research papers Global indicators of X-ray data quality
Journal of
Applied Crystallography ISSN 0021-8898
Manfred S. Weiss
Received 6 September 2000 Accepted 22 November 2000
Institute of Molecular Biotechnology, Department of Structural Biology and Crystallography, PO Box 100813, D-07708 Jena, Germany. Correspondence e-mail:
[email protected]
# 2001 International Union of Crystallography Printed in Great Britain ± all rights reserved
Global indicators of the quality of diffraction data are presented and discussed, and are evaluated in terms of their performance with respect to various tasks. Based on the results obtained, it is suggested that some of the conventional indicators still in use in the crystallographic community should be abandoned, such as the nominal resolution dmin or the merging R factor Rmerge, and replaced by more objective and more meaningful numbers, such as the effective optical resolution deff,opt and the redundancy-independent merging R factor Rr.i.m.. Furthermore, it is recommended that the precision-indicating merging R factor Rp.i.m. should be reported with every diffraction data set published, because it describes the precision of the averaged measurements, which are the quantities normally used in crystallography as observables.
1. Introduction The quality or the information content of a crystallographically determined three-dimensional structure is ultimately dependent on the quality or the information content of the underlying diffraction data. The importance of ensuring that the highest quality data set of a given crystal is obtained cannot be overstressed, especially in light of the fact that the diffraction experiment is the last real experimental step in the course of a structure determination. Literally dozens of papers have been published in the past decade on the evaluation of the quality of structures determined by X-ray crystallographic methods [see the recent review by Kleywegt (2000) and the references therein]. By contrast, comparatively few studies on the evaluation of the quality of the underlying diffraction data have been published (Weiss & Hilgenfeld, 1997; Diederichs & Karplus, 1997; Vaguine et al., 1999). In these papers a number of new and improved quality indicators have been proposed, but the crystallographic community has been slow to adopt these new indicators. Therefore, the present situation is still less than satisfactory. In this paper, an overview of the global indicators of the quality of diffraction data that have been described to date will be given, their strengths and their weaknesses will be discussed, and their usefulness will be demonstrated using two practical examples.
2. The quality indicators Assuming that the crystal specimen preparation, the data collection strategy, the actual intensity measurements, the indexing and integration of the raw data as well as the scaling of the diffraction data, were carried out in a meaningful and sensible way, the question arises `what is the quality of the obtained diffraction data and how is the quality assessed as
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objectively as possible?' For the sake of simplifying the discussion, it appears advantageous to dissect this question into two: `how can we describe the quality of the data set (i) when only averaged intensities/amplitudes are available and (ii) when equivalent re¯ections have not yet been merged together?' A list of criteria that can be calculated to answer the ®rst part of the question is presented in Table 1. The ®rst and also the most important indicator is the resolution of the data set dtrue, which is de®ned as the minimum distance at which two features in the corresponding electron density map can be resolved. In practice, dtrue is usually approximated by dmin, which will in the following be referred to as the nominal resolution. Unfortunately, dmin is not unambiguously de®ned by the diffraction data. It is a rather subjective term as different authors or different laboratories adopt different criteria for how to de®ne the physical diffraction limit of the crystal and the nominal resolution of a data set. In most cases, these criteria include some kind of signal-to-noise ratio, but again, the estimation of the standard deviations (or uncertainties) depends to some extent on the computer program chosen for data processing and scaling; therefore, data sets collected at different laboratories cannot be compared directly with one another based solely on the quantity dmin. Given the fact, however, that sensible choices have been made for the de®nition of dmin, it has been noted that dmin is a slightly conservative estimate of dtrue (Blundell & Johnson, 1976; Stenkamp & Jensen, 1984). A second important number is the completeness of the data set. It has long been realised that any missing re¯ection, be it one that has not been measured or one that has been deliberately excluded to lower the ®nal R factor of the structure, leads to a deterioration of the model parameters (Hirshfeld & Rabinovich, 1973). It seems therefore justi®ed to de®ne the term `effective resolution' based on the nominal resolution and the cube root of the completeness of the data set. For J. Appl. Cryst. (2001). 34, 130±135
research papers Prior to merging equivalent re¯ections, the merging statistics provide a valuable source of information about the quality of a data set. A list of indicators that can be calculated in this (1) True resolution dtrue situation is presented in Table 2. The ®rst criterion is the (2) Nominal resolution dmin redundancy of the data. Since X-ray diffraction data collection (3) Signal-to-noise ratio I/(I) is to some extent in¯uenced by counting statistics, the aver(4) Completeness C (5) Effective resolution deff = deffCÿ1/3 aged measurement should become more accurate as more 2 2 (6) Optical resolution dopt = 2
Patt sph 1=2 (' Patt21/2) individual measurements are made. A highly redundant data ÿ1/3 (7) Effective optical resolution deff,opt = doptC set will therefore be intrinsically of higher quality than a data (8) Wilson plot appearance set in which every re¯ection has only been measured once. Ê data set which is only 70% complete instance, a nominal 1.9 A Naturally, data collected from crystals exhibiting low spaceÊ data set. This relation is of course will effectively be a 2.1 A group symmetry potentially suffer most from this problem. not correct in a strict mathematical sense, but it accounts for The most frequently reported descriptor of data quality is the fact that missing re¯ections lead to a deterioration of the the conventional merging R factor Rmerge (Stout & Jensen, ®nal model parameters much in the same way as reduced 1968; Blundell & Johnson, 1976; Drenth, 1994). This is resolution does. A positive side effect of a requirement for unfortunate, because Rmerge is intrinsically dependent on the authors to report the effective resolution of their data set redundancy of the data, as has been noted by a number of instead of the nominal resolution might also be an increased authors (Weiss & Hilgenfeld, 1997; Diederichs & Karplus, motivation to collect more complete diffraction data sets. 1997). Low redundancy will always yield a lower Rmerge, but at A very interesting criterion is the optical resolution dopt of a the same time result in less accurate data. This dependence of diffraction data set, calculated based on the standard deviaRmerge on the redundancy can be remedied by the introduction tion of a Gaussian ®tted to the origin peak of the Patterson of the redundancy-independent merging R factor Rr.i.m. (Weiss function ( Patt) and the standard deviation of another Gaus& Hilgenfeld, 1997; Weiss et al., 1998). This R factor has also sian ®tted to the origin peak of a spherical interference been called Rmeas by Diederichs & Karplus (1997). It describes function ( sph) as de®ned by Vaguine et al. (1999) and as the precision of the individual measurements, independent of implemented in the program SFCHECK. This indicator takes how often a given re¯ection has been measured, and could into account errors in the data, atomic displacement factors, therefore be used as a substitute for the conventional Rmerge quality of the crystal and series termination effects (Blundell factor. Diederichs & Karplus (1997) have also introduced the & Johnson, 1976; Vaguine et al., 1999). It is relatively indepooled coef®cient of variation, PCV, an indicator which is pendent of the choice of the nominal resolution of the data set basically described by the sum of the standard deviations and should therefore be more objective. It is also relatively divided by the sum of the re¯ection intensities. It can be shown independent of the completeness of the data set. Nevertheless, that PCV is related to Rr.i.m. by the equation PCV = Vaguine et al. (1999) de®ned an effective optical resolution (/2)1/2Rr.i.m. ' 1.25Rr.i.m. (Diederichs & Karplus, 1997). based on a complete data set by estimating the amplitudes of Another indicator that can be calculated is the so-called the missing re¯ections. Similarly, an effective optical resoluprecision-indicating merging R factor Rp.i.m. (Weiss & tion may be de®ned in the same way as the effective resolution Hilgenfeld, 1997; Weiss et al., 1998), which describes the (Table 1). precision of the averaged measurement. Since in the course of In some cases, an overestimation of the nominal resolution structure determination and re®nement, averaged intensities can also be identi®ed based on the appearance of the Wilson or amplitudes are normally used, this R factor should deliver (1949) plot of the data set. In particular, a clear deviation of the most information when it comes to predicting the the plot from linearity at the high-resolution side is often a performance of a given data set in structure determination. good indication. However, quantitative studies have not been The precision of the averaged measurement can also be carried out along this line. estimated by the quantity Rmrgd-I as de®ned by Diederichs & Karplus (1997). The total set of re¯ections is divided into two Table 2 parts, then each part is merged Global quality indicators that can be derived from a diffraction data set in which equivalent re¯ections have and averaged separately and not been merged. ®nally an R factor is calculated (1) Redundancy N .PP PP between the two subsets. jIi
hkl ÿ I
hklj Ii
hkl (2) Merging R factor Rmerge = hkl i hkl i .PP However, this R factor does not P P (3) Redundancy-independent merging R factor Rr.i.m. = N=
N ÿ 11=2 jIi
hkl ÿ I
hklj Ii
hkl describe the precision of the hkl i hkl i ®nal averaged intensity directly, (4) Pooled coef®cient of variation PCV = . P P PP Ii
hkl 1=
N ÿ 1 jIi
hkl ÿ I
hklj2 1=2 because the additional merging hkl i hkl i . P PP step leads to a further reduction 1=2 P (5) Precision-indicating merging R factor Rp.i.m. = 1=
N ÿ 1 jIi
hkl ÿ I
hklj Ii
hkl hkl i hkl i in error. In principle, Rp.i.m. and . P P (6) R factor between data subsets I1 and I2 Rmrgd-I = jI1
hkl ÿ I2
hklj 0:5 I1
hkl I2
hkl Rmrgd-I are related in that they hkl hkl both exhibit a 1/N1/2 behaviour, Table 1
Global quality indicators that can be derived from a diffraction data set in which equivalent re¯ections have been merged.
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research papers and, in fact, Rp.i.m. is an analytical version of Rmrgd-I. However, whereas Rp.i.m. describes the precision of the averaged measurement directly, Rmrgd-I introduces a multiplicative factor of 2. Another potential problem with Rmrgd-I is that it depends to some extent on how the subsets I1 and I2 of the total data set were de®ned. There are of course many other ways to assess how well a given set of unmerged re¯ections can be merged together. Extrapolating from the Diederichs & Karplus (1997) de®nition of Rmrgd-I, one can for instance divide the whole set of re¯ections into two parts of unequal size [similar to the concept of the free R factor in crystallographic re®nement (BruÈnger, 1992)], then scale and merge the subsets separately and report the ®nal merging statistics when the two parts of the data set are merged together. Doing that, one has to be aware, however, as was noted for Rmrgd-I in the previous paragraph, that any additional merging will increase the accuracy of the averaged measurement. This will of course be less severe when one of the subsets is chosen to be very small (e.g. 10%) compared with the other. Such a procedure has not yet been implemented by us in a computer program and will therefore not be discussed any further in this paper. In the following, the usefulness of the discussed quality indicators, with the main focus on the statistical parameters obtained by merging equivalent re¯ections, will be assessed by comparing their performance in two sets of experiments.
Table 3
Data collection and re®nement statistics for factor XIII data. Ê , resolution limits 100.0± Space group P21, a = 133.85, b = 70.62, c = 100.53 A Ê. 1.95 A Data set Data collection Rotation range ( ) Total No. of re¯ections No. of unique re¯ections Redundancy Ê (%) Completeness, 100.0±1.95 A Ê (%) Completeness, 1.98±1.95 A Ê I/(I), 100.0±1.95 A Ê I/(I), 1.98±1.95 A Ê (%) Rmerge, 100.0±1.95 A Ê (%) Rmerge, 1.98±1.95 A Ê (%) Rr.i.m., 100.0±1.95 A Ê (%) Rr.i.m., 1.98±1.95 A Ê (%) Rp.i.m., 100.0±1.95 A Ê (%) Rp.i.m., 1.98±1.95 A Ê (%) PCV, 100.0±1.95 A Ê (%) PCV, 1.98±1.95 A Ê 2) Overall B (A Ê) deff,opt (A Re®nement Final R (%) Final Rfree (%) No. of water molecules Total CC²
1
2
3
4
90 243669 124657 1.95 94.6 85.1 12.3 1.1 5.3 54.8 7.0 71.3 4.5 45.2 7.3 79.6 30.8 1.63
180 480070 129513 3.71 98.3 95.3 17.3 1.3 6.9 74.6 8.1 89.1 4.2 48.1 9.2 104.8 31.1 1.61
270 725675 129683 5.60 98.5 96.5 21.4 1.7 7.1 75.6 7.9 84.6 3.3 36.9 9.5 109.9 31.0 1.61
360 964098 129800 7.43 98.5 97.3 25.0 2.0 7.3 79.6 7.8 86.7 2.8 33.6 9.4 111.1 31.3 1.62
21.69 26.89 898 0.8486
21.15 25.25 932 0.8508
20.93 24.81 930 0.8510
20.80 24.76 936 0.8516
² Correlation coef®cient between electron density maps calculated based on the coef®cients 2Fobs ÿ Fcalc and Fcalc, and the phases of the re®ned model.
3. Experimental Crystals of coagulation factor XIII were grown as described by Weiss et al. (1998) except that 20 mM of ethylenediaminetetraacetic acid (EDTA) was also present in the crystallization solution. A crystal was then soaked for 1 h in reservoir solution containing 30% (v/v) glycerol, and then ¯ash-cooled in a nitrogen stream at 100 K. 360 of diffraction data were collected at the X31 beamline of the EMBL outstation (DESY, Hamburg) in 600 images of 0.6 each. The wavelength Ê and the detector was a 345 mm MAR Research was 1.044 A imaging plate (MarResearch Hamburg, Germany) set at a distance of 270 mm from the crystal. The data were integrated using the program DENZO (Otwinowski & Minor, 1997). Four consecutive 90 wedges of raw data were de®ned, which resulted in four data sets consisting of 90, 180, 270 and 360 . Using the NO MERGE ORIGINAL INDEX option in SCALEPACK (Otwinowski & Minor, 1997), scaled but unmerged data were written out so that the redundancy N, Rr.i.m., Rp.i.m. and PCV could be calculated for the four data sets. (The program RMERGE which was written for that purpose is available via http://www.imb-jena.de/www_sbx/projects/ sbx_qual.html or from the author upon request.) A summary of the statistics of the four data sets is presented in Table 3. Structure-factor amplitudes were calculated from the intensities using the method of French & Wilson (1978) as implemented in the program TRUNCATE (Collaborative Computational Project, Number 4, 1994). Re®nement against these data was carried out using the programs PROTIN, REFMAC and ARP (Lamzin & Wilson, 1993). The starting
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structure was the structure of the coagulation factor XIII/Ca2+ Ê resolution (unpublished results), complex re®ned at 1.6 A devoid of the calcium ions and the water positions. A standard re®nement protocol included 20 cycles of rigid-body re®neÊ , 20 cycles of ment in the resolution range 10.0±2.5 A Ê and restrained re®nement in the resolution range 40.0±1.95 A 20 grand cycles of iterating re®nement and water building, with a maximum of 50 new water molecules added in each grand cycle. The ®nal R and Rfree values of the re®nements of the four data sets are shown in Table 3. Hen egg-white lysozyme crystals were grown as described previously (Weiss et al., 2000). A total of 360 of diffraction data were collected from one crystal at T = 100 K in a nitrogen stream. The rotation range per image was 1 and the detector (30 cm MAR Research imaging plate) was set at a distance of 100 mm. Each image was exposed for 750 s using a rotatinganode X-ray source (FR591, Nonius, Delft, The Netherlands) operated at 40 kV and 90 mA. The maximum resolution at the Ê . This data collection proved edge of the detector was 1.63 A somewhat problematic since severe ice formation on the goniometer head and on the crystal made frequent interruption of the data collection necessary in order to clean the crystal of growing ice. The whole rotation range was divided into eight separate wedges of 45 each. Care was taken to assure the highest possible completeness for the ®rst 45 wedge of data. Using the program STRATEGY (Ravelli et al., 1997), the starting ' angle for the ®rst wedge was picked. This ' angle was then taken as the starting angle of all the other J. Appl. Cryst. (2001). 34, 130±135
research papers data sets, which comprise rotation ranges of 45, 90, 135, 180, 225, 270, 315 and 360 , respectively. The raw data were integrated using the program DENZO (Otwinowski & Minor, 1997) and scaled using the program SCALEPACK (Otwinowski & Minor, 1997). The redundancy N, Rr.i.m., Rp.i.m. and PCV were calculated as above. A summary of the data sets is presented in Table 4. The test of the accuracy of the averaged intensities of these data sets was whether it would be possible to determine the positions of the 18 anomalously scattering atoms [10 S, 8 Clÿ (Dauter et al., 1999)] using the program Shake-and-Bake (SnB) version 2.0 (Weeks & Miller, 1999). The expected anomalous signal at the Ê , can be estimated to h|F|i/hFi Cu K wavelength, = 1.54 A 00 ' 1.5±1.9%, using F values of 0.56 electrons (for S) and 0.70 electrons (for Clÿ). h|I|i/hIi should then be expected to amount to 3.0±3.8%. SnB was initially run with default parameters (Howell et al., 2000). Because of the small signal, the restrictions in Xmin and Ymin for the calculation of the difference-E values were reduced to 0.1 in both cases, and the number of re¯ections to be phased was set at 1800, i.e. 100 times the expected number of sites. The default here was 30 times the number of sites, but it was found that the discrimination between correct solutions and incorrect solutions was
enhanced when more re¯ections were phased, despite the fact that the total number of successful trials was to some extent smaller. An even better discrimination between correct and incorrect solutions was achieved when the number of re¯ections to be phased was increased to 200 times the number of sites (data not shown).
4. Results and discussion The results for the two practical examples tested are summarized in Tables 3 and 4 and in Figs. 1 and 2. An interesting feature was observed by comparing the four data sets of coagulation factor XIII. According to a conventional I/(I) 2.0 criterion, the nominal resolution limits of Ê , respecthe four data sets would be 2.1, 2.0, 1.98 and 1.95 A tively (data not shown). The increased redundancy N of the measurements leads to smaller standard deviations (by a factor N1/2) and an apparent higher resolution limit. It was noted, however, that the optical resolution dopt and the effective optical resolution deff,opt were hardly at all in¯uenced by the redundancy of the data. The differences in the averaged measurements are obviously too small to have a signi®cant effect on dopt or deff,opt, and the standard deviations are not
Table 4
Data collection and Shake-and-Bake run statistics for lysozyme data. Ê , resolution limits 100.0±1.64 A Ê. Space group P43212, a = 77.05, c = 37.21 A Data set 1
2
3
4
5
6
7
8
Data collection Rotation range ( ) Total No. of re¯ections No. of unique re¯ections Redundancy Ê (%) Completeness, 100.0±1.64 A Ê (%) Completeness, 1.67±1.64 A Ê I/(I), 100.0±1.64 A Ê I/(I), 1.67±1.64 A Ê (%) Rmerge, 100.0±1.64 A Ê (%) Rmerge, 1.67±1.64 A Ê (%) Rr.i.m., 100.0±1.64 A Ê (%) Rr.i.m., 1.67±1.64 A Ê (%) Rp.i.m., 100.0±1.64 A Ê (%) Rp.i.m., 1.67±1.64 A Ê (%) PCV, 100.0±1.64 A Ê (%) PCV, 1.67±1.64 A Ê (%) Ranom,² 100.0±1.64 A Ê (%) Ranom,² 1.67±1.64 A Ê 2) Overall B (A Ê) deff,opt (A
45 45493 13712 3.3 96.0 96.5 24.9 8.1 3.8 12.6 4.5 14.9 2.4 7.8 5.9 16.8 2.7 8.9 18.7 1.35
90 89015 14055 6.3 98.4 100.0 32.2 10.9 3.9 14.8 4.3 16.2 1.6 6.2 6.1 19.0 2.2 7.4 18.6 1.33
135 136348 14112 9.7 98.8 100.0 40.7 14.1 4.0 15.6 4.2 16.5 1.3 5.2 6.1 20.3 1.7 5.4 18.5 1.33
180 181974 14265 12.8 99.8 100.0 49.5 16.5 4.4 17.0 4.6 17.7 1.3 5.0 6.8 22.0 1.5 4.7 18.6 1.32
225 229114 14268 16.1 99.8 100.0 55.2 19.2 4.5 17.1 4.7 17.7 1.2 4.5 7.2 22.4 1.5 4.2 18.6 1.32
270 273083 14272 19.2 99.9 100.0 59.4 20.7 4.5 16.9 4.6 17.3 1.1 4.0 7.2 22.2 1.4 4.0 18.6 1.32
315 318779 14273 22.4 99.9 100.0 63.9 22.7 4.4 16.8 4.5 17.2 1.0 3.7 7.0 21.9 1.4 3.7 18.6 1.32
360 364260 14278 25.6 99.9 100.0 70.5 24.2 4.4 16.8 4.5 17.2 0.9 3.4 6.8 21.8 1.4 3.5 18.5 1.32
SnB runs Total No. of trials No. of re¯ections phased No. of successful trials Rmin³ (correct) Rmin³ (next) Rcryst³ (correct) Rcryst³ (next) CC³ (correct) CC³ (next)
5000 1643 0 ± 0.76 ± 0.41 ± 0.28
5000 1605 0 ± 0.80 ± 0.39 ± 0.28
5000 1695 0 ± 0.79 ± 0.38 ± 0.29
5000 1753 0 ± 0.78 ± 0.37 ± 0.40
5000 1748 1 0.75 0.76 0.37 0.37 0.31 0.35
5000 1735 7 0.72±0.74 0.78 0.35±0.37 0.37 0.37±0.29 0.36
5000 1737 13 0.69±0.71 0.77 0.34±0.36 0.37 0.41±0.34 0.37
5000 1782 40 0.66±0.70 0.77 0.32±0.38 0.37 0.43±0.33 0.40
.P P ² Ranom = hkl jI
hkl ÿ I
h k lj ³ Rmin, Rcryst and CC are the minimal function, a crystallographic R factor based on E values and the correlation coef®cient based on E hkl I
hkl. values, respectively. These parameters are the ®gures of merit provided by the program SnB. The range of the values for the correct solutions is given along with the values for the best (`next') incorrect solution.
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research papers
6 5
6
4 3
25 20
6 R factors
8
R anom
Redundancy PCV R r.i.m. R merge R p.i.m.
8
Redundancy
R factors
10
7
15 4 10
4 2 2
2
5
1 1
2
3
1
4
2
3
26
24
22
20
6
7
8
6
7
8
35 30 25 20 15 10 5 1
2
5
40 Successful trials / 5000 SnB runs
R free R
1
4
Data set
Data set
R factors
Redundancy
Redundancy PCV R r.i.m. R merge R p.i.m.
3
4
2
3
4
5
Data set
Data set
Figure 2
Figure 1
Global diffraction data quality indicators (top panel) and re®nement results (bottom panel) for the four data sets constructed for factor XIII.
used in the calculation of dopt or deff,opt. Therefore, if dopt or deff,opt were reported instead of dmin, there would be no more need to rely on estimated standard deviations in order to de®ne the resolution of the data set. As was noted above, the standard deviations are estimated differently by different programs, which makes comparisons between different data sets based solely on dmin dif®cult. It is therefore concluded that dopt or deff,opt constitute a more objective, hence superior, description of the physical diffraction limit of the crystal. The test applied to the four coagulation factor XIII data sets considered re®nement and automated water building. Merging and re®nement statistics for the four data sets are listed in Table 3 and displayed graphically in Fig. 1. It can be seen that both the R factor and the free R factor (bottom panel) follow the same trend as the precision-indicating merging R factor Rp.i.m. (top panel). The correlation coef®cient between R and Rp.i.m. is 0.90 and that between Rfree and Rp.i.m. is 0.82, whereas the correlation coef®cients between R and Rmerge and between Rfree and Rmerge are ÿ0.98 and ÿ1.00, respectively. Apparently, the increased accuracy in the averaged measurements as described by Rp.i.m. leads to a better behaviour of the re®nement, which is re¯ected in both better re®nement statistics and also in a larger number of water
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Global diffraction data quality indicators (top panel) and SnB results (bottom panel) for the eight data sets constructed for lysozyme. In the top panel, the observed anomalous difference Ranom for the data sets is also shown.
molecules detected. In order to dissect these two effects, two additional experiments have been performed. In the ®rst one, the model re®ned against data set 1 is re®ned against all four data sets for another 20 cycles without additional water building. The resulting values for Rfree are 26.81% (R = 21.67%), 25.96% (R = 21.42%), 25.87% (R = 21.23%) and 25.78% (R = 21.11%), for re®nement against data sets 1, 2, 3 and 4, respectively. This demonstrates convincingly that about half of the total improvement is solely due to the increased accuracy of the averaged measurement on the re®nement statistics. The other half seems to result from the additional water molecules that can be detected as the quality of the data sets improves. The second experiment was the calculation of R factors of the model derived from re®nement against data set 4 against the data sets 1 to 3. The resulting numbers are 25.25%, 24.80% and 24.77% for Rfree and 21.83%, 21.28% and 20.98% for R, respectively. This demonstrates that the model re®ned against data set 4 is also a better explanation of the data in data sets 1 to 3. In the top panel of Fig. 1, both Rr.i.m. and PCV are increased when going from the ®rst to the second data set, whereas theoretically they should be constant and independent of the J. Appl. Cryst. (2001). 34, 130±135
research papers redundancy. This increase re¯ects a deterioration of the raw data quality, maybe by ice formation during collection. This is also manifested by a smaller than expected drop in Rp.i.m. between the ®rst two data sets. In any case, despite these problems the averaged measurement becomes more accurate and the re®nement proceeds more effectively. For the lysozyme data, a different test was chosen, because the overall quality of the lysozyme data, even for the ®rst data set with a redundancy of 3.3, is already so good that automated re®nement could not reveal signi®cant trends. Therefore, a much more sensitive test was chosen. Dauter et al. (1999) have shown recently that tetragonal hen egg-white lysozyme crystals grown in the presence of NaCl contain 18 atoms (10 S, 8 Clÿ) that scatter X-rays anomalously at a wavelength of Ê . The test was now whether these 18 anomalous scat1.54 A terers could be found based on the anomalous differences in the eight data sets, using the program SnB (Weeks & Miller, 1999). The results are shown in Table 4 and the bottom panel of Fig. 2. The plot of the minimal function values versus the number of SnB trials exhibits a clear bimodal distribution indicative of a correct solution for the data sets 5±8 only. Data sets 1±4 did not exhibit any bimodality. Again, this trend proceeds parallel to the redundancy of the data set or inversely with Rp.i.m.. It is interesting to note that the anomalous R factor based on intensities (Ranom) decreases with increased redundancy of the data and reaches a minimum value at some point at about 1.4%. An explanation for this could be that statistical errors in¯uencing the small differences |I + ÿ I ÿ| are averaged out the more often these differences are observed. In the higherredundancy data sets, the anomalous differences are then as close to the true differences as they can get. It even seems to be the case that Rp.i.m. can predict whether the positions of the anomalous scatterers can be identi®ed: as soon as Rp.i.m. drops below the value for Ranom at high redundancy (1.4%), correct solutions appear among the trials performed.
5. Conclusion and recommendations The results can be summarized in the following four statements. (i) The optical resolution dopt or the effective optical resolution deff,opt are better and more objective approximations for the true resolution dtrue than the conventionally used subjective term dmin. Therefore, it would be an informative addition if dopt or deff,opt were reported with each diffraction data set published. (ii) The redundancy-independent merging R factor Rr.i.m. is a better indicator of the precision of the single intensity measurements than the conventional Rmerge. As a conse-
J. Appl. Cryst. (2001). 34, 130±135
quence, in accordance with the suggestion made by Diederichs & Karplus (1997), Rmerge needs to be replaced by Rr.i.m.. (iii) The precision-indicating merging R factor Rp.i.m. constitutes an indicator of the precision of the averaged measurements and would be a useful additional statistical descriptor for published diffraction data sets. (iv) Rp.i.m. seems to be able to predict the performance of diffraction data sets with respect to the determination of the anomalously scattering substructure based on I values and with respect to standard crystallographic re®nement.
I would like to thank Tom Sicker for the FXIII crystal and his help in collecting the FXIII diffraction data, Dr Gottfried J. Palm for discussions, and the crew of the EMBL outstation (DESY, Hamburg) for their data collection facilities. I am also grateful to the two referees who helped to make the paper much clearer in many places by providing many stimulating thoughts. The work was in part supported by BMBF (through DESY-HS), grant 05SH8BJA1 to Rolf Hilgenfeld, Head of the Department of Structural Biology and Crystallography at the IMB Jena.
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