Ab initio structure determination of low-molecular-weight compounds using synchrotron radiation Laue diffraction

Ravelli, R.B.G.; Raves, M.L.; Scheres, S.H.W.; Schouten, A.; Kroon, J.

doi:10.1107/S0909049598011595

research papers

JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 6| Part 1| January 1999| Pages 19-28

https://doi.org/10.1107/S0909049598011595

Ab initio structure determination of low-molecular-weight compounds using synchrotron radiation Laue diffraction

Raimond B. G. Ravelli,^a ^*‡ Mia L. Raves,^b Sjors H. W. Scheres,^a Arie Schouten ^a and Jan Kroon ^a

^aDepartment of Crystal and Structural Chemistry, Bijvoet Center for Biomolecular Research, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands, and ^bDepartment of Structural Biology, Weizmann Institute of Science, 76100 Rehovot, Israel
^*Correspondence e-mail: ravelli@embl-grenoble.fr

(Received 20 March 1998; accepted 7 September 1998)

The potential of using polychromatic synchrotron radiation for ab initio structure determination of low-molecular-weight compounds is investigated. Three different organic compounds were studied. For each of the structures the cell volume was determined with the aid of a Zr attenuator that was placed in the direct beam to obtain a sharp $[\lambda_{\rm min}]$ edge. Space-group determination did not prove to be complicated; it was aided by deconvolution of the multiple spots. Two low-temperature data sets were collected, and it appeared that spot streaking owing to the inherent increase of mosaicity due to freezing was not too detrimental to the application of the Laue method. Although both the $[R_{\rm merge}]$ values (around 12%) and the final [R_1] values (around 10%) are higher than usually found in monochromatic experiments, all structures show reasonable geometry. Comparison of one of the structures with a reference structure obtained with monochromatic data shows an r.m.s. deviation between the 21 non-H-atom positions of 0.019 Å.

Keywords: Laue diffraction; ab initio structure determination; white radiation.

1. Introduction

The Laue diffraction method makes use of the full spectrum of X-rays produced by a synchrotron as opposed to the monochromatic technique. This enables the collection of a large amount of intensity measurements in very short exposure times. Two important applications that exploit these benefits have proved to be successful: structure determination on very small crystals (see, for example, Snell et al., 1995 ; Kariuki & Harding, 1995 ), and time-resolved studies on proteins (Schlichting et al., 1990 ; Bolduc et al., 1995 ; Šrajer et al., 1996 ; Genick et al., 1997 ).

However, these advantages are nowadays challenged by different applications of monochromatic radiation. The extreme brilliance of third-generation synchrotrons permits a reduction in the size of the beam to the micrometre scale, and monochromatic structure determinations on extremely small protein crystals (a few micrometres thick) have been reported (Xiao et al., 1995 ). Time-resolved studies on (enzymatic) reactions that occur on a time scale of minutes can also be followed using fast monochromatic techniques like the large-oscillation-angle Weissenberg method (Gouet et al., 1996 ; Hajdu & Andersson, 1993 ). Intermediates that are formed during these reactions can be trapped either chemically (Verschueren et al., 1993 ) or by freezing techniques, which eliminates the necessity for rapid data collection. The advantage of monochromatic techniques is that they can circumvent problems associated with the Laue diffraction experiment, like sensitivity to mosaicity of the crystals, harmonic and spatial overlap, low-resolution hole, radiation damage, unfavourable signal-to-background ratio and labour-intensive data processing. Nevertheless, the Laue technique remains the only technique fast enough to collect structural information on a millisecond and even nanosecond time scale in order to follow reactions (Šrajer et al., 1996; Genick et al., 1997).

Here we present the use of the Laue technique in a field in which the disadvantages are less problematic: ab initio structure determination of low-molecular-weight compounds. The mosaicity of crystals of these compounds is more likely to be acceptable as far as spatial overlap is concerned. Even if it is high (as expressed in a radial smearing of the spots), the relatively small number of spots on each frame limits the risk of overlap. As with macromolecules, both the low-resolution hole and the harmonic overlap problem can, at least partly, be overcome by collecting a large number of frames on a crystal and using a reliable deconvolution algorithm. Radiation damage can be prevented by cooling the crystal at liquid-nitrogen temperature. The gain in intensity by using a synchrotron beam compared with an in-house rotating anode can cancel out the disadvantage of the unfavourable signal-to-background ratio inherent to the Laue technique. Finally, during the last few years, important progress has been made in both hardware (CCD and IP detectors) and software to simplify the data collection and processing.

Three features presented in this paper are relatively new and thus far not used in combination with each other in the Laue field: freezing crystals, deconvolution of multiples, and absolute scaling of the unit cell with the use of an attenuator. Some important steps in the data reduction are discussed in the next section. The sensitivity of direct methods to solve structures with the kind of systematic incompleteness that often occurs with Laue data is quantified using the C(z) formalism of Giacovazzo et al. (1994 ). Although the quality of the data presented in this paper is still not ideal, it is shown that the Laue technique can be a fast and efficient alternative for traditional monochromatic structure determination.

The relevance of carrying out ab initio structure determinations of low-molecular-weight compounds using Laue data not only lies in the field of tiny crystal studies (Kariuki & Harding, 1995) or possibly high-throughput structure determinations. These well defined systems can also be considered as appropriate pilot models to improve on data collection and processing. This is expected to be beneficial to the applications of the Laue method that make full use of the intense white beam produced by a synchrotron: time-resolved (perturbation) studies, both on low-molecular-weight compounds (Cheetham et al., 1995 ; H. Graafsma, personal communication) and proteins (Šrajer et al., 1996; Genick et al., 1997).

2. Selected theoretical aspects

A good overview of Laue data collection and reduction methods can be found in Helliwell, Habash et al. (1989 ), Ren & Moffat (1995a ), Yang et al. (1997 ) and Ravelli (1998 ). Here, only a small number of steps are discussed that proved to be the most important for a reliable ab initio structure determination of the compounds investigated.

2.1. Profile fitting

Laue diffraction patterns are very sensitive to the mosaic spread of the crystal (see, for example, Bartunik & Borchert, 1989 ; Hajdu et al., 1991 ). A high mosaicity will cause the diffraction spots to be smeared and, eventually, to disappear into the background. Freezing the crystal is likely to increase the mosaicity: it is therefore important to be able to account for streakiness during integration. An analytical profile-fitting method is able to fit even severely streaked spots (Ren & Moffat, 1995a). However, in the unfavourable case that freezing has slightly cracked the crystal, causing some spots to be doubled, it might be better to use an empirical profile-fitting algorithm (Rossmann, 1979 ) like that recently implemented in the program LAUEGEN (Campbell, 1995 ).

2.2. Determination of the unit-cell parameters on an absolute scale

The determination of the unit-cell and crystal-orientation parameters is crucial for the indexing of a single-crystal X-ray diffraction pattern. These parameters can be used to construct two matrices; [A ] for a standard setting of the (reciprocal) unit cell, and [C ] to rotate the unit cell to its actual setting. The two matrices relate the indices of each reflection $[{\bf h} \left(h,k,l \right)]$ to the reciprocal-lattice vector $[{\bf x} \left(x,y,z \right)]$ by (Helliwell, Habash et al., 1989; Ravelli et al., 1996 )

$[{\bf x} = \left[C\right]\left[A\right]{\bf h}.\eqno(1)]$

The spot positions (X_f, Y_f ) for a reflection $[{\bf h}]$ in a Laue diffraction pattern can now be calculated, assuming a flat detector perpendicular to the outgoing beam at a distance d_f from the crystal, and a beam that hits the crystal coming from the negative Z axis, by

$[\eqalignno{&X_f=d_f[xz/(z^2-x^2-y^2)],\cr& Y_f=d_f[yz/(z^2-x^2-y^2)].&(2)}]$

These equations clearly show the scaling problem in the Laue diffraction experiment: the matrix [A ] can be multiplied by an arbitrary constant s without affecting the predicted spot positions. There is only one restriction on $[{\bf x}]$ : it should lie in an accessible part of reciprocal space, bounded by the $[\lambda_{\rm min}]$ , $[\lambda_{\rm max}]$ and d_min spheres. However, by scaling these spheres by s, these boundary restrictions will remain exactly identical.

Fortunately, an approximate value of $[\lambda_{\rm min}]$ is usually known for each beamline set-up, and can be used to determine the value of s within roughly 20% (Kariuki & Harding, 1995), leading to an initial estimate of the cell volume. Furthermore, in most Laue diffraction experiments the cell parameters and volume are already known from monochromatic experiments, and can be used directly to solve the scaling problem. For ab initio structure determination using only Laue data, however, a start-off point is needed to determine the cell volume more accurately. It has been suggested (Carr et al., 1992 ; Kariuki & Harding, 1995) and demonstrated for a known structure (Carr et al., 1993 ) that an attenuator with an absorption edge (just) above the expected minimum wavelength of the X-ray beam can be used to obtain a sharp wavelength cut-off. The Pt absorption edges from a Pt-coated focusing mirror (Ren & Moffat, 1995a) or the $[K\alpha]$ or $[K\beta]$ radiation produced by a (rotating) anode in a laboratory Laue experiment (Ravelli et al., 1996), however, can also be used. Here, we report on the use of a thin foil of Zr (thickness 25 µm) placed in the direct beam.

2.3. Deconvolution of multiples

Several deconvolution algorithms have been described in the literature (Hao et al., 1993 , 1995 ; Ren & Moffat, 1995b ; Bourenkov et al., 1996 ; Xie & Hao, 1997 ). Deconvolution of multiples is not only very important for the completeness, especially in the low-resolution shells, but also for the assignment of screw axes and glide planes by identification of systematic absences.

The deconvolution algorithm that is incorporated in LAUEVIEW (Ren & Moffat, 1995b) solves the equation

$[I_j \left({\bf h}_0\right) = \textstyle\sum\limits_{n = n_{\rm min}}^{n_{\rm max}} F^2 \left(n{\bf h}_0\right) / f_{{\rm general}\,j} \left(n{\bf h}_0\right),\eqno(3)]$

where $[I_j \left({\bf h}_0\right)]$ is the total intensity of the multiple spot, $[f_{{\rm general}\,j} \left(n{\bf h}_0\right)]$ is the general scale factor of the nth harmonic $[n{\bf h}_0]$ for the multiple spot j, and $[F \left(n{\bf h}_0\right)]$ represents the structure-factor amplitude. Equation (3) can be solved for $[F^2 \left(n{\bf h}_0\right)]$ by traditional least-squares methods if the number of measurements of $[I_j \left({\bf h}_0\right)]$ (including singles and symmetry-related measurements) is equal to or greater than the number of unknown values of $[F \left(n{\bf h}_0\right)]$ . This can be accomplished by increasing the number of frames collected on one crystal.

2.4. Direct methods

It has been discussed (Harding, 1988 ) that direct methods might lose their power to solve low-molecular-weight structures in the case of the Laue diffraction experiment. Not only does the overall completeness tend to be lower compared with a monochromatic experiment, but especially low-resolution reflections, important in forming triplets, suffer from either the low-resolution hole or the harmonic overlap problem (not all multiples can be deconvoluted; see §2.3).

To quantify the sensitivity of direct methods to the incompleteness of an individual Laue data set, the theoretical approach of Giacovazzo et al. (1994) can be used. It utilizes the reliability parameter $[\alpha_{\bf h}]$ of the Cochran distribution (Cochran, 1955 ). In the absence of any information on the phases, the expectation value of $[\alpha]$ , $[\left\langle \alpha \right\rangle]$ , and its variance $[\sigma^2_{\alpha_{\bf h}}]$ can be calculated (Cascarano et al., 1984 ) by

$[\eqalignno{&\left\langle\alpha_{\bf h} \right\rangle=\textstyle\sum\limits_{j=1}^{r}G_jD_1 \left(G_j\right),\cr&\sigma^2_{\alpha_{\bf h}} \simeq(1/2)\textstyle\sum\limits_{j=1}^{r}G_j^2 \left[1+D_2\left(G_j\right)-2D_1^2\left(G_j\right) \right],&(4)}]$

where $[G_j = 2 \left| E_{\bf h} E_{{\bf k}_j} E_{-{\bf h}-{\bf k}_j} \right| /N^{1/2}]$ , $[E_{\bf h}]$ is a normalized structure factor, N is the number of atoms in the unit cell, r is the number of triplets selected for reflection $[{\bf h}]$ , and D_j = I_j / I₀ with I_j being a modified Bessel function of order j. It has been shown by Giacovazzo et al. (1994), that the parameter $[z_{\bf h} = \left\langle \alpha_{\bf h} \right\rangle / \sigma_{\alpha_{\bf h}}]$ can be considered as a signal-to-noise ratio. Diffraction data for which z values are greater than 2 or 3 for most strong reflections constitute a data set which is suitable for the successful application of direct methods.

Although the individual values of G are expected to be large for low-molecular-weight molecules in general, r might become critically low in the case of Laue structure determination, resulting in smaller values for z. By calculating the cumulative distribution function C(z ) for each Laue data set, one can check if problems are to be expected when trying to solve the structure.

3. Data collection

Four data sets were collected on three different compounds: bis[1,8(dimethylamino)naphthalene]mellitate dihydrate (DMANM), trans-bicyclo[4.4.0]decane-3,8-dione (tBDD) and 5-(N,N-dimethylamino)-3,3-dimethyl-2-(4-nitrophenyl)-3H-indole (1DA). Two of the data sets were collected at a temperature of 105 K. The compound codes and structural formulae are listed in Table 1. All Laue data were collected at beamline X26C at the National Synchrotron Light Source (NSLS) in Brookhaven, USA, using a BioCARS Laue bench camera, home-built freezing equipment, a 0.2 mm collimator and Fuji HR-III_N imaging plates, 20.1 × 25.2 cm in size. The Fuji plates were digitized off-line with a BAS2000 scanner at 100 µm raster size. Three of the four data sets were collected with the white beam focused by a Pt-coated cylindrical mirror; one data set, 1DA, was collected without focusing. A 25 µm Zr foil was placed in the direct beam between the shutter and the collimator to obtain a sharp $[\lambda_{\rm min}]$ cut-off in the white X-ray spectrum.

Table 1
Details of the data collection

Compound names are given in the text. The same crystal was used for the two data sets DMANM and DMANMlnt.

Compound code	DMANM	DMANMlnt	tBDD	1DA
Structural formula	C₄₀H₄₆N₄O₁₄	C₄₀H₄₆N₄O₁₄	C₁₀H₁₄O₂	C₁₈H₁₉N₃O₂
Crystal size (mm)	0.6 × 0.5 × 0.1	0.6 × 0.5 × 0.1	0.8 × 0.5 × 0.5	0.3 × 0.3 × 0.4
Temperature (K)	295	105	105	295
Synchrotron mode	25 bunch	25 bunch	25 bunch	Single bunch
Current (mA)	145	145	170	80
Beam	Focused	Focused	Focused	Unfocused
Distance (mm)	52	52	52	37
Exposure time (s)	4.0	8.0	2.0	0.5
Number of frames	11	11	10	7
Spindle angles (°)	0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100	0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100	5, 10, 15, 20, 25, 30, 35, 40, 45, 50	0, 15, 22.5, 30.0, 37.5, 45, 52.5

The exposure times varied between 0.5 and 8 s, the crystal-to-detector distances between 37 and 52 mm (see Table 1). Although the purpose was to collect a minimum of ten frames for each data set, a number of frames had to be rejected for reasons such as scanning problems, crystal cracking, bad image plates etc. For each data set, a total of ∼1.5 h was spent on crystal selection and mounting, collecting and scanning test frames, actual data collection and frame scanning. A carousel could not be used due to the extremely short crystal-to-detector distance, so that the image plates had to be changed manually after each exposure. Table 1 lists the final number of frames used for each data set, and the spindle angles at which the frames were collected.

4. Data processing

All test frames were indexed immediately after collection and scanning using the program LAUECELL (Ravelli et al., 1996), which is able to determine the crystal orientation as well as the relative cell parameters. In the case of serious indexing problems, the program LAUEGEN (Campbell, 1995) was used first to determine the direct-beam position more accurately, after which LAUECELL was able to find the correct cell. Cell and orientation parameters were imported into LAUEGEN (Campbell, 1995). The LAUEGEN program was used to check whether the predicted pattern could be refined to acceptable values of the root-mean-square deviation between predicted and measured spot positions, whether the $[\lambda_{\rm min}]$ Zr edge could be observed, and whether there were enough high-resolution spots. The latter two checks can be performed by the histogram analysis developed by Hao et al. (1995). A different crystal was mounted in case of any indication of problems.

Processing of the full data sets was started by indexing using LAUECELL. Cell volumes were estimated by using 0.55 and 1.8 Å for $[\lambda_{\rm min}]$ and $[\lambda_{\rm max}]$ , respectively, and by comparing the number of predicted spots with the number of observed spots as a function of cell volume and resolution. The Bravais lattice of the crystal was chosen after running the program LEPAGE (LePage, 1982 ). Full refinement and integration were performed independently with LAUEGEN (Campbell, 1995) and LAUEVIEW (Ren & Moffat, 1995a). The refinement gave comparable results for both programs, in the range 0.03–0.08 mm for the r.m.s. deviation between the observed and predicted spot positions for a maximum number of spots. Integration using the empirical-profile fitting method implemented in LAUEGEN not only proved to be much faster than the analytical profile-fitting method used in LAUEVIEW; in some unfavourable cases of split spot profiles at high values of $[2\theta]$ it also appeared to be more accurate. The histogram analysis of LAUEGEN was used to derive the absolute scale of the unit-cell parameters, by inspecting the ratio between the number of spots with significant intensity and the number of predicted spots, versus the wavelength. The histograms of the individual frames were averaged for each data set; Fig. 1 shows two examples of histograms thus obtained. Values of the starting cell volume, the wavelength found for the Zr edge at the sharp decline in the histograms, scaled cell parameters and final cell volumes are listed in Table 2.

Table 2
Crystallographic data I: (scaled) cell dimensions

Estimated standard deviations are given in brackets. The absorption edge of Zr, used to calculate the scale factors, is at 0.6889 Å. Compound names are given in the text.

Compound code	DMANM	DMANMlnt	tBDD	1DA
Initial cell volume (Å³)	2000	2000	600	2000
$[\lambda_{\rm edge}^{\rm observed}]$ (Å)	0.700 (10)	0.700 (10)	0.765 (15)	0.745 (5)
Scale factor on volume	0.95 (2)	0.95 (2)	0.73 (4)	0.791 (16)
Scaled cell parameters
a (Å)	9.46 (14)	9.45 (14)	5.24 (10)	8.98 (6)
b (Å)	12.56 (18)	12.59 (18)	10.6 (2)	11.20 (8)
c (Å)	17.0 (2)	16.9 (2)	7.99 (16)	16.36 (11)
α (°)	106.4 (2)	105.70 (7)	90	90
β (°)	97.71 (3)	97.74 (5)	96.758 (10)	105.967 (16)
γ (°)	93.797 (5)	93.70 (5)	90	90
Scaled cell volume (Å³)	1910 (80)	1910 (80)	440 (25)	1580 (30)
Bravais lattice	P triclinic	P triclinic	P monoclinic	P monoclinic

Figure 1
Histograms of the ratio of the number of significant intensities to the number of measured intensities (non-overlapping singles) as a function of the wavelength. The wavelength is on a scale obtained from an initial estimate (based on $[\lambda_{\rm min}]$ of the X-ray beam) of the cell volume (Table 2

). The absorption edge of Zr, indicated by an arrow, is located at the sharp decline in the histograms. All the frames within one data set are used. (a) Histogram for DMANM. (b) Histogram for 1DA.

All frames were re-integrated using the scaled cell dimensions and resolution limits. A value of 0.40 Å was taken for $[\lambda_{\rm min}]$ in the case of an unfocused beam, and 0.60 Å in the case of a focused beam. It was necessary to use $[\lambda_{\rm min}]$ values lower than the wavelength of the Zr edge ( $[\lambda_{\rm edge}]$ ) to account for the few spots that still had significant intensity below the absorption edge (see Fig. 1). Since only a few spots showed overlap, a resolution-dependent filter could be applied after the integration (performed by LAUEGEN; Campbell, 1995): all spots that were predicted (based on a known starting $[\lambda]$ curve and Wilson statistics) to have an intensity lower than a certain threshold were discarded. The singles were merged based on the highest-symmetry Laue group belonging to the Bravais lattice that was chosen in LAUECELL. The values of $[R_{\rm merge}]$ and the overall appearance of the wavelength-normalization curve were carefully inspected during the scaling to check if the correct Laue group had been selected.

The scaling of the data was performed using the program LAUEVIEW (Ren & Moffat, 1995a). Low-order Chebyshev polynomials (third or fourth order) were fitted against the singles diffracting at a wavelength between 0.72 and 1.6 Å in order to minimize $[R_{\rm merge}]$ . The small number of singles at higher wavelengths made it necessary to refine the wavelength-normalization curve extremely carefully. Only the parts at lower wavelengths (below ∼1.1 Å) could be refined with higher-order Chebyshev terms (8) without showing overfitting. The refinement of the $[\lambda]$ curve was alternated with refinement of isotropic scale and temperature factors. The combined wavelength-normalization curves are shown in Fig. 2. Only details in the curves that are visible between 0.8 and 1.2 Å are likely to be realistic: details in the curves beyond 1.2 Å are artifacts caused by appending different parts of the curves. In the case of the unfocused beam, it was possible to extend the $[\lambda]$ curve below the absorption edge of Zr.

Figure 2
Wavelength-normalization curves. An unfocused beam was used for 1DA with the synchrotron running in single-bunch mode. A focused beam was used for DMANM, DMANMlnt and tBDD, while the synchrotron was running in 25-bunch mode.

The wavelength-normalization curves were used to deconvolute the multiples (Ren & Moffat, 1995b), which were scaled and included in the final data sets. All $[R_{\rm merge}]$ factors between the deconvoluted multiples and the singles are good, between 2.7 and 7.8%, which is not surprising since the singles are actively used in the energy-deconvolution procedure (Ren & Moffat, 1995b). Table 3 gives an overview of the statistics of the final data sets.

Table 3
Crystallographic data II: data set statistics

Compound names are given in the text.

Compound code	DMANM	DMANMlnt	tBDD	1DA
d_min (Å)	0.85	0.75	0.75	0.70
Singles
Number of measurements	10181	11877	2352	9571
Number of unique reflections	4581	5920	628	3228
Redundancy	2.2	2.0	3.7	3.0
R_merge†	12.3	11.6	13.2	11.5
Multiples
Number of multiples	762	686	188	751
Total
Number of unique reflections	4799	6168	709	3478
Completeness (%)
∞–3d_min	30.0	34.0	47.1	39.3
3d_min–2d_min (∞–2d_min)	73.5 (60.5)	76.4 (63.8)	85.6 (72.9)	70.5 (60.8)
2d_min–d_min (∞–d_min)	75.6 (73.7)	65.2 (65.0)	59.7 (61.4)	69.7 (68.6)

†R_merge is defined as $[\textstyle\sum_{\bf h}\textstyle\sum_{i} (|F^2_{{\bf h}_i}-\langle F^2_{\bf h} \rangle|)/\textstyle\sum_{\bf h}\textstyle\sum_{i}F^2_{{\bf h}_i}]$ , with $[\langle F^2_{\bf h}\rangle = (1/N)\sum_{i = 1}^{N}F^2_{{\bf h}_i}]$ , where the summation is performed over all identical or equivalent reflections.

5. Structure determination

The program PLATON (Spek, 1990 ) was used to determine the different possible space groups for each compound. Glide planes were identified for tBDD and 1DA, but no data was available to discriminate between screw and rotation axes. SIR92 (Altomare et al., 1994 ) was used for each data set to determine the C(z ) distribution (as defined in §2.4). These distributions (shown in Fig. 3) show that all data sets have values of z larger than 3 for most of their strong reflections, indicating that the (systematic) incompleteness of these data sets should not impose serious limitations on the application of direct methods. If only the singles are used in the C(z ) distribution, the values are less promising.

Figure 3
Cumulative distribution function of $[z = \left\langle \alpha \right\rangle /\sigma_{\alpha}]$ , as defined by Giacovazzo et al. (1994

). For all our Laue data sets, z values larger than 3 are found for most of the strong reflections, indicating that the systematic incompleteness of these data sets would not impose serious limitations on application of direct methods.

The structures were solved by direct methods revealing most of the non-H atoms [using both SHELXS97 (Sheldrick, 1997 ) and SIR92 (Altomare et al., 1994)], followed by difference Fourier syntheses. All possible space groups were checked: P2/n and P2₁/n for tBDD, $[P\bar{1}]$ and P1 for DMANM and DMANMlnt, and P2/c and P2₁/c for 1DA. The correct space groups were corroborated by the structure determination and refinement; they are listed in Table 4.

Table 4
Details of the structure determinations of the low-molecular-weight compounds

The number of unique reflections used in the refinement is often lower than the total number of reflections in each data set, as given in Table 3. This is due to rejection of systematically absent and inconsistent equivalent reflections.

Compound code	DMANM	DMANMlnt	tBDD	1DA
Space group	$[P\bar{1}]$	$[P\bar{1}]$	P2₁/n	P2₁/c
Number of molecules per unit cell	2	2	2	4
F(000)	852	852	180	656
Number of unique reflections	4780	6168	686	3411
with F^obs> 4σ(F^obs)	4032	5374	629	2852
Number of parameters	553	545	55	211
Weighting scheme†
w₁	0.0870	0.1704	0.1665	0.1546
w₂	1.32	1.56	0.19	0.13
Final R₁‡ [F^obs > 4σ(F^obs)]	0.0906	0.1060	0.1012	0.1096
R₁ (all data)	0.1129	0.1233	0.1209	0.1358
wR₂	0.2091	0.2671	0.2573	0.2774
S	1.041	1.071	1.133	1.130
Minimum residual density (e Å⁻³)	-0.31	-0.46	-0.55	-0.63
Maximum residual density (e Å⁻³)	0.29	0.47	0.49	0.58

†Weighting scheme is defined as w⁻¹ ≡ σ²(F^obs)² + (w₁P)² + w₂P, where the parameter P ≡ $[\{\mbox{\rm max}({F_{\bf h}^{\rm obs}}^2,0)+\,\,]$ $[2{F_{\bf h}^{\rm calc}}^2\} / 3]$ .
‡ $[R_1 = \sum_{\bf h} (| F_{\bf h}|^{\rm obs} - | F_{\bf h}|^{\rm calc}) / \sum_{\bf h}| F_{\bf h}|^{\rm obs}.]$ $[wR_2 = [\sum_{\bf h}\{ w ({F_{\bf h}^{\rm obs}}^2 - {F_{\bf h}^{\rm calc}}^2)^2\} / \sum_{\bf h} \{w({F_{\bf h}^{\rm obs}}^2)^2\}]^{1/2}.]$ The goodness of fit, S, is based on F²: $[S = [\sum_{\bf h} \{ w({F_{\bf h}^{\rm obs}}^2 -]$ $[{F_{\bf h}^{\rm calc}}^2)^2\} / (n - p)]^{1/2}]$ , where n is the number of reflections and p is the total number of parameters refined.

Refinement on F² with all unique reflections was carried out by full-matrix least-squares techniques (SHELXL93; Sheldrick, 1993 ). H atoms not belonging to hydroxyl and methyl groups were introduced on calculated positions and included in the refinement riding on their carrier atoms with isotropic thermal parameters related to U_eq of the carrier atoms. Only those hydroxyl and methyl H atoms that could be located from inspection of the electron-density difference maps were included in the refinement with geometry constraints. All non-H atoms were refined with anisotropic thermal parameters. Weights were introduced in the final refinement cycles. Numerical details of the structure determination are given in Table 4. Thermal-motion ellipsoid plots are shown in Fig. 4.

Figure 4
Thermal-motion ellipsoid plots (50% probability) with the adopted atom labelling of the X-ray structures of (a) DMANM, (b) DMANlnt, (c) tBDD and (d) 1DA.

The structure of 1DA was the only one that could be compared with that of a published monochromatic study (van Walree et al., 1997 ), carried out independently on the same batch of crystals. The cell volume determined by the traditional monochromatic method was 1550.0 (2) Å³ (measured at 150 K), which verified the correctness of the scaling of the unit cell obtained from the Laue data [volume 1580 (30) Å³, measured at room temperature]. For tBDD, cell parameters were published previously (Miesen et al., 1994 ). As in the case of 1DA, the volume obtained from the Laue data [440 (25) Å³] is in good agreement with the volume determined by monochromatic experiment (442.4 Å³).

6. Discussion and conclusions

In this paper it is shown that ab initio structure determination of low-molecular-weight compounds using Laue data is feasible. The step that makes the structure determination independent from monochromatic experiment is the scaling of the unit cell to an absolute volume. The use of a Zr foil gives a sharp edge in the wavelength-normalization curve. This edge can be used to scale the cell volume with an accuracy of a few percent. This method of determining the absolute cell volume has been demonstrated before (Carr et al., 1992, 1993; Kariuki & Harding, 1995), but not applied to unknown structures thus far.

For the compounds presented here, it proved to be possible to solve the structure by means of direct methods. Although our data sets are far from complete, enough strong triplets could be found. The same observation was made by Helliwell, Gomez de Anderez et al. (1989 ), who solved an organic structure with Laue data using the Sayre tangent formula. In connection with the discussion by Harding (1988) about the risk that direct methods lose their power to solve the structure of low-molecular-weight compounds using only Laue data, we used the C(z) distribution devised by Giacovazzo et al. (1994) to evaluate this risk.

The potential of using Laue data for structure refinement has been demonstrated before, both for low-molecular-weight compounds (e.g. Wood et al., 1983 ; Gomez de Anderez et al., 1989 ; Dodd et al., 1994 ; Kariuki & Harding, 1995) and for proteins (Bourgeois et al., 1997 ; Yang et al., 1997; Ravelli et al., 1998 ). The Laue data presented in this paper are of modest quality in terms of $[R_{\rm merge}]$ and completeness. A partial explanation for the high R factors is related to the long exposure times for DMANM, DMANMlnt and tBDD given a focused beam and the synchrotron running in 25 bunch mode. These times are indicative of some problems with the beamline at the time of the measurement of these three data sets. Nevertheless, the structures obtained show good geometry;¹ all temperature factors appear to be physically reasonable, and show the expected differences between the room-temperature and the liquid-nitrogen temperature data sets. Comparison of one structure (1DA) with the published structure determined in a monochromatic experiment showed a low r.m.s. deviation for all non-H-atom positions: 0.019 Å. For all structures a significant decrease in the R factors was observed upon introduction of the non-hydroxyl and non-methyl H atoms. Furthermore, a number of hydroxyl and methyl H atoms showed up in the electron-density difference maps.

The Laue technique could be used more routinely for structure determination of low-molecular-weight compounds, provided that a number of improvements are made:

(i) Our data showed a severe $[\theta]$ cut-off. This has serious implications on the sampling of the reflections at longer wavelengths at which the highest percentage of singles can be found (Cruickshank et al., 1987 ; Ravelli, 1998), causing problems in the accuracy of the determination of the wavelength-normalization curve. An inaccurate $[\lambda]$ curve affects the deconvolution of all multiples, which constitute up to 17% of our data. Refinement of the $[\lambda]$ curve was unstable at higher wavelengths, and could only be performed using very low-order Chebyshev polynomials in that region. Since the crystal-to-detector distance was very short, a $[\theta]$ cut-off could only be prevented by eliminating the low-energy radiation from the X-ray spectrum. This could have been obtained by absorption of all longer wavelengths using thick Cu or Al attenuators.

(ii) The data completeness can be improved by collecting a larger number of frames for each compound. For unknown unit cells, a sampling of reciprocal space of 180° is advisable, and an angular width of 6° is needed to overlap the accessible part of reciprocal space from different exposures at lower resolutions [the maximum angular width depends on the X-ray spectrum and the desired resolution at which the accessible regions should overlap (see Moffat, 1997 ; Yang et al., 1997; Ravelli, 1998)]. More frames also increase the redundancy of the single measurements, which stabilizes the refinement of the $[\lambda]$ curve and the frame scaling factors.

(iii) Absorption correction can be important, especially in the case of Laue diffraction experiments on inorganic and organometallic compounds. Since the crystal is stationary in the Laue method, we may assume that the path length through the crystal and the air between crystal and detector varies smoothly, for different beams collected on one frame, as a function of the position of the spot. Maginn et al. (1993 ) demonstrated that the refinement of an absorption surface for each frame can give a significant reduction both in $[R_{\rm merge}]$ and R factor. For our data sets, we tried to refine an absorption surface by using the approach of Ren & Moffat (1995a): refinement of anisotropic scale factors. This led to overfitting; although the $[R_{\rm merge}]$ values could be reduced by 2 to 3%, the structures started to show non-realistic temperature movements.

(iv) The scaling of the unit-cell parameters requires a sharp and clear edge in the X-ray spectrum. Obviously, a clear edge well above the minimum wavelength of the X-ray spectrum, as was obtained in the $[\lambda]$ curve of 1DA (Fig. 2), is to be preferred over the edge in the other three curves, where it is located just above the minimum wavelength of the incident X-rays. The tail in the $[\lambda]$ curve of 1DA at lower wavelengths, however, caused some difficulties with the deconvolution, which requires the full $[\lambda]$ curve to be known accurately. Combination of different attenuators should prevent this problem. Alternatively, absorption edges of Pt, present in some coatings of focusing mirrors, may be used to scale the unit-cell parameters, provided that the measured data are sufficiently accurate and redundant.

(v) The integration software for Laue data collected on crystals of low-molecular-weight compounds is still amenable to improvements. The small number of spots per frames, the extremely low background for some spots and the large dynamical range of the spot intensity constitute a challenge for spot-profile fitting methods. The introduction of an extinction coefficient during the structure refinement showed a reduction of R factors for all our data sets, but only for unrealistically high values of this coefficient. This indicates either a systematic overestimation of the high-resolution reflections or a systematic underestimation of the low-resolution reflections. The latter may be caused by the deconvolution algorithm, or by an improper integration of intense spots (D. Bourgeois, personal communication).

We believe that, once the conditions listed above are fulfilled and structures can be obtained as easily and accurately as in the case of the monochromatic experiment using in-house laboratory equipment and standard software, one will be in a better position to exploit more widely the most important application of the Laue method: time-resolved (pertubation) studies.

Footnotes

‡Present address: European Molecular Biology Laboratory (EMBL), Grenoble Outstation, c/o ILL, BP 156, 38042 Grenoble CEDEX 9, France.

¹Deposited in the Cambridge Structural Database.

Acknowledgements

The investigations were supported by the Netherlands Foundation for Chemical Research (SON) with financial aid from the Netherlands Organization for Scientific Research (NWO) and by the European Union. We would like to thank Grace Shea and Ling Peng for assistance in data collection, Dritan Siliqi for help in obtaining Fig. 3, and Joel Sussman for supporting these experiments and giving us the opportunity to collect data at X26C.

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JOURNAL OF
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Volume 6| Part 1| January 1999| Pages 19-28

https://doi.org/10.1107/S0909049598011595

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

Ab initio structure determination of low-molecular-weight compounds using synchrotron radiation Laue diffraction

1. Introduction

2. Selected theoretical aspects

2.1. Profile fitting

2.2. Determination of the unit-cell parameters on an absolute scale

2.3. Deconvolution of multiples

2.4. Direct methods

3. Data collection

4. Data processing

5. Structure determination

6. Discussion and conclusions

Footnotes

Acknowledgements

References

research papers