Nocek et al. (unpublished); (7) El-Kabbani et al. (2004); (8) Vojtěchovský et al. (1999); (9) Fanfrlik et al. (2013); (10) McAuley et al. (2001); (11) Parisini et al. (1999); (12) Kondrashov et al. (2004); (13) Dauter et al. (1997); (14) Sato et al. (2009); (15) Kurisu et al. (2000); (16) Avvaru et al. (2010); (17) Meijers et al. (2001).

†Four disordered Se positions.

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

The t_ρ parameter controlling the threshold of the m_ρ mask is always 2.50. N/c as in Table 1; N_p = number of peaks showing up in the final E map above the n σ_ρ threshold; CC_M = correlation coefficient between experimental and calculated modulus function; N_iter = number of iterations to achieve convergence (n.c. = no convergence in 1000 iterations); t_η is the parameter controlling the number N_η of strongest η peaks; Q = N_η(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 S_M-ipp and S_M 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; N_p = number of peaks showing up in the final E map above the n σ_ρ threshold; CC_M = correlation coefficient between experimental and calculated modulus function; N_iter = number of iterations to achieve convergence (n.c. = no convergence in 1000 iterations); t_ρ, t_η = parameters controlling, respectively, the threshold of the m_ρ mask and the number N_η of strongest η peaks; Q = N_η(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 d_min = 0.78 and 0.85 Å, respectively. N_iter(max) = 200.

Table 5
Application of S_M-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; N_p, CC_M, N_iter, n.c., t_ρ, t_η and Q as in Table 3. [t_ρ = 2.80 except for 1w3m (2.60), 3d1p (2.70), 1a6m (2.75), 41au (2.70) and 3fsa (2.70); 〈B_Wilson〉 is 6.8 (1.1) Ų 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 N_iter values are given; for B1, 〈N_iter〉 corresponds to N_iter values in Table 5. 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; 〈N_iter〉 = 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). 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). 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).

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. 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 Ų) 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 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 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). 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 symmetry-independent 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 crystal structure. 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 refinement 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 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 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 (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 S_M-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 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).

Figure 5 Unit-cell content of aldose reductase (Zhao et al., 2008; 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 S_M-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 S_M-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 S_M 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 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 light-atom 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 refinement 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 moderate-resolution data sets. This `structure-factor extrapolation' technique (Caliandro et al., 2005a,b, 2007; see also Jia-xing 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.